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<!DOCTYPE html> <html lang="en"> <head> <meta content="text/html; charset=utf-8" http-equiv="content-type"/> <title>A Hamiltonian approach to the gradient-flow equations in information geometry</title>  <meta content="width=device-width, initial-scale=1, shrink-to-fit=no" name="viewport"/> <link href="https://cdn.jsdelivr.net/npm/bootstrap@5.3.0/dist/css/bootstrap.min.css" rel="stylesheet" type="text/css"/> <link href="/static/browse/0.3.4/css/ar5iv.0.7.9.min.css" rel="stylesheet" type="text/css"/> <link href="/static/browse/0.3.4/css/ar5iv-fonts.0.7.9.min.css" rel="stylesheet" type="text/css"/> <link href="/static/browse/0.3.4/css/latexml_styles.css" rel="stylesheet" type="text/css"/> <script src="https://cdn.jsdelivr.net/npm/bootstrap@5.3.0/dist/js/bootstrap.bundle.min.js"></script> <script src="https://cdnjs.cloudflare.com/ajax/libs/html2canvas/1.3.3/html2canvas.min.js"></script> <script src="/static/browse/0.3.4/js/addons_new.js"></script> <script src="/static/browse/0.3.4/js/feedbackOverlay.js"></script> <base href="/html/2406.11224v2/"/></head> <body> <nav class="ltx_page_navbar"> <nav class="ltx_TOC"> <ol class="ltx_toclist"> <li class="ltx_tocentry ltx_tocentry_section"><a class="ltx_ref" href="https://arxiv.org/html/2406.11224v2#S1" title="In A Hamiltonian approach to the gradient-flow equations in information geometry"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">1 </span>Introduction</span></a></li> <li class="ltx_tocentry ltx_tocentry_section"> <a class="ltx_ref" href="https://arxiv.org/html/2406.11224v2#S2" title="In A Hamiltonian approach to the gradient-flow equations in information geometry"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">2 </span>Information Geometry and Gradient-Flow Equations</span></a> <ol class="ltx_toclist ltx_toclist_section"> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2406.11224v2#S2.SS1" title="In 2 Information Geometry and Gradient-Flow Equations ‣ A Hamiltonian approach to the gradient-flow equations in information geometry"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">2.1 </span>Information Geometry</span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2406.11224v2#S2.SS2" title="In 2 Information Geometry and Gradient-Flow Equations ‣ A Hamiltonian approach to the gradient-flow equations in information geometry"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">2.2 </span>Gradient-Flow Equations</span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2406.11224v2#S2.SS3" title="In 2 Information Geometry and Gradient-Flow Equations ‣ A Hamiltonian approach to the gradient-flow equations in information geometry"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">2.3 </span>Randers-Finsler deformation of the gradient-flow equations</span></a></li> </ol> </li> <li class="ltx_tocentry ltx_tocentry_section"><a class="ltx_ref" href="https://arxiv.org/html/2406.11224v2#S3" title="In A Hamiltonian approach to the gradient-flow equations in information geometry"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">3 </span>Complete integrability and geodesic Hamiltonian</span></a></li> <li class="ltx_tocentry ltx_tocentry_section"> <a class="ltx_ref" href="https://arxiv.org/html/2406.11224v2#S4" title="In A Hamiltonian approach to the gradient-flow equations in information geometry"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">4 </span>The motions of a light-like particle in a pseudo Riemann space</span></a> <ol class="ltx_toclist ltx_toclist_section"> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2406.11224v2#S4.SS1" title="In 4 The motions of a light-like particle in a pseudo Riemann space ‣ A Hamiltonian approach to the gradient-flow equations in information geometry"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">4.1 </span>Relation to the Randers-Finsler Lagrangian</span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2406.11224v2#S4.SS2" title="In 4 The motions of a light-like particle in a pseudo Riemann space ‣ A Hamiltonian approach to the gradient-flow equations in information geometry"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">4.2 </span>Applications to the gradient-flow equations</span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2406.11224v2#S4.SS3" title="In 4 The motions of a light-like particle in a pseudo Riemann space ‣ A Hamiltonian approach to the gradient-flow equations in information geometry"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">4.3 </span>Gaussian model</span></a></li> </ol> </li> <li class="ltx_tocentry ltx_tocentry_section"><a class="ltx_ref" href="https://arxiv.org/html/2406.11224v2#S5" title="In A Hamiltonian approach to the gradient-flow equations in information geometry"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">5 </span>Conclusions and perspectives</span></a></li> <li class="ltx_tocentry ltx_tocentry_appendix"><a class="ltx_ref" href="https://arxiv.org/html/2406.11224v2#A1" title="In A Hamiltonian approach to the gradient-flow equations in information geometry"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">A </span>The proof of (<span class="ltx_text ltx_ref_tag">38</span>)</span></a></li> <li class="ltx_tocentry ltx_tocentry_appendix"><a class="ltx_ref" href="https://arxiv.org/html/2406.11224v2#A2" title="In A Hamiltonian approach to the gradient-flow equations in information geometry"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">B </span>Conformal rescaling as reparametrization</span></a></li> <li class="ltx_tocentry ltx_tocentry_appendix"><a class="ltx_ref" href="https://arxiv.org/html/2406.11224v2#A3" title="In A Hamiltonian approach to the gradient-flow equations in information geometry"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">C </span>Zermelo form</span></a></li> </ol></nav> </nav> <div class="ltx_page_main"> <div class="ltx_page_content"> <article class="ltx_document ltx_authors_1line ltx_fleqn"> <div class="ltx_para" id="p1"> <p class="ltx_p" id="p1.1">[1]<span class="ltx_ERROR undefined" id="p1.1.1">\fnm</span>Tatsuaki <span class="ltx_ERROR undefined" id="p1.1.2">\sur</span>Wada</p> </div> <div class="ltx_para" id="p2"> <p class="ltx_p" id="p2.1">[1]<span class="ltx_ERROR undefined" id="p2.1.1">\orgdiv</span>Region of Electrical and Electronic Systems Engineering, <span class="ltx_ERROR undefined" id="p2.1.2">\orgname</span>Ibaraki University, <span class="ltx_ERROR undefined" id="p2.1.3">\orgaddress</span><span class="ltx_ERROR undefined" id="p2.1.4">\street</span>Nakanarusawa-cho, <span class="ltx_ERROR undefined" id="p2.1.5">\city</span>Hitachi, <span class="ltx_ERROR undefined" id="p2.1.6">\postcode</span>316-8511, <span class="ltx_ERROR undefined" id="p2.1.7">\country</span>Japan</p> </div> <div class="ltx_para" id="p3"> <p class="ltx_p" id="p3.1">2]<span class="ltx_ERROR undefined" id="p3.1.1">\orgname</span>Istituto dei Sistemi Complessi, Consiglio Nazionale delle Ricerche (ISC-CNR), c/o Politecnico di Torino, <span class="ltx_ERROR undefined" id="p3.1.2">\orgaddress</span><span class="ltx_ERROR undefined" id="p3.1.3">\street</span>Corso Duca degli Abruzzi 24, <span class="ltx_ERROR undefined" id="p3.1.4">\city</span>Torino, <span class="ltx_ERROR undefined" id="p3.1.5">\postcode</span>I-10129, <span class="ltx_ERROR undefined" id="p3.1.6">\country</span>Italy</p> </div> <h1 class="ltx_title ltx_title_document">A Hamiltonian approach to the gradient-flow equations in information geometry</h1> <div class="ltx_authors"> <span class="ltx_creator ltx_role_author"> <span class="ltx_personname"> </span><span class="ltx_author_notes"> <span class="ltx_contact ltx_role_email"><a href="mailto:tatsuaki.wada.to@vc.ibaraki.ac.jp">tatsuaki.wada.to@vc.ibaraki.ac.jp</a> </span></span></span> <span class="ltx_author_before"> </span><span class="ltx_creator ltx_role_author"> <span class="ltx_personname"><span class="ltx_ERROR undefined" id="id1.1.id1">\fnm</span>Antonio M. <span class="ltx_ERROR undefined" id="id2.2.id2">\sur</span>Scarfone </span><span class="ltx_author_notes"> <span class="ltx_contact ltx_role_email"><a href="mailto:antonio.scarfone@polito.it">antonio.scarfone@polito.it</a> </span> <span class="ltx_contact ltx_role_affiliation">* </span> <span class="ltx_contact ltx_role_affiliation">[ </span></span></span> </div> <div class="ltx_abstract"> <h6 class="ltx_title ltx_title_abstract">Abstract</h6> <p class="ltx_p" id="id3.id1">We have studied the gradient-flow equations in information geometry from a point-particle perspective. Based on the motion of a null (or light-like) particle in a curved space, we have rederived the Hamiltonians which describe the gradient-flows in information geometry.</p> </div> <div class="ltx_classification"> <h6 class="ltx_title ltx_title_classification">keywords: </h6>gradient-flow, information geometry, null geodesics, Finsler metric, Randers function </div> <section class="ltx_section" id="S1"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">1 </span>Introduction</h2> <div class="ltx_para" id="S1.p1"> <p class="ltx_p" id="S1.p1.1">Information geometry (IG) <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2406.11224v2#bib.bib1" title="">1</a>]</cite> is a useful and powerful framework for studying some families of probability distributions by identifying the space of probability distributions with a differentiable manifold endowed with Fisher metric as a Riemann metric and <math alttext="\alpha" class="ltx_Math" display="inline" id="S1.p1.1.m1.1"><semantics id="S1.p1.1.m1.1a"><mi id="S1.p1.1.m1.1.1" xref="S1.p1.1.m1.1.1.cmml">α</mi><annotation-xml encoding="MathML-Content" id="S1.p1.1.m1.1b"><ci id="S1.p1.1.m1.1.1.cmml" xref="S1.p1.1.m1.1.1">𝛼</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p1.1.m1.1c">\alpha</annotation><annotation encoding="application/x-llamapun" id="S1.p1.1.m1.1d">italic_α</annotation></semantics></math>-connection as an affine connection. In IG, the Riemann metric is obtained from the Hessian of a potential function, and the fluctuations play important role. Especially the so-called fluctuation-response relations are related with the Hessian metric <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2406.11224v2#bib.bib2" title="">2</a>]</cite>. On the other hand, the gradient-flow equations are useful for some optimization problems. The gradient flows on a Riemann manifold follow the direction of gradient descent (or ascent) in the landscape of a potential functional, with respect to the curved structure of the underlying metric space. The IG studies on the gradient systems were originally performed independently by Nakamura <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2406.11224v2#bib.bib3" title="">3</a>]</cite> and Fujiwara-Amari <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2406.11224v2#bib.bib4" title="">4</a>]</cite>. A remarkable feature of their works is that a certain kind of gradient flow on a dually flat space can be expressed as a Hamilton flow. Later, several works on this issue have been done from the different perspectives. Malagó and Pistone <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2406.11224v2#bib.bib5" title="">5</a>]</cite> studied the natural gradient flows in the mixture geometry of a discrete exponential family. Boumuki and Noda <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2406.11224v2#bib.bib6" title="">6</a>]</cite> studied the relationship between the Hamiltonian-flows and gradient-flows from the perspective of symplectic geometries. Chirico et al. <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2406.11224v2#bib.bib7" title="">7</a>]</cite> provided an information geometric formulation of classical mechanics on the statistical manifold. Furthermore Pistone <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2406.11224v2#bib.bib8" title="">8</a>]</cite> studied Lagrangian function on the finite state space statistical bundle. Together with our collaborators, we also studied the same issue by some heuristic approaches and related to some different fields. In Ref. <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2406.11224v2#bib.bib9" title="">9</a>]</cite>, we studied the gradient-flow equations based on the generalized eikonal equation for a simple thermodynamic system and introduced a mock time evolution in IG as a Hamilton-Jacobi dynamics. We studied <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2406.11224v2#bib.bib10" title="">10</a>]</cite> the same issue from the perspective of geometric optics and related the gradient-flows in IG to the light trajectories in anisotropic optical media by using Huygens’ equations. Furthermore the analytical mechanical properties concerning the gradient-flow equations in IG are studied <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2406.11224v2#bib.bib11" title="">11</a>]</cite> and discussed the deformations of the gradient-flow equations which lead to Randers-Finsler metrics <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2406.11224v2#bib.bib12" title="">12</a>]</cite>. Ref. <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2406.11224v2#bib.bib13" title="">13</a>]</cite> provided a Weyl geometric approach. Through these studies we realize the importance of treating space and time on equal footing, which is an essence of Einstein’s relativity <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2406.11224v2#bib.bib14" title="">14</a>]</cite>. In addition, it is known in general relativity that in a suitable coordinate system, the physical equations have simple forms and clear physical meanings <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2406.11224v2#bib.bib15" title="">15</a>]</cite>.</p> </div> <div class="ltx_para" id="S1.p2"> <p class="ltx_p" id="S1.p2.2">Through our previous studies <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2406.11224v2#bib.bib9" title="">9</a>, <a class="ltx_ref" href="https://arxiv.org/html/2406.11224v2#bib.bib10" title="">10</a>, <a class="ltx_ref" href="https://arxiv.org/html/2406.11224v2#bib.bib11" title="">11</a>]</cite>, we already related the gradient-flows in IG to the Hamilton-flows based on the physical concepts such as a light-ray, refractive index in the geometrical optics, in which optical (or Fermat) metrics play a central role. It is noted <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2406.11224v2#bib.bib16" title="">16</a>]</cite> that based on Fermat’s principle, the spatial part of null geodesic in <math alttext="(N+1)" class="ltx_Math" display="inline" id="S1.p2.1.m1.1"><semantics id="S1.p2.1.m1.1a"><mrow id="S1.p2.1.m1.1.1.1" xref="S1.p2.1.m1.1.1.1.1.cmml"><mo id="S1.p2.1.m1.1.1.1.2" stretchy="false" xref="S1.p2.1.m1.1.1.1.1.cmml">(</mo><mrow id="S1.p2.1.m1.1.1.1.1" xref="S1.p2.1.m1.1.1.1.1.cmml"><mi id="S1.p2.1.m1.1.1.1.1.2" xref="S1.p2.1.m1.1.1.1.1.2.cmml">N</mi><mo id="S1.p2.1.m1.1.1.1.1.1" xref="S1.p2.1.m1.1.1.1.1.1.cmml">+</mo><mn id="S1.p2.1.m1.1.1.1.1.3" xref="S1.p2.1.m1.1.1.1.1.3.cmml">1</mn></mrow><mo id="S1.p2.1.m1.1.1.1.3" stretchy="false" xref="S1.p2.1.m1.1.1.1.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S1.p2.1.m1.1b"><apply id="S1.p2.1.m1.1.1.1.1.cmml" xref="S1.p2.1.m1.1.1.1"><plus id="S1.p2.1.m1.1.1.1.1.1.cmml" xref="S1.p2.1.m1.1.1.1.1.1"></plus><ci id="S1.p2.1.m1.1.1.1.1.2.cmml" xref="S1.p2.1.m1.1.1.1.1.2">𝑁</ci><cn id="S1.p2.1.m1.1.1.1.1.3.cmml" type="integer" xref="S1.p2.1.m1.1.1.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p2.1.m1.1c">(N+1)</annotation><annotation encoding="application/x-llamapun" id="S1.p2.1.m1.1d">( italic_N + 1 )</annotation></semantics></math>-dimensional spacetime is regarded as the geodesic of the corresponding <math alttext="N" class="ltx_Math" display="inline" id="S1.p2.2.m2.1"><semantics id="S1.p2.2.m2.1a"><mi id="S1.p2.2.m2.1.1" xref="S1.p2.2.m2.1.1.cmml">N</mi><annotation-xml encoding="MathML-Content" id="S1.p2.2.m2.1b"><ci id="S1.p2.2.m2.1.1.cmml" xref="S1.p2.2.m2.1.1">𝑁</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p2.2.m2.1c">N</annotation><annotation encoding="application/x-llamapun" id="S1.p2.2.m2.1d">italic_N</annotation></semantics></math>-dimensional optical geometry. A null geodesic equation in pseudo-Riemann space can be considered as a Hamilton-Jacobi equation, and the complete integrability is a key for solving the Hamilton-Jacobi equations. In this contribution we take a different approach based on Hamiltonian systems to the gradient-flows in IG by considering the complete integrability and by analyzing the motion of a null (or light-like) particle in a pseudo-Riemann metric.</p> </div> <div class="ltx_para" id="S1.p3"> <p class="ltx_p" id="S1.p3.1">The rest of the paper consists as follows. In Section <a class="ltx_ref" href="https://arxiv.org/html/2406.11224v2#S2" title="2 Information Geometry and Gradient-Flow Equations ‣ A Hamiltonian approach to the gradient-flow equations in information geometry"><span class="ltx_text ltx_ref_tag">2</span></a>, we first review some basics of IG and the associated gradient-flow equations. In Section <a class="ltx_ref" href="https://arxiv.org/html/2406.11224v2#S3" title="3 Complete integrability and geodesic Hamiltonian ‣ A Hamiltonian approach to the gradient-flow equations in information geometry"><span class="ltx_text ltx_ref_tag">3</span></a> we consider the analytical mechanics based on Cartan’s theory <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2406.11224v2#bib.bib17" title="">17</a>]</cite> on the complete integrability of Pfaffian systems. We obtain the form of a Hamiltonian which satisfies the complete integrability condition and relate it to the Hamiltonian describing the gradient-flows in IG. Section <a class="ltx_ref" href="https://arxiv.org/html/2406.11224v2#S4" title="4 The motions of a light-like particle in a pseudo Riemann space ‣ A Hamiltonian approach to the gradient-flow equations in information geometry"><span class="ltx_text ltx_ref_tag">4</span></a> shows that the gradient-flows in IG are related to the motions of a light-like particle in a curved space characterized by a pseudo-Riemann metric, which are analyzed in the fields of general relativity. The final Section <a class="ltx_ref" href="https://arxiv.org/html/2406.11224v2#S5" title="5 Conclusions and perspectives ‣ A Hamiltonian approach to the gradient-flow equations in information geometry"><span class="ltx_text ltx_ref_tag">5</span></a> is devoted to our conclusions and perspective. Appendix <a class="ltx_ref" href="https://arxiv.org/html/2406.11224v2#A1" title="Appendix A The proof of (38) ‣ A Hamiltonian approach to the gradient-flow equations in information geometry"><span class="ltx_text ltx_ref_tag">A</span></a> shows the proof of the condition (<a class="ltx_ref" href="https://arxiv.org/html/2406.11224v2#S3.E38" title="In 3 Complete integrability and geodesic Hamiltonian ‣ A Hamiltonian approach to the gradient-flow equations in information geometry"><span class="ltx_text ltx_ref_tag">38</span></a>) for the complete integrability of Pfaffian equation. Appendix <a class="ltx_ref" href="https://arxiv.org/html/2406.11224v2#A2" title="Appendix B Conformal rescaling as reparametrization ‣ A Hamiltonian approach to the gradient-flow equations in information geometry"><span class="ltx_text ltx_ref_tag">B</span></a> provides the relation between the conformal scaling and reparametrization. Appendix <a class="ltx_ref" href="https://arxiv.org/html/2406.11224v2#A3" title="Appendix C Zermelo form ‣ A Hamiltonian approach to the gradient-flow equations in information geometry"><span class="ltx_text ltx_ref_tag">C</span></a> shows the explicit relation of the stationary metric discussed in Section <a class="ltx_ref" href="https://arxiv.org/html/2406.11224v2#S4" title="4 The motions of a light-like particle in a pseudo Riemann space ‣ A Hamiltonian approach to the gradient-flow equations in information geometry"><span class="ltx_text ltx_ref_tag">4</span></a> to the Zermelo form.</p> </div> <div class="ltx_para" id="S1.p4"> <p class="ltx_p" id="S1.p4.17">We would like to emphasize that our approaches to IG are different from the conventional method as follows. In conventional method <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2406.11224v2#bib.bib1" title="">1</a>]</cite> of IG, the natural (<math alttext="\theta" class="ltx_Math" display="inline" id="S1.p4.1.m1.1"><semantics id="S1.p4.1.m1.1a"><mi id="S1.p4.1.m1.1.1" xref="S1.p4.1.m1.1.1.cmml">θ</mi><annotation-xml encoding="MathML-Content" id="S1.p4.1.m1.1b"><ci id="S1.p4.1.m1.1.1.cmml" xref="S1.p4.1.m1.1.1">𝜃</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p4.1.m1.1c">\theta</annotation><annotation encoding="application/x-llamapun" id="S1.p4.1.m1.1d">italic_θ</annotation></semantics></math>- or <math alttext="\eta" class="ltx_Math" display="inline" id="S1.p4.2.m2.1"><semantics id="S1.p4.2.m2.1a"><mi id="S1.p4.2.m2.1.1" xref="S1.p4.2.m2.1.1.cmml">η</mi><annotation-xml encoding="MathML-Content" id="S1.p4.2.m2.1b"><ci id="S1.p4.2.m2.1.1.cmml" xref="S1.p4.2.m2.1.1">𝜂</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p4.2.m2.1c">\eta</annotation><annotation encoding="application/x-llamapun" id="S1.p4.2.m2.1d">italic_η</annotation></semantics></math>-) coordinate space (or dually-flat manifold) is characterized with Fisher metric <math alttext="g" class="ltx_Math" display="inline" id="S1.p4.3.m3.1"><semantics id="S1.p4.3.m3.1a"><mi id="S1.p4.3.m3.1.1" xref="S1.p4.3.m3.1.1.cmml">g</mi><annotation-xml encoding="MathML-Content" id="S1.p4.3.m3.1b"><ci id="S1.p4.3.m3.1.1.cmml" xref="S1.p4.3.m3.1.1">𝑔</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p4.3.m3.1c">g</annotation><annotation encoding="application/x-llamapun" id="S1.p4.3.m3.1d">italic_g</annotation></semantics></math> and the <math alttext="\alpha" class="ltx_Math" display="inline" id="S1.p4.4.m4.1"><semantics id="S1.p4.4.m4.1a"><mi id="S1.p4.4.m4.1.1" xref="S1.p4.4.m4.1.1.cmml">α</mi><annotation-xml encoding="MathML-Content" id="S1.p4.4.m4.1b"><ci id="S1.p4.4.m4.1.1.cmml" xref="S1.p4.4.m4.1.1">𝛼</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p4.4.m4.1c">\alpha</annotation><annotation encoding="application/x-llamapun" id="S1.p4.4.m4.1d">italic_α</annotation></semantics></math>-connections, which provide the parallel translation rule. In addition, unlike Riemann geometry, the metric <math alttext="g" class="ltx_Math" display="inline" id="S1.p4.5.m5.1"><semantics id="S1.p4.5.m5.1a"><mi id="S1.p4.5.m5.1.1" xref="S1.p4.5.m5.1.1.cmml">g</mi><annotation-xml encoding="MathML-Content" id="S1.p4.5.m5.1b"><ci id="S1.p4.5.m5.1.1.cmml" xref="S1.p4.5.m5.1.1">𝑔</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p4.5.m5.1c">g</annotation><annotation encoding="application/x-llamapun" id="S1.p4.5.m5.1d">italic_g</annotation></semantics></math> is only used to determine the orthogonality but not used to determine a distance in the natural coordinate spaces. The <math alttext="\theta" class="ltx_Math" display="inline" id="S1.p4.6.m6.1"><semantics id="S1.p4.6.m6.1a"><mi id="S1.p4.6.m6.1.1" xref="S1.p4.6.m6.1.1.cmml">θ</mi><annotation-xml encoding="MathML-Content" id="S1.p4.6.m6.1b"><ci id="S1.p4.6.m6.1.1.cmml" xref="S1.p4.6.m6.1.1">𝜃</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p4.6.m6.1c">\theta</annotation><annotation encoding="application/x-llamapun" id="S1.p4.6.m6.1d">italic_θ</annotation></semantics></math>- and <math alttext="\eta" class="ltx_Math" display="inline" id="S1.p4.7.m7.1"><semantics id="S1.p4.7.m7.1a"><mi id="S1.p4.7.m7.1.1" xref="S1.p4.7.m7.1.1.cmml">η</mi><annotation-xml encoding="MathML-Content" id="S1.p4.7.m7.1b"><ci id="S1.p4.7.m7.1.1.cmml" xref="S1.p4.7.m7.1.1">𝜂</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p4.7.m7.1c">\eta</annotation><annotation encoding="application/x-llamapun" id="S1.p4.7.m7.1d">italic_η</annotation></semantics></math>-coordinate systems are regarded as the dual coordinates on the same manifold. In contrast, in our perspective, the <math alttext="\theta" class="ltx_Math" display="inline" id="S1.p4.8.m8.1"><semantics id="S1.p4.8.m8.1a"><mi id="S1.p4.8.m8.1.1" xref="S1.p4.8.m8.1.1.cmml">θ</mi><annotation-xml encoding="MathML-Content" id="S1.p4.8.m8.1b"><ci id="S1.p4.8.m8.1.1.cmml" xref="S1.p4.8.m8.1.1">𝜃</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p4.8.m8.1c">\theta</annotation><annotation encoding="application/x-llamapun" id="S1.p4.8.m8.1d">italic_θ</annotation></semantics></math>- and <math alttext="\eta" class="ltx_Math" display="inline" id="S1.p4.9.m9.1"><semantics id="S1.p4.9.m9.1a"><mi id="S1.p4.9.m9.1.1" xref="S1.p4.9.m9.1.1.cmml">η</mi><annotation-xml encoding="MathML-Content" id="S1.p4.9.m9.1b"><ci id="S1.p4.9.m9.1.1.cmml" xref="S1.p4.9.m9.1.1">𝜂</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p4.9.m9.1c">\eta</annotation><annotation encoding="application/x-llamapun" id="S1.p4.9.m9.1d">italic_η</annotation></semantics></math>-coordinate spaces are regarded as the two different spaces (or basis manifolds) in general. When the <math alttext="\theta" class="ltx_Math" display="inline" id="S1.p4.10.m10.1"><semantics id="S1.p4.10.m10.1a"><mi id="S1.p4.10.m10.1.1" xref="S1.p4.10.m10.1.1.cmml">θ</mi><annotation-xml encoding="MathML-Content" id="S1.p4.10.m10.1b"><ci id="S1.p4.10.m10.1.1.cmml" xref="S1.p4.10.m10.1.1">𝜃</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p4.10.m10.1c">\theta</annotation><annotation encoding="application/x-llamapun" id="S1.p4.10.m10.1d">italic_θ</annotation></semantics></math>-space belongs to a curved space which is regarded as a basis manifold <math alttext="\mathcal{M}" class="ltx_Math" display="inline" id="S1.p4.11.m11.1"><semantics id="S1.p4.11.m11.1a"><mi class="ltx_font_mathcaligraphic" id="S1.p4.11.m11.1.1" xref="S1.p4.11.m11.1.1.cmml">ℳ</mi><annotation-xml encoding="MathML-Content" id="S1.p4.11.m11.1b"><ci id="S1.p4.11.m11.1.1.cmml" xref="S1.p4.11.m11.1.1">ℳ</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p4.11.m11.1c">\mathcal{M}</annotation><annotation encoding="application/x-llamapun" id="S1.p4.11.m11.1d">caligraphic_M</annotation></semantics></math>, the corresponding <math alttext="\eta" class="ltx_Math" display="inline" id="S1.p4.12.m12.1"><semantics id="S1.p4.12.m12.1a"><mi id="S1.p4.12.m12.1.1" xref="S1.p4.12.m12.1.1.cmml">η</mi><annotation-xml encoding="MathML-Content" id="S1.p4.12.m12.1b"><ci id="S1.p4.12.m12.1.1.cmml" xref="S1.p4.12.m12.1.1">𝜂</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p4.12.m12.1c">\eta</annotation><annotation encoding="application/x-llamapun" id="S1.p4.12.m12.1d">italic_η</annotation></semantics></math>-space belongs to the cotangent space <math alttext="T^{\star}\mathcal{M}" class="ltx_Math" display="inline" id="S1.p4.13.m13.1"><semantics id="S1.p4.13.m13.1a"><mrow id="S1.p4.13.m13.1.1" xref="S1.p4.13.m13.1.1.cmml"><msup id="S1.p4.13.m13.1.1.2" xref="S1.p4.13.m13.1.1.2.cmml"><mi id="S1.p4.13.m13.1.1.2.2" xref="S1.p4.13.m13.1.1.2.2.cmml">T</mi><mo id="S1.p4.13.m13.1.1.2.3" xref="S1.p4.13.m13.1.1.2.3.cmml">⋆</mo></msup><mo id="S1.p4.13.m13.1.1.1" xref="S1.p4.13.m13.1.1.1.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="S1.p4.13.m13.1.1.3" xref="S1.p4.13.m13.1.1.3.cmml">ℳ</mi></mrow><annotation-xml encoding="MathML-Content" id="S1.p4.13.m13.1b"><apply id="S1.p4.13.m13.1.1.cmml" xref="S1.p4.13.m13.1.1"><times id="S1.p4.13.m13.1.1.1.cmml" xref="S1.p4.13.m13.1.1.1"></times><apply id="S1.p4.13.m13.1.1.2.cmml" xref="S1.p4.13.m13.1.1.2"><csymbol cd="ambiguous" id="S1.p4.13.m13.1.1.2.1.cmml" xref="S1.p4.13.m13.1.1.2">superscript</csymbol><ci id="S1.p4.13.m13.1.1.2.2.cmml" xref="S1.p4.13.m13.1.1.2.2">𝑇</ci><ci id="S1.p4.13.m13.1.1.2.3.cmml" xref="S1.p4.13.m13.1.1.2.3">⋆</ci></apply><ci id="S1.p4.13.m13.1.1.3.cmml" xref="S1.p4.13.m13.1.1.3">ℳ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p4.13.m13.1c">T^{\star}\mathcal{M}</annotation><annotation encoding="application/x-llamapun" id="S1.p4.13.m13.1d">italic_T start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT caligraphic_M</annotation></semantics></math>. On the other side, when the <math alttext="\eta" class="ltx_Math" display="inline" id="S1.p4.14.m14.1"><semantics id="S1.p4.14.m14.1a"><mi id="S1.p4.14.m14.1.1" xref="S1.p4.14.m14.1.1.cmml">η</mi><annotation-xml encoding="MathML-Content" id="S1.p4.14.m14.1b"><ci id="S1.p4.14.m14.1.1.cmml" xref="S1.p4.14.m14.1.1">𝜂</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p4.14.m14.1c">\eta</annotation><annotation encoding="application/x-llamapun" id="S1.p4.14.m14.1d">italic_η</annotation></semantics></math>-space belongs to a curved space <math alttext="\mathcal{N}" class="ltx_Math" display="inline" id="S1.p4.15.m15.1"><semantics id="S1.p4.15.m15.1a"><mi class="ltx_font_mathcaligraphic" id="S1.p4.15.m15.1.1" xref="S1.p4.15.m15.1.1.cmml">𝒩</mi><annotation-xml encoding="MathML-Content" id="S1.p4.15.m15.1b"><ci id="S1.p4.15.m15.1.1.cmml" xref="S1.p4.15.m15.1.1">𝒩</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p4.15.m15.1c">\mathcal{N}</annotation><annotation encoding="application/x-llamapun" id="S1.p4.15.m15.1d">caligraphic_N</annotation></semantics></math>, the corresponding <math alttext="\theta" class="ltx_Math" display="inline" id="S1.p4.16.m16.1"><semantics id="S1.p4.16.m16.1a"><mi id="S1.p4.16.m16.1.1" xref="S1.p4.16.m16.1.1.cmml">θ</mi><annotation-xml encoding="MathML-Content" id="S1.p4.16.m16.1b"><ci id="S1.p4.16.m16.1.1.cmml" xref="S1.p4.16.m16.1.1">𝜃</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p4.16.m16.1c">\theta</annotation><annotation encoding="application/x-llamapun" id="S1.p4.16.m16.1d">italic_θ</annotation></semantics></math>-space belongs to the cotangent space <math alttext="T^{\star}\mathcal{N}" class="ltx_Math" display="inline" id="S1.p4.17.m17.1"><semantics id="S1.p4.17.m17.1a"><mrow id="S1.p4.17.m17.1.1" xref="S1.p4.17.m17.1.1.cmml"><msup id="S1.p4.17.m17.1.1.2" xref="S1.p4.17.m17.1.1.2.cmml"><mi id="S1.p4.17.m17.1.1.2.2" xref="S1.p4.17.m17.1.1.2.2.cmml">T</mi><mo id="S1.p4.17.m17.1.1.2.3" xref="S1.p4.17.m17.1.1.2.3.cmml">⋆</mo></msup><mo id="S1.p4.17.m17.1.1.1" xref="S1.p4.17.m17.1.1.1.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="S1.p4.17.m17.1.1.3" xref="S1.p4.17.m17.1.1.3.cmml">𝒩</mi></mrow><annotation-xml encoding="MathML-Content" id="S1.p4.17.m17.1b"><apply id="S1.p4.17.m17.1.1.cmml" xref="S1.p4.17.m17.1.1"><times id="S1.p4.17.m17.1.1.1.cmml" xref="S1.p4.17.m17.1.1.1"></times><apply id="S1.p4.17.m17.1.1.2.cmml" xref="S1.p4.17.m17.1.1.2"><csymbol cd="ambiguous" id="S1.p4.17.m17.1.1.2.1.cmml" xref="S1.p4.17.m17.1.1.2">superscript</csymbol><ci id="S1.p4.17.m17.1.1.2.2.cmml" xref="S1.p4.17.m17.1.1.2.2">𝑇</ci><ci id="S1.p4.17.m17.1.1.2.3.cmml" xref="S1.p4.17.m17.1.1.2.3">⋆</ci></apply><ci id="S1.p4.17.m17.1.1.3.cmml" xref="S1.p4.17.m17.1.1.3">𝒩</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p4.17.m17.1c">T^{\star}\mathcal{N}</annotation><annotation encoding="application/x-llamapun" id="S1.p4.17.m17.1d">italic_T start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT caligraphic_N</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S1.p5"> <p class="ltx_p" id="S1.p5.6">Throughout the paper, we use Einstein’s summation convention and assuming that a Latin index (e.g., <math alttext="i,j,k,\dots" class="ltx_Math" display="inline" id="S1.p5.1.m1.4"><semantics id="S1.p5.1.m1.4a"><mrow id="S1.p5.1.m1.4.5.2" xref="S1.p5.1.m1.4.5.1.cmml"><mi id="S1.p5.1.m1.1.1" xref="S1.p5.1.m1.1.1.cmml">i</mi><mo id="S1.p5.1.m1.4.5.2.1" xref="S1.p5.1.m1.4.5.1.cmml">,</mo><mi id="S1.p5.1.m1.2.2" xref="S1.p5.1.m1.2.2.cmml">j</mi><mo id="S1.p5.1.m1.4.5.2.2" xref="S1.p5.1.m1.4.5.1.cmml">,</mo><mi id="S1.p5.1.m1.3.3" xref="S1.p5.1.m1.3.3.cmml">k</mi><mo id="S1.p5.1.m1.4.5.2.3" xref="S1.p5.1.m1.4.5.1.cmml">,</mo><mi id="S1.p5.1.m1.4.4" mathvariant="normal" xref="S1.p5.1.m1.4.4.cmml">…</mi></mrow><annotation-xml encoding="MathML-Content" id="S1.p5.1.m1.4b"><list id="S1.p5.1.m1.4.5.1.cmml" xref="S1.p5.1.m1.4.5.2"><ci id="S1.p5.1.m1.1.1.cmml" xref="S1.p5.1.m1.1.1">𝑖</ci><ci id="S1.p5.1.m1.2.2.cmml" xref="S1.p5.1.m1.2.2">𝑗</ci><ci id="S1.p5.1.m1.3.3.cmml" xref="S1.p5.1.m1.3.3">𝑘</ci><ci id="S1.p5.1.m1.4.4.cmml" xref="S1.p5.1.m1.4.4">…</ci></list></annotation-xml><annotation encoding="application/x-tex" id="S1.p5.1.m1.4c">i,j,k,\dots</annotation><annotation encoding="application/x-llamapun" id="S1.p5.1.m1.4d">italic_i , italic_j , italic_k , …</annotation></semantics></math>) runs from <math alttext="1" class="ltx_Math" display="inline" id="S1.p5.2.m2.1"><semantics id="S1.p5.2.m2.1a"><mn id="S1.p5.2.m2.1.1" xref="S1.p5.2.m2.1.1.cmml">1</mn><annotation-xml encoding="MathML-Content" id="S1.p5.2.m2.1b"><cn id="S1.p5.2.m2.1.1.cmml" type="integer" xref="S1.p5.2.m2.1.1">1</cn></annotation-xml><annotation encoding="application/x-tex" id="S1.p5.2.m2.1c">1</annotation><annotation encoding="application/x-llamapun" id="S1.p5.2.m2.1d">1</annotation></semantics></math> to <math alttext="N" class="ltx_Math" display="inline" id="S1.p5.3.m3.1"><semantics id="S1.p5.3.m3.1a"><mi id="S1.p5.3.m3.1.1" xref="S1.p5.3.m3.1.1.cmml">N</mi><annotation-xml encoding="MathML-Content" id="S1.p5.3.m3.1b"><ci id="S1.p5.3.m3.1.1.cmml" xref="S1.p5.3.m3.1.1">𝑁</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p5.3.m3.1c">N</annotation><annotation encoding="application/x-llamapun" id="S1.p5.3.m3.1d">italic_N</annotation></semantics></math>, while a Greek index (e.g., <math alttext="\mu,\nu,\dots" class="ltx_Math" display="inline" id="S1.p5.4.m4.3"><semantics id="S1.p5.4.m4.3a"><mrow id="S1.p5.4.m4.3.4.2" xref="S1.p5.4.m4.3.4.1.cmml"><mi id="S1.p5.4.m4.1.1" xref="S1.p5.4.m4.1.1.cmml">μ</mi><mo id="S1.p5.4.m4.3.4.2.1" xref="S1.p5.4.m4.3.4.1.cmml">,</mo><mi id="S1.p5.4.m4.2.2" xref="S1.p5.4.m4.2.2.cmml">ν</mi><mo id="S1.p5.4.m4.3.4.2.2" xref="S1.p5.4.m4.3.4.1.cmml">,</mo><mi id="S1.p5.4.m4.3.3" mathvariant="normal" xref="S1.p5.4.m4.3.3.cmml">…</mi></mrow><annotation-xml encoding="MathML-Content" id="S1.p5.4.m4.3b"><list id="S1.p5.4.m4.3.4.1.cmml" xref="S1.p5.4.m4.3.4.2"><ci id="S1.p5.4.m4.1.1.cmml" xref="S1.p5.4.m4.1.1">𝜇</ci><ci id="S1.p5.4.m4.2.2.cmml" xref="S1.p5.4.m4.2.2">𝜈</ci><ci id="S1.p5.4.m4.3.3.cmml" xref="S1.p5.4.m4.3.3">…</ci></list></annotation-xml><annotation encoding="application/x-tex" id="S1.p5.4.m4.3c">\mu,\nu,\dots</annotation><annotation encoding="application/x-llamapun" id="S1.p5.4.m4.3d">italic_μ , italic_ν , …</annotation></semantics></math>) runs from <math alttext="0" class="ltx_Math" display="inline" id="S1.p5.5.m5.1"><semantics id="S1.p5.5.m5.1a"><mn id="S1.p5.5.m5.1.1" xref="S1.p5.5.m5.1.1.cmml">0</mn><annotation-xml encoding="MathML-Content" id="S1.p5.5.m5.1b"><cn id="S1.p5.5.m5.1.1.cmml" type="integer" xref="S1.p5.5.m5.1.1">0</cn></annotation-xml></semantics></math> to <math alttext="N" class="ltx_Math" display="inline" id="S1.p5.6.m6.1"><semantics id="S1.p5.6.m6.1a"><mi id="S1.p5.6.m6.1.1" xref="S1.p5.6.m6.1.1.cmml">N</mi><annotation-xml encoding="MathML-Content" id="S1.p5.6.m6.1b"><ci id="S1.p5.6.m6.1.1.cmml" xref="S1.p5.6.m6.1.1">𝑁</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p5.6.m6.1c">N</annotation><annotation encoding="application/x-llamapun" id="S1.p5.6.m6.1d">italic_N</annotation></semantics></math>.</p> </div> </section> <section class="ltx_section" id="S2"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">2 </span>Information Geometry and Gradient-Flow Equations</h2> <div class="ltx_para" id="S2.p1"> <p class="ltx_p" id="S2.p1.1">Here, some basics of IG and the gradient-flow equations are reviewed.</p> </div> <section class="ltx_subsection" id="S2.SS1"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">2.1 </span>Information Geometry</h3> <div class="ltx_para" id="S2.SS1.p1"> <p class="ltx_p" id="S2.SS1.p1.2">In IG <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2406.11224v2#bib.bib1" title="">1</a>]</cite>, the so-called <span class="ltx_text ltx_font_italic" id="S2.SS1.p1.2.1">dually-flat structures</span> are important. For a given set of some functions <math alttext="F_{i}(x),i=1,\dots,N" class="ltx_Math" display="inline" id="S2.SS1.p1.1.m1.6"><semantics id="S2.SS1.p1.1.m1.6a"><mrow id="S2.SS1.p1.1.m1.6.6.2" xref="S2.SS1.p1.1.m1.6.6.3.cmml"><mrow id="S2.SS1.p1.1.m1.5.5.1.1" xref="S2.SS1.p1.1.m1.5.5.1.1.cmml"><mrow id="S2.SS1.p1.1.m1.5.5.1.1.1.1" xref="S2.SS1.p1.1.m1.5.5.1.1.1.2.cmml"><mrow id="S2.SS1.p1.1.m1.5.5.1.1.1.1.1" xref="S2.SS1.p1.1.m1.5.5.1.1.1.1.1.cmml"><msub id="S2.SS1.p1.1.m1.5.5.1.1.1.1.1.2" xref="S2.SS1.p1.1.m1.5.5.1.1.1.1.1.2.cmml"><mi id="S2.SS1.p1.1.m1.5.5.1.1.1.1.1.2.2" xref="S2.SS1.p1.1.m1.5.5.1.1.1.1.1.2.2.cmml">F</mi><mi id="S2.SS1.p1.1.m1.5.5.1.1.1.1.1.2.3" xref="S2.SS1.p1.1.m1.5.5.1.1.1.1.1.2.3.cmml">i</mi></msub><mo id="S2.SS1.p1.1.m1.5.5.1.1.1.1.1.1" xref="S2.SS1.p1.1.m1.5.5.1.1.1.1.1.1.cmml">⁢</mo><mrow id="S2.SS1.p1.1.m1.5.5.1.1.1.1.1.3.2" xref="S2.SS1.p1.1.m1.5.5.1.1.1.1.1.cmml"><mo id="S2.SS1.p1.1.m1.5.5.1.1.1.1.1.3.2.1" stretchy="false" 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xref="S2.SS1.p1.1.m1.5.5.1.1.1.1.1.2.3">𝑖</ci></apply><ci id="S2.SS1.p1.1.m1.1.1.cmml" xref="S2.SS1.p1.1.m1.1.1">𝑥</ci></apply><ci id="S2.SS1.p1.1.m1.2.2.cmml" xref="S2.SS1.p1.1.m1.2.2">𝑖</ci></list><cn id="S2.SS1.p1.1.m1.5.5.1.1.3.cmml" type="integer" xref="S2.SS1.p1.1.m1.5.5.1.1.3">1</cn></apply><list id="S2.SS1.p1.1.m1.6.6.2.2.1.cmml" xref="S2.SS1.p1.1.m1.6.6.2.2.2"><ci id="S2.SS1.p1.1.m1.3.3.cmml" xref="S2.SS1.p1.1.m1.3.3">…</ci><ci id="S2.SS1.p1.1.m1.4.4.cmml" xref="S2.SS1.p1.1.m1.4.4">𝑁</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p1.1.m1.6c">F_{i}(x),i=1,\dots,N</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p1.1.m1.6d">italic_F start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_x ) , italic_i = 1 , … , italic_N</annotation></semantics></math>, the <math alttext="\theta" class="ltx_Math" display="inline" id="S2.SS1.p1.2.m2.1"><semantics id="S2.SS1.p1.2.m2.1a"><mi id="S2.SS1.p1.2.m2.1.1" xref="S2.SS1.p1.2.m2.1.1.cmml">θ</mi><annotation-xml encoding="MathML-Content" id="S2.SS1.p1.2.m2.1b"><ci id="S2.SS1.p1.2.m2.1.1.cmml" xref="S2.SS1.p1.2.m2.1.1">𝜃</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p1.2.m2.1c">\theta</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p1.2.m2.1d">italic_θ</annotation></semantics></math>-parametrized probability distribution function (pdf)</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="S3.EGx1"> <tbody id="S2.E1"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_left_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle p_{\theta}(x)=\exp\left[\theta^{i}F_{i}(x)-\Psi(\theta)\right]," class="ltx_Math" display="inline" id="S2.E1.m1.5"><semantics id="S2.E1.m1.5a"><mrow id="S2.E1.m1.5.5.1" xref="S2.E1.m1.5.5.1.1.cmml"><mrow id="S2.E1.m1.5.5.1.1" xref="S2.E1.m1.5.5.1.1.cmml"><mrow id="S2.E1.m1.5.5.1.1.3" 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xref="S2.E1.m1.5.5.1.1.1.1.1.1.2.3.2">𝐹</ci><ci id="S2.E1.m1.5.5.1.1.1.1.1.1.2.3.3.cmml" xref="S2.E1.m1.5.5.1.1.1.1.1.1.2.3.3">𝑖</ci></apply><ci id="S2.E1.m1.2.2.cmml" xref="S2.E1.m1.2.2">𝑥</ci></apply><apply id="S2.E1.m1.5.5.1.1.1.1.1.1.3.cmml" xref="S2.E1.m1.5.5.1.1.1.1.1.1.3"><times id="S2.E1.m1.5.5.1.1.1.1.1.1.3.1.cmml" xref="S2.E1.m1.5.5.1.1.1.1.1.1.3.1"></times><ci id="S2.E1.m1.5.5.1.1.1.1.1.1.3.2.cmml" xref="S2.E1.m1.5.5.1.1.1.1.1.1.3.2">Ψ</ci><ci id="S2.E1.m1.3.3.cmml" xref="S2.E1.m1.3.3">𝜃</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.E1.m1.5c">\displaystyle p_{\theta}(x)=\exp\left[\theta^{i}F_{i}(x)-\Psi(\theta)\right],</annotation><annotation encoding="application/x-llamapun" id="S2.E1.m1.5d">italic_p start_POSTSUBSCRIPT italic_θ end_POSTSUBSCRIPT ( italic_x ) = roman_exp [ italic_θ start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT italic_F start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_x ) - roman_Ψ ( italic_θ ) ] ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_left_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(1)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S2.SS1.p1.15">is called an exponential pdf. Here <math alttext="\Psi(\theta)" class="ltx_Math" display="inline" id="S2.SS1.p1.3.m1.1"><semantics id="S2.SS1.p1.3.m1.1a"><mrow id="S2.SS1.p1.3.m1.1.2" xref="S2.SS1.p1.3.m1.1.2.cmml"><mi id="S2.SS1.p1.3.m1.1.2.2" mathvariant="normal" xref="S2.SS1.p1.3.m1.1.2.2.cmml">Ψ</mi><mo id="S2.SS1.p1.3.m1.1.2.1" xref="S2.SS1.p1.3.m1.1.2.1.cmml">⁢</mo><mrow id="S2.SS1.p1.3.m1.1.2.3.2" xref="S2.SS1.p1.3.m1.1.2.cmml"><mo id="S2.SS1.p1.3.m1.1.2.3.2.1" stretchy="false" xref="S2.SS1.p1.3.m1.1.2.cmml">(</mo><mi id="S2.SS1.p1.3.m1.1.1" xref="S2.SS1.p1.3.m1.1.1.cmml">θ</mi><mo id="S2.SS1.p1.3.m1.1.2.3.2.2" stretchy="false" xref="S2.SS1.p1.3.m1.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p1.3.m1.1b"><apply id="S2.SS1.p1.3.m1.1.2.cmml" xref="S2.SS1.p1.3.m1.1.2"><times id="S2.SS1.p1.3.m1.1.2.1.cmml" xref="S2.SS1.p1.3.m1.1.2.1"></times><ci id="S2.SS1.p1.3.m1.1.2.2.cmml" xref="S2.SS1.p1.3.m1.1.2.2">Ψ</ci><ci id="S2.SS1.p1.3.m1.1.1.cmml" xref="S2.SS1.p1.3.m1.1.1">𝜃</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p1.3.m1.1c">\Psi(\theta)</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p1.3.m1.1d">roman_Ψ ( italic_θ )</annotation></semantics></math> is determined from the normalization of <math alttext="p_{\theta}(x)" class="ltx_Math" display="inline" id="S2.SS1.p1.4.m2.1"><semantics id="S2.SS1.p1.4.m2.1a"><mrow id="S2.SS1.p1.4.m2.1.2" xref="S2.SS1.p1.4.m2.1.2.cmml"><msub id="S2.SS1.p1.4.m2.1.2.2" xref="S2.SS1.p1.4.m2.1.2.2.cmml"><mi id="S2.SS1.p1.4.m2.1.2.2.2" xref="S2.SS1.p1.4.m2.1.2.2.2.cmml">p</mi><mi id="S2.SS1.p1.4.m2.1.2.2.3" xref="S2.SS1.p1.4.m2.1.2.2.3.cmml">θ</mi></msub><mo id="S2.SS1.p1.4.m2.1.2.1" xref="S2.SS1.p1.4.m2.1.2.1.cmml">⁢</mo><mrow id="S2.SS1.p1.4.m2.1.2.3.2" xref="S2.SS1.p1.4.m2.1.2.cmml"><mo id="S2.SS1.p1.4.m2.1.2.3.2.1" stretchy="false" xref="S2.SS1.p1.4.m2.1.2.cmml">(</mo><mi id="S2.SS1.p1.4.m2.1.1" xref="S2.SS1.p1.4.m2.1.1.cmml">x</mi><mo id="S2.SS1.p1.4.m2.1.2.3.2.2" stretchy="false" xref="S2.SS1.p1.4.m2.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p1.4.m2.1b"><apply id="S2.SS1.p1.4.m2.1.2.cmml" xref="S2.SS1.p1.4.m2.1.2"><times id="S2.SS1.p1.4.m2.1.2.1.cmml" xref="S2.SS1.p1.4.m2.1.2.1"></times><apply id="S2.SS1.p1.4.m2.1.2.2.cmml" xref="S2.SS1.p1.4.m2.1.2.2"><csymbol cd="ambiguous" id="S2.SS1.p1.4.m2.1.2.2.1.cmml" xref="S2.SS1.p1.4.m2.1.2.2">subscript</csymbol><ci id="S2.SS1.p1.4.m2.1.2.2.2.cmml" xref="S2.SS1.p1.4.m2.1.2.2.2">𝑝</ci><ci id="S2.SS1.p1.4.m2.1.2.2.3.cmml" xref="S2.SS1.p1.4.m2.1.2.2.3">𝜃</ci></apply><ci id="S2.SS1.p1.4.m2.1.1.cmml" xref="S2.SS1.p1.4.m2.1.1">𝑥</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p1.4.m2.1c">p_{\theta}(x)</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p1.4.m2.1d">italic_p start_POSTSUBSCRIPT italic_θ end_POSTSUBSCRIPT ( italic_x )</annotation></semantics></math> as <math 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encoding="application/x-llamapun" id="S2.SS1.p1.5.m3.5d">roman_Ψ ( italic_θ ) = roman_ln [ ∫ italic_d italic_x roman_exp ( italic_θ start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT italic_F start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_x ) ) ]</annotation></semantics></math>. A manifold of probability distribution, which is called <span class="ltx_text ltx_font_italic" id="S2.SS1.p1.15.1">statistical manifold</span> <math alttext="(\mathcal{M},g,\nabla,\nabla^{\star})" class="ltx_Math" display="inline" id="S2.SS1.p1.6.m4.4"><semantics id="S2.SS1.p1.6.m4.4a"><mrow id="S2.SS1.p1.6.m4.4.4.1" xref="S2.SS1.p1.6.m4.4.4.2.cmml"><mo id="S2.SS1.p1.6.m4.4.4.1.2" stretchy="false" xref="S2.SS1.p1.6.m4.4.4.2.cmml">(</mo><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p1.6.m4.1.1" xref="S2.SS1.p1.6.m4.1.1.cmml">ℳ</mi><mo id="S2.SS1.p1.6.m4.4.4.1.3" xref="S2.SS1.p1.6.m4.4.4.2.cmml">,</mo><mi id="S2.SS1.p1.6.m4.2.2" xref="S2.SS1.p1.6.m4.2.2.cmml">g</mi><mo id="S2.SS1.p1.6.m4.4.4.1.4" xref="S2.SS1.p1.6.m4.4.4.2.cmml">,</mo><mo id="S2.SS1.p1.6.m4.3.3" xref="S2.SS1.p1.6.m4.3.3.cmml">∇</mo><mo id="S2.SS1.p1.6.m4.4.4.1.5" xref="S2.SS1.p1.6.m4.4.4.2.cmml">,</mo><msup id="S2.SS1.p1.6.m4.4.4.1.1" xref="S2.SS1.p1.6.m4.4.4.1.1.cmml"><mo id="S2.SS1.p1.6.m4.4.4.1.1.2" xref="S2.SS1.p1.6.m4.4.4.1.1.2.cmml">∇</mo><mo id="S2.SS1.p1.6.m4.4.4.1.1.3" xref="S2.SS1.p1.6.m4.4.4.1.1.3.cmml">⋆</mo></msup><mo id="S2.SS1.p1.6.m4.4.4.1.6" stretchy="false" xref="S2.SS1.p1.6.m4.4.4.2.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p1.6.m4.4b"><vector id="S2.SS1.p1.6.m4.4.4.2.cmml" xref="S2.SS1.p1.6.m4.4.4.1"><ci id="S2.SS1.p1.6.m4.1.1.cmml" xref="S2.SS1.p1.6.m4.1.1">ℳ</ci><ci id="S2.SS1.p1.6.m4.2.2.cmml" xref="S2.SS1.p1.6.m4.2.2">𝑔</ci><ci id="S2.SS1.p1.6.m4.3.3.cmml" xref="S2.SS1.p1.6.m4.3.3">∇</ci><apply id="S2.SS1.p1.6.m4.4.4.1.1.cmml" xref="S2.SS1.p1.6.m4.4.4.1.1"><csymbol cd="ambiguous" id="S2.SS1.p1.6.m4.4.4.1.1.1.cmml" xref="S2.SS1.p1.6.m4.4.4.1.1">superscript</csymbol><ci id="S2.SS1.p1.6.m4.4.4.1.1.2.cmml" xref="S2.SS1.p1.6.m4.4.4.1.1.2">∇</ci><ci id="S2.SS1.p1.6.m4.4.4.1.1.3.cmml" xref="S2.SS1.p1.6.m4.4.4.1.1.3">⋆</ci></apply></vector></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p1.6.m4.4c">(\mathcal{M},g,\nabla,\nabla^{\star})</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p1.6.m4.4d">( caligraphic_M , italic_g , ∇ , ∇ start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT )</annotation></semantics></math>, is characterized by a pseudo-Riemannian metric <math alttext="g" class="ltx_Math" display="inline" id="S2.SS1.p1.7.m5.1"><semantics id="S2.SS1.p1.7.m5.1a"><mi id="S2.SS1.p1.7.m5.1.1" xref="S2.SS1.p1.7.m5.1.1.cmml">g</mi><annotation-xml encoding="MathML-Content" id="S2.SS1.p1.7.m5.1b"><ci id="S2.SS1.p1.7.m5.1.1.cmml" xref="S2.SS1.p1.7.m5.1.1">𝑔</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p1.7.m5.1c">g</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p1.7.m5.1d">italic_g</annotation></semantics></math>, and torsion-less dual affine connections <math alttext="\nabla" class="ltx_Math" display="inline" id="S2.SS1.p1.8.m6.1"><semantics id="S2.SS1.p1.8.m6.1a"><mo id="S2.SS1.p1.8.m6.1.1" xref="S2.SS1.p1.8.m6.1.1.cmml">∇</mo><annotation-xml encoding="MathML-Content" id="S2.SS1.p1.8.m6.1b"><ci id="S2.SS1.p1.8.m6.1.1.cmml" xref="S2.SS1.p1.8.m6.1.1">∇</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p1.8.m6.1c">\nabla</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p1.8.m6.1d">∇</annotation></semantics></math> and <math alttext="\nabla^{\star}" class="ltx_Math" display="inline" id="S2.SS1.p1.9.m7.1"><semantics id="S2.SS1.p1.9.m7.1a"><msup id="S2.SS1.p1.9.m7.1.1" xref="S2.SS1.p1.9.m7.1.1.cmml"><mo id="S2.SS1.p1.9.m7.1.1.2" xref="S2.SS1.p1.9.m7.1.1.2.cmml">∇</mo><mo id="S2.SS1.p1.9.m7.1.1.3" xref="S2.SS1.p1.9.m7.1.1.3.cmml">⋆</mo></msup><annotation-xml encoding="MathML-Content" id="S2.SS1.p1.9.m7.1b"><apply id="S2.SS1.p1.9.m7.1.1.cmml" xref="S2.SS1.p1.9.m7.1.1"><csymbol cd="ambiguous" id="S2.SS1.p1.9.m7.1.1.1.cmml" xref="S2.SS1.p1.9.m7.1.1">superscript</csymbol><ci id="S2.SS1.p1.9.m7.1.1.2.cmml" xref="S2.SS1.p1.9.m7.1.1.2">∇</ci><ci id="S2.SS1.p1.9.m7.1.1.3.cmml" xref="S2.SS1.p1.9.m7.1.1.3">⋆</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p1.9.m7.1c">\nabla^{\star}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p1.9.m7.1d">∇ start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT</annotation></semantics></math>. For a given convex function <math alttext="\Psi(\theta)" class="ltx_Math" display="inline" id="S2.SS1.p1.10.m8.1"><semantics id="S2.SS1.p1.10.m8.1a"><mrow id="S2.SS1.p1.10.m8.1.2" xref="S2.SS1.p1.10.m8.1.2.cmml"><mi id="S2.SS1.p1.10.m8.1.2.2" mathvariant="normal" xref="S2.SS1.p1.10.m8.1.2.2.cmml">Ψ</mi><mo id="S2.SS1.p1.10.m8.1.2.1" xref="S2.SS1.p1.10.m8.1.2.1.cmml">⁢</mo><mrow id="S2.SS1.p1.10.m8.1.2.3.2" xref="S2.SS1.p1.10.m8.1.2.cmml"><mo id="S2.SS1.p1.10.m8.1.2.3.2.1" stretchy="false" xref="S2.SS1.p1.10.m8.1.2.cmml">(</mo><mi id="S2.SS1.p1.10.m8.1.1" xref="S2.SS1.p1.10.m8.1.1.cmml">θ</mi><mo id="S2.SS1.p1.10.m8.1.2.3.2.2" stretchy="false" xref="S2.SS1.p1.10.m8.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p1.10.m8.1b"><apply id="S2.SS1.p1.10.m8.1.2.cmml" xref="S2.SS1.p1.10.m8.1.2"><times id="S2.SS1.p1.10.m8.1.2.1.cmml" xref="S2.SS1.p1.10.m8.1.2.1"></times><ci id="S2.SS1.p1.10.m8.1.2.2.cmml" xref="S2.SS1.p1.10.m8.1.2.2">Ψ</ci><ci id="S2.SS1.p1.10.m8.1.1.cmml" xref="S2.SS1.p1.10.m8.1.1">𝜃</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p1.10.m8.1c">\Psi(\theta)</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p1.10.m8.1d">roman_Ψ ( italic_θ )</annotation></semantics></math> together with its dual convex function <math alttext="\Psi^{\star}(\eta)" class="ltx_Math" display="inline" id="S2.SS1.p1.11.m9.1"><semantics id="S2.SS1.p1.11.m9.1a"><mrow id="S2.SS1.p1.11.m9.1.2" xref="S2.SS1.p1.11.m9.1.2.cmml"><msup id="S2.SS1.p1.11.m9.1.2.2" xref="S2.SS1.p1.11.m9.1.2.2.cmml"><mi id="S2.SS1.p1.11.m9.1.2.2.2" mathvariant="normal" xref="S2.SS1.p1.11.m9.1.2.2.2.cmml">Ψ</mi><mo id="S2.SS1.p1.11.m9.1.2.2.3" xref="S2.SS1.p1.11.m9.1.2.2.3.cmml">⋆</mo></msup><mo id="S2.SS1.p1.11.m9.1.2.1" xref="S2.SS1.p1.11.m9.1.2.1.cmml">⁢</mo><mrow id="S2.SS1.p1.11.m9.1.2.3.2" xref="S2.SS1.p1.11.m9.1.2.cmml"><mo id="S2.SS1.p1.11.m9.1.2.3.2.1" stretchy="false" xref="S2.SS1.p1.11.m9.1.2.cmml">(</mo><mi id="S2.SS1.p1.11.m9.1.1" xref="S2.SS1.p1.11.m9.1.1.cmml">η</mi><mo id="S2.SS1.p1.11.m9.1.2.3.2.2" stretchy="false" xref="S2.SS1.p1.11.m9.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p1.11.m9.1b"><apply id="S2.SS1.p1.11.m9.1.2.cmml" xref="S2.SS1.p1.11.m9.1.2"><times id="S2.SS1.p1.11.m9.1.2.1.cmml" xref="S2.SS1.p1.11.m9.1.2.1"></times><apply id="S2.SS1.p1.11.m9.1.2.2.cmml" xref="S2.SS1.p1.11.m9.1.2.2"><csymbol cd="ambiguous" id="S2.SS1.p1.11.m9.1.2.2.1.cmml" xref="S2.SS1.p1.11.m9.1.2.2">superscript</csymbol><ci id="S2.SS1.p1.11.m9.1.2.2.2.cmml" xref="S2.SS1.p1.11.m9.1.2.2.2">Ψ</ci><ci id="S2.SS1.p1.11.m9.1.2.2.3.cmml" xref="S2.SS1.p1.11.m9.1.2.2.3">⋆</ci></apply><ci id="S2.SS1.p1.11.m9.1.1.cmml" xref="S2.SS1.p1.11.m9.1.1">𝜂</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p1.11.m9.1c">\Psi^{\star}(\eta)</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p1.11.m9.1d">roman_Ψ start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT ( italic_η )</annotation></semantics></math>, one can construct the dually-flat structure as follows. From the dual convex functions <math alttext="\Psi^{\star}(\eta)" class="ltx_Math" display="inline" id="S2.SS1.p1.12.m10.1"><semantics id="S2.SS1.p1.12.m10.1a"><mrow id="S2.SS1.p1.12.m10.1.2" xref="S2.SS1.p1.12.m10.1.2.cmml"><msup id="S2.SS1.p1.12.m10.1.2.2" xref="S2.SS1.p1.12.m10.1.2.2.cmml"><mi id="S2.SS1.p1.12.m10.1.2.2.2" mathvariant="normal" xref="S2.SS1.p1.12.m10.1.2.2.2.cmml">Ψ</mi><mo id="S2.SS1.p1.12.m10.1.2.2.3" xref="S2.SS1.p1.12.m10.1.2.2.3.cmml">⋆</mo></msup><mo id="S2.SS1.p1.12.m10.1.2.1" xref="S2.SS1.p1.12.m10.1.2.1.cmml">⁢</mo><mrow id="S2.SS1.p1.12.m10.1.2.3.2" xref="S2.SS1.p1.12.m10.1.2.cmml"><mo id="S2.SS1.p1.12.m10.1.2.3.2.1" stretchy="false" xref="S2.SS1.p1.12.m10.1.2.cmml">(</mo><mi id="S2.SS1.p1.12.m10.1.1" xref="S2.SS1.p1.12.m10.1.1.cmml">η</mi><mo id="S2.SS1.p1.12.m10.1.2.3.2.2" stretchy="false" xref="S2.SS1.p1.12.m10.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p1.12.m10.1b"><apply id="S2.SS1.p1.12.m10.1.2.cmml" xref="S2.SS1.p1.12.m10.1.2"><times id="S2.SS1.p1.12.m10.1.2.1.cmml" xref="S2.SS1.p1.12.m10.1.2.1"></times><apply id="S2.SS1.p1.12.m10.1.2.2.cmml" xref="S2.SS1.p1.12.m10.1.2.2"><csymbol cd="ambiguous" id="S2.SS1.p1.12.m10.1.2.2.1.cmml" xref="S2.SS1.p1.12.m10.1.2.2">superscript</csymbol><ci id="S2.SS1.p1.12.m10.1.2.2.2.cmml" xref="S2.SS1.p1.12.m10.1.2.2.2">Ψ</ci><ci id="S2.SS1.p1.12.m10.1.2.2.3.cmml" xref="S2.SS1.p1.12.m10.1.2.2.3">⋆</ci></apply><ci id="S2.SS1.p1.12.m10.1.1.cmml" xref="S2.SS1.p1.12.m10.1.1">𝜂</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p1.12.m10.1c">\Psi^{\star}(\eta)</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p1.12.m10.1d">roman_Ψ start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT ( italic_η )</annotation></semantics></math> and <math alttext="\Psi(\theta)" class="ltx_Math" display="inline" id="S2.SS1.p1.13.m11.1"><semantics id="S2.SS1.p1.13.m11.1a"><mrow id="S2.SS1.p1.13.m11.1.2" xref="S2.SS1.p1.13.m11.1.2.cmml"><mi id="S2.SS1.p1.13.m11.1.2.2" mathvariant="normal" xref="S2.SS1.p1.13.m11.1.2.2.cmml">Ψ</mi><mo id="S2.SS1.p1.13.m11.1.2.1" xref="S2.SS1.p1.13.m11.1.2.1.cmml">⁢</mo><mrow id="S2.SS1.p1.13.m11.1.2.3.2" xref="S2.SS1.p1.13.m11.1.2.cmml"><mo id="S2.SS1.p1.13.m11.1.2.3.2.1" stretchy="false" xref="S2.SS1.p1.13.m11.1.2.cmml">(</mo><mi id="S2.SS1.p1.13.m11.1.1" xref="S2.SS1.p1.13.m11.1.1.cmml">θ</mi><mo id="S2.SS1.p1.13.m11.1.2.3.2.2" stretchy="false" xref="S2.SS1.p1.13.m11.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p1.13.m11.1b"><apply id="S2.SS1.p1.13.m11.1.2.cmml" xref="S2.SS1.p1.13.m11.1.2"><times id="S2.SS1.p1.13.m11.1.2.1.cmml" xref="S2.SS1.p1.13.m11.1.2.1"></times><ci id="S2.SS1.p1.13.m11.1.2.2.cmml" xref="S2.SS1.p1.13.m11.1.2.2">Ψ</ci><ci id="S2.SS1.p1.13.m11.1.1.cmml" xref="S2.SS1.p1.13.m11.1.1">𝜃</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p1.13.m11.1c">\Psi(\theta)</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p1.13.m11.1d">roman_Ψ ( italic_θ )</annotation></semantics></math>, the associated dual affine coordinates <math alttext="\theta^{i}" class="ltx_Math" display="inline" id="S2.SS1.p1.14.m12.1"><semantics id="S2.SS1.p1.14.m12.1a"><msup id="S2.SS1.p1.14.m12.1.1" xref="S2.SS1.p1.14.m12.1.1.cmml"><mi id="S2.SS1.p1.14.m12.1.1.2" xref="S2.SS1.p1.14.m12.1.1.2.cmml">θ</mi><mi id="S2.SS1.p1.14.m12.1.1.3" xref="S2.SS1.p1.14.m12.1.1.3.cmml">i</mi></msup><annotation-xml encoding="MathML-Content" id="S2.SS1.p1.14.m12.1b"><apply id="S2.SS1.p1.14.m12.1.1.cmml" xref="S2.SS1.p1.14.m12.1.1"><csymbol cd="ambiguous" id="S2.SS1.p1.14.m12.1.1.1.cmml" xref="S2.SS1.p1.14.m12.1.1">superscript</csymbol><ci id="S2.SS1.p1.14.m12.1.1.2.cmml" xref="S2.SS1.p1.14.m12.1.1.2">𝜃</ci><ci id="S2.SS1.p1.14.m12.1.1.3.cmml" xref="S2.SS1.p1.14.m12.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p1.14.m12.1c">\theta^{i}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p1.14.m12.1d">italic_θ start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT</annotation></semantics></math> and <math alttext="\eta_{i}" class="ltx_Math" display="inline" id="S2.SS1.p1.15.m13.1"><semantics id="S2.SS1.p1.15.m13.1a"><msub id="S2.SS1.p1.15.m13.1.1" xref="S2.SS1.p1.15.m13.1.1.cmml"><mi id="S2.SS1.p1.15.m13.1.1.2" xref="S2.SS1.p1.15.m13.1.1.2.cmml">η</mi><mi id="S2.SS1.p1.15.m13.1.1.3" xref="S2.SS1.p1.15.m13.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S2.SS1.p1.15.m13.1b"><apply id="S2.SS1.p1.15.m13.1.1.cmml" xref="S2.SS1.p1.15.m13.1.1"><csymbol cd="ambiguous" id="S2.SS1.p1.15.m13.1.1.1.cmml" xref="S2.SS1.p1.15.m13.1.1">subscript</csymbol><ci id="S2.SS1.p1.15.m13.1.1.2.cmml" xref="S2.SS1.p1.15.m13.1.1.2">𝜂</ci><ci id="S2.SS1.p1.15.m13.1.1.3.cmml" xref="S2.SS1.p1.15.m13.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p1.15.m13.1c">\eta_{i}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p1.15.m13.1d">italic_η start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> are obtained as</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="S3.EGx2"> <tbody id="S2.E2"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_left_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\theta^{i}=\frac{\partial\Psi^{\star}(\eta)}{\partial\eta_{i}},% \quad\eta_{i}=\frac{\partial\Psi(\theta)}{\partial\theta^{i}}," class="ltx_Math" display="inline" id="S2.E2.m1.3"><semantics id="S2.E2.m1.3a"><mrow id="S2.E2.m1.3.3.1"><mrow id="S2.E2.m1.3.3.1.1.2" xref="S2.E2.m1.3.3.1.1.3.cmml"><mrow id="S2.E2.m1.3.3.1.1.1.1" xref="S2.E2.m1.3.3.1.1.1.1.cmml"><msup id="S2.E2.m1.3.3.1.1.1.1.2" xref="S2.E2.m1.3.3.1.1.1.1.2.cmml"><mi id="S2.E2.m1.3.3.1.1.1.1.2.2" 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xref="S2.E2.m1.1.1.1.1.cmml">η</mi><mo id="S2.E2.m1.1.1.1.3.3.2.2" stretchy="false" xref="S2.E2.m1.1.1.1.3.cmml">)</mo></mrow></mrow></mrow><mrow id="S2.E2.m1.1.1.3" xref="S2.E2.m1.1.1.3.cmml"><mo id="S2.E2.m1.1.1.3.1" rspace="0em" xref="S2.E2.m1.1.1.3.1.cmml">∂</mo><msub id="S2.E2.m1.1.1.3.2" xref="S2.E2.m1.1.1.3.2.cmml"><mi id="S2.E2.m1.1.1.3.2.2" xref="S2.E2.m1.1.1.3.2.2.cmml">η</mi><mi id="S2.E2.m1.1.1.3.2.3" xref="S2.E2.m1.1.1.3.2.3.cmml">i</mi></msub></mrow></mfrac></mstyle></mrow><mo id="S2.E2.m1.3.3.1.1.2.3" rspace="1.167em" xref="S2.E2.m1.3.3.1.1.3a.cmml">,</mo><mrow id="S2.E2.m1.3.3.1.1.2.2" xref="S2.E2.m1.3.3.1.1.2.2.cmml"><msub id="S2.E2.m1.3.3.1.1.2.2.2" xref="S2.E2.m1.3.3.1.1.2.2.2.cmml"><mi id="S2.E2.m1.3.3.1.1.2.2.2.2" xref="S2.E2.m1.3.3.1.1.2.2.2.2.cmml">η</mi><mi id="S2.E2.m1.3.3.1.1.2.2.2.3" xref="S2.E2.m1.3.3.1.1.2.2.2.3.cmml">i</mi></msub><mo id="S2.E2.m1.3.3.1.1.2.2.1" xref="S2.E2.m1.3.3.1.1.2.2.1.cmml">=</mo><mstyle displaystyle="true" id="S2.E2.m1.2.2" 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xref="S2.E2.m1.2.2.3.2.3.cmml">i</mi></msup></mrow></mfrac></mstyle></mrow></mrow><mo id="S2.E2.m1.3.3.1.2">,</mo></mrow><annotation-xml encoding="MathML-Content" id="S2.E2.m1.3b"><apply id="S2.E2.m1.3.3.1.1.3.cmml" xref="S2.E2.m1.3.3.1.1.2"><csymbol cd="ambiguous" id="S2.E2.m1.3.3.1.1.3a.cmml" xref="S2.E2.m1.3.3.1.1.2.3">formulae-sequence</csymbol><apply id="S2.E2.m1.3.3.1.1.1.1.cmml" xref="S2.E2.m1.3.3.1.1.1.1"><eq id="S2.E2.m1.3.3.1.1.1.1.1.cmml" xref="S2.E2.m1.3.3.1.1.1.1.1"></eq><apply id="S2.E2.m1.3.3.1.1.1.1.2.cmml" xref="S2.E2.m1.3.3.1.1.1.1.2"><csymbol cd="ambiguous" id="S2.E2.m1.3.3.1.1.1.1.2.1.cmml" xref="S2.E2.m1.3.3.1.1.1.1.2">superscript</csymbol><ci id="S2.E2.m1.3.3.1.1.1.1.2.2.cmml" xref="S2.E2.m1.3.3.1.1.1.1.2.2">𝜃</ci><ci id="S2.E2.m1.3.3.1.1.1.1.2.3.cmml" xref="S2.E2.m1.3.3.1.1.1.1.2.3">𝑖</ci></apply><apply id="S2.E2.m1.1.1.cmml" xref="S2.E2.m1.1.1"><divide id="S2.E2.m1.1.1.2.cmml" xref="S2.E2.m1.1.1"></divide><apply id="S2.E2.m1.1.1.1.cmml" 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id="S2.E2.m1.2.2.1.1.cmml" xref="S2.E2.m1.2.2.1.1">𝜃</ci></apply></apply><apply id="S2.E2.m1.2.2.3.cmml" xref="S2.E2.m1.2.2.3"><partialdiff id="S2.E2.m1.2.2.3.1.cmml" xref="S2.E2.m1.2.2.3.1"></partialdiff><apply id="S2.E2.m1.2.2.3.2.cmml" xref="S2.E2.m1.2.2.3.2"><csymbol cd="ambiguous" id="S2.E2.m1.2.2.3.2.1.cmml" xref="S2.E2.m1.2.2.3.2">superscript</csymbol><ci id="S2.E2.m1.2.2.3.2.2.cmml" xref="S2.E2.m1.2.2.3.2.2">𝜃</ci><ci id="S2.E2.m1.2.2.3.2.3.cmml" xref="S2.E2.m1.2.2.3.2.3">𝑖</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.E2.m1.3c">\displaystyle\theta^{i}=\frac{\partial\Psi^{\star}(\eta)}{\partial\eta_{i}},% \quad\eta_{i}=\frac{\partial\Psi(\theta)}{\partial\theta^{i}},</annotation><annotation encoding="application/x-llamapun" id="S2.E2.m1.3d">italic_θ start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT = divide start_ARG ∂ roman_Ψ start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT ( italic_η ) end_ARG start_ARG ∂ italic_η start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_ARG , italic_η start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = divide start_ARG ∂ roman_Ψ ( italic_θ ) end_ARG start_ARG ∂ italic_θ start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT end_ARG ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_left_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(2)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S2.SS1.p1.16">respectively. These convex functions are Legendre dual to each other</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="S3.EGx3"> <tbody id="S2.E3"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_left_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\Psi^{\star}(\eta)=\theta^{i}\,\eta_{i}-\Psi(\theta)." class="ltx_Math" display="inline" id="S2.E3.m1.3"><semantics id="S2.E3.m1.3a"><mrow id="S2.E3.m1.3.3.1" xref="S2.E3.m1.3.3.1.1.cmml"><mrow id="S2.E3.m1.3.3.1.1" xref="S2.E3.m1.3.3.1.1.cmml"><mrow id="S2.E3.m1.3.3.1.1.2" xref="S2.E3.m1.3.3.1.1.2.cmml"><msup id="S2.E3.m1.3.3.1.1.2.2" xref="S2.E3.m1.3.3.1.1.2.2.cmml"><mi id="S2.E3.m1.3.3.1.1.2.2.2" mathvariant="normal" xref="S2.E3.m1.3.3.1.1.2.2.2.cmml">Ψ</mi><mo id="S2.E3.m1.3.3.1.1.2.2.3" xref="S2.E3.m1.3.3.1.1.2.2.3.cmml">⋆</mo></msup><mo id="S2.E3.m1.3.3.1.1.2.1" xref="S2.E3.m1.3.3.1.1.2.1.cmml">⁢</mo><mrow id="S2.E3.m1.3.3.1.1.2.3.2" 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xref="S2.E3.m1.3.3.1.1.3.2.2.2">𝜃</ci><ci id="S2.E3.m1.3.3.1.1.3.2.2.3.cmml" xref="S2.E3.m1.3.3.1.1.3.2.2.3">𝑖</ci></apply><apply id="S2.E3.m1.3.3.1.1.3.2.3.cmml" xref="S2.E3.m1.3.3.1.1.3.2.3"><csymbol cd="ambiguous" id="S2.E3.m1.3.3.1.1.3.2.3.1.cmml" xref="S2.E3.m1.3.3.1.1.3.2.3">subscript</csymbol><ci id="S2.E3.m1.3.3.1.1.3.2.3.2.cmml" xref="S2.E3.m1.3.3.1.1.3.2.3.2">𝜂</ci><ci id="S2.E3.m1.3.3.1.1.3.2.3.3.cmml" xref="S2.E3.m1.3.3.1.1.3.2.3.3">𝑖</ci></apply></apply><apply id="S2.E3.m1.3.3.1.1.3.3.cmml" xref="S2.E3.m1.3.3.1.1.3.3"><times id="S2.E3.m1.3.3.1.1.3.3.1.cmml" xref="S2.E3.m1.3.3.1.1.3.3.1"></times><ci id="S2.E3.m1.3.3.1.1.3.3.2.cmml" xref="S2.E3.m1.3.3.1.1.3.3.2">Ψ</ci><ci id="S2.E3.m1.2.2.cmml" xref="S2.E3.m1.2.2">𝜃</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.E3.m1.3c">\displaystyle\Psi^{\star}(\eta)=\theta^{i}\,\eta_{i}-\Psi(\theta).</annotation><annotation encoding="application/x-llamapun" id="S2.E3.m1.3d">roman_Ψ start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT ( italic_η ) = italic_θ start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT italic_η start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT - roman_Ψ ( italic_θ ) .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_left_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(3)</span></td> </tr></tbody> </table> </div> <div class="ltx_para" id="S2.SS1.p2"> <p class="ltx_p" id="S2.SS1.p2.10">Taking logarithm of both sides of (<a class="ltx_ref" href="https://arxiv.org/html/2406.11224v2#S2.E1" title="In 2.1 Information Geometry ‣ 2 Information Geometry and Gradient-Flow Equations ‣ A Hamiltonian approach to the gradient-flow equations in information geometry"><span class="ltx_text ltx_ref_tag">1</span></a>) and taking expectation, we have</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="S3.EGx4"> <tbody id="S2.E4"><tr 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italic_p start_POSTSUBSCRIPT italic_θ end_POSTSUBSCRIPT end_POSTSUBSCRIPT [ italic_F start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_x ) ] - roman_Ψ ( italic_θ ) ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_left_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(4)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S2.SS1.p2.11">where</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="S3.EGx5"> <tbody id="S2.E5"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_left_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle{\rm E}_{p_{\theta}}[f(x)]:=\int dx\,p_{\theta}(x)f(x)," class="ltx_Math" display="inline" id="S2.E5.m1.4"><semantics id="S2.E5.m1.4a"><mrow id="S2.E5.m1.4.4.1" xref="S2.E5.m1.4.4.1.1.cmml"><mrow id="S2.E5.m1.4.4.1.1" 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encoding="application/x-tex" id="S2.E5.m1.4c">\displaystyle{\rm E}_{p_{\theta}}[f(x)]:=\int dx\,p_{\theta}(x)f(x),</annotation><annotation encoding="application/x-llamapun" id="S2.E5.m1.4d">roman_E start_POSTSUBSCRIPT italic_p start_POSTSUBSCRIPT italic_θ end_POSTSUBSCRIPT end_POSTSUBSCRIPT [ italic_f ( italic_x ) ] := ∫ italic_d italic_x italic_p start_POSTSUBSCRIPT italic_θ end_POSTSUBSCRIPT ( italic_x ) italic_f ( italic_x ) ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_left_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(5)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S2.SS1.p2.2">denotes the expectation value of a function <math alttext="f(x)" class="ltx_Math" display="inline" id="S2.SS1.p2.1.m1.1"><semantics id="S2.SS1.p2.1.m1.1a"><mrow id="S2.SS1.p2.1.m1.1.2" xref="S2.SS1.p2.1.m1.1.2.cmml"><mi id="S2.SS1.p2.1.m1.1.2.2" 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respect to <math alttext="p_{\theta}(x)" class="ltx_Math" display="inline" id="S2.SS1.p2.2.m2.1"><semantics id="S2.SS1.p2.2.m2.1a"><mrow id="S2.SS1.p2.2.m2.1.2" xref="S2.SS1.p2.2.m2.1.2.cmml"><msub id="S2.SS1.p2.2.m2.1.2.2" xref="S2.SS1.p2.2.m2.1.2.2.cmml"><mi id="S2.SS1.p2.2.m2.1.2.2.2" xref="S2.SS1.p2.2.m2.1.2.2.2.cmml">p</mi><mi id="S2.SS1.p2.2.m2.1.2.2.3" xref="S2.SS1.p2.2.m2.1.2.2.3.cmml">θ</mi></msub><mo id="S2.SS1.p2.2.m2.1.2.1" xref="S2.SS1.p2.2.m2.1.2.1.cmml">⁢</mo><mrow id="S2.SS1.p2.2.m2.1.2.3.2" xref="S2.SS1.p2.2.m2.1.2.cmml"><mo id="S2.SS1.p2.2.m2.1.2.3.2.1" stretchy="false" xref="S2.SS1.p2.2.m2.1.2.cmml">(</mo><mi id="S2.SS1.p2.2.m2.1.1" xref="S2.SS1.p2.2.m2.1.1.cmml">x</mi><mo id="S2.SS1.p2.2.m2.1.2.3.2.2" stretchy="false" xref="S2.SS1.p2.2.m2.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p2.2.m2.1b"><apply id="S2.SS1.p2.2.m2.1.2.cmml" xref="S2.SS1.p2.2.m2.1.2"><times id="S2.SS1.p2.2.m2.1.2.1.cmml" xref="S2.SS1.p2.2.m2.1.2.1"></times><apply id="S2.SS1.p2.2.m2.1.2.2.cmml" xref="S2.SS1.p2.2.m2.1.2.2"><csymbol cd="ambiguous" id="S2.SS1.p2.2.m2.1.2.2.1.cmml" xref="S2.SS1.p2.2.m2.1.2.2">subscript</csymbol><ci id="S2.SS1.p2.2.m2.1.2.2.2.cmml" xref="S2.SS1.p2.2.m2.1.2.2.2">𝑝</ci><ci id="S2.SS1.p2.2.m2.1.2.2.3.cmml" xref="S2.SS1.p2.2.m2.1.2.2.3">𝜃</ci></apply><ci id="S2.SS1.p2.2.m2.1.1.cmml" xref="S2.SS1.p2.2.m2.1.1">𝑥</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p2.2.m2.1c">p_{\theta}(x)</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p2.2.m2.1d">italic_p start_POSTSUBSCRIPT italic_θ end_POSTSUBSCRIPT ( italic_x )</annotation></semantics></math>. Comparing (<a class="ltx_ref" href="https://arxiv.org/html/2406.11224v2#S2.E3" title="In 2.1 Information Geometry ‣ 2 Information Geometry and Gradient-Flow Equations ‣ A Hamiltonian approach to the gradient-flow equations in information geometry"><span class="ltx_text ltx_ref_tag">3</span></a>) to (<a class="ltx_ref" href="https://arxiv.org/html/2406.11224v2#S2.E4" title="In 2.1 Information Geometry ‣ 2 Information Geometry and Gradient-Flow Equations ‣ A Hamiltonian approach to the gradient-flow equations in information geometry"><span class="ltx_text ltx_ref_tag">4</span></a>), we see that</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="S3.EGx6"> <tbody id="S2.E6"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_left_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\eta_{i}={\rm E}_{p_{\theta}}[F_{i}(x)]," class="ltx_Math" display="inline" id="S2.E6.m1.2"><semantics 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id="S2.E6.m1.1.1.cmml" xref="S2.E6.m1.1.1">𝑥</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.E6.m1.2c">\displaystyle\eta_{i}={\rm E}_{p_{\theta}}[F_{i}(x)],</annotation><annotation encoding="application/x-llamapun" id="S2.E6.m1.2d">italic_η start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = roman_E start_POSTSUBSCRIPT italic_p start_POSTSUBSCRIPT italic_θ end_POSTSUBSCRIPT end_POSTSUBSCRIPT [ italic_F start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_x ) ] ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_left_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(6)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S2.SS1.p2.12">and</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="S3.EGx7"> <tbody id="S2.E7"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_left_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\Psi^{\star}(\eta)={\rm E}_{p_{\theta}}[\ln p_{\theta}(x)]=:-S(% \eta)," class="ltx_math_unparsed" display="inline" id="S2.E7.m1.2"><semantics id="S2.E7.m1.2a"><mrow id="S2.E7.m1.2b"><msup id="S2.E7.m1.2.3"><mi id="S2.E7.m1.2.3.2" mathvariant="normal">Ψ</mi><mo id="S2.E7.m1.2.3.3">⋆</mo></msup><mrow id="S2.E7.m1.2.4"><mo id="S2.E7.m1.2.4.1" stretchy="false">(</mo><mi id="S2.E7.m1.1.1">η</mi><mo id="S2.E7.m1.2.4.2" stretchy="false">)</mo></mrow><mo id="S2.E7.m1.2.5">=</mo><msub id="S2.E7.m1.2.6"><mi id="S2.E7.m1.2.6.2" mathvariant="normal">E</mi><msub id="S2.E7.m1.2.6.3"><mi id="S2.E7.m1.2.6.3.2">p</mi><mi id="S2.E7.m1.2.6.3.3">θ</mi></msub></msub><mrow id="S2.E7.m1.2.7"><mo id="S2.E7.m1.2.7.1" stretchy="false">[</mo><mi id="S2.E7.m1.2.7.2">ln</mi><msub id="S2.E7.m1.2.7.3"><mi id="S2.E7.m1.2.7.3.2">p</mi><mi id="S2.E7.m1.2.7.3.3">θ</mi></msub><mrow id="S2.E7.m1.2.7.4"><mo id="S2.E7.m1.2.7.4.1" stretchy="false">(</mo><mi id="S2.E7.m1.2.2">x</mi><mo id="S2.E7.m1.2.7.4.2" stretchy="false">)</mo></mrow><mo id="S2.E7.m1.2.7.5" stretchy="false">]</mo></mrow><mo id="S2.E7.m1.2.8" rspace="0em">=</mo><mo id="S2.E7.m1.2.9" rspace="0em">:</mo><mo id="S2.E7.m1.2.10" lspace="0em">−</mo><mi id="S2.E7.m1.2.11">S</mi><mrow id="S2.E7.m1.2.12"><mo id="S2.E7.m1.2.12.1" stretchy="false">(</mo><mi id="S2.E7.m1.2.12.2">η</mi><mo id="S2.E7.m1.2.12.3" stretchy="false">)</mo></mrow><mo id="S2.E7.m1.2.13">,</mo></mrow><annotation encoding="application/x-tex" id="S2.E7.m1.2c">\displaystyle\Psi^{\star}(\eta)={\rm E}_{p_{\theta}}[\ln p_{\theta}(x)]=:-S(% \eta),</annotation><annotation encoding="application/x-llamapun" id="S2.E7.m1.2d">roman_Ψ start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT ( italic_η ) = roman_E start_POSTSUBSCRIPT italic_p start_POSTSUBSCRIPT italic_θ end_POSTSUBSCRIPT end_POSTSUBSCRIPT [ roman_ln italic_p start_POSTSUBSCRIPT italic_θ end_POSTSUBSCRIPT ( italic_x ) ] = : - italic_S ( italic_η ) ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_left_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(7)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S2.SS1.p2.7">where <math alttext="S(\eta)" class="ltx_Math" display="inline" id="S2.SS1.p2.3.m1.1"><semantics id="S2.SS1.p2.3.m1.1a"><mrow id="S2.SS1.p2.3.m1.1.2" xref="S2.SS1.p2.3.m1.1.2.cmml"><mi id="S2.SS1.p2.3.m1.1.2.2" xref="S2.SS1.p2.3.m1.1.2.2.cmml">S</mi><mo id="S2.SS1.p2.3.m1.1.2.1" xref="S2.SS1.p2.3.m1.1.2.1.cmml">⁢</mo><mrow id="S2.SS1.p2.3.m1.1.2.3.2" xref="S2.SS1.p2.3.m1.1.2.cmml"><mo id="S2.SS1.p2.3.m1.1.2.3.2.1" stretchy="false" xref="S2.SS1.p2.3.m1.1.2.cmml">(</mo><mi id="S2.SS1.p2.3.m1.1.1" xref="S2.SS1.p2.3.m1.1.1.cmml">η</mi><mo id="S2.SS1.p2.3.m1.1.2.3.2.2" stretchy="false" xref="S2.SS1.p2.3.m1.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p2.3.m1.1b"><apply id="S2.SS1.p2.3.m1.1.2.cmml" xref="S2.SS1.p2.3.m1.1.2"><times id="S2.SS1.p2.3.m1.1.2.1.cmml" xref="S2.SS1.p2.3.m1.1.2.1"></times><ci id="S2.SS1.p2.3.m1.1.2.2.cmml" xref="S2.SS1.p2.3.m1.1.2.2">𝑆</ci><ci id="S2.SS1.p2.3.m1.1.1.cmml" xref="S2.SS1.p2.3.m1.1.1">𝜂</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p2.3.m1.1c">S(\eta)</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p2.3.m1.1d">italic_S ( italic_η )</annotation></semantics></math> is the entropy. The positive definite matrices <math alttext="g_{ij}(\theta)" class="ltx_Math" display="inline" id="S2.SS1.p2.4.m2.1"><semantics id="S2.SS1.p2.4.m2.1a"><mrow id="S2.SS1.p2.4.m2.1.2" xref="S2.SS1.p2.4.m2.1.2.cmml"><msub id="S2.SS1.p2.4.m2.1.2.2" xref="S2.SS1.p2.4.m2.1.2.2.cmml"><mi id="S2.SS1.p2.4.m2.1.2.2.2" xref="S2.SS1.p2.4.m2.1.2.2.2.cmml">g</mi><mrow id="S2.SS1.p2.4.m2.1.2.2.3" xref="S2.SS1.p2.4.m2.1.2.2.3.cmml"><mi id="S2.SS1.p2.4.m2.1.2.2.3.2" xref="S2.SS1.p2.4.m2.1.2.2.3.2.cmml">i</mi><mo id="S2.SS1.p2.4.m2.1.2.2.3.1" xref="S2.SS1.p2.4.m2.1.2.2.3.1.cmml">⁢</mo><mi id="S2.SS1.p2.4.m2.1.2.2.3.3" xref="S2.SS1.p2.4.m2.1.2.2.3.3.cmml">j</mi></mrow></msub><mo id="S2.SS1.p2.4.m2.1.2.1" xref="S2.SS1.p2.4.m2.1.2.1.cmml">⁢</mo><mrow id="S2.SS1.p2.4.m2.1.2.3.2" xref="S2.SS1.p2.4.m2.1.2.cmml"><mo id="S2.SS1.p2.4.m2.1.2.3.2.1" stretchy="false" xref="S2.SS1.p2.4.m2.1.2.cmml">(</mo><mi id="S2.SS1.p2.4.m2.1.1" xref="S2.SS1.p2.4.m2.1.1.cmml">θ</mi><mo id="S2.SS1.p2.4.m2.1.2.3.2.2" stretchy="false" xref="S2.SS1.p2.4.m2.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p2.4.m2.1b"><apply id="S2.SS1.p2.4.m2.1.2.cmml" xref="S2.SS1.p2.4.m2.1.2"><times id="S2.SS1.p2.4.m2.1.2.1.cmml" xref="S2.SS1.p2.4.m2.1.2.1"></times><apply id="S2.SS1.p2.4.m2.1.2.2.cmml" xref="S2.SS1.p2.4.m2.1.2.2"><csymbol cd="ambiguous" id="S2.SS1.p2.4.m2.1.2.2.1.cmml" xref="S2.SS1.p2.4.m2.1.2.2">subscript</csymbol><ci id="S2.SS1.p2.4.m2.1.2.2.2.cmml" xref="S2.SS1.p2.4.m2.1.2.2.2">𝑔</ci><apply id="S2.SS1.p2.4.m2.1.2.2.3.cmml" xref="S2.SS1.p2.4.m2.1.2.2.3"><times id="S2.SS1.p2.4.m2.1.2.2.3.1.cmml" xref="S2.SS1.p2.4.m2.1.2.2.3.1"></times><ci id="S2.SS1.p2.4.m2.1.2.2.3.2.cmml" xref="S2.SS1.p2.4.m2.1.2.2.3.2">𝑖</ci><ci id="S2.SS1.p2.4.m2.1.2.2.3.3.cmml" xref="S2.SS1.p2.4.m2.1.2.2.3.3">𝑗</ci></apply></apply><ci id="S2.SS1.p2.4.m2.1.1.cmml" xref="S2.SS1.p2.4.m2.1.1">𝜃</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p2.4.m2.1c">g_{ij}(\theta)</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p2.4.m2.1d">italic_g start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT ( italic_θ )</annotation></semantics></math> and <math alttext="g^{ij}(\eta)" class="ltx_Math" display="inline" id="S2.SS1.p2.5.m3.1"><semantics id="S2.SS1.p2.5.m3.1a"><mrow id="S2.SS1.p2.5.m3.1.2" xref="S2.SS1.p2.5.m3.1.2.cmml"><msup id="S2.SS1.p2.5.m3.1.2.2" xref="S2.SS1.p2.5.m3.1.2.2.cmml"><mi id="S2.SS1.p2.5.m3.1.2.2.2" xref="S2.SS1.p2.5.m3.1.2.2.2.cmml">g</mi><mrow id="S2.SS1.p2.5.m3.1.2.2.3" xref="S2.SS1.p2.5.m3.1.2.2.3.cmml"><mi id="S2.SS1.p2.5.m3.1.2.2.3.2" xref="S2.SS1.p2.5.m3.1.2.2.3.2.cmml">i</mi><mo id="S2.SS1.p2.5.m3.1.2.2.3.1" xref="S2.SS1.p2.5.m3.1.2.2.3.1.cmml">⁢</mo><mi id="S2.SS1.p2.5.m3.1.2.2.3.3" xref="S2.SS1.p2.5.m3.1.2.2.3.3.cmml">j</mi></mrow></msup><mo id="S2.SS1.p2.5.m3.1.2.1" xref="S2.SS1.p2.5.m3.1.2.1.cmml">⁢</mo><mrow id="S2.SS1.p2.5.m3.1.2.3.2" xref="S2.SS1.p2.5.m3.1.2.cmml"><mo id="S2.SS1.p2.5.m3.1.2.3.2.1" stretchy="false" xref="S2.SS1.p2.5.m3.1.2.cmml">(</mo><mi id="S2.SS1.p2.5.m3.1.1" xref="S2.SS1.p2.5.m3.1.1.cmml">η</mi><mo id="S2.SS1.p2.5.m3.1.2.3.2.2" stretchy="false" xref="S2.SS1.p2.5.m3.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p2.5.m3.1b"><apply id="S2.SS1.p2.5.m3.1.2.cmml" xref="S2.SS1.p2.5.m3.1.2"><times id="S2.SS1.p2.5.m3.1.2.1.cmml" xref="S2.SS1.p2.5.m3.1.2.1"></times><apply id="S2.SS1.p2.5.m3.1.2.2.cmml" xref="S2.SS1.p2.5.m3.1.2.2"><csymbol cd="ambiguous" id="S2.SS1.p2.5.m3.1.2.2.1.cmml" xref="S2.SS1.p2.5.m3.1.2.2">superscript</csymbol><ci id="S2.SS1.p2.5.m3.1.2.2.2.cmml" xref="S2.SS1.p2.5.m3.1.2.2.2">𝑔</ci><apply id="S2.SS1.p2.5.m3.1.2.2.3.cmml" xref="S2.SS1.p2.5.m3.1.2.2.3"><times id="S2.SS1.p2.5.m3.1.2.2.3.1.cmml" xref="S2.SS1.p2.5.m3.1.2.2.3.1"></times><ci id="S2.SS1.p2.5.m3.1.2.2.3.2.cmml" xref="S2.SS1.p2.5.m3.1.2.2.3.2">𝑖</ci><ci id="S2.SS1.p2.5.m3.1.2.2.3.3.cmml" xref="S2.SS1.p2.5.m3.1.2.2.3.3">𝑗</ci></apply></apply><ci id="S2.SS1.p2.5.m3.1.1.cmml" xref="S2.SS1.p2.5.m3.1.1">𝜂</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p2.5.m3.1c">g^{ij}(\eta)</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p2.5.m3.1d">italic_g start_POSTSUPERSCRIPT italic_i italic_j end_POSTSUPERSCRIPT ( italic_η )</annotation></semantics></math> are obtained from the Hessian matrices of the convex function <math alttext="\Psi(\theta)" class="ltx_Math" display="inline" id="S2.SS1.p2.6.m4.1"><semantics id="S2.SS1.p2.6.m4.1a"><mrow id="S2.SS1.p2.6.m4.1.2" xref="S2.SS1.p2.6.m4.1.2.cmml"><mi id="S2.SS1.p2.6.m4.1.2.2" mathvariant="normal" xref="S2.SS1.p2.6.m4.1.2.2.cmml">Ψ</mi><mo id="S2.SS1.p2.6.m4.1.2.1" xref="S2.SS1.p2.6.m4.1.2.1.cmml">⁢</mo><mrow id="S2.SS1.p2.6.m4.1.2.3.2" xref="S2.SS1.p2.6.m4.1.2.cmml"><mo id="S2.SS1.p2.6.m4.1.2.3.2.1" stretchy="false" xref="S2.SS1.p2.6.m4.1.2.cmml">(</mo><mi id="S2.SS1.p2.6.m4.1.1" xref="S2.SS1.p2.6.m4.1.1.cmml">θ</mi><mo id="S2.SS1.p2.6.m4.1.2.3.2.2" stretchy="false" xref="S2.SS1.p2.6.m4.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p2.6.m4.1b"><apply id="S2.SS1.p2.6.m4.1.2.cmml" xref="S2.SS1.p2.6.m4.1.2"><times id="S2.SS1.p2.6.m4.1.2.1.cmml" xref="S2.SS1.p2.6.m4.1.2.1"></times><ci id="S2.SS1.p2.6.m4.1.2.2.cmml" xref="S2.SS1.p2.6.m4.1.2.2">Ψ</ci><ci id="S2.SS1.p2.6.m4.1.1.cmml" xref="S2.SS1.p2.6.m4.1.1">𝜃</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p2.6.m4.1c">\Psi(\theta)</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p2.6.m4.1d">roman_Ψ ( italic_θ )</annotation></semantics></math> and <math alttext="\Psi^{\star}(\eta)" class="ltx_Math" display="inline" id="S2.SS1.p2.7.m5.1"><semantics id="S2.SS1.p2.7.m5.1a"><mrow id="S2.SS1.p2.7.m5.1.2" xref="S2.SS1.p2.7.m5.1.2.cmml"><msup id="S2.SS1.p2.7.m5.1.2.2" xref="S2.SS1.p2.7.m5.1.2.2.cmml"><mi id="S2.SS1.p2.7.m5.1.2.2.2" mathvariant="normal" xref="S2.SS1.p2.7.m5.1.2.2.2.cmml">Ψ</mi><mo id="S2.SS1.p2.7.m5.1.2.2.3" xref="S2.SS1.p2.7.m5.1.2.2.3.cmml">⋆</mo></msup><mo id="S2.SS1.p2.7.m5.1.2.1" xref="S2.SS1.p2.7.m5.1.2.1.cmml">⁢</mo><mrow id="S2.SS1.p2.7.m5.1.2.3.2" xref="S2.SS1.p2.7.m5.1.2.cmml"><mo id="S2.SS1.p2.7.m5.1.2.3.2.1" stretchy="false" xref="S2.SS1.p2.7.m5.1.2.cmml">(</mo><mi id="S2.SS1.p2.7.m5.1.1" xref="S2.SS1.p2.7.m5.1.1.cmml">η</mi><mo id="S2.SS1.p2.7.m5.1.2.3.2.2" stretchy="false" xref="S2.SS1.p2.7.m5.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p2.7.m5.1b"><apply id="S2.SS1.p2.7.m5.1.2.cmml" xref="S2.SS1.p2.7.m5.1.2"><times id="S2.SS1.p2.7.m5.1.2.1.cmml" xref="S2.SS1.p2.7.m5.1.2.1"></times><apply id="S2.SS1.p2.7.m5.1.2.2.cmml" xref="S2.SS1.p2.7.m5.1.2.2"><csymbol cd="ambiguous" id="S2.SS1.p2.7.m5.1.2.2.1.cmml" xref="S2.SS1.p2.7.m5.1.2.2">superscript</csymbol><ci id="S2.SS1.p2.7.m5.1.2.2.2.cmml" xref="S2.SS1.p2.7.m5.1.2.2.2">Ψ</ci><ci id="S2.SS1.p2.7.m5.1.2.2.3.cmml" xref="S2.SS1.p2.7.m5.1.2.2.3">⋆</ci></apply><ci id="S2.SS1.p2.7.m5.1.1.cmml" xref="S2.SS1.p2.7.m5.1.1">𝜂</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p2.7.m5.1c">\Psi^{\star}(\eta)</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p2.7.m5.1d">roman_Ψ start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT ( italic_η )</annotation></semantics></math> as</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="S3.EGx8"> <tbody id="S2.Ex1"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_left_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle g_{ij}(\theta)" class="ltx_Math" display="inline" id="S2.Ex1.m1.1"><semantics id="S2.Ex1.m1.1a"><mrow id="S2.Ex1.m1.1.2" xref="S2.Ex1.m1.1.2.cmml"><msub id="S2.Ex1.m1.1.2.2" xref="S2.Ex1.m1.1.2.2.cmml"><mi id="S2.Ex1.m1.1.2.2.2" xref="S2.Ex1.m1.1.2.2.2.cmml">g</mi><mrow id="S2.Ex1.m1.1.2.2.3" xref="S2.Ex1.m1.1.2.2.3.cmml"><mi id="S2.Ex1.m1.1.2.2.3.2" xref="S2.Ex1.m1.1.2.2.3.2.cmml">i</mi><mo id="S2.Ex1.m1.1.2.2.3.1" xref="S2.Ex1.m1.1.2.2.3.1.cmml">⁢</mo><mi id="S2.Ex1.m1.1.2.2.3.3" xref="S2.Ex1.m1.1.2.2.3.3.cmml">j</mi></mrow></msub><mo id="S2.Ex1.m1.1.2.1" xref="S2.Ex1.m1.1.2.1.cmml">⁢</mo><mrow id="S2.Ex1.m1.1.2.3.2" xref="S2.Ex1.m1.1.2.cmml"><mo id="S2.Ex1.m1.1.2.3.2.1" stretchy="false" xref="S2.Ex1.m1.1.2.cmml">(</mo><mi id="S2.Ex1.m1.1.1" xref="S2.Ex1.m1.1.1.cmml">θ</mi><mo id="S2.Ex1.m1.1.2.3.2.2" stretchy="false" xref="S2.Ex1.m1.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.Ex1.m1.1b"><apply id="S2.Ex1.m1.1.2.cmml" xref="S2.Ex1.m1.1.2"><times id="S2.Ex1.m1.1.2.1.cmml" xref="S2.Ex1.m1.1.2.1"></times><apply id="S2.Ex1.m1.1.2.2.cmml" xref="S2.Ex1.m1.1.2.2"><csymbol cd="ambiguous" id="S2.Ex1.m1.1.2.2.1.cmml" xref="S2.Ex1.m1.1.2.2">subscript</csymbol><ci id="S2.Ex1.m1.1.2.2.2.cmml" xref="S2.Ex1.m1.1.2.2.2">𝑔</ci><apply id="S2.Ex1.m1.1.2.2.3.cmml" xref="S2.Ex1.m1.1.2.2.3"><times id="S2.Ex1.m1.1.2.2.3.1.cmml" xref="S2.Ex1.m1.1.2.2.3.1"></times><ci id="S2.Ex1.m1.1.2.2.3.2.cmml" xref="S2.Ex1.m1.1.2.2.3.2">𝑖</ci><ci id="S2.Ex1.m1.1.2.2.3.3.cmml" xref="S2.Ex1.m1.1.2.2.3.3">𝑗</ci></apply></apply><ci id="S2.Ex1.m1.1.1.cmml" xref="S2.Ex1.m1.1.1">𝜃</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Ex1.m1.1c">\displaystyle g_{ij}(\theta)</annotation><annotation encoding="application/x-llamapun" id="S2.Ex1.m1.1d">italic_g start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT ( italic_θ )</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle=\frac{\partial\eta_{i}}{\partial\theta^{j}}=\frac{\partial^{2}% \Psi(\theta)}{\partial\theta^{i}\partial\theta^{j}}," class="ltx_Math" display="inline" id="S2.Ex1.m2.2"><semantics id="S2.Ex1.m2.2a"><mrow id="S2.Ex1.m2.2.2.1" 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end_POSTSUBSCRIPT ∂ italic_η start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT end_ARG ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_left_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(8)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S2.SS1.p2.9">respectively. 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id="S2.SS1.p2.8.m1.2c">g^{ij}(\eta)\,g_{jk}(\theta)=\delta^{i}_{k}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p2.8.m1.2d">italic_g start_POSTSUPERSCRIPT italic_i italic_j end_POSTSUPERSCRIPT ( italic_η ) italic_g start_POSTSUBSCRIPT italic_j italic_k end_POSTSUBSCRIPT ( italic_θ ) = italic_δ start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT</annotation></semantics></math>, where <math alttext="\delta^{i}_{k}" class="ltx_Math" display="inline" id="S2.SS1.p2.9.m2.1"><semantics id="S2.SS1.p2.9.m2.1a"><msubsup id="S2.SS1.p2.9.m2.1.1" xref="S2.SS1.p2.9.m2.1.1.cmml"><mi id="S2.SS1.p2.9.m2.1.1.2.2" xref="S2.SS1.p2.9.m2.1.1.2.2.cmml">δ</mi><mi id="S2.SS1.p2.9.m2.1.1.3" xref="S2.SS1.p2.9.m2.1.1.3.cmml">k</mi><mi id="S2.SS1.p2.9.m2.1.1.2.3" xref="S2.SS1.p2.9.m2.1.1.2.3.cmml">i</mi></msubsup><annotation-xml encoding="MathML-Content" id="S2.SS1.p2.9.m2.1b"><apply id="S2.SS1.p2.9.m2.1.1.cmml" 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alttext="\operatorname{\Gamma}" class="ltx_Math" display="inline" id="S2.SS1.p3.1.m1.1"><semantics id="S2.SS1.p3.1.m1.1a"><mi id="S2.SS1.p3.1.m1.1.1" mathvariant="normal" xref="S2.SS1.p3.1.m1.1.1.cmml">Γ</mi><annotation-xml encoding="MathML-Content" id="S2.SS1.p3.1.m1.1b"><ci id="S2.SS1.p3.1.m1.1.1.cmml" xref="S2.SS1.p3.1.m1.1.1">Γ</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p3.1.m1.1c">\operatorname{\Gamma}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p3.1.m1.1d">roman_Γ</annotation></semantics></math> are not tensors, there exists a coordinate system in which all connection coefficients become zero and such a coordinate system is called an <span class="ltx_text ltx_font_italic" id="S2.SS1.p3.6.1">affine coordinate</span>. The <math alttext="\alpha" class="ltx_Math" display="inline" id="S2.SS1.p3.2.m2.1"><semantics id="S2.SS1.p3.2.m2.1a"><mi id="S2.SS1.p3.2.m2.1.1" xref="S2.SS1.p3.2.m2.1.1.cmml">α</mi><annotation-xml encoding="MathML-Content" id="S2.SS1.p3.2.m2.1b"><ci id="S2.SS1.p3.2.m2.1.1.cmml" xref="S2.SS1.p3.2.m2.1.1">𝛼</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p3.2.m2.1c">\alpha</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p3.2.m2.1d">italic_α</annotation></semantics></math>-connection <math alttext="\nabla^{(\alpha)}" class="ltx_Math" display="inline" id="S2.SS1.p3.3.m3.1"><semantics id="S2.SS1.p3.3.m3.1a"><msup id="S2.SS1.p3.3.m3.1.2" xref="S2.SS1.p3.3.m3.1.2.cmml"><mo id="S2.SS1.p3.3.m3.1.2.2" xref="S2.SS1.p3.3.m3.1.2.2.cmml">∇</mo><mrow id="S2.SS1.p3.3.m3.1.1.1.3" xref="S2.SS1.p3.3.m3.1.2.cmml"><mo id="S2.SS1.p3.3.m3.1.1.1.3.1" stretchy="false" xref="S2.SS1.p3.3.m3.1.2.cmml">(</mo><mi id="S2.SS1.p3.3.m3.1.1.1.1" xref="S2.SS1.p3.3.m3.1.1.1.1.cmml">α</mi><mo id="S2.SS1.p3.3.m3.1.1.1.3.2" stretchy="false" xref="S2.SS1.p3.3.m3.1.2.cmml">)</mo></mrow></msup><annotation-xml encoding="MathML-Content" id="S2.SS1.p3.3.m3.1b"><apply id="S2.SS1.p3.3.m3.1.2.cmml" xref="S2.SS1.p3.3.m3.1.2"><csymbol cd="ambiguous" id="S2.SS1.p3.3.m3.1.2.1.cmml" xref="S2.SS1.p3.3.m3.1.2">superscript</csymbol><ci id="S2.SS1.p3.3.m3.1.2.2.cmml" xref="S2.SS1.p3.3.m3.1.2.2">∇</ci><ci id="S2.SS1.p3.3.m3.1.1.1.1.cmml" xref="S2.SS1.p3.3.m3.1.1.1.1">𝛼</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p3.3.m3.1c">\nabla^{(\alpha)}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p3.3.m3.1d">∇ start_POSTSUPERSCRIPT ( italic_α ) end_POSTSUPERSCRIPT</annotation></semantics></math> <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2406.11224v2#bib.bib1" title="">1</a>]</cite>, which is a one-parameter extension <math alttext="\left\{\nabla^{\alpha}\right\}_{\alpha\in\mathbb{R}}" class="ltx_Math" display="inline" id="S2.SS1.p3.4.m4.1"><semantics id="S2.SS1.p3.4.m4.1a"><msub id="S2.SS1.p3.4.m4.1.1" xref="S2.SS1.p3.4.m4.1.1.cmml"><mrow id="S2.SS1.p3.4.m4.1.1.1.1" xref="S2.SS1.p3.4.m4.1.1.1.2.cmml"><mo id="S2.SS1.p3.4.m4.1.1.1.1.2" xref="S2.SS1.p3.4.m4.1.1.1.2.cmml">{</mo><msup id="S2.SS1.p3.4.m4.1.1.1.1.1" xref="S2.SS1.p3.4.m4.1.1.1.1.1.cmml"><mo id="S2.SS1.p3.4.m4.1.1.1.1.1.2" xref="S2.SS1.p3.4.m4.1.1.1.1.1.2.cmml">∇</mo><mi id="S2.SS1.p3.4.m4.1.1.1.1.1.3" xref="S2.SS1.p3.4.m4.1.1.1.1.1.3.cmml">α</mi></msup><mo id="S2.SS1.p3.4.m4.1.1.1.1.3" xref="S2.SS1.p3.4.m4.1.1.1.2.cmml">}</mo></mrow><mrow id="S2.SS1.p3.4.m4.1.1.3" xref="S2.SS1.p3.4.m4.1.1.3.cmml"><mi id="S2.SS1.p3.4.m4.1.1.3.2" xref="S2.SS1.p3.4.m4.1.1.3.2.cmml">α</mi><mo id="S2.SS1.p3.4.m4.1.1.3.1" xref="S2.SS1.p3.4.m4.1.1.3.1.cmml">∈</mo><mi id="S2.SS1.p3.4.m4.1.1.3.3" xref="S2.SS1.p3.4.m4.1.1.3.3.cmml">ℝ</mi></mrow></msub><annotation-xml 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id="S2.SS1.p3.4.m4.1c">\left\{\nabla^{\alpha}\right\}_{\alpha\in\mathbb{R}}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p3.4.m4.1d">{ ∇ start_POSTSUPERSCRIPT italic_α end_POSTSUPERSCRIPT } start_POSTSUBSCRIPT italic_α ∈ blackboard_R end_POSTSUBSCRIPT</annotation></semantics></math> of Levi-Civita’s connection <math alttext="\nabla^{(0)}" class="ltx_Math" display="inline" id="S2.SS1.p3.5.m5.1"><semantics id="S2.SS1.p3.5.m5.1a"><msup id="S2.SS1.p3.5.m5.1.2" xref="S2.SS1.p3.5.m5.1.2.cmml"><mo id="S2.SS1.p3.5.m5.1.2.2" xref="S2.SS1.p3.5.m5.1.2.2.cmml">∇</mo><mrow id="S2.SS1.p3.5.m5.1.1.1.3" xref="S2.SS1.p3.5.m5.1.2.cmml"><mo id="S2.SS1.p3.5.m5.1.1.1.3.1" stretchy="false" xref="S2.SS1.p3.5.m5.1.2.cmml">(</mo><mn id="S2.SS1.p3.5.m5.1.1.1.1" xref="S2.SS1.p3.5.m5.1.1.1.1.cmml">0</mn><mo id="S2.SS1.p3.5.m5.1.1.1.3.2" stretchy="false" xref="S2.SS1.p3.5.m5.1.2.cmml">)</mo></mrow></msup><annotation-xml encoding="MathML-Content" id="S2.SS1.p3.5.m5.1b"><apply id="S2.SS1.p3.5.m5.1.2.cmml" xref="S2.SS1.p3.5.m5.1.2"><csymbol cd="ambiguous" id="S2.SS1.p3.5.m5.1.2.1.cmml" xref="S2.SS1.p3.5.m5.1.2">superscript</csymbol><ci id="S2.SS1.p3.5.m5.1.2.2.cmml" xref="S2.SS1.p3.5.m5.1.2.2">∇</ci><cn id="S2.SS1.p3.5.m5.1.1.1.1.cmml" type="integer" xref="S2.SS1.p3.5.m5.1.1.1.1">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p3.5.m5.1c">\nabla^{(0)}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p3.5.m5.1d">∇ start_POSTSUPERSCRIPT ( 0 ) end_POSTSUPERSCRIPT</annotation></semantics></math>, and its dual <math alttext="\nabla^{\star(\alpha)}" class="ltx_Math" display="inline" id="S2.SS1.p3.6.m6.1"><semantics id="S2.SS1.p3.6.m6.1a"><msup id="S2.SS1.p3.6.m6.1.2" xref="S2.SS1.p3.6.m6.1.2.cmml"><mo id="S2.SS1.p3.6.m6.1.2.2" xref="S2.SS1.p3.6.m6.1.2.2.cmml">∇</mo><mrow id="S2.SS1.p3.6.m6.1.1.1" xref="S2.SS1.p3.6.m6.1.1.1.cmml"><mi id="S2.SS1.p3.6.m6.1.1.1.3" xref="S2.SS1.p3.6.m6.1.1.1.3.cmml"></mi><mo id="S2.SS1.p3.6.m6.1.1.1.2" lspace="0.222em" rspace="0.222em" xref="S2.SS1.p3.6.m6.1.1.1.2.cmml">⋆</mo><mrow id="S2.SS1.p3.6.m6.1.1.1.4.2" xref="S2.SS1.p3.6.m6.1.1.1.cmml"><mo id="S2.SS1.p3.6.m6.1.1.1.4.2.1" stretchy="false" xref="S2.SS1.p3.6.m6.1.1.1.cmml">(</mo><mi id="S2.SS1.p3.6.m6.1.1.1.1" xref="S2.SS1.p3.6.m6.1.1.1.1.cmml">α</mi><mo id="S2.SS1.p3.6.m6.1.1.1.4.2.2" stretchy="false" xref="S2.SS1.p3.6.m6.1.1.1.cmml">)</mo></mrow></mrow></msup><annotation-xml encoding="MathML-Content" id="S2.SS1.p3.6.m6.1b"><apply id="S2.SS1.p3.6.m6.1.2.cmml" xref="S2.SS1.p3.6.m6.1.2"><csymbol cd="ambiguous" id="S2.SS1.p3.6.m6.1.2.1.cmml" xref="S2.SS1.p3.6.m6.1.2">superscript</csymbol><ci id="S2.SS1.p3.6.m6.1.2.2.cmml" xref="S2.SS1.p3.6.m6.1.2.2">∇</ci><apply id="S2.SS1.p3.6.m6.1.1.1.cmml" xref="S2.SS1.p3.6.m6.1.1.1"><ci id="S2.SS1.p3.6.m6.1.1.1.2.cmml" xref="S2.SS1.p3.6.m6.1.1.1.2">⋆</ci><csymbol cd="latexml" id="S2.SS1.p3.6.m6.1.1.1.3.cmml" xref="S2.SS1.p3.6.m6.1.1.1.3">absent</csymbol><ci id="S2.SS1.p3.6.m6.1.1.1.1.cmml" xref="S2.SS1.p3.6.m6.1.1.1.1">𝛼</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p3.6.m6.1c">\nabla^{\star(\alpha)}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p3.6.m6.1d">∇ start_POSTSUPERSCRIPT ⋆ ( italic_α ) end_POSTSUPERSCRIPT</annotation></semantics></math> are defined by their coefficients as</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="S3.EGx9"> <tbody id="S2.Ex2"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_left_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\operatorname{\Gamma}^{(\alpha)}{}_{ijk}(\theta)" class="ltx_math_unparsed" display="inline" id="S2.Ex2.m1.1"><semantics id="S2.Ex2.m1.1a"><mrow id="S2.Ex2.m1.1b"><msup id="S2.Ex2.m1.1.2"><mi id="S2.Ex2.m1.1.2.2" mathvariant="normal">Γ</mi><mrow id="S2.Ex2.m1.1.1.1.3"><mo id="S2.Ex2.m1.1.1.1.3.1" stretchy="false">(</mo><mi id="S2.Ex2.m1.1.1.1.1">α</mi><mo id="S2.Ex2.m1.1.1.1.3.2" stretchy="false">)</mo></mrow></msup><mmultiscripts id="S2.Ex2.m1.1.3"><mrow id="S2.Ex2.m1.1.3.2"><mo id="S2.Ex2.m1.1.3.2.1" stretchy="false">(</mo><mi id="S2.Ex2.m1.1.3.2.2">θ</mi><mo id="S2.Ex2.m1.1.3.2.3" stretchy="false">)</mo></mrow><mprescripts id="S2.Ex2.m1.1.3a"></mprescripts><mrow id="S2.Ex2.m1.1.3.3"><mi id="S2.Ex2.m1.1.3.3.2">i</mi><mo id="S2.Ex2.m1.1.3.3.1">⁢</mo><mi id="S2.Ex2.m1.1.3.3.3">j</mi><mo id="S2.Ex2.m1.1.3.3.1a">⁢</mo><mi id="S2.Ex2.m1.1.3.3.4">k</mi></mrow><mrow id="S2.Ex2.m1.1.3b"></mrow></mmultiscripts></mrow><annotation encoding="application/x-tex" id="S2.Ex2.m1.1c">\displaystyle\operatorname{\Gamma}^{(\alpha)}{}_{ijk}(\theta)</annotation><annotation encoding="application/x-llamapun" id="S2.Ex2.m1.1d">roman_Γ start_POSTSUPERSCRIPT ( italic_α ) end_POSTSUPERSCRIPT start_FLOATSUBSCRIPT italic_i italic_j italic_k end_FLOATSUBSCRIPT ( italic_θ )</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle:=\frac{(1-\alpha)}{2}C_{ijk}(\theta)," class="ltx_Math" display="inline" id="S2.Ex2.m2.3"><semantics id="S2.Ex2.m2.3a"><mrow id="S2.Ex2.m2.3.3.1" xref="S2.Ex2.m2.3.3.1.1.cmml"><mrow id="S2.Ex2.m2.3.3.1.1" xref="S2.Ex2.m2.3.3.1.1.cmml"><mi id="S2.Ex2.m2.3.3.1.1.2" xref="S2.Ex2.m2.3.3.1.1.2.cmml"></mi><mo id="S2.Ex2.m2.3.3.1.1.1" lspace="0.278em" rspace="0.278em" xref="S2.Ex2.m2.3.3.1.1.1.cmml">:=</mo><mrow id="S2.Ex2.m2.3.3.1.1.3" xref="S2.Ex2.m2.3.3.1.1.3.cmml"><mstyle displaystyle="true" id="S2.Ex2.m2.1.1" xref="S2.Ex2.m2.1.1.cmml"><mfrac id="S2.Ex2.m2.1.1a" xref="S2.Ex2.m2.1.1.cmml"><mrow id="S2.Ex2.m2.1.1.1.1" xref="S2.Ex2.m2.1.1.1.1.1.cmml"><mo id="S2.Ex2.m2.1.1.1.1.2" stretchy="false" xref="S2.Ex2.m2.1.1.1.1.1.cmml">(</mo><mrow id="S2.Ex2.m2.1.1.1.1.1" xref="S2.Ex2.m2.1.1.1.1.1.cmml"><mn id="S2.Ex2.m2.1.1.1.1.1.2" xref="S2.Ex2.m2.1.1.1.1.1.2.cmml">1</mn><mo 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xref="S2.Ex2.m2.3.3.1.1.3.2.3.4.cmml">k</mi></mrow></msub><mo id="S2.Ex2.m2.3.3.1.1.3.1a" xref="S2.Ex2.m2.3.3.1.1.3.1.cmml">⁢</mo><mrow id="S2.Ex2.m2.3.3.1.1.3.3.2" xref="S2.Ex2.m2.3.3.1.1.3.cmml"><mo id="S2.Ex2.m2.3.3.1.1.3.3.2.1" stretchy="false" xref="S2.Ex2.m2.3.3.1.1.3.cmml">(</mo><mi id="S2.Ex2.m2.2.2" xref="S2.Ex2.m2.2.2.cmml">θ</mi><mo id="S2.Ex2.m2.3.3.1.1.3.3.2.2" stretchy="false" xref="S2.Ex2.m2.3.3.1.1.3.cmml">)</mo></mrow></mrow></mrow><mo id="S2.Ex2.m2.3.3.1.2" xref="S2.Ex2.m2.3.3.1.1.cmml">,</mo></mrow><annotation-xml encoding="MathML-Content" id="S2.Ex2.m2.3b"><apply id="S2.Ex2.m2.3.3.1.1.cmml" xref="S2.Ex2.m2.3.3.1"><csymbol cd="latexml" id="S2.Ex2.m2.3.3.1.1.1.cmml" xref="S2.Ex2.m2.3.3.1.1.1">assign</csymbol><csymbol cd="latexml" id="S2.Ex2.m2.3.3.1.1.2.cmml" xref="S2.Ex2.m2.3.3.1.1.2">absent</csymbol><apply id="S2.Ex2.m2.3.3.1.1.3.cmml" xref="S2.Ex2.m2.3.3.1.1.3"><times id="S2.Ex2.m2.3.3.1.1.3.1.cmml" xref="S2.Ex2.m2.3.3.1.1.3.1"></times><apply 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id="S2.Ex2.m2.3.3.1.1.3.2.3.3.cmml" xref="S2.Ex2.m2.3.3.1.1.3.2.3.3">𝑗</ci><ci id="S2.Ex2.m2.3.3.1.1.3.2.3.4.cmml" xref="S2.Ex2.m2.3.3.1.1.3.2.3.4">𝑘</ci></apply></apply><ci id="S2.Ex2.m2.2.2.cmml" xref="S2.Ex2.m2.2.2">𝜃</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Ex2.m2.3c">\displaystyle:=\frac{(1-\alpha)}{2}C_{ijk}(\theta),</annotation><annotation encoding="application/x-llamapun" id="S2.Ex2.m2.3d">:= divide start_ARG ( 1 - italic_α ) end_ARG start_ARG 2 end_ARG italic_C start_POSTSUBSCRIPT italic_i italic_j italic_k end_POSTSUBSCRIPT ( italic_θ ) ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_left_padright"></td> </tr></tbody> <tbody id="S2.E9"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_left_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\operatorname{\Gamma}^{\star(\alpha)\;ijk}(\eta)" class="ltx_Math" display="inline" 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xref="S2.E9.m2.2.2">𝜂</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.E9.m2.3c">\displaystyle:=\frac{(1+\alpha)}{2}C^{ijk}(\eta),</annotation><annotation encoding="application/x-llamapun" id="S2.E9.m2.3d">:= divide start_ARG ( 1 + italic_α ) end_ARG start_ARG 2 end_ARG italic_C start_POSTSUPERSCRIPT italic_i italic_j italic_k end_POSTSUPERSCRIPT ( italic_η ) ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_left_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(9)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S2.SS1.p3.8">respectively. Here <math alttext="C_{ijk}(\theta)" class="ltx_Math" display="inline" id="S2.SS1.p3.7.m1.1"><semantics id="S2.SS1.p3.7.m1.1a"><mrow id="S2.SS1.p3.7.m1.1.2" xref="S2.SS1.p3.7.m1.1.2.cmml"><msub id="S2.SS1.p3.7.m1.1.2.2" xref="S2.SS1.p3.7.m1.1.2.2.cmml"><mi id="S2.SS1.p3.7.m1.1.2.2.2" xref="S2.SS1.p3.7.m1.1.2.2.2.cmml">C</mi><mrow id="S2.SS1.p3.7.m1.1.2.2.3" xref="S2.SS1.p3.7.m1.1.2.2.3.cmml"><mi id="S2.SS1.p3.7.m1.1.2.2.3.2" xref="S2.SS1.p3.7.m1.1.2.2.3.2.cmml">i</mi><mo id="S2.SS1.p3.7.m1.1.2.2.3.1" xref="S2.SS1.p3.7.m1.1.2.2.3.1.cmml">⁢</mo><mi id="S2.SS1.p3.7.m1.1.2.2.3.3" xref="S2.SS1.p3.7.m1.1.2.2.3.3.cmml">j</mi><mo id="S2.SS1.p3.7.m1.1.2.2.3.1a" xref="S2.SS1.p3.7.m1.1.2.2.3.1.cmml">⁢</mo><mi id="S2.SS1.p3.7.m1.1.2.2.3.4" xref="S2.SS1.p3.7.m1.1.2.2.3.4.cmml">k</mi></mrow></msub><mo id="S2.SS1.p3.7.m1.1.2.1" xref="S2.SS1.p3.7.m1.1.2.1.cmml">⁢</mo><mrow id="S2.SS1.p3.7.m1.1.2.3.2" xref="S2.SS1.p3.7.m1.1.2.cmml"><mo id="S2.SS1.p3.7.m1.1.2.3.2.1" stretchy="false" xref="S2.SS1.p3.7.m1.1.2.cmml">(</mo><mi id="S2.SS1.p3.7.m1.1.1" xref="S2.SS1.p3.7.m1.1.1.cmml">θ</mi><mo id="S2.SS1.p3.7.m1.1.2.3.2.2" stretchy="false" xref="S2.SS1.p3.7.m1.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p3.7.m1.1b"><apply id="S2.SS1.p3.7.m1.1.2.cmml" xref="S2.SS1.p3.7.m1.1.2"><times id="S2.SS1.p3.7.m1.1.2.1.cmml" xref="S2.SS1.p3.7.m1.1.2.1"></times><apply id="S2.SS1.p3.7.m1.1.2.2.cmml" xref="S2.SS1.p3.7.m1.1.2.2"><csymbol cd="ambiguous" id="S2.SS1.p3.7.m1.1.2.2.1.cmml" xref="S2.SS1.p3.7.m1.1.2.2">subscript</csymbol><ci id="S2.SS1.p3.7.m1.1.2.2.2.cmml" xref="S2.SS1.p3.7.m1.1.2.2.2">𝐶</ci><apply id="S2.SS1.p3.7.m1.1.2.2.3.cmml" xref="S2.SS1.p3.7.m1.1.2.2.3"><times id="S2.SS1.p3.7.m1.1.2.2.3.1.cmml" xref="S2.SS1.p3.7.m1.1.2.2.3.1"></times><ci id="S2.SS1.p3.7.m1.1.2.2.3.2.cmml" xref="S2.SS1.p3.7.m1.1.2.2.3.2">𝑖</ci><ci id="S2.SS1.p3.7.m1.1.2.2.3.3.cmml" xref="S2.SS1.p3.7.m1.1.2.2.3.3">𝑗</ci><ci id="S2.SS1.p3.7.m1.1.2.2.3.4.cmml" xref="S2.SS1.p3.7.m1.1.2.2.3.4">𝑘</ci></apply></apply><ci id="S2.SS1.p3.7.m1.1.1.cmml" xref="S2.SS1.p3.7.m1.1.1">𝜃</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p3.7.m1.1c">C_{ijk}(\theta)</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p3.7.m1.1d">italic_C start_POSTSUBSCRIPT italic_i italic_j italic_k end_POSTSUBSCRIPT ( italic_θ )</annotation></semantics></math> and <math alttext="C^{ijk}(\eta)" class="ltx_Math" display="inline" id="S2.SS1.p3.8.m2.1"><semantics id="S2.SS1.p3.8.m2.1a"><mrow id="S2.SS1.p3.8.m2.1.2" xref="S2.SS1.p3.8.m2.1.2.cmml"><msup id="S2.SS1.p3.8.m2.1.2.2" xref="S2.SS1.p3.8.m2.1.2.2.cmml"><mi id="S2.SS1.p3.8.m2.1.2.2.2" xref="S2.SS1.p3.8.m2.1.2.2.2.cmml">C</mi><mrow id="S2.SS1.p3.8.m2.1.2.2.3" xref="S2.SS1.p3.8.m2.1.2.2.3.cmml"><mi id="S2.SS1.p3.8.m2.1.2.2.3.2" xref="S2.SS1.p3.8.m2.1.2.2.3.2.cmml">i</mi><mo id="S2.SS1.p3.8.m2.1.2.2.3.1" xref="S2.SS1.p3.8.m2.1.2.2.3.1.cmml">⁢</mo><mi id="S2.SS1.p3.8.m2.1.2.2.3.3" xref="S2.SS1.p3.8.m2.1.2.2.3.3.cmml">j</mi><mo id="S2.SS1.p3.8.m2.1.2.2.3.1a" xref="S2.SS1.p3.8.m2.1.2.2.3.1.cmml">⁢</mo><mi id="S2.SS1.p3.8.m2.1.2.2.3.4" xref="S2.SS1.p3.8.m2.1.2.2.3.4.cmml">k</mi></mrow></msup><mo id="S2.SS1.p3.8.m2.1.2.1" xref="S2.SS1.p3.8.m2.1.2.1.cmml">⁢</mo><mrow id="S2.SS1.p3.8.m2.1.2.3.2" xref="S2.SS1.p3.8.m2.1.2.cmml"><mo id="S2.SS1.p3.8.m2.1.2.3.2.1" stretchy="false" xref="S2.SS1.p3.8.m2.1.2.cmml">(</mo><mi id="S2.SS1.p3.8.m2.1.1" xref="S2.SS1.p3.8.m2.1.1.cmml">η</mi><mo id="S2.SS1.p3.8.m2.1.2.3.2.2" stretchy="false" xref="S2.SS1.p3.8.m2.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p3.8.m2.1b"><apply id="S2.SS1.p3.8.m2.1.2.cmml" xref="S2.SS1.p3.8.m2.1.2"><times id="S2.SS1.p3.8.m2.1.2.1.cmml" xref="S2.SS1.p3.8.m2.1.2.1"></times><apply id="S2.SS1.p3.8.m2.1.2.2.cmml" xref="S2.SS1.p3.8.m2.1.2.2"><csymbol cd="ambiguous" id="S2.SS1.p3.8.m2.1.2.2.1.cmml" xref="S2.SS1.p3.8.m2.1.2.2">superscript</csymbol><ci id="S2.SS1.p3.8.m2.1.2.2.2.cmml" xref="S2.SS1.p3.8.m2.1.2.2.2">𝐶</ci><apply id="S2.SS1.p3.8.m2.1.2.2.3.cmml" xref="S2.SS1.p3.8.m2.1.2.2.3"><times id="S2.SS1.p3.8.m2.1.2.2.3.1.cmml" xref="S2.SS1.p3.8.m2.1.2.2.3.1"></times><ci id="S2.SS1.p3.8.m2.1.2.2.3.2.cmml" xref="S2.SS1.p3.8.m2.1.2.2.3.2">𝑖</ci><ci id="S2.SS1.p3.8.m2.1.2.2.3.3.cmml" xref="S2.SS1.p3.8.m2.1.2.2.3.3">𝑗</ci><ci id="S2.SS1.p3.8.m2.1.2.2.3.4.cmml" xref="S2.SS1.p3.8.m2.1.2.2.3.4">𝑘</ci></apply></apply><ci id="S2.SS1.p3.8.m2.1.1.cmml" xref="S2.SS1.p3.8.m2.1.1">𝜂</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p3.8.m2.1c">C^{ijk}(\eta)</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p3.8.m2.1d">italic_C start_POSTSUPERSCRIPT italic_i italic_j italic_k end_POSTSUPERSCRIPT ( italic_η )</annotation></semantics></math> are the total symmetric cubic tensors (<span class="ltx_text ltx_font_italic" id="S2.SS1.p3.8.1">Amari-Chentsov</span> tensors)</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="S3.EGx10"> <tbody id="S2.Ex3"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_left_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle C_{ijk}(\theta)" class="ltx_Math" display="inline" id="S2.Ex3.m1.1"><semantics id="S2.Ex3.m1.1a"><mrow id="S2.Ex3.m1.1.2" xref="S2.Ex3.m1.1.2.cmml"><msub id="S2.Ex3.m1.1.2.2" xref="S2.Ex3.m1.1.2.2.cmml"><mi id="S2.Ex3.m1.1.2.2.2" xref="S2.Ex3.m1.1.2.2.2.cmml">C</mi><mrow id="S2.Ex3.m1.1.2.2.3" xref="S2.Ex3.m1.1.2.2.3.cmml"><mi id="S2.Ex3.m1.1.2.2.3.2" xref="S2.Ex3.m1.1.2.2.3.2.cmml">i</mi><mo id="S2.Ex3.m1.1.2.2.3.1" xref="S2.Ex3.m1.1.2.2.3.1.cmml">⁢</mo><mi id="S2.Ex3.m1.1.2.2.3.3" xref="S2.Ex3.m1.1.2.2.3.3.cmml">j</mi><mo id="S2.Ex3.m1.1.2.2.3.1a" xref="S2.Ex3.m1.1.2.2.3.1.cmml">⁢</mo><mi id="S2.Ex3.m1.1.2.2.3.4" xref="S2.Ex3.m1.1.2.2.3.4.cmml">k</mi></mrow></msub><mo id="S2.Ex3.m1.1.2.1" xref="S2.Ex3.m1.1.2.1.cmml">⁢</mo><mrow id="S2.Ex3.m1.1.2.3.2" xref="S2.Ex3.m1.1.2.cmml"><mo id="S2.Ex3.m1.1.2.3.2.1" stretchy="false" xref="S2.Ex3.m1.1.2.cmml">(</mo><mi id="S2.Ex3.m1.1.1" xref="S2.Ex3.m1.1.1.cmml">θ</mi><mo id="S2.Ex3.m1.1.2.3.2.2" stretchy="false" xref="S2.Ex3.m1.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.Ex3.m1.1b"><apply id="S2.Ex3.m1.1.2.cmml" xref="S2.Ex3.m1.1.2"><times id="S2.Ex3.m1.1.2.1.cmml" xref="S2.Ex3.m1.1.2.1"></times><apply id="S2.Ex3.m1.1.2.2.cmml" xref="S2.Ex3.m1.1.2.2"><csymbol cd="ambiguous" id="S2.Ex3.m1.1.2.2.1.cmml" xref="S2.Ex3.m1.1.2.2">subscript</csymbol><ci id="S2.Ex3.m1.1.2.2.2.cmml" xref="S2.Ex3.m1.1.2.2.2">𝐶</ci><apply id="S2.Ex3.m1.1.2.2.3.cmml" xref="S2.Ex3.m1.1.2.2.3"><times id="S2.Ex3.m1.1.2.2.3.1.cmml" xref="S2.Ex3.m1.1.2.2.3.1"></times><ci id="S2.Ex3.m1.1.2.2.3.2.cmml" xref="S2.Ex3.m1.1.2.2.3.2">𝑖</ci><ci id="S2.Ex3.m1.1.2.2.3.3.cmml" xref="S2.Ex3.m1.1.2.2.3.3">𝑗</ci><ci id="S2.Ex3.m1.1.2.2.3.4.cmml" xref="S2.Ex3.m1.1.2.2.3.4">𝑘</ci></apply></apply><ci id="S2.Ex3.m1.1.1.cmml" xref="S2.Ex3.m1.1.1">𝜃</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Ex3.m1.1c">\displaystyle C_{ijk}(\theta)</annotation><annotation encoding="application/x-llamapun" id="S2.Ex3.m1.1d">italic_C start_POSTSUBSCRIPT italic_i italic_j italic_k end_POSTSUBSCRIPT ( italic_θ )</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle:=\frac{\partial^{3}\Psi(\theta)}{\partial\theta^{i}\partial% \theta^{j}\partial\theta^{k}}," class="ltx_Math" display="inline" id="S2.Ex3.m2.2"><semantics id="S2.Ex3.m2.2a"><mrow id="S2.Ex3.m2.2.2.1" xref="S2.Ex3.m2.2.2.1.1.cmml"><mrow id="S2.Ex3.m2.2.2.1.1" xref="S2.Ex3.m2.2.2.1.1.cmml"><mi id="S2.Ex3.m2.2.2.1.1.2" xref="S2.Ex3.m2.2.2.1.1.2.cmml"></mi><mo id="S2.Ex3.m2.2.2.1.1.1" lspace="0.278em" rspace="0.278em" xref="S2.Ex3.m2.2.2.1.1.1.cmml">:=</mo><mstyle displaystyle="true" id="S2.Ex3.m2.1.1" xref="S2.Ex3.m2.1.1.cmml"><mfrac id="S2.Ex3.m2.1.1a" xref="S2.Ex3.m2.1.1.cmml"><mrow id="S2.Ex3.m2.1.1.1" xref="S2.Ex3.m2.1.1.1.cmml"><msup id="S2.Ex3.m2.1.1.1.2" xref="S2.Ex3.m2.1.1.1.2.cmml"><mo id="S2.Ex3.m2.1.1.1.2.2" xref="S2.Ex3.m2.1.1.1.2.2.cmml">∂</mo><mn id="S2.Ex3.m2.1.1.1.2.3" xref="S2.Ex3.m2.1.1.1.2.3.cmml">3</mn></msup><mrow id="S2.Ex3.m2.1.1.1.3" xref="S2.Ex3.m2.1.1.1.3.cmml"><mi id="S2.Ex3.m2.1.1.1.3.2" mathvariant="normal" xref="S2.Ex3.m2.1.1.1.3.2.cmml">Ψ</mi><mo id="S2.Ex3.m2.1.1.1.3.1" xref="S2.Ex3.m2.1.1.1.3.1.cmml">⁢</mo><mrow id="S2.Ex3.m2.1.1.1.3.3.2" xref="S2.Ex3.m2.1.1.1.3.cmml"><mo id="S2.Ex3.m2.1.1.1.3.3.2.1" stretchy="false" xref="S2.Ex3.m2.1.1.1.3.cmml">(</mo><mi id="S2.Ex3.m2.1.1.1.1" xref="S2.Ex3.m2.1.1.1.1.cmml">θ</mi><mo id="S2.Ex3.m2.1.1.1.3.3.2.2" stretchy="false" xref="S2.Ex3.m2.1.1.1.3.cmml">)</mo></mrow></mrow></mrow><mrow id="S2.Ex3.m2.1.1.3" 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xref="S2.Ex3.m2.1.1.3.2.3.2.3.1"></partialdiff><apply id="S2.Ex3.m2.1.1.3.2.3.2.3.2.cmml" xref="S2.Ex3.m2.1.1.3.2.3.2.3.2"><csymbol cd="ambiguous" id="S2.Ex3.m2.1.1.3.2.3.2.3.2.1.cmml" xref="S2.Ex3.m2.1.1.3.2.3.2.3.2">superscript</csymbol><ci id="S2.Ex3.m2.1.1.3.2.3.2.3.2.2.cmml" xref="S2.Ex3.m2.1.1.3.2.3.2.3.2.2">𝜃</ci><ci id="S2.Ex3.m2.1.1.3.2.3.2.3.2.3.cmml" xref="S2.Ex3.m2.1.1.3.2.3.2.3.2.3">𝑘</ci></apply></apply></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Ex3.m2.2c">\displaystyle:=\frac{\partial^{3}\Psi(\theta)}{\partial\theta^{i}\partial% \theta^{j}\partial\theta^{k}},</annotation><annotation encoding="application/x-llamapun" id="S2.Ex3.m2.2d">:= divide start_ARG ∂ start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT roman_Ψ ( italic_θ ) end_ARG start_ARG ∂ italic_θ start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT ∂ italic_θ start_POSTSUPERSCRIPT italic_j end_POSTSUPERSCRIPT ∂ italic_θ start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT end_ARG ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_left_padright"></td> </tr></tbody> <tbody id="S2.E10"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_left_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle C^{ijk}(\eta)" class="ltx_Math" display="inline" id="S2.E10.m1.1"><semantics id="S2.E10.m1.1a"><mrow id="S2.E10.m1.1.2" xref="S2.E10.m1.1.2.cmml"><msup id="S2.E10.m1.1.2.2" xref="S2.E10.m1.1.2.2.cmml"><mi id="S2.E10.m1.1.2.2.2" xref="S2.E10.m1.1.2.2.2.cmml">C</mi><mrow id="S2.E10.m1.1.2.2.3" xref="S2.E10.m1.1.2.2.3.cmml"><mi id="S2.E10.m1.1.2.2.3.2" xref="S2.E10.m1.1.2.2.3.2.cmml">i</mi><mo id="S2.E10.m1.1.2.2.3.1" xref="S2.E10.m1.1.2.2.3.1.cmml">⁢</mo><mi id="S2.E10.m1.1.2.2.3.3" xref="S2.E10.m1.1.2.2.3.3.cmml">j</mi><mo id="S2.E10.m1.1.2.2.3.1a" xref="S2.E10.m1.1.2.2.3.1.cmml">⁢</mo><mi id="S2.E10.m1.1.2.2.3.4" xref="S2.E10.m1.1.2.2.3.4.cmml">k</mi></mrow></msup><mo id="S2.E10.m1.1.2.1" xref="S2.E10.m1.1.2.1.cmml">⁢</mo><mrow id="S2.E10.m1.1.2.3.2" xref="S2.E10.m1.1.2.cmml"><mo id="S2.E10.m1.1.2.3.2.1" stretchy="false" xref="S2.E10.m1.1.2.cmml">(</mo><mi id="S2.E10.m1.1.1" xref="S2.E10.m1.1.1.cmml">η</mi><mo id="S2.E10.m1.1.2.3.2.2" stretchy="false" xref="S2.E10.m1.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.E10.m1.1b"><apply id="S2.E10.m1.1.2.cmml" xref="S2.E10.m1.1.2"><times id="S2.E10.m1.1.2.1.cmml" xref="S2.E10.m1.1.2.1"></times><apply id="S2.E10.m1.1.2.2.cmml" xref="S2.E10.m1.1.2.2"><csymbol cd="ambiguous" id="S2.E10.m1.1.2.2.1.cmml" xref="S2.E10.m1.1.2.2">superscript</csymbol><ci id="S2.E10.m1.1.2.2.2.cmml" xref="S2.E10.m1.1.2.2.2">𝐶</ci><apply id="S2.E10.m1.1.2.2.3.cmml" xref="S2.E10.m1.1.2.2.3"><times id="S2.E10.m1.1.2.2.3.1.cmml" xref="S2.E10.m1.1.2.2.3.1"></times><ci id="S2.E10.m1.1.2.2.3.2.cmml" xref="S2.E10.m1.1.2.2.3.2">𝑖</ci><ci id="S2.E10.m1.1.2.2.3.3.cmml" xref="S2.E10.m1.1.2.2.3.3">𝑗</ci><ci id="S2.E10.m1.1.2.2.3.4.cmml" xref="S2.E10.m1.1.2.2.3.4">𝑘</ci></apply></apply><ci id="S2.E10.m1.1.1.cmml" xref="S2.E10.m1.1.1">𝜂</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.E10.m1.1c">\displaystyle C^{ijk}(\eta)</annotation><annotation encoding="application/x-llamapun" id="S2.E10.m1.1d">italic_C start_POSTSUPERSCRIPT italic_i italic_j italic_k end_POSTSUPERSCRIPT ( italic_η )</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle:=\frac{\partial^{3}\Psi^{\star}(\eta)}{\partial\eta_{i}\partial% \eta_{j}\partial\eta_{k}}." class="ltx_Math" display="inline" id="S2.E10.m2.2"><semantics id="S2.E10.m2.2a"><mrow id="S2.E10.m2.2.2.1" xref="S2.E10.m2.2.2.1.1.cmml"><mrow id="S2.E10.m2.2.2.1.1" xref="S2.E10.m2.2.2.1.1.cmml"><mi id="S2.E10.m2.2.2.1.1.2" xref="S2.E10.m2.2.2.1.1.2.cmml"></mi><mo id="S2.E10.m2.2.2.1.1.1" lspace="0.278em" 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id="S2.SS1.p3.18">Among the <math alttext="\alpha" class="ltx_Math" display="inline" id="S2.SS1.p3.9.m1.1"><semantics id="S2.SS1.p3.9.m1.1a"><mi id="S2.SS1.p3.9.m1.1.1" xref="S2.SS1.p3.9.m1.1.1.cmml">α</mi><annotation-xml encoding="MathML-Content" id="S2.SS1.p3.9.m1.1b"><ci id="S2.SS1.p3.9.m1.1.1.cmml" xref="S2.SS1.p3.9.m1.1.1">𝛼</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p3.9.m1.1c">\alpha</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p3.9.m1.1d">italic_α</annotation></semantics></math>-connections, <math alttext="\alpha=\pm 1" class="ltx_Math" display="inline" id="S2.SS1.p3.10.m2.1"><semantics id="S2.SS1.p3.10.m2.1a"><mrow id="S2.SS1.p3.10.m2.1.1" xref="S2.SS1.p3.10.m2.1.1.cmml"><mi id="S2.SS1.p3.10.m2.1.1.2" xref="S2.SS1.p3.10.m2.1.1.2.cmml">α</mi><mo id="S2.SS1.p3.10.m2.1.1.1" xref="S2.SS1.p3.10.m2.1.1.1.cmml">=</mo><mrow id="S2.SS1.p3.10.m2.1.1.3" xref="S2.SS1.p3.10.m2.1.1.3.cmml"><mo id="S2.SS1.p3.10.m2.1.1.3a" 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href="https://arxiv.org/html/2406.11224v2#bib.bib1" title="">1</a>]</cite>. One readily sees, from (<a class="ltx_ref" href="https://arxiv.org/html/2406.11224v2#S2.E9" title="In 2.1 Information Geometry ‣ 2 Information Geometry and Gradient-Flow Equations ‣ A Hamiltonian approach to the gradient-flow equations in information geometry"><span class="ltx_text ltx_ref_tag">9</span></a>), that the connection coefficients <math alttext="\operatorname{\Gamma}^{(1)}{}_{ijk}(\theta)" class="ltx_math_unparsed" display="inline" id="S2.SS1.p3.11.m3.1"><semantics id="S2.SS1.p3.11.m3.1a"><mrow id="S2.SS1.p3.11.m3.1b"><msup id="S2.SS1.p3.11.m3.1.2"><mi id="S2.SS1.p3.11.m3.1.2.2" mathvariant="normal">Γ</mi><mrow id="S2.SS1.p3.11.m3.1.1.1.3"><mo id="S2.SS1.p3.11.m3.1.1.1.3.1" stretchy="false">(</mo><mn id="S2.SS1.p3.11.m3.1.1.1.1">1</mn><mo id="S2.SS1.p3.11.m3.1.1.1.3.2" stretchy="false">)</mo></mrow></msup><mmultiscripts id="S2.SS1.p3.11.m3.1.3"><mrow id="S2.SS1.p3.11.m3.1.3.2"><mo id="S2.SS1.p3.11.m3.1.3.2.1" stretchy="false">(</mo><mi id="S2.SS1.p3.11.m3.1.3.2.2">θ</mi><mo id="S2.SS1.p3.11.m3.1.3.2.3" stretchy="false">)</mo></mrow><mprescripts id="S2.SS1.p3.11.m3.1.3a"></mprescripts><mrow id="S2.SS1.p3.11.m3.1.3.3"><mi id="S2.SS1.p3.11.m3.1.3.3.2">i</mi><mo id="S2.SS1.p3.11.m3.1.3.3.1">⁢</mo><mi id="S2.SS1.p3.11.m3.1.3.3.3">j</mi><mo id="S2.SS1.p3.11.m3.1.3.3.1a">⁢</mo><mi id="S2.SS1.p3.11.m3.1.3.3.4">k</mi></mrow><mrow id="S2.SS1.p3.11.m3.1.3b"></mrow></mmultiscripts></mrow><annotation encoding="application/x-tex" id="S2.SS1.p3.11.m3.1c">\operatorname{\Gamma}^{(1)}{}_{ijk}(\theta)</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p3.11.m3.1d">roman_Γ start_POSTSUPERSCRIPT ( 1 ) end_POSTSUPERSCRIPT start_FLOATSUBSCRIPT italic_i italic_j italic_k end_FLOATSUBSCRIPT ( italic_θ )</annotation></semantics></math> of <math alttext="\nabla^{(1)}" class="ltx_Math" display="inline" id="S2.SS1.p3.12.m4.1"><semantics id="S2.SS1.p3.12.m4.1a"><msup id="S2.SS1.p3.12.m4.1.2" xref="S2.SS1.p3.12.m4.1.2.cmml"><mo id="S2.SS1.p3.12.m4.1.2.2" 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italic_η )</annotation></semantics></math> of <math alttext="\nabla^{\star(-1)}" class="ltx_Math" display="inline" id="S2.SS1.p3.14.m6.1"><semantics id="S2.SS1.p3.14.m6.1a"><msup id="S2.SS1.p3.14.m6.1.2" xref="S2.SS1.p3.14.m6.1.2.cmml"><mo id="S2.SS1.p3.14.m6.1.2.2" xref="S2.SS1.p3.14.m6.1.2.2.cmml">∇</mo><mrow id="S2.SS1.p3.14.m6.1.1.1" xref="S2.SS1.p3.14.m6.1.1.1.cmml"><mi id="S2.SS1.p3.14.m6.1.1.1.3" xref="S2.SS1.p3.14.m6.1.1.1.3.cmml"></mi><mo id="S2.SS1.p3.14.m6.1.1.1.2" lspace="0.222em" rspace="0.222em" xref="S2.SS1.p3.14.m6.1.1.1.2.cmml">⋆</mo><mrow id="S2.SS1.p3.14.m6.1.1.1.1.1" xref="S2.SS1.p3.14.m6.1.1.1.1.1.1.cmml"><mo id="S2.SS1.p3.14.m6.1.1.1.1.1.2" stretchy="false" xref="S2.SS1.p3.14.m6.1.1.1.1.1.1.cmml">(</mo><mrow id="S2.SS1.p3.14.m6.1.1.1.1.1.1" xref="S2.SS1.p3.14.m6.1.1.1.1.1.1.cmml"><mo id="S2.SS1.p3.14.m6.1.1.1.1.1.1a" xref="S2.SS1.p3.14.m6.1.1.1.1.1.1.cmml">−</mo><mn id="S2.SS1.p3.14.m6.1.1.1.1.1.1.2" xref="S2.SS1.p3.14.m6.1.1.1.1.1.1.2.cmml">1</mn></mrow><mo 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encoding="application/x-tex" id="S2.SS1.p3.14.m6.1c">\nabla^{\star(-1)}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p3.14.m6.1d">∇ start_POSTSUPERSCRIPT ⋆ ( - 1 ) end_POSTSUPERSCRIPT</annotation></semantics></math>) vanish and hence the <math alttext="\theta" class="ltx_Math" display="inline" id="S2.SS1.p3.15.m7.1"><semantics id="S2.SS1.p3.15.m7.1a"><mi id="S2.SS1.p3.15.m7.1.1" xref="S2.SS1.p3.15.m7.1.1.cmml">θ</mi><annotation-xml encoding="MathML-Content" id="S2.SS1.p3.15.m7.1b"><ci id="S2.SS1.p3.15.m7.1.1.cmml" xref="S2.SS1.p3.15.m7.1.1">𝜃</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p3.15.m7.1c">\theta</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p3.15.m7.1d">italic_θ</annotation></semantics></math>-coordinates (<math alttext="\eta" class="ltx_Math" display="inline" id="S2.SS1.p3.16.m8.1"><semantics id="S2.SS1.p3.16.m8.1a"><mi id="S2.SS1.p3.16.m8.1.1" xref="S2.SS1.p3.16.m8.1.1.cmml">η</mi><annotation-xml 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id="S2.SS1.p3.18.m10.1.1.1.3" xref="S2.SS1.p3.18.m10.1.1.1.3.cmml"></mi><mo id="S2.SS1.p3.18.m10.1.1.1.2" lspace="0.222em" rspace="0.222em" xref="S2.SS1.p3.18.m10.1.1.1.2.cmml">⋆</mo><mrow id="S2.SS1.p3.18.m10.1.1.1.1.1" xref="S2.SS1.p3.18.m10.1.1.1.1.1.1.cmml"><mo id="S2.SS1.p3.18.m10.1.1.1.1.1.2" stretchy="false" xref="S2.SS1.p3.18.m10.1.1.1.1.1.1.cmml">(</mo><mrow id="S2.SS1.p3.18.m10.1.1.1.1.1.1" xref="S2.SS1.p3.18.m10.1.1.1.1.1.1.cmml"><mo id="S2.SS1.p3.18.m10.1.1.1.1.1.1a" xref="S2.SS1.p3.18.m10.1.1.1.1.1.1.cmml">−</mo><mn id="S2.SS1.p3.18.m10.1.1.1.1.1.1.2" xref="S2.SS1.p3.18.m10.1.1.1.1.1.1.2.cmml">1</mn></mrow><mo id="S2.SS1.p3.18.m10.1.1.1.1.1.3" stretchy="false" xref="S2.SS1.p3.18.m10.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></msup><annotation-xml encoding="MathML-Content" id="S2.SS1.p3.18.m10.1b"><apply id="S2.SS1.p3.18.m10.1.2.cmml" xref="S2.SS1.p3.18.m10.1.2"><csymbol cd="ambiguous" id="S2.SS1.p3.18.m10.1.2.1.cmml" xref="S2.SS1.p3.18.m10.1.2">superscript</csymbol><ci id="S2.SS1.p3.18.m10.1.2.2.cmml" xref="S2.SS1.p3.18.m10.1.2.2">∇</ci><apply id="S2.SS1.p3.18.m10.1.1.1.cmml" xref="S2.SS1.p3.18.m10.1.1.1"><ci id="S2.SS1.p3.18.m10.1.1.1.2.cmml" xref="S2.SS1.p3.18.m10.1.1.1.2">⋆</ci><csymbol cd="latexml" id="S2.SS1.p3.18.m10.1.1.1.3.cmml" xref="S2.SS1.p3.18.m10.1.1.1.3">absent</csymbol><apply id="S2.SS1.p3.18.m10.1.1.1.1.1.1.cmml" xref="S2.SS1.p3.18.m10.1.1.1.1.1"><minus id="S2.SS1.p3.18.m10.1.1.1.1.1.1.1.cmml" xref="S2.SS1.p3.18.m10.1.1.1.1.1"></minus><cn id="S2.SS1.p3.18.m10.1.1.1.1.1.1.2.cmml" type="integer" xref="S2.SS1.p3.18.m10.1.1.1.1.1.1.2">1</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p3.18.m10.1c">\nabla^{\star(-1)}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p3.18.m10.1d">∇ start_POSTSUPERSCRIPT ⋆ ( - 1 ) end_POSTSUPERSCRIPT</annotation></semantics></math>) .</p> </div> <div class="ltx_para" id="S2.SS1.p4"> <p class="ltx_p" id="S2.SS1.p4.6">A divergence <math alttext="D(p,q)" class="ltx_Math" display="inline" id="S2.SS1.p4.1.m1.2"><semantics id="S2.SS1.p4.1.m1.2a"><mrow id="S2.SS1.p4.1.m1.2.3" xref="S2.SS1.p4.1.m1.2.3.cmml"><mi id="S2.SS1.p4.1.m1.2.3.2" xref="S2.SS1.p4.1.m1.2.3.2.cmml">D</mi><mo id="S2.SS1.p4.1.m1.2.3.1" xref="S2.SS1.p4.1.m1.2.3.1.cmml">⁢</mo><mrow id="S2.SS1.p4.1.m1.2.3.3.2" xref="S2.SS1.p4.1.m1.2.3.3.1.cmml"><mo id="S2.SS1.p4.1.m1.2.3.3.2.1" stretchy="false" xref="S2.SS1.p4.1.m1.2.3.3.1.cmml">(</mo><mi id="S2.SS1.p4.1.m1.1.1" xref="S2.SS1.p4.1.m1.1.1.cmml">p</mi><mo id="S2.SS1.p4.1.m1.2.3.3.2.2" xref="S2.SS1.p4.1.m1.2.3.3.1.cmml">,</mo><mi id="S2.SS1.p4.1.m1.2.2" xref="S2.SS1.p4.1.m1.2.2.cmml">q</mi><mo id="S2.SS1.p4.1.m1.2.3.3.2.3" stretchy="false" xref="S2.SS1.p4.1.m1.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p4.1.m1.2b"><apply id="S2.SS1.p4.1.m1.2.3.cmml" xref="S2.SS1.p4.1.m1.2.3"><times id="S2.SS1.p4.1.m1.2.3.1.cmml" xref="S2.SS1.p4.1.m1.2.3.1"></times><ci id="S2.SS1.p4.1.m1.2.3.2.cmml" xref="S2.SS1.p4.1.m1.2.3.2">𝐷</ci><interval closure="open" id="S2.SS1.p4.1.m1.2.3.3.1.cmml" xref="S2.SS1.p4.1.m1.2.3.3.2"><ci id="S2.SS1.p4.1.m1.1.1.cmml" xref="S2.SS1.p4.1.m1.1.1">𝑝</ci><ci id="S2.SS1.p4.1.m1.2.2.cmml" xref="S2.SS1.p4.1.m1.2.2">𝑞</ci></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p4.1.m1.2c">D(p,q)</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p4.1.m1.2d">italic_D ( italic_p , italic_q )</annotation></semantics></math> of a set of two states <math alttext="p" class="ltx_Math" display="inline" id="S2.SS1.p4.2.m2.1"><semantics id="S2.SS1.p4.2.m2.1a"><mi id="S2.SS1.p4.2.m2.1.1" xref="S2.SS1.p4.2.m2.1.1.cmml">p</mi><annotation-xml encoding="MathML-Content" id="S2.SS1.p4.2.m2.1b"><ci id="S2.SS1.p4.2.m2.1.1.cmml" xref="S2.SS1.p4.2.m2.1.1">𝑝</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p4.2.m2.1c">p</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p4.2.m2.1d">italic_p</annotation></semantics></math> and <math alttext="q" class="ltx_Math" display="inline" id="S2.SS1.p4.3.m3.1"><semantics id="S2.SS1.p4.3.m3.1a"><mi id="S2.SS1.p4.3.m3.1.1" xref="S2.SS1.p4.3.m3.1.1.cmml">q</mi><annotation-xml encoding="MathML-Content" id="S2.SS1.p4.3.m3.1b"><ci id="S2.SS1.p4.3.m3.1.1.cmml" xref="S2.SS1.p4.3.m3.1.1">𝑞</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p4.3.m3.1c">q</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p4.3.m3.1d">italic_q</annotation></semantics></math> is a non-negative function providing a measure how much they differ. Some known examples of the divergences are relative entropy (or Kullback-Leibler divergence) and <math alttext="f" class="ltx_Math" display="inline" id="S2.SS1.p4.4.m4.1"><semantics id="S2.SS1.p4.4.m4.1a"><mi id="S2.SS1.p4.4.m4.1.1" xref="S2.SS1.p4.4.m4.1.1.cmml">f</mi><annotation-xml encoding="MathML-Content" id="S2.SS1.p4.4.m4.1b"><ci id="S2.SS1.p4.4.m4.1.1.cmml" xref="S2.SS1.p4.4.m4.1.1">𝑓</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p4.4.m4.1c">f</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p4.4.m4.1d">italic_f</annotation></semantics></math>-divergence. In IG the <math alttext="\theta" class="ltx_Math" display="inline" id="S2.SS1.p4.5.m5.1"><semantics id="S2.SS1.p4.5.m5.1a"><mi id="S2.SS1.p4.5.m5.1.1" xref="S2.SS1.p4.5.m5.1.1.cmml">θ</mi><annotation-xml encoding="MathML-Content" id="S2.SS1.p4.5.m5.1b"><ci id="S2.SS1.p4.5.m5.1.1.cmml" xref="S2.SS1.p4.5.m5.1.1">𝜃</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p4.5.m5.1c">\theta</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p4.5.m5.1d">italic_θ</annotation></semantics></math>- and <math alttext="\eta" class="ltx_Math" display="inline" id="S2.SS1.p4.6.m6.1"><semantics id="S2.SS1.p4.6.m6.1a"><mi id="S2.SS1.p4.6.m6.1.1" xref="S2.SS1.p4.6.m6.1.1.cmml">η</mi><annotation-xml encoding="MathML-Content" id="S2.SS1.p4.6.m6.1b"><ci id="S2.SS1.p4.6.m6.1.1.cmml" xref="S2.SS1.p4.6.m6.1.1">𝜂</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p4.6.m6.1c">\eta</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p4.6.m6.1d">italic_η</annotation></semantics></math>-divergence functions are</p> <table class="ltx_equationgroup ltx_eqn_table" id="S2.E11"> <tbody> <tr class="ltx_eqn_row" id="S3.EGx11"><td class="ltx_eqn_cell" colspan="5"></td></tr> <tr class="ltx_equation ltx_eqn_row ltx_align_baseline" id="S2.E11.1"> <td class="ltx_eqn_cell ltx_eqn_left_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle D(\theta,\theta_{\rm r})" class="ltx_Math" display="inline" id="S2.E11.1.m1.2"><semantics id="S2.E11.1.m1.2a"><mrow id="S2.E11.1.m1.2.2" xref="S2.E11.1.m1.2.2.cmml"><mi id="S2.E11.1.m1.2.2.3" xref="S2.E11.1.m1.2.2.3.cmml">D</mi><mo id="S2.E11.1.m1.2.2.2" xref="S2.E11.1.m1.2.2.2.cmml">⁢</mo><mrow id="S2.E11.1.m1.2.2.1.1" xref="S2.E11.1.m1.2.2.1.2.cmml"><mo id="S2.E11.1.m1.2.2.1.1.2" stretchy="false" xref="S2.E11.1.m1.2.2.1.2.cmml">(</mo><mi id="S2.E11.1.m1.1.1" xref="S2.E11.1.m1.1.1.cmml">θ</mi><mo id="S2.E11.1.m1.2.2.1.1.3" xref="S2.E11.1.m1.2.2.1.2.cmml">,</mo><msub id="S2.E11.1.m1.2.2.1.1.1" xref="S2.E11.1.m1.2.2.1.1.1.cmml"><mi id="S2.E11.1.m1.2.2.1.1.1.2" xref="S2.E11.1.m1.2.2.1.1.1.2.cmml">θ</mi><mi id="S2.E11.1.m1.2.2.1.1.1.3" mathvariant="normal" xref="S2.E11.1.m1.2.2.1.1.1.3.cmml">r</mi></msub><mo id="S2.E11.1.m1.2.2.1.1.4" stretchy="false" xref="S2.E11.1.m1.2.2.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.E11.1.m1.2b"><apply id="S2.E11.1.m1.2.2.cmml" xref="S2.E11.1.m1.2.2"><times id="S2.E11.1.m1.2.2.2.cmml" xref="S2.E11.1.m1.2.2.2"></times><ci id="S2.E11.1.m1.2.2.3.cmml" xref="S2.E11.1.m1.2.2.3">𝐷</ci><interval closure="open" id="S2.E11.1.m1.2.2.1.2.cmml" xref="S2.E11.1.m1.2.2.1.1"><ci id="S2.E11.1.m1.1.1.cmml" xref="S2.E11.1.m1.1.1">𝜃</ci><apply id="S2.E11.1.m1.2.2.1.1.1.cmml" xref="S2.E11.1.m1.2.2.1.1.1"><csymbol cd="ambiguous" id="S2.E11.1.m1.2.2.1.1.1.1.cmml" xref="S2.E11.1.m1.2.2.1.1.1">subscript</csymbol><ci id="S2.E11.1.m1.2.2.1.1.1.2.cmml" xref="S2.E11.1.m1.2.2.1.1.1.2">𝜃</ci><ci id="S2.E11.1.m1.2.2.1.1.1.3.cmml" xref="S2.E11.1.m1.2.2.1.1.1.3">r</ci></apply></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.E11.1.m1.2c">\displaystyle D(\theta,\theta_{\rm r})</annotation><annotation encoding="application/x-llamapun" id="S2.E11.1.m1.2d">italic_D ( italic_θ , italic_θ start_POSTSUBSCRIPT roman_r end_POSTSUBSCRIPT )</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle:=\Psi(\theta)-\Psi(\theta_{\rm r})-\eta_{i}^{\rm r}(\theta^{i}-% \theta^{i}_{\rm r})," class="ltx_Math" display="inline" id="S2.E11.1.m2.2"><semantics id="S2.E11.1.m2.2a"><mrow id="S2.E11.1.m2.2.2.1" xref="S2.E11.1.m2.2.2.1.1.cmml"><mrow id="S2.E11.1.m2.2.2.1.1" xref="S2.E11.1.m2.2.2.1.1.cmml"><mi id="S2.E11.1.m2.2.2.1.1.4" xref="S2.E11.1.m2.2.2.1.1.4.cmml"></mi><mo id="S2.E11.1.m2.2.2.1.1.3" lspace="0.278em" rspace="0.278em" xref="S2.E11.1.m2.2.2.1.1.3.cmml">:=</mo><mrow id="S2.E11.1.m2.2.2.1.1.2" xref="S2.E11.1.m2.2.2.1.1.2.cmml"><mrow id="S2.E11.1.m2.2.2.1.1.2.4" xref="S2.E11.1.m2.2.2.1.1.2.4.cmml"><mi id="S2.E11.1.m2.2.2.1.1.2.4.2" mathvariant="normal" xref="S2.E11.1.m2.2.2.1.1.2.4.2.cmml">Ψ</mi><mo id="S2.E11.1.m2.2.2.1.1.2.4.1" xref="S2.E11.1.m2.2.2.1.1.2.4.1.cmml">⁢</mo><mrow id="S2.E11.1.m2.2.2.1.1.2.4.3.2" xref="S2.E11.1.m2.2.2.1.1.2.4.cmml"><mo id="S2.E11.1.m2.2.2.1.1.2.4.3.2.1" stretchy="false" xref="S2.E11.1.m2.2.2.1.1.2.4.cmml">(</mo><mi id="S2.E11.1.m2.1.1" xref="S2.E11.1.m2.1.1.cmml">θ</mi><mo id="S2.E11.1.m2.2.2.1.1.2.4.3.2.2" stretchy="false" xref="S2.E11.1.m2.2.2.1.1.2.4.cmml">)</mo></mrow></mrow><mo id="S2.E11.1.m2.2.2.1.1.2.3" xref="S2.E11.1.m2.2.2.1.1.2.3.cmml">−</mo><mrow id="S2.E11.1.m2.2.2.1.1.1.1" xref="S2.E11.1.m2.2.2.1.1.1.1.cmml"><mi id="S2.E11.1.m2.2.2.1.1.1.1.3" mathvariant="normal" xref="S2.E11.1.m2.2.2.1.1.1.1.3.cmml">Ψ</mi><mo id="S2.E11.1.m2.2.2.1.1.1.1.2" 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xref="S2.E11.1.m2.2.2.1.1.2.2.3.2.3.cmml">i</mi><mi id="S2.E11.1.m2.2.2.1.1.2.2.3.3" mathvariant="normal" xref="S2.E11.1.m2.2.2.1.1.2.2.3.3.cmml">r</mi></msubsup><mo id="S2.E11.1.m2.2.2.1.1.2.2.2" xref="S2.E11.1.m2.2.2.1.1.2.2.2.cmml">⁢</mo><mrow id="S2.E11.1.m2.2.2.1.1.2.2.1.1" xref="S2.E11.1.m2.2.2.1.1.2.2.1.1.1.cmml"><mo id="S2.E11.1.m2.2.2.1.1.2.2.1.1.2" stretchy="false" xref="S2.E11.1.m2.2.2.1.1.2.2.1.1.1.cmml">(</mo><mrow id="S2.E11.1.m2.2.2.1.1.2.2.1.1.1" xref="S2.E11.1.m2.2.2.1.1.2.2.1.1.1.cmml"><msup id="S2.E11.1.m2.2.2.1.1.2.2.1.1.1.2" xref="S2.E11.1.m2.2.2.1.1.2.2.1.1.1.2.cmml"><mi id="S2.E11.1.m2.2.2.1.1.2.2.1.1.1.2.2" xref="S2.E11.1.m2.2.2.1.1.2.2.1.1.1.2.2.cmml">θ</mi><mi id="S2.E11.1.m2.2.2.1.1.2.2.1.1.1.2.3" xref="S2.E11.1.m2.2.2.1.1.2.2.1.1.1.2.3.cmml">i</mi></msup><mo id="S2.E11.1.m2.2.2.1.1.2.2.1.1.1.1" xref="S2.E11.1.m2.2.2.1.1.2.2.1.1.1.1.cmml">−</mo><msubsup id="S2.E11.1.m2.2.2.1.1.2.2.1.1.1.3" xref="S2.E11.1.m2.2.2.1.1.2.2.1.1.1.3.cmml"><mi 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id="S2.E11.1.m2.2.2.1.1.2.2.1.1.1.2.cmml" xref="S2.E11.1.m2.2.2.1.1.2.2.1.1.1.2"><csymbol cd="ambiguous" id="S2.E11.1.m2.2.2.1.1.2.2.1.1.1.2.1.cmml" xref="S2.E11.1.m2.2.2.1.1.2.2.1.1.1.2">superscript</csymbol><ci id="S2.E11.1.m2.2.2.1.1.2.2.1.1.1.2.2.cmml" xref="S2.E11.1.m2.2.2.1.1.2.2.1.1.1.2.2">𝜃</ci><ci id="S2.E11.1.m2.2.2.1.1.2.2.1.1.1.2.3.cmml" xref="S2.E11.1.m2.2.2.1.1.2.2.1.1.1.2.3">𝑖</ci></apply><apply id="S2.E11.1.m2.2.2.1.1.2.2.1.1.1.3.cmml" xref="S2.E11.1.m2.2.2.1.1.2.2.1.1.1.3"><csymbol cd="ambiguous" id="S2.E11.1.m2.2.2.1.1.2.2.1.1.1.3.1.cmml" xref="S2.E11.1.m2.2.2.1.1.2.2.1.1.1.3">subscript</csymbol><apply id="S2.E11.1.m2.2.2.1.1.2.2.1.1.1.3.2.cmml" xref="S2.E11.1.m2.2.2.1.1.2.2.1.1.1.3"><csymbol cd="ambiguous" id="S2.E11.1.m2.2.2.1.1.2.2.1.1.1.3.2.1.cmml" xref="S2.E11.1.m2.2.2.1.1.2.2.1.1.1.3">superscript</csymbol><ci id="S2.E11.1.m2.2.2.1.1.2.2.1.1.1.3.2.2.cmml" xref="S2.E11.1.m2.2.2.1.1.2.2.1.1.1.3.2.2">𝜃</ci><ci id="S2.E11.1.m2.2.2.1.1.2.2.1.1.1.3.2.3.cmml" xref="S2.E11.1.m2.2.2.1.1.2.2.1.1.1.3.2.3">𝑖</ci></apply><ci id="S2.E11.1.m2.2.2.1.1.2.2.1.1.1.3.3.cmml" xref="S2.E11.1.m2.2.2.1.1.2.2.1.1.1.3.3">r</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.E11.1.m2.2c">\displaystyle:=\Psi(\theta)-\Psi(\theta_{\rm r})-\eta_{i}^{\rm r}(\theta^{i}-% \theta^{i}_{\rm r}),</annotation><annotation encoding="application/x-llamapun" id="S2.E11.1.m2.2d">:= roman_Ψ ( italic_θ ) - roman_Ψ ( italic_θ start_POSTSUBSCRIPT roman_r end_POSTSUBSCRIPT ) - italic_η start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT roman_r end_POSTSUPERSCRIPT ( italic_θ start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT - italic_θ start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT start_POSTSUBSCRIPT roman_r end_POSTSUBSCRIPT ) ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_left_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(11a)</span></td> </tr> <tr class="ltx_equation ltx_eqn_row ltx_align_baseline" id="S2.E11.2"> <td class="ltx_eqn_cell ltx_eqn_left_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle D(\eta,\eta^{\rm r})" class="ltx_Math" display="inline" id="S2.E11.2.m1.2"><semantics id="S2.E11.2.m1.2a"><mrow id="S2.E11.2.m1.2.2" xref="S2.E11.2.m1.2.2.cmml"><mi id="S2.E11.2.m1.2.2.3" xref="S2.E11.2.m1.2.2.3.cmml">D</mi><mo id="S2.E11.2.m1.2.2.2" xref="S2.E11.2.m1.2.2.2.cmml">⁢</mo><mrow id="S2.E11.2.m1.2.2.1.1" xref="S2.E11.2.m1.2.2.1.2.cmml"><mo id="S2.E11.2.m1.2.2.1.1.2" stretchy="false" xref="S2.E11.2.m1.2.2.1.2.cmml">(</mo><mi id="S2.E11.2.m1.1.1" xref="S2.E11.2.m1.1.1.cmml">η</mi><mo id="S2.E11.2.m1.2.2.1.1.3" xref="S2.E11.2.m1.2.2.1.2.cmml">,</mo><msup id="S2.E11.2.m1.2.2.1.1.1" xref="S2.E11.2.m1.2.2.1.1.1.cmml"><mi id="S2.E11.2.m1.2.2.1.1.1.2" xref="S2.E11.2.m1.2.2.1.1.1.2.cmml">η</mi><mi id="S2.E11.2.m1.2.2.1.1.1.3" mathvariant="normal" xref="S2.E11.2.m1.2.2.1.1.1.3.cmml">r</mi></msup><mo id="S2.E11.2.m1.2.2.1.1.4" stretchy="false" xref="S2.E11.2.m1.2.2.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.E11.2.m1.2b"><apply id="S2.E11.2.m1.2.2.cmml" xref="S2.E11.2.m1.2.2"><times id="S2.E11.2.m1.2.2.2.cmml" xref="S2.E11.2.m1.2.2.2"></times><ci id="S2.E11.2.m1.2.2.3.cmml" xref="S2.E11.2.m1.2.2.3">𝐷</ci><interval closure="open" id="S2.E11.2.m1.2.2.1.2.cmml" xref="S2.E11.2.m1.2.2.1.1"><ci id="S2.E11.2.m1.1.1.cmml" xref="S2.E11.2.m1.1.1">𝜂</ci><apply id="S2.E11.2.m1.2.2.1.1.1.cmml" xref="S2.E11.2.m1.2.2.1.1.1"><csymbol cd="ambiguous" id="S2.E11.2.m1.2.2.1.1.1.1.cmml" xref="S2.E11.2.m1.2.2.1.1.1">superscript</csymbol><ci id="S2.E11.2.m1.2.2.1.1.1.2.cmml" xref="S2.E11.2.m1.2.2.1.1.1.2">𝜂</ci><ci id="S2.E11.2.m1.2.2.1.1.1.3.cmml" xref="S2.E11.2.m1.2.2.1.1.1.3">r</ci></apply></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.E11.2.m1.2c">\displaystyle D(\eta,\eta^{\rm r})</annotation><annotation encoding="application/x-llamapun" id="S2.E11.2.m1.2d">italic_D ( italic_η , italic_η start_POSTSUPERSCRIPT roman_r end_POSTSUPERSCRIPT )</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle:=\Psi^{\star}(\eta)-\Psi^{\star}(\eta_{\rm r})-\theta^{i}_{\rm r% }(\eta_{i}-\eta_{i}^{\rm r})," class="ltx_Math" display="inline" id="S2.E11.2.m2.2"><semantics id="S2.E11.2.m2.2a"><mrow id="S2.E11.2.m2.2.2.1" xref="S2.E11.2.m2.2.2.1.1.cmml"><mrow id="S2.E11.2.m2.2.2.1.1" xref="S2.E11.2.m2.2.2.1.1.cmml"><mi id="S2.E11.2.m2.2.2.1.1.4" xref="S2.E11.2.m2.2.2.1.1.4.cmml"></mi><mo id="S2.E11.2.m2.2.2.1.1.3" lspace="0.278em" rspace="0.278em" xref="S2.E11.2.m2.2.2.1.1.3.cmml">:=</mo><mrow id="S2.E11.2.m2.2.2.1.1.2" xref="S2.E11.2.m2.2.2.1.1.2.cmml"><mrow id="S2.E11.2.m2.2.2.1.1.2.4" xref="S2.E11.2.m2.2.2.1.1.2.4.cmml"><msup id="S2.E11.2.m2.2.2.1.1.2.4.2" 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id="S2.E11.2.m2.2c">\displaystyle:=\Psi^{\star}(\eta)-\Psi^{\star}(\eta_{\rm r})-\theta^{i}_{\rm r% }(\eta_{i}-\eta_{i}^{\rm r}),</annotation><annotation encoding="application/x-llamapun" id="S2.E11.2.m2.2d">:= roman_Ψ start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT ( italic_η ) - roman_Ψ start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT ( italic_η start_POSTSUBSCRIPT roman_r end_POSTSUBSCRIPT ) - italic_θ start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT start_POSTSUBSCRIPT roman_r end_POSTSUBSCRIPT ( italic_η start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT - italic_η start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT roman_r end_POSTSUPERSCRIPT ) ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_left_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(11b)</span></td> </tr> </tbody> </table> <p class="ltx_p" id="S2.SS1.p4.16">respectively. Here the <math alttext="\theta_{\rm r}" class="ltx_Math" display="inline" id="S2.SS1.p4.7.m1.1"><semantics id="S2.SS1.p4.7.m1.1a"><msub id="S2.SS1.p4.7.m1.1.1" xref="S2.SS1.p4.7.m1.1.1.cmml"><mi id="S2.SS1.p4.7.m1.1.1.2" xref="S2.SS1.p4.7.m1.1.1.2.cmml">θ</mi><mi id="S2.SS1.p4.7.m1.1.1.3" mathvariant="normal" xref="S2.SS1.p4.7.m1.1.1.3.cmml">r</mi></msub><annotation-xml encoding="MathML-Content" id="S2.SS1.p4.7.m1.1b"><apply id="S2.SS1.p4.7.m1.1.1.cmml" xref="S2.SS1.p4.7.m1.1.1"><csymbol cd="ambiguous" id="S2.SS1.p4.7.m1.1.1.1.cmml" xref="S2.SS1.p4.7.m1.1.1">subscript</csymbol><ci id="S2.SS1.p4.7.m1.1.1.2.cmml" xref="S2.SS1.p4.7.m1.1.1.2">𝜃</ci><ci id="S2.SS1.p4.7.m1.1.1.3.cmml" xref="S2.SS1.p4.7.m1.1.1.3">r</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p4.7.m1.1c">\theta_{\rm r}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p4.7.m1.1d">italic_θ start_POSTSUBSCRIPT roman_r end_POSTSUBSCRIPT</annotation></semantics></math> (or <math alttext="\eta^{\rm r}" class="ltx_Math" display="inline" id="S2.SS1.p4.8.m2.1"><semantics id="S2.SS1.p4.8.m2.1a"><msup id="S2.SS1.p4.8.m2.1.1" xref="S2.SS1.p4.8.m2.1.1.cmml"><mi id="S2.SS1.p4.8.m2.1.1.2" xref="S2.SS1.p4.8.m2.1.1.2.cmml">η</mi><mi id="S2.SS1.p4.8.m2.1.1.3" mathvariant="normal" xref="S2.SS1.p4.8.m2.1.1.3.cmml">r</mi></msup><annotation-xml encoding="MathML-Content" id="S2.SS1.p4.8.m2.1b"><apply id="S2.SS1.p4.8.m2.1.1.cmml" xref="S2.SS1.p4.8.m2.1.1"><csymbol cd="ambiguous" id="S2.SS1.p4.8.m2.1.1.1.cmml" xref="S2.SS1.p4.8.m2.1.1">superscript</csymbol><ci id="S2.SS1.p4.8.m2.1.1.2.cmml" xref="S2.SS1.p4.8.m2.1.1.2">𝜂</ci><ci id="S2.SS1.p4.8.m2.1.1.3.cmml" xref="S2.SS1.p4.8.m2.1.1.3">r</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p4.8.m2.1c">\eta^{\rm r}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p4.8.m2.1d">italic_η start_POSTSUPERSCRIPT roman_r end_POSTSUPERSCRIPT</annotation></semantics></math>) denotes the <math alttext="\theta" class="ltx_Math" display="inline" id="S2.SS1.p4.9.m3.1"><semantics id="S2.SS1.p4.9.m3.1a"><mi id="S2.SS1.p4.9.m3.1.1" xref="S2.SS1.p4.9.m3.1.1.cmml">θ</mi><annotation-xml encoding="MathML-Content" id="S2.SS1.p4.9.m3.1b"><ci id="S2.SS1.p4.9.m3.1.1.cmml" xref="S2.SS1.p4.9.m3.1.1">𝜃</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p4.9.m3.1c">\theta</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p4.9.m3.1d">italic_θ</annotation></semantics></math>- (or <math alttext="\eta" class="ltx_Math" display="inline" id="S2.SS1.p4.10.m4.1"><semantics id="S2.SS1.p4.10.m4.1a"><mi id="S2.SS1.p4.10.m4.1.1" xref="S2.SS1.p4.10.m4.1.1.cmml">η</mi><annotation-xml encoding="MathML-Content" id="S2.SS1.p4.10.m4.1b"><ci id="S2.SS1.p4.10.m4.1.1.cmml" xref="S2.SS1.p4.10.m4.1.1">𝜂</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p4.10.m4.1c">\eta</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p4.10.m4.1d">italic_η</annotation></semantics></math>-) vector of a reference state. When <math alttext="\theta=\theta_{\rm r}" class="ltx_Math" display="inline" id="S2.SS1.p4.11.m5.1"><semantics id="S2.SS1.p4.11.m5.1a"><mrow id="S2.SS1.p4.11.m5.1.1" xref="S2.SS1.p4.11.m5.1.1.cmml"><mi id="S2.SS1.p4.11.m5.1.1.2" xref="S2.SS1.p4.11.m5.1.1.2.cmml">θ</mi><mo id="S2.SS1.p4.11.m5.1.1.1" xref="S2.SS1.p4.11.m5.1.1.1.cmml">=</mo><msub id="S2.SS1.p4.11.m5.1.1.3" xref="S2.SS1.p4.11.m5.1.1.3.cmml"><mi id="S2.SS1.p4.11.m5.1.1.3.2" xref="S2.SS1.p4.11.m5.1.1.3.2.cmml">θ</mi><mi id="S2.SS1.p4.11.m5.1.1.3.3" mathvariant="normal" xref="S2.SS1.p4.11.m5.1.1.3.3.cmml">r</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p4.11.m5.1b"><apply id="S2.SS1.p4.11.m5.1.1.cmml" xref="S2.SS1.p4.11.m5.1.1"><eq id="S2.SS1.p4.11.m5.1.1.1.cmml" xref="S2.SS1.p4.11.m5.1.1.1"></eq><ci id="S2.SS1.p4.11.m5.1.1.2.cmml" xref="S2.SS1.p4.11.m5.1.1.2">𝜃</ci><apply id="S2.SS1.p4.11.m5.1.1.3.cmml" xref="S2.SS1.p4.11.m5.1.1.3"><csymbol cd="ambiguous" id="S2.SS1.p4.11.m5.1.1.3.1.cmml" xref="S2.SS1.p4.11.m5.1.1.3">subscript</csymbol><ci id="S2.SS1.p4.11.m5.1.1.3.2.cmml" xref="S2.SS1.p4.11.m5.1.1.3.2">𝜃</ci><ci id="S2.SS1.p4.11.m5.1.1.3.3.cmml" xref="S2.SS1.p4.11.m5.1.1.3.3">r</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p4.11.m5.1c">\theta=\theta_{\rm r}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p4.11.m5.1d">italic_θ = italic_θ start_POSTSUBSCRIPT roman_r end_POSTSUBSCRIPT</annotation></semantics></math>, the <math alttext="\theta" class="ltx_Math" display="inline" id="S2.SS1.p4.12.m6.1"><semantics id="S2.SS1.p4.12.m6.1a"><mi id="S2.SS1.p4.12.m6.1.1" xref="S2.SS1.p4.12.m6.1.1.cmml">θ</mi><annotation-xml encoding="MathML-Content" id="S2.SS1.p4.12.m6.1b"><ci id="S2.SS1.p4.12.m6.1.1.cmml" xref="S2.SS1.p4.12.m6.1.1">𝜃</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p4.12.m6.1c">\theta</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p4.12.m6.1d">italic_θ</annotation></semantics></math>-divergence <math alttext="D(\theta,\theta_{\rm r})" class="ltx_Math" display="inline" id="S2.SS1.p4.13.m7.2"><semantics id="S2.SS1.p4.13.m7.2a"><mrow id="S2.SS1.p4.13.m7.2.2" xref="S2.SS1.p4.13.m7.2.2.cmml"><mi id="S2.SS1.p4.13.m7.2.2.3" xref="S2.SS1.p4.13.m7.2.2.3.cmml">D</mi><mo id="S2.SS1.p4.13.m7.2.2.2" xref="S2.SS1.p4.13.m7.2.2.2.cmml">⁢</mo><mrow id="S2.SS1.p4.13.m7.2.2.1.1" xref="S2.SS1.p4.13.m7.2.2.1.2.cmml"><mo id="S2.SS1.p4.13.m7.2.2.1.1.2" stretchy="false" xref="S2.SS1.p4.13.m7.2.2.1.2.cmml">(</mo><mi id="S2.SS1.p4.13.m7.1.1" xref="S2.SS1.p4.13.m7.1.1.cmml">θ</mi><mo id="S2.SS1.p4.13.m7.2.2.1.1.3" xref="S2.SS1.p4.13.m7.2.2.1.2.cmml">,</mo><msub id="S2.SS1.p4.13.m7.2.2.1.1.1" xref="S2.SS1.p4.13.m7.2.2.1.1.1.cmml"><mi id="S2.SS1.p4.13.m7.2.2.1.1.1.2" xref="S2.SS1.p4.13.m7.2.2.1.1.1.2.cmml">θ</mi><mi id="S2.SS1.p4.13.m7.2.2.1.1.1.3" mathvariant="normal" xref="S2.SS1.p4.13.m7.2.2.1.1.1.3.cmml">r</mi></msub><mo id="S2.SS1.p4.13.m7.2.2.1.1.4" stretchy="false" xref="S2.SS1.p4.13.m7.2.2.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p4.13.m7.2b"><apply id="S2.SS1.p4.13.m7.2.2.cmml" xref="S2.SS1.p4.13.m7.2.2"><times id="S2.SS1.p4.13.m7.2.2.2.cmml" xref="S2.SS1.p4.13.m7.2.2.2"></times><ci id="S2.SS1.p4.13.m7.2.2.3.cmml" xref="S2.SS1.p4.13.m7.2.2.3">𝐷</ci><interval closure="open" id="S2.SS1.p4.13.m7.2.2.1.2.cmml" xref="S2.SS1.p4.13.m7.2.2.1.1"><ci id="S2.SS1.p4.13.m7.1.1.cmml" xref="S2.SS1.p4.13.m7.1.1">𝜃</ci><apply id="S2.SS1.p4.13.m7.2.2.1.1.1.cmml" xref="S2.SS1.p4.13.m7.2.2.1.1.1"><csymbol cd="ambiguous" id="S2.SS1.p4.13.m7.2.2.1.1.1.1.cmml" xref="S2.SS1.p4.13.m7.2.2.1.1.1">subscript</csymbol><ci id="S2.SS1.p4.13.m7.2.2.1.1.1.2.cmml" xref="S2.SS1.p4.13.m7.2.2.1.1.1.2">𝜃</ci><ci id="S2.SS1.p4.13.m7.2.2.1.1.1.3.cmml" xref="S2.SS1.p4.13.m7.2.2.1.1.1.3">r</ci></apply></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p4.13.m7.2c">D(\theta,\theta_{\rm r})</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p4.13.m7.2d">italic_D ( italic_θ , italic_θ start_POSTSUBSCRIPT roman_r end_POSTSUBSCRIPT )</annotation></semantics></math> vanishes and similarly the <math alttext="\eta" class="ltx_Math" display="inline" id="S2.SS1.p4.14.m8.1"><semantics id="S2.SS1.p4.14.m8.1a"><mi id="S2.SS1.p4.14.m8.1.1" xref="S2.SS1.p4.14.m8.1.1.cmml">η</mi><annotation-xml encoding="MathML-Content" id="S2.SS1.p4.14.m8.1b"><ci id="S2.SS1.p4.14.m8.1.1.cmml" xref="S2.SS1.p4.14.m8.1.1">𝜂</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p4.14.m8.1c">\eta</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p4.14.m8.1d">italic_η</annotation></semantics></math>-divergence <math alttext="D(\eta,\eta^{\rm r})" class="ltx_Math" display="inline" id="S2.SS1.p4.15.m9.2"><semantics id="S2.SS1.p4.15.m9.2a"><mrow id="S2.SS1.p4.15.m9.2.2" xref="S2.SS1.p4.15.m9.2.2.cmml"><mi id="S2.SS1.p4.15.m9.2.2.3" xref="S2.SS1.p4.15.m9.2.2.3.cmml">D</mi><mo id="S2.SS1.p4.15.m9.2.2.2" xref="S2.SS1.p4.15.m9.2.2.2.cmml">⁢</mo><mrow id="S2.SS1.p4.15.m9.2.2.1.1" xref="S2.SS1.p4.15.m9.2.2.1.2.cmml"><mo id="S2.SS1.p4.15.m9.2.2.1.1.2" stretchy="false" xref="S2.SS1.p4.15.m9.2.2.1.2.cmml">(</mo><mi id="S2.SS1.p4.15.m9.1.1" xref="S2.SS1.p4.15.m9.1.1.cmml">η</mi><mo id="S2.SS1.p4.15.m9.2.2.1.1.3" xref="S2.SS1.p4.15.m9.2.2.1.2.cmml">,</mo><msup id="S2.SS1.p4.15.m9.2.2.1.1.1" xref="S2.SS1.p4.15.m9.2.2.1.1.1.cmml"><mi id="S2.SS1.p4.15.m9.2.2.1.1.1.2" xref="S2.SS1.p4.15.m9.2.2.1.1.1.2.cmml">η</mi><mi id="S2.SS1.p4.15.m9.2.2.1.1.1.3" mathvariant="normal" xref="S2.SS1.p4.15.m9.2.2.1.1.1.3.cmml">r</mi></msup><mo id="S2.SS1.p4.15.m9.2.2.1.1.4" stretchy="false" xref="S2.SS1.p4.15.m9.2.2.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p4.15.m9.2b"><apply id="S2.SS1.p4.15.m9.2.2.cmml" xref="S2.SS1.p4.15.m9.2.2"><times id="S2.SS1.p4.15.m9.2.2.2.cmml" xref="S2.SS1.p4.15.m9.2.2.2"></times><ci id="S2.SS1.p4.15.m9.2.2.3.cmml" xref="S2.SS1.p4.15.m9.2.2.3">𝐷</ci><interval closure="open" id="S2.SS1.p4.15.m9.2.2.1.2.cmml" xref="S2.SS1.p4.15.m9.2.2.1.1"><ci id="S2.SS1.p4.15.m9.1.1.cmml" xref="S2.SS1.p4.15.m9.1.1">𝜂</ci><apply id="S2.SS1.p4.15.m9.2.2.1.1.1.cmml" xref="S2.SS1.p4.15.m9.2.2.1.1.1"><csymbol cd="ambiguous" id="S2.SS1.p4.15.m9.2.2.1.1.1.1.cmml" xref="S2.SS1.p4.15.m9.2.2.1.1.1">superscript</csymbol><ci id="S2.SS1.p4.15.m9.2.2.1.1.1.2.cmml" xref="S2.SS1.p4.15.m9.2.2.1.1.1.2">𝜂</ci><ci id="S2.SS1.p4.15.m9.2.2.1.1.1.3.cmml" xref="S2.SS1.p4.15.m9.2.2.1.1.1.3">r</ci></apply></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p4.15.m9.2c">D(\eta,\eta^{\rm r})</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p4.15.m9.2d">italic_D ( italic_η , italic_η start_POSTSUPERSCRIPT roman_r end_POSTSUPERSCRIPT )</annotation></semantics></math> vanishes when <math alttext="\eta=\eta^{\rm r}" class="ltx_Math" display="inline" id="S2.SS1.p4.16.m10.1"><semantics id="S2.SS1.p4.16.m10.1a"><mrow id="S2.SS1.p4.16.m10.1.1" xref="S2.SS1.p4.16.m10.1.1.cmml"><mi id="S2.SS1.p4.16.m10.1.1.2" xref="S2.SS1.p4.16.m10.1.1.2.cmml">η</mi><mo id="S2.SS1.p4.16.m10.1.1.1" xref="S2.SS1.p4.16.m10.1.1.1.cmml">=</mo><msup id="S2.SS1.p4.16.m10.1.1.3" xref="S2.SS1.p4.16.m10.1.1.3.cmml"><mi id="S2.SS1.p4.16.m10.1.1.3.2" xref="S2.SS1.p4.16.m10.1.1.3.2.cmml">η</mi><mi id="S2.SS1.p4.16.m10.1.1.3.3" mathvariant="normal" xref="S2.SS1.p4.16.m10.1.1.3.3.cmml">r</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p4.16.m10.1b"><apply id="S2.SS1.p4.16.m10.1.1.cmml" xref="S2.SS1.p4.16.m10.1.1"><eq id="S2.SS1.p4.16.m10.1.1.1.cmml" xref="S2.SS1.p4.16.m10.1.1.1"></eq><ci id="S2.SS1.p4.16.m10.1.1.2.cmml" xref="S2.SS1.p4.16.m10.1.1.2">𝜂</ci><apply id="S2.SS1.p4.16.m10.1.1.3.cmml" xref="S2.SS1.p4.16.m10.1.1.3"><csymbol cd="ambiguous" id="S2.SS1.p4.16.m10.1.1.3.1.cmml" xref="S2.SS1.p4.16.m10.1.1.3">superscript</csymbol><ci id="S2.SS1.p4.16.m10.1.1.3.2.cmml" xref="S2.SS1.p4.16.m10.1.1.3.2">𝜂</ci><ci id="S2.SS1.p4.16.m10.1.1.3.3.cmml" xref="S2.SS1.p4.16.m10.1.1.3.3">r</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p4.16.m10.1c">\eta=\eta^{\rm r}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p4.16.m10.1d">italic_η = italic_η start_POSTSUPERSCRIPT roman_r end_POSTSUPERSCRIPT</annotation></semantics></math>.</p> </div> </section> <section class="ltx_subsection" id="S2.SS2"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">2.2 </span>Gradient-Flow Equations</h3> <div class="ltx_para" id="S2.SS2.p1"> <p class="ltx_p" id="S2.SS2.p1.3">The gradient-flow equations <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2406.11224v2#bib.bib3" title="">3</a>, <a class="ltx_ref" href="https://arxiv.org/html/2406.11224v2#bib.bib4" title="">4</a>, <a class="ltx_ref" href="https://arxiv.org/html/2406.11224v2#bib.bib6" title="">6</a>]</cite> in IG are briefly explained here. The gradient-flow equations with respect to the <math alttext="\theta" class="ltx_Math" display="inline" id="S2.SS2.p1.1.m1.1"><semantics id="S2.SS2.p1.1.m1.1a"><mi id="S2.SS2.p1.1.m1.1.1" xref="S2.SS2.p1.1.m1.1.1.cmml">θ</mi><annotation-xml encoding="MathML-Content" id="S2.SS2.p1.1.m1.1b"><ci id="S2.SS2.p1.1.m1.1.1.cmml" xref="S2.SS2.p1.1.m1.1.1">𝜃</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p1.1.m1.1c">\theta</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p1.1.m1.1d">italic_θ</annotation></semantics></math>-divergence function <math alttext="D(\theta,\theta_{\rm r})" class="ltx_Math" display="inline" id="S2.SS2.p1.2.m2.2"><semantics id="S2.SS2.p1.2.m2.2a"><mrow id="S2.SS2.p1.2.m2.2.2" xref="S2.SS2.p1.2.m2.2.2.cmml"><mi id="S2.SS2.p1.2.m2.2.2.3" xref="S2.SS2.p1.2.m2.2.2.3.cmml">D</mi><mo id="S2.SS2.p1.2.m2.2.2.2" xref="S2.SS2.p1.2.m2.2.2.2.cmml">⁢</mo><mrow id="S2.SS2.p1.2.m2.2.2.1.1" xref="S2.SS2.p1.2.m2.2.2.1.2.cmml"><mo 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id="S2.Ex1a.m1.1c">\displaystyle\frac{d\theta^{i}}{dt}</annotation><annotation encoding="application/x-llamapun" id="S2.Ex1a.m1.1d">divide start_ARG italic_d italic_θ start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT end_ARG start_ARG italic_d italic_t end_ARG</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle=g^{ij}(\theta)\,\frac{\partial D(\theta,\theta_{\rm r})}{% \partial\theta^{j}}" class="ltx_Math" display="inline" id="S2.Ex1a.m2.3"><semantics id="S2.Ex1a.m2.3a"><mrow id="S2.Ex1a.m2.3.4" xref="S2.Ex1a.m2.3.4.cmml"><mi id="S2.Ex1a.m2.3.4.2" xref="S2.Ex1a.m2.3.4.2.cmml"></mi><mo id="S2.Ex1a.m2.3.4.1" xref="S2.Ex1a.m2.3.4.1.cmml">=</mo><mrow id="S2.Ex1a.m2.3.4.3" xref="S2.Ex1a.m2.3.4.3.cmml"><msup id="S2.Ex1a.m2.3.4.3.2" xref="S2.Ex1a.m2.3.4.3.2.cmml"><mi id="S2.Ex1a.m2.3.4.3.2.2" xref="S2.Ex1a.m2.3.4.3.2.2.cmml">g</mi><mrow id="S2.Ex1a.m2.3.4.3.2.3" xref="S2.Ex1a.m2.3.4.3.2.3.cmml"><mi id="S2.Ex1a.m2.3.4.3.2.3.2" 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start_POSTSUBSCRIPT italic_θ start_POSTSUPERSCRIPT italic_j end_POSTSUPERSCRIPT = italic_θ start_POSTSUBSCRIPT roman_r end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_j end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ) ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_left_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(12)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S2.SS2.p1.4">in the <math alttext="\theta" class="ltx_Math" display="inline" id="S2.SS2.p1.4.m1.1"><semantics id="S2.SS2.p1.4.m1.1a"><mi id="S2.SS2.p1.4.m1.1.1" xref="S2.SS2.p1.4.m1.1.1.cmml">θ</mi><annotation-xml encoding="MathML-Content" id="S2.SS2.p1.4.m1.1b"><ci id="S2.SS2.p1.4.m1.1.1.cmml" xref="S2.SS2.p1.4.m1.1.1">𝜃</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p1.4.m1.1c">\theta</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p1.4.m1.1d">italic_θ</annotation></semantics></math>-coordinate system. By using the properties (<a class="ltx_ref" href="https://arxiv.org/html/2406.11224v2#S2.E2" title="In 2.1 Information Geometry ‣ 2 Information Geometry and Gradient-Flow Equations ‣ A Hamiltonian approach to the gradient-flow equations in information geometry"><span class="ltx_text ltx_ref_tag">2</span></a>) and (<a class="ltx_ref" href="https://arxiv.org/html/2406.11224v2#S2.E8" title="In 2.1 Information Geometry ‣ 2 Information Geometry and Gradient-Flow Equations ‣ A Hamiltonian approach to the gradient-flow equations in information geometry"><span class="ltx_text ltx_ref_tag">8</span></a>), the left-hand side of (<a class="ltx_ref" href="https://arxiv.org/html/2406.11224v2#S2.E12" title="In 2.2 Gradient-Flow Equations ‣ 2 Information Geometry and Gradient-Flow Equations ‣ A Hamiltonian approach to the gradient-flow equations in information geometry"><span class="ltx_text ltx_ref_tag">12</span></a>) is rewritten by</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" 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id="S2.E13.m1.1.1.3.3.cmml" xref="S2.E13.m1.1.1.3.3">𝑡</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.E13.m1.1c">\displaystyle\frac{d\theta^{i}}{dt}</annotation><annotation encoding="application/x-llamapun" id="S2.E13.m1.1d">divide start_ARG italic_d italic_θ start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT end_ARG start_ARG italic_d italic_t end_ARG</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle=\frac{\partial\theta^{i}}{\partial\eta_{j}}\frac{d\eta_{j}}{dt}=% g^{ij}(\theta)\frac{d\eta_{j}}{dt}," class="ltx_Math" display="inline" id="S2.E13.m2.2"><semantics id="S2.E13.m2.2a"><mrow id="S2.E13.m2.2.2.1" xref="S2.E13.m2.2.2.1.1.cmml"><mrow id="S2.E13.m2.2.2.1.1" xref="S2.E13.m2.2.2.1.1.cmml"><mi id="S2.E13.m2.2.2.1.1.2" xref="S2.E13.m2.2.2.1.1.2.cmml"></mi><mo id="S2.E13.m2.2.2.1.1.3" xref="S2.E13.m2.2.2.1.1.3.cmml">=</mo><mrow id="S2.E13.m2.2.2.1.1.4" 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id="S2.E13.m2.2.2.1.1.6.4.3.cmml" xref="S2.E13.m2.2.2.1.1.6.4.3"><times id="S2.E13.m2.2.2.1.1.6.4.3.1.cmml" xref="S2.E13.m2.2.2.1.1.6.4.3.1"></times><ci id="S2.E13.m2.2.2.1.1.6.4.3.2.cmml" xref="S2.E13.m2.2.2.1.1.6.4.3.2">𝑑</ci><ci id="S2.E13.m2.2.2.1.1.6.4.3.3.cmml" xref="S2.E13.m2.2.2.1.1.6.4.3.3">𝑡</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.E13.m2.2c">\displaystyle=\frac{\partial\theta^{i}}{\partial\eta_{j}}\frac{d\eta_{j}}{dt}=% g^{ij}(\theta)\frac{d\eta_{j}}{dt},</annotation><annotation encoding="application/x-llamapun" id="S2.E13.m2.2d">= divide start_ARG ∂ italic_θ start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT end_ARG start_ARG ∂ italic_η start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT end_ARG divide start_ARG italic_d italic_η start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT end_ARG start_ARG italic_d italic_t end_ARG = italic_g start_POSTSUPERSCRIPT italic_i italic_j end_POSTSUPERSCRIPT ( italic_θ ) divide start_ARG italic_d italic_η start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT end_ARG start_ARG italic_d italic_t end_ARG ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_left_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(13)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S2.SS2.p1.6">and applying (<a class="ltx_ref" href="https://arxiv.org/html/2406.11224v2#S2.E2" title="In 2.1 Information Geometry ‣ 2 Information Geometry and Gradient-Flow Equations ‣ A Hamiltonian approach to the gradient-flow equations in information geometry"><span class="ltx_text ltx_ref_tag">2</span></a>) to the right-hand side of (<a class="ltx_ref" href="https://arxiv.org/html/2406.11224v2#S2.E12" title="In 2.2 Gradient-Flow Equations ‣ 2 Information Geometry and Gradient-Flow Equations ‣ A Hamiltonian approach to the gradient-flow equations in information geometry"><span class="ltx_text ltx_ref_tag">12</span></a>) leads to <math alttext="g^{ij}(\theta)\,(\eta_{j}-\eta^{\rm r}_{j})" class="ltx_Math" display="inline" id="S2.SS2.p1.5.m1.2"><semantics id="S2.SS2.p1.5.m1.2a"><mrow id="S2.SS2.p1.5.m1.2.2" xref="S2.SS2.p1.5.m1.2.2.cmml"><msup id="S2.SS2.p1.5.m1.2.2.3" xref="S2.SS2.p1.5.m1.2.2.3.cmml"><mi id="S2.SS2.p1.5.m1.2.2.3.2" xref="S2.SS2.p1.5.m1.2.2.3.2.cmml">g</mi><mrow id="S2.SS2.p1.5.m1.2.2.3.3" xref="S2.SS2.p1.5.m1.2.2.3.3.cmml"><mi id="S2.SS2.p1.5.m1.2.2.3.3.2" xref="S2.SS2.p1.5.m1.2.2.3.3.2.cmml">i</mi><mo id="S2.SS2.p1.5.m1.2.2.3.3.1" xref="S2.SS2.p1.5.m1.2.2.3.3.1.cmml">⁢</mo><mi id="S2.SS2.p1.5.m1.2.2.3.3.3" xref="S2.SS2.p1.5.m1.2.2.3.3.3.cmml">j</mi></mrow></msup><mo id="S2.SS2.p1.5.m1.2.2.2" xref="S2.SS2.p1.5.m1.2.2.2.cmml">⁢</mo><mrow id="S2.SS2.p1.5.m1.2.2.4.2" xref="S2.SS2.p1.5.m1.2.2.cmml"><mo id="S2.SS2.p1.5.m1.2.2.4.2.1" stretchy="false" xref="S2.SS2.p1.5.m1.2.2.cmml">(</mo><mi id="S2.SS2.p1.5.m1.1.1" xref="S2.SS2.p1.5.m1.1.1.cmml">θ</mi><mo id="S2.SS2.p1.5.m1.2.2.4.2.2" stretchy="false" xref="S2.SS2.p1.5.m1.2.2.cmml">)</mo></mrow><mo id="S2.SS2.p1.5.m1.2.2.2a" lspace="0.170em" xref="S2.SS2.p1.5.m1.2.2.2.cmml">⁢</mo><mrow id="S2.SS2.p1.5.m1.2.2.1.1" xref="S2.SS2.p1.5.m1.2.2.1.1.1.cmml"><mo id="S2.SS2.p1.5.m1.2.2.1.1.2" stretchy="false" xref="S2.SS2.p1.5.m1.2.2.1.1.1.cmml">(</mo><mrow id="S2.SS2.p1.5.m1.2.2.1.1.1" xref="S2.SS2.p1.5.m1.2.2.1.1.1.cmml"><msub id="S2.SS2.p1.5.m1.2.2.1.1.1.2" xref="S2.SS2.p1.5.m1.2.2.1.1.1.2.cmml"><mi id="S2.SS2.p1.5.m1.2.2.1.1.1.2.2" xref="S2.SS2.p1.5.m1.2.2.1.1.1.2.2.cmml">η</mi><mi id="S2.SS2.p1.5.m1.2.2.1.1.1.2.3" xref="S2.SS2.p1.5.m1.2.2.1.1.1.2.3.cmml">j</mi></msub><mo id="S2.SS2.p1.5.m1.2.2.1.1.1.1" xref="S2.SS2.p1.5.m1.2.2.1.1.1.1.cmml">−</mo><msubsup id="S2.SS2.p1.5.m1.2.2.1.1.1.3" xref="S2.SS2.p1.5.m1.2.2.1.1.1.3.cmml"><mi id="S2.SS2.p1.5.m1.2.2.1.1.1.3.2.2" xref="S2.SS2.p1.5.m1.2.2.1.1.1.3.2.2.cmml">η</mi><mi id="S2.SS2.p1.5.m1.2.2.1.1.1.3.3" xref="S2.SS2.p1.5.m1.2.2.1.1.1.3.3.cmml">j</mi><mi id="S2.SS2.p1.5.m1.2.2.1.1.1.3.2.3" mathvariant="normal" xref="S2.SS2.p1.5.m1.2.2.1.1.1.3.2.3.cmml">r</mi></msubsup></mrow><mo id="S2.SS2.p1.5.m1.2.2.1.1.3" stretchy="false" xref="S2.SS2.p1.5.m1.2.2.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p1.5.m1.2b"><apply id="S2.SS2.p1.5.m1.2.2.cmml" xref="S2.SS2.p1.5.m1.2.2"><times id="S2.SS2.p1.5.m1.2.2.2.cmml" xref="S2.SS2.p1.5.m1.2.2.2"></times><apply id="S2.SS2.p1.5.m1.2.2.3.cmml" xref="S2.SS2.p1.5.m1.2.2.3"><csymbol cd="ambiguous" id="S2.SS2.p1.5.m1.2.2.3.1.cmml" xref="S2.SS2.p1.5.m1.2.2.3">superscript</csymbol><ci id="S2.SS2.p1.5.m1.2.2.3.2.cmml" xref="S2.SS2.p1.5.m1.2.2.3.2">𝑔</ci><apply id="S2.SS2.p1.5.m1.2.2.3.3.cmml" xref="S2.SS2.p1.5.m1.2.2.3.3"><times id="S2.SS2.p1.5.m1.2.2.3.3.1.cmml" xref="S2.SS2.p1.5.m1.2.2.3.3.1"></times><ci id="S2.SS2.p1.5.m1.2.2.3.3.2.cmml" xref="S2.SS2.p1.5.m1.2.2.3.3.2">𝑖</ci><ci id="S2.SS2.p1.5.m1.2.2.3.3.3.cmml" xref="S2.SS2.p1.5.m1.2.2.3.3.3">𝑗</ci></apply></apply><ci id="S2.SS2.p1.5.m1.1.1.cmml" xref="S2.SS2.p1.5.m1.1.1">𝜃</ci><apply id="S2.SS2.p1.5.m1.2.2.1.1.1.cmml" xref="S2.SS2.p1.5.m1.2.2.1.1"><minus id="S2.SS2.p1.5.m1.2.2.1.1.1.1.cmml" xref="S2.SS2.p1.5.m1.2.2.1.1.1.1"></minus><apply id="S2.SS2.p1.5.m1.2.2.1.1.1.2.cmml" xref="S2.SS2.p1.5.m1.2.2.1.1.1.2"><csymbol cd="ambiguous" id="S2.SS2.p1.5.m1.2.2.1.1.1.2.1.cmml" xref="S2.SS2.p1.5.m1.2.2.1.1.1.2">subscript</csymbol><ci id="S2.SS2.p1.5.m1.2.2.1.1.1.2.2.cmml" xref="S2.SS2.p1.5.m1.2.2.1.1.1.2.2">𝜂</ci><ci id="S2.SS2.p1.5.m1.2.2.1.1.1.2.3.cmml" xref="S2.SS2.p1.5.m1.2.2.1.1.1.2.3">𝑗</ci></apply><apply id="S2.SS2.p1.5.m1.2.2.1.1.1.3.cmml" xref="S2.SS2.p1.5.m1.2.2.1.1.1.3"><csymbol cd="ambiguous" id="S2.SS2.p1.5.m1.2.2.1.1.1.3.1.cmml" xref="S2.SS2.p1.5.m1.2.2.1.1.1.3">subscript</csymbol><apply id="S2.SS2.p1.5.m1.2.2.1.1.1.3.2.cmml" xref="S2.SS2.p1.5.m1.2.2.1.1.1.3"><csymbol cd="ambiguous" id="S2.SS2.p1.5.m1.2.2.1.1.1.3.2.1.cmml" xref="S2.SS2.p1.5.m1.2.2.1.1.1.3">superscript</csymbol><ci id="S2.SS2.p1.5.m1.2.2.1.1.1.3.2.2.cmml" xref="S2.SS2.p1.5.m1.2.2.1.1.1.3.2.2">𝜂</ci><ci id="S2.SS2.p1.5.m1.2.2.1.1.1.3.2.3.cmml" xref="S2.SS2.p1.5.m1.2.2.1.1.1.3.2.3">r</ci></apply><ci id="S2.SS2.p1.5.m1.2.2.1.1.1.3.3.cmml" xref="S2.SS2.p1.5.m1.2.2.1.1.1.3.3">𝑗</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p1.5.m1.2c">g^{ij}(\theta)\,(\eta_{j}-\eta^{\rm r}_{j})</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p1.5.m1.2d">italic_g start_POSTSUPERSCRIPT italic_i italic_j end_POSTSUPERSCRIPT ( italic_θ ) ( italic_η start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT - italic_η start_POSTSUPERSCRIPT roman_r end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT )</annotation></semantics></math>. Consequently, the gradient-flow equations (<a class="ltx_ref" href="https://arxiv.org/html/2406.11224v2#S2.E12" title="In 2.2 Gradient-Flow Equations ‣ 2 Information Geometry and Gradient-Flow Equations ‣ A Hamiltonian approach to the gradient-flow equations in information geometry"><span class="ltx_text ltx_ref_tag">12</span></a>) in the <math alttext="\theta" class="ltx_Math" display="inline" id="S2.SS2.p1.6.m2.1"><semantics id="S2.SS2.p1.6.m2.1a"><mi id="S2.SS2.p1.6.m2.1.1" xref="S2.SS2.p1.6.m2.1.1.cmml">θ</mi><annotation-xml encoding="MathML-Content" id="S2.SS2.p1.6.m2.1b"><ci id="S2.SS2.p1.6.m2.1.1.cmml" xref="S2.SS2.p1.6.m2.1.1">𝜃</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p1.6.m2.1c">\theta</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p1.6.m2.1d">italic_θ</annotation></semantics></math>-coordinate system are equivalent to the linear differential equations</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="S3.EGx14"> <tbody id="S2.E14"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_left_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\frac{d\eta_{i}(t)}{dt}=\eta_{i}(t)-\eta^{\rm r}_{i}," class="ltx_Math" display="inline" id="S2.E14.m1.3"><semantics id="S2.E14.m1.3a"><mrow id="S2.E14.m1.3.3.1" xref="S2.E14.m1.3.3.1.1.cmml"><mrow id="S2.E14.m1.3.3.1.1" xref="S2.E14.m1.3.3.1.1.cmml"><mstyle displaystyle="true" id="S2.E14.m1.1.1" xref="S2.E14.m1.1.1.cmml"><mfrac id="S2.E14.m1.1.1a" xref="S2.E14.m1.1.1.cmml"><mrow id="S2.E14.m1.1.1.1" xref="S2.E14.m1.1.1.1.cmml"><mi id="S2.E14.m1.1.1.1.3" xref="S2.E14.m1.1.1.1.3.cmml">d</mi><mo id="S2.E14.m1.1.1.1.2" xref="S2.E14.m1.1.1.1.2.cmml">⁢</mo><msub id="S2.E14.m1.1.1.1.4" xref="S2.E14.m1.1.1.1.4.cmml"><mi id="S2.E14.m1.1.1.1.4.2" xref="S2.E14.m1.1.1.1.4.2.cmml">η</mi><mi id="S2.E14.m1.1.1.1.4.3" xref="S2.E14.m1.1.1.1.4.3.cmml">i</mi></msub><mo id="S2.E14.m1.1.1.1.2a" xref="S2.E14.m1.1.1.1.2.cmml">⁢</mo><mrow id="S2.E14.m1.1.1.1.5.2" xref="S2.E14.m1.1.1.1.cmml"><mo id="S2.E14.m1.1.1.1.5.2.1" stretchy="false" xref="S2.E14.m1.1.1.1.cmml">(</mo><mi id="S2.E14.m1.1.1.1.1" xref="S2.E14.m1.1.1.1.1.cmml">t</mi><mo id="S2.E14.m1.1.1.1.5.2.2" stretchy="false" xref="S2.E14.m1.1.1.1.cmml">)</mo></mrow></mrow><mrow id="S2.E14.m1.1.1.3" xref="S2.E14.m1.1.1.3.cmml"><mi id="S2.E14.m1.1.1.3.2" xref="S2.E14.m1.1.1.3.2.cmml">d</mi><mo id="S2.E14.m1.1.1.3.1" xref="S2.E14.m1.1.1.3.1.cmml">⁢</mo><mi id="S2.E14.m1.1.1.3.3" xref="S2.E14.m1.1.1.3.3.cmml">t</mi></mrow></mfrac></mstyle><mo id="S2.E14.m1.3.3.1.1.1" xref="S2.E14.m1.3.3.1.1.1.cmml">=</mo><mrow id="S2.E14.m1.3.3.1.1.2" xref="S2.E14.m1.3.3.1.1.2.cmml"><mrow id="S2.E14.m1.3.3.1.1.2.2" xref="S2.E14.m1.3.3.1.1.2.2.cmml"><msub id="S2.E14.m1.3.3.1.1.2.2.2" xref="S2.E14.m1.3.3.1.1.2.2.2.cmml"><mi id="S2.E14.m1.3.3.1.1.2.2.2.2" xref="S2.E14.m1.3.3.1.1.2.2.2.2.cmml">η</mi><mi id="S2.E14.m1.3.3.1.1.2.2.2.3" xref="S2.E14.m1.3.3.1.1.2.2.2.3.cmml">i</mi></msub><mo id="S2.E14.m1.3.3.1.1.2.2.1" xref="S2.E14.m1.3.3.1.1.2.2.1.cmml">⁢</mo><mrow id="S2.E14.m1.3.3.1.1.2.2.3.2" xref="S2.E14.m1.3.3.1.1.2.2.cmml"><mo id="S2.E14.m1.3.3.1.1.2.2.3.2.1" stretchy="false" xref="S2.E14.m1.3.3.1.1.2.2.cmml">(</mo><mi id="S2.E14.m1.2.2" xref="S2.E14.m1.2.2.cmml">t</mi><mo id="S2.E14.m1.3.3.1.1.2.2.3.2.2" stretchy="false" xref="S2.E14.m1.3.3.1.1.2.2.cmml">)</mo></mrow></mrow><mo id="S2.E14.m1.3.3.1.1.2.1" xref="S2.E14.m1.3.3.1.1.2.1.cmml">−</mo><msubsup id="S2.E14.m1.3.3.1.1.2.3" xref="S2.E14.m1.3.3.1.1.2.3.cmml"><mi id="S2.E14.m1.3.3.1.1.2.3.2.2" xref="S2.E14.m1.3.3.1.1.2.3.2.2.cmml">η</mi><mi id="S2.E14.m1.3.3.1.1.2.3.3" xref="S2.E14.m1.3.3.1.1.2.3.3.cmml">i</mi><mi id="S2.E14.m1.3.3.1.1.2.3.2.3" mathvariant="normal" xref="S2.E14.m1.3.3.1.1.2.3.2.3.cmml">r</mi></msubsup></mrow></mrow><mo id="S2.E14.m1.3.3.1.2" xref="S2.E14.m1.3.3.1.1.cmml">,</mo></mrow><annotation-xml encoding="MathML-Content" id="S2.E14.m1.3b"><apply id="S2.E14.m1.3.3.1.1.cmml" xref="S2.E14.m1.3.3.1"><eq id="S2.E14.m1.3.3.1.1.1.cmml" xref="S2.E14.m1.3.3.1.1.1"></eq><apply id="S2.E14.m1.1.1.cmml" xref="S2.E14.m1.1.1"><divide id="S2.E14.m1.1.1.2.cmml" xref="S2.E14.m1.1.1"></divide><apply id="S2.E14.m1.1.1.1.cmml" xref="S2.E14.m1.1.1.1"><times id="S2.E14.m1.1.1.1.2.cmml" xref="S2.E14.m1.1.1.1.2"></times><ci id="S2.E14.m1.1.1.1.3.cmml" xref="S2.E14.m1.1.1.1.3">𝑑</ci><apply id="S2.E14.m1.1.1.1.4.cmml" xref="S2.E14.m1.1.1.1.4"><csymbol cd="ambiguous" id="S2.E14.m1.1.1.1.4.1.cmml" xref="S2.E14.m1.1.1.1.4">subscript</csymbol><ci id="S2.E14.m1.1.1.1.4.2.cmml" xref="S2.E14.m1.1.1.1.4.2">𝜂</ci><ci id="S2.E14.m1.1.1.1.4.3.cmml" xref="S2.E14.m1.1.1.1.4.3">𝑖</ci></apply><ci id="S2.E14.m1.1.1.1.1.cmml" xref="S2.E14.m1.1.1.1.1">𝑡</ci></apply><apply id="S2.E14.m1.1.1.3.cmml" xref="S2.E14.m1.1.1.3"><times id="S2.E14.m1.1.1.3.1.cmml" xref="S2.E14.m1.1.1.3.1"></times><ci id="S2.E14.m1.1.1.3.2.cmml" 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id="S2.E14.m1.3.3.1.1.2.3.2.cmml" xref="S2.E14.m1.3.3.1.1.2.3"><csymbol cd="ambiguous" id="S2.E14.m1.3.3.1.1.2.3.2.1.cmml" xref="S2.E14.m1.3.3.1.1.2.3">superscript</csymbol><ci id="S2.E14.m1.3.3.1.1.2.3.2.2.cmml" xref="S2.E14.m1.3.3.1.1.2.3.2.2">𝜂</ci><ci id="S2.E14.m1.3.3.1.1.2.3.2.3.cmml" xref="S2.E14.m1.3.3.1.1.2.3.2.3">r</ci></apply><ci id="S2.E14.m1.3.3.1.1.2.3.3.cmml" xref="S2.E14.m1.3.3.1.1.2.3.3">𝑖</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.E14.m1.3c">\displaystyle\frac{d\eta_{i}(t)}{dt}=\eta_{i}(t)-\eta^{\rm r}_{i},</annotation><annotation encoding="application/x-llamapun" id="S2.E14.m1.3d">divide start_ARG italic_d italic_η start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_t ) end_ARG start_ARG italic_d italic_t end_ARG = italic_η start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_t ) - italic_η start_POSTSUPERSCRIPT roman_r end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_left_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(14)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S2.SS2.p1.7">in the <math alttext="\eta" class="ltx_Math" display="inline" id="S2.SS2.p1.7.m1.1"><semantics id="S2.SS2.p1.7.m1.1a"><mi id="S2.SS2.p1.7.m1.1.1" xref="S2.SS2.p1.7.m1.1.1.cmml">η</mi><annotation-xml encoding="MathML-Content" id="S2.SS2.p1.7.m1.1b"><ci id="S2.SS2.p1.7.m1.1.1.cmml" xref="S2.SS2.p1.7.m1.1.1">𝜂</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p1.7.m1.1c">\eta</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p1.7.m1.1d">italic_η</annotation></semantics></math>-coordinate system. This linearization is one of the merits due to the dually-flat structure <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2406.11224v2#bib.bib1" title="">1</a>]</cite> in IG.</p> </div> <div class="ltx_para" id="S2.SS2.p2"> <p class="ltx_p" id="S2.SS2.p2.8">The other set of gradient-flow equations are given by</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="S3.EGx15"> <tbody id="S2.Ex2a"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_left_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\frac{d\eta_{i}}{dt}" class="ltx_Math" display="inline" id="S2.Ex2a.m1.1"><semantics id="S2.Ex2a.m1.1a"><mstyle displaystyle="true" id="S2.Ex2a.m1.1.1" xref="S2.Ex2a.m1.1.1.cmml"><mfrac id="S2.Ex2a.m1.1.1a" xref="S2.Ex2a.m1.1.1.cmml"><mrow id="S2.Ex2a.m1.1.1.2" xref="S2.Ex2a.m1.1.1.2.cmml"><mi id="S2.Ex2a.m1.1.1.2.2" xref="S2.Ex2a.m1.1.1.2.2.cmml">d</mi><mo 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cd="ambiguous" id="S2.Ex2a.m1.1.1.2.3.1.cmml" xref="S2.Ex2a.m1.1.1.2.3">subscript</csymbol><ci id="S2.Ex2a.m1.1.1.2.3.2.cmml" xref="S2.Ex2a.m1.1.1.2.3.2">𝜂</ci><ci id="S2.Ex2a.m1.1.1.2.3.3.cmml" xref="S2.Ex2a.m1.1.1.2.3.3">𝑖</ci></apply></apply><apply id="S2.Ex2a.m1.1.1.3.cmml" xref="S2.Ex2a.m1.1.1.3"><times id="S2.Ex2a.m1.1.1.3.1.cmml" xref="S2.Ex2a.m1.1.1.3.1"></times><ci id="S2.Ex2a.m1.1.1.3.2.cmml" xref="S2.Ex2a.m1.1.1.3.2">𝑑</ci><ci id="S2.Ex2a.m1.1.1.3.3.cmml" xref="S2.Ex2a.m1.1.1.3.3">𝑡</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Ex2a.m1.1c">\displaystyle\frac{d\eta_{i}}{dt}</annotation><annotation encoding="application/x-llamapun" id="S2.Ex2a.m1.1d">divide start_ARG italic_d italic_η start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_ARG start_ARG italic_d italic_t end_ARG</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle=-g_{ij}(\eta)\frac{\partial D(\eta,\eta^{\rm 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roman_Ψ start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT ( italic_η ) end_ARG start_ARG ∂ italic_η start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT end_ARG - divide start_ARG ∂ roman_Ψ start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT ( italic_η ) end_ARG start_ARG ∂ italic_η start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT end_ARG | start_POSTSUBSCRIPT italic_η start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT = italic_η start_POSTSUPERSCRIPT roman_r end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT end_POSTSUBSCRIPT ) ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_left_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(15)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S2.SS2.p2.1">in the <math alttext="\eta" class="ltx_Math" display="inline" id="S2.SS2.p2.1.m1.1"><semantics id="S2.SS2.p2.1.m1.1a"><mi id="S2.SS2.p2.1.m1.1.1" xref="S2.SS2.p2.1.m1.1.1.cmml">η</mi><annotation-xml encoding="MathML-Content" id="S2.SS2.p2.1.m1.1b"><ci id="S2.SS2.p2.1.m1.1.1.cmml" xref="S2.SS2.p2.1.m1.1.1">𝜂</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p2.1.m1.1c">\eta</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p2.1.m1.1d">italic_η</annotation></semantics></math>-coordinate system. Similarly, they are equivalent to the linear differential equations</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="S3.EGx16"> <tbody id="S2.E16"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_left_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\frac{d\theta^{i}(t)}{dt}=-\theta^{i}(t)+\theta_{\rm r}^{j}," class="ltx_Math" display="inline" id="S2.E16.m1.3"><semantics id="S2.E16.m1.3a"><mrow id="S2.E16.m1.3.3.1" xref="S2.E16.m1.3.3.1.1.cmml"><mrow id="S2.E16.m1.3.3.1.1" xref="S2.E16.m1.3.3.1.1.cmml"><mstyle displaystyle="true" id="S2.E16.m1.1.1" xref="S2.E16.m1.1.1.cmml"><mfrac id="S2.E16.m1.1.1a" xref="S2.E16.m1.1.1.cmml"><mrow id="S2.E16.m1.1.1.1" xref="S2.E16.m1.1.1.1.cmml"><mi id="S2.E16.m1.1.1.1.3" xref="S2.E16.m1.1.1.1.3.cmml">d</mi><mo id="S2.E16.m1.1.1.1.2" xref="S2.E16.m1.1.1.1.2.cmml">⁢</mo><msup id="S2.E16.m1.1.1.1.4" xref="S2.E16.m1.1.1.1.4.cmml"><mi 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encoding="application/x-tex" id="S2.E16.m1.3c">\displaystyle\frac{d\theta^{i}(t)}{dt}=-\theta^{i}(t)+\theta_{\rm r}^{j},</annotation><annotation encoding="application/x-llamapun" id="S2.E16.m1.3d">divide start_ARG italic_d italic_θ start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT ( italic_t ) end_ARG start_ARG italic_d italic_t end_ARG = - italic_θ start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT ( italic_t ) + italic_θ start_POSTSUBSCRIPT roman_r end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_j end_POSTSUPERSCRIPT ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_left_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(16)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S2.SS2.p2.7">in the <math alttext="\theta" class="ltx_Math" display="inline" id="S2.SS2.p2.2.m1.1"><semantics id="S2.SS2.p2.2.m1.1a"><mi id="S2.SS2.p2.2.m1.1.1" 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In the previous works <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2406.11224v2#bib.bib9" title="">9</a>, <a class="ltx_ref" href="https://arxiv.org/html/2406.11224v2#bib.bib10" title="">10</a>, <a class="ltx_ref" href="https://arxiv.org/html/2406.11224v2#bib.bib11" title="">11</a>, <a class="ltx_ref" href="https://arxiv.org/html/2406.11224v2#bib.bib13" title="">13</a>]</cite>, the gradient-flow equations with respect to the <math alttext="\theta" class="ltx_Math" display="inline" id="S2.SS2.p2.3.m2.1"><semantics id="S2.SS2.p2.3.m2.1a"><mi id="S2.SS2.p2.3.m2.1.1" xref="S2.SS2.p2.3.m2.1.1.cmml">θ</mi><annotation-xml encoding="MathML-Content" id="S2.SS2.p2.3.m2.1b"><ci id="S2.SS2.p2.3.m2.1.1.cmml" xref="S2.SS2.p2.3.m2.1.1">𝜃</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p2.3.m2.1c">\theta</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p2.3.m2.1d">italic_θ</annotation></semantics></math>- or <math alttext="\eta" class="ltx_Math" display="inline" id="S2.SS2.p2.4.m3.1"><semantics id="S2.SS2.p2.4.m3.1a"><mi id="S2.SS2.p2.4.m3.1.1" xref="S2.SS2.p2.4.m3.1.1.cmml">η</mi><annotation-xml encoding="MathML-Content" id="S2.SS2.p2.4.m3.1b"><ci id="S2.SS2.p2.4.m3.1.1.cmml" xref="S2.SS2.p2.4.m3.1.1">𝜂</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p2.4.m3.1c">\eta</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p2.4.m3.1d">italic_η</annotation></semantics></math>-potential functions were considered. These cases correspond to <math alttext="\theta_{\rm r}^{i}=\eta^{\rm r}_{i}=0" class="ltx_Math" display="inline" id="S2.SS2.p2.5.m4.1"><semantics id="S2.SS2.p2.5.m4.1a"><mrow id="S2.SS2.p2.5.m4.1.1" xref="S2.SS2.p2.5.m4.1.1.cmml"><msubsup id="S2.SS2.p2.5.m4.1.1.2" xref="S2.SS2.p2.5.m4.1.1.2.cmml"><mi id="S2.SS2.p2.5.m4.1.1.2.2.2" xref="S2.SS2.p2.5.m4.1.1.2.2.2.cmml">θ</mi><mi id="S2.SS2.p2.5.m4.1.1.2.2.3" mathvariant="normal" xref="S2.SS2.p2.5.m4.1.1.2.2.3.cmml">r</mi><mi id="S2.SS2.p2.5.m4.1.1.2.3" xref="S2.SS2.p2.5.m4.1.1.2.3.cmml">i</mi></msubsup><mo id="S2.SS2.p2.5.m4.1.1.3" xref="S2.SS2.p2.5.m4.1.1.3.cmml">=</mo><msubsup id="S2.SS2.p2.5.m4.1.1.4" xref="S2.SS2.p2.5.m4.1.1.4.cmml"><mi id="S2.SS2.p2.5.m4.1.1.4.2.2" xref="S2.SS2.p2.5.m4.1.1.4.2.2.cmml">η</mi><mi id="S2.SS2.p2.5.m4.1.1.4.3" xref="S2.SS2.p2.5.m4.1.1.4.3.cmml">i</mi><mi id="S2.SS2.p2.5.m4.1.1.4.2.3" mathvariant="normal" xref="S2.SS2.p2.5.m4.1.1.4.2.3.cmml">r</mi></msubsup><mo id="S2.SS2.p2.5.m4.1.1.5" xref="S2.SS2.p2.5.m4.1.1.5.cmml">=</mo><mn id="S2.SS2.p2.5.m4.1.1.6" xref="S2.SS2.p2.5.m4.1.1.6.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p2.5.m4.1b"><apply id="S2.SS2.p2.5.m4.1.1.cmml" xref="S2.SS2.p2.5.m4.1.1"><and id="S2.SS2.p2.5.m4.1.1a.cmml" xref="S2.SS2.p2.5.m4.1.1"></and><apply id="S2.SS2.p2.5.m4.1.1b.cmml" xref="S2.SS2.p2.5.m4.1.1"><eq id="S2.SS2.p2.5.m4.1.1.3.cmml" xref="S2.SS2.p2.5.m4.1.1.3"></eq><apply id="S2.SS2.p2.5.m4.1.1.2.cmml" xref="S2.SS2.p2.5.m4.1.1.2"><csymbol cd="ambiguous" id="S2.SS2.p2.5.m4.1.1.2.1.cmml" xref="S2.SS2.p2.5.m4.1.1.2">superscript</csymbol><apply id="S2.SS2.p2.5.m4.1.1.2.2.cmml" xref="S2.SS2.p2.5.m4.1.1.2"><csymbol cd="ambiguous" id="S2.SS2.p2.5.m4.1.1.2.2.1.cmml" xref="S2.SS2.p2.5.m4.1.1.2">subscript</csymbol><ci id="S2.SS2.p2.5.m4.1.1.2.2.2.cmml" xref="S2.SS2.p2.5.m4.1.1.2.2.2">𝜃</ci><ci id="S2.SS2.p2.5.m4.1.1.2.2.3.cmml" xref="S2.SS2.p2.5.m4.1.1.2.2.3">r</ci></apply><ci id="S2.SS2.p2.5.m4.1.1.2.3.cmml" xref="S2.SS2.p2.5.m4.1.1.2.3">𝑖</ci></apply><apply id="S2.SS2.p2.5.m4.1.1.4.cmml" xref="S2.SS2.p2.5.m4.1.1.4"><csymbol cd="ambiguous" id="S2.SS2.p2.5.m4.1.1.4.1.cmml" xref="S2.SS2.p2.5.m4.1.1.4">subscript</csymbol><apply id="S2.SS2.p2.5.m4.1.1.4.2.cmml" xref="S2.SS2.p2.5.m4.1.1.4"><csymbol cd="ambiguous" id="S2.SS2.p2.5.m4.1.1.4.2.1.cmml" xref="S2.SS2.p2.5.m4.1.1.4">superscript</csymbol><ci id="S2.SS2.p2.5.m4.1.1.4.2.2.cmml" xref="S2.SS2.p2.5.m4.1.1.4.2.2">𝜂</ci><ci id="S2.SS2.p2.5.m4.1.1.4.2.3.cmml" xref="S2.SS2.p2.5.m4.1.1.4.2.3">r</ci></apply><ci id="S2.SS2.p2.5.m4.1.1.4.3.cmml" xref="S2.SS2.p2.5.m4.1.1.4.3">𝑖</ci></apply></apply><apply id="S2.SS2.p2.5.m4.1.1c.cmml" xref="S2.SS2.p2.5.m4.1.1"><eq id="S2.SS2.p2.5.m4.1.1.5.cmml" xref="S2.SS2.p2.5.m4.1.1.5"></eq><share href="https://arxiv.org/html/2406.11224v2#S2.SS2.p2.5.m4.1.1.4.cmml" id="S2.SS2.p2.5.m4.1.1d.cmml" xref="S2.SS2.p2.5.m4.1.1"></share><cn id="S2.SS2.p2.5.m4.1.1.6.cmml" type="integer" xref="S2.SS2.p2.5.m4.1.1.6">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p2.5.m4.1c">\theta_{\rm r}^{i}=\eta^{\rm r}_{i}=0</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p2.5.m4.1d">italic_θ start_POSTSUBSCRIPT roman_r end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT = italic_η start_POSTSUPERSCRIPT roman_r end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = 0</annotation></semantics></math>, since, for example, the gradients of the <math alttext="\theta" class="ltx_Math" display="inline" id="S2.SS2.p2.6.m5.1"><semantics id="S2.SS2.p2.6.m5.1a"><mi id="S2.SS2.p2.6.m5.1.1" xref="S2.SS2.p2.6.m5.1.1.cmml">θ</mi><annotation-xml encoding="MathML-Content" id="S2.SS2.p2.6.m5.1b"><ci id="S2.SS2.p2.6.m5.1.1.cmml" xref="S2.SS2.p2.6.m5.1.1">𝜃</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p2.6.m5.1c">\theta</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p2.6.m5.1d">italic_θ</annotation></semantics></math>-divergence (<a class="ltx_ref" href="https://arxiv.org/html/2406.11224v2#S2.E11.1" title="In 11 ‣ 2.1 Information Geometry ‣ 2 Information Geometry and Gradient-Flow Equations ‣ A Hamiltonian approach to the gradient-flow equations in information geometry"><span class="ltx_text ltx_ref_tag">11a</span></a>) in the cases are equal to the gradients of the <math alttext="\theta" class="ltx_Math" display="inline" id="S2.SS2.p2.7.m6.1"><semantics id="S2.SS2.p2.7.m6.1a"><mi id="S2.SS2.p2.7.m6.1.1" xref="S2.SS2.p2.7.m6.1.1.cmml">θ</mi><annotation-xml encoding="MathML-Content" id="S2.SS2.p2.7.m6.1b"><ci id="S2.SS2.p2.7.m6.1.1.cmml" xref="S2.SS2.p2.7.m6.1.1">𝜃</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p2.7.m6.1c">\theta</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p2.7.m6.1d">italic_θ</annotation></semantics></math>-potential function. i.e.,</p> <table class="ltx_equationgroup 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xref="S2.E17.m1.1.1.1"><partialdiff id="S2.E17.m1.1.1.1.2.cmml" xref="S2.E17.m1.1.1.1.2"></partialdiff><apply id="S2.E17.m1.1.1.1.3.cmml" xref="S2.E17.m1.1.1.1.3"><times id="S2.E17.m1.1.1.1.3.1.cmml" xref="S2.E17.m1.1.1.1.3.1"></times><ci id="S2.E17.m1.1.1.1.3.2.cmml" xref="S2.E17.m1.1.1.1.3.2">Ψ</ci><ci id="S2.E17.m1.1.1.1.1.cmml" xref="S2.E17.m1.1.1.1.1">𝜃</ci></apply></apply><apply id="S2.E17.m1.1.1.3.cmml" xref="S2.E17.m1.1.1.3"><partialdiff id="S2.E17.m1.1.1.3.1.cmml" xref="S2.E17.m1.1.1.3.1"></partialdiff><apply id="S2.E17.m1.1.1.3.2.cmml" xref="S2.E17.m1.1.1.3.2"><csymbol cd="ambiguous" id="S2.E17.m1.1.1.3.2.1.cmml" xref="S2.E17.m1.1.1.3.2">superscript</csymbol><ci id="S2.E17.m1.1.1.3.2.2.cmml" xref="S2.E17.m1.1.1.3.2.2">𝜃</ci><ci id="S2.E17.m1.1.1.3.2.3.cmml" xref="S2.E17.m1.1.1.3.2.3">𝑖</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.E17.m1.4c">\displaystyle\frac{\partial}{\partial\theta^{i}}D(\theta,\theta_{\rm r})\Big{|% }_{\theta_{\rm r}^{i}=\eta^{\rm r}_{i}=0}=\frac{\partial\Psi(\theta)}{\partial% \theta^{i}}.</annotation><annotation encoding="application/x-llamapun" id="S2.E17.m1.4d">divide start_ARG ∂ end_ARG start_ARG ∂ italic_θ start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT end_ARG italic_D ( italic_θ , italic_θ start_POSTSUBSCRIPT roman_r end_POSTSUBSCRIPT ) | start_POSTSUBSCRIPT italic_θ start_POSTSUBSCRIPT roman_r end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT = italic_η start_POSTSUPERSCRIPT roman_r end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = 0 end_POSTSUBSCRIPT = divide start_ARG ∂ roman_Ψ ( italic_θ ) end_ARG start_ARG ∂ italic_θ start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT end_ARG .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_left_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(17)</span></td> </tr></tbody> </table> </div> <div class="ltx_para" id="S2.SS2.p3"> <p class="ltx_p" id="S2.SS2.p3.3">It is worth emphasizing that the two sets of differential equations (<a class="ltx_ref" href="https://arxiv.org/html/2406.11224v2#S2.E12" title="In 2.2 Gradient-Flow Equations ‣ 2 Information Geometry and Gradient-Flow Equations ‣ A Hamiltonian approach to the gradient-flow equations in information geometry"><span class="ltx_text ltx_ref_tag">12</span></a>) and (<a class="ltx_ref" href="https://arxiv.org/html/2406.11224v2#S2.E15" title="In 2.2 Gradient-Flow Equations ‣ 2 Information Geometry and Gradient-Flow Equations ‣ A Hamiltonian approach to the gradient-flow equations in information geometry"><span class="ltx_text ltx_ref_tag">15</span></a>) describe different processes in general <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2406.11224v2#bib.bib10" title="">10</a>, <a class="ltx_ref" href="https://arxiv.org/html/2406.11224v2#bib.bib11" title="">11</a>]</cite>. In addition, the evolutional parameter <math alttext="t" class="ltx_Math" display="inline" id="S2.SS2.p3.1.m1.1"><semantics id="S2.SS2.p3.1.m1.1a"><mi id="S2.SS2.p3.1.m1.1.1" xref="S2.SS2.p3.1.m1.1.1.cmml">t</mi><annotation-xml encoding="MathML-Content" id="S2.SS2.p3.1.m1.1b"><ci id="S2.SS2.p3.1.m1.1.1.cmml" xref="S2.SS2.p3.1.m1.1.1">𝑡</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p3.1.m1.1c">t</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p3.1.m1.1d">italic_t</annotation></semantics></math> in the gradient-flow equations (<a class="ltx_ref" href="https://arxiv.org/html/2406.11224v2#S2.E12" title="In 2.2 Gradient-Flow Equations ‣ 2 Information Geometry and Gradient-Flow Equations ‣ A Hamiltonian approach to the gradient-flow equations in information geometry"><span class="ltx_text ltx_ref_tag">12</span></a>) and (<a class="ltx_ref" href="https://arxiv.org/html/2406.11224v2#S2.E15" title="In 2.2 Gradient-Flow Equations ‣ 2 Information Geometry and Gradient-Flow Equations ‣ A Hamiltonian approach to the gradient-flow equations in information geometry"><span class="ltx_text ltx_ref_tag">15</span></a>) is a non-affine parameter. Recall that a parameter <math alttext="s" class="ltx_Math" display="inline" id="S2.SS2.p3.2.m2.1"><semantics id="S2.SS2.p3.2.m2.1a"><mi id="S2.SS2.p3.2.m2.1.1" xref="S2.SS2.p3.2.m2.1.1.cmml">s</mi><annotation-xml encoding="MathML-Content" id="S2.SS2.p3.2.m2.1b"><ci id="S2.SS2.p3.2.m2.1.1.cmml" xref="S2.SS2.p3.2.m2.1.1">𝑠</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p3.2.m2.1c">s</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p3.2.m2.1d">italic_s</annotation></semantics></math> is affine if the geodesic equations of a curve <math alttext="x^{i}=x^{i}(s)" class="ltx_Math" display="inline" id="S2.SS2.p3.3.m3.1"><semantics id="S2.SS2.p3.3.m3.1a"><mrow id="S2.SS2.p3.3.m3.1.2" xref="S2.SS2.p3.3.m3.1.2.cmml"><msup id="S2.SS2.p3.3.m3.1.2.2" xref="S2.SS2.p3.3.m3.1.2.2.cmml"><mi id="S2.SS2.p3.3.m3.1.2.2.2" xref="S2.SS2.p3.3.m3.1.2.2.2.cmml">x</mi><mi id="S2.SS2.p3.3.m3.1.2.2.3" xref="S2.SS2.p3.3.m3.1.2.2.3.cmml">i</mi></msup><mo id="S2.SS2.p3.3.m3.1.2.1" xref="S2.SS2.p3.3.m3.1.2.1.cmml">=</mo><mrow id="S2.SS2.p3.3.m3.1.2.3" xref="S2.SS2.p3.3.m3.1.2.3.cmml"><msup id="S2.SS2.p3.3.m3.1.2.3.2" xref="S2.SS2.p3.3.m3.1.2.3.2.cmml"><mi id="S2.SS2.p3.3.m3.1.2.3.2.2" xref="S2.SS2.p3.3.m3.1.2.3.2.2.cmml">x</mi><mi id="S2.SS2.p3.3.m3.1.2.3.2.3" xref="S2.SS2.p3.3.m3.1.2.3.2.3.cmml">i</mi></msup><mo id="S2.SS2.p3.3.m3.1.2.3.1" xref="S2.SS2.p3.3.m3.1.2.3.1.cmml">⁢</mo><mrow id="S2.SS2.p3.3.m3.1.2.3.3.2" xref="S2.SS2.p3.3.m3.1.2.3.cmml"><mo id="S2.SS2.p3.3.m3.1.2.3.3.2.1" stretchy="false" xref="S2.SS2.p3.3.m3.1.2.3.cmml">(</mo><mi id="S2.SS2.p3.3.m3.1.1" xref="S2.SS2.p3.3.m3.1.1.cmml">s</mi><mo id="S2.SS2.p3.3.m3.1.2.3.3.2.2" stretchy="false" xref="S2.SS2.p3.3.m3.1.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p3.3.m3.1b"><apply id="S2.SS2.p3.3.m3.1.2.cmml" xref="S2.SS2.p3.3.m3.1.2"><eq id="S2.SS2.p3.3.m3.1.2.1.cmml" xref="S2.SS2.p3.3.m3.1.2.1"></eq><apply id="S2.SS2.p3.3.m3.1.2.2.cmml" xref="S2.SS2.p3.3.m3.1.2.2"><csymbol cd="ambiguous" id="S2.SS2.p3.3.m3.1.2.2.1.cmml" xref="S2.SS2.p3.3.m3.1.2.2">superscript</csymbol><ci id="S2.SS2.p3.3.m3.1.2.2.2.cmml" xref="S2.SS2.p3.3.m3.1.2.2.2">𝑥</ci><ci id="S2.SS2.p3.3.m3.1.2.2.3.cmml" xref="S2.SS2.p3.3.m3.1.2.2.3">𝑖</ci></apply><apply id="S2.SS2.p3.3.m3.1.2.3.cmml" xref="S2.SS2.p3.3.m3.1.2.3"><times id="S2.SS2.p3.3.m3.1.2.3.1.cmml" xref="S2.SS2.p3.3.m3.1.2.3.1"></times><apply id="S2.SS2.p3.3.m3.1.2.3.2.cmml" xref="S2.SS2.p3.3.m3.1.2.3.2"><csymbol cd="ambiguous" id="S2.SS2.p3.3.m3.1.2.3.2.1.cmml" xref="S2.SS2.p3.3.m3.1.2.3.2">superscript</csymbol><ci id="S2.SS2.p3.3.m3.1.2.3.2.2.cmml" xref="S2.SS2.p3.3.m3.1.2.3.2.2">𝑥</ci><ci id="S2.SS2.p3.3.m3.1.2.3.2.3.cmml" xref="S2.SS2.p3.3.m3.1.2.3.2.3">𝑖</ci></apply><ci id="S2.SS2.p3.3.m3.1.1.cmml" xref="S2.SS2.p3.3.m3.1.1">𝑠</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p3.3.m3.1c">x^{i}=x^{i}(s)</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p3.3.m3.1d">italic_x start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT = italic_x start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT ( italic_s )</annotation></semantics></math> are in the form:</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="S3.EGx18"> <tbody id="S2.E18"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_left_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\frac{d^{2}x^{i}(s)}{ds^{2}}+\operatorname{\Gamma}^{i}{}_{jk}(x)% \frac{dx^{j}(s)}{ds}\frac{dx^{j}(s)}{ds}=0." class="ltx_math_unparsed" display="inline" id="S2.E18.m1.3"><semantics id="S2.E18.m1.3a"><mrow id="S2.E18.m1.3b"><mstyle displaystyle="true" id="S2.E18.m1.1.1"><mfrac id="S2.E18.m1.1.1a"><mrow id="S2.E18.m1.1.1.1"><msup id="S2.E18.m1.1.1.1.3"><mi id="S2.E18.m1.1.1.1.3.2">d</mi><mn id="S2.E18.m1.1.1.1.3.3">2</mn></msup><mo id="S2.E18.m1.1.1.1.2">⁢</mo><msup id="S2.E18.m1.1.1.1.4"><mi id="S2.E18.m1.1.1.1.4.2">x</mi><mi id="S2.E18.m1.1.1.1.4.3">i</mi></msup><mo id="S2.E18.m1.1.1.1.2a">⁢</mo><mrow id="S2.E18.m1.1.1.1.5.2"><mo id="S2.E18.m1.1.1.1.5.2.1" stretchy="false">(</mo><mi id="S2.E18.m1.1.1.1.1">s</mi><mo id="S2.E18.m1.1.1.1.5.2.2" stretchy="false">)</mo></mrow></mrow><mrow id="S2.E18.m1.1.1.3"><mi id="S2.E18.m1.1.1.3.2">d</mi><mo id="S2.E18.m1.1.1.3.1">⁢</mo><msup id="S2.E18.m1.1.1.3.3"><mi id="S2.E18.m1.1.1.3.3.2">s</mi><mn id="S2.E18.m1.1.1.3.3.3">2</mn></msup></mrow></mfrac></mstyle><mo id="S2.E18.m1.3.4">+</mo><msup id="S2.E18.m1.3.5"><mi id="S2.E18.m1.3.5.2" mathvariant="normal">Γ</mi><mi id="S2.E18.m1.3.5.3">i</mi></msup><mmultiscripts id="S2.E18.m1.3.6"><mrow id="S2.E18.m1.3.6.2"><mo id="S2.E18.m1.3.6.2.1" stretchy="false">(</mo><mi id="S2.E18.m1.3.6.2.2">x</mi><mo id="S2.E18.m1.3.6.2.3" stretchy="false">)</mo></mrow><mprescripts id="S2.E18.m1.3.6a"></mprescripts><mrow id="S2.E18.m1.3.6.3"><mi id="S2.E18.m1.3.6.3.2">j</mi><mo id="S2.E18.m1.3.6.3.1">⁢</mo><mi id="S2.E18.m1.3.6.3.3">k</mi></mrow><mrow id="S2.E18.m1.3.6b"></mrow></mmultiscripts><mstyle displaystyle="true" id="S2.E18.m1.2.2"><mfrac id="S2.E18.m1.2.2a"><mrow id="S2.E18.m1.2.2.1"><mi id="S2.E18.m1.2.2.1.3">d</mi><mo id="S2.E18.m1.2.2.1.2">⁢</mo><msup id="S2.E18.m1.2.2.1.4"><mi id="S2.E18.m1.2.2.1.4.2">x</mi><mi id="S2.E18.m1.2.2.1.4.3">j</mi></msup><mo id="S2.E18.m1.2.2.1.2a">⁢</mo><mrow id="S2.E18.m1.2.2.1.5.2"><mo id="S2.E18.m1.2.2.1.5.2.1" stretchy="false">(</mo><mi id="S2.E18.m1.2.2.1.1">s</mi><mo id="S2.E18.m1.2.2.1.5.2.2" stretchy="false">)</mo></mrow></mrow><mrow id="S2.E18.m1.2.2.3"><mi id="S2.E18.m1.2.2.3.2">d</mi><mo id="S2.E18.m1.2.2.3.1">⁢</mo><mi id="S2.E18.m1.2.2.3.3">s</mi></mrow></mfrac></mstyle><mstyle displaystyle="true" id="S2.E18.m1.3.3"><mfrac id="S2.E18.m1.3.3a"><mrow id="S2.E18.m1.3.3.1"><mi id="S2.E18.m1.3.3.1.3">d</mi><mo id="S2.E18.m1.3.3.1.2">⁢</mo><msup id="S2.E18.m1.3.3.1.4"><mi id="S2.E18.m1.3.3.1.4.2">x</mi><mi id="S2.E18.m1.3.3.1.4.3">j</mi></msup><mo id="S2.E18.m1.3.3.1.2a">⁢</mo><mrow id="S2.E18.m1.3.3.1.5.2"><mo id="S2.E18.m1.3.3.1.5.2.1" stretchy="false">(</mo><mi id="S2.E18.m1.3.3.1.1">s</mi><mo id="S2.E18.m1.3.3.1.5.2.2" stretchy="false">)</mo></mrow></mrow><mrow id="S2.E18.m1.3.3.3"><mi id="S2.E18.m1.3.3.3.2">d</mi><mo id="S2.E18.m1.3.3.3.1">⁢</mo><mi id="S2.E18.m1.3.3.3.3">s</mi></mrow></mfrac></mstyle><mo id="S2.E18.m1.3.7">=</mo><mn id="S2.E18.m1.3.8">0</mn><mo id="S2.E18.m1.3.9" lspace="0em">.</mo></mrow><annotation encoding="application/x-tex" id="S2.E18.m1.3c">\displaystyle\frac{d^{2}x^{i}(s)}{ds^{2}}+\operatorname{\Gamma}^{i}{}_{jk}(x)% \frac{dx^{j}(s)}{ds}\frac{dx^{j}(s)}{ds}=0.</annotation><annotation encoding="application/x-llamapun" id="S2.E18.m1.3d">divide start_ARG italic_d start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_x start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT ( italic_s ) end_ARG start_ARG italic_d italic_s start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG + roman_Γ start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT start_FLOATSUBSCRIPT italic_j italic_k end_FLOATSUBSCRIPT ( italic_x ) divide start_ARG italic_d italic_x start_POSTSUPERSCRIPT italic_j end_POSTSUPERSCRIPT ( italic_s ) end_ARG start_ARG italic_d italic_s end_ARG divide start_ARG italic_d italic_x start_POSTSUPERSCRIPT italic_j end_POSTSUPERSCRIPT ( italic_s ) end_ARG start_ARG italic_d italic_s end_ARG = 0 .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_left_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(18)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S2.SS2.p3.7">For example, we see from (<a class="ltx_ref" href="https://arxiv.org/html/2406.11224v2#S2.E16" title="In 2.2 Gradient-Flow Equations ‣ 2 Information Geometry and Gradient-Flow Equations ‣ A Hamiltonian approach to the gradient-flow equations in information geometry"><span class="ltx_text ltx_ref_tag">16</span></a>) that</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="S3.EGx19"> <tbody id="S2.E19"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_left_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\frac{d^{2}\theta^{i}}{dt^{2}}=-\frac{d\theta^{i}}{dt}." class="ltx_Math" display="inline" id="S2.E19.m1.1"><semantics id="S2.E19.m1.1a"><mrow id="S2.E19.m1.1.1.1" xref="S2.E19.m1.1.1.1.1.cmml"><mrow id="S2.E19.m1.1.1.1.1" xref="S2.E19.m1.1.1.1.1.cmml"><mstyle displaystyle="true" id="S2.E19.m1.1.1.1.1.2" xref="S2.E19.m1.1.1.1.1.2.cmml"><mfrac id="S2.E19.m1.1.1.1.1.2a" xref="S2.E19.m1.1.1.1.1.2.cmml"><mrow id="S2.E19.m1.1.1.1.1.2.2" xref="S2.E19.m1.1.1.1.1.2.2.cmml"><msup id="S2.E19.m1.1.1.1.1.2.2.2" xref="S2.E19.m1.1.1.1.1.2.2.2.cmml"><mi id="S2.E19.m1.1.1.1.1.2.2.2.2" xref="S2.E19.m1.1.1.1.1.2.2.2.2.cmml">d</mi><mn id="S2.E19.m1.1.1.1.1.2.2.2.3" xref="S2.E19.m1.1.1.1.1.2.2.2.3.cmml">2</mn></msup><mo id="S2.E19.m1.1.1.1.1.2.2.1" xref="S2.E19.m1.1.1.1.1.2.2.1.cmml">⁢</mo><msup id="S2.E19.m1.1.1.1.1.2.2.3" xref="S2.E19.m1.1.1.1.1.2.2.3.cmml"><mi id="S2.E19.m1.1.1.1.1.2.2.3.2" xref="S2.E19.m1.1.1.1.1.2.2.3.2.cmml">θ</mi><mi id="S2.E19.m1.1.1.1.1.2.2.3.3" xref="S2.E19.m1.1.1.1.1.2.2.3.3.cmml">i</mi></msup></mrow><mrow id="S2.E19.m1.1.1.1.1.2.3" xref="S2.E19.m1.1.1.1.1.2.3.cmml"><mi id="S2.E19.m1.1.1.1.1.2.3.2" xref="S2.E19.m1.1.1.1.1.2.3.2.cmml">d</mi><mo id="S2.E19.m1.1.1.1.1.2.3.1" xref="S2.E19.m1.1.1.1.1.2.3.1.cmml">⁢</mo><msup id="S2.E19.m1.1.1.1.1.2.3.3" xref="S2.E19.m1.1.1.1.1.2.3.3.cmml"><mi id="S2.E19.m1.1.1.1.1.2.3.3.2" xref="S2.E19.m1.1.1.1.1.2.3.3.2.cmml">t</mi><mn id="S2.E19.m1.1.1.1.1.2.3.3.3" xref="S2.E19.m1.1.1.1.1.2.3.3.3.cmml">2</mn></msup></mrow></mfrac></mstyle><mo id="S2.E19.m1.1.1.1.1.1" xref="S2.E19.m1.1.1.1.1.1.cmml">=</mo><mrow id="S2.E19.m1.1.1.1.1.3" xref="S2.E19.m1.1.1.1.1.3.cmml"><mo id="S2.E19.m1.1.1.1.1.3a" xref="S2.E19.m1.1.1.1.1.3.cmml">−</mo><mstyle displaystyle="true" id="S2.E19.m1.1.1.1.1.3.2" xref="S2.E19.m1.1.1.1.1.3.2.cmml"><mfrac id="S2.E19.m1.1.1.1.1.3.2a" xref="S2.E19.m1.1.1.1.1.3.2.cmml"><mrow id="S2.E19.m1.1.1.1.1.3.2.2" xref="S2.E19.m1.1.1.1.1.3.2.2.cmml"><mi id="S2.E19.m1.1.1.1.1.3.2.2.2" xref="S2.E19.m1.1.1.1.1.3.2.2.2.cmml">d</mi><mo id="S2.E19.m1.1.1.1.1.3.2.2.1" xref="S2.E19.m1.1.1.1.1.3.2.2.1.cmml">⁢</mo><msup id="S2.E19.m1.1.1.1.1.3.2.2.3" xref="S2.E19.m1.1.1.1.1.3.2.2.3.cmml"><mi id="S2.E19.m1.1.1.1.1.3.2.2.3.2" xref="S2.E19.m1.1.1.1.1.3.2.2.3.2.cmml">θ</mi><mi id="S2.E19.m1.1.1.1.1.3.2.2.3.3" xref="S2.E19.m1.1.1.1.1.3.2.2.3.3.cmml">i</mi></msup></mrow><mrow id="S2.E19.m1.1.1.1.1.3.2.3" xref="S2.E19.m1.1.1.1.1.3.2.3.cmml"><mi id="S2.E19.m1.1.1.1.1.3.2.3.2" xref="S2.E19.m1.1.1.1.1.3.2.3.2.cmml">d</mi><mo id="S2.E19.m1.1.1.1.1.3.2.3.1" xref="S2.E19.m1.1.1.1.1.3.2.3.1.cmml">⁢</mo><mi id="S2.E19.m1.1.1.1.1.3.2.3.3" xref="S2.E19.m1.1.1.1.1.3.2.3.3.cmml">t</mi></mrow></mfrac></mstyle></mrow></mrow><mo id="S2.E19.m1.1.1.1.2" lspace="0em" xref="S2.E19.m1.1.1.1.1.cmml">.</mo></mrow><annotation-xml encoding="MathML-Content" id="S2.E19.m1.1b"><apply id="S2.E19.m1.1.1.1.1.cmml" xref="S2.E19.m1.1.1.1"><eq id="S2.E19.m1.1.1.1.1.1.cmml" xref="S2.E19.m1.1.1.1.1.1"></eq><apply id="S2.E19.m1.1.1.1.1.2.cmml" xref="S2.E19.m1.1.1.1.1.2"><divide id="S2.E19.m1.1.1.1.1.2.1.cmml" xref="S2.E19.m1.1.1.1.1.2"></divide><apply id="S2.E19.m1.1.1.1.1.2.2.cmml" xref="S2.E19.m1.1.1.1.1.2.2"><times id="S2.E19.m1.1.1.1.1.2.2.1.cmml" xref="S2.E19.m1.1.1.1.1.2.2.1"></times><apply id="S2.E19.m1.1.1.1.1.2.2.2.cmml" xref="S2.E19.m1.1.1.1.1.2.2.2"><csymbol cd="ambiguous" id="S2.E19.m1.1.1.1.1.2.2.2.1.cmml" xref="S2.E19.m1.1.1.1.1.2.2.2">superscript</csymbol><ci id="S2.E19.m1.1.1.1.1.2.2.2.2.cmml" xref="S2.E19.m1.1.1.1.1.2.2.2.2">𝑑</ci><cn 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xref="S2.E19.m1.1.1.1.1.3.2.3"><times id="S2.E19.m1.1.1.1.1.3.2.3.1.cmml" xref="S2.E19.m1.1.1.1.1.3.2.3.1"></times><ci id="S2.E19.m1.1.1.1.1.3.2.3.2.cmml" xref="S2.E19.m1.1.1.1.1.3.2.3.2">𝑑</ci><ci id="S2.E19.m1.1.1.1.1.3.2.3.3.cmml" xref="S2.E19.m1.1.1.1.1.3.2.3.3">𝑡</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.E19.m1.1c">\displaystyle\frac{d^{2}\theta^{i}}{dt^{2}}=-\frac{d\theta^{i}}{dt}.</annotation><annotation encoding="application/x-llamapun" id="S2.E19.m1.1d">divide start_ARG italic_d start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_θ start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT end_ARG start_ARG italic_d italic_t start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG = - divide start_ARG italic_d italic_θ start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT end_ARG start_ARG italic_d italic_t end_ARG .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_left_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(19)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S2.SS2.p3.6">This is the non-affinely parametrized geodesic (or pre-geodesic) equations in the <math alttext="\theta" class="ltx_Math" display="inline" id="S2.SS2.p3.4.m1.1"><semantics id="S2.SS2.p3.4.m1.1a"><mi id="S2.SS2.p3.4.m1.1.1" xref="S2.SS2.p3.4.m1.1.1.cmml">θ</mi><annotation-xml encoding="MathML-Content" id="S2.SS2.p3.4.m1.1b"><ci id="S2.SS2.p3.4.m1.1.1.cmml" xref="S2.SS2.p3.4.m1.1.1">𝜃</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p3.4.m1.1c">\theta</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p3.4.m1.1d">italic_θ</annotation></semantics></math>-space, in which the <math alttext="\theta" class="ltx_Math" display="inline" id="S2.SS2.p3.5.m2.1"><semantics id="S2.SS2.p3.5.m2.1a"><mi id="S2.SS2.p3.5.m2.1.1" xref="S2.SS2.p3.5.m2.1.1.cmml">θ</mi><annotation-xml encoding="MathML-Content" id="S2.SS2.p3.5.m2.1b"><ci id="S2.SS2.p3.5.m2.1.1.cmml" xref="S2.SS2.p3.5.m2.1.1">𝜃</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p3.5.m2.1c">\theta</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p3.5.m2.1d">italic_θ</annotation></semantics></math>-coordinates are of course affine and <math alttext="\operatorname{\Gamma}^{i}{}_{jk}(\theta)=0" class="ltx_math_unparsed" display="inline" id="S2.SS2.p3.6.m3.1"><semantics id="S2.SS2.p3.6.m3.1a"><mrow id="S2.SS2.p3.6.m3.1b"><msup id="S2.SS2.p3.6.m3.1.1"><mi id="S2.SS2.p3.6.m3.1.1.2" mathvariant="normal">Γ</mi><mi id="S2.SS2.p3.6.m3.1.1.3">i</mi></msup><mmultiscripts id="S2.SS2.p3.6.m3.1.2"><mrow id="S2.SS2.p3.6.m3.1.2.2"><mo id="S2.SS2.p3.6.m3.1.2.2.1" stretchy="false">(</mo><mi id="S2.SS2.p3.6.m3.1.2.2.2">θ</mi><mo id="S2.SS2.p3.6.m3.1.2.2.3" stretchy="false">)</mo></mrow><mprescripts id="S2.SS2.p3.6.m3.1.2a"></mprescripts><mrow id="S2.SS2.p3.6.m3.1.2.3"><mi id="S2.SS2.p3.6.m3.1.2.3.2">j</mi><mo id="S2.SS2.p3.6.m3.1.2.3.1">⁢</mo><mi id="S2.SS2.p3.6.m3.1.2.3.3">k</mi></mrow><mrow id="S2.SS2.p3.6.m3.1.2b"></mrow></mmultiscripts><mo id="S2.SS2.p3.6.m3.1.3">=</mo><mn id="S2.SS2.p3.6.m3.1.4">0</mn></mrow><annotation encoding="application/x-tex" id="S2.SS2.p3.6.m3.1c">\operatorname{\Gamma}^{i}{}_{jk}(\theta)=0</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p3.6.m3.1d">roman_Γ start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT start_FLOATSUBSCRIPT italic_j italic_k end_FLOATSUBSCRIPT ( italic_θ ) = 0</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S2.SS2.p4"> <p class="ltx_p" id="S2.SS2.p4.1">As we mentioned in Introduction, Ref. <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2406.11224v2#bib.bib10" title="">10</a>, <a class="ltx_ref" href="https://arxiv.org/html/2406.11224v2#bib.bib11" title="">11</a>]</cite> have related the gradient-flows (<a class="ltx_ref" href="https://arxiv.org/html/2406.11224v2#S2.E12" title="In 2.2 Gradient-Flow Equations ‣ 2 Information Geometry and Gradient-Flow Equations ‣ A Hamiltonian approach to the gradient-flow equations in information geometry"><span class="ltx_text ltx_ref_tag">12</span></a>) and (<a class="ltx_ref" href="https://arxiv.org/html/2406.11224v2#S2.E15" title="In 2.2 Gradient-Flow Equations ‣ 2 Information Geometry and Gradient-Flow Equations ‣ A Hamiltonian approach to the gradient-flow equations in information geometry"><span class="ltx_text ltx_ref_tag">15</span></a>) in the case of <math alttext="\theta_{\rm r}^{i}=\eta^{\rm r}_{i}=0" class="ltx_Math" display="inline" id="S2.SS2.p4.1.m1.1"><semantics id="S2.SS2.p4.1.m1.1a"><mrow id="S2.SS2.p4.1.m1.1.1" xref="S2.SS2.p4.1.m1.1.1.cmml"><msubsup id="S2.SS2.p4.1.m1.1.1.2" xref="S2.SS2.p4.1.m1.1.1.2.cmml"><mi id="S2.SS2.p4.1.m1.1.1.2.2.2" xref="S2.SS2.p4.1.m1.1.1.2.2.2.cmml">θ</mi><mi id="S2.SS2.p4.1.m1.1.1.2.2.3" mathvariant="normal" xref="S2.SS2.p4.1.m1.1.1.2.2.3.cmml">r</mi><mi id="S2.SS2.p4.1.m1.1.1.2.3" xref="S2.SS2.p4.1.m1.1.1.2.3.cmml">i</mi></msubsup><mo id="S2.SS2.p4.1.m1.1.1.3" xref="S2.SS2.p4.1.m1.1.1.3.cmml">=</mo><msubsup id="S2.SS2.p4.1.m1.1.1.4" xref="S2.SS2.p4.1.m1.1.1.4.cmml"><mi id="S2.SS2.p4.1.m1.1.1.4.2.2" xref="S2.SS2.p4.1.m1.1.1.4.2.2.cmml">η</mi><mi id="S2.SS2.p4.1.m1.1.1.4.3" xref="S2.SS2.p4.1.m1.1.1.4.3.cmml">i</mi><mi id="S2.SS2.p4.1.m1.1.1.4.2.3" mathvariant="normal" xref="S2.SS2.p4.1.m1.1.1.4.2.3.cmml">r</mi></msubsup><mo id="S2.SS2.p4.1.m1.1.1.5" xref="S2.SS2.p4.1.m1.1.1.5.cmml">=</mo><mn id="S2.SS2.p4.1.m1.1.1.6" xref="S2.SS2.p4.1.m1.1.1.6.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p4.1.m1.1b"><apply id="S2.SS2.p4.1.m1.1.1.cmml" xref="S2.SS2.p4.1.m1.1.1"><and id="S2.SS2.p4.1.m1.1.1a.cmml" xref="S2.SS2.p4.1.m1.1.1"></and><apply id="S2.SS2.p4.1.m1.1.1b.cmml" 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start_POSTSUPERSCRIPT roman_r end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = 0</annotation></semantics></math> to the Hamilton-flows characterized by the Hamiltonians</p> <table class="ltx_equationgroup ltx_eqn_table" id="S2.E20"> <tbody> <tr class="ltx_eqn_row" id="S3.EGx20"><td class="ltx_eqn_cell" colspan="5"></td></tr> <tr class="ltx_equation ltx_eqn_row ltx_align_baseline" id="S2.E20.x1"> <td class="ltx_eqn_cell ltx_eqn_left_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle H(\theta,\eta)" class="ltx_Math" display="inline" id="S2.E20.x1.m1.2"><semantics id="S2.E20.x1.m1.2a"><mrow id="S2.E20.x1.m1.2.3" xref="S2.E20.x1.m1.2.3.cmml"><mi id="S2.E20.x1.m1.2.3.2" xref="S2.E20.x1.m1.2.3.2.cmml">H</mi><mo id="S2.E20.x1.m1.2.3.1" xref="S2.E20.x1.m1.2.3.1.cmml">⁢</mo><mrow id="S2.E20.x1.m1.2.3.3.2" xref="S2.E20.x1.m1.2.3.3.1.cmml"><mo id="S2.E20.x1.m1.2.3.3.2.1" stretchy="false" xref="S2.E20.x1.m1.2.3.3.1.cmml">(</mo><mi id="S2.E20.x1.m1.1.1" xref="S2.E20.x1.m1.1.1.cmml">θ</mi><mo id="S2.E20.x1.m1.2.3.3.2.2" xref="S2.E20.x1.m1.2.3.3.1.cmml">,</mo><mi id="S2.E20.x1.m1.2.2" xref="S2.E20.x1.m1.2.2.cmml">η</mi><mo id="S2.E20.x1.m1.2.3.3.2.3" stretchy="false" xref="S2.E20.x1.m1.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.E20.x1.m1.2b"><apply id="S2.E20.x1.m1.2.3.cmml" xref="S2.E20.x1.m1.2.3"><times id="S2.E20.x1.m1.2.3.1.cmml" xref="S2.E20.x1.m1.2.3.1"></times><ci id="S2.E20.x1.m1.2.3.2.cmml" xref="S2.E20.x1.m1.2.3.2">𝐻</ci><interval closure="open" id="S2.E20.x1.m1.2.3.3.1.cmml" xref="S2.E20.x1.m1.2.3.3.2"><ci id="S2.E20.x1.m1.1.1.cmml" xref="S2.E20.x1.m1.1.1">𝜃</ci><ci id="S2.E20.x1.m1.2.2.cmml" xref="S2.E20.x1.m1.2.2">𝜂</ci></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.E20.x1.m1.2c">\displaystyle H(\theta,\eta)</annotation><annotation encoding="application/x-llamapun" id="S2.E20.x1.m1.2d">italic_H ( italic_θ , italic_η )</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle=\sqrt{g^{ij}(\theta)\eta_{i}\eta_{j}}-\sqrt{\eta^{2}(\theta)}," class="ltx_Math" display="inline" id="S2.E20.x1.m2.3"><semantics id="S2.E20.x1.m2.3a"><mrow id="S2.E20.x1.m2.3.3.1" xref="S2.E20.x1.m2.3.3.1.1.cmml"><mrow id="S2.E20.x1.m2.3.3.1.1" xref="S2.E20.x1.m2.3.3.1.1.cmml"><mi id="S2.E20.x1.m2.3.3.1.1.2" xref="S2.E20.x1.m2.3.3.1.1.2.cmml"></mi><mo id="S2.E20.x1.m2.3.3.1.1.1" xref="S2.E20.x1.m2.3.3.1.1.1.cmml">=</mo><mrow id="S2.E20.x1.m2.3.3.1.1.3" xref="S2.E20.x1.m2.3.3.1.1.3.cmml"><msqrt id="S2.E20.x1.m2.1.1" xref="S2.E20.x1.m2.1.1.cmml"><mrow id="S2.E20.x1.m2.1.1.1" xref="S2.E20.x1.m2.1.1.1.cmml"><msup id="S2.E20.x1.m2.1.1.1.3" xref="S2.E20.x1.m2.1.1.1.3.cmml"><mi id="S2.E20.x1.m2.1.1.1.3.2" xref="S2.E20.x1.m2.1.1.1.3.2.cmml">g</mi><mrow id="S2.E20.x1.m2.1.1.1.3.3" xref="S2.E20.x1.m2.1.1.1.3.3.cmml"><mi id="S2.E20.x1.m2.1.1.1.3.3.2" 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xref="S2.E20.x1.m2.1.1.1.6.cmml"><mi id="S2.E20.x1.m2.1.1.1.6.2" xref="S2.E20.x1.m2.1.1.1.6.2.cmml">η</mi><mi id="S2.E20.x1.m2.1.1.1.6.3" xref="S2.E20.x1.m2.1.1.1.6.3.cmml">j</mi></msub></mrow></msqrt><mo id="S2.E20.x1.m2.3.3.1.1.3.1" xref="S2.E20.x1.m2.3.3.1.1.3.1.cmml">−</mo><msqrt id="S2.E20.x1.m2.2.2" xref="S2.E20.x1.m2.2.2.cmml"><mrow id="S2.E20.x1.m2.2.2.1" xref="S2.E20.x1.m2.2.2.1.cmml"><msup id="S2.E20.x1.m2.2.2.1.3" xref="S2.E20.x1.m2.2.2.1.3.cmml"><mi id="S2.E20.x1.m2.2.2.1.3.2" xref="S2.E20.x1.m2.2.2.1.3.2.cmml">η</mi><mn id="S2.E20.x1.m2.2.2.1.3.3" xref="S2.E20.x1.m2.2.2.1.3.3.cmml">2</mn></msup><mo id="S2.E20.x1.m2.2.2.1.2" xref="S2.E20.x1.m2.2.2.1.2.cmml">⁢</mo><mrow id="S2.E20.x1.m2.2.2.1.4.2" xref="S2.E20.x1.m2.2.2.1.cmml"><mo id="S2.E20.x1.m2.2.2.1.4.2.1" stretchy="false" xref="S2.E20.x1.m2.2.2.1.cmml">(</mo><mi id="S2.E20.x1.m2.2.2.1.1" xref="S2.E20.x1.m2.2.2.1.1.cmml">θ</mi><mo id="S2.E20.x1.m2.2.2.1.4.2.2" stretchy="false" 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start_POSTSUPERSCRIPT italic_i italic_j end_POSTSUPERSCRIPT ( italic_θ ) divide start_ARG ∂ roman_Ψ ( italic_θ ) end_ARG start_ARG ∂ italic_θ start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT end_ARG divide start_ARG ∂ roman_Ψ ( italic_θ ) end_ARG start_ARG ∂ italic_θ start_POSTSUPERSCRIPT italic_j end_POSTSUPERSCRIPT end_ARG ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_left_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(20a)</span></td> </tr> <tr class="ltx_equation ltx_eqn_row ltx_align_baseline" id="S2.E20.x2"> <td class="ltx_eqn_cell ltx_eqn_left_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle H(\eta,-\theta)" class="ltx_Math" display="inline" id="S2.E20.x2.m1.2"><semantics id="S2.E20.x2.m1.2a"><mrow id="S2.E20.x2.m1.2.2" xref="S2.E20.x2.m1.2.2.cmml"><mi id="S2.E20.x2.m1.2.2.3" xref="S2.E20.x2.m1.2.2.3.cmml">H</mi><mo id="S2.E20.x2.m1.2.2.2" xref="S2.E20.x2.m1.2.2.2.cmml">⁢</mo><mrow id="S2.E20.x2.m1.2.2.1.1" xref="S2.E20.x2.m1.2.2.1.2.cmml"><mo id="S2.E20.x2.m1.2.2.1.1.2" stretchy="false" xref="S2.E20.x2.m1.2.2.1.2.cmml">(</mo><mi id="S2.E20.x2.m1.1.1" xref="S2.E20.x2.m1.1.1.cmml">η</mi><mo id="S2.E20.x2.m1.2.2.1.1.3" xref="S2.E20.x2.m1.2.2.1.2.cmml">,</mo><mrow id="S2.E20.x2.m1.2.2.1.1.1" xref="S2.E20.x2.m1.2.2.1.1.1.cmml"><mo id="S2.E20.x2.m1.2.2.1.1.1a" xref="S2.E20.x2.m1.2.2.1.1.1.cmml">−</mo><mi id="S2.E20.x2.m1.2.2.1.1.1.2" xref="S2.E20.x2.m1.2.2.1.1.1.2.cmml">θ</mi></mrow><mo id="S2.E20.x2.m1.2.2.1.1.4" stretchy="false" xref="S2.E20.x2.m1.2.2.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.E20.x2.m1.2b"><apply id="S2.E20.x2.m1.2.2.cmml" xref="S2.E20.x2.m1.2.2"><times id="S2.E20.x2.m1.2.2.2.cmml" xref="S2.E20.x2.m1.2.2.2"></times><ci id="S2.E20.x2.m1.2.2.3.cmml" xref="S2.E20.x2.m1.2.2.3">𝐻</ci><interval closure="open" 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id="S2.E20.x2.m2.2.2.1.2.cmml" xref="S2.E20.x2.m2.2.2.1.2"></times><apply id="S2.E20.x2.m2.2.2.1.3.cmml" xref="S2.E20.x2.m2.2.2.1.3"><csymbol cd="ambiguous" id="S2.E20.x2.m2.2.2.1.3.1.cmml" xref="S2.E20.x2.m2.2.2.1.3">superscript</csymbol><ci id="S2.E20.x2.m2.2.2.1.3.2.cmml" xref="S2.E20.x2.m2.2.2.1.3.2">𝜃</ci><cn id="S2.E20.x2.m2.2.2.1.3.3.cmml" type="integer" xref="S2.E20.x2.m2.2.2.1.3.3">2</cn></apply><ci id="S2.E20.x2.m2.2.2.1.1.cmml" xref="S2.E20.x2.m2.2.2.1.1">𝜂</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.E20.x2.m2.3c">\displaystyle=\sqrt{g_{ij}(\eta)\theta^{i}\theta^{j}}-\sqrt{\theta^{2}(\eta)},</annotation><annotation encoding="application/x-llamapun" id="S2.E20.x2.m2.3d">= square-root start_ARG italic_g start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT ( italic_η ) italic_θ start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT italic_θ start_POSTSUPERSCRIPT italic_j end_POSTSUPERSCRIPT end_ARG - square-root start_ARG italic_θ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( italic_η ) end_ARG ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_left_padright"></td> </tr> <tr class="ltx_equation ltx_eqn_row ltx_align_baseline" id="S2.E20.2"> <td class="ltx_eqn_cell ltx_eqn_left_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><span class="ltx_text ltx_markedasmath" id="S2.E20.2.2.1.1.1">with</span></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\theta^{2}(\eta):=g_{ij}(\eta)\frac{\partial\Psi^{\star}(\eta)}{% \partial\eta_{i}}\frac{\partial\Psi^{\star}(\eta)}{\partial\eta_{j}}," class="ltx_Math" display="inline" id="S2.E20.2.m2.5"><semantics id="S2.E20.2.m2.5a"><mrow id="S2.E20.2.m2.5.5.1" xref="S2.E20.2.m2.5.5.1.1.cmml"><mrow id="S2.E20.2.m2.5.5.1.1" xref="S2.E20.2.m2.5.5.1.1.cmml"><mrow id="S2.E20.2.m2.5.5.1.1.2" xref="S2.E20.2.m2.5.5.1.1.2.cmml"><msup id="S2.E20.2.m2.5.5.1.1.2.2" xref="S2.E20.2.m2.5.5.1.1.2.2.cmml"><mi 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xref="S2.E20.2.m2.2.2.3.2.3">𝑗</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.E20.2.m2.5c">\displaystyle\theta^{2}(\eta):=g_{ij}(\eta)\frac{\partial\Psi^{\star}(\eta)}{% \partial\eta_{i}}\frac{\partial\Psi^{\star}(\eta)}{\partial\eta_{j}},</annotation><annotation encoding="application/x-llamapun" id="S2.E20.2.m2.5d">italic_θ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( italic_η ) := italic_g start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT ( italic_η ) divide start_ARG ∂ roman_Ψ start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT ( italic_η ) end_ARG start_ARG ∂ italic_η start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_ARG divide start_ARG ∂ roman_Ψ start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT ( italic_η ) end_ARG start_ARG ∂ italic_η start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT end_ARG ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_left_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(20b)</span></td> </tr> </tbody> </table> <p class="ltx_p" id="S2.SS2.p4.6">or equivalently</p> <table class="ltx_equationgroup ltx_eqn_table" id="S2.E21"> <tbody> <tr class="ltx_eqn_row" id="S3.EGx21"><td class="ltx_eqn_cell" colspan="5"></td></tr> <tr class="ltx_equation ltx_eqn_row ltx_align_baseline" id="S2.E21.1"> <td class="ltx_eqn_cell ltx_eqn_left_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle H(\theta,\eta)" class="ltx_Math" display="inline" id="S2.E21.1.m1.2"><semantics id="S2.E21.1.m1.2a"><mrow id="S2.E21.1.m1.2.3" xref="S2.E21.1.m1.2.3.cmml"><mi id="S2.E21.1.m1.2.3.2" xref="S2.E21.1.m1.2.3.2.cmml">H</mi><mo id="S2.E21.1.m1.2.3.1" xref="S2.E21.1.m1.2.3.1.cmml">⁢</mo><mrow id="S2.E21.1.m1.2.3.3.2" xref="S2.E21.1.m1.2.3.3.1.cmml"><mo id="S2.E21.1.m1.2.3.3.2.1" stretchy="false" xref="S2.E21.1.m1.2.3.3.1.cmml">(</mo><mi id="S2.E21.1.m1.1.1" xref="S2.E21.1.m1.1.1.cmml">θ</mi><mo id="S2.E21.1.m1.2.3.3.2.2" xref="S2.E21.1.m1.2.3.3.1.cmml">,</mo><mi id="S2.E21.1.m1.2.2" xref="S2.E21.1.m1.2.2.cmml">η</mi><mo id="S2.E21.1.m1.2.3.3.2.3" stretchy="false" xref="S2.E21.1.m1.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.E21.1.m1.2b"><apply id="S2.E21.1.m1.2.3.cmml" xref="S2.E21.1.m1.2.3"><times id="S2.E21.1.m1.2.3.1.cmml" xref="S2.E21.1.m1.2.3.1"></times><ci id="S2.E21.1.m1.2.3.2.cmml" xref="S2.E21.1.m1.2.3.2">𝐻</ci><interval closure="open" id="S2.E21.1.m1.2.3.3.1.cmml" xref="S2.E21.1.m1.2.3.3.2"><ci id="S2.E21.1.m1.1.1.cmml" xref="S2.E21.1.m1.1.1">𝜃</ci><ci id="S2.E21.1.m1.2.2.cmml" xref="S2.E21.1.m1.2.2">𝜂</ci></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.E21.1.m1.2c">\displaystyle H(\theta,\eta)</annotation><annotation encoding="application/x-llamapun" id="S2.E21.1.m1.2d">italic_H ( italic_θ , italic_η 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start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_η start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT - divide start_ARG 1 end_ARG start_ARG 2 end_ARG italic_η start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( italic_θ ) ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_left_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(21a)</span></td> </tr> <tr class="ltx_equation ltx_eqn_row ltx_align_baseline" id="S2.E21.2"> <td class="ltx_eqn_cell ltx_eqn_left_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle H(\eta,-\theta)" class="ltx_Math" display="inline" id="S2.E21.2.m1.2"><semantics id="S2.E21.2.m1.2a"><mrow id="S2.E21.2.m1.2.2" xref="S2.E21.2.m1.2.2.cmml"><mi id="S2.E21.2.m1.2.2.3" xref="S2.E21.2.m1.2.2.3.cmml">H</mi><mo id="S2.E21.2.m1.2.2.2" xref="S2.E21.2.m1.2.2.2.cmml">⁢</mo><mrow id="S2.E21.2.m1.2.2.1.1" xref="S2.E21.2.m1.2.2.1.2.cmml"><mo id="S2.E21.2.m1.2.2.1.1.2" stretchy="false" xref="S2.E21.2.m1.2.2.1.2.cmml">(</mo><mi id="S2.E21.2.m1.1.1" xref="S2.E21.2.m1.1.1.cmml">η</mi><mo id="S2.E21.2.m1.2.2.1.1.3" xref="S2.E21.2.m1.2.2.1.2.cmml">,</mo><mrow id="S2.E21.2.m1.2.2.1.1.1" xref="S2.E21.2.m1.2.2.1.1.1.cmml"><mo id="S2.E21.2.m1.2.2.1.1.1a" xref="S2.E21.2.m1.2.2.1.1.1.cmml">−</mo><mi id="S2.E21.2.m1.2.2.1.1.1.2" xref="S2.E21.2.m1.2.2.1.1.1.2.cmml">θ</mi></mrow><mo id="S2.E21.2.m1.2.2.1.1.4" stretchy="false" xref="S2.E21.2.m1.2.2.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.E21.2.m1.2b"><apply id="S2.E21.2.m1.2.2.cmml" xref="S2.E21.2.m1.2.2"><times id="S2.E21.2.m1.2.2.2.cmml" xref="S2.E21.2.m1.2.2.2"></times><ci id="S2.E21.2.m1.2.2.3.cmml" xref="S2.E21.2.m1.2.2.3">𝐻</ci><interval closure="open" id="S2.E21.2.m1.2.2.1.2.cmml" xref="S2.E21.2.m1.2.2.1.1"><ci id="S2.E21.2.m1.1.1.cmml" xref="S2.E21.2.m1.1.1">𝜂</ci><apply id="S2.E21.2.m1.2.2.1.1.1.cmml" xref="S2.E21.2.m1.2.2.1.1.1"><minus id="S2.E21.2.m1.2.2.1.1.1.1.cmml" xref="S2.E21.2.m1.2.2.1.1.1"></minus><ci id="S2.E21.2.m1.2.2.1.1.1.2.cmml" xref="S2.E21.2.m1.2.2.1.1.1.2">𝜃</ci></apply></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.E21.2.m1.2c">\displaystyle H(\eta,-\theta)</annotation><annotation encoding="application/x-llamapun" id="S2.E21.2.m1.2d">italic_H ( italic_η , - italic_θ )</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle=\frac{1}{2}g_{ij}(\eta)\theta^{i}\theta^{j}-\frac{1}{2}\theta^{2% }(\eta)," class="ltx_Math" display="inline" id="S2.E21.2.m2.3"><semantics id="S2.E21.2.m2.3a"><mrow id="S2.E21.2.m2.3.3.1" xref="S2.E21.2.m2.3.3.1.1.cmml"><mrow id="S2.E21.2.m2.3.3.1.1" xref="S2.E21.2.m2.3.3.1.1.cmml"><mi id="S2.E21.2.m2.3.3.1.1.2" xref="S2.E21.2.m2.3.3.1.1.2.cmml"></mi><mo id="S2.E21.2.m2.3.3.1.1.1" xref="S2.E21.2.m2.3.3.1.1.1.cmml">=</mo><mrow id="S2.E21.2.m2.3.3.1.1.3" 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encoding="application/x-tex" id="S2.E21.2.m2.3c">\displaystyle=\frac{1}{2}g_{ij}(\eta)\theta^{i}\theta^{j}-\frac{1}{2}\theta^{2% }(\eta),</annotation><annotation encoding="application/x-llamapun" id="S2.E21.2.m2.3d">= divide start_ARG 1 end_ARG start_ARG 2 end_ARG italic_g start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT ( italic_η ) italic_θ start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT italic_θ start_POSTSUPERSCRIPT italic_j end_POSTSUPERSCRIPT - divide start_ARG 1 end_ARG start_ARG 2 end_ARG italic_θ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( italic_η ) ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_left_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(21b)</span></td> </tr> </tbody> </table> <p class="ltx_p" id="S2.SS2.p4.5">respectively<span class="ltx_note ltx_role_footnote" id="footnote1"><sup class="ltx_note_mark">1</sup><span class="ltx_note_outer"><span class="ltx_note_content"><sup class="ltx_note_mark">1</sup><span class="ltx_tag ltx_tag_note">1</span>The quantity <math alttext="\theta^{2}(\eta)" class="ltx_Math" display="inline" id="footnote1.m1.1"><semantics id="footnote1.m1.1b"><mrow id="footnote1.m1.1.2" xref="footnote1.m1.1.2.cmml"><msup id="footnote1.m1.1.2.2" xref="footnote1.m1.1.2.2.cmml"><mi id="footnote1.m1.1.2.2.2" xref="footnote1.m1.1.2.2.2.cmml">θ</mi><mn id="footnote1.m1.1.2.2.3" xref="footnote1.m1.1.2.2.3.cmml">2</mn></msup><mo id="footnote1.m1.1.2.1" xref="footnote1.m1.1.2.1.cmml">⁢</mo><mrow id="footnote1.m1.1.2.3.2" xref="footnote1.m1.1.2.cmml"><mo id="footnote1.m1.1.2.3.2.1" stretchy="false" xref="footnote1.m1.1.2.cmml">(</mo><mi id="footnote1.m1.1.1" xref="footnote1.m1.1.1.cmml">η</mi><mo id="footnote1.m1.1.2.3.2.2" stretchy="false" xref="footnote1.m1.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="footnote1.m1.1c"><apply id="footnote1.m1.1.2.cmml" xref="footnote1.m1.1.2"><times id="footnote1.m1.1.2.1.cmml" xref="footnote1.m1.1.2.1"></times><apply id="footnote1.m1.1.2.2.cmml" xref="footnote1.m1.1.2.2"><csymbol cd="ambiguous" id="footnote1.m1.1.2.2.1.cmml" xref="footnote1.m1.1.2.2">superscript</csymbol><ci id="footnote1.m1.1.2.2.2.cmml" xref="footnote1.m1.1.2.2.2">𝜃</ci><cn id="footnote1.m1.1.2.2.3.cmml" type="integer" xref="footnote1.m1.1.2.2.3">2</cn></apply><ci id="footnote1.m1.1.1.cmml" xref="footnote1.m1.1.1">𝜂</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="footnote1.m1.1d">\theta^{2}(\eta)</annotation><annotation encoding="application/x-llamapun" id="footnote1.m1.1e">italic_θ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( italic_η )</annotation></semantics></math> and <math alttext="\eta^{2}(\theta)" class="ltx_Math" display="inline" id="footnote1.m2.1"><semantics id="footnote1.m2.1b"><mrow id="footnote1.m2.1.2" xref="footnote1.m2.1.2.cmml"><msup id="footnote1.m2.1.2.2" xref="footnote1.m2.1.2.2.cmml"><mi id="footnote1.m2.1.2.2.2" xref="footnote1.m2.1.2.2.2.cmml">η</mi><mn id="footnote1.m2.1.2.2.3" xref="footnote1.m2.1.2.2.3.cmml">2</mn></msup><mo id="footnote1.m2.1.2.1" xref="footnote1.m2.1.2.1.cmml">⁢</mo><mrow id="footnote1.m2.1.2.3.2" xref="footnote1.m2.1.2.cmml"><mo id="footnote1.m2.1.2.3.2.1" stretchy="false" xref="footnote1.m2.1.2.cmml">(</mo><mi id="footnote1.m2.1.1" xref="footnote1.m2.1.1.cmml">θ</mi><mo id="footnote1.m2.1.2.3.2.2" stretchy="false" xref="footnote1.m2.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="footnote1.m2.1c"><apply id="footnote1.m2.1.2.cmml" xref="footnote1.m2.1.2"><times id="footnote1.m2.1.2.1.cmml" xref="footnote1.m2.1.2.1"></times><apply id="footnote1.m2.1.2.2.cmml" xref="footnote1.m2.1.2.2"><csymbol cd="ambiguous" id="footnote1.m2.1.2.2.1.cmml" xref="footnote1.m2.1.2.2">superscript</csymbol><ci id="footnote1.m2.1.2.2.2.cmml" xref="footnote1.m2.1.2.2.2">𝜂</ci><cn id="footnote1.m2.1.2.2.3.cmml" type="integer" xref="footnote1.m2.1.2.2.3">2</cn></apply><ci id="footnote1.m2.1.1.cmml" xref="footnote1.m2.1.1">𝜃</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="footnote1.m2.1d">\eta^{2}(\theta)</annotation><annotation encoding="application/x-llamapun" id="footnote1.m2.1e">italic_η start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( italic_θ )</annotation></semantics></math> were denoted as <math alttext="n^{2}" class="ltx_Math" display="inline" id="footnote1.m3.1"><semantics id="footnote1.m3.1b"><msup id="footnote1.m3.1.1" xref="footnote1.m3.1.1.cmml"><mi id="footnote1.m3.1.1.2" xref="footnote1.m3.1.1.2.cmml">n</mi><mn id="footnote1.m3.1.1.3" xref="footnote1.m3.1.1.3.cmml">2</mn></msup><annotation-xml encoding="MathML-Content" id="footnote1.m3.1c"><apply id="footnote1.m3.1.1.cmml" xref="footnote1.m3.1.1"><csymbol cd="ambiguous" id="footnote1.m3.1.1.1.cmml" xref="footnote1.m3.1.1">superscript</csymbol><ci id="footnote1.m3.1.1.2.cmml" xref="footnote1.m3.1.1.2">𝑛</ci><cn id="footnote1.m3.1.1.3.cmml" type="integer" xref="footnote1.m3.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="footnote1.m3.1d">n^{2}</annotation><annotation encoding="application/x-llamapun" id="footnote1.m3.1e">italic_n start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT</annotation></semantics></math> and <math alttext="(n^{\star})^{2}" class="ltx_Math" display="inline" id="footnote1.m4.1"><semantics id="footnote1.m4.1b"><msup id="footnote1.m4.1.1" xref="footnote1.m4.1.1.cmml"><mrow id="footnote1.m4.1.1.1.1" xref="footnote1.m4.1.1.1.1.1.cmml"><mo id="footnote1.m4.1.1.1.1.2" stretchy="false" xref="footnote1.m4.1.1.1.1.1.cmml">(</mo><msup id="footnote1.m4.1.1.1.1.1" xref="footnote1.m4.1.1.1.1.1.cmml"><mi id="footnote1.m4.1.1.1.1.1.2" xref="footnote1.m4.1.1.1.1.1.2.cmml">n</mi><mo id="footnote1.m4.1.1.1.1.1.3" xref="footnote1.m4.1.1.1.1.1.3.cmml">⋆</mo></msup><mo id="footnote1.m4.1.1.1.1.3" stretchy="false" xref="footnote1.m4.1.1.1.1.1.cmml">)</mo></mrow><mn id="footnote1.m4.1.1.3" xref="footnote1.m4.1.1.3.cmml">2</mn></msup><annotation-xml encoding="MathML-Content" id="footnote1.m4.1c"><apply id="footnote1.m4.1.1.cmml" xref="footnote1.m4.1.1"><csymbol cd="ambiguous" id="footnote1.m4.1.1.2.cmml" xref="footnote1.m4.1.1">superscript</csymbol><apply id="footnote1.m4.1.1.1.1.1.cmml" xref="footnote1.m4.1.1.1.1"><csymbol cd="ambiguous" id="footnote1.m4.1.1.1.1.1.1.cmml" xref="footnote1.m4.1.1.1.1">superscript</csymbol><ci id="footnote1.m4.1.1.1.1.1.2.cmml" xref="footnote1.m4.1.1.1.1.1.2">𝑛</ci><ci id="footnote1.m4.1.1.1.1.1.3.cmml" xref="footnote1.m4.1.1.1.1.1.3">⋆</ci></apply><cn id="footnote1.m4.1.1.3.cmml" type="integer" xref="footnote1.m4.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="footnote1.m4.1d">(n^{\star})^{2}</annotation><annotation encoding="application/x-llamapun" id="footnote1.m4.1e">( italic_n start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT</annotation></semantics></math>, respectively, in our previous study <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2406.11224v2#bib.bib10" title="">10</a>]</cite>, and <math alttext="n" class="ltx_Math" display="inline" id="footnote1.m5.1"><semantics id="footnote1.m5.1b"><mi id="footnote1.m5.1.1" xref="footnote1.m5.1.1.cmml">n</mi><annotation-xml encoding="MathML-Content" id="footnote1.m5.1c"><ci id="footnote1.m5.1.1.cmml" xref="footnote1.m5.1.1">𝑛</ci></annotation-xml><annotation encoding="application/x-tex" id="footnote1.m5.1d">n</annotation><annotation encoding="application/x-llamapun" id="footnote1.m5.1e">italic_n</annotation></semantics></math> or <math alttext="n^{\star}" class="ltx_Math" display="inline" id="footnote1.m6.1"><semantics id="footnote1.m6.1b"><msup id="footnote1.m6.1.1" xref="footnote1.m6.1.1.cmml"><mi id="footnote1.m6.1.1.2" xref="footnote1.m6.1.1.2.cmml">n</mi><mo id="footnote1.m6.1.1.3" xref="footnote1.m6.1.1.3.cmml">⋆</mo></msup><annotation-xml encoding="MathML-Content" id="footnote1.m6.1c"><apply id="footnote1.m6.1.1.cmml" xref="footnote1.m6.1.1"><csymbol cd="ambiguous" id="footnote1.m6.1.1.1.cmml" xref="footnote1.m6.1.1">superscript</csymbol><ci id="footnote1.m6.1.1.2.cmml" xref="footnote1.m6.1.1.2">𝑛</ci><ci id="footnote1.m6.1.1.3.cmml" xref="footnote1.m6.1.1.3">⋆</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="footnote1.m6.1d">n^{\star}</annotation><annotation encoding="application/x-llamapun" id="footnote1.m6.1e">italic_n start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT</annotation></semantics></math> was regarded as the refractive index of an optical medium.</span></span></span>. It must be noted that <math alttext="\eta^{2}(\theta)" class="ltx_Math" display="inline" id="S2.SS2.p4.2.m1.1"><semantics id="S2.SS2.p4.2.m1.1a"><mrow id="S2.SS2.p4.2.m1.1.2" xref="S2.SS2.p4.2.m1.1.2.cmml"><msup id="S2.SS2.p4.2.m1.1.2.2" xref="S2.SS2.p4.2.m1.1.2.2.cmml"><mi id="S2.SS2.p4.2.m1.1.2.2.2" xref="S2.SS2.p4.2.m1.1.2.2.2.cmml">η</mi><mn id="S2.SS2.p4.2.m1.1.2.2.3" xref="S2.SS2.p4.2.m1.1.2.2.3.cmml">2</mn></msup><mo id="S2.SS2.p4.2.m1.1.2.1" xref="S2.SS2.p4.2.m1.1.2.1.cmml">⁢</mo><mrow id="S2.SS2.p4.2.m1.1.2.3.2" xref="S2.SS2.p4.2.m1.1.2.cmml"><mo id="S2.SS2.p4.2.m1.1.2.3.2.1" stretchy="false" xref="S2.SS2.p4.2.m1.1.2.cmml">(</mo><mi id="S2.SS2.p4.2.m1.1.1" xref="S2.SS2.p4.2.m1.1.1.cmml">θ</mi><mo id="S2.SS2.p4.2.m1.1.2.3.2.2" stretchy="false" xref="S2.SS2.p4.2.m1.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p4.2.m1.1b"><apply id="S2.SS2.p4.2.m1.1.2.cmml" xref="S2.SS2.p4.2.m1.1.2"><times id="S2.SS2.p4.2.m1.1.2.1.cmml" xref="S2.SS2.p4.2.m1.1.2.1"></times><apply id="S2.SS2.p4.2.m1.1.2.2.cmml" xref="S2.SS2.p4.2.m1.1.2.2"><csymbol cd="ambiguous" id="S2.SS2.p4.2.m1.1.2.2.1.cmml" xref="S2.SS2.p4.2.m1.1.2.2">superscript</csymbol><ci id="S2.SS2.p4.2.m1.1.2.2.2.cmml" xref="S2.SS2.p4.2.m1.1.2.2.2">𝜂</ci><cn id="S2.SS2.p4.2.m1.1.2.2.3.cmml" type="integer" xref="S2.SS2.p4.2.m1.1.2.2.3">2</cn></apply><ci id="S2.SS2.p4.2.m1.1.1.cmml" xref="S2.SS2.p4.2.m1.1.1">𝜃</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p4.2.m1.1c">\eta^{2}(\theta)</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p4.2.m1.1d">italic_η start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( italic_θ )</annotation></semantics></math> is a function of <math alttext="\theta^{i}" class="ltx_Math" display="inline" id="S2.SS2.p4.3.m2.1"><semantics id="S2.SS2.p4.3.m2.1a"><msup id="S2.SS2.p4.3.m2.1.1" xref="S2.SS2.p4.3.m2.1.1.cmml"><mi id="S2.SS2.p4.3.m2.1.1.2" xref="S2.SS2.p4.3.m2.1.1.2.cmml">θ</mi><mi id="S2.SS2.p4.3.m2.1.1.3" xref="S2.SS2.p4.3.m2.1.1.3.cmml">i</mi></msup><annotation-xml encoding="MathML-Content" id="S2.SS2.p4.3.m2.1b"><apply id="S2.SS2.p4.3.m2.1.1.cmml" xref="S2.SS2.p4.3.m2.1.1"><csymbol cd="ambiguous" id="S2.SS2.p4.3.m2.1.1.1.cmml" xref="S2.SS2.p4.3.m2.1.1">superscript</csymbol><ci id="S2.SS2.p4.3.m2.1.1.2.cmml" xref="S2.SS2.p4.3.m2.1.1.2">𝜃</ci><ci id="S2.SS2.p4.3.m2.1.1.3.cmml" xref="S2.SS2.p4.3.m2.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p4.3.m2.1c">\theta^{i}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p4.3.m2.1d">italic_θ start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT</annotation></semantics></math> only and <math alttext="\theta^{2}(\eta)" class="ltx_Math" display="inline" id="S2.SS2.p4.4.m3.1"><semantics id="S2.SS2.p4.4.m3.1a"><mrow id="S2.SS2.p4.4.m3.1.2" xref="S2.SS2.p4.4.m3.1.2.cmml"><msup id="S2.SS2.p4.4.m3.1.2.2" xref="S2.SS2.p4.4.m3.1.2.2.cmml"><mi id="S2.SS2.p4.4.m3.1.2.2.2" xref="S2.SS2.p4.4.m3.1.2.2.2.cmml">θ</mi><mn id="S2.SS2.p4.4.m3.1.2.2.3" xref="S2.SS2.p4.4.m3.1.2.2.3.cmml">2</mn></msup><mo id="S2.SS2.p4.4.m3.1.2.1" xref="S2.SS2.p4.4.m3.1.2.1.cmml">⁢</mo><mrow id="S2.SS2.p4.4.m3.1.2.3.2" xref="S2.SS2.p4.4.m3.1.2.cmml"><mo id="S2.SS2.p4.4.m3.1.2.3.2.1" stretchy="false" xref="S2.SS2.p4.4.m3.1.2.cmml">(</mo><mi id="S2.SS2.p4.4.m3.1.1" xref="S2.SS2.p4.4.m3.1.1.cmml">η</mi><mo id="S2.SS2.p4.4.m3.1.2.3.2.2" stretchy="false" xref="S2.SS2.p4.4.m3.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p4.4.m3.1b"><apply id="S2.SS2.p4.4.m3.1.2.cmml" xref="S2.SS2.p4.4.m3.1.2"><times id="S2.SS2.p4.4.m3.1.2.1.cmml" xref="S2.SS2.p4.4.m3.1.2.1"></times><apply id="S2.SS2.p4.4.m3.1.2.2.cmml" xref="S2.SS2.p4.4.m3.1.2.2"><csymbol cd="ambiguous" id="S2.SS2.p4.4.m3.1.2.2.1.cmml" xref="S2.SS2.p4.4.m3.1.2.2">superscript</csymbol><ci id="S2.SS2.p4.4.m3.1.2.2.2.cmml" xref="S2.SS2.p4.4.m3.1.2.2.2">𝜃</ci><cn id="S2.SS2.p4.4.m3.1.2.2.3.cmml" type="integer" xref="S2.SS2.p4.4.m3.1.2.2.3">2</cn></apply><ci id="S2.SS2.p4.4.m3.1.1.cmml" xref="S2.SS2.p4.4.m3.1.1">𝜂</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p4.4.m3.1c">\theta^{2}(\eta)</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p4.4.m3.1d">italic_θ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( italic_η )</annotation></semantics></math> is a function of <math alttext="\eta_{i}" class="ltx_Math" display="inline" id="S2.SS2.p4.5.m4.1"><semantics id="S2.SS2.p4.5.m4.1a"><msub id="S2.SS2.p4.5.m4.1.1" xref="S2.SS2.p4.5.m4.1.1.cmml"><mi id="S2.SS2.p4.5.m4.1.1.2" xref="S2.SS2.p4.5.m4.1.1.2.cmml">η</mi><mi id="S2.SS2.p4.5.m4.1.1.3" xref="S2.SS2.p4.5.m4.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S2.SS2.p4.5.m4.1b"><apply id="S2.SS2.p4.5.m4.1.1.cmml" xref="S2.SS2.p4.5.m4.1.1"><csymbol cd="ambiguous" id="S2.SS2.p4.5.m4.1.1.1.cmml" xref="S2.SS2.p4.5.m4.1.1">subscript</csymbol><ci id="S2.SS2.p4.5.m4.1.1.2.cmml" xref="S2.SS2.p4.5.m4.1.1.2">𝜂</ci><ci id="S2.SS2.p4.5.m4.1.1.3.cmml" xref="S2.SS2.p4.5.m4.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p4.5.m4.1c">\eta_{i}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p4.5.m4.1d">italic_η start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> only.</p> </div> <div class="ltx_para" id="S2.SS2.p5"> <p class="ltx_p" id="S2.SS2.p5.7">The associated evolutional parameter is <math alttext="t" class="ltx_Math" display="inline" id="S2.SS2.p5.1.m1.1"><semantics id="S2.SS2.p5.1.m1.1a"><mi id="S2.SS2.p5.1.m1.1.1" xref="S2.SS2.p5.1.m1.1.1.cmml">t</mi><annotation-xml encoding="MathML-Content" id="S2.SS2.p5.1.m1.1b"><ci id="S2.SS2.p5.1.m1.1.1.cmml" xref="S2.SS2.p5.1.m1.1.1">𝑡</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p5.1.m1.1c">t</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p5.1.m1.1d">italic_t</annotation></semantics></math>, which is a non-affine parameter. Note also that <math alttext="H(\theta,\eta)" class="ltx_Math" display="inline" id="S2.SS2.p5.2.m2.2"><semantics id="S2.SS2.p5.2.m2.2a"><mrow id="S2.SS2.p5.2.m2.2.3" xref="S2.SS2.p5.2.m2.2.3.cmml"><mi id="S2.SS2.p5.2.m2.2.3.2" xref="S2.SS2.p5.2.m2.2.3.2.cmml">H</mi><mo id="S2.SS2.p5.2.m2.2.3.1" xref="S2.SS2.p5.2.m2.2.3.1.cmml">⁢</mo><mrow id="S2.SS2.p5.2.m2.2.3.3.2" xref="S2.SS2.p5.2.m2.2.3.3.1.cmml"><mo id="S2.SS2.p5.2.m2.2.3.3.2.1" stretchy="false" xref="S2.SS2.p5.2.m2.2.3.3.1.cmml">(</mo><mi id="S2.SS2.p5.2.m2.1.1" xref="S2.SS2.p5.2.m2.1.1.cmml">θ</mi><mo id="S2.SS2.p5.2.m2.2.3.3.2.2" xref="S2.SS2.p5.2.m2.2.3.3.1.cmml">,</mo><mi id="S2.SS2.p5.2.m2.2.2" xref="S2.SS2.p5.2.m2.2.2.cmml">η</mi><mo id="S2.SS2.p5.2.m2.2.3.3.2.3" stretchy="false" xref="S2.SS2.p5.2.m2.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p5.2.m2.2b"><apply id="S2.SS2.p5.2.m2.2.3.cmml" xref="S2.SS2.p5.2.m2.2.3"><times id="S2.SS2.p5.2.m2.2.3.1.cmml" xref="S2.SS2.p5.2.m2.2.3.1"></times><ci id="S2.SS2.p5.2.m2.2.3.2.cmml" xref="S2.SS2.p5.2.m2.2.3.2">𝐻</ci><interval closure="open" id="S2.SS2.p5.2.m2.2.3.3.1.cmml" xref="S2.SS2.p5.2.m2.2.3.3.2"><ci id="S2.SS2.p5.2.m2.1.1.cmml" xref="S2.SS2.p5.2.m2.1.1">𝜃</ci><ci id="S2.SS2.p5.2.m2.2.2.cmml" xref="S2.SS2.p5.2.m2.2.2">𝜂</ci></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p5.2.m2.2c">H(\theta,\eta)</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p5.2.m2.2d">italic_H ( italic_θ , italic_η )</annotation></semantics></math> and <math alttext="H(\eta,-\theta)" class="ltx_Math" display="inline" id="S2.SS2.p5.3.m3.2"><semantics id="S2.SS2.p5.3.m3.2a"><mrow id="S2.SS2.p5.3.m3.2.2" xref="S2.SS2.p5.3.m3.2.2.cmml"><mi id="S2.SS2.p5.3.m3.2.2.3" xref="S2.SS2.p5.3.m3.2.2.3.cmml">H</mi><mo id="S2.SS2.p5.3.m3.2.2.2" xref="S2.SS2.p5.3.m3.2.2.2.cmml">⁢</mo><mrow id="S2.SS2.p5.3.m3.2.2.1.1" xref="S2.SS2.p5.3.m3.2.2.1.2.cmml"><mo id="S2.SS2.p5.3.m3.2.2.1.1.2" stretchy="false" xref="S2.SS2.p5.3.m3.2.2.1.2.cmml">(</mo><mi id="S2.SS2.p5.3.m3.1.1" xref="S2.SS2.p5.3.m3.1.1.cmml">η</mi><mo id="S2.SS2.p5.3.m3.2.2.1.1.3" xref="S2.SS2.p5.3.m3.2.2.1.2.cmml">,</mo><mrow id="S2.SS2.p5.3.m3.2.2.1.1.1" xref="S2.SS2.p5.3.m3.2.2.1.1.1.cmml"><mo id="S2.SS2.p5.3.m3.2.2.1.1.1a" xref="S2.SS2.p5.3.m3.2.2.1.1.1.cmml">−</mo><mi id="S2.SS2.p5.3.m3.2.2.1.1.1.2" xref="S2.SS2.p5.3.m3.2.2.1.1.1.2.cmml">θ</mi></mrow><mo id="S2.SS2.p5.3.m3.2.2.1.1.4" stretchy="false" xref="S2.SS2.p5.3.m3.2.2.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p5.3.m3.2b"><apply id="S2.SS2.p5.3.m3.2.2.cmml" xref="S2.SS2.p5.3.m3.2.2"><times id="S2.SS2.p5.3.m3.2.2.2.cmml" xref="S2.SS2.p5.3.m3.2.2.2"></times><ci id="S2.SS2.p5.3.m3.2.2.3.cmml" xref="S2.SS2.p5.3.m3.2.2.3">𝐻</ci><interval closure="open" id="S2.SS2.p5.3.m3.2.2.1.2.cmml" xref="S2.SS2.p5.3.m3.2.2.1.1"><ci id="S2.SS2.p5.3.m3.1.1.cmml" xref="S2.SS2.p5.3.m3.1.1">𝜂</ci><apply id="S2.SS2.p5.3.m3.2.2.1.1.1.cmml" xref="S2.SS2.p5.3.m3.2.2.1.1.1"><minus id="S2.SS2.p5.3.m3.2.2.1.1.1.1.cmml" xref="S2.SS2.p5.3.m3.2.2.1.1.1"></minus><ci id="S2.SS2.p5.3.m3.2.2.1.1.1.2.cmml" xref="S2.SS2.p5.3.m3.2.2.1.1.1.2">𝜃</ci></apply></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p5.3.m3.2c">H(\eta,-\theta)</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p5.3.m3.2d">italic_H ( italic_η , - italic_θ )</annotation></semantics></math> are related through the canonical transformation <math alttext="(\theta^{i},\eta_{i})" class="ltx_Math" display="inline" id="S2.SS2.p5.4.m4.2"><semantics id="S2.SS2.p5.4.m4.2a"><mrow id="S2.SS2.p5.4.m4.2.2.2" xref="S2.SS2.p5.4.m4.2.2.3.cmml"><mo id="S2.SS2.p5.4.m4.2.2.2.3" stretchy="false" xref="S2.SS2.p5.4.m4.2.2.3.cmml">(</mo><msup id="S2.SS2.p5.4.m4.1.1.1.1" xref="S2.SS2.p5.4.m4.1.1.1.1.cmml"><mi id="S2.SS2.p5.4.m4.1.1.1.1.2" xref="S2.SS2.p5.4.m4.1.1.1.1.2.cmml">θ</mi><mi id="S2.SS2.p5.4.m4.1.1.1.1.3" xref="S2.SS2.p5.4.m4.1.1.1.1.3.cmml">i</mi></msup><mo id="S2.SS2.p5.4.m4.2.2.2.4" xref="S2.SS2.p5.4.m4.2.2.3.cmml">,</mo><msub id="S2.SS2.p5.4.m4.2.2.2.2" xref="S2.SS2.p5.4.m4.2.2.2.2.cmml"><mi id="S2.SS2.p5.4.m4.2.2.2.2.2" xref="S2.SS2.p5.4.m4.2.2.2.2.2.cmml">η</mi><mi id="S2.SS2.p5.4.m4.2.2.2.2.3" xref="S2.SS2.p5.4.m4.2.2.2.2.3.cmml">i</mi></msub><mo id="S2.SS2.p5.4.m4.2.2.2.5" stretchy="false" xref="S2.SS2.p5.4.m4.2.2.3.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p5.4.m4.2b"><interval closure="open" id="S2.SS2.p5.4.m4.2.2.3.cmml" xref="S2.SS2.p5.4.m4.2.2.2"><apply id="S2.SS2.p5.4.m4.1.1.1.1.cmml" xref="S2.SS2.p5.4.m4.1.1.1.1"><csymbol cd="ambiguous" id="S2.SS2.p5.4.m4.1.1.1.1.1.cmml" xref="S2.SS2.p5.4.m4.1.1.1.1">superscript</csymbol><ci id="S2.SS2.p5.4.m4.1.1.1.1.2.cmml" xref="S2.SS2.p5.4.m4.1.1.1.1.2">𝜃</ci><ci id="S2.SS2.p5.4.m4.1.1.1.1.3.cmml" xref="S2.SS2.p5.4.m4.1.1.1.1.3">𝑖</ci></apply><apply id="S2.SS2.p5.4.m4.2.2.2.2.cmml" xref="S2.SS2.p5.4.m4.2.2.2.2"><csymbol cd="ambiguous" id="S2.SS2.p5.4.m4.2.2.2.2.1.cmml" xref="S2.SS2.p5.4.m4.2.2.2.2">subscript</csymbol><ci id="S2.SS2.p5.4.m4.2.2.2.2.2.cmml" xref="S2.SS2.p5.4.m4.2.2.2.2.2">𝜂</ci><ci id="S2.SS2.p5.4.m4.2.2.2.2.3.cmml" xref="S2.SS2.p5.4.m4.2.2.2.2.3">𝑖</ci></apply></interval></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p5.4.m4.2c">(\theta^{i},\eta_{i})</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p5.4.m4.2d">( italic_θ start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT , italic_η start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT )</annotation></semantics></math> to <math alttext="(\eta_{i},-\theta^{i})" class="ltx_Math" display="inline" id="S2.SS2.p5.5.m5.2"><semantics id="S2.SS2.p5.5.m5.2a"><mrow id="S2.SS2.p5.5.m5.2.2.2" xref="S2.SS2.p5.5.m5.2.2.3.cmml"><mo id="S2.SS2.p5.5.m5.2.2.2.3" stretchy="false" xref="S2.SS2.p5.5.m5.2.2.3.cmml">(</mo><msub id="S2.SS2.p5.5.m5.1.1.1.1" xref="S2.SS2.p5.5.m5.1.1.1.1.cmml"><mi id="S2.SS2.p5.5.m5.1.1.1.1.2" xref="S2.SS2.p5.5.m5.1.1.1.1.2.cmml">η</mi><mi id="S2.SS2.p5.5.m5.1.1.1.1.3" xref="S2.SS2.p5.5.m5.1.1.1.1.3.cmml">i</mi></msub><mo id="S2.SS2.p5.5.m5.2.2.2.4" xref="S2.SS2.p5.5.m5.2.2.3.cmml">,</mo><mrow id="S2.SS2.p5.5.m5.2.2.2.2" xref="S2.SS2.p5.5.m5.2.2.2.2.cmml"><mo id="S2.SS2.p5.5.m5.2.2.2.2a" xref="S2.SS2.p5.5.m5.2.2.2.2.cmml">−</mo><msup id="S2.SS2.p5.5.m5.2.2.2.2.2" xref="S2.SS2.p5.5.m5.2.2.2.2.2.cmml"><mi id="S2.SS2.p5.5.m5.2.2.2.2.2.2" xref="S2.SS2.p5.5.m5.2.2.2.2.2.2.cmml">θ</mi><mi id="S2.SS2.p5.5.m5.2.2.2.2.2.3" xref="S2.SS2.p5.5.m5.2.2.2.2.2.3.cmml">i</mi></msup></mrow><mo id="S2.SS2.p5.5.m5.2.2.2.5" stretchy="false" xref="S2.SS2.p5.5.m5.2.2.3.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p5.5.m5.2b"><interval closure="open" id="S2.SS2.p5.5.m5.2.2.3.cmml" xref="S2.SS2.p5.5.m5.2.2.2"><apply id="S2.SS2.p5.5.m5.1.1.1.1.cmml" xref="S2.SS2.p5.5.m5.1.1.1.1"><csymbol cd="ambiguous" id="S2.SS2.p5.5.m5.1.1.1.1.1.cmml" xref="S2.SS2.p5.5.m5.1.1.1.1">subscript</csymbol><ci id="S2.SS2.p5.5.m5.1.1.1.1.2.cmml" xref="S2.SS2.p5.5.m5.1.1.1.1.2">𝜂</ci><ci id="S2.SS2.p5.5.m5.1.1.1.1.3.cmml" xref="S2.SS2.p5.5.m5.1.1.1.1.3">𝑖</ci></apply><apply id="S2.SS2.p5.5.m5.2.2.2.2.cmml" xref="S2.SS2.p5.5.m5.2.2.2.2"><minus id="S2.SS2.p5.5.m5.2.2.2.2.1.cmml" xref="S2.SS2.p5.5.m5.2.2.2.2"></minus><apply id="S2.SS2.p5.5.m5.2.2.2.2.2.cmml" xref="S2.SS2.p5.5.m5.2.2.2.2.2"><csymbol cd="ambiguous" id="S2.SS2.p5.5.m5.2.2.2.2.2.1.cmml" xref="S2.SS2.p5.5.m5.2.2.2.2.2">superscript</csymbol><ci id="S2.SS2.p5.5.m5.2.2.2.2.2.2.cmml" xref="S2.SS2.p5.5.m5.2.2.2.2.2.2">𝜃</ci><ci id="S2.SS2.p5.5.m5.2.2.2.2.2.3.cmml" xref="S2.SS2.p5.5.m5.2.2.2.2.2.3">𝑖</ci></apply></apply></interval></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p5.5.m5.2c">(\eta_{i},-\theta^{i})</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p5.5.m5.2d">( italic_η start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT , - italic_θ start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT )</annotation></semantics></math> <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2406.11224v2#bib.bib11" title="">11</a>]</cite>. In Section <a class="ltx_ref" href="https://arxiv.org/html/2406.11224v2#S3" title="3 Complete integrability and geodesic Hamiltonian ‣ A Hamiltonian approach to the gradient-flow equations in information geometry"><span class="ltx_text ltx_ref_tag">3</span></a> we will show the complete integrability of Pfaffian systems leads to a Hamiltonian <math alttext="H(x,p)" class="ltx_Math" display="inline" id="S2.SS2.p5.6.m6.2"><semantics id="S2.SS2.p5.6.m6.2a"><mrow id="S2.SS2.p5.6.m6.2.3" xref="S2.SS2.p5.6.m6.2.3.cmml"><mi id="S2.SS2.p5.6.m6.2.3.2" xref="S2.SS2.p5.6.m6.2.3.2.cmml">H</mi><mo id="S2.SS2.p5.6.m6.2.3.1" xref="S2.SS2.p5.6.m6.2.3.1.cmml">⁢</mo><mrow id="S2.SS2.p5.6.m6.2.3.3.2" xref="S2.SS2.p5.6.m6.2.3.3.1.cmml"><mo id="S2.SS2.p5.6.m6.2.3.3.2.1" stretchy="false" xref="S2.SS2.p5.6.m6.2.3.3.1.cmml">(</mo><mi id="S2.SS2.p5.6.m6.1.1" xref="S2.SS2.p5.6.m6.1.1.cmml">x</mi><mo id="S2.SS2.p5.6.m6.2.3.3.2.2" xref="S2.SS2.p5.6.m6.2.3.3.1.cmml">,</mo><mi id="S2.SS2.p5.6.m6.2.2" xref="S2.SS2.p5.6.m6.2.2.cmml">p</mi><mo id="S2.SS2.p5.6.m6.2.3.3.2.3" stretchy="false" xref="S2.SS2.p5.6.m6.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p5.6.m6.2b"><apply id="S2.SS2.p5.6.m6.2.3.cmml" xref="S2.SS2.p5.6.m6.2.3"><times id="S2.SS2.p5.6.m6.2.3.1.cmml" xref="S2.SS2.p5.6.m6.2.3.1"></times><ci id="S2.SS2.p5.6.m6.2.3.2.cmml" xref="S2.SS2.p5.6.m6.2.3.2">𝐻</ci><interval closure="open" id="S2.SS2.p5.6.m6.2.3.3.1.cmml" xref="S2.SS2.p5.6.m6.2.3.3.2"><ci id="S2.SS2.p5.6.m6.1.1.cmml" xref="S2.SS2.p5.6.m6.1.1">𝑥</ci><ci id="S2.SS2.p5.6.m6.2.2.cmml" xref="S2.SS2.p5.6.m6.2.2">𝑝</ci></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p5.6.m6.2c">H(x,p)</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p5.6.m6.2d">italic_H ( italic_x , italic_p )</annotation></semantics></math> which is homogeneous of first order in the variable <math alttext="p" class="ltx_Math" display="inline" id="S2.SS2.p5.7.m7.1"><semantics id="S2.SS2.p5.7.m7.1a"><mi id="S2.SS2.p5.7.m7.1.1" xref="S2.SS2.p5.7.m7.1.1.cmml">p</mi><annotation-xml encoding="MathML-Content" id="S2.SS2.p5.7.m7.1b"><ci id="S2.SS2.p5.7.m7.1.1.cmml" xref="S2.SS2.p5.7.m7.1.1">𝑝</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p5.7.m7.1c">p</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p5.7.m7.1d">italic_p</annotation></semantics></math>. In Section <a class="ltx_ref" href="https://arxiv.org/html/2406.11224v2#S4" title="4 The motions of a light-like particle in a pseudo Riemann space ‣ A Hamiltonian approach to the gradient-flow equations in information geometry"><span class="ltx_text ltx_ref_tag">4</span></a> we will rederive the above Hamiltonians by considering the motion of a null (or light-like) particle in a curved space.</p> </div> <div class="ltx_para" id="S2.SS2.p6"> <p class="ltx_p" id="S2.SS2.p6.4">It is worth noting that the scalar field <math alttext="\eta^{2}(\theta)" class="ltx_Math" display="inline" id="S2.SS2.p6.1.m1.1"><semantics id="S2.SS2.p6.1.m1.1a"><mrow id="S2.SS2.p6.1.m1.1.2" xref="S2.SS2.p6.1.m1.1.2.cmml"><msup id="S2.SS2.p6.1.m1.1.2.2" xref="S2.SS2.p6.1.m1.1.2.2.cmml"><mi id="S2.SS2.p6.1.m1.1.2.2.2" xref="S2.SS2.p6.1.m1.1.2.2.2.cmml">η</mi><mn id="S2.SS2.p6.1.m1.1.2.2.3" xref="S2.SS2.p6.1.m1.1.2.2.3.cmml">2</mn></msup><mo id="S2.SS2.p6.1.m1.1.2.1" xref="S2.SS2.p6.1.m1.1.2.1.cmml">⁢</mo><mrow 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id="S2.SS2.p6.1.m1.1c">\eta^{2}(\theta)</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p6.1.m1.1d">italic_η start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( italic_θ )</annotation></semantics></math> characterizes the rate of the <math alttext="\theta" class="ltx_Math" display="inline" id="S2.SS2.p6.2.m2.1"><semantics id="S2.SS2.p6.2.m2.1a"><mi id="S2.SS2.p6.2.m2.1.1" xref="S2.SS2.p6.2.m2.1.1.cmml">θ</mi><annotation-xml encoding="MathML-Content" id="S2.SS2.p6.2.m2.1b"><ci id="S2.SS2.p6.2.m2.1.1.cmml" xref="S2.SS2.p6.2.m2.1.1">𝜃</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p6.2.m2.1c">\theta</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p6.2.m2.1d">italic_θ</annotation></semantics></math>-potential, since it is related to the <math alttext="\theta" class="ltx_Math" display="inline" id="S2.SS2.p6.3.m3.1"><semantics id="S2.SS2.p6.3.m3.1a"><mi id="S2.SS2.p6.3.m3.1.1" xref="S2.SS2.p6.3.m3.1.1.cmml">θ</mi><annotation-xml 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xref="S2.SS2.p6.4.m4.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p6.4.m4.1b"><apply id="S2.SS2.p6.4.m4.1.2.cmml" xref="S2.SS2.p6.4.m4.1.2"><times id="S2.SS2.p6.4.m4.1.2.1.cmml" xref="S2.SS2.p6.4.m4.1.2.1"></times><ci id="S2.SS2.p6.4.m4.1.2.2.cmml" xref="S2.SS2.p6.4.m4.1.2.2">Ψ</ci><ci id="S2.SS2.p6.4.m4.1.1.cmml" xref="S2.SS2.p6.4.m4.1.1">𝜃</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p6.4.m4.1c">\Psi(\theta)</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p6.4.m4.1d">roman_Ψ ( italic_θ )</annotation></semantics></math> as follows.</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="S3.EGx22"> <tbody id="S2.Ex1b"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_left_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\frac{d\Psi(\theta)}{dt}" class="ltx_Math" display="inline" 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id="S2.Ex1b.m1.1c">\displaystyle\frac{d\Psi(\theta)}{dt}</annotation><annotation encoding="application/x-llamapun" id="S2.Ex1b.m1.1d">divide start_ARG italic_d roman_Ψ ( italic_θ ) end_ARG start_ARG italic_d italic_t end_ARG</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle=\frac{\partial\Psi(\theta)}{\partial\theta^{i}}\frac{d\theta^{i}% }{dt}" class="ltx_Math" display="inline" id="S2.Ex1b.m2.1"><semantics id="S2.Ex1b.m2.1a"><mrow id="S2.Ex1b.m2.1.2" xref="S2.Ex1b.m2.1.2.cmml"><mi id="S2.Ex1b.m2.1.2.2" xref="S2.Ex1b.m2.1.2.2.cmml"></mi><mo id="S2.Ex1b.m2.1.2.1" xref="S2.Ex1b.m2.1.2.1.cmml">=</mo><mrow id="S2.Ex1b.m2.1.2.3" xref="S2.Ex1b.m2.1.2.3.cmml"><mstyle displaystyle="true" id="S2.Ex1b.m2.1.1" xref="S2.Ex1b.m2.1.1.cmml"><mfrac id="S2.Ex1b.m2.1.1a" xref="S2.Ex1b.m2.1.1.cmml"><mrow id="S2.Ex1b.m2.1.1.1" xref="S2.Ex1b.m2.1.1.1.cmml"><mo id="S2.Ex1b.m2.1.1.1.2" rspace="0em" xref="S2.Ex1b.m2.1.1.1.2.cmml">∂</mo><mrow 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id="S2.E22.m1.3.3.1.1.3.3.5.1.cmml" xref="S2.E22.m1.3.3.1.1.3.3.5">subscript</csymbol><apply id="S2.E22.m1.3.3.1.1.3.3.5.2.cmml" xref="S2.E22.m1.3.3.1.1.3.3.5"><csymbol cd="ambiguous" id="S2.E22.m1.3.3.1.1.3.3.5.2.1.cmml" xref="S2.E22.m1.3.3.1.1.3.3.5">superscript</csymbol><ci id="S2.E22.m1.3.3.1.1.3.3.5.2.2.cmml" xref="S2.E22.m1.3.3.1.1.3.3.5.2.2">𝜂</ci><ci id="S2.E22.m1.3.3.1.1.3.3.5.2.3.cmml" xref="S2.E22.m1.3.3.1.1.3.3.5.2.3">r</ci></apply><ci id="S2.E22.m1.3.3.1.1.3.3.5.3.cmml" xref="S2.E22.m1.3.3.1.1.3.3.5.3">𝑗</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.E22.m1.3c">\displaystyle=\eta^{2}(\theta)-g^{ij}(\theta)\eta_{i}\eta^{\rm r}_{j}.</annotation><annotation encoding="application/x-llamapun" id="S2.E22.m1.3d">= italic_η start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( italic_θ ) - italic_g start_POSTSUPERSCRIPT italic_i italic_j end_POSTSUPERSCRIPT ( italic_θ ) italic_η start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_η start_POSTSUPERSCRIPT roman_r end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_left_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(22)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S2.SS2.p6.6">where the relations (<a class="ltx_ref" href="https://arxiv.org/html/2406.11224v2#S2.E2" title="In 2.1 Information Geometry ‣ 2 Information Geometry and Gradient-Flow Equations ‣ A Hamiltonian approach to the gradient-flow equations in information geometry"><span class="ltx_text ltx_ref_tag">2</span></a>) and (<a class="ltx_ref" href="https://arxiv.org/html/2406.11224v2#S2.E12" title="In 2.2 Gradient-Flow Equations ‣ 2 Information Geometry and Gradient-Flow Equations ‣ A Hamiltonian approach to the gradient-flow equations in information geometry"><span class="ltx_text ltx_ref_tag">12</span></a>) are used. Similarly, the scalar field <math alttext="\theta^{2}(\eta)" class="ltx_Math" display="inline" id="S2.SS2.p6.5.m1.1"><semantics id="S2.SS2.p6.5.m1.1a"><mrow id="S2.SS2.p6.5.m1.1.2" xref="S2.SS2.p6.5.m1.1.2.cmml"><msup id="S2.SS2.p6.5.m1.1.2.2" xref="S2.SS2.p6.5.m1.1.2.2.cmml"><mi id="S2.SS2.p6.5.m1.1.2.2.2" xref="S2.SS2.p6.5.m1.1.2.2.2.cmml">θ</mi><mn id="S2.SS2.p6.5.m1.1.2.2.3" xref="S2.SS2.p6.5.m1.1.2.2.3.cmml">2</mn></msup><mo id="S2.SS2.p6.5.m1.1.2.1" xref="S2.SS2.p6.5.m1.1.2.1.cmml">⁢</mo><mrow id="S2.SS2.p6.5.m1.1.2.3.2" xref="S2.SS2.p6.5.m1.1.2.cmml"><mo id="S2.SS2.p6.5.m1.1.2.3.2.1" stretchy="false" xref="S2.SS2.p6.5.m1.1.2.cmml">(</mo><mi id="S2.SS2.p6.5.m1.1.1" xref="S2.SS2.p6.5.m1.1.1.cmml">η</mi><mo id="S2.SS2.p6.5.m1.1.2.3.2.2" stretchy="false" xref="S2.SS2.p6.5.m1.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p6.5.m1.1b"><apply id="S2.SS2.p6.5.m1.1.2.cmml" xref="S2.SS2.p6.5.m1.1.2"><times id="S2.SS2.p6.5.m1.1.2.1.cmml" 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id="S2.SS2.p6.6.m2.1b"><ci id="S2.SS2.p6.6.m2.1.1.cmml" xref="S2.SS2.p6.6.m2.1.1">𝜂</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p6.6.m2.1c">\eta</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p6.6.m2.1d">italic_η</annotation></semantics></math>-potential as</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="S3.EGx23"> <tbody id="S2.E23"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_left_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\frac{d\Psi^{\star}(\eta)}{dt}=-\theta^{2}(\eta)+g_{ij}(\eta)% \theta^{i}\theta_{\rm r}^{j}." class="ltx_Math" display="inline" id="S2.E23.m1.4"><semantics id="S2.E23.m1.4a"><mrow id="S2.E23.m1.4.4.1" xref="S2.E23.m1.4.4.1.1.cmml"><mrow id="S2.E23.m1.4.4.1.1" xref="S2.E23.m1.4.4.1.1.cmml"><mstyle displaystyle="true" id="S2.E23.m1.1.1" xref="S2.E23.m1.1.1.cmml"><mfrac id="S2.E23.m1.1.1a" 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id="S2.E23.m1.4c">\displaystyle\frac{d\Psi^{\star}(\eta)}{dt}=-\theta^{2}(\eta)+g_{ij}(\eta)% \theta^{i}\theta_{\rm r}^{j}.</annotation><annotation encoding="application/x-llamapun" id="S2.E23.m1.4d">divide start_ARG italic_d roman_Ψ start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT ( italic_η ) end_ARG start_ARG italic_d italic_t end_ARG = - italic_θ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( italic_η ) + italic_g start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT ( italic_η ) italic_θ start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT italic_θ start_POSTSUBSCRIPT roman_r end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_j end_POSTSUPERSCRIPT .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_left_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(23)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S2.SS2.p6.10">Since <math alttext="-\Psi^{\star}(\eta)" class="ltx_Math" display="inline" id="S2.SS2.p6.7.m1.1"><semantics id="S2.SS2.p6.7.m1.1a"><mrow id="S2.SS2.p6.7.m1.1.2" xref="S2.SS2.p6.7.m1.1.2.cmml"><mo id="S2.SS2.p6.7.m1.1.2a" xref="S2.SS2.p6.7.m1.1.2.cmml">−</mo><mrow id="S2.SS2.p6.7.m1.1.2.2" xref="S2.SS2.p6.7.m1.1.2.2.cmml"><msup id="S2.SS2.p6.7.m1.1.2.2.2" xref="S2.SS2.p6.7.m1.1.2.2.2.cmml"><mi id="S2.SS2.p6.7.m1.1.2.2.2.2" mathvariant="normal" xref="S2.SS2.p6.7.m1.1.2.2.2.2.cmml">Ψ</mi><mo id="S2.SS2.p6.7.m1.1.2.2.2.3" xref="S2.SS2.p6.7.m1.1.2.2.2.3.cmml">⋆</mo></msup><mo id="S2.SS2.p6.7.m1.1.2.2.1" xref="S2.SS2.p6.7.m1.1.2.2.1.cmml">⁢</mo><mrow id="S2.SS2.p6.7.m1.1.2.2.3.2" xref="S2.SS2.p6.7.m1.1.2.2.cmml"><mo id="S2.SS2.p6.7.m1.1.2.2.3.2.1" stretchy="false" xref="S2.SS2.p6.7.m1.1.2.2.cmml">(</mo><mi id="S2.SS2.p6.7.m1.1.1" xref="S2.SS2.p6.7.m1.1.1.cmml">η</mi><mo id="S2.SS2.p6.7.m1.1.2.2.3.2.2" stretchy="false" xref="S2.SS2.p6.7.m1.1.2.2.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p6.7.m1.1b"><apply id="S2.SS2.p6.7.m1.1.2.cmml" xref="S2.SS2.p6.7.m1.1.2"><minus id="S2.SS2.p6.7.m1.1.2.1.cmml" xref="S2.SS2.p6.7.m1.1.2"></minus><apply id="S2.SS2.p6.7.m1.1.2.2.cmml" xref="S2.SS2.p6.7.m1.1.2.2"><times id="S2.SS2.p6.7.m1.1.2.2.1.cmml" xref="S2.SS2.p6.7.m1.1.2.2.1"></times><apply id="S2.SS2.p6.7.m1.1.2.2.2.cmml" xref="S2.SS2.p6.7.m1.1.2.2.2"><csymbol cd="ambiguous" id="S2.SS2.p6.7.m1.1.2.2.2.1.cmml" xref="S2.SS2.p6.7.m1.1.2.2.2">superscript</csymbol><ci id="S2.SS2.p6.7.m1.1.2.2.2.2.cmml" xref="S2.SS2.p6.7.m1.1.2.2.2.2">Ψ</ci><ci id="S2.SS2.p6.7.m1.1.2.2.2.3.cmml" xref="S2.SS2.p6.7.m1.1.2.2.2.3">⋆</ci></apply><ci id="S2.SS2.p6.7.m1.1.1.cmml" xref="S2.SS2.p6.7.m1.1.1">𝜂</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p6.7.m1.1c">-\Psi^{\star}(\eta)</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p6.7.m1.1d">- roman_Ψ start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT ( italic_η )</annotation></semantics></math> is the entropy <math alttext="S(\eta)" class="ltx_Math" display="inline" id="S2.SS2.p6.8.m2.1"><semantics id="S2.SS2.p6.8.m2.1a"><mrow id="S2.SS2.p6.8.m2.1.2" xref="S2.SS2.p6.8.m2.1.2.cmml"><mi id="S2.SS2.p6.8.m2.1.2.2" xref="S2.SS2.p6.8.m2.1.2.2.cmml">S</mi><mo id="S2.SS2.p6.8.m2.1.2.1" xref="S2.SS2.p6.8.m2.1.2.1.cmml">⁢</mo><mrow id="S2.SS2.p6.8.m2.1.2.3.2" xref="S2.SS2.p6.8.m2.1.2.cmml"><mo id="S2.SS2.p6.8.m2.1.2.3.2.1" stretchy="false" xref="S2.SS2.p6.8.m2.1.2.cmml">(</mo><mi id="S2.SS2.p6.8.m2.1.1" xref="S2.SS2.p6.8.m2.1.1.cmml">η</mi><mo id="S2.SS2.p6.8.m2.1.2.3.2.2" stretchy="false" xref="S2.SS2.p6.8.m2.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p6.8.m2.1b"><apply id="S2.SS2.p6.8.m2.1.2.cmml" xref="S2.SS2.p6.8.m2.1.2"><times id="S2.SS2.p6.8.m2.1.2.1.cmml" xref="S2.SS2.p6.8.m2.1.2.1"></times><ci id="S2.SS2.p6.8.m2.1.2.2.cmml" xref="S2.SS2.p6.8.m2.1.2.2">𝑆</ci><ci id="S2.SS2.p6.8.m2.1.1.cmml" xref="S2.SS2.p6.8.m2.1.1">𝜂</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p6.8.m2.1c">S(\eta)</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p6.8.m2.1d">italic_S ( italic_η )</annotation></semantics></math> in (<a class="ltx_ref" href="https://arxiv.org/html/2406.11224v2#S2.E7" title="In 2.1 Information Geometry ‣ 2 Information Geometry and Gradient-Flow Equations ‣ A Hamiltonian approach to the gradient-flow equations in information geometry"><span class="ltx_text ltx_ref_tag">7</span></a>), the scalar field <math alttext="\theta^{2}(\eta)" class="ltx_Math" display="inline" id="S2.SS2.p6.9.m3.1"><semantics id="S2.SS2.p6.9.m3.1a"><mrow id="S2.SS2.p6.9.m3.1.2" xref="S2.SS2.p6.9.m3.1.2.cmml"><msup id="S2.SS2.p6.9.m3.1.2.2" xref="S2.SS2.p6.9.m3.1.2.2.cmml"><mi id="S2.SS2.p6.9.m3.1.2.2.2" xref="S2.SS2.p6.9.m3.1.2.2.2.cmml">θ</mi><mn id="S2.SS2.p6.9.m3.1.2.2.3" xref="S2.SS2.p6.9.m3.1.2.2.3.cmml">2</mn></msup><mo id="S2.SS2.p6.9.m3.1.2.1" xref="S2.SS2.p6.9.m3.1.2.1.cmml">⁢</mo><mrow id="S2.SS2.p6.9.m3.1.2.3.2" xref="S2.SS2.p6.9.m3.1.2.cmml"><mo id="S2.SS2.p6.9.m3.1.2.3.2.1" stretchy="false" xref="S2.SS2.p6.9.m3.1.2.cmml">(</mo><mi id="S2.SS2.p6.9.m3.1.1" xref="S2.SS2.p6.9.m3.1.1.cmml">η</mi><mo id="S2.SS2.p6.9.m3.1.2.3.2.2" stretchy="false" xref="S2.SS2.p6.9.m3.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p6.9.m3.1b"><apply id="S2.SS2.p6.9.m3.1.2.cmml" xref="S2.SS2.p6.9.m3.1.2"><times id="S2.SS2.p6.9.m3.1.2.1.cmml" xref="S2.SS2.p6.9.m3.1.2.1"></times><apply id="S2.SS2.p6.9.m3.1.2.2.cmml" xref="S2.SS2.p6.9.m3.1.2.2"><csymbol cd="ambiguous" id="S2.SS2.p6.9.m3.1.2.2.1.cmml" xref="S2.SS2.p6.9.m3.1.2.2">superscript</csymbol><ci id="S2.SS2.p6.9.m3.1.2.2.2.cmml" xref="S2.SS2.p6.9.m3.1.2.2.2">𝜃</ci><cn id="S2.SS2.p6.9.m3.1.2.2.3.cmml" type="integer" xref="S2.SS2.p6.9.m3.1.2.2.3">2</cn></apply><ci id="S2.SS2.p6.9.m3.1.1.cmml" xref="S2.SS2.p6.9.m3.1.1">𝜂</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p6.9.m3.1c">\theta^{2}(\eta)</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p6.9.m3.1d">italic_θ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( italic_η )</annotation></semantics></math> characterizes the rate of the entropy <math alttext="dS(\eta)/dt" class="ltx_Math" display="inline" id="S2.SS2.p6.10.m4.1"><semantics id="S2.SS2.p6.10.m4.1a"><mrow id="S2.SS2.p6.10.m4.1.2" xref="S2.SS2.p6.10.m4.1.2.cmml"><mrow id="S2.SS2.p6.10.m4.1.2.2" xref="S2.SS2.p6.10.m4.1.2.2.cmml"><mrow id="S2.SS2.p6.10.m4.1.2.2.2" xref="S2.SS2.p6.10.m4.1.2.2.2.cmml"><mi id="S2.SS2.p6.10.m4.1.2.2.2.2" xref="S2.SS2.p6.10.m4.1.2.2.2.2.cmml">d</mi><mo id="S2.SS2.p6.10.m4.1.2.2.2.1" xref="S2.SS2.p6.10.m4.1.2.2.2.1.cmml">⁢</mo><mi id="S2.SS2.p6.10.m4.1.2.2.2.3" xref="S2.SS2.p6.10.m4.1.2.2.2.3.cmml">S</mi><mo id="S2.SS2.p6.10.m4.1.2.2.2.1a" xref="S2.SS2.p6.10.m4.1.2.2.2.1.cmml">⁢</mo><mrow 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xref="S2.SS2.p6.10.m4.1.2.2.1"></divide><apply id="S2.SS2.p6.10.m4.1.2.2.2.cmml" xref="S2.SS2.p6.10.m4.1.2.2.2"><times id="S2.SS2.p6.10.m4.1.2.2.2.1.cmml" xref="S2.SS2.p6.10.m4.1.2.2.2.1"></times><ci id="S2.SS2.p6.10.m4.1.2.2.2.2.cmml" xref="S2.SS2.p6.10.m4.1.2.2.2.2">𝑑</ci><ci id="S2.SS2.p6.10.m4.1.2.2.2.3.cmml" xref="S2.SS2.p6.10.m4.1.2.2.2.3">𝑆</ci><ci id="S2.SS2.p6.10.m4.1.1.cmml" xref="S2.SS2.p6.10.m4.1.1">𝜂</ci></apply><ci id="S2.SS2.p6.10.m4.1.2.2.3.cmml" xref="S2.SS2.p6.10.m4.1.2.2.3">𝑑</ci></apply><ci id="S2.SS2.p6.10.m4.1.2.3.cmml" xref="S2.SS2.p6.10.m4.1.2.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p6.10.m4.1c">dS(\eta)/dt</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p6.10.m4.1d">italic_d italic_S ( italic_η ) / italic_d italic_t</annotation></semantics></math> in the gradient-flows.</p> </div> </section> <section class="ltx_subsection" id="S2.SS3"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">2.3 </span>Randers-Finsler deformation of the gradient-flow equations</h3> <div class="ltx_para" id="S2.SS3.p1"> <p class="ltx_p" id="S2.SS3.p1.1">Here we briefly review the Randers-Finsler (RF) deformation <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2406.11224v2#bib.bib11" title="">11</a>]</cite> of the gradient-flow equations.</p> </div> <div class="ltx_para" id="S2.SS3.p2"> <p class="ltx_p" id="S2.SS3.p2.6">Finsler space is a general space based on the line-element <math alttext="d\ell=F(x,dx)" class="ltx_Math" display="inline" id="S2.SS3.p2.1.m1.2"><semantics id="S2.SS3.p2.1.m1.2a"><mrow id="S2.SS3.p2.1.m1.2.2" xref="S2.SS3.p2.1.m1.2.2.cmml"><mrow id="S2.SS3.p2.1.m1.2.2.3" xref="S2.SS3.p2.1.m1.2.2.3.cmml"><mi id="S2.SS3.p2.1.m1.2.2.3.2" xref="S2.SS3.p2.1.m1.2.2.3.2.cmml">d</mi><mo id="S2.SS3.p2.1.m1.2.2.3.1" xref="S2.SS3.p2.1.m1.2.2.3.1.cmml">⁢</mo><mi id="S2.SS3.p2.1.m1.2.2.3.3" mathvariant="normal" xref="S2.SS3.p2.1.m1.2.2.3.3.cmml">ℓ</mi></mrow><mo id="S2.SS3.p2.1.m1.2.2.2" xref="S2.SS3.p2.1.m1.2.2.2.cmml">=</mo><mrow id="S2.SS3.p2.1.m1.2.2.1" xref="S2.SS3.p2.1.m1.2.2.1.cmml"><mi id="S2.SS3.p2.1.m1.2.2.1.3" xref="S2.SS3.p2.1.m1.2.2.1.3.cmml">F</mi><mo id="S2.SS3.p2.1.m1.2.2.1.2" xref="S2.SS3.p2.1.m1.2.2.1.2.cmml">⁢</mo><mrow id="S2.SS3.p2.1.m1.2.2.1.1.1" xref="S2.SS3.p2.1.m1.2.2.1.1.2.cmml"><mo id="S2.SS3.p2.1.m1.2.2.1.1.1.2" stretchy="false" xref="S2.SS3.p2.1.m1.2.2.1.1.2.cmml">(</mo><mi id="S2.SS3.p2.1.m1.1.1" xref="S2.SS3.p2.1.m1.1.1.cmml">x</mi><mo id="S2.SS3.p2.1.m1.2.2.1.1.1.3" xref="S2.SS3.p2.1.m1.2.2.1.1.2.cmml">,</mo><mrow id="S2.SS3.p2.1.m1.2.2.1.1.1.1" xref="S2.SS3.p2.1.m1.2.2.1.1.1.1.cmml"><mi id="S2.SS3.p2.1.m1.2.2.1.1.1.1.2" xref="S2.SS3.p2.1.m1.2.2.1.1.1.1.2.cmml">d</mi><mo id="S2.SS3.p2.1.m1.2.2.1.1.1.1.1" xref="S2.SS3.p2.1.m1.2.2.1.1.1.1.1.cmml">⁢</mo><mi id="S2.SS3.p2.1.m1.2.2.1.1.1.1.3" xref="S2.SS3.p2.1.m1.2.2.1.1.1.1.3.cmml">x</mi></mrow><mo id="S2.SS3.p2.1.m1.2.2.1.1.1.4" stretchy="false" xref="S2.SS3.p2.1.m1.2.2.1.1.2.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS3.p2.1.m1.2b"><apply id="S2.SS3.p2.1.m1.2.2.cmml" xref="S2.SS3.p2.1.m1.2.2"><eq id="S2.SS3.p2.1.m1.2.2.2.cmml" xref="S2.SS3.p2.1.m1.2.2.2"></eq><apply id="S2.SS3.p2.1.m1.2.2.3.cmml" xref="S2.SS3.p2.1.m1.2.2.3"><times id="S2.SS3.p2.1.m1.2.2.3.1.cmml" xref="S2.SS3.p2.1.m1.2.2.3.1"></times><ci id="S2.SS3.p2.1.m1.2.2.3.2.cmml" xref="S2.SS3.p2.1.m1.2.2.3.2">𝑑</ci><ci id="S2.SS3.p2.1.m1.2.2.3.3.cmml" xref="S2.SS3.p2.1.m1.2.2.3.3">ℓ</ci></apply><apply id="S2.SS3.p2.1.m1.2.2.1.cmml" xref="S2.SS3.p2.1.m1.2.2.1"><times id="S2.SS3.p2.1.m1.2.2.1.2.cmml" xref="S2.SS3.p2.1.m1.2.2.1.2"></times><ci id="S2.SS3.p2.1.m1.2.2.1.3.cmml" xref="S2.SS3.p2.1.m1.2.2.1.3">𝐹</ci><interval closure="open" id="S2.SS3.p2.1.m1.2.2.1.1.2.cmml" xref="S2.SS3.p2.1.m1.2.2.1.1.1"><ci id="S2.SS3.p2.1.m1.1.1.cmml" xref="S2.SS3.p2.1.m1.1.1">𝑥</ci><apply id="S2.SS3.p2.1.m1.2.2.1.1.1.1.cmml" xref="S2.SS3.p2.1.m1.2.2.1.1.1.1"><times id="S2.SS3.p2.1.m1.2.2.1.1.1.1.1.cmml" xref="S2.SS3.p2.1.m1.2.2.1.1.1.1.1"></times><ci id="S2.SS3.p2.1.m1.2.2.1.1.1.1.2.cmml" xref="S2.SS3.p2.1.m1.2.2.1.1.1.1.2">𝑑</ci><ci id="S2.SS3.p2.1.m1.2.2.1.1.1.1.3.cmml" xref="S2.SS3.p2.1.m1.2.2.1.1.1.1.3">𝑥</ci></apply></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p2.1.m1.2c">d\ell=F(x,dx)</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p2.1.m1.2d">italic_d roman_ℓ = italic_F ( italic_x , italic_d italic_x )</annotation></semantics></math>, where <math alttext="F(x,dx)>0" class="ltx_Math" display="inline" id="S2.SS3.p2.2.m2.2"><semantics id="S2.SS3.p2.2.m2.2a"><mrow id="S2.SS3.p2.2.m2.2.2" xref="S2.SS3.p2.2.m2.2.2.cmml"><mrow id="S2.SS3.p2.2.m2.2.2.1" xref="S2.SS3.p2.2.m2.2.2.1.cmml"><mi id="S2.SS3.p2.2.m2.2.2.1.3" xref="S2.SS3.p2.2.m2.2.2.1.3.cmml">F</mi><mo id="S2.SS3.p2.2.m2.2.2.1.2" xref="S2.SS3.p2.2.m2.2.2.1.2.cmml">⁢</mo><mrow id="S2.SS3.p2.2.m2.2.2.1.1.1" xref="S2.SS3.p2.2.m2.2.2.1.1.2.cmml"><mo id="S2.SS3.p2.2.m2.2.2.1.1.1.2" stretchy="false" xref="S2.SS3.p2.2.m2.2.2.1.1.2.cmml">(</mo><mi id="S2.SS3.p2.2.m2.1.1" xref="S2.SS3.p2.2.m2.1.1.cmml">x</mi><mo id="S2.SS3.p2.2.m2.2.2.1.1.1.3" xref="S2.SS3.p2.2.m2.2.2.1.1.2.cmml">,</mo><mrow id="S2.SS3.p2.2.m2.2.2.1.1.1.1" xref="S2.SS3.p2.2.m2.2.2.1.1.1.1.cmml"><mi id="S2.SS3.p2.2.m2.2.2.1.1.1.1.2" xref="S2.SS3.p2.2.m2.2.2.1.1.1.1.2.cmml">d</mi><mo id="S2.SS3.p2.2.m2.2.2.1.1.1.1.1" xref="S2.SS3.p2.2.m2.2.2.1.1.1.1.1.cmml">⁢</mo><mi id="S2.SS3.p2.2.m2.2.2.1.1.1.1.3" xref="S2.SS3.p2.2.m2.2.2.1.1.1.1.3.cmml">x</mi></mrow><mo id="S2.SS3.p2.2.m2.2.2.1.1.1.4" stretchy="false" xref="S2.SS3.p2.2.m2.2.2.1.1.2.cmml">)</mo></mrow></mrow><mo id="S2.SS3.p2.2.m2.2.2.2" xref="S2.SS3.p2.2.m2.2.2.2.cmml">></mo><mn id="S2.SS3.p2.2.m2.2.2.3" xref="S2.SS3.p2.2.m2.2.2.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.SS3.p2.2.m2.2b"><apply id="S2.SS3.p2.2.m2.2.2.cmml" xref="S2.SS3.p2.2.m2.2.2"><gt id="S2.SS3.p2.2.m2.2.2.2.cmml" xref="S2.SS3.p2.2.m2.2.2.2"></gt><apply id="S2.SS3.p2.2.m2.2.2.1.cmml" xref="S2.SS3.p2.2.m2.2.2.1"><times id="S2.SS3.p2.2.m2.2.2.1.2.cmml" xref="S2.SS3.p2.2.m2.2.2.1.2"></times><ci id="S2.SS3.p2.2.m2.2.2.1.3.cmml" xref="S2.SS3.p2.2.m2.2.2.1.3">𝐹</ci><interval closure="open" id="S2.SS3.p2.2.m2.2.2.1.1.2.cmml" xref="S2.SS3.p2.2.m2.2.2.1.1.1"><ci id="S2.SS3.p2.2.m2.1.1.cmml" xref="S2.SS3.p2.2.m2.1.1">𝑥</ci><apply id="S2.SS3.p2.2.m2.2.2.1.1.1.1.cmml" xref="S2.SS3.p2.2.m2.2.2.1.1.1.1"><times id="S2.SS3.p2.2.m2.2.2.1.1.1.1.1.cmml" xref="S2.SS3.p2.2.m2.2.2.1.1.1.1.1"></times><ci id="S2.SS3.p2.2.m2.2.2.1.1.1.1.2.cmml" xref="S2.SS3.p2.2.m2.2.2.1.1.1.1.2">𝑑</ci><ci id="S2.SS3.p2.2.m2.2.2.1.1.1.1.3.cmml" xref="S2.SS3.p2.2.m2.2.2.1.1.1.1.3">𝑥</ci></apply></interval></apply><cn id="S2.SS3.p2.2.m2.2.2.3.cmml" type="integer" xref="S2.SS3.p2.2.m2.2.2.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p2.2.m2.2c">F(x,dx)>0</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p2.2.m2.2d">italic_F ( italic_x , italic_d italic_x ) > 0</annotation></semantics></math> for <math alttext="dx\neq 0" class="ltx_Math" display="inline" id="S2.SS3.p2.3.m3.1"><semantics id="S2.SS3.p2.3.m3.1a"><mrow id="S2.SS3.p2.3.m3.1.1" xref="S2.SS3.p2.3.m3.1.1.cmml"><mrow id="S2.SS3.p2.3.m3.1.1.2" xref="S2.SS3.p2.3.m3.1.1.2.cmml"><mi id="S2.SS3.p2.3.m3.1.1.2.2" xref="S2.SS3.p2.3.m3.1.1.2.2.cmml">d</mi><mo id="S2.SS3.p2.3.m3.1.1.2.1" xref="S2.SS3.p2.3.m3.1.1.2.1.cmml">⁢</mo><mi id="S2.SS3.p2.3.m3.1.1.2.3" xref="S2.SS3.p2.3.m3.1.1.2.3.cmml">x</mi></mrow><mo id="S2.SS3.p2.3.m3.1.1.1" xref="S2.SS3.p2.3.m3.1.1.1.cmml">≠</mo><mn id="S2.SS3.p2.3.m3.1.1.3" xref="S2.SS3.p2.3.m3.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.SS3.p2.3.m3.1b"><apply id="S2.SS3.p2.3.m3.1.1.cmml" xref="S2.SS3.p2.3.m3.1.1"><neq id="S2.SS3.p2.3.m3.1.1.1.cmml" xref="S2.SS3.p2.3.m3.1.1.1"></neq><apply id="S2.SS3.p2.3.m3.1.1.2.cmml" xref="S2.SS3.p2.3.m3.1.1.2"><times id="S2.SS3.p2.3.m3.1.1.2.1.cmml" xref="S2.SS3.p2.3.m3.1.1.2.1"></times><ci id="S2.SS3.p2.3.m3.1.1.2.2.cmml" xref="S2.SS3.p2.3.m3.1.1.2.2">𝑑</ci><ci id="S2.SS3.p2.3.m3.1.1.2.3.cmml" xref="S2.SS3.p2.3.m3.1.1.2.3">𝑥</ci></apply><cn id="S2.SS3.p2.3.m3.1.1.3.cmml" type="integer" xref="S2.SS3.p2.3.m3.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p2.3.m3.1c">dx\neq 0</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p2.3.m3.1d">italic_d italic_x ≠ 0</annotation></semantics></math> is a function in the tangent space <math alttext="T_{x}\mathcal{M}" class="ltx_Math" display="inline" id="S2.SS3.p2.4.m4.1"><semantics id="S2.SS3.p2.4.m4.1a"><mrow id="S2.SS3.p2.4.m4.1.1" xref="S2.SS3.p2.4.m4.1.1.cmml"><msub id="S2.SS3.p2.4.m4.1.1.2" xref="S2.SS3.p2.4.m4.1.1.2.cmml"><mi id="S2.SS3.p2.4.m4.1.1.2.2" xref="S2.SS3.p2.4.m4.1.1.2.2.cmml">T</mi><mi id="S2.SS3.p2.4.m4.1.1.2.3" xref="S2.SS3.p2.4.m4.1.1.2.3.cmml">x</mi></msub><mo id="S2.SS3.p2.4.m4.1.1.1" xref="S2.SS3.p2.4.m4.1.1.1.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="S2.SS3.p2.4.m4.1.1.3" xref="S2.SS3.p2.4.m4.1.1.3.cmml">ℳ</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.SS3.p2.4.m4.1b"><apply id="S2.SS3.p2.4.m4.1.1.cmml" xref="S2.SS3.p2.4.m4.1.1"><times id="S2.SS3.p2.4.m4.1.1.1.cmml" xref="S2.SS3.p2.4.m4.1.1.1"></times><apply id="S2.SS3.p2.4.m4.1.1.2.cmml" xref="S2.SS3.p2.4.m4.1.1.2"><csymbol cd="ambiguous" id="S2.SS3.p2.4.m4.1.1.2.1.cmml" xref="S2.SS3.p2.4.m4.1.1.2">subscript</csymbol><ci id="S2.SS3.p2.4.m4.1.1.2.2.cmml" xref="S2.SS3.p2.4.m4.1.1.2.2">𝑇</ci><ci id="S2.SS3.p2.4.m4.1.1.2.3.cmml" xref="S2.SS3.p2.4.m4.1.1.2.3">𝑥</ci></apply><ci id="S2.SS3.p2.4.m4.1.1.3.cmml" xref="S2.SS3.p2.4.m4.1.1.3">ℳ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p2.4.m4.1c">T_{x}\mathcal{M}</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p2.4.m4.1d">italic_T start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT caligraphic_M</annotation></semantics></math>, and is a homogeneous function of first order in <math alttext="dx" class="ltx_Math" display="inline" id="S2.SS3.p2.5.m5.1"><semantics id="S2.SS3.p2.5.m5.1a"><mrow id="S2.SS3.p2.5.m5.1.1" xref="S2.SS3.p2.5.m5.1.1.cmml"><mi id="S2.SS3.p2.5.m5.1.1.2" xref="S2.SS3.p2.5.m5.1.1.2.cmml">d</mi><mo id="S2.SS3.p2.5.m5.1.1.1" xref="S2.SS3.p2.5.m5.1.1.1.cmml">⁢</mo><mi id="S2.SS3.p2.5.m5.1.1.3" xref="S2.SS3.p2.5.m5.1.1.3.cmml">x</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.SS3.p2.5.m5.1b"><apply id="S2.SS3.p2.5.m5.1.1.cmml" xref="S2.SS3.p2.5.m5.1.1"><times id="S2.SS3.p2.5.m5.1.1.1.cmml" xref="S2.SS3.p2.5.m5.1.1.1"></times><ci id="S2.SS3.p2.5.m5.1.1.2.cmml" xref="S2.SS3.p2.5.m5.1.1.2">𝑑</ci><ci id="S2.SS3.p2.5.m5.1.1.3.cmml" xref="S2.SS3.p2.5.m5.1.1.3">𝑥</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p2.5.m5.1c">dx</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p2.5.m5.1d">italic_d italic_x</annotation></semantics></math>. The function <math alttext="F(x,dx)" class="ltx_Math" display="inline" id="S2.SS3.p2.6.m6.2"><semantics id="S2.SS3.p2.6.m6.2a"><mrow id="S2.SS3.p2.6.m6.2.2" xref="S2.SS3.p2.6.m6.2.2.cmml"><mi id="S2.SS3.p2.6.m6.2.2.3" xref="S2.SS3.p2.6.m6.2.2.3.cmml">F</mi><mo id="S2.SS3.p2.6.m6.2.2.2" xref="S2.SS3.p2.6.m6.2.2.2.cmml">⁢</mo><mrow id="S2.SS3.p2.6.m6.2.2.1.1" xref="S2.SS3.p2.6.m6.2.2.1.2.cmml"><mo id="S2.SS3.p2.6.m6.2.2.1.1.2" stretchy="false" xref="S2.SS3.p2.6.m6.2.2.1.2.cmml">(</mo><mi id="S2.SS3.p2.6.m6.1.1" xref="S2.SS3.p2.6.m6.1.1.cmml">x</mi><mo id="S2.SS3.p2.6.m6.2.2.1.1.3" xref="S2.SS3.p2.6.m6.2.2.1.2.cmml">,</mo><mrow id="S2.SS3.p2.6.m6.2.2.1.1.1" xref="S2.SS3.p2.6.m6.2.2.1.1.1.cmml"><mi id="S2.SS3.p2.6.m6.2.2.1.1.1.2" xref="S2.SS3.p2.6.m6.2.2.1.1.1.2.cmml">d</mi><mo id="S2.SS3.p2.6.m6.2.2.1.1.1.1" xref="S2.SS3.p2.6.m6.2.2.1.1.1.1.cmml">⁢</mo><mi id="S2.SS3.p2.6.m6.2.2.1.1.1.3" xref="S2.SS3.p2.6.m6.2.2.1.1.1.3.cmml">x</mi></mrow><mo id="S2.SS3.p2.6.m6.2.2.1.1.4" stretchy="false" xref="S2.SS3.p2.6.m6.2.2.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS3.p2.6.m6.2b"><apply id="S2.SS3.p2.6.m6.2.2.cmml" xref="S2.SS3.p2.6.m6.2.2"><times id="S2.SS3.p2.6.m6.2.2.2.cmml" xref="S2.SS3.p2.6.m6.2.2.2"></times><ci id="S2.SS3.p2.6.m6.2.2.3.cmml" xref="S2.SS3.p2.6.m6.2.2.3">𝐹</ci><interval closure="open" id="S2.SS3.p2.6.m6.2.2.1.2.cmml" xref="S2.SS3.p2.6.m6.2.2.1.1"><ci id="S2.SS3.p2.6.m6.1.1.cmml" xref="S2.SS3.p2.6.m6.1.1">𝑥</ci><apply id="S2.SS3.p2.6.m6.2.2.1.1.1.cmml" xref="S2.SS3.p2.6.m6.2.2.1.1.1"><times id="S2.SS3.p2.6.m6.2.2.1.1.1.1.cmml" xref="S2.SS3.p2.6.m6.2.2.1.1.1.1"></times><ci id="S2.SS3.p2.6.m6.2.2.1.1.1.2.cmml" xref="S2.SS3.p2.6.m6.2.2.1.1.1.2">𝑑</ci><ci id="S2.SS3.p2.6.m6.2.2.1.1.1.3.cmml" xref="S2.SS3.p2.6.m6.2.2.1.1.1.3">𝑥</ci></apply></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p2.6.m6.2c">F(x,dx)</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p2.6.m6.2d">italic_F ( italic_x , italic_d italic_x )</annotation></semantics></math> is called <span class="ltx_text ltx_font_italic" id="S2.SS3.p2.6.1">Finsler function</span>, which provides the metric tensor</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="S3.EGx24"> <tbody id="S2.E24"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_left_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle g_{ij}(x,dx)=\frac{1}{2}\frac{\partial F^{2}(x,dx)}{\partial dx^% {i}\partial dx^{j}}," class="ltx_Math" display="inline" id="S2.E24.m1.4"><semantics id="S2.E24.m1.4a"><mrow id="S2.E24.m1.4.4.1" xref="S2.E24.m1.4.4.1.1.cmml"><mrow id="S2.E24.m1.4.4.1.1" xref="S2.E24.m1.4.4.1.1.cmml"><mrow id="S2.E24.m1.4.4.1.1.1" xref="S2.E24.m1.4.4.1.1.1.cmml"><msub id="S2.E24.m1.4.4.1.1.1.3" xref="S2.E24.m1.4.4.1.1.1.3.cmml"><mi id="S2.E24.m1.4.4.1.1.1.3.2" 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xref="S2.E24.m1.2.2.4.2.4.2.3.3">𝑗</ci></apply></apply></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.E24.m1.4c">\displaystyle g_{ij}(x,dx)=\frac{1}{2}\frac{\partial F^{2}(x,dx)}{\partial dx^% {i}\partial dx^{j}},</annotation><annotation encoding="application/x-llamapun" id="S2.E24.m1.4d">italic_g start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT ( italic_x , italic_d italic_x ) = divide start_ARG 1 end_ARG start_ARG 2 end_ARG divide start_ARG ∂ italic_F start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( italic_x , italic_d italic_x ) end_ARG start_ARG ∂ italic_d italic_x start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT ∂ italic_d italic_x start_POSTSUPERSCRIPT italic_j end_POSTSUPERSCRIPT end_ARG ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_left_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(24)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S2.SS3.p2.7">in the tangent space. 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cd="ambiguous" id="S2.SS3.p2.7.m1.1.2.3.5.1.cmml" xref="S2.SS3.p2.7.m1.1.2.3.5">superscript</csymbol><ci id="S2.SS3.p2.7.m1.1.2.3.5.2.cmml" xref="S2.SS3.p2.7.m1.1.2.3.5.2">𝑥</ci><ci id="S2.SS3.p2.7.m1.1.2.3.5.3.cmml" xref="S2.SS3.p2.7.m1.1.2.3.5.3">𝑖</ci></apply><ci id="S2.SS3.p2.7.m1.1.2.3.6.cmml" xref="S2.SS3.p2.7.m1.1.2.3.6">𝑑</ci><apply id="S2.SS3.p2.7.m1.1.2.3.7.cmml" xref="S2.SS3.p2.7.m1.1.2.3.7"><csymbol cd="ambiguous" id="S2.SS3.p2.7.m1.1.2.3.7.1.cmml" xref="S2.SS3.p2.7.m1.1.2.3.7">superscript</csymbol><ci id="S2.SS3.p2.7.m1.1.2.3.7.2.cmml" xref="S2.SS3.p2.7.m1.1.2.3.7.2">𝑥</ci><ci id="S2.SS3.p2.7.m1.1.2.3.7.3.cmml" xref="S2.SS3.p2.7.m1.1.2.3.7.3">𝑗</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p2.7.m1.1c">F^{2}=g_{ij}(x)dx^{i}dx^{j}</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p2.7.m1.1d">italic_F start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT = italic_g start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT ( italic_x ) italic_d italic_x start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT italic_d italic_x start_POSTSUPERSCRIPT italic_j end_POSTSUPERSCRIPT</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S2.SS3.p3"> <p class="ltx_p" id="S2.SS3.p3.3">Randers <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2406.11224v2#bib.bib12" title="">12</a>]</cite> functions were derived from his research on general relativity and have been applied in many fields of sciences. Randers function <math alttext="F(x,dx)" class="ltx_Math" display="inline" id="S2.SS3.p3.1.m1.2"><semantics id="S2.SS3.p3.1.m1.2a"><mrow id="S2.SS3.p3.1.m1.2.2" xref="S2.SS3.p3.1.m1.2.2.cmml"><mi id="S2.SS3.p3.1.m1.2.2.3" xref="S2.SS3.p3.1.m1.2.2.3.cmml">F</mi><mo id="S2.SS3.p3.1.m1.2.2.2" xref="S2.SS3.p3.1.m1.2.2.2.cmml">⁢</mo><mrow id="S2.SS3.p3.1.m1.2.2.1.1" xref="S2.SS3.p3.1.m1.2.2.1.2.cmml"><mo id="S2.SS3.p3.1.m1.2.2.1.1.2" stretchy="false" xref="S2.SS3.p3.1.m1.2.2.1.2.cmml">(</mo><mi id="S2.SS3.p3.1.m1.1.1" xref="S2.SS3.p3.1.m1.1.1.cmml">x</mi><mo id="S2.SS3.p3.1.m1.2.2.1.1.3" xref="S2.SS3.p3.1.m1.2.2.1.2.cmml">,</mo><mrow id="S2.SS3.p3.1.m1.2.2.1.1.1" xref="S2.SS3.p3.1.m1.2.2.1.1.1.cmml"><mi id="S2.SS3.p3.1.m1.2.2.1.1.1.2" xref="S2.SS3.p3.1.m1.2.2.1.1.1.2.cmml">d</mi><mo id="S2.SS3.p3.1.m1.2.2.1.1.1.1" xref="S2.SS3.p3.1.m1.2.2.1.1.1.1.cmml">⁢</mo><mi id="S2.SS3.p3.1.m1.2.2.1.1.1.3" xref="S2.SS3.p3.1.m1.2.2.1.1.1.3.cmml">x</mi></mrow><mo id="S2.SS3.p3.1.m1.2.2.1.1.4" stretchy="false" xref="S2.SS3.p3.1.m1.2.2.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS3.p3.1.m1.2b"><apply id="S2.SS3.p3.1.m1.2.2.cmml" xref="S2.SS3.p3.1.m1.2.2"><times id="S2.SS3.p3.1.m1.2.2.2.cmml" xref="S2.SS3.p3.1.m1.2.2.2"></times><ci id="S2.SS3.p3.1.m1.2.2.3.cmml" xref="S2.SS3.p3.1.m1.2.2.3">𝐹</ci><interval closure="open" id="S2.SS3.p3.1.m1.2.2.1.2.cmml" xref="S2.SS3.p3.1.m1.2.2.1.1"><ci id="S2.SS3.p3.1.m1.1.1.cmml" xref="S2.SS3.p3.1.m1.1.1">𝑥</ci><apply id="S2.SS3.p3.1.m1.2.2.1.1.1.cmml" xref="S2.SS3.p3.1.m1.2.2.1.1.1"><times id="S2.SS3.p3.1.m1.2.2.1.1.1.1.cmml" xref="S2.SS3.p3.1.m1.2.2.1.1.1.1"></times><ci id="S2.SS3.p3.1.m1.2.2.1.1.1.2.cmml" xref="S2.SS3.p3.1.m1.2.2.1.1.1.2">𝑑</ci><ci id="S2.SS3.p3.1.m1.2.2.1.1.1.3.cmml" xref="S2.SS3.p3.1.m1.2.2.1.1.1.3">𝑥</ci></apply></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p3.1.m1.2c">F(x,dx)</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p3.1.m1.2d">italic_F ( italic_x , italic_d italic_x )</annotation></semantics></math> is a special class of Finsler function and is composed of a Riemannian line-element <math alttext="\sqrt{a_{ij}(x)dx^{i}dx^{j}}" class="ltx_Math" display="inline" id="S2.SS3.p3.2.m2.1"><semantics id="S2.SS3.p3.2.m2.1a"><msqrt id="S2.SS3.p3.2.m2.1.1" xref="S2.SS3.p3.2.m2.1.1.cmml"><mrow id="S2.SS3.p3.2.m2.1.1.1" xref="S2.SS3.p3.2.m2.1.1.1.cmml"><msub id="S2.SS3.p3.2.m2.1.1.1.3" xref="S2.SS3.p3.2.m2.1.1.1.3.cmml"><mi id="S2.SS3.p3.2.m2.1.1.1.3.2" xref="S2.SS3.p3.2.m2.1.1.1.3.2.cmml">a</mi><mrow id="S2.SS3.p3.2.m2.1.1.1.3.3" xref="S2.SS3.p3.2.m2.1.1.1.3.3.cmml"><mi id="S2.SS3.p3.2.m2.1.1.1.3.3.2" xref="S2.SS3.p3.2.m2.1.1.1.3.3.2.cmml">i</mi><mo id="S2.SS3.p3.2.m2.1.1.1.3.3.1" xref="S2.SS3.p3.2.m2.1.1.1.3.3.1.cmml">⁢</mo><mi id="S2.SS3.p3.2.m2.1.1.1.3.3.3" xref="S2.SS3.p3.2.m2.1.1.1.3.3.3.cmml">j</mi></mrow></msub><mo id="S2.SS3.p3.2.m2.1.1.1.2" xref="S2.SS3.p3.2.m2.1.1.1.2.cmml">⁢</mo><mrow 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xref="S2.SS3.p3.2.m2.1.1.1.8.2">𝑥</ci><ci id="S2.SS3.p3.2.m2.1.1.1.8.3.cmml" xref="S2.SS3.p3.2.m2.1.1.1.8.3">𝑗</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p3.2.m2.1c">\sqrt{a_{ij}(x)dx^{i}dx^{j}}</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p3.2.m2.1d">square-root start_ARG italic_a start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT ( italic_x ) italic_d italic_x start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT italic_d italic_x start_POSTSUPERSCRIPT italic_j end_POSTSUPERSCRIPT end_ARG</annotation></semantics></math> and one-form <math alttext="b_{i}(x)dx^{i}" class="ltx_Math" display="inline" id="S2.SS3.p3.3.m3.1"><semantics id="S2.SS3.p3.3.m3.1a"><mrow id="S2.SS3.p3.3.m3.1.2" xref="S2.SS3.p3.3.m3.1.2.cmml"><msub id="S2.SS3.p3.3.m3.1.2.2" xref="S2.SS3.p3.3.m3.1.2.2.cmml"><mi id="S2.SS3.p3.3.m3.1.2.2.2" xref="S2.SS3.p3.3.m3.1.2.2.2.cmml">b</mi><mi id="S2.SS3.p3.3.m3.1.2.2.3" 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id="S2.E25.m1.3.3.cmml" xref="S2.E25.m1.3.3">𝑥</ci><ci id="S2.E25.m1.4.4.1.1.3.2.4.cmml" xref="S2.E25.m1.4.4.1.1.3.2.4">𝑑</ci><apply id="S2.E25.m1.4.4.1.1.3.2.5.cmml" xref="S2.E25.m1.4.4.1.1.3.2.5"><csymbol cd="ambiguous" id="S2.E25.m1.4.4.1.1.3.2.5.1.cmml" xref="S2.E25.m1.4.4.1.1.3.2.5">superscript</csymbol><ci id="S2.E25.m1.4.4.1.1.3.2.5.2.cmml" xref="S2.E25.m1.4.4.1.1.3.2.5.2">𝑥</ci><ci id="S2.E25.m1.4.4.1.1.3.2.5.3.cmml" xref="S2.E25.m1.4.4.1.1.3.2.5.3">𝑖</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.E25.m1.4c">\displaystyle F(x,dx)=\sqrt{a_{ij}(x)dx^{i}dx^{j}}+b_{i}(x)dx^{i},</annotation><annotation encoding="application/x-llamapun" id="S2.E25.m1.4d">italic_F ( italic_x , italic_d italic_x ) = square-root start_ARG italic_a start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT ( italic_x ) italic_d italic_x start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT italic_d italic_x start_POSTSUPERSCRIPT italic_j end_POSTSUPERSCRIPT end_ARG + italic_b start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_x ) italic_d italic_x start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_left_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(25)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S2.SS3.p3.6">which is homogeneous of first order in <math alttext="dx^{i}" class="ltx_Math" display="inline" id="S2.SS3.p3.4.m1.1"><semantics id="S2.SS3.p3.4.m1.1a"><mrow id="S2.SS3.p3.4.m1.1.1" xref="S2.SS3.p3.4.m1.1.1.cmml"><mi id="S2.SS3.p3.4.m1.1.1.2" xref="S2.SS3.p3.4.m1.1.1.2.cmml">d</mi><mo id="S2.SS3.p3.4.m1.1.1.1" xref="S2.SS3.p3.4.m1.1.1.1.cmml">⁢</mo><msup id="S2.SS3.p3.4.m1.1.1.3" xref="S2.SS3.p3.4.m1.1.1.3.cmml"><mi id="S2.SS3.p3.4.m1.1.1.3.2" xref="S2.SS3.p3.4.m1.1.1.3.2.cmml">x</mi><mi id="S2.SS3.p3.4.m1.1.1.3.3" xref="S2.SS3.p3.4.m1.1.1.3.3.cmml">i</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="S2.SS3.p3.4.m1.1b"><apply id="S2.SS3.p3.4.m1.1.1.cmml" xref="S2.SS3.p3.4.m1.1.1"><times id="S2.SS3.p3.4.m1.1.1.1.cmml" xref="S2.SS3.p3.4.m1.1.1.1"></times><ci id="S2.SS3.p3.4.m1.1.1.2.cmml" xref="S2.SS3.p3.4.m1.1.1.2">𝑑</ci><apply id="S2.SS3.p3.4.m1.1.1.3.cmml" xref="S2.SS3.p3.4.m1.1.1.3"><csymbol cd="ambiguous" id="S2.SS3.p3.4.m1.1.1.3.1.cmml" xref="S2.SS3.p3.4.m1.1.1.3">superscript</csymbol><ci id="S2.SS3.p3.4.m1.1.1.3.2.cmml" xref="S2.SS3.p3.4.m1.1.1.3.2">𝑥</ci><ci id="S2.SS3.p3.4.m1.1.1.3.3.cmml" xref="S2.SS3.p3.4.m1.1.1.3.3">𝑖</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p3.4.m1.1c">dx^{i}</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p3.4.m1.1d">italic_d italic_x start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT</annotation></semantics></math>. For the arc length <math alttext="s" class="ltx_Math" display="inline" id="S2.SS3.p3.5.m2.1"><semantics id="S2.SS3.p3.5.m2.1a"><mi id="S2.SS3.p3.5.m2.1.1" xref="S2.SS3.p3.5.m2.1.1.cmml">s</mi><annotation-xml encoding="MathML-Content" id="S2.SS3.p3.5.m2.1b"><ci id="S2.SS3.p3.5.m2.1.1.cmml" xref="S2.SS3.p3.5.m2.1.1">𝑠</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p3.5.m2.1c">s</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p3.5.m2.1d">italic_s</annotation></semantics></math> of a curve parameterized by <math alttext="\tau" class="ltx_Math" display="inline" id="S2.SS3.p3.6.m3.1"><semantics id="S2.SS3.p3.6.m3.1a"><mi id="S2.SS3.p3.6.m3.1.1" xref="S2.SS3.p3.6.m3.1.1.cmml">τ</mi><annotation-xml encoding="MathML-Content" id="S2.SS3.p3.6.m3.1b"><ci id="S2.SS3.p3.6.m3.1.1.cmml" xref="S2.SS3.p3.6.m3.1.1">𝜏</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p3.6.m3.1c">\tau</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p3.6.m3.1d">italic_τ</annotation></semantics></math> between two points given by</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="S3.EGx26"> <tbody id="S2.E26"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_left_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle s=\int_{A}^{B}F(x,dx)=\int_{A}^{B}L_{\rm RF}\left(x,\frac{dx}{d% \tau}\right)d\tau," class="ltx_Math" display="inline" id="S2.E26.m1.4"><semantics id="S2.E26.m1.4a"><mrow id="S2.E26.m1.4.4.1" xref="S2.E26.m1.4.4.1.1.cmml"><mrow id="S2.E26.m1.4.4.1.1" xref="S2.E26.m1.4.4.1.1.cmml"><mi id="S2.E26.m1.4.4.1.1.3" xref="S2.E26.m1.4.4.1.1.3.cmml">s</mi><mo id="S2.E26.m1.4.4.1.1.4" xref="S2.E26.m1.4.4.1.1.4.cmml">=</mo><mrow id="S2.E26.m1.4.4.1.1.1" xref="S2.E26.m1.4.4.1.1.1.cmml"><mstyle displaystyle="true" id="S2.E26.m1.4.4.1.1.1.2" xref="S2.E26.m1.4.4.1.1.1.2.cmml"><msubsup 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ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(26)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S2.SS3.p3.7">the corresponding RF Lagrangian is</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="S3.EGx27"> <tbody id="S2.E27"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_left_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle L_{\rm RF}\!\left(x,\frac{dx}{d\tau}\right)\!=\!\sqrt{a_{ij}(x)% \frac{dx^{i}}{d\tau}\frac{dx^{j}}{d\tau}}\!+\!b_{i}(x)\frac{dx^{i}}{d\tau}." class="ltx_Math" display="inline" id="S2.E27.m1.5"><semantics id="S2.E27.m1.5a"><mrow id="S2.E27.m1.5.5.1" xref="S2.E27.m1.5.5.1.1.cmml"><mrow id="S2.E27.m1.5.5.1.1" xref="S2.E27.m1.5.5.1.1.cmml"><mrow id="S2.E27.m1.5.5.1.1.2" xref="S2.E27.m1.5.5.1.1.2.cmml"><msub id="S2.E27.m1.5.5.1.1.2.2" xref="S2.E27.m1.5.5.1.1.2.2.cmml"><mi 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start_ARG italic_d italic_x end_ARG start_ARG italic_d italic_τ end_ARG ) = square-root start_ARG italic_a start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT ( italic_x ) divide start_ARG italic_d italic_x start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT end_ARG start_ARG italic_d italic_τ end_ARG divide start_ARG italic_d italic_x start_POSTSUPERSCRIPT italic_j end_POSTSUPERSCRIPT end_ARG start_ARG italic_d italic_τ end_ARG end_ARG + italic_b start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_x ) divide start_ARG italic_d italic_x start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT end_ARG start_ARG italic_d italic_τ end_ARG .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_left_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(27)</span></td> </tr></tbody> </table> </div> <div class="ltx_para" id="S2.SS3.p4"> <p class="ltx_p" id="S2.SS3.p4.2">In Ref. <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2406.11224v2#bib.bib11" title="">11</a>]</cite>, based on the Randers functions, the gradient-flow equations with respect to the <math alttext="\theta" class="ltx_Math" display="inline" id="S2.SS3.p4.1.m1.1"><semantics id="S2.SS3.p4.1.m1.1a"><mi id="S2.SS3.p4.1.m1.1.1" xref="S2.SS3.p4.1.m1.1.1.cmml">θ</mi><annotation-xml encoding="MathML-Content" id="S2.SS3.p4.1.m1.1b"><ci id="S2.SS3.p4.1.m1.1.1.cmml" xref="S2.SS3.p4.1.m1.1.1">𝜃</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p4.1.m1.1c">\theta</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p4.1.m1.1d">italic_θ</annotation></semantics></math>-potential function <math alttext="\Psi(\theta)" class="ltx_Math" display="inline" id="S2.SS3.p4.2.m2.1"><semantics id="S2.SS3.p4.2.m2.1a"><mrow id="S2.SS3.p4.2.m2.1.2" xref="S2.SS3.p4.2.m2.1.2.cmml"><mi id="S2.SS3.p4.2.m2.1.2.2" mathvariant="normal" 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were deformed as</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="S3.EGx28"> <tbody id="S2.Ex3a"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_left_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\frac{d\theta^{i}}{dt}" class="ltx_Math" display="inline" id="S2.Ex3a.m1.1"><semantics id="S2.Ex3a.m1.1a"><mstyle displaystyle="true" id="S2.Ex3a.m1.1.1" xref="S2.Ex3a.m1.1.1.cmml"><mfrac id="S2.Ex3a.m1.1.1a" xref="S2.Ex3a.m1.1.1.cmml"><mrow id="S2.Ex3a.m1.1.1.2" xref="S2.Ex3a.m1.1.1.2.cmml"><mi id="S2.Ex3a.m1.1.1.2.2" xref="S2.Ex3a.m1.1.1.2.2.cmml">d</mi><mo id="S2.Ex3a.m1.1.1.2.1" xref="S2.Ex3a.m1.1.1.2.1.cmml">⁢</mo><msup id="S2.Ex3a.m1.1.1.2.3" xref="S2.Ex3a.m1.1.1.2.3.cmml"><mi id="S2.Ex3a.m1.1.1.2.3.2" xref="S2.Ex3a.m1.1.1.2.3.2.cmml">θ</mi><mi id="S2.Ex3a.m1.1.1.2.3.3" xref="S2.Ex3a.m1.1.1.2.3.3.cmml">i</mi></msup></mrow><mrow id="S2.Ex3a.m1.1.1.3" xref="S2.Ex3a.m1.1.1.3.cmml"><mi 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xref="S2.Ex3a.m1.1.1.3.1"></times><ci id="S2.Ex3a.m1.1.1.3.2.cmml" xref="S2.Ex3a.m1.1.1.3.2">𝑑</ci><ci id="S2.Ex3a.m1.1.1.3.3.cmml" xref="S2.Ex3a.m1.1.1.3.3">𝑡</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Ex3a.m1.1c">\displaystyle\frac{d\theta^{i}}{dt}</annotation><annotation encoding="application/x-llamapun" id="S2.Ex3a.m1.1d">divide start_ARG italic_d italic_θ start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT end_ARG start_ARG italic_d italic_t end_ARG</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle=g^{ij}(\theta)\left(\frac{\partial\Psi(\theta)}{\partial\theta^{% j}}-A_{j}(\theta)\right)" class="ltx_Math" display="inline" id="S2.Ex3a.m2.4"><semantics id="S2.Ex3a.m2.4a"><mrow id="S2.Ex3a.m2.4.4" xref="S2.Ex3a.m2.4.4.cmml"><mi id="S2.Ex3a.m2.4.4.3" xref="S2.Ex3a.m2.4.4.3.cmml"></mi><mo id="S2.Ex3a.m2.4.4.2" xref="S2.Ex3a.m2.4.4.2.cmml">=</mo><mrow id="S2.Ex3a.m2.4.4.1" 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id="S2.E28.m1.3.3.1.1.1.1.1.1.3.2.1.cmml" xref="S2.E28.m1.3.3.1.1.1.1.1.1.3.2">subscript</csymbol><ci id="S2.E28.m1.3.3.1.1.1.1.1.1.3.2.2.cmml" xref="S2.E28.m1.3.3.1.1.1.1.1.1.3.2.2">𝐴</ci><ci id="S2.E28.m1.3.3.1.1.1.1.1.1.3.2.3.cmml" xref="S2.E28.m1.3.3.1.1.1.1.1.1.3.2.3">𝑗</ci></apply><ci id="S2.E28.m1.2.2.cmml" xref="S2.E28.m1.2.2">𝜃</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.E28.m1.3c">\displaystyle=g^{ij}(\theta)\Big{(}\eta_{j}-A_{j}(\theta)\Big{)},</annotation><annotation encoding="application/x-llamapun" id="S2.E28.m1.3d">= italic_g start_POSTSUPERSCRIPT italic_i italic_j end_POSTSUPERSCRIPT ( italic_θ ) ( italic_η start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT - italic_A start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ( italic_θ ) ) ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_left_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(28)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S2.SS3.p4.6">where each <math alttext="A_{j}(\theta)" class="ltx_Math" display="inline" id="S2.SS3.p4.3.m1.1"><semantics id="S2.SS3.p4.3.m1.1a"><mrow id="S2.SS3.p4.3.m1.1.2" xref="S2.SS3.p4.3.m1.1.2.cmml"><msub id="S2.SS3.p4.3.m1.1.2.2" xref="S2.SS3.p4.3.m1.1.2.2.cmml"><mi id="S2.SS3.p4.3.m1.1.2.2.2" xref="S2.SS3.p4.3.m1.1.2.2.2.cmml">A</mi><mi id="S2.SS3.p4.3.m1.1.2.2.3" xref="S2.SS3.p4.3.m1.1.2.2.3.cmml">j</mi></msub><mo id="S2.SS3.p4.3.m1.1.2.1" xref="S2.SS3.p4.3.m1.1.2.1.cmml">⁢</mo><mrow id="S2.SS3.p4.3.m1.1.2.3.2" xref="S2.SS3.p4.3.m1.1.2.cmml"><mo id="S2.SS3.p4.3.m1.1.2.3.2.1" stretchy="false" xref="S2.SS3.p4.3.m1.1.2.cmml">(</mo><mi id="S2.SS3.p4.3.m1.1.1" xref="S2.SS3.p4.3.m1.1.1.cmml">θ</mi><mo id="S2.SS3.p4.3.m1.1.2.3.2.2" stretchy="false" xref="S2.SS3.p4.3.m1.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS3.p4.3.m1.1b"><apply id="S2.SS3.p4.3.m1.1.2.cmml" xref="S2.SS3.p4.3.m1.1.2"><times id="S2.SS3.p4.3.m1.1.2.1.cmml" xref="S2.SS3.p4.3.m1.1.2.1"></times><apply id="S2.SS3.p4.3.m1.1.2.2.cmml" xref="S2.SS3.p4.3.m1.1.2.2"><csymbol cd="ambiguous" id="S2.SS3.p4.3.m1.1.2.2.1.cmml" xref="S2.SS3.p4.3.m1.1.2.2">subscript</csymbol><ci id="S2.SS3.p4.3.m1.1.2.2.2.cmml" xref="S2.SS3.p4.3.m1.1.2.2.2">𝐴</ci><ci id="S2.SS3.p4.3.m1.1.2.2.3.cmml" xref="S2.SS3.p4.3.m1.1.2.2.3">𝑗</ci></apply><ci id="S2.SS3.p4.3.m1.1.1.cmml" xref="S2.SS3.p4.3.m1.1.1">𝜃</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p4.3.m1.1c">A_{j}(\theta)</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p4.3.m1.1d">italic_A start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ( italic_θ )</annotation></semantics></math> denotes a function of <math alttext="\theta" class="ltx_Math" display="inline" id="S2.SS3.p4.4.m2.1"><semantics id="S2.SS3.p4.4.m2.1a"><mi id="S2.SS3.p4.4.m2.1.1" xref="S2.SS3.p4.4.m2.1.1.cmml">θ</mi><annotation-xml encoding="MathML-Content" id="S2.SS3.p4.4.m2.1b"><ci id="S2.SS3.p4.4.m2.1.1.cmml" xref="S2.SS3.p4.4.m2.1.1">𝜃</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p4.4.m2.1c">\theta</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p4.4.m2.1d">italic_θ</annotation></semantics></math> due to this deformation. It is worth noting that the gradient-flow equations (<a class="ltx_ref" href="https://arxiv.org/html/2406.11224v2#S2.E12" title="In 2.2 Gradient-Flow Equations ‣ 2 Information Geometry and Gradient-Flow Equations ‣ A Hamiltonian approach to the gradient-flow equations in information geometry"><span class="ltx_text ltx_ref_tag">12</span></a>) with respect to the divergence <math alttext="D(\theta,\theta_{r})" class="ltx_Math" display="inline" id="S2.SS3.p4.5.m3.2"><semantics id="S2.SS3.p4.5.m3.2a"><mrow id="S2.SS3.p4.5.m3.2.2" xref="S2.SS3.p4.5.m3.2.2.cmml"><mi id="S2.SS3.p4.5.m3.2.2.3" xref="S2.SS3.p4.5.m3.2.2.3.cmml">D</mi><mo id="S2.SS3.p4.5.m3.2.2.2" xref="S2.SS3.p4.5.m3.2.2.2.cmml">⁢</mo><mrow id="S2.SS3.p4.5.m3.2.2.1.1" xref="S2.SS3.p4.5.m3.2.2.1.2.cmml"><mo id="S2.SS3.p4.5.m3.2.2.1.1.2" stretchy="false" xref="S2.SS3.p4.5.m3.2.2.1.2.cmml">(</mo><mi id="S2.SS3.p4.5.m3.1.1" xref="S2.SS3.p4.5.m3.1.1.cmml">θ</mi><mo id="S2.SS3.p4.5.m3.2.2.1.1.3" xref="S2.SS3.p4.5.m3.2.2.1.2.cmml">,</mo><msub id="S2.SS3.p4.5.m3.2.2.1.1.1" xref="S2.SS3.p4.5.m3.2.2.1.1.1.cmml"><mi id="S2.SS3.p4.5.m3.2.2.1.1.1.2" xref="S2.SS3.p4.5.m3.2.2.1.1.1.2.cmml">θ</mi><mi id="S2.SS3.p4.5.m3.2.2.1.1.1.3" xref="S2.SS3.p4.5.m3.2.2.1.1.1.3.cmml">r</mi></msub><mo id="S2.SS3.p4.5.m3.2.2.1.1.4" stretchy="false" xref="S2.SS3.p4.5.m3.2.2.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS3.p4.5.m3.2b"><apply id="S2.SS3.p4.5.m3.2.2.cmml" xref="S2.SS3.p4.5.m3.2.2"><times id="S2.SS3.p4.5.m3.2.2.2.cmml" xref="S2.SS3.p4.5.m3.2.2.2"></times><ci id="S2.SS3.p4.5.m3.2.2.3.cmml" xref="S2.SS3.p4.5.m3.2.2.3">𝐷</ci><interval closure="open" id="S2.SS3.p4.5.m3.2.2.1.2.cmml" xref="S2.SS3.p4.5.m3.2.2.1.1"><ci id="S2.SS3.p4.5.m3.1.1.cmml" xref="S2.SS3.p4.5.m3.1.1">𝜃</ci><apply id="S2.SS3.p4.5.m3.2.2.1.1.1.cmml" xref="S2.SS3.p4.5.m3.2.2.1.1.1"><csymbol cd="ambiguous" id="S2.SS3.p4.5.m3.2.2.1.1.1.1.cmml" 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id="S2.SS3.p4.6.m4.1.2.3.cmml" xref="S2.SS3.p4.6.m4.1.2.3"><csymbol cd="ambiguous" id="S2.SS3.p4.6.m4.1.2.3.1.cmml" xref="S2.SS3.p4.6.m4.1.2.3">subscript</csymbol><apply id="S2.SS3.p4.6.m4.1.2.3.2.cmml" xref="S2.SS3.p4.6.m4.1.2.3"><csymbol cd="ambiguous" id="S2.SS3.p4.6.m4.1.2.3.2.1.cmml" xref="S2.SS3.p4.6.m4.1.2.3">superscript</csymbol><ci id="S2.SS3.p4.6.m4.1.2.3.2.2.cmml" xref="S2.SS3.p4.6.m4.1.2.3.2.2">𝜂</ci><ci id="S2.SS3.p4.6.m4.1.2.3.2.3.cmml" xref="S2.SS3.p4.6.m4.1.2.3.2.3">r</ci></apply><ci id="S2.SS3.p4.6.m4.1.2.3.3.cmml" xref="S2.SS3.p4.6.m4.1.2.3.3">𝑗</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p4.6.m4.1c">A_{j}(\theta)=\eta^{\rm r}_{j}</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p4.6.m4.1d">italic_A start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ( italic_θ ) = italic_η start_POSTSUPERSCRIPT roman_r end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S2.SS3.p5"> <p class="ltx_p" id="S2.SS3.p5.1">Now introducing the quantity <math alttext="\chi^{2}(\theta)" class="ltx_Math" display="inline" id="S2.SS3.p5.1.m1.1"><semantics id="S2.SS3.p5.1.m1.1a"><mrow id="S2.SS3.p5.1.m1.1.2" xref="S2.SS3.p5.1.m1.1.2.cmml"><msup id="S2.SS3.p5.1.m1.1.2.2" xref="S2.SS3.p5.1.m1.1.2.2.cmml"><mi id="S2.SS3.p5.1.m1.1.2.2.2" xref="S2.SS3.p5.1.m1.1.2.2.2.cmml">χ</mi><mn id="S2.SS3.p5.1.m1.1.2.2.3" xref="S2.SS3.p5.1.m1.1.2.2.3.cmml">2</mn></msup><mo id="S2.SS3.p5.1.m1.1.2.1" xref="S2.SS3.p5.1.m1.1.2.1.cmml">⁢</mo><mrow id="S2.SS3.p5.1.m1.1.2.3.2" xref="S2.SS3.p5.1.m1.1.2.cmml"><mo id="S2.SS3.p5.1.m1.1.2.3.2.1" stretchy="false" xref="S2.SS3.p5.1.m1.1.2.cmml">(</mo><mi id="S2.SS3.p5.1.m1.1.1" xref="S2.SS3.p5.1.m1.1.1.cmml">θ</mi><mo id="S2.SS3.p5.1.m1.1.2.3.2.2" stretchy="false" xref="S2.SS3.p5.1.m1.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS3.p5.1.m1.1b"><apply id="S2.SS3.p5.1.m1.1.2.cmml" xref="S2.SS3.p5.1.m1.1.2"><times id="S2.SS3.p5.1.m1.1.2.1.cmml" xref="S2.SS3.p5.1.m1.1.2.1"></times><apply id="S2.SS3.p5.1.m1.1.2.2.cmml" xref="S2.SS3.p5.1.m1.1.2.2"><csymbol cd="ambiguous" id="S2.SS3.p5.1.m1.1.2.2.1.cmml" xref="S2.SS3.p5.1.m1.1.2.2">superscript</csymbol><ci id="S2.SS3.p5.1.m1.1.2.2.2.cmml" xref="S2.SS3.p5.1.m1.1.2.2.2">𝜒</ci><cn id="S2.SS3.p5.1.m1.1.2.2.3.cmml" type="integer" xref="S2.SS3.p5.1.m1.1.2.2.3">2</cn></apply><ci id="S2.SS3.p5.1.m1.1.1.cmml" xref="S2.SS3.p5.1.m1.1.1">𝜃</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p5.1.m1.1c">\chi^{2}(\theta)</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p5.1.m1.1d">italic_χ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( italic_θ )</annotation></semantics></math> as</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="S3.EGx29"> <tbody id="S2.E29"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_left_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\chi^{2}(\theta):=g_{ij}(\theta)\frac{d\theta^{i}}{dt}\frac{d% \theta^{j}}{dt}," class="ltx_Math" display="inline" id="S2.E29.m1.3"><semantics id="S2.E29.m1.3a"><mrow id="S2.E29.m1.3.3.1" xref="S2.E29.m1.3.3.1.1.cmml"><mrow id="S2.E29.m1.3.3.1.1" xref="S2.E29.m1.3.3.1.1.cmml"><mrow id="S2.E29.m1.3.3.1.1.2" xref="S2.E29.m1.3.3.1.1.2.cmml"><msup id="S2.E29.m1.3.3.1.1.2.2" xref="S2.E29.m1.3.3.1.1.2.2.cmml"><mi id="S2.E29.m1.3.3.1.1.2.2.2" xref="S2.E29.m1.3.3.1.1.2.2.2.cmml">χ</mi><mn id="S2.E29.m1.3.3.1.1.2.2.3" xref="S2.E29.m1.3.3.1.1.2.2.3.cmml">2</mn></msup><mo id="S2.E29.m1.3.3.1.1.2.1" xref="S2.E29.m1.3.3.1.1.2.1.cmml">⁢</mo><mrow id="S2.E29.m1.3.3.1.1.2.3.2" xref="S2.E29.m1.3.3.1.1.2.cmml"><mo id="S2.E29.m1.3.3.1.1.2.3.2.1" stretchy="false" xref="S2.E29.m1.3.3.1.1.2.cmml">(</mo><mi id="S2.E29.m1.1.1" xref="S2.E29.m1.1.1.cmml">θ</mi><mo id="S2.E29.m1.3.3.1.1.2.3.2.2" rspace="0.278em" stretchy="false" xref="S2.E29.m1.3.3.1.1.2.cmml">)</mo></mrow></mrow><mo id="S2.E29.m1.3.3.1.1.1" rspace="0.278em" xref="S2.E29.m1.3.3.1.1.1.cmml">:=</mo><mrow id="S2.E29.m1.3.3.1.1.3" xref="S2.E29.m1.3.3.1.1.3.cmml"><msub id="S2.E29.m1.3.3.1.1.3.2" xref="S2.E29.m1.3.3.1.1.3.2.cmml"><mi id="S2.E29.m1.3.3.1.1.3.2.2" xref="S2.E29.m1.3.3.1.1.3.2.2.cmml">g</mi><mrow id="S2.E29.m1.3.3.1.1.3.2.3" xref="S2.E29.m1.3.3.1.1.3.2.3.cmml"><mi id="S2.E29.m1.3.3.1.1.3.2.3.2" xref="S2.E29.m1.3.3.1.1.3.2.3.2.cmml">i</mi><mo id="S2.E29.m1.3.3.1.1.3.2.3.1" xref="S2.E29.m1.3.3.1.1.3.2.3.1.cmml">⁢</mo><mi id="S2.E29.m1.3.3.1.1.3.2.3.3" xref="S2.E29.m1.3.3.1.1.3.2.3.3.cmml">j</mi></mrow></msub><mo id="S2.E29.m1.3.3.1.1.3.1" xref="S2.E29.m1.3.3.1.1.3.1.cmml">⁢</mo><mrow id="S2.E29.m1.3.3.1.1.3.3.2" xref="S2.E29.m1.3.3.1.1.3.cmml"><mo id="S2.E29.m1.3.3.1.1.3.3.2.1" stretchy="false" xref="S2.E29.m1.3.3.1.1.3.cmml">(</mo><mi id="S2.E29.m1.2.2" xref="S2.E29.m1.2.2.cmml">θ</mi><mo 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id="S2.E29.m1.3.3.1.1.3.5.cmml" xref="S2.E29.m1.3.3.1.1.3.5"><divide id="S2.E29.m1.3.3.1.1.3.5.1.cmml" xref="S2.E29.m1.3.3.1.1.3.5"></divide><apply id="S2.E29.m1.3.3.1.1.3.5.2.cmml" xref="S2.E29.m1.3.3.1.1.3.5.2"><times id="S2.E29.m1.3.3.1.1.3.5.2.1.cmml" xref="S2.E29.m1.3.3.1.1.3.5.2.1"></times><ci id="S2.E29.m1.3.3.1.1.3.5.2.2.cmml" xref="S2.E29.m1.3.3.1.1.3.5.2.2">𝑑</ci><apply id="S2.E29.m1.3.3.1.1.3.5.2.3.cmml" xref="S2.E29.m1.3.3.1.1.3.5.2.3"><csymbol cd="ambiguous" id="S2.E29.m1.3.3.1.1.3.5.2.3.1.cmml" xref="S2.E29.m1.3.3.1.1.3.5.2.3">superscript</csymbol><ci id="S2.E29.m1.3.3.1.1.3.5.2.3.2.cmml" xref="S2.E29.m1.3.3.1.1.3.5.2.3.2">𝜃</ci><ci id="S2.E29.m1.3.3.1.1.3.5.2.3.3.cmml" xref="S2.E29.m1.3.3.1.1.3.5.2.3.3">𝑗</ci></apply></apply><apply id="S2.E29.m1.3.3.1.1.3.5.3.cmml" xref="S2.E29.m1.3.3.1.1.3.5.3"><times id="S2.E29.m1.3.3.1.1.3.5.3.1.cmml" xref="S2.E29.m1.3.3.1.1.3.5.3.1"></times><ci id="S2.E29.m1.3.3.1.1.3.5.3.2.cmml" xref="S2.E29.m1.3.3.1.1.3.5.3.2">𝑑</ci><ci id="S2.E29.m1.3.3.1.1.3.5.3.3.cmml" xref="S2.E29.m1.3.3.1.1.3.5.3.3">𝑡</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.E29.m1.3c">\displaystyle\chi^{2}(\theta):=g_{ij}(\theta)\frac{d\theta^{i}}{dt}\frac{d% \theta^{j}}{dt},</annotation><annotation encoding="application/x-llamapun" id="S2.E29.m1.3d">italic_χ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( italic_θ ) := italic_g start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT ( italic_θ ) divide start_ARG italic_d italic_θ start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT end_ARG start_ARG italic_d italic_t end_ARG divide start_ARG italic_d italic_θ start_POSTSUPERSCRIPT italic_j end_POSTSUPERSCRIPT end_ARG start_ARG italic_d italic_t end_ARG ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_left_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(29)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S2.SS3.p5.2">which is the deformation of <math alttext="\eta^{2}(\theta)" class="ltx_Math" display="inline" id="S2.SS3.p5.2.m1.1"><semantics id="S2.SS3.p5.2.m1.1a"><mrow id="S2.SS3.p5.2.m1.1.2" xref="S2.SS3.p5.2.m1.1.2.cmml"><msup id="S2.SS3.p5.2.m1.1.2.2" xref="S2.SS3.p5.2.m1.1.2.2.cmml"><mi id="S2.SS3.p5.2.m1.1.2.2.2" xref="S2.SS3.p5.2.m1.1.2.2.2.cmml">η</mi><mn id="S2.SS3.p5.2.m1.1.2.2.3" xref="S2.SS3.p5.2.m1.1.2.2.3.cmml">2</mn></msup><mo id="S2.SS3.p5.2.m1.1.2.1" xref="S2.SS3.p5.2.m1.1.2.1.cmml">⁢</mo><mrow id="S2.SS3.p5.2.m1.1.2.3.2" xref="S2.SS3.p5.2.m1.1.2.cmml"><mo id="S2.SS3.p5.2.m1.1.2.3.2.1" stretchy="false" xref="S2.SS3.p5.2.m1.1.2.cmml">(</mo><mi id="S2.SS3.p5.2.m1.1.1" xref="S2.SS3.p5.2.m1.1.1.cmml">θ</mi><mo id="S2.SS3.p5.2.m1.1.2.3.2.2" stretchy="false" xref="S2.SS3.p5.2.m1.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS3.p5.2.m1.1b"><apply id="S2.SS3.p5.2.m1.1.2.cmml" xref="S2.SS3.p5.2.m1.1.2"><times id="S2.SS3.p5.2.m1.1.2.1.cmml" xref="S2.SS3.p5.2.m1.1.2.1"></times><apply id="S2.SS3.p5.2.m1.1.2.2.cmml" xref="S2.SS3.p5.2.m1.1.2.2"><csymbol cd="ambiguous" id="S2.SS3.p5.2.m1.1.2.2.1.cmml" xref="S2.SS3.p5.2.m1.1.2.2">superscript</csymbol><ci id="S2.SS3.p5.2.m1.1.2.2.2.cmml" xref="S2.SS3.p5.2.m1.1.2.2.2">𝜂</ci><cn id="S2.SS3.p5.2.m1.1.2.2.3.cmml" type="integer" xref="S2.SS3.p5.2.m1.1.2.2.3">2</cn></apply><ci id="S2.SS3.p5.2.m1.1.1.cmml" xref="S2.SS3.p5.2.m1.1.1">𝜃</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p5.2.m1.1c">\eta^{2}(\theta)</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p5.2.m1.1d">italic_η start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( italic_θ )</annotation></semantics></math> in (<a class="ltx_ref" href="https://arxiv.org/html/2406.11224v2#S2.E20.1" title="In 20 ‣ 2.2 Gradient-Flow Equations ‣ 2 Information Geometry and Gradient-Flow Equations ‣ A Hamiltonian approach to the gradient-flow equations in information geometry"><span class="ltx_text ltx_ref_tag">20a</span></a>). Indeed, by utilizing (<a class="ltx_ref" href="https://arxiv.org/html/2406.11224v2#S2.E28" title="In 2.3 Randers-Finsler deformation of the gradient-flow equations ‣ 2 Information Geometry and Gradient-Flow Equations ‣ A Hamiltonian approach to the gradient-flow equations in information geometry"><span class="ltx_text ltx_ref_tag">28</span></a>), we see that</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="S3.EGx30"> <tbody id="S2.E30"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_left_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\chi^{2}(\theta)=g^{ij}(\theta)\Big{(}\eta_{i}-A_{i}(\theta)\Big{% )}\Big{(}\eta_{j}-A_{j}(\theta)\Big{)}," class="ltx_Math" display="inline" id="S2.E30.m1.5"><semantics id="S2.E30.m1.5a"><mrow id="S2.E30.m1.5.5.1" xref="S2.E30.m1.5.5.1.1.cmml"><mrow id="S2.E30.m1.5.5.1.1" xref="S2.E30.m1.5.5.1.1.cmml"><mrow id="S2.E30.m1.5.5.1.1.4" 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id="S2.E30.m1.5.5.1.1.2.4.2.cmml" xref="S2.E30.m1.5.5.1.1.2.4.2">𝑔</ci><apply id="S2.E30.m1.5.5.1.1.2.4.3.cmml" xref="S2.E30.m1.5.5.1.1.2.4.3"><times id="S2.E30.m1.5.5.1.1.2.4.3.1.cmml" xref="S2.E30.m1.5.5.1.1.2.4.3.1"></times><ci id="S2.E30.m1.5.5.1.1.2.4.3.2.cmml" xref="S2.E30.m1.5.5.1.1.2.4.3.2">𝑖</ci><ci id="S2.E30.m1.5.5.1.1.2.4.3.3.cmml" xref="S2.E30.m1.5.5.1.1.2.4.3.3">𝑗</ci></apply></apply><ci id="S2.E30.m1.2.2.cmml" xref="S2.E30.m1.2.2">𝜃</ci><apply id="S2.E30.m1.5.5.1.1.1.1.1.1.cmml" xref="S2.E30.m1.5.5.1.1.1.1.1"><minus id="S2.E30.m1.5.5.1.1.1.1.1.1.1.cmml" xref="S2.E30.m1.5.5.1.1.1.1.1.1.1"></minus><apply id="S2.E30.m1.5.5.1.1.1.1.1.1.2.cmml" xref="S2.E30.m1.5.5.1.1.1.1.1.1.2"><csymbol cd="ambiguous" id="S2.E30.m1.5.5.1.1.1.1.1.1.2.1.cmml" xref="S2.E30.m1.5.5.1.1.1.1.1.1.2">subscript</csymbol><ci id="S2.E30.m1.5.5.1.1.1.1.1.1.2.2.cmml" xref="S2.E30.m1.5.5.1.1.1.1.1.1.2.2">𝜂</ci><ci id="S2.E30.m1.5.5.1.1.1.1.1.1.2.3.cmml" xref="S2.E30.m1.5.5.1.1.1.1.1.1.2.3">𝑖</ci></apply><apply id="S2.E30.m1.5.5.1.1.1.1.1.1.3.cmml" xref="S2.E30.m1.5.5.1.1.1.1.1.1.3"><times id="S2.E30.m1.5.5.1.1.1.1.1.1.3.1.cmml" xref="S2.E30.m1.5.5.1.1.1.1.1.1.3.1"></times><apply id="S2.E30.m1.5.5.1.1.1.1.1.1.3.2.cmml" xref="S2.E30.m1.5.5.1.1.1.1.1.1.3.2"><csymbol cd="ambiguous" id="S2.E30.m1.5.5.1.1.1.1.1.1.3.2.1.cmml" xref="S2.E30.m1.5.5.1.1.1.1.1.1.3.2">subscript</csymbol><ci id="S2.E30.m1.5.5.1.1.1.1.1.1.3.2.2.cmml" xref="S2.E30.m1.5.5.1.1.1.1.1.1.3.2.2">𝐴</ci><ci id="S2.E30.m1.5.5.1.1.1.1.1.1.3.2.3.cmml" xref="S2.E30.m1.5.5.1.1.1.1.1.1.3.2.3">𝑖</ci></apply><ci id="S2.E30.m1.3.3.cmml" xref="S2.E30.m1.3.3">𝜃</ci></apply></apply><apply id="S2.E30.m1.5.5.1.1.2.2.1.1.cmml" xref="S2.E30.m1.5.5.1.1.2.2.1"><minus id="S2.E30.m1.5.5.1.1.2.2.1.1.1.cmml" xref="S2.E30.m1.5.5.1.1.2.2.1.1.1"></minus><apply id="S2.E30.m1.5.5.1.1.2.2.1.1.2.cmml" xref="S2.E30.m1.5.5.1.1.2.2.1.1.2"><csymbol cd="ambiguous" id="S2.E30.m1.5.5.1.1.2.2.1.1.2.1.cmml" 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id="S2.E30.m1.5c">\displaystyle\chi^{2}(\theta)=g^{ij}(\theta)\Big{(}\eta_{i}-A_{i}(\theta)\Big{% )}\Big{(}\eta_{j}-A_{j}(\theta)\Big{)},</annotation><annotation encoding="application/x-llamapun" id="S2.E30.m1.5d">italic_χ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( italic_θ ) = italic_g start_POSTSUPERSCRIPT italic_i italic_j end_POSTSUPERSCRIPT ( italic_θ ) ( italic_η start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT - italic_A start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_θ ) ) ( italic_η start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT - italic_A start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ( italic_θ ) ) ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_left_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(30)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S2.SS3.p5.7">which reduces to <math alttext="\eta^{2}(\theta)" class="ltx_Math" display="inline" id="S2.SS3.p5.3.m1.1"><semantics id="S2.SS3.p5.3.m1.1a"><mrow id="S2.SS3.p5.3.m1.1.2" xref="S2.SS3.p5.3.m1.1.2.cmml"><msup id="S2.SS3.p5.3.m1.1.2.2" xref="S2.SS3.p5.3.m1.1.2.2.cmml"><mi id="S2.SS3.p5.3.m1.1.2.2.2" xref="S2.SS3.p5.3.m1.1.2.2.2.cmml">η</mi><mn id="S2.SS3.p5.3.m1.1.2.2.3" xref="S2.SS3.p5.3.m1.1.2.2.3.cmml">2</mn></msup><mo id="S2.SS3.p5.3.m1.1.2.1" xref="S2.SS3.p5.3.m1.1.2.1.cmml">⁢</mo><mrow id="S2.SS3.p5.3.m1.1.2.3.2" xref="S2.SS3.p5.3.m1.1.2.cmml"><mo id="S2.SS3.p5.3.m1.1.2.3.2.1" stretchy="false" xref="S2.SS3.p5.3.m1.1.2.cmml">(</mo><mi id="S2.SS3.p5.3.m1.1.1" xref="S2.SS3.p5.3.m1.1.1.cmml">θ</mi><mo id="S2.SS3.p5.3.m1.1.2.3.2.2" stretchy="false" xref="S2.SS3.p5.3.m1.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS3.p5.3.m1.1b"><apply id="S2.SS3.p5.3.m1.1.2.cmml" xref="S2.SS3.p5.3.m1.1.2"><times id="S2.SS3.p5.3.m1.1.2.1.cmml" xref="S2.SS3.p5.3.m1.1.2.1"></times><apply id="S2.SS3.p5.3.m1.1.2.2.cmml" xref="S2.SS3.p5.3.m1.1.2.2"><csymbol cd="ambiguous" id="S2.SS3.p5.3.m1.1.2.2.1.cmml" xref="S2.SS3.p5.3.m1.1.2.2">superscript</csymbol><ci id="S2.SS3.p5.3.m1.1.2.2.2.cmml" xref="S2.SS3.p5.3.m1.1.2.2.2">𝜂</ci><cn id="S2.SS3.p5.3.m1.1.2.2.3.cmml" type="integer" xref="S2.SS3.p5.3.m1.1.2.2.3">2</cn></apply><ci id="S2.SS3.p5.3.m1.1.1.cmml" xref="S2.SS3.p5.3.m1.1.1">𝜃</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p5.3.m1.1c">\eta^{2}(\theta)</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p5.3.m1.1d">italic_η start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( italic_θ )</annotation></semantics></math> when <math alttext="A_{j}(\theta)\to 0" class="ltx_Math" display="inline" id="S2.SS3.p5.4.m2.1"><semantics id="S2.SS3.p5.4.m2.1a"><mrow id="S2.SS3.p5.4.m2.1.2" xref="S2.SS3.p5.4.m2.1.2.cmml"><mrow id="S2.SS3.p5.4.m2.1.2.2" xref="S2.SS3.p5.4.m2.1.2.2.cmml"><msub id="S2.SS3.p5.4.m2.1.2.2.2" xref="S2.SS3.p5.4.m2.1.2.2.2.cmml"><mi id="S2.SS3.p5.4.m2.1.2.2.2.2" xref="S2.SS3.p5.4.m2.1.2.2.2.2.cmml">A</mi><mi id="S2.SS3.p5.4.m2.1.2.2.2.3" xref="S2.SS3.p5.4.m2.1.2.2.2.3.cmml">j</mi></msub><mo id="S2.SS3.p5.4.m2.1.2.2.1" xref="S2.SS3.p5.4.m2.1.2.2.1.cmml">⁢</mo><mrow id="S2.SS3.p5.4.m2.1.2.2.3.2" xref="S2.SS3.p5.4.m2.1.2.2.cmml"><mo id="S2.SS3.p5.4.m2.1.2.2.3.2.1" stretchy="false" xref="S2.SS3.p5.4.m2.1.2.2.cmml">(</mo><mi id="S2.SS3.p5.4.m2.1.1" xref="S2.SS3.p5.4.m2.1.1.cmml">θ</mi><mo id="S2.SS3.p5.4.m2.1.2.2.3.2.2" stretchy="false" xref="S2.SS3.p5.4.m2.1.2.2.cmml">)</mo></mrow></mrow><mo id="S2.SS3.p5.4.m2.1.2.1" stretchy="false" xref="S2.SS3.p5.4.m2.1.2.1.cmml">→</mo><mn id="S2.SS3.p5.4.m2.1.2.3" xref="S2.SS3.p5.4.m2.1.2.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.SS3.p5.4.m2.1b"><apply id="S2.SS3.p5.4.m2.1.2.cmml" xref="S2.SS3.p5.4.m2.1.2"><ci id="S2.SS3.p5.4.m2.1.2.1.cmml" xref="S2.SS3.p5.4.m2.1.2.1">→</ci><apply id="S2.SS3.p5.4.m2.1.2.2.cmml" xref="S2.SS3.p5.4.m2.1.2.2"><times id="S2.SS3.p5.4.m2.1.2.2.1.cmml" xref="S2.SS3.p5.4.m2.1.2.2.1"></times><apply id="S2.SS3.p5.4.m2.1.2.2.2.cmml" xref="S2.SS3.p5.4.m2.1.2.2.2"><csymbol cd="ambiguous" id="S2.SS3.p5.4.m2.1.2.2.2.1.cmml" xref="S2.SS3.p5.4.m2.1.2.2.2">subscript</csymbol><ci id="S2.SS3.p5.4.m2.1.2.2.2.2.cmml" xref="S2.SS3.p5.4.m2.1.2.2.2.2">𝐴</ci><ci id="S2.SS3.p5.4.m2.1.2.2.2.3.cmml" xref="S2.SS3.p5.4.m2.1.2.2.2.3">𝑗</ci></apply><ci id="S2.SS3.p5.4.m2.1.1.cmml" xref="S2.SS3.p5.4.m2.1.1">𝜃</ci></apply><cn id="S2.SS3.p5.4.m2.1.2.3.cmml" type="integer" xref="S2.SS3.p5.4.m2.1.2.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p5.4.m2.1c">A_{j}(\theta)\to 0</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p5.4.m2.1d">italic_A start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ( italic_θ ) → 0</annotation></semantics></math>. Note that the quantity <math alttext="\chi^{2}(\theta)" class="ltx_Math" display="inline" id="S2.SS3.p5.5.m3.1"><semantics id="S2.SS3.p5.5.m3.1a"><mrow id="S2.SS3.p5.5.m3.1.2" xref="S2.SS3.p5.5.m3.1.2.cmml"><msup id="S2.SS3.p5.5.m3.1.2.2" xref="S2.SS3.p5.5.m3.1.2.2.cmml"><mi id="S2.SS3.p5.5.m3.1.2.2.2" xref="S2.SS3.p5.5.m3.1.2.2.2.cmml">χ</mi><mn id="S2.SS3.p5.5.m3.1.2.2.3" xref="S2.SS3.p5.5.m3.1.2.2.3.cmml">2</mn></msup><mo id="S2.SS3.p5.5.m3.1.2.1" xref="S2.SS3.p5.5.m3.1.2.1.cmml">⁢</mo><mrow id="S2.SS3.p5.5.m3.1.2.3.2" xref="S2.SS3.p5.5.m3.1.2.cmml"><mo id="S2.SS3.p5.5.m3.1.2.3.2.1" stretchy="false" xref="S2.SS3.p5.5.m3.1.2.cmml">(</mo><mi id="S2.SS3.p5.5.m3.1.1" xref="S2.SS3.p5.5.m3.1.1.cmml">θ</mi><mo id="S2.SS3.p5.5.m3.1.2.3.2.2" stretchy="false" xref="S2.SS3.p5.5.m3.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS3.p5.5.m3.1b"><apply id="S2.SS3.p5.5.m3.1.2.cmml" xref="S2.SS3.p5.5.m3.1.2"><times id="S2.SS3.p5.5.m3.1.2.1.cmml" xref="S2.SS3.p5.5.m3.1.2.1"></times><apply id="S2.SS3.p5.5.m3.1.2.2.cmml" xref="S2.SS3.p5.5.m3.1.2.2"><csymbol cd="ambiguous" id="S2.SS3.p5.5.m3.1.2.2.1.cmml" xref="S2.SS3.p5.5.m3.1.2.2">superscript</csymbol><ci id="S2.SS3.p5.5.m3.1.2.2.2.cmml" xref="S2.SS3.p5.5.m3.1.2.2.2">𝜒</ci><cn id="S2.SS3.p5.5.m3.1.2.2.3.cmml" type="integer" xref="S2.SS3.p5.5.m3.1.2.2.3">2</cn></apply><ci id="S2.SS3.p5.5.m3.1.1.cmml" xref="S2.SS3.p5.5.m3.1.1">𝜃</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p5.5.m3.1c">\chi^{2}(\theta)</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p5.5.m3.1d">italic_χ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( italic_θ )</annotation></semantics></math> characterizes the ratio of the infinitesimal arc-length square <math alttext="ds^{2}=g_{ij}(\theta)d\theta^{i}d\theta^{j}" class="ltx_Math" display="inline" id="S2.SS3.p5.6.m4.1"><semantics id="S2.SS3.p5.6.m4.1a"><mrow id="S2.SS3.p5.6.m4.1.2" xref="S2.SS3.p5.6.m4.1.2.cmml"><mrow id="S2.SS3.p5.6.m4.1.2.2" xref="S2.SS3.p5.6.m4.1.2.2.cmml"><mi id="S2.SS3.p5.6.m4.1.2.2.2" xref="S2.SS3.p5.6.m4.1.2.2.2.cmml">d</mi><mo id="S2.SS3.p5.6.m4.1.2.2.1" xref="S2.SS3.p5.6.m4.1.2.2.1.cmml">⁢</mo><msup id="S2.SS3.p5.6.m4.1.2.2.3" xref="S2.SS3.p5.6.m4.1.2.2.3.cmml"><mi id="S2.SS3.p5.6.m4.1.2.2.3.2" xref="S2.SS3.p5.6.m4.1.2.2.3.2.cmml">s</mi><mn id="S2.SS3.p5.6.m4.1.2.2.3.3" xref="S2.SS3.p5.6.m4.1.2.2.3.3.cmml">2</mn></msup></mrow><mo id="S2.SS3.p5.6.m4.1.2.1" xref="S2.SS3.p5.6.m4.1.2.1.cmml">=</mo><mrow id="S2.SS3.p5.6.m4.1.2.3" xref="S2.SS3.p5.6.m4.1.2.3.cmml"><msub id="S2.SS3.p5.6.m4.1.2.3.2" xref="S2.SS3.p5.6.m4.1.2.3.2.cmml"><mi id="S2.SS3.p5.6.m4.1.2.3.2.2" xref="S2.SS3.p5.6.m4.1.2.3.2.2.cmml">g</mi><mrow id="S2.SS3.p5.6.m4.1.2.3.2.3" xref="S2.SS3.p5.6.m4.1.2.3.2.3.cmml"><mi id="S2.SS3.p5.6.m4.1.2.3.2.3.2" xref="S2.SS3.p5.6.m4.1.2.3.2.3.2.cmml">i</mi><mo id="S2.SS3.p5.6.m4.1.2.3.2.3.1" xref="S2.SS3.p5.6.m4.1.2.3.2.3.1.cmml">⁢</mo><mi id="S2.SS3.p5.6.m4.1.2.3.2.3.3" xref="S2.SS3.p5.6.m4.1.2.3.2.3.3.cmml">j</mi></mrow></msub><mo id="S2.SS3.p5.6.m4.1.2.3.1" xref="S2.SS3.p5.6.m4.1.2.3.1.cmml">⁢</mo><mrow id="S2.SS3.p5.6.m4.1.2.3.3.2" xref="S2.SS3.p5.6.m4.1.2.3.cmml"><mo id="S2.SS3.p5.6.m4.1.2.3.3.2.1" stretchy="false" xref="S2.SS3.p5.6.m4.1.2.3.cmml">(</mo><mi id="S2.SS3.p5.6.m4.1.1" xref="S2.SS3.p5.6.m4.1.1.cmml">θ</mi><mo id="S2.SS3.p5.6.m4.1.2.3.3.2.2" stretchy="false" xref="S2.SS3.p5.6.m4.1.2.3.cmml">)</mo></mrow><mo id="S2.SS3.p5.6.m4.1.2.3.1a" xref="S2.SS3.p5.6.m4.1.2.3.1.cmml">⁢</mo><mi id="S2.SS3.p5.6.m4.1.2.3.4" xref="S2.SS3.p5.6.m4.1.2.3.4.cmml">d</mi><mo id="S2.SS3.p5.6.m4.1.2.3.1b" xref="S2.SS3.p5.6.m4.1.2.3.1.cmml">⁢</mo><msup id="S2.SS3.p5.6.m4.1.2.3.5" xref="S2.SS3.p5.6.m4.1.2.3.5.cmml"><mi id="S2.SS3.p5.6.m4.1.2.3.5.2" xref="S2.SS3.p5.6.m4.1.2.3.5.2.cmml">θ</mi><mi id="S2.SS3.p5.6.m4.1.2.3.5.3" xref="S2.SS3.p5.6.m4.1.2.3.5.3.cmml">i</mi></msup><mo id="S2.SS3.p5.6.m4.1.2.3.1c" xref="S2.SS3.p5.6.m4.1.2.3.1.cmml">⁢</mo><mi id="S2.SS3.p5.6.m4.1.2.3.6" xref="S2.SS3.p5.6.m4.1.2.3.6.cmml">d</mi><mo id="S2.SS3.p5.6.m4.1.2.3.1d" xref="S2.SS3.p5.6.m4.1.2.3.1.cmml">⁢</mo><msup id="S2.SS3.p5.6.m4.1.2.3.7" xref="S2.SS3.p5.6.m4.1.2.3.7.cmml"><mi id="S2.SS3.p5.6.m4.1.2.3.7.2" xref="S2.SS3.p5.6.m4.1.2.3.7.2.cmml">θ</mi><mi id="S2.SS3.p5.6.m4.1.2.3.7.3" xref="S2.SS3.p5.6.m4.1.2.3.7.3.cmml">j</mi></msup></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS3.p5.6.m4.1b"><apply id="S2.SS3.p5.6.m4.1.2.cmml" xref="S2.SS3.p5.6.m4.1.2"><eq id="S2.SS3.p5.6.m4.1.2.1.cmml" xref="S2.SS3.p5.6.m4.1.2.1"></eq><apply id="S2.SS3.p5.6.m4.1.2.2.cmml" xref="S2.SS3.p5.6.m4.1.2.2"><times id="S2.SS3.p5.6.m4.1.2.2.1.cmml" xref="S2.SS3.p5.6.m4.1.2.2.1"></times><ci id="S2.SS3.p5.6.m4.1.2.2.2.cmml" xref="S2.SS3.p5.6.m4.1.2.2.2">𝑑</ci><apply id="S2.SS3.p5.6.m4.1.2.2.3.cmml" xref="S2.SS3.p5.6.m4.1.2.2.3"><csymbol cd="ambiguous" id="S2.SS3.p5.6.m4.1.2.2.3.1.cmml" xref="S2.SS3.p5.6.m4.1.2.2.3">superscript</csymbol><ci id="S2.SS3.p5.6.m4.1.2.2.3.2.cmml" xref="S2.SS3.p5.6.m4.1.2.2.3.2">𝑠</ci><cn id="S2.SS3.p5.6.m4.1.2.2.3.3.cmml" type="integer" xref="S2.SS3.p5.6.m4.1.2.2.3.3">2</cn></apply></apply><apply id="S2.SS3.p5.6.m4.1.2.3.cmml" xref="S2.SS3.p5.6.m4.1.2.3"><times id="S2.SS3.p5.6.m4.1.2.3.1.cmml" xref="S2.SS3.p5.6.m4.1.2.3.1"></times><apply id="S2.SS3.p5.6.m4.1.2.3.2.cmml" xref="S2.SS3.p5.6.m4.1.2.3.2"><csymbol cd="ambiguous" id="S2.SS3.p5.6.m4.1.2.3.2.1.cmml" xref="S2.SS3.p5.6.m4.1.2.3.2">subscript</csymbol><ci id="S2.SS3.p5.6.m4.1.2.3.2.2.cmml" xref="S2.SS3.p5.6.m4.1.2.3.2.2">𝑔</ci><apply id="S2.SS3.p5.6.m4.1.2.3.2.3.cmml" xref="S2.SS3.p5.6.m4.1.2.3.2.3"><times id="S2.SS3.p5.6.m4.1.2.3.2.3.1.cmml" xref="S2.SS3.p5.6.m4.1.2.3.2.3.1"></times><ci id="S2.SS3.p5.6.m4.1.2.3.2.3.2.cmml" xref="S2.SS3.p5.6.m4.1.2.3.2.3.2">𝑖</ci><ci id="S2.SS3.p5.6.m4.1.2.3.2.3.3.cmml" xref="S2.SS3.p5.6.m4.1.2.3.2.3.3">𝑗</ci></apply></apply><ci id="S2.SS3.p5.6.m4.1.1.cmml" xref="S2.SS3.p5.6.m4.1.1">𝜃</ci><ci id="S2.SS3.p5.6.m4.1.2.3.4.cmml" xref="S2.SS3.p5.6.m4.1.2.3.4">𝑑</ci><apply id="S2.SS3.p5.6.m4.1.2.3.5.cmml" xref="S2.SS3.p5.6.m4.1.2.3.5"><csymbol cd="ambiguous" id="S2.SS3.p5.6.m4.1.2.3.5.1.cmml" xref="S2.SS3.p5.6.m4.1.2.3.5">superscript</csymbol><ci id="S2.SS3.p5.6.m4.1.2.3.5.2.cmml" xref="S2.SS3.p5.6.m4.1.2.3.5.2">𝜃</ci><ci id="S2.SS3.p5.6.m4.1.2.3.5.3.cmml" xref="S2.SS3.p5.6.m4.1.2.3.5.3">𝑖</ci></apply><ci id="S2.SS3.p5.6.m4.1.2.3.6.cmml" xref="S2.SS3.p5.6.m4.1.2.3.6">𝑑</ci><apply id="S2.SS3.p5.6.m4.1.2.3.7.cmml" xref="S2.SS3.p5.6.m4.1.2.3.7"><csymbol cd="ambiguous" id="S2.SS3.p5.6.m4.1.2.3.7.1.cmml" xref="S2.SS3.p5.6.m4.1.2.3.7">superscript</csymbol><ci id="S2.SS3.p5.6.m4.1.2.3.7.2.cmml" xref="S2.SS3.p5.6.m4.1.2.3.7.2">𝜃</ci><ci id="S2.SS3.p5.6.m4.1.2.3.7.3.cmml" xref="S2.SS3.p5.6.m4.1.2.3.7.3">𝑗</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p5.6.m4.1c">ds^{2}=g_{ij}(\theta)d\theta^{i}d\theta^{j}</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p5.6.m4.1d">italic_d italic_s start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT = italic_g start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT ( italic_θ ) italic_d italic_θ start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT italic_d italic_θ start_POSTSUPERSCRIPT italic_j end_POSTSUPERSCRIPT</annotation></semantics></math> to <math alttext="dt^{2}" class="ltx_Math" display="inline" id="S2.SS3.p5.7.m5.1"><semantics id="S2.SS3.p5.7.m5.1a"><mrow id="S2.SS3.p5.7.m5.1.1" xref="S2.SS3.p5.7.m5.1.1.cmml"><mi id="S2.SS3.p5.7.m5.1.1.2" xref="S2.SS3.p5.7.m5.1.1.2.cmml">d</mi><mo id="S2.SS3.p5.7.m5.1.1.1" xref="S2.SS3.p5.7.m5.1.1.1.cmml">⁢</mo><msup id="S2.SS3.p5.7.m5.1.1.3" xref="S2.SS3.p5.7.m5.1.1.3.cmml"><mi id="S2.SS3.p5.7.m5.1.1.3.2" xref="S2.SS3.p5.7.m5.1.1.3.2.cmml">t</mi><mn id="S2.SS3.p5.7.m5.1.1.3.3" xref="S2.SS3.p5.7.m5.1.1.3.3.cmml">2</mn></msup></mrow><annotation-xml encoding="MathML-Content" id="S2.SS3.p5.7.m5.1b"><apply id="S2.SS3.p5.7.m5.1.1.cmml" xref="S2.SS3.p5.7.m5.1.1"><times id="S2.SS3.p5.7.m5.1.1.1.cmml" xref="S2.SS3.p5.7.m5.1.1.1"></times><ci id="S2.SS3.p5.7.m5.1.1.2.cmml" xref="S2.SS3.p5.7.m5.1.1.2">𝑑</ci><apply id="S2.SS3.p5.7.m5.1.1.3.cmml" xref="S2.SS3.p5.7.m5.1.1.3"><csymbol cd="ambiguous" id="S2.SS3.p5.7.m5.1.1.3.1.cmml" xref="S2.SS3.p5.7.m5.1.1.3">superscript</csymbol><ci id="S2.SS3.p5.7.m5.1.1.3.2.cmml" xref="S2.SS3.p5.7.m5.1.1.3.2">𝑡</ci><cn id="S2.SS3.p5.7.m5.1.1.3.3.cmml" type="integer" xref="S2.SS3.p5.7.m5.1.1.3.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p5.7.m5.1c">dt^{2}</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p5.7.m5.1d">italic_d italic_t start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT</annotation></semantics></math>, i.e.,</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="S3.EGx31"> <tbody id="S2.E31"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_left_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\chi^{2}(\theta)=\frac{ds^{2}}{dt^{2}}." class="ltx_Math" display="inline" id="S2.E31.m1.2"><semantics id="S2.E31.m1.2a"><mrow id="S2.E31.m1.2.2.1" xref="S2.E31.m1.2.2.1.1.cmml"><mrow id="S2.E31.m1.2.2.1.1" xref="S2.E31.m1.2.2.1.1.cmml"><mrow id="S2.E31.m1.2.2.1.1.2" xref="S2.E31.m1.2.2.1.1.2.cmml"><msup id="S2.E31.m1.2.2.1.1.2.2" xref="S2.E31.m1.2.2.1.1.2.2.cmml"><mi id="S2.E31.m1.2.2.1.1.2.2.2" xref="S2.E31.m1.2.2.1.1.2.2.2.cmml">χ</mi><mn id="S2.E31.m1.2.2.1.1.2.2.3" xref="S2.E31.m1.2.2.1.1.2.2.3.cmml">2</mn></msup><mo id="S2.E31.m1.2.2.1.1.2.1" xref="S2.E31.m1.2.2.1.1.2.1.cmml">⁢</mo><mrow id="S2.E31.m1.2.2.1.1.2.3.2" xref="S2.E31.m1.2.2.1.1.2.cmml"><mo id="S2.E31.m1.2.2.1.1.2.3.2.1" stretchy="false" xref="S2.E31.m1.2.2.1.1.2.cmml">(</mo><mi id="S2.E31.m1.1.1" xref="S2.E31.m1.1.1.cmml">θ</mi><mo id="S2.E31.m1.2.2.1.1.2.3.2.2" stretchy="false" xref="S2.E31.m1.2.2.1.1.2.cmml">)</mo></mrow></mrow><mo id="S2.E31.m1.2.2.1.1.1" xref="S2.E31.m1.2.2.1.1.1.cmml">=</mo><mstyle displaystyle="true" id="S2.E31.m1.2.2.1.1.3" xref="S2.E31.m1.2.2.1.1.3.cmml"><mfrac id="S2.E31.m1.2.2.1.1.3a" xref="S2.E31.m1.2.2.1.1.3.cmml"><mrow id="S2.E31.m1.2.2.1.1.3.2" xref="S2.E31.m1.2.2.1.1.3.2.cmml"><mi id="S2.E31.m1.2.2.1.1.3.2.2" xref="S2.E31.m1.2.2.1.1.3.2.2.cmml">d</mi><mo id="S2.E31.m1.2.2.1.1.3.2.1" xref="S2.E31.m1.2.2.1.1.3.2.1.cmml">⁢</mo><msup id="S2.E31.m1.2.2.1.1.3.2.3" xref="S2.E31.m1.2.2.1.1.3.2.3.cmml"><mi id="S2.E31.m1.2.2.1.1.3.2.3.2" xref="S2.E31.m1.2.2.1.1.3.2.3.2.cmml">s</mi><mn id="S2.E31.m1.2.2.1.1.3.2.3.3" xref="S2.E31.m1.2.2.1.1.3.2.3.3.cmml">2</mn></msup></mrow><mrow id="S2.E31.m1.2.2.1.1.3.3" xref="S2.E31.m1.2.2.1.1.3.3.cmml"><mi id="S2.E31.m1.2.2.1.1.3.3.2" xref="S2.E31.m1.2.2.1.1.3.3.2.cmml">d</mi><mo id="S2.E31.m1.2.2.1.1.3.3.1" xref="S2.E31.m1.2.2.1.1.3.3.1.cmml">⁢</mo><msup id="S2.E31.m1.2.2.1.1.3.3.3" xref="S2.E31.m1.2.2.1.1.3.3.3.cmml"><mi id="S2.E31.m1.2.2.1.1.3.3.3.2" xref="S2.E31.m1.2.2.1.1.3.3.3.2.cmml">t</mi><mn id="S2.E31.m1.2.2.1.1.3.3.3.3" xref="S2.E31.m1.2.2.1.1.3.3.3.3.cmml">2</mn></msup></mrow></mfrac></mstyle></mrow><mo id="S2.E31.m1.2.2.1.2" lspace="0em" xref="S2.E31.m1.2.2.1.1.cmml">.</mo></mrow><annotation-xml encoding="MathML-Content" id="S2.E31.m1.2b"><apply id="S2.E31.m1.2.2.1.1.cmml" xref="S2.E31.m1.2.2.1"><eq id="S2.E31.m1.2.2.1.1.1.cmml" xref="S2.E31.m1.2.2.1.1.1"></eq><apply id="S2.E31.m1.2.2.1.1.2.cmml" xref="S2.E31.m1.2.2.1.1.2"><times id="S2.E31.m1.2.2.1.1.2.1.cmml" xref="S2.E31.m1.2.2.1.1.2.1"></times><apply id="S2.E31.m1.2.2.1.1.2.2.cmml" xref="S2.E31.m1.2.2.1.1.2.2"><csymbol cd="ambiguous" id="S2.E31.m1.2.2.1.1.2.2.1.cmml" xref="S2.E31.m1.2.2.1.1.2.2">superscript</csymbol><ci id="S2.E31.m1.2.2.1.1.2.2.2.cmml" xref="S2.E31.m1.2.2.1.1.2.2.2">𝜒</ci><cn id="S2.E31.m1.2.2.1.1.2.2.3.cmml" type="integer" xref="S2.E31.m1.2.2.1.1.2.2.3">2</cn></apply><ci id="S2.E31.m1.1.1.cmml" xref="S2.E31.m1.1.1">𝜃</ci></apply><apply id="S2.E31.m1.2.2.1.1.3.cmml" xref="S2.E31.m1.2.2.1.1.3"><divide id="S2.E31.m1.2.2.1.1.3.1.cmml" xref="S2.E31.m1.2.2.1.1.3"></divide><apply id="S2.E31.m1.2.2.1.1.3.2.cmml" xref="S2.E31.m1.2.2.1.1.3.2"><times id="S2.E31.m1.2.2.1.1.3.2.1.cmml" xref="S2.E31.m1.2.2.1.1.3.2.1"></times><ci id="S2.E31.m1.2.2.1.1.3.2.2.cmml" xref="S2.E31.m1.2.2.1.1.3.2.2">𝑑</ci><apply id="S2.E31.m1.2.2.1.1.3.2.3.cmml" xref="S2.E31.m1.2.2.1.1.3.2.3"><csymbol cd="ambiguous" id="S2.E31.m1.2.2.1.1.3.2.3.1.cmml" xref="S2.E31.m1.2.2.1.1.3.2.3">superscript</csymbol><ci id="S2.E31.m1.2.2.1.1.3.2.3.2.cmml" xref="S2.E31.m1.2.2.1.1.3.2.3.2">𝑠</ci><cn id="S2.E31.m1.2.2.1.1.3.2.3.3.cmml" type="integer" xref="S2.E31.m1.2.2.1.1.3.2.3.3">2</cn></apply></apply><apply id="S2.E31.m1.2.2.1.1.3.3.cmml" xref="S2.E31.m1.2.2.1.1.3.3"><times id="S2.E31.m1.2.2.1.1.3.3.1.cmml" xref="S2.E31.m1.2.2.1.1.3.3.1"></times><ci id="S2.E31.m1.2.2.1.1.3.3.2.cmml" xref="S2.E31.m1.2.2.1.1.3.3.2">𝑑</ci><apply id="S2.E31.m1.2.2.1.1.3.3.3.cmml" xref="S2.E31.m1.2.2.1.1.3.3.3"><csymbol cd="ambiguous" id="S2.E31.m1.2.2.1.1.3.3.3.1.cmml" xref="S2.E31.m1.2.2.1.1.3.3.3">superscript</csymbol><ci id="S2.E31.m1.2.2.1.1.3.3.3.2.cmml" xref="S2.E31.m1.2.2.1.1.3.3.3.2">𝑡</ci><cn id="S2.E31.m1.2.2.1.1.3.3.3.3.cmml" type="integer" xref="S2.E31.m1.2.2.1.1.3.3.3.3">2</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.E31.m1.2c">\displaystyle\chi^{2}(\theta)=\frac{ds^{2}}{dt^{2}}.</annotation><annotation encoding="application/x-llamapun" id="S2.E31.m1.2d">italic_χ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( italic_θ ) = divide start_ARG italic_d italic_s start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG italic_d italic_t start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_left_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(31)</span></td> </tr></tbody> </table> </div> <div class="ltx_para" id="S2.SS3.p6"> <p class="ltx_p" id="S2.SS3.p6.1">Next we introduce <math alttext="d\tilde{t}" class="ltx_Math" display="inline" id="S2.SS3.p6.1.m1.1"><semantics id="S2.SS3.p6.1.m1.1a"><mrow id="S2.SS3.p6.1.m1.1.1" xref="S2.SS3.p6.1.m1.1.1.cmml"><mi id="S2.SS3.p6.1.m1.1.1.2" xref="S2.SS3.p6.1.m1.1.1.2.cmml">d</mi><mo id="S2.SS3.p6.1.m1.1.1.1" xref="S2.SS3.p6.1.m1.1.1.1.cmml">⁢</mo><mover accent="true" id="S2.SS3.p6.1.m1.1.1.3" xref="S2.SS3.p6.1.m1.1.1.3.cmml"><mi id="S2.SS3.p6.1.m1.1.1.3.2" xref="S2.SS3.p6.1.m1.1.1.3.2.cmml">t</mi><mo id="S2.SS3.p6.1.m1.1.1.3.1" xref="S2.SS3.p6.1.m1.1.1.3.1.cmml">~</mo></mover></mrow><annotation-xml encoding="MathML-Content" id="S2.SS3.p6.1.m1.1b"><apply id="S2.SS3.p6.1.m1.1.1.cmml" xref="S2.SS3.p6.1.m1.1.1"><times id="S2.SS3.p6.1.m1.1.1.1.cmml" xref="S2.SS3.p6.1.m1.1.1.1"></times><ci id="S2.SS3.p6.1.m1.1.1.2.cmml" xref="S2.SS3.p6.1.m1.1.1.2">𝑑</ci><apply id="S2.SS3.p6.1.m1.1.1.3.cmml" xref="S2.SS3.p6.1.m1.1.1.3"><ci id="S2.SS3.p6.1.m1.1.1.3.1.cmml" xref="S2.SS3.p6.1.m1.1.1.3.1">~</ci><ci id="S2.SS3.p6.1.m1.1.1.3.2.cmml" xref="S2.SS3.p6.1.m1.1.1.3.2">𝑡</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p6.1.m1.1c">d\tilde{t}</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p6.1.m1.1d">italic_d over~ start_ARG italic_t end_ARG</annotation></semantics></math> such as</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="S3.EGx32"> <tbody id="S2.E32"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_left_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\chi^{2}(\theta)=\frac{d\Psi(\theta)}{d\tilde{t}}." class="ltx_Math" display="inline" id="S2.E32.m1.3"><semantics id="S2.E32.m1.3a"><mrow id="S2.E32.m1.3.3.1" xref="S2.E32.m1.3.3.1.1.cmml"><mrow id="S2.E32.m1.3.3.1.1" xref="S2.E32.m1.3.3.1.1.cmml"><mrow id="S2.E32.m1.3.3.1.1.2" xref="S2.E32.m1.3.3.1.1.2.cmml"><msup id="S2.E32.m1.3.3.1.1.2.2" xref="S2.E32.m1.3.3.1.1.2.2.cmml"><mi id="S2.E32.m1.3.3.1.1.2.2.2" xref="S2.E32.m1.3.3.1.1.2.2.2.cmml">χ</mi><mn id="S2.E32.m1.3.3.1.1.2.2.3" xref="S2.E32.m1.3.3.1.1.2.2.3.cmml">2</mn></msup><mo id="S2.E32.m1.3.3.1.1.2.1" xref="S2.E32.m1.3.3.1.1.2.1.cmml">⁢</mo><mrow id="S2.E32.m1.3.3.1.1.2.3.2" xref="S2.E32.m1.3.3.1.1.2.cmml"><mo id="S2.E32.m1.3.3.1.1.2.3.2.1" stretchy="false" xref="S2.E32.m1.3.3.1.1.2.cmml">(</mo><mi id="S2.E32.m1.2.2" xref="S2.E32.m1.2.2.cmml">θ</mi><mo id="S2.E32.m1.3.3.1.1.2.3.2.2" stretchy="false" xref="S2.E32.m1.3.3.1.1.2.cmml">)</mo></mrow></mrow><mo id="S2.E32.m1.3.3.1.1.1" xref="S2.E32.m1.3.3.1.1.1.cmml">=</mo><mstyle displaystyle="true" id="S2.E32.m1.1.1" xref="S2.E32.m1.1.1.cmml"><mfrac id="S2.E32.m1.1.1a" xref="S2.E32.m1.1.1.cmml"><mrow id="S2.E32.m1.1.1.1" xref="S2.E32.m1.1.1.1.cmml"><mi id="S2.E32.m1.1.1.1.3" xref="S2.E32.m1.1.1.1.3.cmml">d</mi><mo id="S2.E32.m1.1.1.1.2" xref="S2.E32.m1.1.1.1.2.cmml">⁢</mo><mi id="S2.E32.m1.1.1.1.4" mathvariant="normal" xref="S2.E32.m1.1.1.1.4.cmml">Ψ</mi><mo id="S2.E32.m1.1.1.1.2a" xref="S2.E32.m1.1.1.1.2.cmml">⁢</mo><mrow id="S2.E32.m1.1.1.1.5.2" xref="S2.E32.m1.1.1.1.cmml"><mo id="S2.E32.m1.1.1.1.5.2.1" stretchy="false" xref="S2.E32.m1.1.1.1.cmml">(</mo><mi id="S2.E32.m1.1.1.1.1" xref="S2.E32.m1.1.1.1.1.cmml">θ</mi><mo id="S2.E32.m1.1.1.1.5.2.2" stretchy="false" xref="S2.E32.m1.1.1.1.cmml">)</mo></mrow></mrow><mrow id="S2.E32.m1.1.1.3" xref="S2.E32.m1.1.1.3.cmml"><mi id="S2.E32.m1.1.1.3.2" xref="S2.E32.m1.1.1.3.2.cmml">d</mi><mo id="S2.E32.m1.1.1.3.1" xref="S2.E32.m1.1.1.3.1.cmml">⁢</mo><mover accent="true" id="S2.E32.m1.1.1.3.3" xref="S2.E32.m1.1.1.3.3.cmml"><mi id="S2.E32.m1.1.1.3.3.2" xref="S2.E32.m1.1.1.3.3.2.cmml">t</mi><mo id="S2.E32.m1.1.1.3.3.1" xref="S2.E32.m1.1.1.3.3.1.cmml">~</mo></mover></mrow></mfrac></mstyle></mrow><mo id="S2.E32.m1.3.3.1.2" lspace="0em" xref="S2.E32.m1.3.3.1.1.cmml">.</mo></mrow><annotation-xml encoding="MathML-Content" id="S2.E32.m1.3b"><apply id="S2.E32.m1.3.3.1.1.cmml" xref="S2.E32.m1.3.3.1"><eq id="S2.E32.m1.3.3.1.1.1.cmml" xref="S2.E32.m1.3.3.1.1.1"></eq><apply id="S2.E32.m1.3.3.1.1.2.cmml" xref="S2.E32.m1.3.3.1.1.2"><times id="S2.E32.m1.3.3.1.1.2.1.cmml" xref="S2.E32.m1.3.3.1.1.2.1"></times><apply id="S2.E32.m1.3.3.1.1.2.2.cmml" xref="S2.E32.m1.3.3.1.1.2.2"><csymbol cd="ambiguous" id="S2.E32.m1.3.3.1.1.2.2.1.cmml" xref="S2.E32.m1.3.3.1.1.2.2">superscript</csymbol><ci id="S2.E32.m1.3.3.1.1.2.2.2.cmml" xref="S2.E32.m1.3.3.1.1.2.2.2">𝜒</ci><cn id="S2.E32.m1.3.3.1.1.2.2.3.cmml" type="integer" xref="S2.E32.m1.3.3.1.1.2.2.3">2</cn></apply><ci id="S2.E32.m1.2.2.cmml" xref="S2.E32.m1.2.2">𝜃</ci></apply><apply id="S2.E32.m1.1.1.cmml" xref="S2.E32.m1.1.1"><divide id="S2.E32.m1.1.1.2.cmml" xref="S2.E32.m1.1.1"></divide><apply id="S2.E32.m1.1.1.1.cmml" xref="S2.E32.m1.1.1.1"><times id="S2.E32.m1.1.1.1.2.cmml" xref="S2.E32.m1.1.1.1.2"></times><ci id="S2.E32.m1.1.1.1.3.cmml" xref="S2.E32.m1.1.1.1.3">𝑑</ci><ci id="S2.E32.m1.1.1.1.4.cmml" xref="S2.E32.m1.1.1.1.4">Ψ</ci><ci id="S2.E32.m1.1.1.1.1.cmml" xref="S2.E32.m1.1.1.1.1">𝜃</ci></apply><apply id="S2.E32.m1.1.1.3.cmml" xref="S2.E32.m1.1.1.3"><times id="S2.E32.m1.1.1.3.1.cmml" xref="S2.E32.m1.1.1.3.1"></times><ci id="S2.E32.m1.1.1.3.2.cmml" xref="S2.E32.m1.1.1.3.2">𝑑</ci><apply id="S2.E32.m1.1.1.3.3.cmml" xref="S2.E32.m1.1.1.3.3"><ci id="S2.E32.m1.1.1.3.3.1.cmml" xref="S2.E32.m1.1.1.3.3.1">~</ci><ci id="S2.E32.m1.1.1.3.3.2.cmml" xref="S2.E32.m1.1.1.3.3.2">𝑡</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.E32.m1.3c">\displaystyle\chi^{2}(\theta)=\frac{d\Psi(\theta)}{d\tilde{t}}.</annotation><annotation encoding="application/x-llamapun" id="S2.E32.m1.3d">italic_χ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( italic_θ ) = divide start_ARG italic_d roman_Ψ ( italic_θ ) end_ARG start_ARG italic_d over~ start_ARG italic_t end_ARG end_ARG .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_left_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(32)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S2.SS3.p6.5">Then, from the RF deformation (<a class="ltx_ref" href="https://arxiv.org/html/2406.11224v2#S2.E28" title="In 2.3 Randers-Finsler deformation of the gradient-flow equations ‣ 2 Information Geometry and Gradient-Flow Equations ‣ A Hamiltonian approach to the gradient-flow equations in information geometry"><span class="ltx_text ltx_ref_tag">28</span></a>) we have</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="S3.EGx33"> <tbody id="S2.Ex4"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_left_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle d\tilde{t}" class="ltx_Math" display="inline" id="S2.Ex4.m1.1"><semantics id="S2.Ex4.m1.1a"><mrow id="S2.Ex4.m1.1.1" xref="S2.Ex4.m1.1.1.cmml"><mi id="S2.Ex4.m1.1.1.2" xref="S2.Ex4.m1.1.1.2.cmml">d</mi><mo id="S2.Ex4.m1.1.1.1" xref="S2.Ex4.m1.1.1.1.cmml">⁢</mo><mover accent="true" id="S2.Ex4.m1.1.1.3" xref="S2.Ex4.m1.1.1.3.cmml"><mi id="S2.Ex4.m1.1.1.3.2" xref="S2.Ex4.m1.1.1.3.2.cmml">t</mi><mo id="S2.Ex4.m1.1.1.3.1" xref="S2.Ex4.m1.1.1.3.1.cmml">~</mo></mover></mrow><annotation-xml encoding="MathML-Content" id="S2.Ex4.m1.1b"><apply id="S2.Ex4.m1.1.1.cmml" xref="S2.Ex4.m1.1.1"><times id="S2.Ex4.m1.1.1.1.cmml" xref="S2.Ex4.m1.1.1.1"></times><ci id="S2.Ex4.m1.1.1.2.cmml" xref="S2.Ex4.m1.1.1.2">𝑑</ci><apply id="S2.Ex4.m1.1.1.3.cmml" xref="S2.Ex4.m1.1.1.3"><ci id="S2.Ex4.m1.1.1.3.1.cmml" xref="S2.Ex4.m1.1.1.3.1">~</ci><ci id="S2.Ex4.m1.1.1.3.2.cmml" xref="S2.Ex4.m1.1.1.3.2">𝑡</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Ex4.m1.1c">\displaystyle d\tilde{t}</annotation><annotation encoding="application/x-llamapun" id="S2.Ex4.m1.1d">italic_d over~ start_ARG italic_t end_ARG</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle=\frac{1}{\chi^{2}(\theta)}\,d\Psi(\theta)=\frac{1}{\chi^{2}(% \theta)}\,\frac{\partial\Psi(\theta)}{\partial\theta^{i}}d\theta^{i}" class="ltx_Math" display="inline" id="S2.Ex4.m2.4"><semantics id="S2.Ex4.m2.4a"><mrow id="S2.Ex4.m2.4.5" xref="S2.Ex4.m2.4.5.cmml"><mi id="S2.Ex4.m2.4.5.2" xref="S2.Ex4.m2.4.5.2.cmml"></mi><mo id="S2.Ex4.m2.4.5.3" xref="S2.Ex4.m2.4.5.3.cmml">=</mo><mrow id="S2.Ex4.m2.4.5.4" 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xref="S2.Ex4.m2.4.5.6.3.3">𝑖</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Ex4.m2.4c">\displaystyle=\frac{1}{\chi^{2}(\theta)}\,d\Psi(\theta)=\frac{1}{\chi^{2}(% \theta)}\,\frac{\partial\Psi(\theta)}{\partial\theta^{i}}d\theta^{i}</annotation><annotation encoding="application/x-llamapun" id="S2.Ex4.m2.4d">= divide start_ARG 1 end_ARG start_ARG italic_χ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( italic_θ ) end_ARG italic_d roman_Ψ ( italic_θ ) = divide start_ARG 1 end_ARG start_ARG italic_χ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( italic_θ ) end_ARG divide start_ARG ∂ roman_Ψ ( italic_θ ) end_ARG start_ARG ∂ italic_θ start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT end_ARG italic_d italic_θ start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_left_padright"></td> </tr></tbody> <tbody id="S2.Ex5"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td 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encoding="application/x-llamapun" id="S2.Ex5.m1.4d">= divide start_ARG italic_g start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT ( italic_θ ) end_ARG start_ARG italic_χ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( italic_θ ) end_ARG divide start_ARG italic_d italic_θ start_POSTSUPERSCRIPT italic_j end_POSTSUPERSCRIPT end_ARG start_ARG italic_d italic_t end_ARG italic_d italic_θ start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT + divide start_ARG italic_A start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_θ ) end_ARG start_ARG italic_χ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( italic_θ ) end_ARG italic_d italic_θ start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_left_padright"></td> </tr></tbody> <tbody id="S2.E33"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_left_padleft"></td> <td class="ltx_td ltx_eqn_cell"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle=dt+\frac{A_{i}(\theta)}{\chi^{2}(\theta)}\,d\theta^{i}." class="ltx_Math" display="inline" id="S2.E33.m1.3"><semantics id="S2.E33.m1.3a"><mrow id="S2.E33.m1.3.3.1" xref="S2.E33.m1.3.3.1.1.cmml"><mrow id="S2.E33.m1.3.3.1.1" xref="S2.E33.m1.3.3.1.1.cmml"><mi id="S2.E33.m1.3.3.1.1.2" xref="S2.E33.m1.3.3.1.1.2.cmml"></mi><mo id="S2.E33.m1.3.3.1.1.1" xref="S2.E33.m1.3.3.1.1.1.cmml">=</mo><mrow id="S2.E33.m1.3.3.1.1.3" xref="S2.E33.m1.3.3.1.1.3.cmml"><mrow id="S2.E33.m1.3.3.1.1.3.2" xref="S2.E33.m1.3.3.1.1.3.2.cmml"><mi id="S2.E33.m1.3.3.1.1.3.2.2" xref="S2.E33.m1.3.3.1.1.3.2.2.cmml">d</mi><mo id="S2.E33.m1.3.3.1.1.3.2.1" xref="S2.E33.m1.3.3.1.1.3.2.1.cmml">⁢</mo><mi id="S2.E33.m1.3.3.1.1.3.2.3" xref="S2.E33.m1.3.3.1.1.3.2.3.cmml">t</mi></mrow><mo id="S2.E33.m1.3.3.1.1.3.1" xref="S2.E33.m1.3.3.1.1.3.1.cmml">+</mo><mrow id="S2.E33.m1.3.3.1.1.3.3" xref="S2.E33.m1.3.3.1.1.3.3.cmml"><mstyle displaystyle="true" id="S2.E33.m1.2.2" xref="S2.E33.m1.2.2.cmml"><mfrac id="S2.E33.m1.2.2a" xref="S2.E33.m1.2.2.cmml"><mrow id="S2.E33.m1.1.1.1" xref="S2.E33.m1.1.1.1.cmml"><msub id="S2.E33.m1.1.1.1.3" xref="S2.E33.m1.1.1.1.3.cmml"><mi id="S2.E33.m1.1.1.1.3.2" xref="S2.E33.m1.1.1.1.3.2.cmml">A</mi><mi id="S2.E33.m1.1.1.1.3.3" xref="S2.E33.m1.1.1.1.3.3.cmml">i</mi></msub><mo id="S2.E33.m1.1.1.1.2" xref="S2.E33.m1.1.1.1.2.cmml">⁢</mo><mrow id="S2.E33.m1.1.1.1.4.2" xref="S2.E33.m1.1.1.1.cmml"><mo id="S2.E33.m1.1.1.1.4.2.1" stretchy="false" xref="S2.E33.m1.1.1.1.cmml">(</mo><mi id="S2.E33.m1.1.1.1.1" xref="S2.E33.m1.1.1.1.1.cmml">θ</mi><mo id="S2.E33.m1.1.1.1.4.2.2" stretchy="false" xref="S2.E33.m1.1.1.1.cmml">)</mo></mrow></mrow><mrow id="S2.E33.m1.2.2.2" xref="S2.E33.m1.2.2.2.cmml"><msup id="S2.E33.m1.2.2.2.3" xref="S2.E33.m1.2.2.2.3.cmml"><mi id="S2.E33.m1.2.2.2.3.2" xref="S2.E33.m1.2.2.2.3.2.cmml">χ</mi><mn id="S2.E33.m1.2.2.2.3.3" xref="S2.E33.m1.2.2.2.3.3.cmml">2</mn></msup><mo id="S2.E33.m1.2.2.2.2" xref="S2.E33.m1.2.2.2.2.cmml">⁢</mo><mrow id="S2.E33.m1.2.2.2.4.2" xref="S2.E33.m1.2.2.2.cmml"><mo id="S2.E33.m1.2.2.2.4.2.1" stretchy="false" xref="S2.E33.m1.2.2.2.cmml">(</mo><mi id="S2.E33.m1.2.2.2.1" xref="S2.E33.m1.2.2.2.1.cmml">θ</mi><mo id="S2.E33.m1.2.2.2.4.2.2" stretchy="false" xref="S2.E33.m1.2.2.2.cmml">)</mo></mrow></mrow></mfrac></mstyle><mo id="S2.E33.m1.3.3.1.1.3.3.1" lspace="0.170em" xref="S2.E33.m1.3.3.1.1.3.3.1.cmml">⁢</mo><mi id="S2.E33.m1.3.3.1.1.3.3.2" xref="S2.E33.m1.3.3.1.1.3.3.2.cmml">d</mi><mo id="S2.E33.m1.3.3.1.1.3.3.1a" xref="S2.E33.m1.3.3.1.1.3.3.1.cmml">⁢</mo><msup id="S2.E33.m1.3.3.1.1.3.3.3" xref="S2.E33.m1.3.3.1.1.3.3.3.cmml"><mi id="S2.E33.m1.3.3.1.1.3.3.3.2" xref="S2.E33.m1.3.3.1.1.3.3.3.2.cmml">θ</mi><mi id="S2.E33.m1.3.3.1.1.3.3.3.3" xref="S2.E33.m1.3.3.1.1.3.3.3.3.cmml">i</mi></msup></mrow></mrow></mrow><mo id="S2.E33.m1.3.3.1.2" lspace="0em" xref="S2.E33.m1.3.3.1.1.cmml">.</mo></mrow><annotation-xml encoding="MathML-Content" id="S2.E33.m1.3b"><apply id="S2.E33.m1.3.3.1.1.cmml" xref="S2.E33.m1.3.3.1"><eq id="S2.E33.m1.3.3.1.1.1.cmml" xref="S2.E33.m1.3.3.1.1.1"></eq><csymbol cd="latexml" id="S2.E33.m1.3.3.1.1.2.cmml" xref="S2.E33.m1.3.3.1.1.2">absent</csymbol><apply id="S2.E33.m1.3.3.1.1.3.cmml" xref="S2.E33.m1.3.3.1.1.3"><plus id="S2.E33.m1.3.3.1.1.3.1.cmml" xref="S2.E33.m1.3.3.1.1.3.1"></plus><apply id="S2.E33.m1.3.3.1.1.3.2.cmml" xref="S2.E33.m1.3.3.1.1.3.2"><times id="S2.E33.m1.3.3.1.1.3.2.1.cmml" xref="S2.E33.m1.3.3.1.1.3.2.1"></times><ci id="S2.E33.m1.3.3.1.1.3.2.2.cmml" xref="S2.E33.m1.3.3.1.1.3.2.2">𝑑</ci><ci id="S2.E33.m1.3.3.1.1.3.2.3.cmml" xref="S2.E33.m1.3.3.1.1.3.2.3">𝑡</ci></apply><apply id="S2.E33.m1.3.3.1.1.3.3.cmml" xref="S2.E33.m1.3.3.1.1.3.3"><times id="S2.E33.m1.3.3.1.1.3.3.1.cmml" xref="S2.E33.m1.3.3.1.1.3.3.1"></times><apply id="S2.E33.m1.2.2.cmml" xref="S2.E33.m1.2.2"><divide id="S2.E33.m1.2.2.3.cmml" xref="S2.E33.m1.2.2"></divide><apply id="S2.E33.m1.1.1.1.cmml" 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id="S2.E33.m1.3.3.1.1.3.3.2.cmml" xref="S2.E33.m1.3.3.1.1.3.3.2">𝑑</ci><apply id="S2.E33.m1.3.3.1.1.3.3.3.cmml" xref="S2.E33.m1.3.3.1.1.3.3.3"><csymbol cd="ambiguous" id="S2.E33.m1.3.3.1.1.3.3.3.1.cmml" xref="S2.E33.m1.3.3.1.1.3.3.3">superscript</csymbol><ci id="S2.E33.m1.3.3.1.1.3.3.3.2.cmml" xref="S2.E33.m1.3.3.1.1.3.3.3.2">𝜃</ci><ci id="S2.E33.m1.3.3.1.1.3.3.3.3.cmml" xref="S2.E33.m1.3.3.1.1.3.3.3.3">𝑖</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.E33.m1.3c">\displaystyle=dt+\frac{A_{i}(\theta)}{\chi^{2}(\theta)}\,d\theta^{i}.</annotation><annotation encoding="application/x-llamapun" id="S2.E33.m1.3d">= italic_d italic_t + divide start_ARG italic_A start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_θ ) end_ARG start_ARG italic_χ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( italic_θ ) end_ARG italic_d italic_θ start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_left_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(33)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S2.SS3.p6.6">Here we used</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="S3.EGx34"> <tbody id="S2.E34"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_left_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle dt=\sqrt{\frac{g_{ij}(\theta)}{\chi^{2}(\theta)}d\theta^{i}d% \theta^{j}}," class="ltx_Math" display="inline" id="S2.E34.m1.3"><semantics id="S2.E34.m1.3a"><mrow id="S2.E34.m1.3.3.1" xref="S2.E34.m1.3.3.1.1.cmml"><mrow id="S2.E34.m1.3.3.1.1" xref="S2.E34.m1.3.3.1.1.cmml"><mrow id="S2.E34.m1.3.3.1.1.2" xref="S2.E34.m1.3.3.1.1.2.cmml"><mi id="S2.E34.m1.3.3.1.1.2.2" xref="S2.E34.m1.3.3.1.1.2.2.cmml">d</mi><mo id="S2.E34.m1.3.3.1.1.2.1" xref="S2.E34.m1.3.3.1.1.2.1.cmml">⁢</mo><mi id="S2.E34.m1.3.3.1.1.2.3" xref="S2.E34.m1.3.3.1.1.2.3.cmml">t</mi></mrow><mo id="S2.E34.m1.3.3.1.1.1" xref="S2.E34.m1.3.3.1.1.1.cmml">=</mo><msqrt id="S2.E34.m1.2.2" xref="S2.E34.m1.2.2.cmml"><mrow id="S2.E34.m1.2.2.2" xref="S2.E34.m1.2.2.2.cmml"><mstyle displaystyle="true" id="S2.E34.m1.2.2.2.2" xref="S2.E34.m1.2.2.2.2.cmml"><mfrac id="S2.E34.m1.2.2.2.2a" xref="S2.E34.m1.2.2.2.2.cmml"><mrow id="S2.E34.m1.1.1.1.1.1" xref="S2.E34.m1.1.1.1.1.1.cmml"><msub id="S2.E34.m1.1.1.1.1.1.3" xref="S2.E34.m1.1.1.1.1.1.3.cmml"><mi id="S2.E34.m1.1.1.1.1.1.3.2" xref="S2.E34.m1.1.1.1.1.1.3.2.cmml">g</mi><mrow id="S2.E34.m1.1.1.1.1.1.3.3" xref="S2.E34.m1.1.1.1.1.1.3.3.cmml"><mi id="S2.E34.m1.1.1.1.1.1.3.3.2" xref="S2.E34.m1.1.1.1.1.1.3.3.2.cmml">i</mi><mo id="S2.E34.m1.1.1.1.1.1.3.3.1" xref="S2.E34.m1.1.1.1.1.1.3.3.1.cmml">⁢</mo><mi id="S2.E34.m1.1.1.1.1.1.3.3.3" xref="S2.E34.m1.1.1.1.1.1.3.3.3.cmml">j</mi></mrow></msub><mo id="S2.E34.m1.1.1.1.1.1.2" xref="S2.E34.m1.1.1.1.1.1.2.cmml">⁢</mo><mrow id="S2.E34.m1.1.1.1.1.1.4.2" xref="S2.E34.m1.1.1.1.1.1.cmml"><mo id="S2.E34.m1.1.1.1.1.1.4.2.1" stretchy="false" xref="S2.E34.m1.1.1.1.1.1.cmml">(</mo><mi id="S2.E34.m1.1.1.1.1.1.1" xref="S2.E34.m1.1.1.1.1.1.1.cmml">θ</mi><mo id="S2.E34.m1.1.1.1.1.1.4.2.2" stretchy="false" xref="S2.E34.m1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mrow id="S2.E34.m1.2.2.2.2.2" xref="S2.E34.m1.2.2.2.2.2.cmml"><msup id="S2.E34.m1.2.2.2.2.2.3" xref="S2.E34.m1.2.2.2.2.2.3.cmml"><mi id="S2.E34.m1.2.2.2.2.2.3.2" xref="S2.E34.m1.2.2.2.2.2.3.2.cmml">χ</mi><mn id="S2.E34.m1.2.2.2.2.2.3.3" xref="S2.E34.m1.2.2.2.2.2.3.3.cmml">2</mn></msup><mo id="S2.E34.m1.2.2.2.2.2.2" xref="S2.E34.m1.2.2.2.2.2.2.cmml">⁢</mo><mrow id="S2.E34.m1.2.2.2.2.2.4.2" xref="S2.E34.m1.2.2.2.2.2.cmml"><mo id="S2.E34.m1.2.2.2.2.2.4.2.1" stretchy="false" xref="S2.E34.m1.2.2.2.2.2.cmml">(</mo><mi id="S2.E34.m1.2.2.2.2.2.1" xref="S2.E34.m1.2.2.2.2.2.1.cmml">θ</mi><mo id="S2.E34.m1.2.2.2.2.2.4.2.2" stretchy="false" xref="S2.E34.m1.2.2.2.2.2.cmml">)</mo></mrow></mrow></mfrac></mstyle><mo id="S2.E34.m1.2.2.2.3" xref="S2.E34.m1.2.2.2.3.cmml">⁢</mo><mi id="S2.E34.m1.2.2.2.4" xref="S2.E34.m1.2.2.2.4.cmml">d</mi><mo id="S2.E34.m1.2.2.2.3a" xref="S2.E34.m1.2.2.2.3.cmml">⁢</mo><msup id="S2.E34.m1.2.2.2.5" xref="S2.E34.m1.2.2.2.5.cmml"><mi id="S2.E34.m1.2.2.2.5.2" xref="S2.E34.m1.2.2.2.5.2.cmml">θ</mi><mi id="S2.E34.m1.2.2.2.5.3" xref="S2.E34.m1.2.2.2.5.3.cmml">i</mi></msup><mo id="S2.E34.m1.2.2.2.3b" xref="S2.E34.m1.2.2.2.3.cmml">⁢</mo><mi id="S2.E34.m1.2.2.2.6" xref="S2.E34.m1.2.2.2.6.cmml">d</mi><mo id="S2.E34.m1.2.2.2.3c" xref="S2.E34.m1.2.2.2.3.cmml">⁢</mo><msup id="S2.E34.m1.2.2.2.7" xref="S2.E34.m1.2.2.2.7.cmml"><mi id="S2.E34.m1.2.2.2.7.2" xref="S2.E34.m1.2.2.2.7.2.cmml">θ</mi><mi id="S2.E34.m1.2.2.2.7.3" xref="S2.E34.m1.2.2.2.7.3.cmml">j</mi></msup></mrow></msqrt></mrow><mo id="S2.E34.m1.3.3.1.2" xref="S2.E34.m1.3.3.1.1.cmml">,</mo></mrow><annotation-xml encoding="MathML-Content" 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xref="S2.E34.m1.2.2.2.2.2.3.2">𝜒</ci><cn id="S2.E34.m1.2.2.2.2.2.3.3.cmml" type="integer" xref="S2.E34.m1.2.2.2.2.2.3.3">2</cn></apply><ci id="S2.E34.m1.2.2.2.2.2.1.cmml" xref="S2.E34.m1.2.2.2.2.2.1">𝜃</ci></apply></apply><ci id="S2.E34.m1.2.2.2.4.cmml" xref="S2.E34.m1.2.2.2.4">𝑑</ci><apply id="S2.E34.m1.2.2.2.5.cmml" xref="S2.E34.m1.2.2.2.5"><csymbol cd="ambiguous" id="S2.E34.m1.2.2.2.5.1.cmml" xref="S2.E34.m1.2.2.2.5">superscript</csymbol><ci id="S2.E34.m1.2.2.2.5.2.cmml" xref="S2.E34.m1.2.2.2.5.2">𝜃</ci><ci id="S2.E34.m1.2.2.2.5.3.cmml" xref="S2.E34.m1.2.2.2.5.3">𝑖</ci></apply><ci id="S2.E34.m1.2.2.2.6.cmml" xref="S2.E34.m1.2.2.2.6">𝑑</ci><apply id="S2.E34.m1.2.2.2.7.cmml" xref="S2.E34.m1.2.2.2.7"><csymbol cd="ambiguous" id="S2.E34.m1.2.2.2.7.1.cmml" xref="S2.E34.m1.2.2.2.7">superscript</csymbol><ci id="S2.E34.m1.2.2.2.7.2.cmml" xref="S2.E34.m1.2.2.2.7.2">𝜃</ci><ci id="S2.E34.m1.2.2.2.7.3.cmml" xref="S2.E34.m1.2.2.2.7.3">𝑗</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.E34.m1.3c">\displaystyle dt=\sqrt{\frac{g_{ij}(\theta)}{\chi^{2}(\theta)}d\theta^{i}d% \theta^{j}},</annotation><annotation encoding="application/x-llamapun" id="S2.E34.m1.3d">italic_d italic_t = square-root start_ARG divide start_ARG italic_g start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT ( italic_θ ) end_ARG start_ARG italic_χ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( italic_θ ) end_ARG italic_d italic_θ start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT italic_d italic_θ start_POSTSUPERSCRIPT italic_j end_POSTSUPERSCRIPT end_ARG ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_left_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(34)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S2.SS3.p6.4">which is obtained from (<a class="ltx_ref" href="https://arxiv.org/html/2406.11224v2#S2.E29" title="In 2.3 Randers-Finsler deformation of the gradient-flow equations ‣ 2 Information Geometry and Gradient-Flow Equations ‣ A Hamiltonian approach to the gradient-flow equations in information geometry"><span class="ltx_text ltx_ref_tag">29</span></a>). The Randers function <math alttext="d\tilde{t}" class="ltx_Math" display="inline" id="S2.SS3.p6.2.m1.1"><semantics id="S2.SS3.p6.2.m1.1a"><mrow id="S2.SS3.p6.2.m1.1.1" xref="S2.SS3.p6.2.m1.1.1.cmml"><mi id="S2.SS3.p6.2.m1.1.1.2" xref="S2.SS3.p6.2.m1.1.1.2.cmml">d</mi><mo id="S2.SS3.p6.2.m1.1.1.1" xref="S2.SS3.p6.2.m1.1.1.1.cmml">⁢</mo><mover accent="true" id="S2.SS3.p6.2.m1.1.1.3" xref="S2.SS3.p6.2.m1.1.1.3.cmml"><mi id="S2.SS3.p6.2.m1.1.1.3.2" xref="S2.SS3.p6.2.m1.1.1.3.2.cmml">t</mi><mo id="S2.SS3.p6.2.m1.1.1.3.1" xref="S2.SS3.p6.2.m1.1.1.3.1.cmml">~</mo></mover></mrow><annotation-xml encoding="MathML-Content" id="S2.SS3.p6.2.m1.1b"><apply id="S2.SS3.p6.2.m1.1.1.cmml" xref="S2.SS3.p6.2.m1.1.1"><times id="S2.SS3.p6.2.m1.1.1.1.cmml" xref="S2.SS3.p6.2.m1.1.1.1"></times><ci id="S2.SS3.p6.2.m1.1.1.2.cmml" xref="S2.SS3.p6.2.m1.1.1.2">𝑑</ci><apply id="S2.SS3.p6.2.m1.1.1.3.cmml" xref="S2.SS3.p6.2.m1.1.1.3"><ci id="S2.SS3.p6.2.m1.1.1.3.1.cmml" xref="S2.SS3.p6.2.m1.1.1.3.1">~</ci><ci id="S2.SS3.p6.2.m1.1.1.3.2.cmml" xref="S2.SS3.p6.2.m1.1.1.3.2">𝑡</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p6.2.m1.1c">d\tilde{t}</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p6.2.m1.1d">italic_d over~ start_ARG italic_t end_ARG</annotation></semantics></math> reduces to <math alttext="dt" class="ltx_Math" display="inline" id="S2.SS3.p6.3.m2.1"><semantics id="S2.SS3.p6.3.m2.1a"><mrow id="S2.SS3.p6.3.m2.1.1" xref="S2.SS3.p6.3.m2.1.1.cmml"><mi id="S2.SS3.p6.3.m2.1.1.2" xref="S2.SS3.p6.3.m2.1.1.2.cmml">d</mi><mo id="S2.SS3.p6.3.m2.1.1.1" xref="S2.SS3.p6.3.m2.1.1.1.cmml">⁢</mo><mi id="S2.SS3.p6.3.m2.1.1.3" xref="S2.SS3.p6.3.m2.1.1.3.cmml">t</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.SS3.p6.3.m2.1b"><apply id="S2.SS3.p6.3.m2.1.1.cmml" xref="S2.SS3.p6.3.m2.1.1"><times id="S2.SS3.p6.3.m2.1.1.1.cmml" xref="S2.SS3.p6.3.m2.1.1.1"></times><ci id="S2.SS3.p6.3.m2.1.1.2.cmml" xref="S2.SS3.p6.3.m2.1.1.2">𝑑</ci><ci id="S2.SS3.p6.3.m2.1.1.3.cmml" xref="S2.SS3.p6.3.m2.1.1.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p6.3.m2.1c">dt</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p6.3.m2.1d">italic_d italic_t</annotation></semantics></math> when <math alttext="A_{i}(\theta)\to 0" class="ltx_Math" display="inline" id="S2.SS3.p6.4.m3.1"><semantics id="S2.SS3.p6.4.m3.1a"><mrow id="S2.SS3.p6.4.m3.1.2" xref="S2.SS3.p6.4.m3.1.2.cmml"><mrow id="S2.SS3.p6.4.m3.1.2.2" xref="S2.SS3.p6.4.m3.1.2.2.cmml"><msub id="S2.SS3.p6.4.m3.1.2.2.2" xref="S2.SS3.p6.4.m3.1.2.2.2.cmml"><mi id="S2.SS3.p6.4.m3.1.2.2.2.2" xref="S2.SS3.p6.4.m3.1.2.2.2.2.cmml">A</mi><mi id="S2.SS3.p6.4.m3.1.2.2.2.3" xref="S2.SS3.p6.4.m3.1.2.2.2.3.cmml">i</mi></msub><mo id="S2.SS3.p6.4.m3.1.2.2.1" xref="S2.SS3.p6.4.m3.1.2.2.1.cmml">⁢</mo><mrow id="S2.SS3.p6.4.m3.1.2.2.3.2" xref="S2.SS3.p6.4.m3.1.2.2.cmml"><mo id="S2.SS3.p6.4.m3.1.2.2.3.2.1" stretchy="false" xref="S2.SS3.p6.4.m3.1.2.2.cmml">(</mo><mi id="S2.SS3.p6.4.m3.1.1" xref="S2.SS3.p6.4.m3.1.1.cmml">θ</mi><mo id="S2.SS3.p6.4.m3.1.2.2.3.2.2" stretchy="false" xref="S2.SS3.p6.4.m3.1.2.2.cmml">)</mo></mrow></mrow><mo id="S2.SS3.p6.4.m3.1.2.1" stretchy="false" xref="S2.SS3.p6.4.m3.1.2.1.cmml">→</mo><mn id="S2.SS3.p6.4.m3.1.2.3" xref="S2.SS3.p6.4.m3.1.2.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.SS3.p6.4.m3.1b"><apply id="S2.SS3.p6.4.m3.1.2.cmml" xref="S2.SS3.p6.4.m3.1.2"><ci id="S2.SS3.p6.4.m3.1.2.1.cmml" xref="S2.SS3.p6.4.m3.1.2.1">→</ci><apply id="S2.SS3.p6.4.m3.1.2.2.cmml" xref="S2.SS3.p6.4.m3.1.2.2"><times id="S2.SS3.p6.4.m3.1.2.2.1.cmml" xref="S2.SS3.p6.4.m3.1.2.2.1"></times><apply id="S2.SS3.p6.4.m3.1.2.2.2.cmml" xref="S2.SS3.p6.4.m3.1.2.2.2"><csymbol cd="ambiguous" id="S2.SS3.p6.4.m3.1.2.2.2.1.cmml" xref="S2.SS3.p6.4.m3.1.2.2.2">subscript</csymbol><ci id="S2.SS3.p6.4.m3.1.2.2.2.2.cmml" xref="S2.SS3.p6.4.m3.1.2.2.2.2">𝐴</ci><ci id="S2.SS3.p6.4.m3.1.2.2.2.3.cmml" xref="S2.SS3.p6.4.m3.1.2.2.2.3">𝑖</ci></apply><ci id="S2.SS3.p6.4.m3.1.1.cmml" xref="S2.SS3.p6.4.m3.1.1">𝜃</ci></apply><cn id="S2.SS3.p6.4.m3.1.2.3.cmml" type="integer" xref="S2.SS3.p6.4.m3.1.2.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p6.4.m3.1c">A_{i}(\theta)\to 0</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p6.4.m3.1d">italic_A start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_θ ) → 0</annotation></semantics></math>. The corresponding RF Lagrangian is</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="S3.EGx35"> <tbody id="S2.E35"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_left_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\mathcal{L}_{\rm RF}\!\left(\theta,\!\frac{d\theta}{dt}\right)\!=% \!\sqrt{\frac{g_{ij}(\theta)}{\chi^{2}(\theta)}\frac{d\theta^{i}}{dt}\frac{d% \theta^{j}}{dt}}\!+\!\frac{A_{i}(\theta)}{\chi^{2}(\theta)}\,\frac{d\theta^{i}% }{dt}." class="ltx_Math" display="inline" id="S2.E35.m1.7"><semantics id="S2.E35.m1.7a"><mrow id="S2.E35.m1.7.7.1" xref="S2.E35.m1.7.7.1.1.cmml"><mrow id="S2.E35.m1.7.7.1.1" xref="S2.E35.m1.7.7.1.1.cmml"><mrow id="S2.E35.m1.7.7.1.1.2" xref="S2.E35.m1.7.7.1.1.2.cmml"><msub id="S2.E35.m1.7.7.1.1.2.2" xref="S2.E35.m1.7.7.1.1.2.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.E35.m1.7.7.1.1.2.2.2" xref="S2.E35.m1.7.7.1.1.2.2.2.cmml">ℒ</mi><mi id="S2.E35.m1.7.7.1.1.2.2.3" xref="S2.E35.m1.7.7.1.1.2.2.3.cmml">RF</mi></msub><mo id="S2.E35.m1.7.7.1.1.2.1" xref="S2.E35.m1.7.7.1.1.2.1.cmml">⁢</mo><mrow id="S2.E35.m1.7.7.1.1.2.3.2" xref="S2.E35.m1.7.7.1.1.2.3.1.cmml"><mo id="S2.E35.m1.7.7.1.1.2.3.2.1" xref="S2.E35.m1.7.7.1.1.2.3.1.cmml">(</mo><mi id="S2.E35.m1.5.5" xref="S2.E35.m1.5.5.cmml">θ</mi><mpadded width="0.275em"><mo id="S2.E35.m1.7.7.1.1.2.3.2.2" xref="S2.E35.m1.7.7.1.1.2.3.1.cmml">,</mo></mpadded><mstyle displaystyle="true" id="S2.E35.m1.6.6" xref="S2.E35.m1.6.6.cmml"><mfrac id="S2.E35.m1.6.6a" xref="S2.E35.m1.6.6.cmml"><mrow id="S2.E35.m1.6.6.2" xref="S2.E35.m1.6.6.2.cmml"><mi id="S2.E35.m1.6.6.2.2" xref="S2.E35.m1.6.6.2.2.cmml">d</mi><mo id="S2.E35.m1.6.6.2.1" xref="S2.E35.m1.6.6.2.1.cmml">⁢</mo><mi id="S2.E35.m1.6.6.2.3" xref="S2.E35.m1.6.6.2.3.cmml">θ</mi></mrow><mrow id="S2.E35.m1.6.6.3" xref="S2.E35.m1.6.6.3.cmml"><mi id="S2.E35.m1.6.6.3.2" xref="S2.E35.m1.6.6.3.2.cmml">d</mi><mo id="S2.E35.m1.6.6.3.1" xref="S2.E35.m1.6.6.3.1.cmml">⁢</mo><mi id="S2.E35.m1.6.6.3.3" xref="S2.E35.m1.6.6.3.3.cmml">t</mi></mrow></mfrac></mstyle><mo id="S2.E35.m1.7.7.1.1.2.3.2.3" xref="S2.E35.m1.7.7.1.1.2.3.1.cmml">)</mo></mrow></mrow><mo id="S2.E35.m1.7.7.1.1.1" lspace="0.108em" rspace="0.108em" xref="S2.E35.m1.7.7.1.1.1.cmml">=</mo><mrow id="S2.E35.m1.7.7.1.1.3" xref="S2.E35.m1.7.7.1.1.3.cmml"><msqrt id="S2.E35.m1.2.2" xref="S2.E35.m1.2.2.cmml"><mrow id="S2.E35.m1.2.2.2" xref="S2.E35.m1.2.2.2.cmml"><mstyle displaystyle="true" id="S2.E35.m1.2.2.2.2" xref="S2.E35.m1.2.2.2.2.cmml"><mfrac id="S2.E35.m1.2.2.2.2a" xref="S2.E35.m1.2.2.2.2.cmml"><mrow id="S2.E35.m1.1.1.1.1.1" xref="S2.E35.m1.1.1.1.1.1.cmml"><msub id="S2.E35.m1.1.1.1.1.1.3" xref="S2.E35.m1.1.1.1.1.1.3.cmml"><mi id="S2.E35.m1.1.1.1.1.1.3.2" xref="S2.E35.m1.1.1.1.1.1.3.2.cmml">g</mi><mrow id="S2.E35.m1.1.1.1.1.1.3.3" xref="S2.E35.m1.1.1.1.1.1.3.3.cmml"><mi id="S2.E35.m1.1.1.1.1.1.3.3.2" xref="S2.E35.m1.1.1.1.1.1.3.3.2.cmml">i</mi><mo id="S2.E35.m1.1.1.1.1.1.3.3.1" xref="S2.E35.m1.1.1.1.1.1.3.3.1.cmml">⁢</mo><mi id="S2.E35.m1.1.1.1.1.1.3.3.3" xref="S2.E35.m1.1.1.1.1.1.3.3.3.cmml">j</mi></mrow></msub><mo id="S2.E35.m1.1.1.1.1.1.2" xref="S2.E35.m1.1.1.1.1.1.2.cmml">⁢</mo><mrow id="S2.E35.m1.1.1.1.1.1.4.2" xref="S2.E35.m1.1.1.1.1.1.cmml"><mo id="S2.E35.m1.1.1.1.1.1.4.2.1" stretchy="false" xref="S2.E35.m1.1.1.1.1.1.cmml">(</mo><mi id="S2.E35.m1.1.1.1.1.1.1" xref="S2.E35.m1.1.1.1.1.1.1.cmml">θ</mi><mo id="S2.E35.m1.1.1.1.1.1.4.2.2" stretchy="false" xref="S2.E35.m1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mrow id="S2.E35.m1.2.2.2.2.2" xref="S2.E35.m1.2.2.2.2.2.cmml"><msup id="S2.E35.m1.2.2.2.2.2.3" xref="S2.E35.m1.2.2.2.2.2.3.cmml"><mi id="S2.E35.m1.2.2.2.2.2.3.2" xref="S2.E35.m1.2.2.2.2.2.3.2.cmml">χ</mi><mn id="S2.E35.m1.2.2.2.2.2.3.3" xref="S2.E35.m1.2.2.2.2.2.3.3.cmml">2</mn></msup><mo id="S2.E35.m1.2.2.2.2.2.2" xref="S2.E35.m1.2.2.2.2.2.2.cmml">⁢</mo><mrow id="S2.E35.m1.2.2.2.2.2.4.2" xref="S2.E35.m1.2.2.2.2.2.cmml"><mo id="S2.E35.m1.2.2.2.2.2.4.2.1" stretchy="false" xref="S2.E35.m1.2.2.2.2.2.cmml">(</mo><mi id="S2.E35.m1.2.2.2.2.2.1" xref="S2.E35.m1.2.2.2.2.2.1.cmml">θ</mi><mo id="S2.E35.m1.2.2.2.2.2.4.2.2" stretchy="false" xref="S2.E35.m1.2.2.2.2.2.cmml">)</mo></mrow></mrow></mfrac></mstyle><mo id="S2.E35.m1.2.2.2.3" xref="S2.E35.m1.2.2.2.3.cmml">⁢</mo><mstyle displaystyle="true" id="S2.E35.m1.2.2.2.4" xref="S2.E35.m1.2.2.2.4.cmml"><mfrac id="S2.E35.m1.2.2.2.4a" xref="S2.E35.m1.2.2.2.4.cmml"><mrow id="S2.E35.m1.2.2.2.4.2" xref="S2.E35.m1.2.2.2.4.2.cmml"><mi id="S2.E35.m1.2.2.2.4.2.2" xref="S2.E35.m1.2.2.2.4.2.2.cmml">d</mi><mo id="S2.E35.m1.2.2.2.4.2.1" xref="S2.E35.m1.2.2.2.4.2.1.cmml">⁢</mo><msup id="S2.E35.m1.2.2.2.4.2.3" xref="S2.E35.m1.2.2.2.4.2.3.cmml"><mi id="S2.E35.m1.2.2.2.4.2.3.2" xref="S2.E35.m1.2.2.2.4.2.3.2.cmml">θ</mi><mi id="S2.E35.m1.2.2.2.4.2.3.3" xref="S2.E35.m1.2.2.2.4.2.3.3.cmml">i</mi></msup></mrow><mrow id="S2.E35.m1.2.2.2.4.3" xref="S2.E35.m1.2.2.2.4.3.cmml"><mi 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id="S2.E35.m1.7c">\displaystyle\mathcal{L}_{\rm RF}\!\left(\theta,\!\frac{d\theta}{dt}\right)\!=% \!\sqrt{\frac{g_{ij}(\theta)}{\chi^{2}(\theta)}\frac{d\theta^{i}}{dt}\frac{d% \theta^{j}}{dt}}\!+\!\frac{A_{i}(\theta)}{\chi^{2}(\theta)}\,\frac{d\theta^{i}% }{dt}.</annotation><annotation encoding="application/x-llamapun" id="S2.E35.m1.7d">caligraphic_L start_POSTSUBSCRIPT roman_RF end_POSTSUBSCRIPT ( italic_θ , divide start_ARG italic_d italic_θ end_ARG start_ARG italic_d italic_t end_ARG ) = square-root start_ARG divide start_ARG italic_g start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT ( italic_θ ) end_ARG start_ARG italic_χ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( italic_θ ) end_ARG divide start_ARG italic_d italic_θ start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT end_ARG start_ARG italic_d italic_t end_ARG divide start_ARG italic_d italic_θ start_POSTSUPERSCRIPT italic_j end_POSTSUPERSCRIPT end_ARG start_ARG italic_d italic_t end_ARG end_ARG + divide start_ARG italic_A start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_θ ) end_ARG start_ARG italic_χ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( italic_θ ) end_ARG divide start_ARG italic_d italic_θ start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT end_ARG start_ARG italic_d italic_t end_ARG .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_left_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(35)</span></td> </tr></tbody> </table> </div> </section> </section> <section class="ltx_section" id="S3"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">3 </span>Complete integrability and geodesic Hamiltonian</h2> <div class="ltx_para" id="S3.p1"> <p class="ltx_p" id="S3.p1.4">Here we discuss the complete integrability concerning a certain kind of Hamiltonian in analytical mechanics. Let us begin with a brief review on the complete integrability of Pfaffian systems by Élie Cartan <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2406.11224v2#bib.bib17" title="">17</a>]</cite>. He extended Poincaré’s theory on the integral invariant, and showed that the one-form</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="S3.EGx36"> <tbody id="S3.E36"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_left_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\omega_{\rm PC}:=p_{j}dx^{j}-Hdt," class="ltx_Math" display="inline" id="S3.E36.m1.1"><semantics id="S3.E36.m1.1a"><mrow id="S3.E36.m1.1.1.1" xref="S3.E36.m1.1.1.1.1.cmml"><mrow id="S3.E36.m1.1.1.1.1" xref="S3.E36.m1.1.1.1.1.cmml"><msub id="S3.E36.m1.1.1.1.1.2" xref="S3.E36.m1.1.1.1.1.2.cmml"><mi id="S3.E36.m1.1.1.1.1.2.2" xref="S3.E36.m1.1.1.1.1.2.2.cmml">ω</mi><mi id="S3.E36.m1.1.1.1.1.2.3" xref="S3.E36.m1.1.1.1.1.2.3.cmml">PC</mi></msub><mo id="S3.E36.m1.1.1.1.1.1" lspace="0.278em" rspace="0.278em" xref="S3.E36.m1.1.1.1.1.1.cmml">:=</mo><mrow id="S3.E36.m1.1.1.1.1.3" xref="S3.E36.m1.1.1.1.1.3.cmml"><mrow id="S3.E36.m1.1.1.1.1.3.2" xref="S3.E36.m1.1.1.1.1.3.2.cmml"><msub id="S3.E36.m1.1.1.1.1.3.2.2" xref="S3.E36.m1.1.1.1.1.3.2.2.cmml"><mi id="S3.E36.m1.1.1.1.1.3.2.2.2" xref="S3.E36.m1.1.1.1.1.3.2.2.2.cmml">p</mi><mi id="S3.E36.m1.1.1.1.1.3.2.2.3" xref="S3.E36.m1.1.1.1.1.3.2.2.3.cmml">j</mi></msub><mo id="S3.E36.m1.1.1.1.1.3.2.1" xref="S3.E36.m1.1.1.1.1.3.2.1.cmml">⁢</mo><mi id="S3.E36.m1.1.1.1.1.3.2.3" xref="S3.E36.m1.1.1.1.1.3.2.3.cmml">d</mi><mo id="S3.E36.m1.1.1.1.1.3.2.1a" xref="S3.E36.m1.1.1.1.1.3.2.1.cmml">⁢</mo><msup id="S3.E36.m1.1.1.1.1.3.2.4" xref="S3.E36.m1.1.1.1.1.3.2.4.cmml"><mi id="S3.E36.m1.1.1.1.1.3.2.4.2" xref="S3.E36.m1.1.1.1.1.3.2.4.2.cmml">x</mi><mi id="S3.E36.m1.1.1.1.1.3.2.4.3" xref="S3.E36.m1.1.1.1.1.3.2.4.3.cmml">j</mi></msup></mrow><mo id="S3.E36.m1.1.1.1.1.3.1" xref="S3.E36.m1.1.1.1.1.3.1.cmml">−</mo><mrow id="S3.E36.m1.1.1.1.1.3.3" xref="S3.E36.m1.1.1.1.1.3.3.cmml"><mi id="S3.E36.m1.1.1.1.1.3.3.2" xref="S3.E36.m1.1.1.1.1.3.3.2.cmml">H</mi><mo id="S3.E36.m1.1.1.1.1.3.3.1" xref="S3.E36.m1.1.1.1.1.3.3.1.cmml">⁢</mo><mi id="S3.E36.m1.1.1.1.1.3.3.3" xref="S3.E36.m1.1.1.1.1.3.3.3.cmml">d</mi><mo id="S3.E36.m1.1.1.1.1.3.3.1a" xref="S3.E36.m1.1.1.1.1.3.3.1.cmml">⁢</mo><mi id="S3.E36.m1.1.1.1.1.3.3.4" xref="S3.E36.m1.1.1.1.1.3.3.4.cmml">t</mi></mrow></mrow></mrow><mo id="S3.E36.m1.1.1.1.2" xref="S3.E36.m1.1.1.1.1.cmml">,</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.E36.m1.1b"><apply id="S3.E36.m1.1.1.1.1.cmml" xref="S3.E36.m1.1.1.1"><csymbol cd="latexml" id="S3.E36.m1.1.1.1.1.1.cmml" xref="S3.E36.m1.1.1.1.1.1">assign</csymbol><apply id="S3.E36.m1.1.1.1.1.2.cmml" xref="S3.E36.m1.1.1.1.1.2"><csymbol cd="ambiguous" id="S3.E36.m1.1.1.1.1.2.1.cmml" xref="S3.E36.m1.1.1.1.1.2">subscript</csymbol><ci id="S3.E36.m1.1.1.1.1.2.2.cmml" xref="S3.E36.m1.1.1.1.1.2.2">𝜔</ci><ci id="S3.E36.m1.1.1.1.1.2.3.cmml" xref="S3.E36.m1.1.1.1.1.2.3">PC</ci></apply><apply id="S3.E36.m1.1.1.1.1.3.cmml" xref="S3.E36.m1.1.1.1.1.3"><minus id="S3.E36.m1.1.1.1.1.3.1.cmml" xref="S3.E36.m1.1.1.1.1.3.1"></minus><apply id="S3.E36.m1.1.1.1.1.3.2.cmml" xref="S3.E36.m1.1.1.1.1.3.2"><times id="S3.E36.m1.1.1.1.1.3.2.1.cmml" xref="S3.E36.m1.1.1.1.1.3.2.1"></times><apply id="S3.E36.m1.1.1.1.1.3.2.2.cmml" xref="S3.E36.m1.1.1.1.1.3.2.2"><csymbol cd="ambiguous" id="S3.E36.m1.1.1.1.1.3.2.2.1.cmml" xref="S3.E36.m1.1.1.1.1.3.2.2">subscript</csymbol><ci id="S3.E36.m1.1.1.1.1.3.2.2.2.cmml" xref="S3.E36.m1.1.1.1.1.3.2.2.2">𝑝</ci><ci id="S3.E36.m1.1.1.1.1.3.2.2.3.cmml" xref="S3.E36.m1.1.1.1.1.3.2.2.3">𝑗</ci></apply><ci id="S3.E36.m1.1.1.1.1.3.2.3.cmml" xref="S3.E36.m1.1.1.1.1.3.2.3">𝑑</ci><apply id="S3.E36.m1.1.1.1.1.3.2.4.cmml" xref="S3.E36.m1.1.1.1.1.3.2.4"><csymbol cd="ambiguous" id="S3.E36.m1.1.1.1.1.3.2.4.1.cmml" xref="S3.E36.m1.1.1.1.1.3.2.4">superscript</csymbol><ci id="S3.E36.m1.1.1.1.1.3.2.4.2.cmml" xref="S3.E36.m1.1.1.1.1.3.2.4.2">𝑥</ci><ci id="S3.E36.m1.1.1.1.1.3.2.4.3.cmml" xref="S3.E36.m1.1.1.1.1.3.2.4.3">𝑗</ci></apply></apply><apply id="S3.E36.m1.1.1.1.1.3.3.cmml" xref="S3.E36.m1.1.1.1.1.3.3"><times id="S3.E36.m1.1.1.1.1.3.3.1.cmml" xref="S3.E36.m1.1.1.1.1.3.3.1"></times><ci id="S3.E36.m1.1.1.1.1.3.3.2.cmml" xref="S3.E36.m1.1.1.1.1.3.3.2">𝐻</ci><ci id="S3.E36.m1.1.1.1.1.3.3.3.cmml" xref="S3.E36.m1.1.1.1.1.3.3.3">𝑑</ci><ci id="S3.E36.m1.1.1.1.1.3.3.4.cmml" xref="S3.E36.m1.1.1.1.1.3.3.4">𝑡</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.E36.m1.1c">\displaystyle\omega_{\rm PC}:=p_{j}dx^{j}-Hdt,</annotation><annotation encoding="application/x-llamapun" id="S3.E36.m1.1d">italic_ω start_POSTSUBSCRIPT roman_PC end_POSTSUBSCRIPT := italic_p start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT italic_d italic_x start_POSTSUPERSCRIPT italic_j end_POSTSUPERSCRIPT - italic_H italic_d italic_t ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_left_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(36)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S3.p1.3">is very useful for studying the time evolution in classical mechanics under the action of a Hamiltonian <math alttext="H=H(x,p,t)" class="ltx_Math" display="inline" id="S3.p1.1.m1.3"><semantics id="S3.p1.1.m1.3a"><mrow id="S3.p1.1.m1.3.4" xref="S3.p1.1.m1.3.4.cmml"><mi id="S3.p1.1.m1.3.4.2" xref="S3.p1.1.m1.3.4.2.cmml">H</mi><mo id="S3.p1.1.m1.3.4.1" xref="S3.p1.1.m1.3.4.1.cmml">=</mo><mrow id="S3.p1.1.m1.3.4.3" xref="S3.p1.1.m1.3.4.3.cmml"><mi id="S3.p1.1.m1.3.4.3.2" xref="S3.p1.1.m1.3.4.3.2.cmml">H</mi><mo id="S3.p1.1.m1.3.4.3.1" xref="S3.p1.1.m1.3.4.3.1.cmml">⁢</mo><mrow id="S3.p1.1.m1.3.4.3.3.2" xref="S3.p1.1.m1.3.4.3.3.1.cmml"><mo id="S3.p1.1.m1.3.4.3.3.2.1" stretchy="false" xref="S3.p1.1.m1.3.4.3.3.1.cmml">(</mo><mi id="S3.p1.1.m1.1.1" xref="S3.p1.1.m1.1.1.cmml">x</mi><mo id="S3.p1.1.m1.3.4.3.3.2.2" xref="S3.p1.1.m1.3.4.3.3.1.cmml">,</mo><mi id="S3.p1.1.m1.2.2" xref="S3.p1.1.m1.2.2.cmml">p</mi><mo id="S3.p1.1.m1.3.4.3.3.2.3" xref="S3.p1.1.m1.3.4.3.3.1.cmml">,</mo><mi id="S3.p1.1.m1.3.3" xref="S3.p1.1.m1.3.3.cmml">t</mi><mo id="S3.p1.1.m1.3.4.3.3.2.4" stretchy="false" xref="S3.p1.1.m1.3.4.3.3.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.p1.1.m1.3b"><apply id="S3.p1.1.m1.3.4.cmml" xref="S3.p1.1.m1.3.4"><eq id="S3.p1.1.m1.3.4.1.cmml" xref="S3.p1.1.m1.3.4.1"></eq><ci id="S3.p1.1.m1.3.4.2.cmml" xref="S3.p1.1.m1.3.4.2">𝐻</ci><apply id="S3.p1.1.m1.3.4.3.cmml" xref="S3.p1.1.m1.3.4.3"><times id="S3.p1.1.m1.3.4.3.1.cmml" xref="S3.p1.1.m1.3.4.3.1"></times><ci id="S3.p1.1.m1.3.4.3.2.cmml" xref="S3.p1.1.m1.3.4.3.2">𝐻</ci><vector id="S3.p1.1.m1.3.4.3.3.1.cmml" xref="S3.p1.1.m1.3.4.3.3.2"><ci id="S3.p1.1.m1.1.1.cmml" xref="S3.p1.1.m1.1.1">𝑥</ci><ci id="S3.p1.1.m1.2.2.cmml" xref="S3.p1.1.m1.2.2">𝑝</ci><ci id="S3.p1.1.m1.3.3.cmml" xref="S3.p1.1.m1.3.3">𝑡</ci></vector></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p1.1.m1.3c">H=H(x,p,t)</annotation><annotation encoding="application/x-llamapun" id="S3.p1.1.m1.3d">italic_H = italic_H ( italic_x , italic_p , italic_t )</annotation></semantics></math>. The one-form <math alttext="\omega_{\rm PC}" class="ltx_Math" display="inline" id="S3.p1.2.m2.1"><semantics id="S3.p1.2.m2.1a"><msub id="S3.p1.2.m2.1.1" xref="S3.p1.2.m2.1.1.cmml"><mi id="S3.p1.2.m2.1.1.2" xref="S3.p1.2.m2.1.1.2.cmml">ω</mi><mi id="S3.p1.2.m2.1.1.3" xref="S3.p1.2.m2.1.1.3.cmml">PC</mi></msub><annotation-xml encoding="MathML-Content" id="S3.p1.2.m2.1b"><apply id="S3.p1.2.m2.1.1.cmml" xref="S3.p1.2.m2.1.1"><csymbol cd="ambiguous" id="S3.p1.2.m2.1.1.1.cmml" xref="S3.p1.2.m2.1.1">subscript</csymbol><ci id="S3.p1.2.m2.1.1.2.cmml" xref="S3.p1.2.m2.1.1.2">𝜔</ci><ci id="S3.p1.2.m2.1.1.3.cmml" xref="S3.p1.2.m2.1.1.3">PC</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p1.2.m2.1c">\omega_{\rm PC}</annotation><annotation encoding="application/x-llamapun" id="S3.p1.2.m2.1d">italic_ω start_POSTSUBSCRIPT roman_PC end_POSTSUBSCRIPT</annotation></semantics></math> is defined in the extended configuration space of <math alttext="(x,t)\in\mathcal{M}\times\mathbb{R}" class="ltx_Math" display="inline" id="S3.p1.3.m3.2"><semantics id="S3.p1.3.m3.2a"><mrow id="S3.p1.3.m3.2.3" xref="S3.p1.3.m3.2.3.cmml"><mrow id="S3.p1.3.m3.2.3.2.2" xref="S3.p1.3.m3.2.3.2.1.cmml"><mo id="S3.p1.3.m3.2.3.2.2.1" stretchy="false" xref="S3.p1.3.m3.2.3.2.1.cmml">(</mo><mi id="S3.p1.3.m3.1.1" xref="S3.p1.3.m3.1.1.cmml">x</mi><mo id="S3.p1.3.m3.2.3.2.2.2" xref="S3.p1.3.m3.2.3.2.1.cmml">,</mo><mi id="S3.p1.3.m3.2.2" xref="S3.p1.3.m3.2.2.cmml">t</mi><mo id="S3.p1.3.m3.2.3.2.2.3" stretchy="false" xref="S3.p1.3.m3.2.3.2.1.cmml">)</mo></mrow><mo id="S3.p1.3.m3.2.3.1" xref="S3.p1.3.m3.2.3.1.cmml">∈</mo><mrow id="S3.p1.3.m3.2.3.3" xref="S3.p1.3.m3.2.3.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.p1.3.m3.2.3.3.2" xref="S3.p1.3.m3.2.3.3.2.cmml">ℳ</mi><mo id="S3.p1.3.m3.2.3.3.1" lspace="0.222em" rspace="0.222em" xref="S3.p1.3.m3.2.3.3.1.cmml">×</mo><mi id="S3.p1.3.m3.2.3.3.3" xref="S3.p1.3.m3.2.3.3.3.cmml">ℝ</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.p1.3.m3.2b"><apply id="S3.p1.3.m3.2.3.cmml" xref="S3.p1.3.m3.2.3"><in id="S3.p1.3.m3.2.3.1.cmml" xref="S3.p1.3.m3.2.3.1"></in><interval closure="open" id="S3.p1.3.m3.2.3.2.1.cmml" xref="S3.p1.3.m3.2.3.2.2"><ci id="S3.p1.3.m3.1.1.cmml" xref="S3.p1.3.m3.1.1">𝑥</ci><ci id="S3.p1.3.m3.2.2.cmml" xref="S3.p1.3.m3.2.2">𝑡</ci></interval><apply id="S3.p1.3.m3.2.3.3.cmml" xref="S3.p1.3.m3.2.3.3"><times id="S3.p1.3.m3.2.3.3.1.cmml" xref="S3.p1.3.m3.2.3.3.1"></times><ci id="S3.p1.3.m3.2.3.3.2.cmml" xref="S3.p1.3.m3.2.3.3.2">ℳ</ci><ci id="S3.p1.3.m3.2.3.3.3.cmml" xref="S3.p1.3.m3.2.3.3.3">ℝ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p1.3.m3.2c">(x,t)\in\mathcal{M}\times\mathbb{R}</annotation><annotation encoding="application/x-llamapun" id="S3.p1.3.m3.2d">( italic_x , italic_t ) ∈ caligraphic_M × blackboard_R</annotation></semantics></math>, and is known as the Poincaré-Cartan one-form <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2406.11224v2#bib.bib18" title="">18</a>]</cite>.</p> </div> <div class="ltx_para" id="S3.p2"> <p class="ltx_p" id="S3.p2.17">Now consider the complete integrability of the Pfaffian equation</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="S3.EGx37"> <tbody id="S3.E37"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_left_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\omega_{\rm PC}=0," class="ltx_Math" display="inline" id="S3.E37.m1.1"><semantics id="S3.E37.m1.1a"><mrow id="S3.E37.m1.1.1.1" xref="S3.E37.m1.1.1.1.1.cmml"><mrow id="S3.E37.m1.1.1.1.1" xref="S3.E37.m1.1.1.1.1.cmml"><msub id="S3.E37.m1.1.1.1.1.2" xref="S3.E37.m1.1.1.1.1.2.cmml"><mi id="S3.E37.m1.1.1.1.1.2.2" xref="S3.E37.m1.1.1.1.1.2.2.cmml">ω</mi><mi id="S3.E37.m1.1.1.1.1.2.3" xref="S3.E37.m1.1.1.1.1.2.3.cmml">PC</mi></msub><mo id="S3.E37.m1.1.1.1.1.1" xref="S3.E37.m1.1.1.1.1.1.cmml">=</mo><mn id="S3.E37.m1.1.1.1.1.3" xref="S3.E37.m1.1.1.1.1.3.cmml">0</mn></mrow><mo id="S3.E37.m1.1.1.1.2" xref="S3.E37.m1.1.1.1.1.cmml">,</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.E37.m1.1b"><apply id="S3.E37.m1.1.1.1.1.cmml" xref="S3.E37.m1.1.1.1"><eq id="S3.E37.m1.1.1.1.1.1.cmml" xref="S3.E37.m1.1.1.1.1.1"></eq><apply id="S3.E37.m1.1.1.1.1.2.cmml" xref="S3.E37.m1.1.1.1.1.2"><csymbol cd="ambiguous" id="S3.E37.m1.1.1.1.1.2.1.cmml" xref="S3.E37.m1.1.1.1.1.2">subscript</csymbol><ci id="S3.E37.m1.1.1.1.1.2.2.cmml" xref="S3.E37.m1.1.1.1.1.2.2">𝜔</ci><ci id="S3.E37.m1.1.1.1.1.2.3.cmml" xref="S3.E37.m1.1.1.1.1.2.3">PC</ci></apply><cn id="S3.E37.m1.1.1.1.1.3.cmml" type="integer" xref="S3.E37.m1.1.1.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.E37.m1.1c">\displaystyle\omega_{\rm PC}=0,</annotation><annotation encoding="application/x-llamapun" id="S3.E37.m1.1d">italic_ω start_POSTSUBSCRIPT roman_PC end_POSTSUBSCRIPT = 0 ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_left_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(37)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S3.p2.8">for the Poincaré-Cartan one-form <math alttext="\omega_{\rm PC}" class="ltx_Math" display="inline" id="S3.p2.1.m1.1"><semantics id="S3.p2.1.m1.1a"><msub id="S3.p2.1.m1.1.1" xref="S3.p2.1.m1.1.1.cmml"><mi id="S3.p2.1.m1.1.1.2" xref="S3.p2.1.m1.1.1.2.cmml">ω</mi><mi id="S3.p2.1.m1.1.1.3" xref="S3.p2.1.m1.1.1.3.cmml">PC</mi></msub><annotation-xml encoding="MathML-Content" id="S3.p2.1.m1.1b"><apply id="S3.p2.1.m1.1.1.cmml" xref="S3.p2.1.m1.1.1"><csymbol cd="ambiguous" id="S3.p2.1.m1.1.1.1.cmml" xref="S3.p2.1.m1.1.1">subscript</csymbol><ci id="S3.p2.1.m1.1.1.2.cmml" xref="S3.p2.1.m1.1.1.2">𝜔</ci><ci id="S3.p2.1.m1.1.1.3.cmml" xref="S3.p2.1.m1.1.1.3">PC</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p2.1.m1.1c">\omega_{\rm PC}</annotation><annotation encoding="application/x-llamapun" id="S3.p2.1.m1.1d">italic_ω start_POSTSUBSCRIPT roman_PC end_POSTSUBSCRIPT</annotation></semantics></math> (<a class="ltx_ref" href="https://arxiv.org/html/2406.11224v2#S3.E36" title="In 3 Complete integrability and geodesic Hamiltonian ‣ A Hamiltonian approach to the gradient-flow equations in information geometry"><span class="ltx_text ltx_ref_tag">36</span></a>). Recall that the Pfaffian system is said to be <span class="ltx_text ltx_font_italic" id="S3.p2.8.1">completely integrable</span> if the integral surface of (<a class="ltx_ref" href="https://arxiv.org/html/2406.11224v2#S3.E37" title="In 3 Complete integrability and geodesic Hamiltonian ‣ A Hamiltonian approach to the gradient-flow equations in information geometry"><span class="ltx_text ltx_ref_tag">37</span></a>) is given by the equations <math alttext="S_{a}=\textrm{constant}" class="ltx_Math" display="inline" id="S3.p2.2.m2.1"><semantics id="S3.p2.2.m2.1a"><mrow id="S3.p2.2.m2.1.1" xref="S3.p2.2.m2.1.1.cmml"><msub id="S3.p2.2.m2.1.1.2" xref="S3.p2.2.m2.1.1.2.cmml"><mi id="S3.p2.2.m2.1.1.2.2" xref="S3.p2.2.m2.1.1.2.2.cmml">S</mi><mi id="S3.p2.2.m2.1.1.2.3" xref="S3.p2.2.m2.1.1.2.3.cmml">a</mi></msub><mo id="S3.p2.2.m2.1.1.1" xref="S3.p2.2.m2.1.1.1.cmml">=</mo><mtext id="S3.p2.2.m2.1.1.3" xref="S3.p2.2.m2.1.1.3a.cmml">constant</mtext></mrow><annotation-xml encoding="MathML-Content" id="S3.p2.2.m2.1b"><apply id="S3.p2.2.m2.1.1.cmml" xref="S3.p2.2.m2.1.1"><eq id="S3.p2.2.m2.1.1.1.cmml" xref="S3.p2.2.m2.1.1.1"></eq><apply id="S3.p2.2.m2.1.1.2.cmml" xref="S3.p2.2.m2.1.1.2"><csymbol cd="ambiguous" id="S3.p2.2.m2.1.1.2.1.cmml" xref="S3.p2.2.m2.1.1.2">subscript</csymbol><ci id="S3.p2.2.m2.1.1.2.2.cmml" xref="S3.p2.2.m2.1.1.2.2">𝑆</ci><ci id="S3.p2.2.m2.1.1.2.3.cmml" xref="S3.p2.2.m2.1.1.2.3">𝑎</ci></apply><ci id="S3.p2.2.m2.1.1.3a.cmml" xref="S3.p2.2.m2.1.1.3"><mtext id="S3.p2.2.m2.1.1.3.cmml" xref="S3.p2.2.m2.1.1.3">constant</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p2.2.m2.1c">S_{a}=\textrm{constant}</annotation><annotation encoding="application/x-llamapun" id="S3.p2.2.m2.1d">italic_S start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT = constant</annotation></semantics></math>, where <math alttext="S_{a}" class="ltx_Math" display="inline" id="S3.p2.3.m3.1"><semantics id="S3.p2.3.m3.1a"><msub id="S3.p2.3.m3.1.1" xref="S3.p2.3.m3.1.1.cmml"><mi id="S3.p2.3.m3.1.1.2" xref="S3.p2.3.m3.1.1.2.cmml">S</mi><mi id="S3.p2.3.m3.1.1.3" xref="S3.p2.3.m3.1.1.3.cmml">a</mi></msub><annotation-xml encoding="MathML-Content" id="S3.p2.3.m3.1b"><apply id="S3.p2.3.m3.1.1.cmml" xref="S3.p2.3.m3.1.1"><csymbol cd="ambiguous" id="S3.p2.3.m3.1.1.1.cmml" xref="S3.p2.3.m3.1.1">subscript</csymbol><ci id="S3.p2.3.m3.1.1.2.cmml" xref="S3.p2.3.m3.1.1.2">𝑆</ci><ci id="S3.p2.3.m3.1.1.3.cmml" xref="S3.p2.3.m3.1.1.3">𝑎</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p2.3.m3.1c">S_{a}</annotation><annotation encoding="application/x-llamapun" id="S3.p2.3.m3.1d">italic_S start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT</annotation></semantics></math> is a potential function and <math alttext="\omega_{\rm PC}=dS_{a}" class="ltx_Math" display="inline" id="S3.p2.4.m4.1"><semantics id="S3.p2.4.m4.1a"><mrow id="S3.p2.4.m4.1.1" xref="S3.p2.4.m4.1.1.cmml"><msub id="S3.p2.4.m4.1.1.2" xref="S3.p2.4.m4.1.1.2.cmml"><mi id="S3.p2.4.m4.1.1.2.2" xref="S3.p2.4.m4.1.1.2.2.cmml">ω</mi><mi id="S3.p2.4.m4.1.1.2.3" xref="S3.p2.4.m4.1.1.2.3.cmml">PC</mi></msub><mo id="S3.p2.4.m4.1.1.1" xref="S3.p2.4.m4.1.1.1.cmml">=</mo><mrow id="S3.p2.4.m4.1.1.3" xref="S3.p2.4.m4.1.1.3.cmml"><mi id="S3.p2.4.m4.1.1.3.2" xref="S3.p2.4.m4.1.1.3.2.cmml">d</mi><mo id="S3.p2.4.m4.1.1.3.1" xref="S3.p2.4.m4.1.1.3.1.cmml">⁢</mo><msub id="S3.p2.4.m4.1.1.3.3" xref="S3.p2.4.m4.1.1.3.3.cmml"><mi id="S3.p2.4.m4.1.1.3.3.2" xref="S3.p2.4.m4.1.1.3.3.2.cmml">S</mi><mi id="S3.p2.4.m4.1.1.3.3.3" xref="S3.p2.4.m4.1.1.3.3.3.cmml">a</mi></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.p2.4.m4.1b"><apply id="S3.p2.4.m4.1.1.cmml" xref="S3.p2.4.m4.1.1"><eq id="S3.p2.4.m4.1.1.1.cmml" xref="S3.p2.4.m4.1.1.1"></eq><apply id="S3.p2.4.m4.1.1.2.cmml" xref="S3.p2.4.m4.1.1.2"><csymbol cd="ambiguous" id="S3.p2.4.m4.1.1.2.1.cmml" xref="S3.p2.4.m4.1.1.2">subscript</csymbol><ci id="S3.p2.4.m4.1.1.2.2.cmml" xref="S3.p2.4.m4.1.1.2.2">𝜔</ci><ci id="S3.p2.4.m4.1.1.2.3.cmml" xref="S3.p2.4.m4.1.1.2.3">PC</ci></apply><apply id="S3.p2.4.m4.1.1.3.cmml" xref="S3.p2.4.m4.1.1.3"><times id="S3.p2.4.m4.1.1.3.1.cmml" xref="S3.p2.4.m4.1.1.3.1"></times><ci id="S3.p2.4.m4.1.1.3.2.cmml" xref="S3.p2.4.m4.1.1.3.2">𝑑</ci><apply id="S3.p2.4.m4.1.1.3.3.cmml" xref="S3.p2.4.m4.1.1.3.3"><csymbol cd="ambiguous" id="S3.p2.4.m4.1.1.3.3.1.cmml" xref="S3.p2.4.m4.1.1.3.3">subscript</csymbol><ci id="S3.p2.4.m4.1.1.3.3.2.cmml" xref="S3.p2.4.m4.1.1.3.3.2">𝑆</ci><ci id="S3.p2.4.m4.1.1.3.3.3.cmml" xref="S3.p2.4.m4.1.1.3.3.3">𝑎</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p2.4.m4.1c">\omega_{\rm PC}=dS_{a}</annotation><annotation encoding="application/x-llamapun" id="S3.p2.4.m4.1d">italic_ω start_POSTSUBSCRIPT roman_PC end_POSTSUBSCRIPT = italic_d italic_S start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT</annotation></semantics></math>. In other words, there exists a differentiable function <math alttext="S_{a}=S_{a}(x,t)" class="ltx_Math" display="inline" id="S3.p2.5.m5.2"><semantics id="S3.p2.5.m5.2a"><mrow id="S3.p2.5.m5.2.3" xref="S3.p2.5.m5.2.3.cmml"><msub id="S3.p2.5.m5.2.3.2" xref="S3.p2.5.m5.2.3.2.cmml"><mi id="S3.p2.5.m5.2.3.2.2" xref="S3.p2.5.m5.2.3.2.2.cmml">S</mi><mi id="S3.p2.5.m5.2.3.2.3" xref="S3.p2.5.m5.2.3.2.3.cmml">a</mi></msub><mo id="S3.p2.5.m5.2.3.1" xref="S3.p2.5.m5.2.3.1.cmml">=</mo><mrow id="S3.p2.5.m5.2.3.3" xref="S3.p2.5.m5.2.3.3.cmml"><msub id="S3.p2.5.m5.2.3.3.2" xref="S3.p2.5.m5.2.3.3.2.cmml"><mi id="S3.p2.5.m5.2.3.3.2.2" xref="S3.p2.5.m5.2.3.3.2.2.cmml">S</mi><mi id="S3.p2.5.m5.2.3.3.2.3" xref="S3.p2.5.m5.2.3.3.2.3.cmml">a</mi></msub><mo id="S3.p2.5.m5.2.3.3.1" xref="S3.p2.5.m5.2.3.3.1.cmml">⁢</mo><mrow id="S3.p2.5.m5.2.3.3.3.2" xref="S3.p2.5.m5.2.3.3.3.1.cmml"><mo id="S3.p2.5.m5.2.3.3.3.2.1" stretchy="false" xref="S3.p2.5.m5.2.3.3.3.1.cmml">(</mo><mi id="S3.p2.5.m5.1.1" xref="S3.p2.5.m5.1.1.cmml">x</mi><mo id="S3.p2.5.m5.2.3.3.3.2.2" xref="S3.p2.5.m5.2.3.3.3.1.cmml">,</mo><mi id="S3.p2.5.m5.2.2" xref="S3.p2.5.m5.2.2.cmml">t</mi><mo id="S3.p2.5.m5.2.3.3.3.2.3" stretchy="false" xref="S3.p2.5.m5.2.3.3.3.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.p2.5.m5.2b"><apply id="S3.p2.5.m5.2.3.cmml" xref="S3.p2.5.m5.2.3"><eq id="S3.p2.5.m5.2.3.1.cmml" xref="S3.p2.5.m5.2.3.1"></eq><apply id="S3.p2.5.m5.2.3.2.cmml" xref="S3.p2.5.m5.2.3.2"><csymbol cd="ambiguous" id="S3.p2.5.m5.2.3.2.1.cmml" xref="S3.p2.5.m5.2.3.2">subscript</csymbol><ci id="S3.p2.5.m5.2.3.2.2.cmml" xref="S3.p2.5.m5.2.3.2.2">𝑆</ci><ci id="S3.p2.5.m5.2.3.2.3.cmml" xref="S3.p2.5.m5.2.3.2.3">𝑎</ci></apply><apply id="S3.p2.5.m5.2.3.3.cmml" xref="S3.p2.5.m5.2.3.3"><times id="S3.p2.5.m5.2.3.3.1.cmml" xref="S3.p2.5.m5.2.3.3.1"></times><apply id="S3.p2.5.m5.2.3.3.2.cmml" xref="S3.p2.5.m5.2.3.3.2"><csymbol cd="ambiguous" id="S3.p2.5.m5.2.3.3.2.1.cmml" xref="S3.p2.5.m5.2.3.3.2">subscript</csymbol><ci id="S3.p2.5.m5.2.3.3.2.2.cmml" xref="S3.p2.5.m5.2.3.3.2.2">𝑆</ci><ci id="S3.p2.5.m5.2.3.3.2.3.cmml" xref="S3.p2.5.m5.2.3.3.2.3">𝑎</ci></apply><interval closure="open" id="S3.p2.5.m5.2.3.3.3.1.cmml" xref="S3.p2.5.m5.2.3.3.3.2"><ci id="S3.p2.5.m5.1.1.cmml" xref="S3.p2.5.m5.1.1">𝑥</ci><ci id="S3.p2.5.m5.2.2.cmml" xref="S3.p2.5.m5.2.2">𝑡</ci></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p2.5.m5.2c">S_{a}=S_{a}(x,t)</annotation><annotation encoding="application/x-llamapun" id="S3.p2.5.m5.2d">italic_S start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT = italic_S start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT ( italic_x , italic_t )</annotation></semantics></math> such that the Pfaffian equation (<a class="ltx_ref" href="https://arxiv.org/html/2406.11224v2#S3.E37" title="In 3 Complete integrability and geodesic Hamiltonian ‣ A Hamiltonian approach to the gradient-flow equations in information geometry"><span class="ltx_text ltx_ref_tag">37</span></a>) is equivalent to <math alttext="dS_{a}=0" class="ltx_Math" display="inline" id="S3.p2.6.m6.1"><semantics id="S3.p2.6.m6.1a"><mrow id="S3.p2.6.m6.1.1" xref="S3.p2.6.m6.1.1.cmml"><mrow id="S3.p2.6.m6.1.1.2" xref="S3.p2.6.m6.1.1.2.cmml"><mi id="S3.p2.6.m6.1.1.2.2" xref="S3.p2.6.m6.1.1.2.2.cmml">d</mi><mo id="S3.p2.6.m6.1.1.2.1" xref="S3.p2.6.m6.1.1.2.1.cmml">⁢</mo><msub id="S3.p2.6.m6.1.1.2.3" xref="S3.p2.6.m6.1.1.2.3.cmml"><mi id="S3.p2.6.m6.1.1.2.3.2" xref="S3.p2.6.m6.1.1.2.3.2.cmml">S</mi><mi id="S3.p2.6.m6.1.1.2.3.3" xref="S3.p2.6.m6.1.1.2.3.3.cmml">a</mi></msub></mrow><mo id="S3.p2.6.m6.1.1.1" xref="S3.p2.6.m6.1.1.1.cmml">=</mo><mn id="S3.p2.6.m6.1.1.3" xref="S3.p2.6.m6.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.p2.6.m6.1b"><apply id="S3.p2.6.m6.1.1.cmml" xref="S3.p2.6.m6.1.1"><eq id="S3.p2.6.m6.1.1.1.cmml" xref="S3.p2.6.m6.1.1.1"></eq><apply id="S3.p2.6.m6.1.1.2.cmml" xref="S3.p2.6.m6.1.1.2"><times id="S3.p2.6.m6.1.1.2.1.cmml" xref="S3.p2.6.m6.1.1.2.1"></times><ci id="S3.p2.6.m6.1.1.2.2.cmml" xref="S3.p2.6.m6.1.1.2.2">𝑑</ci><apply id="S3.p2.6.m6.1.1.2.3.cmml" xref="S3.p2.6.m6.1.1.2.3"><csymbol cd="ambiguous" id="S3.p2.6.m6.1.1.2.3.1.cmml" xref="S3.p2.6.m6.1.1.2.3">subscript</csymbol><ci id="S3.p2.6.m6.1.1.2.3.2.cmml" xref="S3.p2.6.m6.1.1.2.3.2">𝑆</ci><ci id="S3.p2.6.m6.1.1.2.3.3.cmml" xref="S3.p2.6.m6.1.1.2.3.3">𝑎</ci></apply></apply><cn id="S3.p2.6.m6.1.1.3.cmml" type="integer" xref="S3.p2.6.m6.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p2.6.m6.1c">dS_{a}=0</annotation><annotation encoding="application/x-llamapun" id="S3.p2.6.m6.1d">italic_d italic_S start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT = 0</annotation></semantics></math>. Soon later, we will see that this function <math alttext="S_{a}" class="ltx_Math" display="inline" id="S3.p2.7.m7.1"><semantics id="S3.p2.7.m7.1a"><msub id="S3.p2.7.m7.1.1" xref="S3.p2.7.m7.1.1.cmml"><mi id="S3.p2.7.m7.1.1.2" xref="S3.p2.7.m7.1.1.2.cmml">S</mi><mi id="S3.p2.7.m7.1.1.3" xref="S3.p2.7.m7.1.1.3.cmml">a</mi></msub><annotation-xml encoding="MathML-Content" id="S3.p2.7.m7.1b"><apply id="S3.p2.7.m7.1.1.cmml" xref="S3.p2.7.m7.1.1"><csymbol cd="ambiguous" id="S3.p2.7.m7.1.1.1.cmml" xref="S3.p2.7.m7.1.1">subscript</csymbol><ci id="S3.p2.7.m7.1.1.2.cmml" xref="S3.p2.7.m7.1.1.2">𝑆</ci><ci id="S3.p2.7.m7.1.1.3.cmml" xref="S3.p2.7.m7.1.1.3">𝑎</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p2.7.m7.1c">S_{a}</annotation><annotation encoding="application/x-llamapun" id="S3.p2.7.m7.1d">italic_S start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT</annotation></semantics></math> is the <span class="ltx_text ltx_font_italic" id="S3.p2.8.2">action</span> in analytical mechanics. According to Frobenius integrability theorem <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2406.11224v2#bib.bib17" title="">17</a>]</cite>, the necessary and sufficient condition of the complete integrability of (<a class="ltx_ref" href="https://arxiv.org/html/2406.11224v2#S3.E37" title="In 3 Complete integrability and geodesic Hamiltonian ‣ A Hamiltonian approach to the gradient-flow equations in information geometry"><span class="ltx_text ltx_ref_tag">37</span></a>) is <math alttext="d\omega_{\rm PC}\wedge\omega_{\rm PC}=0" class="ltx_Math" display="inline" id="S3.p2.8.m8.1"><semantics id="S3.p2.8.m8.1a"><mrow id="S3.p2.8.m8.1.1" xref="S3.p2.8.m8.1.1.cmml"><mrow id="S3.p2.8.m8.1.1.2" xref="S3.p2.8.m8.1.1.2.cmml"><mrow id="S3.p2.8.m8.1.1.2.2" xref="S3.p2.8.m8.1.1.2.2.cmml"><mi id="S3.p2.8.m8.1.1.2.2.2" xref="S3.p2.8.m8.1.1.2.2.2.cmml">d</mi><mo id="S3.p2.8.m8.1.1.2.2.1" xref="S3.p2.8.m8.1.1.2.2.1.cmml">⁢</mo><msub id="S3.p2.8.m8.1.1.2.2.3" xref="S3.p2.8.m8.1.1.2.2.3.cmml"><mi id="S3.p2.8.m8.1.1.2.2.3.2" xref="S3.p2.8.m8.1.1.2.2.3.2.cmml">ω</mi><mi id="S3.p2.8.m8.1.1.2.2.3.3" xref="S3.p2.8.m8.1.1.2.2.3.3.cmml">PC</mi></msub></mrow><mo id="S3.p2.8.m8.1.1.2.1" xref="S3.p2.8.m8.1.1.2.1.cmml">∧</mo><msub id="S3.p2.8.m8.1.1.2.3" xref="S3.p2.8.m8.1.1.2.3.cmml"><mi id="S3.p2.8.m8.1.1.2.3.2" xref="S3.p2.8.m8.1.1.2.3.2.cmml">ω</mi><mi id="S3.p2.8.m8.1.1.2.3.3" xref="S3.p2.8.m8.1.1.2.3.3.cmml">PC</mi></msub></mrow><mo id="S3.p2.8.m8.1.1.1" xref="S3.p2.8.m8.1.1.1.cmml">=</mo><mn id="S3.p2.8.m8.1.1.3" xref="S3.p2.8.m8.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.p2.8.m8.1b"><apply id="S3.p2.8.m8.1.1.cmml" xref="S3.p2.8.m8.1.1"><eq id="S3.p2.8.m8.1.1.1.cmml" xref="S3.p2.8.m8.1.1.1"></eq><apply id="S3.p2.8.m8.1.1.2.cmml" xref="S3.p2.8.m8.1.1.2"><and id="S3.p2.8.m8.1.1.2.1.cmml" xref="S3.p2.8.m8.1.1.2.1"></and><apply id="S3.p2.8.m8.1.1.2.2.cmml" xref="S3.p2.8.m8.1.1.2.2"><times id="S3.p2.8.m8.1.1.2.2.1.cmml" xref="S3.p2.8.m8.1.1.2.2.1"></times><ci id="S3.p2.8.m8.1.1.2.2.2.cmml" xref="S3.p2.8.m8.1.1.2.2.2">𝑑</ci><apply id="S3.p2.8.m8.1.1.2.2.3.cmml" xref="S3.p2.8.m8.1.1.2.2.3"><csymbol cd="ambiguous" id="S3.p2.8.m8.1.1.2.2.3.1.cmml" xref="S3.p2.8.m8.1.1.2.2.3">subscript</csymbol><ci id="S3.p2.8.m8.1.1.2.2.3.2.cmml" xref="S3.p2.8.m8.1.1.2.2.3.2">𝜔</ci><ci id="S3.p2.8.m8.1.1.2.2.3.3.cmml" xref="S3.p2.8.m8.1.1.2.2.3.3">PC</ci></apply></apply><apply id="S3.p2.8.m8.1.1.2.3.cmml" xref="S3.p2.8.m8.1.1.2.3"><csymbol cd="ambiguous" id="S3.p2.8.m8.1.1.2.3.1.cmml" xref="S3.p2.8.m8.1.1.2.3">subscript</csymbol><ci id="S3.p2.8.m8.1.1.2.3.2.cmml" xref="S3.p2.8.m8.1.1.2.3.2">𝜔</ci><ci id="S3.p2.8.m8.1.1.2.3.3.cmml" xref="S3.p2.8.m8.1.1.2.3.3">PC</ci></apply></apply><cn id="S3.p2.8.m8.1.1.3.cmml" type="integer" xref="S3.p2.8.m8.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p2.8.m8.1c">d\omega_{\rm PC}\wedge\omega_{\rm PC}=0</annotation><annotation encoding="application/x-llamapun" id="S3.p2.8.m8.1d">italic_d italic_ω start_POSTSUBSCRIPT roman_PC end_POSTSUBSCRIPT ∧ italic_ω start_POSTSUBSCRIPT roman_PC end_POSTSUBSCRIPT = 0</annotation></semantics></math>. With the help of Hamilton’s equations of motion, this condition becomes <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2406.11224v2#bib.bib18" title="">18</a>]</cite></p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="S3.EGx38"> <tbody id="S3.Ex6"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_left_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle d" class="ltx_Math" display="inline" id="S3.Ex6.m1.1"><semantics id="S3.Ex6.m1.1a"><mi id="S3.Ex6.m1.1.1" xref="S3.Ex6.m1.1.1.cmml">d</mi><annotation-xml encoding="MathML-Content" id="S3.Ex6.m1.1b"><ci id="S3.Ex6.m1.1.1.cmml" xref="S3.Ex6.m1.1.1">𝑑</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Ex6.m1.1c">\displaystyle d</annotation><annotation encoding="application/x-llamapun" id="S3.Ex6.m1.1d">italic_d</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\omega_{\rm PC}\wedge\omega_{\rm PC}" class="ltx_Math" display="inline" id="S3.Ex6.m2.1"><semantics id="S3.Ex6.m2.1a"><mrow id="S3.Ex6.m2.1.1" xref="S3.Ex6.m2.1.1.cmml"><msub id="S3.Ex6.m2.1.1.2" xref="S3.Ex6.m2.1.1.2.cmml"><mi id="S3.Ex6.m2.1.1.2.2" xref="S3.Ex6.m2.1.1.2.2.cmml">ω</mi><mi id="S3.Ex6.m2.1.1.2.3" xref="S3.Ex6.m2.1.1.2.3.cmml">PC</mi></msub><mo id="S3.Ex6.m2.1.1.1" xref="S3.Ex6.m2.1.1.1.cmml">∧</mo><msub id="S3.Ex6.m2.1.1.3" xref="S3.Ex6.m2.1.1.3.cmml"><mi id="S3.Ex6.m2.1.1.3.2" xref="S3.Ex6.m2.1.1.3.2.cmml">ω</mi><mi id="S3.Ex6.m2.1.1.3.3" xref="S3.Ex6.m2.1.1.3.3.cmml">PC</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.Ex6.m2.1b"><apply id="S3.Ex6.m2.1.1.cmml" xref="S3.Ex6.m2.1.1"><and id="S3.Ex6.m2.1.1.1.cmml" xref="S3.Ex6.m2.1.1.1"></and><apply id="S3.Ex6.m2.1.1.2.cmml" xref="S3.Ex6.m2.1.1.2"><csymbol cd="ambiguous" id="S3.Ex6.m2.1.1.2.1.cmml" xref="S3.Ex6.m2.1.1.2">subscript</csymbol><ci id="S3.Ex6.m2.1.1.2.2.cmml" xref="S3.Ex6.m2.1.1.2.2">𝜔</ci><ci id="S3.Ex6.m2.1.1.2.3.cmml" xref="S3.Ex6.m2.1.1.2.3">PC</ci></apply><apply id="S3.Ex6.m2.1.1.3.cmml" xref="S3.Ex6.m2.1.1.3"><csymbol cd="ambiguous" id="S3.Ex6.m2.1.1.3.1.cmml" xref="S3.Ex6.m2.1.1.3">subscript</csymbol><ci id="S3.Ex6.m2.1.1.3.2.cmml" xref="S3.Ex6.m2.1.1.3.2">𝜔</ci><ci id="S3.Ex6.m2.1.1.3.3.cmml" xref="S3.Ex6.m2.1.1.3.3">PC</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Ex6.m2.1c">\displaystyle\omega_{\rm PC}\wedge\omega_{\rm PC}</annotation><annotation encoding="application/x-llamapun" id="S3.Ex6.m2.1d">italic_ω start_POSTSUBSCRIPT roman_PC end_POSTSUBSCRIPT ∧ italic_ω start_POSTSUBSCRIPT roman_PC end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_left_padright"></td> </tr></tbody> <tbody id="S3.E38"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_left_padleft"></td> <td class="ltx_td ltx_eqn_cell"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\operatorname{\overset{\textrm{ \tiny EOM}}{=}}\left(H-p_{i}\frac% {\partial H}{\partial p_{i}}\right)dp_{j}\wedge dx^{j}\wedge dt=0." class="ltx_Math" display="inline" id="S3.E38.m1.2"><semantics id="S3.E38.m1.2a"><mrow id="S3.E38.m1.2.2.1" xref="S3.E38.m1.2.2.1.1.cmml"><mrow id="S3.E38.m1.2.2.1.1" xref="S3.E38.m1.2.2.1.1.cmml"><mrow id="S3.E38.m1.2.2.1.1.1" xref="S3.E38.m1.2.2.1.1.1.cmml"><mrow id="S3.E38.m1.2.2.1.1.1.1" xref="S3.E38.m1.2.2.1.1.1.1.cmml"><mrow id="S3.E38.m1.2.2.1.1.1.1.1.1" xref="S3.E38.m1.2.2.1.1.1.1.1.2.cmml"><mover accent="true" id="S3.E38.m1.1.1" xref="S3.E38.m1.1.1.cmml"><mo id="S3.E38.m1.1.1.2" xref="S3.E38.m1.1.1.2.cmml">=</mo><mtext id="S3.E38.m1.1.1.1" mathsize="50%" xref="S3.E38.m1.1.1.1a.cmml">EOM</mtext></mover><mrow id="S3.E38.m1.2.2.1.1.1.1.1.1.1" xref="S3.E38.m1.2.2.1.1.1.1.1.2.cmml"><mo id="S3.E38.m1.2.2.1.1.1.1.1.1.1.2" 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xref="S3.E38.m1.2.2.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.E38.m1.2c">\displaystyle\operatorname{\overset{\textrm{ \tiny EOM}}{=}}\left(H-p_{i}\frac% {\partial H}{\partial p_{i}}\right)dp_{j}\wedge dx^{j}\wedge dt=0.</annotation><annotation encoding="application/x-llamapun" id="S3.E38.m1.2d">start_OPFUNCTION overEOM start_ARG = end_ARG end_OPFUNCTION ( italic_H - italic_p start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT divide start_ARG ∂ italic_H end_ARG start_ARG ∂ italic_p start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_ARG ) italic_d italic_p start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ∧ italic_d italic_x start_POSTSUPERSCRIPT italic_j end_POSTSUPERSCRIPT ∧ italic_d italic_t = 0 .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_left_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(38)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S3.p2.9">Here the symbol <math alttext="\operatorname{\overset{\textrm{ \tiny EOM}}{=}}" class="ltx_Math" display="inline" id="S3.p2.9.m1.1"><semantics id="S3.p2.9.m1.1a"><mover accent="true" id="S3.p2.9.m1.1.1" xref="S3.p2.9.m1.1.1.cmml"><mo id="S3.p2.9.m1.1.1.2" xref="S3.p2.9.m1.1.1.2.cmml">=</mo><mrow id="S3.p2.9.m1.1.1.1" xref="S3.p2.9.m1.1.1.1c.cmml"><mtext id="S3.p2.9.m1.1.1.1a" xref="S3.p2.9.m1.1.1.1c.cmml"> </mtext><mtext id="S3.p2.9.m1.1.1.1b" mathsize="50%" xref="S3.p2.9.m1.1.1.1c.cmml">EOM</mtext></mrow></mover><annotation-xml encoding="MathML-Content" id="S3.p2.9.m1.1b"><apply id="S3.p2.9.m1.1.1.cmml" xref="S3.p2.9.m1.1.1"><ci id="S3.p2.9.m1.1.1.1c.cmml" xref="S3.p2.9.m1.1.1.1"><mrow id="S3.p2.9.m1.1.1.1.cmml" xref="S3.p2.9.m1.1.1.1"><mtext id="S3.p2.9.m1.1.1.1a.cmml" xref="S3.p2.9.m1.1.1.1"> </mtext><mtext id="S3.p2.9.m1.1.1.1b.cmml" mathsize="50%" xref="S3.p2.9.m1.1.1.1">EOM</mtext></mrow></ci><eq id="S3.p2.9.m1.1.1.2.cmml" xref="S3.p2.9.m1.1.1.2"></eq></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p2.9.m1.1c">\operatorname{\overset{\textrm{ \tiny EOM}}{=}}</annotation><annotation encoding="application/x-llamapun" id="S3.p2.9.m1.1d">start_OPFUNCTION over EOM start_ARG = end_ARG end_OPFUNCTION</annotation></semantics></math> means the equality modulo the equation of motion (EOM) and used in this section in order to avoid possible confusions. A simple proof of (<a class="ltx_ref" href="https://arxiv.org/html/2406.11224v2#S3.E38" title="In 3 Complete integrability and geodesic Hamiltonian ‣ A Hamiltonian approach to the gradient-flow equations in information geometry"><span class="ltx_text ltx_ref_tag">38</span></a>) is given in Appendix <a class="ltx_ref" href="https://arxiv.org/html/2406.11224v2#A1" title="Appendix A The proof of (38) ‣ A Hamiltonian approach to the gradient-flow equations in information geometry"><span class="ltx_text ltx_ref_tag">A</span></a>. Consequently, the Pfaffian system (<a class="ltx_ref" href="https://arxiv.org/html/2406.11224v2#S3.E37" title="In 3 Complete integrability and geodesic Hamiltonian ‣ A Hamiltonian approach to the gradient-flow equations in information geometry"><span class="ltx_text ltx_ref_tag">37</span></a>) is completely integrable if the condition</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="S3.EGx39"> <tbody id="S3.E39"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_left_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle H\operatorname{\overset{\textrm{ \tiny EOM}}{=}}p_{i}\,\frac{% \partial H}{\partial p_{i}}," class="ltx_Math" display="inline" id="S3.E39.m1.1"><semantics id="S3.E39.m1.1a"><mrow id="S3.E39.m1.1.1.1" xref="S3.E39.m1.1.1.1.1.cmml"><mrow id="S3.E39.m1.1.1.1.1" xref="S3.E39.m1.1.1.1.1.cmml"><mi id="S3.E39.m1.1.1.1.1.2" xref="S3.E39.m1.1.1.1.1.2.cmml">H</mi><mo id="S3.E39.m1.1.1.1.1.1" xref="S3.E39.m1.1.1.1.1.1.cmml">⁢</mo><mrow id="S3.E39.m1.1.1.1.1.3" xref="S3.E39.m1.1.1.1.1.3.cmml"><mover accent="true" id="S3.E39.m1.1.1.1.1.3.1" xref="S3.E39.m1.1.1.1.1.3.1.cmml"><mo id="S3.E39.m1.1.1.1.1.3.1.2" xref="S3.E39.m1.1.1.1.1.3.1.2.cmml">=</mo><mrow id="S3.E39.m1.1.1.1.1.3.1.1" xref="S3.E39.m1.1.1.1.1.3.1.1c.cmml"><mtext id="S3.E39.m1.1.1.1.1.3.1.1a" xref="S3.E39.m1.1.1.1.1.3.1.1c.cmml"> </mtext><mtext id="S3.E39.m1.1.1.1.1.3.1.1b" mathsize="50%" xref="S3.E39.m1.1.1.1.1.3.1.1c.cmml">EOM</mtext></mrow></mover><mrow id="S3.E39.m1.1.1.1.1.3.2" xref="S3.E39.m1.1.1.1.1.3.2.cmml"><msub id="S3.E39.m1.1.1.1.1.3.2.2" xref="S3.E39.m1.1.1.1.1.3.2.2.cmml"><mi id="S3.E39.m1.1.1.1.1.3.2.2.2" xref="S3.E39.m1.1.1.1.1.3.2.2.2.cmml">p</mi><mi id="S3.E39.m1.1.1.1.1.3.2.2.3" xref="S3.E39.m1.1.1.1.1.3.2.2.3.cmml">i</mi></msub><mo id="S3.E39.m1.1.1.1.1.3.2.1" xref="S3.E39.m1.1.1.1.1.3.2.1.cmml">⁢</mo><mstyle displaystyle="true" id="S3.E39.m1.1.1.1.1.3.2.3" xref="S3.E39.m1.1.1.1.1.3.2.3.cmml"><mfrac id="S3.E39.m1.1.1.1.1.3.2.3a" xref="S3.E39.m1.1.1.1.1.3.2.3.cmml"><mrow id="S3.E39.m1.1.1.1.1.3.2.3.2" xref="S3.E39.m1.1.1.1.1.3.2.3.2.cmml"><mo id="S3.E39.m1.1.1.1.1.3.2.3.2.1" rspace="0em" xref="S3.E39.m1.1.1.1.1.3.2.3.2.1.cmml">∂</mo><mi id="S3.E39.m1.1.1.1.1.3.2.3.2.2" xref="S3.E39.m1.1.1.1.1.3.2.3.2.2.cmml">H</mi></mrow><mrow id="S3.E39.m1.1.1.1.1.3.2.3.3" xref="S3.E39.m1.1.1.1.1.3.2.3.3.cmml"><mo id="S3.E39.m1.1.1.1.1.3.2.3.3.1" rspace="0em" xref="S3.E39.m1.1.1.1.1.3.2.3.3.1.cmml">∂</mo><msub id="S3.E39.m1.1.1.1.1.3.2.3.3.2" xref="S3.E39.m1.1.1.1.1.3.2.3.3.2.cmml"><mi id="S3.E39.m1.1.1.1.1.3.2.3.3.2.2" xref="S3.E39.m1.1.1.1.1.3.2.3.3.2.2.cmml">p</mi><mi id="S3.E39.m1.1.1.1.1.3.2.3.3.2.3" xref="S3.E39.m1.1.1.1.1.3.2.3.3.2.3.cmml">i</mi></msub></mrow></mfrac></mstyle></mrow></mrow></mrow><mo id="S3.E39.m1.1.1.1.2" xref="S3.E39.m1.1.1.1.1.cmml">,</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.E39.m1.1b"><apply 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id="S3.E39.m1.1.1.1.1.3.2.2.1.cmml" xref="S3.E39.m1.1.1.1.1.3.2.2">subscript</csymbol><ci id="S3.E39.m1.1.1.1.1.3.2.2.2.cmml" xref="S3.E39.m1.1.1.1.1.3.2.2.2">𝑝</ci><ci id="S3.E39.m1.1.1.1.1.3.2.2.3.cmml" xref="S3.E39.m1.1.1.1.1.3.2.2.3">𝑖</ci></apply><apply id="S3.E39.m1.1.1.1.1.3.2.3.cmml" xref="S3.E39.m1.1.1.1.1.3.2.3"><divide id="S3.E39.m1.1.1.1.1.3.2.3.1.cmml" xref="S3.E39.m1.1.1.1.1.3.2.3"></divide><apply id="S3.E39.m1.1.1.1.1.3.2.3.2.cmml" xref="S3.E39.m1.1.1.1.1.3.2.3.2"><partialdiff id="S3.E39.m1.1.1.1.1.3.2.3.2.1.cmml" xref="S3.E39.m1.1.1.1.1.3.2.3.2.1"></partialdiff><ci id="S3.E39.m1.1.1.1.1.3.2.3.2.2.cmml" xref="S3.E39.m1.1.1.1.1.3.2.3.2.2">𝐻</ci></apply><apply id="S3.E39.m1.1.1.1.1.3.2.3.3.cmml" xref="S3.E39.m1.1.1.1.1.3.2.3.3"><partialdiff id="S3.E39.m1.1.1.1.1.3.2.3.3.1.cmml" xref="S3.E39.m1.1.1.1.1.3.2.3.3.1"></partialdiff><apply id="S3.E39.m1.1.1.1.1.3.2.3.3.2.cmml" xref="S3.E39.m1.1.1.1.1.3.2.3.3.2"><csymbol cd="ambiguous" id="S3.E39.m1.1.1.1.1.3.2.3.3.2.1.cmml" xref="S3.E39.m1.1.1.1.1.3.2.3.3.2">subscript</csymbol><ci id="S3.E39.m1.1.1.1.1.3.2.3.3.2.2.cmml" xref="S3.E39.m1.1.1.1.1.3.2.3.3.2.2">𝑝</ci><ci id="S3.E39.m1.1.1.1.1.3.2.3.3.2.3.cmml" xref="S3.E39.m1.1.1.1.1.3.2.3.3.2.3">𝑖</ci></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.E39.m1.1c">\displaystyle H\operatorname{\overset{\textrm{ \tiny EOM}}{=}}p_{i}\,\frac{% \partial H}{\partial p_{i}},</annotation><annotation encoding="application/x-llamapun" id="S3.E39.m1.1d">italic_H start_OPFUNCTION over EOM start_ARG = end_ARG end_OPFUNCTION italic_p start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT divide start_ARG ∂ italic_H end_ARG start_ARG ∂ italic_p start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_ARG ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_left_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(39)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S3.p2.11">is satisfied. From Euler’s theorem on homogeneous functions, this condition means that the Hamiltonian <math alttext="H" class="ltx_Math" display="inline" id="S3.p2.10.m1.1"><semantics id="S3.p2.10.m1.1a"><mi id="S3.p2.10.m1.1.1" xref="S3.p2.10.m1.1.1.cmml">H</mi><annotation-xml encoding="MathML-Content" id="S3.p2.10.m1.1b"><ci id="S3.p2.10.m1.1.1.cmml" xref="S3.p2.10.m1.1.1">𝐻</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.p2.10.m1.1c">H</annotation><annotation encoding="application/x-llamapun" id="S3.p2.10.m1.1d">italic_H</annotation></semantics></math> is a homogeneous function of first order in the variables <math alttext="p_{i}" class="ltx_Math" display="inline" id="S3.p2.11.m2.1"><semantics id="S3.p2.11.m2.1a"><msub id="S3.p2.11.m2.1.1" xref="S3.p2.11.m2.1.1.cmml"><mi id="S3.p2.11.m2.1.1.2" xref="S3.p2.11.m2.1.1.2.cmml">p</mi><mi id="S3.p2.11.m2.1.1.3" xref="S3.p2.11.m2.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S3.p2.11.m2.1b"><apply id="S3.p2.11.m2.1.1.cmml" xref="S3.p2.11.m2.1.1"><csymbol cd="ambiguous" id="S3.p2.11.m2.1.1.1.cmml" xref="S3.p2.11.m2.1.1">subscript</csymbol><ci id="S3.p2.11.m2.1.1.2.cmml" xref="S3.p2.11.m2.1.1.2">𝑝</ci><ci id="S3.p2.11.m2.1.1.3.cmml" xref="S3.p2.11.m2.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p2.11.m2.1c">p_{i}</annotation><annotation encoding="application/x-llamapun" id="S3.p2.11.m2.1d">italic_p start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math>, i.e.,</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="S3.EGx40"> <tbody id="S3.Ex7"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_left_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle H(x,\lambda\,p,t)" class="ltx_Math" display="inline" id="S3.Ex7.m1.3"><semantics id="S3.Ex7.m1.3a"><mrow id="S3.Ex7.m1.3.3" xref="S3.Ex7.m1.3.3.cmml"><mi id="S3.Ex7.m1.3.3.3" xref="S3.Ex7.m1.3.3.3.cmml">H</mi><mo id="S3.Ex7.m1.3.3.2" xref="S3.Ex7.m1.3.3.2.cmml">⁢</mo><mrow id="S3.Ex7.m1.3.3.1.1" xref="S3.Ex7.m1.3.3.1.2.cmml"><mo id="S3.Ex7.m1.3.3.1.1.2" stretchy="false" xref="S3.Ex7.m1.3.3.1.2.cmml">(</mo><mi id="S3.Ex7.m1.1.1" xref="S3.Ex7.m1.1.1.cmml">x</mi><mo id="S3.Ex7.m1.3.3.1.1.3" xref="S3.Ex7.m1.3.3.1.2.cmml">,</mo><mrow id="S3.Ex7.m1.3.3.1.1.1" xref="S3.Ex7.m1.3.3.1.1.1.cmml"><mi id="S3.Ex7.m1.3.3.1.1.1.2" xref="S3.Ex7.m1.3.3.1.1.1.2.cmml">λ</mi><mo id="S3.Ex7.m1.3.3.1.1.1.1" lspace="0.170em" xref="S3.Ex7.m1.3.3.1.1.1.1.cmml">⁢</mo><mi id="S3.Ex7.m1.3.3.1.1.1.3" xref="S3.Ex7.m1.3.3.1.1.1.3.cmml">p</mi></mrow><mo id="S3.Ex7.m1.3.3.1.1.4" xref="S3.Ex7.m1.3.3.1.2.cmml">,</mo><mi id="S3.Ex7.m1.2.2" xref="S3.Ex7.m1.2.2.cmml">t</mi><mo id="S3.Ex7.m1.3.3.1.1.5" stretchy="false" xref="S3.Ex7.m1.3.3.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Ex7.m1.3b"><apply id="S3.Ex7.m1.3.3.cmml" xref="S3.Ex7.m1.3.3"><times id="S3.Ex7.m1.3.3.2.cmml" xref="S3.Ex7.m1.3.3.2"></times><ci id="S3.Ex7.m1.3.3.3.cmml" xref="S3.Ex7.m1.3.3.3">𝐻</ci><vector id="S3.Ex7.m1.3.3.1.2.cmml" xref="S3.Ex7.m1.3.3.1.1"><ci id="S3.Ex7.m1.1.1.cmml" xref="S3.Ex7.m1.1.1">𝑥</ci><apply id="S3.Ex7.m1.3.3.1.1.1.cmml" xref="S3.Ex7.m1.3.3.1.1.1"><times id="S3.Ex7.m1.3.3.1.1.1.1.cmml" xref="S3.Ex7.m1.3.3.1.1.1.1"></times><ci id="S3.Ex7.m1.3.3.1.1.1.2.cmml" xref="S3.Ex7.m1.3.3.1.1.1.2">𝜆</ci><ci id="S3.Ex7.m1.3.3.1.1.1.3.cmml" xref="S3.Ex7.m1.3.3.1.1.1.3">𝑝</ci></apply><ci id="S3.Ex7.m1.2.2.cmml" xref="S3.Ex7.m1.2.2">𝑡</ci></vector></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Ex7.m1.3c">\displaystyle H(x,\lambda\,p,t)</annotation><annotation encoding="application/x-llamapun" id="S3.Ex7.m1.3d">italic_H ( italic_x , italic_λ italic_p , italic_t )</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle=H(x,\lambda\,p_{1},\lambda\ p_{2},\ldots,\lambda\,p_{N},t)" class="ltx_Math" display="inline" id="S3.Ex7.m2.6"><semantics id="S3.Ex7.m2.6a"><mrow id="S3.Ex7.m2.6.6" xref="S3.Ex7.m2.6.6.cmml"><mi id="S3.Ex7.m2.6.6.5" xref="S3.Ex7.m2.6.6.5.cmml"></mi><mo id="S3.Ex7.m2.6.6.4" xref="S3.Ex7.m2.6.6.4.cmml">=</mo><mrow id="S3.Ex7.m2.6.6.3" xref="S3.Ex7.m2.6.6.3.cmml"><mi id="S3.Ex7.m2.6.6.3.5" xref="S3.Ex7.m2.6.6.3.5.cmml">H</mi><mo id="S3.Ex7.m2.6.6.3.4" xref="S3.Ex7.m2.6.6.3.4.cmml">⁢</mo><mrow id="S3.Ex7.m2.6.6.3.3.3" xref="S3.Ex7.m2.6.6.3.3.4.cmml"><mo id="S3.Ex7.m2.6.6.3.3.3.4" stretchy="false" xref="S3.Ex7.m2.6.6.3.3.4.cmml">(</mo><mi id="S3.Ex7.m2.1.1" xref="S3.Ex7.m2.1.1.cmml">x</mi><mo id="S3.Ex7.m2.6.6.3.3.3.5" xref="S3.Ex7.m2.6.6.3.3.4.cmml">,</mo><mrow id="S3.Ex7.m2.4.4.1.1.1.1" xref="S3.Ex7.m2.4.4.1.1.1.1.cmml"><mi id="S3.Ex7.m2.4.4.1.1.1.1.2" xref="S3.Ex7.m2.4.4.1.1.1.1.2.cmml">λ</mi><mo id="S3.Ex7.m2.4.4.1.1.1.1.1" lspace="0.170em" xref="S3.Ex7.m2.4.4.1.1.1.1.1.cmml">⁢</mo><msub 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xref="S3.Ex7.m2.6.6.3.3.3.3.3">subscript</csymbol><ci id="S3.Ex7.m2.6.6.3.3.3.3.3.2.cmml" xref="S3.Ex7.m2.6.6.3.3.3.3.3.2">𝑝</ci><ci id="S3.Ex7.m2.6.6.3.3.3.3.3.3.cmml" xref="S3.Ex7.m2.6.6.3.3.3.3.3.3">𝑁</ci></apply></apply><ci id="S3.Ex7.m2.3.3.cmml" xref="S3.Ex7.m2.3.3">𝑡</ci></vector></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Ex7.m2.6c">\displaystyle=H(x,\lambda\,p_{1},\lambda\ p_{2},\ldots,\lambda\,p_{N},t)</annotation><annotation encoding="application/x-llamapun" id="S3.Ex7.m2.6d">= italic_H ( italic_x , italic_λ italic_p start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_λ italic_p start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , … , italic_λ italic_p start_POSTSUBSCRIPT italic_N end_POSTSUBSCRIPT , italic_t )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_left_padright"></td> </tr></tbody> <tbody id="S3.E40"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_left_padleft"></td> <td class="ltx_td ltx_eqn_cell"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle=\lambda H(x,p,t)," class="ltx_Math" display="inline" id="S3.E40.m1.4"><semantics id="S3.E40.m1.4a"><mrow id="S3.E40.m1.4.4.1" xref="S3.E40.m1.4.4.1.1.cmml"><mrow id="S3.E40.m1.4.4.1.1" xref="S3.E40.m1.4.4.1.1.cmml"><mi id="S3.E40.m1.4.4.1.1.2" xref="S3.E40.m1.4.4.1.1.2.cmml"></mi><mo id="S3.E40.m1.4.4.1.1.1" xref="S3.E40.m1.4.4.1.1.1.cmml">=</mo><mrow id="S3.E40.m1.4.4.1.1.3" xref="S3.E40.m1.4.4.1.1.3.cmml"><mi id="S3.E40.m1.4.4.1.1.3.2" xref="S3.E40.m1.4.4.1.1.3.2.cmml">λ</mi><mo id="S3.E40.m1.4.4.1.1.3.1" xref="S3.E40.m1.4.4.1.1.3.1.cmml">⁢</mo><mi id="S3.E40.m1.4.4.1.1.3.3" xref="S3.E40.m1.4.4.1.1.3.3.cmml">H</mi><mo id="S3.E40.m1.4.4.1.1.3.1a" xref="S3.E40.m1.4.4.1.1.3.1.cmml">⁢</mo><mrow id="S3.E40.m1.4.4.1.1.3.4.2" xref="S3.E40.m1.4.4.1.1.3.4.1.cmml"><mo id="S3.E40.m1.4.4.1.1.3.4.2.1" stretchy="false" xref="S3.E40.m1.4.4.1.1.3.4.1.cmml">(</mo><mi id="S3.E40.m1.1.1" xref="S3.E40.m1.1.1.cmml">x</mi><mo id="S3.E40.m1.4.4.1.1.3.4.2.2" xref="S3.E40.m1.4.4.1.1.3.4.1.cmml">,</mo><mi id="S3.E40.m1.2.2" xref="S3.E40.m1.2.2.cmml">p</mi><mo id="S3.E40.m1.4.4.1.1.3.4.2.3" xref="S3.E40.m1.4.4.1.1.3.4.1.cmml">,</mo><mi id="S3.E40.m1.3.3" xref="S3.E40.m1.3.3.cmml">t</mi><mo id="S3.E40.m1.4.4.1.1.3.4.2.4" stretchy="false" xref="S3.E40.m1.4.4.1.1.3.4.1.cmml">)</mo></mrow></mrow></mrow><mo id="S3.E40.m1.4.4.1.2" xref="S3.E40.m1.4.4.1.1.cmml">,</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.E40.m1.4b"><apply id="S3.E40.m1.4.4.1.1.cmml" xref="S3.E40.m1.4.4.1"><eq id="S3.E40.m1.4.4.1.1.1.cmml" xref="S3.E40.m1.4.4.1.1.1"></eq><csymbol cd="latexml" id="S3.E40.m1.4.4.1.1.2.cmml" xref="S3.E40.m1.4.4.1.1.2">absent</csymbol><apply id="S3.E40.m1.4.4.1.1.3.cmml" xref="S3.E40.m1.4.4.1.1.3"><times id="S3.E40.m1.4.4.1.1.3.1.cmml" xref="S3.E40.m1.4.4.1.1.3.1"></times><ci id="S3.E40.m1.4.4.1.1.3.2.cmml" xref="S3.E40.m1.4.4.1.1.3.2">𝜆</ci><ci id="S3.E40.m1.4.4.1.1.3.3.cmml" xref="S3.E40.m1.4.4.1.1.3.3">𝐻</ci><vector id="S3.E40.m1.4.4.1.1.3.4.1.cmml" xref="S3.E40.m1.4.4.1.1.3.4.2"><ci id="S3.E40.m1.1.1.cmml" xref="S3.E40.m1.1.1">𝑥</ci><ci id="S3.E40.m1.2.2.cmml" xref="S3.E40.m1.2.2">𝑝</ci><ci id="S3.E40.m1.3.3.cmml" xref="S3.E40.m1.3.3">𝑡</ci></vector></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.E40.m1.4c">\displaystyle=\lambda H(x,p,t),</annotation><annotation encoding="application/x-llamapun" id="S3.E40.m1.4d">= italic_λ italic_H ( italic_x , italic_p , italic_t ) ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_left_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(40)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S3.p2.13">for a real <math alttext="\lambda>0" class="ltx_Math" display="inline" id="S3.p2.12.m1.1"><semantics id="S3.p2.12.m1.1a"><mrow id="S3.p2.12.m1.1.1" xref="S3.p2.12.m1.1.1.cmml"><mi id="S3.p2.12.m1.1.1.2" xref="S3.p2.12.m1.1.1.2.cmml">λ</mi><mo id="S3.p2.12.m1.1.1.1" xref="S3.p2.12.m1.1.1.1.cmml">></mo><mn id="S3.p2.12.m1.1.1.3" xref="S3.p2.12.m1.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.p2.12.m1.1b"><apply id="S3.p2.12.m1.1.1.cmml" xref="S3.p2.12.m1.1.1"><gt id="S3.p2.12.m1.1.1.1.cmml" xref="S3.p2.12.m1.1.1.1"></gt><ci id="S3.p2.12.m1.1.1.2.cmml" xref="S3.p2.12.m1.1.1.2">𝜆</ci><cn id="S3.p2.12.m1.1.1.3.cmml" type="integer" xref="S3.p2.12.m1.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p2.12.m1.1c">\lambda>0</annotation><annotation encoding="application/x-llamapun" id="S3.p2.12.m1.1d">italic_λ > 0</annotation></semantics></math>. With the help of the Hamilton equations <math alttext="dx^{i}/dt=\partial H/\partial p_{i}" class="ltx_Math" display="inline" id="S3.p2.13.m2.1"><semantics id="S3.p2.13.m2.1a"><mrow id="S3.p2.13.m2.1.1" xref="S3.p2.13.m2.1.1.cmml"><mrow id="S3.p2.13.m2.1.1.2" xref="S3.p2.13.m2.1.1.2.cmml"><mrow id="S3.p2.13.m2.1.1.2.2" xref="S3.p2.13.m2.1.1.2.2.cmml"><mrow id="S3.p2.13.m2.1.1.2.2.2" xref="S3.p2.13.m2.1.1.2.2.2.cmml"><mi id="S3.p2.13.m2.1.1.2.2.2.2" xref="S3.p2.13.m2.1.1.2.2.2.2.cmml">d</mi><mo id="S3.p2.13.m2.1.1.2.2.2.1" xref="S3.p2.13.m2.1.1.2.2.2.1.cmml">⁢</mo><msup id="S3.p2.13.m2.1.1.2.2.2.3" xref="S3.p2.13.m2.1.1.2.2.2.3.cmml"><mi id="S3.p2.13.m2.1.1.2.2.2.3.2" xref="S3.p2.13.m2.1.1.2.2.2.3.2.cmml">x</mi><mi id="S3.p2.13.m2.1.1.2.2.2.3.3" xref="S3.p2.13.m2.1.1.2.2.2.3.3.cmml">i</mi></msup></mrow><mo id="S3.p2.13.m2.1.1.2.2.1" xref="S3.p2.13.m2.1.1.2.2.1.cmml">/</mo><mi id="S3.p2.13.m2.1.1.2.2.3" xref="S3.p2.13.m2.1.1.2.2.3.cmml">d</mi></mrow><mo id="S3.p2.13.m2.1.1.2.1" xref="S3.p2.13.m2.1.1.2.1.cmml">⁢</mo><mi id="S3.p2.13.m2.1.1.2.3" xref="S3.p2.13.m2.1.1.2.3.cmml">t</mi></mrow><mo id="S3.p2.13.m2.1.1.1" rspace="0.1389em" xref="S3.p2.13.m2.1.1.1.cmml">=</mo><mrow id="S3.p2.13.m2.1.1.3" xref="S3.p2.13.m2.1.1.3.cmml"><mo id="S3.p2.13.m2.1.1.3.1" lspace="0.1389em" rspace="0em" xref="S3.p2.13.m2.1.1.3.1.cmml">∂</mo><mrow id="S3.p2.13.m2.1.1.3.2" xref="S3.p2.13.m2.1.1.3.2.cmml"><mi id="S3.p2.13.m2.1.1.3.2.2" xref="S3.p2.13.m2.1.1.3.2.2.cmml">H</mi><mo id="S3.p2.13.m2.1.1.3.2.1" xref="S3.p2.13.m2.1.1.3.2.1.cmml">/</mo><mrow id="S3.p2.13.m2.1.1.3.2.3" xref="S3.p2.13.m2.1.1.3.2.3.cmml"><mo id="S3.p2.13.m2.1.1.3.2.3.1" lspace="0em" rspace="0em" xref="S3.p2.13.m2.1.1.3.2.3.1.cmml">∂</mo><msub id="S3.p2.13.m2.1.1.3.2.3.2" xref="S3.p2.13.m2.1.1.3.2.3.2.cmml"><mi id="S3.p2.13.m2.1.1.3.2.3.2.2" xref="S3.p2.13.m2.1.1.3.2.3.2.2.cmml">p</mi><mi id="S3.p2.13.m2.1.1.3.2.3.2.3" 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xref="S3.p2.13.m2.1.1.2.2.2.3.2">𝑥</ci><ci id="S3.p2.13.m2.1.1.2.2.2.3.3.cmml" xref="S3.p2.13.m2.1.1.2.2.2.3.3">𝑖</ci></apply></apply><ci id="S3.p2.13.m2.1.1.2.2.3.cmml" xref="S3.p2.13.m2.1.1.2.2.3">𝑑</ci></apply><ci id="S3.p2.13.m2.1.1.2.3.cmml" xref="S3.p2.13.m2.1.1.2.3">𝑡</ci></apply><apply id="S3.p2.13.m2.1.1.3.cmml" xref="S3.p2.13.m2.1.1.3"><partialdiff id="S3.p2.13.m2.1.1.3.1.cmml" xref="S3.p2.13.m2.1.1.3.1"></partialdiff><apply id="S3.p2.13.m2.1.1.3.2.cmml" xref="S3.p2.13.m2.1.1.3.2"><divide id="S3.p2.13.m2.1.1.3.2.1.cmml" xref="S3.p2.13.m2.1.1.3.2.1"></divide><ci id="S3.p2.13.m2.1.1.3.2.2.cmml" xref="S3.p2.13.m2.1.1.3.2.2">𝐻</ci><apply id="S3.p2.13.m2.1.1.3.2.3.cmml" xref="S3.p2.13.m2.1.1.3.2.3"><partialdiff id="S3.p2.13.m2.1.1.3.2.3.1.cmml" xref="S3.p2.13.m2.1.1.3.2.3.1"></partialdiff><apply id="S3.p2.13.m2.1.1.3.2.3.2.cmml" xref="S3.p2.13.m2.1.1.3.2.3.2"><csymbol cd="ambiguous" id="S3.p2.13.m2.1.1.3.2.3.2.1.cmml" xref="S3.p2.13.m2.1.1.3.2.3.2">subscript</csymbol><ci id="S3.p2.13.m2.1.1.3.2.3.2.2.cmml" xref="S3.p2.13.m2.1.1.3.2.3.2.2">𝑝</ci><ci id="S3.p2.13.m2.1.1.3.2.3.2.3.cmml" xref="S3.p2.13.m2.1.1.3.2.3.2.3">𝑖</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p2.13.m2.1c">dx^{i}/dt=\partial H/\partial p_{i}</annotation><annotation encoding="application/x-llamapun" id="S3.p2.13.m2.1d">italic_d italic_x start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT / italic_d italic_t = ∂ italic_H / ∂ italic_p start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math>, the associated Lagrangian</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="S3.EGx41"> <tbody id="S3.E41"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_left_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle L\left(x,\frac{dx}{dt},t\right):=p_{i}\,\frac{dx^{i}}{dt}-H(x,p,% t)," class="ltx_Math" display="inline" 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id="footnote2.m2.5d">L(x,dx/dt,t)+df(x,t)/dt</annotation><annotation encoding="application/x-llamapun" id="footnote2.m2.5e">italic_L ( italic_x , italic_d italic_x / italic_d italic_t , italic_t ) + italic_d italic_f ( italic_x , italic_t ) / italic_d italic_t</annotation></semantics></math> and those for <math alttext="L(x,dx/dt,t)" class="ltx_Math" display="inline" id="footnote2.m3.3"><semantics id="footnote2.m3.3b"><mrow id="footnote2.m3.3.3" xref="footnote2.m3.3.3.cmml"><mi id="footnote2.m3.3.3.3" xref="footnote2.m3.3.3.3.cmml">L</mi><mo id="footnote2.m3.3.3.2" xref="footnote2.m3.3.3.2.cmml">⁢</mo><mrow id="footnote2.m3.3.3.1.1" xref="footnote2.m3.3.3.1.2.cmml"><mo id="footnote2.m3.3.3.1.1.2" stretchy="false" xref="footnote2.m3.3.3.1.2.cmml">(</mo><mi id="footnote2.m3.1.1" xref="footnote2.m3.1.1.cmml">x</mi><mo id="footnote2.m3.3.3.1.1.3" xref="footnote2.m3.3.3.1.2.cmml">,</mo><mrow id="footnote2.m3.3.3.1.1.1" xref="footnote2.m3.3.3.1.1.1.cmml"><mrow id="footnote2.m3.3.3.1.1.1.2" 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This does not mean the Lagrangian <math alttext="L" class="ltx_Math" display="inline" id="S3.p2.15.m2.1"><semantics id="S3.p2.15.m2.1a"><mi id="S3.p2.15.m2.1.1" xref="S3.p2.15.m2.1.1.cmml">L</mi><annotation-xml encoding="MathML-Content" id="S3.p2.15.m2.1b"><ci id="S3.p2.15.m2.1.1.cmml" xref="S3.p2.15.m2.1.1">𝐿</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.p2.15.m2.1c">L</annotation><annotation encoding="application/x-llamapun" id="S3.p2.15.m2.1d">italic_L</annotation></semantics></math> is algebraically null. It means that in the sense of the equality modulo EOM, i.e., for a solution of the Euler-Langrange equations (or Hamilton’s equations of motion), the value of this <math alttext="L" class="ltx_Math" display="inline" id="S3.p2.16.m3.1"><semantics id="S3.p2.16.m3.1a"><mi id="S3.p2.16.m3.1.1" xref="S3.p2.16.m3.1.1.cmml">L</mi><annotation-xml encoding="MathML-Content" id="S3.p2.16.m3.1b"><ci id="S3.p2.16.m3.1.1.cmml" xref="S3.p2.16.m3.1.1">𝐿</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.p2.16.m3.1c">L</annotation><annotation encoding="application/x-llamapun" id="S3.p2.16.m3.1d">italic_L</annotation></semantics></math> becomes zero.</p> </div> <div class="ltx_para" id="S3.p3"> <p class="ltx_p" id="S3.p3.1">It is known that the action <math alttext="S_{a}(x,t)" class="ltx_Math" display="inline" id="S3.p3.1.m1.2"><semantics id="S3.p3.1.m1.2a"><mrow id="S3.p3.1.m1.2.3" xref="S3.p3.1.m1.2.3.cmml"><msub id="S3.p3.1.m1.2.3.2" xref="S3.p3.1.m1.2.3.2.cmml"><mi 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xref="S3.p3.1.m1.2.3.2.2">𝑆</ci><ci id="S3.p3.1.m1.2.3.2.3.cmml" xref="S3.p3.1.m1.2.3.2.3">𝑎</ci></apply><interval closure="open" id="S3.p3.1.m1.2.3.3.1.cmml" xref="S3.p3.1.m1.2.3.3.2"><ci id="S3.p3.1.m1.1.1.cmml" xref="S3.p3.1.m1.1.1">𝑥</ci><ci id="S3.p3.1.m1.2.2.cmml" xref="S3.p3.1.m1.2.2">𝑡</ci></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p3.1.m1.2c">S_{a}(x,t)</annotation><annotation encoding="application/x-llamapun" id="S3.p3.1.m1.2d">italic_S start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT ( italic_x , italic_t )</annotation></semantics></math> satisfies the following relations <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2406.11224v2#bib.bib18" title="">18</a>]</cite>,</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="S3.EGx42"> <tbody id="S3.E42"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_left_padleft"></td> <td class="ltx_td 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start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT ( italic_x , italic_t ) end_ARG start_ARG ∂ italic_t end_ARG = - italic_E ( italic_t ) ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_left_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(42)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S3.p3.3">where <math alttext="E(t)" class="ltx_Math" display="inline" id="S3.p3.2.m1.1"><semantics id="S3.p3.2.m1.1a"><mrow id="S3.p3.2.m1.1.2" xref="S3.p3.2.m1.1.2.cmml"><mi id="S3.p3.2.m1.1.2.2" xref="S3.p3.2.m1.1.2.2.cmml">E</mi><mo id="S3.p3.2.m1.1.2.1" xref="S3.p3.2.m1.1.2.1.cmml">⁢</mo><mrow id="S3.p3.2.m1.1.2.3.2" xref="S3.p3.2.m1.1.2.cmml"><mo id="S3.p3.2.m1.1.2.3.2.1" stretchy="false" xref="S3.p3.2.m1.1.2.cmml">(</mo><mi id="S3.p3.2.m1.1.1" xref="S3.p3.2.m1.1.1.cmml">t</mi><mo id="S3.p3.2.m1.1.2.3.2.2" stretchy="false" xref="S3.p3.2.m1.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.p3.2.m1.1b"><apply id="S3.p3.2.m1.1.2.cmml" xref="S3.p3.2.m1.1.2"><times id="S3.p3.2.m1.1.2.1.cmml" xref="S3.p3.2.m1.1.2.1"></times><ci id="S3.p3.2.m1.1.2.2.cmml" xref="S3.p3.2.m1.1.2.2">𝐸</ci><ci id="S3.p3.2.m1.1.1.cmml" xref="S3.p3.2.m1.1.1">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p3.2.m1.1c">E(t)</annotation><annotation encoding="application/x-llamapun" id="S3.p3.2.m1.1d">italic_E ( italic_t )</annotation></semantics></math> is the total energy of the system at a time <math alttext="t" class="ltx_Math" display="inline" id="S3.p3.3.m2.1"><semantics id="S3.p3.3.m2.1a"><mi id="S3.p3.3.m2.1.1" xref="S3.p3.3.m2.1.1.cmml">t</mi><annotation-xml encoding="MathML-Content" id="S3.p3.3.m2.1b"><ci id="S3.p3.3.m2.1.1.cmml" xref="S3.p3.3.m2.1.1">𝑡</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.p3.3.m2.1c">t</annotation><annotation encoding="application/x-llamapun" id="S3.p3.3.m2.1d">italic_t</annotation></semantics></math>. It follows that</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="S3.EGx43"> <tbody id="S3.Ex8"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_left_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle dS_{a}(x,t)" class="ltx_Math" display="inline" id="S3.Ex8.m1.2"><semantics id="S3.Ex8.m1.2a"><mrow id="S3.Ex8.m1.2.3" xref="S3.Ex8.m1.2.3.cmml"><mi id="S3.Ex8.m1.2.3.2" xref="S3.Ex8.m1.2.3.2.cmml">d</mi><mo id="S3.Ex8.m1.2.3.1" xref="S3.Ex8.m1.2.3.1.cmml">⁢</mo><msub id="S3.Ex8.m1.2.3.3" xref="S3.Ex8.m1.2.3.3.cmml"><mi id="S3.Ex8.m1.2.3.3.2" xref="S3.Ex8.m1.2.3.3.2.cmml">S</mi><mi id="S3.Ex8.m1.2.3.3.3" xref="S3.Ex8.m1.2.3.3.3.cmml">a</mi></msub><mo id="S3.Ex8.m1.2.3.1a" xref="S3.Ex8.m1.2.3.1.cmml">⁢</mo><mrow id="S3.Ex8.m1.2.3.4.2" xref="S3.Ex8.m1.2.3.4.1.cmml"><mo id="S3.Ex8.m1.2.3.4.2.1" stretchy="false" xref="S3.Ex8.m1.2.3.4.1.cmml">(</mo><mi id="S3.Ex8.m1.1.1" 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xref="S3.Ex8.m1.2.2">𝑡</ci></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Ex8.m1.2c">\displaystyle dS_{a}(x,t)</annotation><annotation encoding="application/x-llamapun" id="S3.Ex8.m1.2d">italic_d italic_S start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT ( italic_x , italic_t )</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle=\frac{\partial S_{a}(x,t)}{\partial x^{i}}\,dx^{i}+\frac{% \partial S_{a}(x,t)}{\partial t}\,dt" class="ltx_Math" display="inline" id="S3.Ex8.m2.4"><semantics id="S3.Ex8.m2.4a"><mrow id="S3.Ex8.m2.4.5" xref="S3.Ex8.m2.4.5.cmml"><mi id="S3.Ex8.m2.4.5.2" xref="S3.Ex8.m2.4.5.2.cmml"></mi><mo id="S3.Ex8.m2.4.5.1" xref="S3.Ex8.m2.4.5.1.cmml">=</mo><mrow id="S3.Ex8.m2.4.5.3" xref="S3.Ex8.m2.4.5.3.cmml"><mrow id="S3.Ex8.m2.4.5.3.2" xref="S3.Ex8.m2.4.5.3.2.cmml"><mstyle displaystyle="true" id="S3.Ex8.m2.2.2" xref="S3.Ex8.m2.2.2.cmml"><mfrac id="S3.Ex8.m2.2.2a" 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id="S3.Ex8.m2.4.4.4.1.cmml" xref="S3.Ex8.m2.4.4.4.1"></partialdiff><ci id="S3.Ex8.m2.4.4.4.2.cmml" xref="S3.Ex8.m2.4.4.4.2">𝑡</ci></apply></apply><ci id="S3.Ex8.m2.4.5.3.3.2.cmml" xref="S3.Ex8.m2.4.5.3.3.2">𝑑</ci><ci id="S3.Ex8.m2.4.5.3.3.3.cmml" xref="S3.Ex8.m2.4.5.3.3.3">𝑡</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Ex8.m2.4c">\displaystyle=\frac{\partial S_{a}(x,t)}{\partial x^{i}}\,dx^{i}+\frac{% \partial S_{a}(x,t)}{\partial t}\,dt</annotation><annotation encoding="application/x-llamapun" id="S3.Ex8.m2.4d">= divide start_ARG ∂ italic_S start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT ( italic_x , italic_t ) end_ARG start_ARG ∂ italic_x start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT end_ARG italic_d italic_x start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT + divide start_ARG ∂ italic_S start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT ( italic_x , italic_t ) end_ARG start_ARG ∂ italic_t end_ARG italic_d italic_t</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_left_padright"></td> </tr></tbody> <tbody id="S3.E43"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_left_padleft"></td> <td class="ltx_td ltx_eqn_cell"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle=p_{i}dx^{i}-E(t)dt," class="ltx_Math" display="inline" id="S3.E43.m1.2"><semantics id="S3.E43.m1.2a"><mrow id="S3.E43.m1.2.2.1" xref="S3.E43.m1.2.2.1.1.cmml"><mrow id="S3.E43.m1.2.2.1.1" xref="S3.E43.m1.2.2.1.1.cmml"><mi id="S3.E43.m1.2.2.1.1.2" xref="S3.E43.m1.2.2.1.1.2.cmml"></mi><mo id="S3.E43.m1.2.2.1.1.1" xref="S3.E43.m1.2.2.1.1.1.cmml">=</mo><mrow id="S3.E43.m1.2.2.1.1.3" xref="S3.E43.m1.2.2.1.1.3.cmml"><mrow id="S3.E43.m1.2.2.1.1.3.2" xref="S3.E43.m1.2.2.1.1.3.2.cmml"><msub id="S3.E43.m1.2.2.1.1.3.2.2" xref="S3.E43.m1.2.2.1.1.3.2.2.cmml"><mi id="S3.E43.m1.2.2.1.1.3.2.2.2" xref="S3.E43.m1.2.2.1.1.3.2.2.2.cmml">p</mi><mi id="S3.E43.m1.2.2.1.1.3.2.2.3" xref="S3.E43.m1.2.2.1.1.3.2.2.3.cmml">i</mi></msub><mo id="S3.E43.m1.2.2.1.1.3.2.1" xref="S3.E43.m1.2.2.1.1.3.2.1.cmml">⁢</mo><mi id="S3.E43.m1.2.2.1.1.3.2.3" xref="S3.E43.m1.2.2.1.1.3.2.3.cmml">d</mi><mo id="S3.E43.m1.2.2.1.1.3.2.1a" xref="S3.E43.m1.2.2.1.1.3.2.1.cmml">⁢</mo><msup id="S3.E43.m1.2.2.1.1.3.2.4" xref="S3.E43.m1.2.2.1.1.3.2.4.cmml"><mi id="S3.E43.m1.2.2.1.1.3.2.4.2" xref="S3.E43.m1.2.2.1.1.3.2.4.2.cmml">x</mi><mi id="S3.E43.m1.2.2.1.1.3.2.4.3" xref="S3.E43.m1.2.2.1.1.3.2.4.3.cmml">i</mi></msup></mrow><mo id="S3.E43.m1.2.2.1.1.3.1" xref="S3.E43.m1.2.2.1.1.3.1.cmml">−</mo><mrow id="S3.E43.m1.2.2.1.1.3.3" xref="S3.E43.m1.2.2.1.1.3.3.cmml"><mi id="S3.E43.m1.2.2.1.1.3.3.2" xref="S3.E43.m1.2.2.1.1.3.3.2.cmml">E</mi><mo id="S3.E43.m1.2.2.1.1.3.3.1" xref="S3.E43.m1.2.2.1.1.3.3.1.cmml">⁢</mo><mrow id="S3.E43.m1.2.2.1.1.3.3.3.2" xref="S3.E43.m1.2.2.1.1.3.3.cmml"><mo id="S3.E43.m1.2.2.1.1.3.3.3.2.1" stretchy="false" 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id="S3.E43.m1.2.2.1.1.3.3.cmml" xref="S3.E43.m1.2.2.1.1.3.3"><times id="S3.E43.m1.2.2.1.1.3.3.1.cmml" xref="S3.E43.m1.2.2.1.1.3.3.1"></times><ci id="S3.E43.m1.2.2.1.1.3.3.2.cmml" xref="S3.E43.m1.2.2.1.1.3.3.2">𝐸</ci><ci id="S3.E43.m1.1.1.cmml" xref="S3.E43.m1.1.1">𝑡</ci><ci id="S3.E43.m1.2.2.1.1.3.3.4.cmml" xref="S3.E43.m1.2.2.1.1.3.3.4">𝑑</ci><ci id="S3.E43.m1.2.2.1.1.3.3.5.cmml" xref="S3.E43.m1.2.2.1.1.3.3.5">𝑡</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.E43.m1.2c">\displaystyle=p_{i}dx^{i}-E(t)dt,</annotation><annotation encoding="application/x-llamapun" id="S3.E43.m1.2d">= italic_p start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_d italic_x start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT - italic_E ( italic_t ) italic_d italic_t ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_left_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(43)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S3.p3.5">It is also known that the action <math alttext="S_{a}" class="ltx_Math" display="inline" id="S3.p3.4.m1.1"><semantics id="S3.p3.4.m1.1a"><msub id="S3.p3.4.m1.1.1" xref="S3.p3.4.m1.1.1.cmml"><mi id="S3.p3.4.m1.1.1.2" xref="S3.p3.4.m1.1.1.2.cmml">S</mi><mi id="S3.p3.4.m1.1.1.3" xref="S3.p3.4.m1.1.1.3.cmml">a</mi></msub><annotation-xml encoding="MathML-Content" id="S3.p3.4.m1.1b"><apply id="S3.p3.4.m1.1.1.cmml" xref="S3.p3.4.m1.1.1"><csymbol cd="ambiguous" id="S3.p3.4.m1.1.1.1.cmml" xref="S3.p3.4.m1.1.1">subscript</csymbol><ci id="S3.p3.4.m1.1.1.2.cmml" xref="S3.p3.4.m1.1.1.2">𝑆</ci><ci id="S3.p3.4.m1.1.1.3.cmml" xref="S3.p3.4.m1.1.1.3">𝑎</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p3.4.m1.1c">S_{a}</annotation><annotation encoding="application/x-llamapun" id="S3.p3.4.m1.1d">italic_S start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT</annotation></semantics></math> and the Lagrangian <math alttext="L" class="ltx_Math" display="inline" id="S3.p3.5.m2.1"><semantics id="S3.p3.5.m2.1a"><mi id="S3.p3.5.m2.1.1" xref="S3.p3.5.m2.1.1.cmml">L</mi><annotation-xml encoding="MathML-Content" id="S3.p3.5.m2.1b"><ci id="S3.p3.5.m2.1.1.cmml" xref="S3.p3.5.m2.1.1">𝐿</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.p3.5.m2.1c">L</annotation><annotation encoding="application/x-llamapun" id="S3.p3.5.m2.1d">italic_L</annotation></semantics></math> are related by</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="S3.EGx44"> <tbody id="S3.E44"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_left_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle dS_{a}(x,t)=L\left(x,\frac{dx}{dt},t\right)\,dt." class="ltx_Math" display="inline" id="S3.E44.m1.6"><semantics id="S3.E44.m1.6a"><mrow id="S3.E44.m1.6.6.1" xref="S3.E44.m1.6.6.1.1.cmml"><mrow id="S3.E44.m1.6.6.1.1" xref="S3.E44.m1.6.6.1.1.cmml"><mrow id="S3.E44.m1.6.6.1.1.2" xref="S3.E44.m1.6.6.1.1.2.cmml"><mi id="S3.E44.m1.6.6.1.1.2.2" xref="S3.E44.m1.6.6.1.1.2.2.cmml">d</mi><mo id="S3.E44.m1.6.6.1.1.2.1" xref="S3.E44.m1.6.6.1.1.2.1.cmml">⁢</mo><msub id="S3.E44.m1.6.6.1.1.2.3" xref="S3.E44.m1.6.6.1.1.2.3.cmml"><mi id="S3.E44.m1.6.6.1.1.2.3.2" xref="S3.E44.m1.6.6.1.1.2.3.2.cmml">S</mi><mi id="S3.E44.m1.6.6.1.1.2.3.3" xref="S3.E44.m1.6.6.1.1.2.3.3.cmml">a</mi></msub><mo id="S3.E44.m1.6.6.1.1.2.1a" xref="S3.E44.m1.6.6.1.1.2.1.cmml">⁢</mo><mrow id="S3.E44.m1.6.6.1.1.2.4.2" xref="S3.E44.m1.6.6.1.1.2.4.1.cmml"><mo id="S3.E44.m1.6.6.1.1.2.4.2.1" stretchy="false" xref="S3.E44.m1.6.6.1.1.2.4.1.cmml">(</mo><mi id="S3.E44.m1.1.1" xref="S3.E44.m1.1.1.cmml">x</mi><mo id="S3.E44.m1.6.6.1.1.2.4.2.2" xref="S3.E44.m1.6.6.1.1.2.4.1.cmml">,</mo><mi id="S3.E44.m1.2.2" xref="S3.E44.m1.2.2.cmml">t</mi><mo id="S3.E44.m1.6.6.1.1.2.4.2.3" stretchy="false" 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xref="S3.E44.m1.6.6.1.1.3.5">𝑡</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.E44.m1.6c">\displaystyle dS_{a}(x,t)=L\left(x,\frac{dx}{dt},t\right)\,dt.</annotation><annotation encoding="application/x-llamapun" id="S3.E44.m1.6d">italic_d italic_S start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT ( italic_x , italic_t ) = italic_L ( italic_x , divide start_ARG italic_d italic_x end_ARG start_ARG italic_d italic_t end_ARG , italic_t ) italic_d italic_t .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_left_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(44)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S3.p3.10">We see that the complete integrability of the Pfaffian system (<a class="ltx_ref" href="https://arxiv.org/html/2406.11224v2#S3.E37" title="In 3 Complete integrability and geodesic Hamiltonian ‣ A Hamiltonian approach to the gradient-flow equations in information geometry"><span class="ltx_text ltx_ref_tag">37</span></a>) leads to</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="S3.EGx45"> <tbody id="S3.E45"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_left_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\omega_{\rm PC}=dS_{a}(x,t)=L\,dt=0." class="ltx_Math" display="inline" id="S3.E45.m1.3"><semantics id="S3.E45.m1.3a"><mrow id="S3.E45.m1.3.3.1" xref="S3.E45.m1.3.3.1.1.cmml"><mrow id="S3.E45.m1.3.3.1.1" xref="S3.E45.m1.3.3.1.1.cmml"><msub id="S3.E45.m1.3.3.1.1.2" xref="S3.E45.m1.3.3.1.1.2.cmml"><mi id="S3.E45.m1.3.3.1.1.2.2" xref="S3.E45.m1.3.3.1.1.2.2.cmml">ω</mi><mi id="S3.E45.m1.3.3.1.1.2.3" xref="S3.E45.m1.3.3.1.1.2.3.cmml">PC</mi></msub><mo id="S3.E45.m1.3.3.1.1.3" xref="S3.E45.m1.3.3.1.1.3.cmml">=</mo><mrow id="S3.E45.m1.3.3.1.1.4" xref="S3.E45.m1.3.3.1.1.4.cmml"><mi id="S3.E45.m1.3.3.1.1.4.2" xref="S3.E45.m1.3.3.1.1.4.2.cmml">d</mi><mo id="S3.E45.m1.3.3.1.1.4.1" xref="S3.E45.m1.3.3.1.1.4.1.cmml">⁢</mo><msub id="S3.E45.m1.3.3.1.1.4.3" xref="S3.E45.m1.3.3.1.1.4.3.cmml"><mi id="S3.E45.m1.3.3.1.1.4.3.2" xref="S3.E45.m1.3.3.1.1.4.3.2.cmml">S</mi><mi id="S3.E45.m1.3.3.1.1.4.3.3" xref="S3.E45.m1.3.3.1.1.4.3.3.cmml">a</mi></msub><mo id="S3.E45.m1.3.3.1.1.4.1a" xref="S3.E45.m1.3.3.1.1.4.1.cmml">⁢</mo><mrow id="S3.E45.m1.3.3.1.1.4.4.2" xref="S3.E45.m1.3.3.1.1.4.4.1.cmml"><mo id="S3.E45.m1.3.3.1.1.4.4.2.1" stretchy="false" xref="S3.E45.m1.3.3.1.1.4.4.1.cmml">(</mo><mi id="S3.E45.m1.1.1" xref="S3.E45.m1.1.1.cmml">x</mi><mo id="S3.E45.m1.3.3.1.1.4.4.2.2" xref="S3.E45.m1.3.3.1.1.4.4.1.cmml">,</mo><mi id="S3.E45.m1.2.2" xref="S3.E45.m1.2.2.cmml">t</mi><mo id="S3.E45.m1.3.3.1.1.4.4.2.3" stretchy="false" xref="S3.E45.m1.3.3.1.1.4.4.1.cmml">)</mo></mrow></mrow><mo id="S3.E45.m1.3.3.1.1.5" xref="S3.E45.m1.3.3.1.1.5.cmml">=</mo><mrow id="S3.E45.m1.3.3.1.1.6" xref="S3.E45.m1.3.3.1.1.6.cmml"><mi id="S3.E45.m1.3.3.1.1.6.2" xref="S3.E45.m1.3.3.1.1.6.2.cmml">L</mi><mo id="S3.E45.m1.3.3.1.1.6.1" lspace="0.170em" xref="S3.E45.m1.3.3.1.1.6.1.cmml">⁢</mo><mi id="S3.E45.m1.3.3.1.1.6.3" xref="S3.E45.m1.3.3.1.1.6.3.cmml">d</mi><mo id="S3.E45.m1.3.3.1.1.6.1a" xref="S3.E45.m1.3.3.1.1.6.1.cmml">⁢</mo><mi id="S3.E45.m1.3.3.1.1.6.4" xref="S3.E45.m1.3.3.1.1.6.4.cmml">t</mi></mrow><mo id="S3.E45.m1.3.3.1.1.7" xref="S3.E45.m1.3.3.1.1.7.cmml">=</mo><mn id="S3.E45.m1.3.3.1.1.8" xref="S3.E45.m1.3.3.1.1.8.cmml">0</mn></mrow><mo id="S3.E45.m1.3.3.1.2" lspace="0em" xref="S3.E45.m1.3.3.1.1.cmml">.</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.E45.m1.3b"><apply id="S3.E45.m1.3.3.1.1.cmml" xref="S3.E45.m1.3.3.1"><and id="S3.E45.m1.3.3.1.1a.cmml" xref="S3.E45.m1.3.3.1"></and><apply id="S3.E45.m1.3.3.1.1b.cmml" xref="S3.E45.m1.3.3.1"><eq id="S3.E45.m1.3.3.1.1.3.cmml" xref="S3.E45.m1.3.3.1.1.3"></eq><apply id="S3.E45.m1.3.3.1.1.2.cmml" xref="S3.E45.m1.3.3.1.1.2"><csymbol cd="ambiguous" id="S3.E45.m1.3.3.1.1.2.1.cmml" xref="S3.E45.m1.3.3.1.1.2">subscript</csymbol><ci id="S3.E45.m1.3.3.1.1.2.2.cmml" xref="S3.E45.m1.3.3.1.1.2.2">𝜔</ci><ci id="S3.E45.m1.3.3.1.1.2.3.cmml" xref="S3.E45.m1.3.3.1.1.2.3">PC</ci></apply><apply id="S3.E45.m1.3.3.1.1.4.cmml" xref="S3.E45.m1.3.3.1.1.4"><times id="S3.E45.m1.3.3.1.1.4.1.cmml" xref="S3.E45.m1.3.3.1.1.4.1"></times><ci id="S3.E45.m1.3.3.1.1.4.2.cmml" xref="S3.E45.m1.3.3.1.1.4.2">𝑑</ci><apply id="S3.E45.m1.3.3.1.1.4.3.cmml" xref="S3.E45.m1.3.3.1.1.4.3"><csymbol cd="ambiguous" id="S3.E45.m1.3.3.1.1.4.3.1.cmml" xref="S3.E45.m1.3.3.1.1.4.3">subscript</csymbol><ci id="S3.E45.m1.3.3.1.1.4.3.2.cmml" xref="S3.E45.m1.3.3.1.1.4.3.2">𝑆</ci><ci id="S3.E45.m1.3.3.1.1.4.3.3.cmml" xref="S3.E45.m1.3.3.1.1.4.3.3">𝑎</ci></apply><interval closure="open" id="S3.E45.m1.3.3.1.1.4.4.1.cmml" xref="S3.E45.m1.3.3.1.1.4.4.2"><ci id="S3.E45.m1.1.1.cmml" xref="S3.E45.m1.1.1">𝑥</ci><ci id="S3.E45.m1.2.2.cmml" xref="S3.E45.m1.2.2">𝑡</ci></interval></apply></apply><apply id="S3.E45.m1.3.3.1.1c.cmml" xref="S3.E45.m1.3.3.1"><eq id="S3.E45.m1.3.3.1.1.5.cmml" xref="S3.E45.m1.3.3.1.1.5"></eq><share href="https://arxiv.org/html/2406.11224v2#S3.E45.m1.3.3.1.1.4.cmml" id="S3.E45.m1.3.3.1.1d.cmml" xref="S3.E45.m1.3.3.1"></share><apply id="S3.E45.m1.3.3.1.1.6.cmml" xref="S3.E45.m1.3.3.1.1.6"><times id="S3.E45.m1.3.3.1.1.6.1.cmml" xref="S3.E45.m1.3.3.1.1.6.1"></times><ci id="S3.E45.m1.3.3.1.1.6.2.cmml" xref="S3.E45.m1.3.3.1.1.6.2">𝐿</ci><ci id="S3.E45.m1.3.3.1.1.6.3.cmml" xref="S3.E45.m1.3.3.1.1.6.3">𝑑</ci><ci id="S3.E45.m1.3.3.1.1.6.4.cmml" xref="S3.E45.m1.3.3.1.1.6.4">𝑡</ci></apply></apply><apply id="S3.E45.m1.3.3.1.1e.cmml" xref="S3.E45.m1.3.3.1"><eq id="S3.E45.m1.3.3.1.1.7.cmml" xref="S3.E45.m1.3.3.1.1.7"></eq><share href="https://arxiv.org/html/2406.11224v2#S3.E45.m1.3.3.1.1.6.cmml" id="S3.E45.m1.3.3.1.1f.cmml" xref="S3.E45.m1.3.3.1"></share><cn id="S3.E45.m1.3.3.1.1.8.cmml" type="integer" xref="S3.E45.m1.3.3.1.1.8">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.E45.m1.3c">\displaystyle\omega_{\rm PC}=dS_{a}(x,t)=L\,dt=0.</annotation><annotation encoding="application/x-llamapun" id="S3.E45.m1.3d">italic_ω start_POSTSUBSCRIPT roman_PC end_POSTSUBSCRIPT = italic_d italic_S start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT ( italic_x , italic_t ) = italic_L italic_d italic_t = 0 .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_left_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(45)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S3.p3.11">From this relation and by using (<a class="ltx_ref" href="https://arxiv.org/html/2406.11224v2#S3.E43" title="In 3 Complete integrability and geodesic Hamiltonian ‣ A Hamiltonian approach to the gradient-flow equations in information geometry"><span class="ltx_text ltx_ref_tag">43</span></a>), we have</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="S3.EGx46"> <tbody id="S3.E46"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_left_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle dS_{a}(x,t)=\left(p_{i}\,\frac{dx^{i}}{dt}-E(t)\right)dt=0." class="ltx_Math" display="inline" id="S3.E46.m1.4"><semantics id="S3.E46.m1.4a"><mrow id="S3.E46.m1.4.4.1" xref="S3.E46.m1.4.4.1.1.cmml"><mrow id="S3.E46.m1.4.4.1.1" xref="S3.E46.m1.4.4.1.1.cmml"><mrow id="S3.E46.m1.4.4.1.1.3" xref="S3.E46.m1.4.4.1.1.3.cmml"><mi id="S3.E46.m1.4.4.1.1.3.2" xref="S3.E46.m1.4.4.1.1.3.2.cmml">d</mi><mo id="S3.E46.m1.4.4.1.1.3.1" xref="S3.E46.m1.4.4.1.1.3.1.cmml">⁢</mo><msub id="S3.E46.m1.4.4.1.1.3.3" xref="S3.E46.m1.4.4.1.1.3.3.cmml"><mi id="S3.E46.m1.4.4.1.1.3.3.2" xref="S3.E46.m1.4.4.1.1.3.3.2.cmml">S</mi><mi id="S3.E46.m1.4.4.1.1.3.3.3" 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xref="S3.E46.m1.4.4.1"></share><cn id="S3.E46.m1.4.4.1.1.6.cmml" type="integer" xref="S3.E46.m1.4.4.1.1.6">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.E46.m1.4c">\displaystyle dS_{a}(x,t)=\left(p_{i}\,\frac{dx^{i}}{dt}-E(t)\right)dt=0.</annotation><annotation encoding="application/x-llamapun" id="S3.E46.m1.4d">italic_d italic_S start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT ( italic_x , italic_t ) = ( italic_p start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT divide start_ARG italic_d italic_x start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT end_ARG start_ARG italic_d italic_t end_ARG - italic_E ( italic_t ) ) italic_d italic_t = 0 .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_left_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(46)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S3.p3.12">Consequently it 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start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT end_ARG start_ARG italic_d italic_t end_ARG start_OPFUNCTION over EOM start_ARG = end_ARG end_OPFUNCTION italic_H ( italic_x ( italic_t ) , italic_p ( italic_t ) , italic_t ) ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_left_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(47)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S3.p3.9">where the last expression means the instantaneous value of the Hamiltonian for a solution of the associated Hamilton’s equations of motion. At this point we emphasize that one needs an explicit expression of <math alttext="H(x,p,t)" class="ltx_Math" display="inline" id="S3.p3.6.m1.3"><semantics id="S3.p3.6.m1.3a"><mrow id="S3.p3.6.m1.3.4" xref="S3.p3.6.m1.3.4.cmml"><mi id="S3.p3.6.m1.3.4.2" xref="S3.p3.6.m1.3.4.2.cmml">H</mi><mo id="S3.p3.6.m1.3.4.1" xref="S3.p3.6.m1.3.4.1.cmml">⁢</mo><mrow id="S3.p3.6.m1.3.4.3.2" xref="S3.p3.6.m1.3.4.3.1.cmml"><mo id="S3.p3.6.m1.3.4.3.2.1" stretchy="false" xref="S3.p3.6.m1.3.4.3.1.cmml">(</mo><mi id="S3.p3.6.m1.1.1" xref="S3.p3.6.m1.1.1.cmml">x</mi><mo id="S3.p3.6.m1.3.4.3.2.2" xref="S3.p3.6.m1.3.4.3.1.cmml">,</mo><mi id="S3.p3.6.m1.2.2" xref="S3.p3.6.m1.2.2.cmml">p</mi><mo id="S3.p3.6.m1.3.4.3.2.3" xref="S3.p3.6.m1.3.4.3.1.cmml">,</mo><mi id="S3.p3.6.m1.3.3" xref="S3.p3.6.m1.3.3.cmml">t</mi><mo id="S3.p3.6.m1.3.4.3.2.4" stretchy="false" xref="S3.p3.6.m1.3.4.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.p3.6.m1.3b"><apply id="S3.p3.6.m1.3.4.cmml" xref="S3.p3.6.m1.3.4"><times id="S3.p3.6.m1.3.4.1.cmml" xref="S3.p3.6.m1.3.4.1"></times><ci id="S3.p3.6.m1.3.4.2.cmml" xref="S3.p3.6.m1.3.4.2">𝐻</ci><vector id="S3.p3.6.m1.3.4.3.1.cmml" xref="S3.p3.6.m1.3.4.3.2"><ci id="S3.p3.6.m1.1.1.cmml" xref="S3.p3.6.m1.1.1">𝑥</ci><ci id="S3.p3.6.m1.2.2.cmml" xref="S3.p3.6.m1.2.2">𝑝</ci><ci id="S3.p3.6.m1.3.3.cmml" xref="S3.p3.6.m1.3.3">𝑡</ci></vector></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p3.6.m1.3c">H(x,p,t)</annotation><annotation encoding="application/x-llamapun" id="S3.p3.6.m1.3d">italic_H ( italic_x , italic_p , italic_t )</annotation></semantics></math> as a function of the canonical variables <math alttext="(x,p)" class="ltx_Math" display="inline" id="S3.p3.7.m2.2"><semantics id="S3.p3.7.m2.2a"><mrow id="S3.p3.7.m2.2.3.2" xref="S3.p3.7.m2.2.3.1.cmml"><mo id="S3.p3.7.m2.2.3.2.1" stretchy="false" xref="S3.p3.7.m2.2.3.1.cmml">(</mo><mi id="S3.p3.7.m2.1.1" xref="S3.p3.7.m2.1.1.cmml">x</mi><mo 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xref="S3.p3.8.m3.1.1">𝑡</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.p3.8.m3.1c">t</annotation><annotation encoding="application/x-llamapun" id="S3.p3.8.m3.1d">italic_t</annotation></semantics></math>, not the value of <math alttext="H" class="ltx_Math" display="inline" id="S3.p3.9.m4.1"><semantics id="S3.p3.9.m4.1a"><mi id="S3.p3.9.m4.1.1" xref="S3.p3.9.m4.1.1.cmml">H</mi><annotation-xml encoding="MathML-Content" id="S3.p3.9.m4.1b"><ci id="S3.p3.9.m4.1.1.cmml" xref="S3.p3.9.m4.1.1">𝐻</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.p3.9.m4.1c">H</annotation><annotation encoding="application/x-llamapun" id="S3.p3.9.m4.1d">italic_H</annotation></semantics></math>, in order to describe the associated Hamilton dynamics.</p> </div> <div class="ltx_para" id="S3.p4"> <p class="ltx_p" id="S3.p4.1">An example of the explicit expressions of Hamiltonians which are homogeneous of first order in the variables <math alttext="p_{i}" class="ltx_Math" display="inline" id="S3.p4.1.m1.1"><semantics id="S3.p4.1.m1.1a"><msub id="S3.p4.1.m1.1.1" xref="S3.p4.1.m1.1.1.cmml"><mi id="S3.p4.1.m1.1.1.2" xref="S3.p4.1.m1.1.1.2.cmml">p</mi><mi id="S3.p4.1.m1.1.1.3" xref="S3.p4.1.m1.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S3.p4.1.m1.1b"><apply id="S3.p4.1.m1.1.1.cmml" xref="S3.p4.1.m1.1.1"><csymbol cd="ambiguous" id="S3.p4.1.m1.1.1.1.cmml" xref="S3.p4.1.m1.1.1">subscript</csymbol><ci id="S3.p4.1.m1.1.1.2.cmml" xref="S3.p4.1.m1.1.1.2">𝑝</ci><ci id="S3.p4.1.m1.1.1.3.cmml" xref="S3.p4.1.m1.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p4.1.m1.1c">p_{i}</annotation><annotation encoding="application/x-llamapun" id="S3.p4.1.m1.1d">italic_p start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> is</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="S3.EGx48"> <tbody id="S3.E48"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td 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italic_p start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT italic_p start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_ARG ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_left_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(48)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S3.p4.8">where <math alttext="c" class="ltx_Math" display="inline" id="S3.p4.2.m1.1"><semantics id="S3.p4.2.m1.1a"><mi id="S3.p4.2.m1.1.1" xref="S3.p4.2.m1.1.1.cmml">c</mi><annotation-xml encoding="MathML-Content" id="S3.p4.2.m1.1b"><ci id="S3.p4.2.m1.1.1.cmml" xref="S3.p4.2.m1.1.1">𝑐</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.p4.2.m1.1c">c</annotation><annotation encoding="application/x-llamapun" id="S3.p4.2.m1.1d">italic_c</annotation></semantics></math> is the speed of light in vacuum, <math alttext="\xi(t)" class="ltx_Math" display="inline" 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)</annotation></semantics></math> is a dimensionless factor depending on <math alttext="t" class="ltx_Math" display="inline" id="S3.p4.4.m3.1"><semantics id="S3.p4.4.m3.1a"><mi id="S3.p4.4.m3.1.1" xref="S3.p4.4.m3.1.1.cmml">t</mi><annotation-xml encoding="MathML-Content" id="S3.p4.4.m3.1b"><ci id="S3.p4.4.m3.1.1.cmml" xref="S3.p4.4.m3.1.1">𝑡</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.p4.4.m3.1c">t</annotation><annotation encoding="application/x-llamapun" id="S3.p4.4.m3.1d">italic_t</annotation></semantics></math>, and <math alttext="g^{jk}(x)" class="ltx_Math" display="inline" id="S3.p4.5.m4.1"><semantics id="S3.p4.5.m4.1a"><mrow id="S3.p4.5.m4.1.2" xref="S3.p4.5.m4.1.2.cmml"><msup id="S3.p4.5.m4.1.2.2" xref="S3.p4.5.m4.1.2.2.cmml"><mi id="S3.p4.5.m4.1.2.2.2" xref="S3.p4.5.m4.1.2.2.2.cmml">g</mi><mrow id="S3.p4.5.m4.1.2.2.3" xref="S3.p4.5.m4.1.2.2.3.cmml"><mi id="S3.p4.5.m4.1.2.2.3.2" xref="S3.p4.5.m4.1.2.2.3.2.cmml">j</mi><mo id="S3.p4.5.m4.1.2.2.3.1" 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xref="S3.p4.8.m7.2.3.2.cmml">)</mo></mrow></mrow><mo id="S3.p4.8.m7.2.3.1" xref="S3.p4.8.m7.2.3.1.cmml">=</mo><msubsup id="S3.p4.8.m7.2.3.3" xref="S3.p4.8.m7.2.3.3.cmml"><mi id="S3.p4.8.m7.2.3.3.2.2" xref="S3.p4.8.m7.2.3.3.2.2.cmml">δ</mi><mi id="S3.p4.8.m7.2.3.3.3" mathvariant="normal" xref="S3.p4.8.m7.2.3.3.3.cmml">ℓ</mi><mi id="S3.p4.8.m7.2.3.3.2.3" xref="S3.p4.8.m7.2.3.3.2.3.cmml">j</mi></msubsup></mrow><annotation-xml encoding="MathML-Content" id="S3.p4.8.m7.2b"><apply id="S3.p4.8.m7.2.3.cmml" xref="S3.p4.8.m7.2.3"><eq id="S3.p4.8.m7.2.3.1.cmml" xref="S3.p4.8.m7.2.3.1"></eq><apply id="S3.p4.8.m7.2.3.2.cmml" xref="S3.p4.8.m7.2.3.2"><times id="S3.p4.8.m7.2.3.2.1.cmml" xref="S3.p4.8.m7.2.3.2.1"></times><apply id="S3.p4.8.m7.2.3.2.2.cmml" xref="S3.p4.8.m7.2.3.2.2"><csymbol cd="ambiguous" id="S3.p4.8.m7.2.3.2.2.1.cmml" xref="S3.p4.8.m7.2.3.2.2">superscript</csymbol><ci id="S3.p4.8.m7.2.3.2.2.2.cmml" xref="S3.p4.8.m7.2.3.2.2.2">𝑔</ci><apply id="S3.p4.8.m7.2.3.2.2.3.cmml" xref="S3.p4.8.m7.2.3.2.2.3"><times id="S3.p4.8.m7.2.3.2.2.3.1.cmml" xref="S3.p4.8.m7.2.3.2.2.3.1"></times><ci id="S3.p4.8.m7.2.3.2.2.3.2.cmml" xref="S3.p4.8.m7.2.3.2.2.3.2">𝑗</ci><ci id="S3.p4.8.m7.2.3.2.2.3.3.cmml" xref="S3.p4.8.m7.2.3.2.2.3.3">𝑘</ci></apply></apply><ci id="S3.p4.8.m7.1.1.cmml" xref="S3.p4.8.m7.1.1">𝑥</ci><apply id="S3.p4.8.m7.2.3.2.4.cmml" xref="S3.p4.8.m7.2.3.2.4"><csymbol cd="ambiguous" id="S3.p4.8.m7.2.3.2.4.1.cmml" xref="S3.p4.8.m7.2.3.2.4">subscript</csymbol><ci id="S3.p4.8.m7.2.3.2.4.2.cmml" xref="S3.p4.8.m7.2.3.2.4.2">𝑔</ci><apply id="S3.p4.8.m7.2.3.2.4.3.cmml" xref="S3.p4.8.m7.2.3.2.4.3"><times id="S3.p4.8.m7.2.3.2.4.3.1.cmml" xref="S3.p4.8.m7.2.3.2.4.3.1"></times><ci id="S3.p4.8.m7.2.3.2.4.3.2.cmml" xref="S3.p4.8.m7.2.3.2.4.3.2">𝑘</ci><ci id="S3.p4.8.m7.2.3.2.4.3.3.cmml" xref="S3.p4.8.m7.2.3.2.4.3.3">ℓ</ci></apply></apply><ci id="S3.p4.8.m7.2.2.cmml" xref="S3.p4.8.m7.2.2">𝑥</ci></apply><apply id="S3.p4.8.m7.2.3.3.cmml" xref="S3.p4.8.m7.2.3.3"><csymbol cd="ambiguous" id="S3.p4.8.m7.2.3.3.1.cmml" xref="S3.p4.8.m7.2.3.3">subscript</csymbol><apply id="S3.p4.8.m7.2.3.3.2.cmml" xref="S3.p4.8.m7.2.3.3"><csymbol cd="ambiguous" id="S3.p4.8.m7.2.3.3.2.1.cmml" xref="S3.p4.8.m7.2.3.3">superscript</csymbol><ci id="S3.p4.8.m7.2.3.3.2.2.cmml" xref="S3.p4.8.m7.2.3.3.2.2">𝛿</ci><ci id="S3.p4.8.m7.2.3.3.2.3.cmml" xref="S3.p4.8.m7.2.3.3.2.3">𝑗</ci></apply><ci id="S3.p4.8.m7.2.3.3.3.cmml" xref="S3.p4.8.m7.2.3.3.3">ℓ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p4.8.m7.2c">g^{jk}(x)\,g_{k\ell}(x)=\delta^{j}_{\ell}</annotation><annotation encoding="application/x-llamapun" id="S3.p4.8.m7.2d">italic_g start_POSTSUPERSCRIPT italic_j italic_k end_POSTSUPERSCRIPT ( italic_x ) italic_g start_POSTSUBSCRIPT italic_k roman_ℓ end_POSTSUBSCRIPT ( italic_x ) = italic_δ start_POSTSUPERSCRIPT italic_j end_POSTSUPERSCRIPT start_POSTSUBSCRIPT roman_ℓ end_POSTSUBSCRIPT</annotation></semantics></math>. Note that since</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="S3.EGx49"> <tbody id="S3.E49"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_left_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle p_{i}\frac{\partial H_{h}}{\partial p_{i}}=\frac{\xi(t)\,c\,g^{% ij}(x)\,p_{i}p_{j},}{\sqrt{g^{k\ell}(x)\,p_{k}p_{\ell}}}=H_{h}," class="ltx_math_unparsed" display="inline" id="S3.E49.m1.4"><semantics id="S3.E49.m1.4a"><mrow id="S3.E49.m1.4.4.1"><mrow id="S3.E49.m1.4.4.1.1"><mrow id="S3.E49.m1.4.4.1.1.2"><msub id="S3.E49.m1.4.4.1.1.2.2"><mi id="S3.E49.m1.4.4.1.1.2.2.2">p</mi><mi id="S3.E49.m1.4.4.1.1.2.2.3">i</mi></msub><mo id="S3.E49.m1.4.4.1.1.2.1">⁢</mo><mstyle displaystyle="true" id="S3.E49.m1.4.4.1.1.2.3"><mfrac id="S3.E49.m1.4.4.1.1.2.3a"><mrow id="S3.E49.m1.4.4.1.1.2.3.2"><mo id="S3.E49.m1.4.4.1.1.2.3.2.1" rspace="0em">∂</mo><msub id="S3.E49.m1.4.4.1.1.2.3.2.2"><mi id="S3.E49.m1.4.4.1.1.2.3.2.2.2">H</mi><mi id="S3.E49.m1.4.4.1.1.2.3.2.2.3">h</mi></msub></mrow><mrow id="S3.E49.m1.4.4.1.1.2.3.3"><mo id="S3.E49.m1.4.4.1.1.2.3.3.1" rspace="0em">∂</mo><msub id="S3.E49.m1.4.4.1.1.2.3.3.2"><mi id="S3.E49.m1.4.4.1.1.2.3.3.2.2">p</mi><mi id="S3.E49.m1.4.4.1.1.2.3.3.2.3">i</mi></msub></mrow></mfrac></mstyle></mrow><mo id="S3.E49.m1.4.4.1.1.3">=</mo><mstyle displaystyle="true" id="S3.E49.m1.3.3"><mfrac id="S3.E49.m1.3.3a"><mrow id="S3.E49.m1.2.2.2"><mi id="S3.E49.m1.2.2.2.3">ξ</mi><mrow id="S3.E49.m1.2.2.2.4"><mo id="S3.E49.m1.2.2.2.4.1" stretchy="false">(</mo><mi id="S3.E49.m1.1.1.1.1">t</mi><mo id="S3.E49.m1.2.2.2.4.2" rspace="0.170em" stretchy="false">)</mo></mrow><mi id="S3.E49.m1.2.2.2.5">c</mi><msup id="S3.E49.m1.2.2.2.6"><mi id="S3.E49.m1.2.2.2.6.2">g</mi><mrow id="S3.E49.m1.2.2.2.6.3"><mi id="S3.E49.m1.2.2.2.6.3.2">i</mi><mo id="S3.E49.m1.2.2.2.6.3.1">⁢</mo><mi id="S3.E49.m1.2.2.2.6.3.3">j</mi></mrow></msup><mrow id="S3.E49.m1.2.2.2.7"><mo id="S3.E49.m1.2.2.2.7.1" stretchy="false">(</mo><mi id="S3.E49.m1.2.2.2.2">x</mi><mo id="S3.E49.m1.2.2.2.7.2" rspace="0.170em" stretchy="false">)</mo></mrow><msub id="S3.E49.m1.2.2.2.8"><mi id="S3.E49.m1.2.2.2.8.2">p</mi><mi id="S3.E49.m1.2.2.2.8.3">i</mi></msub><msub id="S3.E49.m1.2.2.2.9"><mi id="S3.E49.m1.2.2.2.9.2">p</mi><mi id="S3.E49.m1.2.2.2.9.3">j</mi></msub><mo id="S3.E49.m1.2.2.2.10">,</mo></mrow><msqrt id="S3.E49.m1.3.3.3"><mrow id="S3.E49.m1.3.3.3.1.1"><msup id="S3.E49.m1.3.3.3.1.1.3"><mi id="S3.E49.m1.3.3.3.1.1.3.2">g</mi><mrow id="S3.E49.m1.3.3.3.1.1.3.3"><mi id="S3.E49.m1.3.3.3.1.1.3.3.2">k</mi><mo id="S3.E49.m1.3.3.3.1.1.3.3.1">⁢</mo><mi id="S3.E49.m1.3.3.3.1.1.3.3.3" mathvariant="normal">ℓ</mi></mrow></msup><mo id="S3.E49.m1.3.3.3.1.1.2">⁢</mo><mrow id="S3.E49.m1.3.3.3.1.1.4.2"><mo id="S3.E49.m1.3.3.3.1.1.4.2.1" stretchy="false">(</mo><mi id="S3.E49.m1.3.3.3.1.1.1">x</mi><mo id="S3.E49.m1.3.3.3.1.1.4.2.2" stretchy="false">)</mo></mrow><mo id="S3.E49.m1.3.3.3.1.1.2a" lspace="0.170em">⁢</mo><msub id="S3.E49.m1.3.3.3.1.1.5"><mi id="S3.E49.m1.3.3.3.1.1.5.2">p</mi><mi id="S3.E49.m1.3.3.3.1.1.5.3">k</mi></msub><mo id="S3.E49.m1.3.3.3.1.1.2b">⁢</mo><msub id="S3.E49.m1.3.3.3.1.1.6"><mi id="S3.E49.m1.3.3.3.1.1.6.2">p</mi><mi id="S3.E49.m1.3.3.3.1.1.6.3" mathvariant="normal">ℓ</mi></msub></mrow></msqrt></mfrac></mstyle><mo id="S3.E49.m1.4.4.1.1.4">=</mo><msub id="S3.E49.m1.4.4.1.1.5"><mi id="S3.E49.m1.4.4.1.1.5.2">H</mi><mi id="S3.E49.m1.4.4.1.1.5.3">h</mi></msub></mrow><mo id="S3.E49.m1.4.4.1.2">,</mo></mrow><annotation encoding="application/x-tex" id="S3.E49.m1.4b">\displaystyle p_{i}\frac{\partial H_{h}}{\partial p_{i}}=\frac{\xi(t)\,c\,g^{% ij}(x)\,p_{i}p_{j},}{\sqrt{g^{k\ell}(x)\,p_{k}p_{\ell}}}=H_{h},</annotation><annotation encoding="application/x-llamapun" id="S3.E49.m1.4c">italic_p start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT divide start_ARG ∂ italic_H start_POSTSUBSCRIPT italic_h end_POSTSUBSCRIPT end_ARG start_ARG ∂ italic_p start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_ARG = divide start_ARG italic_ξ ( italic_t ) italic_c italic_g start_POSTSUPERSCRIPT italic_i italic_j end_POSTSUPERSCRIPT ( italic_x ) italic_p start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_p start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT , end_ARG start_ARG square-root start_ARG italic_g start_POSTSUPERSCRIPT italic_k roman_ℓ end_POSTSUPERSCRIPT ( italic_x ) italic_p start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT italic_p start_POSTSUBSCRIPT roman_ℓ end_POSTSUBSCRIPT end_ARG end_ARG = italic_H start_POSTSUBSCRIPT italic_h end_POSTSUBSCRIPT ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_left_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(49)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S3.p4.10">the Hamiltonian (<a class="ltx_ref" href="https://arxiv.org/html/2406.11224v2#S3.E48" title="In 3 Complete integrability and geodesic Hamiltonian ‣ A Hamiltonian approach to the gradient-flow equations in information geometry"><span class="ltx_text ltx_ref_tag">48</span></a>) satisfies the condition (<a class="ltx_ref" href="https://arxiv.org/html/2406.11224v2#S3.E39" title="In 3 Complete integrability and geodesic Hamiltonian ‣ A Hamiltonian approach to the gradient-flow equations in information geometry"><span class="ltx_text ltx_ref_tag">39</span></a>) for any <math alttext="\xi(t)" class="ltx_Math" display="inline" id="S3.p4.9.m1.1"><semantics id="S3.p4.9.m1.1a"><mrow id="S3.p4.9.m1.1.2" xref="S3.p4.9.m1.1.2.cmml"><mi id="S3.p4.9.m1.1.2.2" xref="S3.p4.9.m1.1.2.2.cmml">ξ</mi><mo id="S3.p4.9.m1.1.2.1" xref="S3.p4.9.m1.1.2.1.cmml">⁢</mo><mrow id="S3.p4.9.m1.1.2.3.2" xref="S3.p4.9.m1.1.2.cmml"><mo id="S3.p4.9.m1.1.2.3.2.1" stretchy="false" xref="S3.p4.9.m1.1.2.cmml">(</mo><mi id="S3.p4.9.m1.1.1" xref="S3.p4.9.m1.1.1.cmml">t</mi><mo id="S3.p4.9.m1.1.2.3.2.2" stretchy="false" xref="S3.p4.9.m1.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.p4.9.m1.1b"><apply id="S3.p4.9.m1.1.2.cmml" xref="S3.p4.9.m1.1.2"><times id="S3.p4.9.m1.1.2.1.cmml" xref="S3.p4.9.m1.1.2.1"></times><ci id="S3.p4.9.m1.1.2.2.cmml" xref="S3.p4.9.m1.1.2.2">𝜉</ci><ci id="S3.p4.9.m1.1.1.cmml" xref="S3.p4.9.m1.1.1">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p4.9.m1.1c">\xi(t)</annotation><annotation encoding="application/x-llamapun" id="S3.p4.9.m1.1d">italic_ξ ( italic_t )</annotation></semantics></math>. In other words, the proportional factor <math alttext="\xi(t)" class="ltx_Math" display="inline" id="S3.p4.10.m2.1"><semantics id="S3.p4.10.m2.1a"><mrow id="S3.p4.10.m2.1.2" xref="S3.p4.10.m2.1.2.cmml"><mi id="S3.p4.10.m2.1.2.2" xref="S3.p4.10.m2.1.2.2.cmml">ξ</mi><mo id="S3.p4.10.m2.1.2.1" xref="S3.p4.10.m2.1.2.1.cmml">⁢</mo><mrow id="S3.p4.10.m2.1.2.3.2" xref="S3.p4.10.m2.1.2.cmml"><mo id="S3.p4.10.m2.1.2.3.2.1" stretchy="false" xref="S3.p4.10.m2.1.2.cmml">(</mo><mi id="S3.p4.10.m2.1.1" xref="S3.p4.10.m2.1.1.cmml">t</mi><mo id="S3.p4.10.m2.1.2.3.2.2" stretchy="false" xref="S3.p4.10.m2.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.p4.10.m2.1b"><apply id="S3.p4.10.m2.1.2.cmml" xref="S3.p4.10.m2.1.2"><times id="S3.p4.10.m2.1.2.1.cmml" xref="S3.p4.10.m2.1.2.1"></times><ci id="S3.p4.10.m2.1.2.2.cmml" xref="S3.p4.10.m2.1.2.2">𝜉</ci><ci id="S3.p4.10.m2.1.1.cmml" xref="S3.p4.10.m2.1.1">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p4.10.m2.1c">\xi(t)</annotation><annotation encoding="application/x-llamapun" id="S3.p4.10.m2.1d">italic_ξ ( italic_t )</annotation></semantics></math> is not determined by the condition (<a class="ltx_ref" href="https://arxiv.org/html/2406.11224v2#S3.E39" title="In 3 Complete integrability and geodesic Hamiltonian ‣ A Hamiltonian approach to the gradient-flow equations in information geometry"><span class="ltx_text ltx_ref_tag">39</span></a>) only.</p> </div> <div class="ltx_para" id="S3.p5"> <p class="ltx_p" id="S3.p5.1">Recall that the energy-momentum relation (or on shell relation) of a particle with a rest mass <math alttext="m" class="ltx_Math" display="inline" id="S3.p5.1.m1.1"><semantics id="S3.p5.1.m1.1a"><mi id="S3.p5.1.m1.1.1" xref="S3.p5.1.m1.1.1.cmml">m</mi><annotation-xml encoding="MathML-Content" id="S3.p5.1.m1.1b"><ci id="S3.p5.1.m1.1.1.cmml" xref="S3.p5.1.m1.1.1">𝑚</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.p5.1.m1.1c">m</annotation><annotation encoding="application/x-llamapun" id="S3.p5.1.m1.1d">italic_m</annotation></semantics></math> is</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="S3.EGx50"> <tbody id="S3.Ex9"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_left_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle E_{\rm rel}(t)" class="ltx_Math" display="inline" id="S3.Ex9.m1.1"><semantics id="S3.Ex9.m1.1a"><mrow id="S3.Ex9.m1.1.2" xref="S3.Ex9.m1.1.2.cmml"><msub id="S3.Ex9.m1.1.2.2" xref="S3.Ex9.m1.1.2.2.cmml"><mi id="S3.Ex9.m1.1.2.2.2" xref="S3.Ex9.m1.1.2.2.2.cmml">E</mi><mi id="S3.Ex9.m1.1.2.2.3" xref="S3.Ex9.m1.1.2.2.3.cmml">rel</mi></msub><mo id="S3.Ex9.m1.1.2.1" xref="S3.Ex9.m1.1.2.1.cmml">⁢</mo><mrow id="S3.Ex9.m1.1.2.3.2" xref="S3.Ex9.m1.1.2.cmml"><mo id="S3.Ex9.m1.1.2.3.2.1" stretchy="false" xref="S3.Ex9.m1.1.2.cmml">(</mo><mi id="S3.Ex9.m1.1.1" xref="S3.Ex9.m1.1.1.cmml">t</mi><mo id="S3.Ex9.m1.1.2.3.2.2" stretchy="false" xref="S3.Ex9.m1.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Ex9.m1.1b"><apply id="S3.Ex9.m1.1.2.cmml" xref="S3.Ex9.m1.1.2"><times id="S3.Ex9.m1.1.2.1.cmml" xref="S3.Ex9.m1.1.2.1"></times><apply id="S3.Ex9.m1.1.2.2.cmml" xref="S3.Ex9.m1.1.2.2"><csymbol cd="ambiguous" id="S3.Ex9.m1.1.2.2.1.cmml" xref="S3.Ex9.m1.1.2.2">subscript</csymbol><ci id="S3.Ex9.m1.1.2.2.2.cmml" xref="S3.Ex9.m1.1.2.2.2">𝐸</ci><ci id="S3.Ex9.m1.1.2.2.3.cmml" xref="S3.Ex9.m1.1.2.2.3">rel</ci></apply><ci id="S3.Ex9.m1.1.1.cmml" xref="S3.Ex9.m1.1.1">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Ex9.m1.1c">\displaystyle E_{\rm rel}(t)</annotation><annotation encoding="application/x-llamapun" id="S3.Ex9.m1.1d">italic_E start_POSTSUBSCRIPT roman_rel end_POSTSUBSCRIPT ( italic_t )</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle=\sqrt{c^{2}p^{2}(t)+m^{2}c^{4}}" class="ltx_Math" display="inline" id="S3.Ex9.m2.1"><semantics id="S3.Ex9.m2.1a"><mrow id="S3.Ex9.m2.1.2" xref="S3.Ex9.m2.1.2.cmml"><mi id="S3.Ex9.m2.1.2.2" xref="S3.Ex9.m2.1.2.2.cmml"></mi><mo id="S3.Ex9.m2.1.2.1" xref="S3.Ex9.m2.1.2.1.cmml">=</mo><msqrt id="S3.Ex9.m2.1.1" xref="S3.Ex9.m2.1.1.cmml"><mrow id="S3.Ex9.m2.1.1.1" xref="S3.Ex9.m2.1.1.1.cmml"><mrow id="S3.Ex9.m2.1.1.1.3" xref="S3.Ex9.m2.1.1.1.3.cmml"><msup id="S3.Ex9.m2.1.1.1.3.2" xref="S3.Ex9.m2.1.1.1.3.2.cmml"><mi id="S3.Ex9.m2.1.1.1.3.2.2" xref="S3.Ex9.m2.1.1.1.3.2.2.cmml">c</mi><mn id="S3.Ex9.m2.1.1.1.3.2.3" xref="S3.Ex9.m2.1.1.1.3.2.3.cmml">2</mn></msup><mo id="S3.Ex9.m2.1.1.1.3.1" xref="S3.Ex9.m2.1.1.1.3.1.cmml">⁢</mo><msup id="S3.Ex9.m2.1.1.1.3.3" xref="S3.Ex9.m2.1.1.1.3.3.cmml"><mi id="S3.Ex9.m2.1.1.1.3.3.2" xref="S3.Ex9.m2.1.1.1.3.3.2.cmml">p</mi><mn id="S3.Ex9.m2.1.1.1.3.3.3" xref="S3.Ex9.m2.1.1.1.3.3.3.cmml">2</mn></msup><mo id="S3.Ex9.m2.1.1.1.3.1a" 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id="S3.E50.m1.2.2.1.1.3.1.cmml" xref="S3.E50.m1.2.2.1.1.3.1"></times><apply id="S3.E50.m1.2.2.1.1.3.2.cmml" xref="S3.E50.m1.2.2.1.1.3.2"><csymbol cd="ambiguous" id="S3.E50.m1.2.2.1.1.3.2.1.cmml" xref="S3.E50.m1.2.2.1.1.3.2">superscript</csymbol><ci id="S3.E50.m1.2.2.1.1.3.2.2.cmml" xref="S3.E50.m1.2.2.1.1.3.2.2">𝑚</ci><cn id="S3.E50.m1.2.2.1.1.3.2.3.cmml" type="integer" xref="S3.E50.m1.2.2.1.1.3.2.3">2</cn></apply><apply id="S3.E50.m1.2.2.1.1.3.3.cmml" xref="S3.E50.m1.2.2.1.1.3.3"><csymbol cd="ambiguous" id="S3.E50.m1.2.2.1.1.3.3.1.cmml" xref="S3.E50.m1.2.2.1.1.3.3">superscript</csymbol><ci id="S3.E50.m1.2.2.1.1.3.3.2.cmml" xref="S3.E50.m1.2.2.1.1.3.3.2">𝑐</ci><cn id="S3.E50.m1.2.2.1.1.3.3.3.cmml" type="integer" xref="S3.E50.m1.2.2.1.1.3.3.3">2</cn></apply></apply><apply id="S3.E50.m1.2.2.1.1.1.cmml" xref="S3.E50.m1.2.2.1.1.1"><times id="S3.E50.m1.2.2.1.1.1.2.cmml" xref="S3.E50.m1.2.2.1.1.1.2"></times><apply id="S3.E50.m1.2.2.1.1.1.3.cmml" xref="S3.E50.m1.2.2.1.1.1.3"><csymbol cd="ambiguous" id="S3.E50.m1.2.2.1.1.1.3.1.cmml" xref="S3.E50.m1.2.2.1.1.1.3">superscript</csymbol><ci id="S3.E50.m1.2.2.1.1.1.3.2.cmml" xref="S3.E50.m1.2.2.1.1.1.3.2">𝑝</ci><cn id="S3.E50.m1.2.2.1.1.1.3.3.cmml" type="integer" xref="S3.E50.m1.2.2.1.1.1.3.3">2</cn></apply><ci id="S3.E50.m1.2.2.1.1.1.1.cmml" xref="S3.E50.m1.2.2.1.1.1.1">𝑡</ci></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.E50.m1.3c">\displaystyle=c\,\sqrt{p^{2}(t)}\,\sqrt{1+\frac{m^{2}c^{2}}{p^{2}(t)}},</annotation><annotation encoding="application/x-llamapun" id="S3.E50.m1.3d">= italic_c square-root start_ARG italic_p start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( italic_t ) end_ARG square-root start_ARG 1 + divide start_ARG italic_m start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_c start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG italic_p start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( italic_t ) end_ARG end_ARG ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_left_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(50)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S3.p5.2">where <math alttext="p^{2}(t)=g^{ij}(x)p_{i}p_{j}" class="ltx_Math" display="inline" id="S3.p5.2.m1.2"><semantics id="S3.p5.2.m1.2a"><mrow id="S3.p5.2.m1.2.3" xref="S3.p5.2.m1.2.3.cmml"><mrow id="S3.p5.2.m1.2.3.2" xref="S3.p5.2.m1.2.3.2.cmml"><msup id="S3.p5.2.m1.2.3.2.2" xref="S3.p5.2.m1.2.3.2.2.cmml"><mi id="S3.p5.2.m1.2.3.2.2.2" xref="S3.p5.2.m1.2.3.2.2.2.cmml">p</mi><mn id="S3.p5.2.m1.2.3.2.2.3" xref="S3.p5.2.m1.2.3.2.2.3.cmml">2</mn></msup><mo id="S3.p5.2.m1.2.3.2.1" xref="S3.p5.2.m1.2.3.2.1.cmml">⁢</mo><mrow id="S3.p5.2.m1.2.3.2.3.2" xref="S3.p5.2.m1.2.3.2.cmml"><mo id="S3.p5.2.m1.2.3.2.3.2.1" stretchy="false" xref="S3.p5.2.m1.2.3.2.cmml">(</mo><mi id="S3.p5.2.m1.1.1" xref="S3.p5.2.m1.1.1.cmml">t</mi><mo id="S3.p5.2.m1.2.3.2.3.2.2" stretchy="false" xref="S3.p5.2.m1.2.3.2.cmml">)</mo></mrow></mrow><mo id="S3.p5.2.m1.2.3.1" xref="S3.p5.2.m1.2.3.1.cmml">=</mo><mrow id="S3.p5.2.m1.2.3.3" xref="S3.p5.2.m1.2.3.3.cmml"><msup id="S3.p5.2.m1.2.3.3.2" xref="S3.p5.2.m1.2.3.3.2.cmml"><mi id="S3.p5.2.m1.2.3.3.2.2" xref="S3.p5.2.m1.2.3.3.2.2.cmml">g</mi><mrow id="S3.p5.2.m1.2.3.3.2.3" xref="S3.p5.2.m1.2.3.3.2.3.cmml"><mi id="S3.p5.2.m1.2.3.3.2.3.2" xref="S3.p5.2.m1.2.3.3.2.3.2.cmml">i</mi><mo id="S3.p5.2.m1.2.3.3.2.3.1" xref="S3.p5.2.m1.2.3.3.2.3.1.cmml">⁢</mo><mi id="S3.p5.2.m1.2.3.3.2.3.3" xref="S3.p5.2.m1.2.3.3.2.3.3.cmml">j</mi></mrow></msup><mo id="S3.p5.2.m1.2.3.3.1" xref="S3.p5.2.m1.2.3.3.1.cmml">⁢</mo><mrow id="S3.p5.2.m1.2.3.3.3.2" xref="S3.p5.2.m1.2.3.3.cmml"><mo id="S3.p5.2.m1.2.3.3.3.2.1" stretchy="false" xref="S3.p5.2.m1.2.3.3.cmml">(</mo><mi id="S3.p5.2.m1.2.2" xref="S3.p5.2.m1.2.2.cmml">x</mi><mo id="S3.p5.2.m1.2.3.3.3.2.2" stretchy="false" xref="S3.p5.2.m1.2.3.3.cmml">)</mo></mrow><mo id="S3.p5.2.m1.2.3.3.1a" xref="S3.p5.2.m1.2.3.3.1.cmml">⁢</mo><msub id="S3.p5.2.m1.2.3.3.4" xref="S3.p5.2.m1.2.3.3.4.cmml"><mi id="S3.p5.2.m1.2.3.3.4.2" xref="S3.p5.2.m1.2.3.3.4.2.cmml">p</mi><mi id="S3.p5.2.m1.2.3.3.4.3" xref="S3.p5.2.m1.2.3.3.4.3.cmml">i</mi></msub><mo id="S3.p5.2.m1.2.3.3.1b" xref="S3.p5.2.m1.2.3.3.1.cmml">⁢</mo><msub id="S3.p5.2.m1.2.3.3.5" xref="S3.p5.2.m1.2.3.3.5.cmml"><mi id="S3.p5.2.m1.2.3.3.5.2" xref="S3.p5.2.m1.2.3.3.5.2.cmml">p</mi><mi id="S3.p5.2.m1.2.3.3.5.3" xref="S3.p5.2.m1.2.3.3.5.3.cmml">j</mi></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.p5.2.m1.2b"><apply id="S3.p5.2.m1.2.3.cmml" xref="S3.p5.2.m1.2.3"><eq id="S3.p5.2.m1.2.3.1.cmml" xref="S3.p5.2.m1.2.3.1"></eq><apply id="S3.p5.2.m1.2.3.2.cmml" xref="S3.p5.2.m1.2.3.2"><times id="S3.p5.2.m1.2.3.2.1.cmml" xref="S3.p5.2.m1.2.3.2.1"></times><apply id="S3.p5.2.m1.2.3.2.2.cmml" xref="S3.p5.2.m1.2.3.2.2"><csymbol cd="ambiguous" id="S3.p5.2.m1.2.3.2.2.1.cmml" xref="S3.p5.2.m1.2.3.2.2">superscript</csymbol><ci id="S3.p5.2.m1.2.3.2.2.2.cmml" xref="S3.p5.2.m1.2.3.2.2.2">𝑝</ci><cn id="S3.p5.2.m1.2.3.2.2.3.cmml" type="integer" xref="S3.p5.2.m1.2.3.2.2.3">2</cn></apply><ci id="S3.p5.2.m1.1.1.cmml" xref="S3.p5.2.m1.1.1">𝑡</ci></apply><apply id="S3.p5.2.m1.2.3.3.cmml" xref="S3.p5.2.m1.2.3.3"><times id="S3.p5.2.m1.2.3.3.1.cmml" xref="S3.p5.2.m1.2.3.3.1"></times><apply id="S3.p5.2.m1.2.3.3.2.cmml" xref="S3.p5.2.m1.2.3.3.2"><csymbol cd="ambiguous" id="S3.p5.2.m1.2.3.3.2.1.cmml" xref="S3.p5.2.m1.2.3.3.2">superscript</csymbol><ci id="S3.p5.2.m1.2.3.3.2.2.cmml" xref="S3.p5.2.m1.2.3.3.2.2">𝑔</ci><apply id="S3.p5.2.m1.2.3.3.2.3.cmml" xref="S3.p5.2.m1.2.3.3.2.3"><times id="S3.p5.2.m1.2.3.3.2.3.1.cmml" xref="S3.p5.2.m1.2.3.3.2.3.1"></times><ci id="S3.p5.2.m1.2.3.3.2.3.2.cmml" xref="S3.p5.2.m1.2.3.3.2.3.2">𝑖</ci><ci id="S3.p5.2.m1.2.3.3.2.3.3.cmml" xref="S3.p5.2.m1.2.3.3.2.3.3">𝑗</ci></apply></apply><ci id="S3.p5.2.m1.2.2.cmml" xref="S3.p5.2.m1.2.2">𝑥</ci><apply id="S3.p5.2.m1.2.3.3.4.cmml" xref="S3.p5.2.m1.2.3.3.4"><csymbol cd="ambiguous" id="S3.p5.2.m1.2.3.3.4.1.cmml" xref="S3.p5.2.m1.2.3.3.4">subscript</csymbol><ci id="S3.p5.2.m1.2.3.3.4.2.cmml" xref="S3.p5.2.m1.2.3.3.4.2">𝑝</ci><ci id="S3.p5.2.m1.2.3.3.4.3.cmml" xref="S3.p5.2.m1.2.3.3.4.3">𝑖</ci></apply><apply id="S3.p5.2.m1.2.3.3.5.cmml" xref="S3.p5.2.m1.2.3.3.5"><csymbol cd="ambiguous" id="S3.p5.2.m1.2.3.3.5.1.cmml" xref="S3.p5.2.m1.2.3.3.5">subscript</csymbol><ci id="S3.p5.2.m1.2.3.3.5.2.cmml" xref="S3.p5.2.m1.2.3.3.5.2">𝑝</ci><ci id="S3.p5.2.m1.2.3.3.5.3.cmml" xref="S3.p5.2.m1.2.3.3.5.3">𝑗</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p5.2.m1.2c">p^{2}(t)=g^{ij}(x)p_{i}p_{j}</annotation><annotation encoding="application/x-llamapun" id="S3.p5.2.m1.2d">italic_p start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( italic_t ) = italic_g start_POSTSUPERSCRIPT italic_i italic_j end_POSTSUPERSCRIPT ( italic_x ) italic_p start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_p start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT</annotation></semantics></math>. By substituting the well known relation</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="S3.EGx51"> <tbody id="S3.E51"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_left_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle p^{2}(t)=\gamma^{2}\,m^{2}v^{2}(t),\;\textrm{with}\;\frac{1}{% \gamma}:=\sqrt{1-\frac{v^{2}(t)}{c^{2}}}," class="ltx_Math" display="inline" id="S3.E51.m1.4"><semantics id="S3.E51.m1.4a"><mrow id="S3.E51.m1.4.4.1"><mrow id="S3.E51.m1.4.4.1.1.2" xref="S3.E51.m1.4.4.1.1.3.cmml"><mrow id="S3.E51.m1.4.4.1.1.1.1" xref="S3.E51.m1.4.4.1.1.1.1.cmml"><mrow id="S3.E51.m1.4.4.1.1.1.1.2" xref="S3.E51.m1.4.4.1.1.1.1.2.cmml"><msup id="S3.E51.m1.4.4.1.1.1.1.2.2" xref="S3.E51.m1.4.4.1.1.1.1.2.2.cmml"><mi id="S3.E51.m1.4.4.1.1.1.1.2.2.2" xref="S3.E51.m1.4.4.1.1.1.1.2.2.2.cmml">p</mi><mn id="S3.E51.m1.4.4.1.1.1.1.2.2.3" xref="S3.E51.m1.4.4.1.1.1.1.2.2.3.cmml">2</mn></msup><mo 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p^{2}(t)=\gamma^{2}\,m^{2}v^{2}(t),\;\textrm{with}\;\frac{1}{% \gamma}:=\sqrt{1-\frac{v^{2}(t)}{c^{2}}},</annotation><annotation encoding="application/x-llamapun" id="S3.E51.m1.4d">italic_p start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( italic_t ) = italic_γ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_m start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_v start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( italic_t ) , with divide start_ARG 1 end_ARG start_ARG italic_γ end_ARG := square-root start_ARG 1 - divide start_ARG italic_v start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( italic_t ) end_ARG start_ARG italic_c start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG end_ARG ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_left_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(51)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S3.p5.11">in the theory of relativity, into (<a class="ltx_ref" href="https://arxiv.org/html/2406.11224v2#S3.E50" title="In 3 Complete integrability and geodesic Hamiltonian ‣ A Hamiltonian approach to the gradient-flow equations in information geometry"><span class="ltx_text ltx_ref_tag">50</span></a>), we see that</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="S3.EGx52"> <tbody id="S3.Ex10"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_left_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle E_{\rm rel}(t)" class="ltx_Math" display="inline" id="S3.Ex10.m1.1"><semantics id="S3.Ex10.m1.1a"><mrow id="S3.Ex10.m1.1.2" xref="S3.Ex10.m1.1.2.cmml"><msub id="S3.Ex10.m1.1.2.2" xref="S3.Ex10.m1.1.2.2.cmml"><mi id="S3.Ex10.m1.1.2.2.2" xref="S3.Ex10.m1.1.2.2.2.cmml">E</mi><mi id="S3.Ex10.m1.1.2.2.3" xref="S3.Ex10.m1.1.2.2.3.cmml">rel</mi></msub><mo id="S3.Ex10.m1.1.2.1" xref="S3.Ex10.m1.1.2.1.cmml">⁢</mo><mrow id="S3.Ex10.m1.1.2.3.2" xref="S3.Ex10.m1.1.2.cmml"><mo id="S3.Ex10.m1.1.2.3.2.1" stretchy="false" xref="S3.Ex10.m1.1.2.cmml">(</mo><mi id="S3.Ex10.m1.1.1" xref="S3.Ex10.m1.1.1.cmml">t</mi><mo id="S3.Ex10.m1.1.2.3.2.2" stretchy="false" xref="S3.Ex10.m1.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Ex10.m1.1b"><apply id="S3.Ex10.m1.1.2.cmml" xref="S3.Ex10.m1.1.2"><times id="S3.Ex10.m1.1.2.1.cmml" xref="S3.Ex10.m1.1.2.1"></times><apply id="S3.Ex10.m1.1.2.2.cmml" xref="S3.Ex10.m1.1.2.2"><csymbol cd="ambiguous" id="S3.Ex10.m1.1.2.2.1.cmml" xref="S3.Ex10.m1.1.2.2">subscript</csymbol><ci id="S3.Ex10.m1.1.2.2.2.cmml" xref="S3.Ex10.m1.1.2.2.2">𝐸</ci><ci id="S3.Ex10.m1.1.2.2.3.cmml" xref="S3.Ex10.m1.1.2.2.3">rel</ci></apply><ci id="S3.Ex10.m1.1.1.cmml" xref="S3.Ex10.m1.1.1">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Ex10.m1.1c">\displaystyle E_{\rm rel}(t)</annotation><annotation encoding="application/x-llamapun" id="S3.Ex10.m1.1d">italic_E start_POSTSUBSCRIPT roman_rel end_POSTSUBSCRIPT ( italic_t )</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle=\sqrt{1+\frac{1}{\gamma^{2}}\,\frac{c^{2}}{v^{2}(t)}}\,c\,\sqrt{% p^{2}(t)}" class="ltx_Math" display="inline" id="S3.Ex10.m2.2"><semantics id="S3.Ex10.m2.2a"><mrow id="S3.Ex10.m2.2.3" xref="S3.Ex10.m2.2.3.cmml"><mi id="S3.Ex10.m2.2.3.2" xref="S3.Ex10.m2.2.3.2.cmml"></mi><mo id="S3.Ex10.m2.2.3.1" xref="S3.Ex10.m2.2.3.1.cmml">=</mo><mrow id="S3.Ex10.m2.2.3.3" xref="S3.Ex10.m2.2.3.3.cmml"><msqrt id="S3.Ex10.m2.1.1" xref="S3.Ex10.m2.1.1.cmml"><mrow id="S3.Ex10.m2.1.1.1" xref="S3.Ex10.m2.1.1.1.cmml"><mn id="S3.Ex10.m2.1.1.1.3" xref="S3.Ex10.m2.1.1.1.3.cmml">1</mn><mo id="S3.Ex10.m2.1.1.1.2" xref="S3.Ex10.m2.1.1.1.2.cmml">+</mo><mrow id="S3.Ex10.m2.1.1.1.4" xref="S3.Ex10.m2.1.1.1.4.cmml"><mstyle displaystyle="true" id="S3.Ex10.m2.1.1.1.4.2" xref="S3.Ex10.m2.1.1.1.4.2.cmml"><mfrac 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p^{2}(t)}</annotation><annotation encoding="application/x-llamapun" id="S3.Ex10.m2.2d">= square-root start_ARG 1 + divide start_ARG 1 end_ARG start_ARG italic_γ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG divide start_ARG italic_c start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG italic_v start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( italic_t ) end_ARG end_ARG italic_c square-root start_ARG italic_p start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( italic_t ) end_ARG</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_left_padright"></td> </tr></tbody> <tbody id="S3.E52"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_left_padleft"></td> <td class="ltx_td ltx_eqn_cell"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle=\frac{c^{2}}{v(t)}\,\sqrt{g^{ij}(x)p_{i}p_{j}}=:H_{\rm rel}(x,p,% t)." class="ltx_math_unparsed" display="inline" id="S3.E52.m1.2"><semantics id="S3.E52.m1.2a"><mrow id="S3.E52.m1.2b"><mo id="S3.E52.m1.2.3">=</mo><mstyle displaystyle="true" id="S3.E52.m1.1.1"><mfrac id="S3.E52.m1.1.1a"><msup id="S3.E52.m1.1.1.3"><mi id="S3.E52.m1.1.1.3.2">c</mi><mn id="S3.E52.m1.1.1.3.3">2</mn></msup><mrow id="S3.E52.m1.1.1.1"><mi id="S3.E52.m1.1.1.1.3">v</mi><mo id="S3.E52.m1.1.1.1.2">⁢</mo><mrow id="S3.E52.m1.1.1.1.4.2"><mo id="S3.E52.m1.1.1.1.4.2.1" stretchy="false">(</mo><mi id="S3.E52.m1.1.1.1.1">t</mi><mo id="S3.E52.m1.1.1.1.4.2.2" stretchy="false">)</mo></mrow></mrow></mfrac></mstyle><msqrt id="S3.E52.m1.2.2"><mrow id="S3.E52.m1.2.2.1"><msup id="S3.E52.m1.2.2.1.3"><mi id="S3.E52.m1.2.2.1.3.2">g</mi><mrow id="S3.E52.m1.2.2.1.3.3"><mi id="S3.E52.m1.2.2.1.3.3.2">i</mi><mo id="S3.E52.m1.2.2.1.3.3.1">⁢</mo><mi id="S3.E52.m1.2.2.1.3.3.3">j</mi></mrow></msup><mo id="S3.E52.m1.2.2.1.2">⁢</mo><mrow id="S3.E52.m1.2.2.1.4.2"><mo id="S3.E52.m1.2.2.1.4.2.1" stretchy="false">(</mo><mi id="S3.E52.m1.2.2.1.1">x</mi><mo id="S3.E52.m1.2.2.1.4.2.2" stretchy="false">)</mo></mrow><mo id="S3.E52.m1.2.2.1.2a">⁢</mo><msub id="S3.E52.m1.2.2.1.5"><mi id="S3.E52.m1.2.2.1.5.2">p</mi><mi id="S3.E52.m1.2.2.1.5.3">i</mi></msub><mo id="S3.E52.m1.2.2.1.2b">⁢</mo><msub id="S3.E52.m1.2.2.1.6"><mi id="S3.E52.m1.2.2.1.6.2">p</mi><mi id="S3.E52.m1.2.2.1.6.3">j</mi></msub></mrow></msqrt><mo id="S3.E52.m1.2.4" rspace="0em">=</mo><mo id="S3.E52.m1.2.5" rspace="0.278em">:</mo><msub id="S3.E52.m1.2.6"><mi id="S3.E52.m1.2.6.2">H</mi><mi id="S3.E52.m1.2.6.3">rel</mi></msub><mrow id="S3.E52.m1.2.7"><mo id="S3.E52.m1.2.7.1" stretchy="false">(</mo><mi id="S3.E52.m1.2.7.2">x</mi><mo id="S3.E52.m1.2.7.3">,</mo><mi id="S3.E52.m1.2.7.4">p</mi><mo id="S3.E52.m1.2.7.5">,</mo><mi id="S3.E52.m1.2.7.6">t</mi><mo id="S3.E52.m1.2.7.7" stretchy="false">)</mo></mrow><mo id="S3.E52.m1.2.8" lspace="0em">.</mo></mrow><annotation encoding="application/x-tex" id="S3.E52.m1.2c">\displaystyle=\frac{c^{2}}{v(t)}\,\sqrt{g^{ij}(x)p_{i}p_{j}}=:H_{\rm rel}(x,p,% t).</annotation><annotation encoding="application/x-llamapun" id="S3.E52.m1.2d">= divide start_ARG italic_c start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG italic_v ( italic_t ) end_ARG square-root start_ARG italic_g start_POSTSUPERSCRIPT italic_i italic_j end_POSTSUPERSCRIPT ( italic_x ) italic_p start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_p start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT end_ARG = : italic_H start_POSTSUBSCRIPT roman_rel end_POSTSUBSCRIPT ( italic_x , italic_p , italic_t ) .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_left_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(52)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S3.p5.6">Comparing this with (<a class="ltx_ref" href="https://arxiv.org/html/2406.11224v2#S3.E48" title="In 3 Complete integrability and geodesic Hamiltonian ‣ A Hamiltonian approach to the gradient-flow equations in information geometry"><span class="ltx_text ltx_ref_tag">48</span></a>), the energy <math alttext="E(t)" class="ltx_Math" display="inline" id="S3.p5.3.m1.1"><semantics id="S3.p5.3.m1.1a"><mrow id="S3.p5.3.m1.1.2" xref="S3.p5.3.m1.1.2.cmml"><mi id="S3.p5.3.m1.1.2.2" xref="S3.p5.3.m1.1.2.2.cmml">E</mi><mo id="S3.p5.3.m1.1.2.1" xref="S3.p5.3.m1.1.2.1.cmml">⁢</mo><mrow id="S3.p5.3.m1.1.2.3.2" xref="S3.p5.3.m1.1.2.cmml"><mo id="S3.p5.3.m1.1.2.3.2.1" stretchy="false" xref="S3.p5.3.m1.1.2.cmml">(</mo><mi id="S3.p5.3.m1.1.1" xref="S3.p5.3.m1.1.1.cmml">t</mi><mo id="S3.p5.3.m1.1.2.3.2.2" stretchy="false" xref="S3.p5.3.m1.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.p5.3.m1.1b"><apply id="S3.p5.3.m1.1.2.cmml" xref="S3.p5.3.m1.1.2"><times id="S3.p5.3.m1.1.2.1.cmml" xref="S3.p5.3.m1.1.2.1"></times><ci id="S3.p5.3.m1.1.2.2.cmml" xref="S3.p5.3.m1.1.2.2">𝐸</ci><ci id="S3.p5.3.m1.1.1.cmml" xref="S3.p5.3.m1.1.1">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p5.3.m1.1c">E(t)</annotation><annotation encoding="application/x-llamapun" id="S3.p5.3.m1.1d">italic_E ( italic_t )</annotation></semantics></math> of the Hamiltonian (<a class="ltx_ref" href="https://arxiv.org/html/2406.11224v2#S3.E48" title="In 3 Complete integrability and geodesic Hamiltonian ‣ A Hamiltonian approach to the gradient-flow equations in information geometry"><span class="ltx_text ltx_ref_tag">48</span></a>) can be considered as the time-dependent relativistic energy of an accelerated (or decelerated) particle whose speed is <math alttext="v(t)=c/\xi(t)" class="ltx_Math" display="inline" id="S3.p5.4.m2.2"><semantics id="S3.p5.4.m2.2a"><mrow id="S3.p5.4.m2.2.3" xref="S3.p5.4.m2.2.3.cmml"><mrow id="S3.p5.4.m2.2.3.2" xref="S3.p5.4.m2.2.3.2.cmml"><mi id="S3.p5.4.m2.2.3.2.2" xref="S3.p5.4.m2.2.3.2.2.cmml">v</mi><mo id="S3.p5.4.m2.2.3.2.1" xref="S3.p5.4.m2.2.3.2.1.cmml">⁢</mo><mrow id="S3.p5.4.m2.2.3.2.3.2" xref="S3.p5.4.m2.2.3.2.cmml"><mo id="S3.p5.4.m2.2.3.2.3.2.1" stretchy="false" xref="S3.p5.4.m2.2.3.2.cmml">(</mo><mi id="S3.p5.4.m2.1.1" xref="S3.p5.4.m2.1.1.cmml">t</mi><mo id="S3.p5.4.m2.2.3.2.3.2.2" stretchy="false" xref="S3.p5.4.m2.2.3.2.cmml">)</mo></mrow></mrow><mo id="S3.p5.4.m2.2.3.1" xref="S3.p5.4.m2.2.3.1.cmml">=</mo><mrow id="S3.p5.4.m2.2.3.3" xref="S3.p5.4.m2.2.3.3.cmml"><mrow id="S3.p5.4.m2.2.3.3.2" xref="S3.p5.4.m2.2.3.3.2.cmml"><mi id="S3.p5.4.m2.2.3.3.2.2" xref="S3.p5.4.m2.2.3.3.2.2.cmml">c</mi><mo id="S3.p5.4.m2.2.3.3.2.1" xref="S3.p5.4.m2.2.3.3.2.1.cmml">/</mo><mi id="S3.p5.4.m2.2.3.3.2.3" xref="S3.p5.4.m2.2.3.3.2.3.cmml">ξ</mi></mrow><mo id="S3.p5.4.m2.2.3.3.1" xref="S3.p5.4.m2.2.3.3.1.cmml">⁢</mo><mrow id="S3.p5.4.m2.2.3.3.3.2" xref="S3.p5.4.m2.2.3.3.cmml"><mo id="S3.p5.4.m2.2.3.3.3.2.1" stretchy="false" xref="S3.p5.4.m2.2.3.3.cmml">(</mo><mi id="S3.p5.4.m2.2.2" xref="S3.p5.4.m2.2.2.cmml">t</mi><mo id="S3.p5.4.m2.2.3.3.3.2.2" stretchy="false" xref="S3.p5.4.m2.2.3.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.p5.4.m2.2b"><apply id="S3.p5.4.m2.2.3.cmml" xref="S3.p5.4.m2.2.3"><eq id="S3.p5.4.m2.2.3.1.cmml" xref="S3.p5.4.m2.2.3.1"></eq><apply id="S3.p5.4.m2.2.3.2.cmml" xref="S3.p5.4.m2.2.3.2"><times id="S3.p5.4.m2.2.3.2.1.cmml" xref="S3.p5.4.m2.2.3.2.1"></times><ci id="S3.p5.4.m2.2.3.2.2.cmml" xref="S3.p5.4.m2.2.3.2.2">𝑣</ci><ci id="S3.p5.4.m2.1.1.cmml" xref="S3.p5.4.m2.1.1">𝑡</ci></apply><apply id="S3.p5.4.m2.2.3.3.cmml" xref="S3.p5.4.m2.2.3.3"><times id="S3.p5.4.m2.2.3.3.1.cmml" xref="S3.p5.4.m2.2.3.3.1"></times><apply id="S3.p5.4.m2.2.3.3.2.cmml" xref="S3.p5.4.m2.2.3.3.2"><divide id="S3.p5.4.m2.2.3.3.2.1.cmml" xref="S3.p5.4.m2.2.3.3.2.1"></divide><ci id="S3.p5.4.m2.2.3.3.2.2.cmml" xref="S3.p5.4.m2.2.3.3.2.2">𝑐</ci><ci id="S3.p5.4.m2.2.3.3.2.3.cmml" xref="S3.p5.4.m2.2.3.3.2.3">𝜉</ci></apply><ci id="S3.p5.4.m2.2.2.cmml" xref="S3.p5.4.m2.2.2">𝑡</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p5.4.m2.2c">v(t)=c/\xi(t)</annotation><annotation encoding="application/x-llamapun" id="S3.p5.4.m2.2d">italic_v ( italic_t ) = italic_c / italic_ξ ( italic_t )</annotation></semantics></math>. From the perspective of geometric optics, the factor <math alttext="\xi(t)=c/v(t)" class="ltx_Math" display="inline" id="S3.p5.5.m3.2"><semantics id="S3.p5.5.m3.2a"><mrow id="S3.p5.5.m3.2.3" xref="S3.p5.5.m3.2.3.cmml"><mrow id="S3.p5.5.m3.2.3.2" xref="S3.p5.5.m3.2.3.2.cmml"><mi id="S3.p5.5.m3.2.3.2.2" xref="S3.p5.5.m3.2.3.2.2.cmml">ξ</mi><mo id="S3.p5.5.m3.2.3.2.1" xref="S3.p5.5.m3.2.3.2.1.cmml">⁢</mo><mrow id="S3.p5.5.m3.2.3.2.3.2" xref="S3.p5.5.m3.2.3.2.cmml"><mo id="S3.p5.5.m3.2.3.2.3.2.1" stretchy="false" xref="S3.p5.5.m3.2.3.2.cmml">(</mo><mi id="S3.p5.5.m3.1.1" xref="S3.p5.5.m3.1.1.cmml">t</mi><mo id="S3.p5.5.m3.2.3.2.3.2.2" stretchy="false" xref="S3.p5.5.m3.2.3.2.cmml">)</mo></mrow></mrow><mo id="S3.p5.5.m3.2.3.1" xref="S3.p5.5.m3.2.3.1.cmml">=</mo><mrow id="S3.p5.5.m3.2.3.3" xref="S3.p5.5.m3.2.3.3.cmml"><mrow id="S3.p5.5.m3.2.3.3.2" xref="S3.p5.5.m3.2.3.3.2.cmml"><mi id="S3.p5.5.m3.2.3.3.2.2" xref="S3.p5.5.m3.2.3.3.2.2.cmml">c</mi><mo id="S3.p5.5.m3.2.3.3.2.1" xref="S3.p5.5.m3.2.3.3.2.1.cmml">/</mo><mi id="S3.p5.5.m3.2.3.3.2.3" xref="S3.p5.5.m3.2.3.3.2.3.cmml">v</mi></mrow><mo id="S3.p5.5.m3.2.3.3.1" xref="S3.p5.5.m3.2.3.3.1.cmml">⁢</mo><mrow id="S3.p5.5.m3.2.3.3.3.2" xref="S3.p5.5.m3.2.3.3.cmml"><mo id="S3.p5.5.m3.2.3.3.3.2.1" stretchy="false" xref="S3.p5.5.m3.2.3.3.cmml">(</mo><mi id="S3.p5.5.m3.2.2" xref="S3.p5.5.m3.2.2.cmml">t</mi><mo id="S3.p5.5.m3.2.3.3.3.2.2" stretchy="false" xref="S3.p5.5.m3.2.3.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.p5.5.m3.2b"><apply id="S3.p5.5.m3.2.3.cmml" xref="S3.p5.5.m3.2.3"><eq id="S3.p5.5.m3.2.3.1.cmml" xref="S3.p5.5.m3.2.3.1"></eq><apply id="S3.p5.5.m3.2.3.2.cmml" xref="S3.p5.5.m3.2.3.2"><times id="S3.p5.5.m3.2.3.2.1.cmml" xref="S3.p5.5.m3.2.3.2.1"></times><ci id="S3.p5.5.m3.2.3.2.2.cmml" xref="S3.p5.5.m3.2.3.2.2">𝜉</ci><ci id="S3.p5.5.m3.1.1.cmml" xref="S3.p5.5.m3.1.1">𝑡</ci></apply><apply id="S3.p5.5.m3.2.3.3.cmml" xref="S3.p5.5.m3.2.3.3"><times id="S3.p5.5.m3.2.3.3.1.cmml" xref="S3.p5.5.m3.2.3.3.1"></times><apply id="S3.p5.5.m3.2.3.3.2.cmml" xref="S3.p5.5.m3.2.3.3.2"><divide id="S3.p5.5.m3.2.3.3.2.1.cmml" xref="S3.p5.5.m3.2.3.3.2.1"></divide><ci id="S3.p5.5.m3.2.3.3.2.2.cmml" xref="S3.p5.5.m3.2.3.3.2.2">𝑐</ci><ci id="S3.p5.5.m3.2.3.3.2.3.cmml" xref="S3.p5.5.m3.2.3.3.2.3">𝑣</ci></apply><ci id="S3.p5.5.m3.2.2.cmml" xref="S3.p5.5.m3.2.2">𝑡</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p5.5.m3.2c">\xi(t)=c/v(t)</annotation><annotation encoding="application/x-llamapun" id="S3.p5.5.m3.2d">italic_ξ ( italic_t ) = italic_c / italic_v ( italic_t )</annotation></semantics></math> can be considered as the refractive index <math alttext="n" class="ltx_Math" display="inline" id="S3.p5.6.m4.1"><semantics id="S3.p5.6.m4.1a"><mi id="S3.p5.6.m4.1.1" xref="S3.p5.6.m4.1.1.cmml">n</mi><annotation-xml encoding="MathML-Content" id="S3.p5.6.m4.1b"><ci id="S3.p5.6.m4.1.1.cmml" xref="S3.p5.6.m4.1.1">𝑛</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.p5.6.m4.1c">n</annotation><annotation encoding="application/x-llamapun" id="S3.p5.6.m4.1d">italic_n</annotation></semantics></math> of an optical medium. In the point-particle viewpoint <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2406.11224v2#bib.bib19" title="">19</a>]</cite>, the refractive index is expressed as</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="S3.EGx53"> <tbody id="S3.E53"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_left_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle n=\frac{c\,p}{E_{\rm ph}}," class="ltx_Math" display="inline" id="S3.E53.m1.1"><semantics id="S3.E53.m1.1a"><mrow id="S3.E53.m1.1.1.1" xref="S3.E53.m1.1.1.1.1.cmml"><mrow id="S3.E53.m1.1.1.1.1" xref="S3.E53.m1.1.1.1.1.cmml"><mi id="S3.E53.m1.1.1.1.1.2" xref="S3.E53.m1.1.1.1.1.2.cmml">n</mi><mo id="S3.E53.m1.1.1.1.1.1" xref="S3.E53.m1.1.1.1.1.1.cmml">=</mo><mstyle displaystyle="true" id="S3.E53.m1.1.1.1.1.3" xref="S3.E53.m1.1.1.1.1.3.cmml"><mfrac id="S3.E53.m1.1.1.1.1.3a" xref="S3.E53.m1.1.1.1.1.3.cmml"><mrow id="S3.E53.m1.1.1.1.1.3.2" xref="S3.E53.m1.1.1.1.1.3.2.cmml"><mi id="S3.E53.m1.1.1.1.1.3.2.2" xref="S3.E53.m1.1.1.1.1.3.2.2.cmml">c</mi><mo id="S3.E53.m1.1.1.1.1.3.2.1" lspace="0.170em" xref="S3.E53.m1.1.1.1.1.3.2.1.cmml">⁢</mo><mi id="S3.E53.m1.1.1.1.1.3.2.3" xref="S3.E53.m1.1.1.1.1.3.2.3.cmml">p</mi></mrow><msub id="S3.E53.m1.1.1.1.1.3.3" xref="S3.E53.m1.1.1.1.1.3.3.cmml"><mi id="S3.E53.m1.1.1.1.1.3.3.2" xref="S3.E53.m1.1.1.1.1.3.3.2.cmml">E</mi><mi id="S3.E53.m1.1.1.1.1.3.3.3" xref="S3.E53.m1.1.1.1.1.3.3.3.cmml">ph</mi></msub></mfrac></mstyle></mrow><mo id="S3.E53.m1.1.1.1.2" xref="S3.E53.m1.1.1.1.1.cmml">,</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.E53.m1.1b"><apply id="S3.E53.m1.1.1.1.1.cmml" xref="S3.E53.m1.1.1.1"><eq id="S3.E53.m1.1.1.1.1.1.cmml" xref="S3.E53.m1.1.1.1.1.1"></eq><ci id="S3.E53.m1.1.1.1.1.2.cmml" xref="S3.E53.m1.1.1.1.1.2">𝑛</ci><apply id="S3.E53.m1.1.1.1.1.3.cmml" xref="S3.E53.m1.1.1.1.1.3"><divide id="S3.E53.m1.1.1.1.1.3.1.cmml" xref="S3.E53.m1.1.1.1.1.3"></divide><apply id="S3.E53.m1.1.1.1.1.3.2.cmml" xref="S3.E53.m1.1.1.1.1.3.2"><times id="S3.E53.m1.1.1.1.1.3.2.1.cmml" xref="S3.E53.m1.1.1.1.1.3.2.1"></times><ci id="S3.E53.m1.1.1.1.1.3.2.2.cmml" xref="S3.E53.m1.1.1.1.1.3.2.2">𝑐</ci><ci id="S3.E53.m1.1.1.1.1.3.2.3.cmml" xref="S3.E53.m1.1.1.1.1.3.2.3">𝑝</ci></apply><apply id="S3.E53.m1.1.1.1.1.3.3.cmml" xref="S3.E53.m1.1.1.1.1.3.3"><csymbol cd="ambiguous" id="S3.E53.m1.1.1.1.1.3.3.1.cmml" xref="S3.E53.m1.1.1.1.1.3.3">subscript</csymbol><ci id="S3.E53.m1.1.1.1.1.3.3.2.cmml" xref="S3.E53.m1.1.1.1.1.3.3.2">𝐸</ci><ci id="S3.E53.m1.1.1.1.1.3.3.3.cmml" xref="S3.E53.m1.1.1.1.1.3.3.3">ph</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.E53.m1.1c">\displaystyle n=\frac{c\,p}{E_{\rm ph}},</annotation><annotation encoding="application/x-llamapun" id="S3.E53.m1.1d">italic_n = divide start_ARG italic_c italic_p end_ARG start_ARG italic_E start_POSTSUBSCRIPT roman_ph end_POSTSUBSCRIPT end_ARG ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_left_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(53)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S3.p5.10">where <math alttext="p" class="ltx_Math" display="inline" id="S3.p5.7.m1.1"><semantics id="S3.p5.7.m1.1a"><mi id="S3.p5.7.m1.1.1" xref="S3.p5.7.m1.1.1.cmml">p</mi><annotation-xml encoding="MathML-Content" id="S3.p5.7.m1.1b"><ci id="S3.p5.7.m1.1.1.cmml" xref="S3.p5.7.m1.1.1">𝑝</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.p5.7.m1.1c">p</annotation><annotation encoding="application/x-llamapun" id="S3.p5.7.m1.1d">italic_p</annotation></semantics></math> and <math alttext="E_{\rm ph}" class="ltx_Math" display="inline" id="S3.p5.8.m2.1"><semantics id="S3.p5.8.m2.1a"><msub id="S3.p5.8.m2.1.1" xref="S3.p5.8.m2.1.1.cmml"><mi id="S3.p5.8.m2.1.1.2" xref="S3.p5.8.m2.1.1.2.cmml">E</mi><mi id="S3.p5.8.m2.1.1.3" xref="S3.p5.8.m2.1.1.3.cmml">ph</mi></msub><annotation-xml encoding="MathML-Content" id="S3.p5.8.m2.1b"><apply id="S3.p5.8.m2.1.1.cmml" xref="S3.p5.8.m2.1.1"><csymbol cd="ambiguous" id="S3.p5.8.m2.1.1.1.cmml" xref="S3.p5.8.m2.1.1">subscript</csymbol><ci id="S3.p5.8.m2.1.1.2.cmml" xref="S3.p5.8.m2.1.1.2">𝐸</ci><ci id="S3.p5.8.m2.1.1.3.cmml" xref="S3.p5.8.m2.1.1.3">ph</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p5.8.m2.1c">E_{\rm ph}</annotation><annotation encoding="application/x-llamapun" id="S3.p5.8.m2.1d">italic_E start_POSTSUBSCRIPT roman_ph end_POSTSUBSCRIPT</annotation></semantics></math> are the photon momentum and energy, respectively. It is worth mentioning that <math alttext="n=c/v_{p}" class="ltx_Math" display="inline" id="S3.p5.9.m3.1"><semantics id="S3.p5.9.m3.1a"><mrow id="S3.p5.9.m3.1.1" xref="S3.p5.9.m3.1.1.cmml"><mi id="S3.p5.9.m3.1.1.2" xref="S3.p5.9.m3.1.1.2.cmml">n</mi><mo id="S3.p5.9.m3.1.1.1" xref="S3.p5.9.m3.1.1.1.cmml">=</mo><mrow id="S3.p5.9.m3.1.1.3" xref="S3.p5.9.m3.1.1.3.cmml"><mi id="S3.p5.9.m3.1.1.3.2" xref="S3.p5.9.m3.1.1.3.2.cmml">c</mi><mo id="S3.p5.9.m3.1.1.3.1" xref="S3.p5.9.m3.1.1.3.1.cmml">/</mo><msub id="S3.p5.9.m3.1.1.3.3" xref="S3.p5.9.m3.1.1.3.3.cmml"><mi id="S3.p5.9.m3.1.1.3.3.2" xref="S3.p5.9.m3.1.1.3.3.2.cmml">v</mi><mi id="S3.p5.9.m3.1.1.3.3.3" xref="S3.p5.9.m3.1.1.3.3.3.cmml">p</mi></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.p5.9.m3.1b"><apply id="S3.p5.9.m3.1.1.cmml" xref="S3.p5.9.m3.1.1"><eq id="S3.p5.9.m3.1.1.1.cmml" xref="S3.p5.9.m3.1.1.1"></eq><ci id="S3.p5.9.m3.1.1.2.cmml" xref="S3.p5.9.m3.1.1.2">𝑛</ci><apply id="S3.p5.9.m3.1.1.3.cmml" xref="S3.p5.9.m3.1.1.3"><divide id="S3.p5.9.m3.1.1.3.1.cmml" xref="S3.p5.9.m3.1.1.3.1"></divide><ci id="S3.p5.9.m3.1.1.3.2.cmml" xref="S3.p5.9.m3.1.1.3.2">𝑐</ci><apply id="S3.p5.9.m3.1.1.3.3.cmml" xref="S3.p5.9.m3.1.1.3.3"><csymbol cd="ambiguous" id="S3.p5.9.m3.1.1.3.3.1.cmml" xref="S3.p5.9.m3.1.1.3.3">subscript</csymbol><ci id="S3.p5.9.m3.1.1.3.3.2.cmml" xref="S3.p5.9.m3.1.1.3.3.2">𝑣</ci><ci id="S3.p5.9.m3.1.1.3.3.3.cmml" xref="S3.p5.9.m3.1.1.3.3.3">𝑝</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p5.9.m3.1c">n=c/v_{p}</annotation><annotation encoding="application/x-llamapun" id="S3.p5.9.m3.1d">italic_n = italic_c / italic_v start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT</annotation></semantics></math> where <math alttext="v_{p}" class="ltx_Math" display="inline" id="S3.p5.10.m4.1"><semantics id="S3.p5.10.m4.1a"><msub id="S3.p5.10.m4.1.1" xref="S3.p5.10.m4.1.1.cmml"><mi id="S3.p5.10.m4.1.1.2" xref="S3.p5.10.m4.1.1.2.cmml">v</mi><mi id="S3.p5.10.m4.1.1.3" xref="S3.p5.10.m4.1.1.3.cmml">p</mi></msub><annotation-xml encoding="MathML-Content" id="S3.p5.10.m4.1b"><apply id="S3.p5.10.m4.1.1.cmml" xref="S3.p5.10.m4.1.1"><csymbol cd="ambiguous" id="S3.p5.10.m4.1.1.1.cmml" xref="S3.p5.10.m4.1.1">subscript</csymbol><ci id="S3.p5.10.m4.1.1.2.cmml" xref="S3.p5.10.m4.1.1.2">𝑣</ci><ci id="S3.p5.10.m4.1.1.3.cmml" xref="S3.p5.10.m4.1.1.3">𝑝</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p5.10.m4.1c">v_{p}</annotation><annotation encoding="application/x-llamapun" id="S3.p5.10.m4.1d">italic_v start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT</annotation></semantics></math> is a particle (photon) velocity, not a phase velocity in the wave theory.</p> </div> <div class="ltx_para" id="S3.p6"> <p class="ltx_p" id="S3.p6.1">Note that in the natural unit <math alttext="c=1" class="ltx_Math" display="inline" id="S3.p6.1.m1.1"><semantics id="S3.p6.1.m1.1a"><mrow id="S3.p6.1.m1.1.1" xref="S3.p6.1.m1.1.1.cmml"><mi id="S3.p6.1.m1.1.1.2" xref="S3.p6.1.m1.1.1.2.cmml">c</mi><mo id="S3.p6.1.m1.1.1.1" xref="S3.p6.1.m1.1.1.1.cmml">=</mo><mn id="S3.p6.1.m1.1.1.3" xref="S3.p6.1.m1.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.p6.1.m1.1b"><apply id="S3.p6.1.m1.1.1.cmml" xref="S3.p6.1.m1.1.1"><eq id="S3.p6.1.m1.1.1.1.cmml" xref="S3.p6.1.m1.1.1.1"></eq><ci id="S3.p6.1.m1.1.1.2.cmml" xref="S3.p6.1.m1.1.1.2">𝑐</ci><cn id="S3.p6.1.m1.1.1.3.cmml" type="integer" xref="S3.p6.1.m1.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p6.1.m1.1c">c=1</annotation><annotation encoding="application/x-llamapun" id="S3.p6.1.m1.1d">italic_c = 1</annotation></semantics></math>, from the above homogeneous Hamiltonian (<a class="ltx_ref" href="https://arxiv.org/html/2406.11224v2#S3.E52" title="In 3 Complete integrability and geodesic Hamiltonian ‣ A Hamiltonian approach to the gradient-flow equations in information geometry"><span class="ltx_text ltx_ref_tag">52</span></a>) we can construct the following null Hamiltonian</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="S3.EGx54"> <tbody id="S3.Ex11"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_left_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle 0" class="ltx_Math" display="inline" id="S3.Ex11.m1.1"><semantics id="S3.Ex11.m1.1a"><mn id="S3.Ex11.m1.1.1" xref="S3.Ex11.m1.1.1.cmml">0</mn><annotation-xml encoding="MathML-Content" id="S3.Ex11.m1.1b"><cn id="S3.Ex11.m1.1.1.cmml" type="integer" xref="S3.Ex11.m1.1.1">0</cn></annotation-xml><annotation encoding="application/x-tex" id="S3.Ex11.m1.1c">\displaystyle 0</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle=v(t)\Big{(}H_{\rm rel}(x,p,t)-E_{\rm rel}(t)\Big{)}" class="ltx_Math" display="inline" id="S3.Ex11.m2.6"><semantics id="S3.Ex11.m2.6a"><mrow 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italic_g start_POSTSUPERSCRIPT italic_i italic_j end_POSTSUPERSCRIPT ( italic_x ) italic_p start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_p start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT end_ARG</annotation></semantics></math> is an instantaneous value of the momentum. The expression in (<a class="ltx_ref" href="https://arxiv.org/html/2406.11224v2#S3.E54" title="In 3 Complete integrability and geodesic Hamiltonian ‣ A Hamiltonian approach to the gradient-flow equations in information geometry"><span class="ltx_text ltx_ref_tag">54</span></a>) is the same form of the Hamiltonian (<a class="ltx_ref" href="https://arxiv.org/html/2406.11224v2#S2.E20" title="In 2.2 Gradient-Flow Equations ‣ 2 Information Geometry and Gradient-Flow Equations ‣ A Hamiltonian approach to the gradient-flow equations in information geometry"><span class="ltx_text ltx_ref_tag">20</span></a>) describing the gradient-flows in IG if we set <math alttext="x^{i}=\theta^{i},p_{i}=\eta_{i},g^{ij}(x)=g^{ij}(\theta)" class="ltx_Math" display="inline" id="S3.p6.3.m2.4"><semantics id="S3.p6.3.m2.4a"><mrow id="S3.p6.3.m2.4.4.2" xref="S3.p6.3.m2.4.4.3.cmml"><mrow id="S3.p6.3.m2.3.3.1.1" xref="S3.p6.3.m2.3.3.1.1.cmml"><msup id="S3.p6.3.m2.3.3.1.1.2" xref="S3.p6.3.m2.3.3.1.1.2.cmml"><mi 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xref="S3.p6.4.m3.2.2">𝑡</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p6.4.m3.3c">p(t)=\eta(\theta(t))</annotation><annotation encoding="application/x-llamapun" id="S3.p6.4.m3.3d">italic_p ( italic_t ) = italic_η ( italic_θ ( italic_t ) )</annotation></semantics></math>.</p> </div> </section> <section class="ltx_section" id="S4"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">4 </span>The motions of a light-like particle in a pseudo Riemann space</h2> <div class="ltx_para" id="S4.p1"> <p class="ltx_p" id="S4.p1.1">In general relativity <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2406.11224v2#bib.bib14" title="">14</a>]</cite>, it is assumed that light propagates along a null geodesic in a pseudo-Riemann space. An eikonal equation is assumed to be satisfied and such a light propagation follows Fermat’s principle <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2406.11224v2#bib.bib16" title="">16</a>]</cite> and is well described in geometric optics <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2406.11224v2#bib.bib20" title="">20</a>]</cite>. The Arnowitt, Deser, Misner (ADM) formalism <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2406.11224v2#bib.bib21" title="">21</a>]</cite> is a Hamiltonian formulation of general relativity. Caveny et al. <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2406.11224v2#bib.bib22" title="">22</a>]</cite> developed the method for tracking black hole event horizon. Their method is based on the hyperbolic eikonal equation and it provides the Hamilton equations of motion for a null (or light-like) geodesic motion in a curved space described by a pseudo-Riemann metric. Belayev <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2406.11224v2#bib.bib23" title="">23</a>]</cite> considered the variation of the energy for a light-like (null) particle in the pseudo-Riemann spacetime. We here first review their method according to Ref. <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2406.11224v2#bib.bib22" title="">22</a>]</cite> and then we apply their method to the gradient-flow equations in IG by taking into account of the role of a conformal factor.</p> </div> <div class="ltx_para" id="S4.p2"> <p class="ltx_p" id="S4.p2.25">Let us consider the following form of a stationary metric</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="S3.EGx55"> <tbody id="S4.Ex12"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_left_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle G_{\mu\nu}(x)" class="ltx_Math" display="inline" id="S4.Ex12.m1.1"><semantics id="S4.Ex12.m1.1a"><mrow id="S4.Ex12.m1.1.2" xref="S4.Ex12.m1.1.2.cmml"><msub id="S4.Ex12.m1.1.2.2" xref="S4.Ex12.m1.1.2.2.cmml"><mi id="S4.Ex12.m1.1.2.2.2" xref="S4.Ex12.m1.1.2.2.2.cmml">G</mi><mrow id="S4.Ex12.m1.1.2.2.3" xref="S4.Ex12.m1.1.2.2.3.cmml"><mi id="S4.Ex12.m1.1.2.2.3.2" xref="S4.Ex12.m1.1.2.2.3.2.cmml">μ</mi><mo id="S4.Ex12.m1.1.2.2.3.1" xref="S4.Ex12.m1.1.2.2.3.1.cmml">⁢</mo><mi id="S4.Ex12.m1.1.2.2.3.3" xref="S4.Ex12.m1.1.2.2.3.3.cmml">ν</mi></mrow></msub><mo id="S4.Ex12.m1.1.2.1" xref="S4.Ex12.m1.1.2.1.cmml">⁢</mo><mrow id="S4.Ex12.m1.1.2.3.2" xref="S4.Ex12.m1.1.2.cmml"><mo id="S4.Ex12.m1.1.2.3.2.1" stretchy="false" xref="S4.Ex12.m1.1.2.cmml">(</mo><mi id="S4.Ex12.m1.1.1" xref="S4.Ex12.m1.1.1.cmml">x</mi><mo id="S4.Ex12.m1.1.2.3.2.2" stretchy="false" xref="S4.Ex12.m1.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Ex12.m1.1b"><apply id="S4.Ex12.m1.1.2.cmml" xref="S4.Ex12.m1.1.2"><times id="S4.Ex12.m1.1.2.1.cmml" xref="S4.Ex12.m1.1.2.1"></times><apply id="S4.Ex12.m1.1.2.2.cmml" xref="S4.Ex12.m1.1.2.2"><csymbol cd="ambiguous" id="S4.Ex12.m1.1.2.2.1.cmml" xref="S4.Ex12.m1.1.2.2">subscript</csymbol><ci id="S4.Ex12.m1.1.2.2.2.cmml" xref="S4.Ex12.m1.1.2.2.2">𝐺</ci><apply id="S4.Ex12.m1.1.2.2.3.cmml" xref="S4.Ex12.m1.1.2.2.3"><times id="S4.Ex12.m1.1.2.2.3.1.cmml" xref="S4.Ex12.m1.1.2.2.3.1"></times><ci id="S4.Ex12.m1.1.2.2.3.2.cmml" xref="S4.Ex12.m1.1.2.2.3.2">𝜇</ci><ci id="S4.Ex12.m1.1.2.2.3.3.cmml" xref="S4.Ex12.m1.1.2.2.3.3">𝜈</ci></apply></apply><ci id="S4.Ex12.m1.1.1.cmml" xref="S4.Ex12.m1.1.1">𝑥</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Ex12.m1.1c">\displaystyle G_{\mu\nu}(x)</annotation><annotation encoding="application/x-llamapun" id="S4.Ex12.m1.1d">italic_G start_POSTSUBSCRIPT italic_μ italic_ν end_POSTSUBSCRIPT ( italic_x )</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle dx^{\mu}dx^{\nu}=-\alpha^{2}(dx^{0})^{2}" class="ltx_Math" display="inline" id="S4.Ex12.m2.1"><semantics id="S4.Ex12.m2.1a"><mrow id="S4.Ex12.m2.1.1" 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xref="S4.Ex12.m2.1.1.1.1.1.3">2</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Ex12.m2.1c">\displaystyle dx^{\mu}dx^{\nu}=-\alpha^{2}(dx^{0})^{2}</annotation><annotation encoding="application/x-llamapun" id="S4.Ex12.m2.1d">italic_d italic_x start_POSTSUPERSCRIPT italic_μ end_POSTSUPERSCRIPT italic_d italic_x start_POSTSUPERSCRIPT italic_ν end_POSTSUPERSCRIPT = - italic_α start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( italic_d italic_x start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_left_padright"></td> </tr></tbody> <tbody id="S4.E55"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_left_padleft"></td> <td class="ltx_td ltx_eqn_cell"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle+\gamma_{ij}(dx^{i}+\beta^{i}dx^{0})(dx^{j}+\beta^{j}dx^{0})," 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id="S4.E55.m1.1.1.1.1.2.2.1.1.3.2.3.cmml" xref="S4.E55.m1.1.1.1.1.2.2.1.1.3.2.3">𝑗</ci></apply><ci id="S4.E55.m1.1.1.1.1.2.2.1.1.3.3.cmml" xref="S4.E55.m1.1.1.1.1.2.2.1.1.3.3">𝑑</ci><apply id="S4.E55.m1.1.1.1.1.2.2.1.1.3.4.cmml" xref="S4.E55.m1.1.1.1.1.2.2.1.1.3.4"><csymbol cd="ambiguous" id="S4.E55.m1.1.1.1.1.2.2.1.1.3.4.1.cmml" xref="S4.E55.m1.1.1.1.1.2.2.1.1.3.4">superscript</csymbol><ci id="S4.E55.m1.1.1.1.1.2.2.1.1.3.4.2.cmml" xref="S4.E55.m1.1.1.1.1.2.2.1.1.3.4.2">𝑥</ci><cn id="S4.E55.m1.1.1.1.1.2.2.1.1.3.4.3.cmml" type="integer" xref="S4.E55.m1.1.1.1.1.2.2.1.1.3.4.3">0</cn></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.E55.m1.1c">\displaystyle+\gamma_{ij}(dx^{i}+\beta^{i}dx^{0})(dx^{j}+\beta^{j}dx^{0}),</annotation><annotation encoding="application/x-llamapun" id="S4.E55.m1.1d">+ italic_γ start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT ( italic_d italic_x start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT + italic_β start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT italic_d italic_x start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT ) ( italic_d italic_x start_POSTSUPERSCRIPT italic_j end_POSTSUPERSCRIPT + italic_β start_POSTSUPERSCRIPT italic_j end_POSTSUPERSCRIPT italic_d italic_x start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT ) ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_left_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(55)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S4.p2.5">where we assume <math alttext="\alpha" class="ltx_Math" display="inline" id="S4.p2.1.m1.1"><semantics id="S4.p2.1.m1.1a"><mi id="S4.p2.1.m1.1.1" xref="S4.p2.1.m1.1.1.cmml">α</mi><annotation-xml encoding="MathML-Content" id="S4.p2.1.m1.1b"><ci id="S4.p2.1.m1.1.1.cmml" xref="S4.p2.1.m1.1.1">𝛼</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.p2.1.m1.1c">\alpha</annotation><annotation encoding="application/x-llamapun" id="S4.p2.1.m1.1d">italic_α</annotation></semantics></math> and <math alttext="\beta^{i}" class="ltx_Math" display="inline" id="S4.p2.2.m2.1"><semantics id="S4.p2.2.m2.1a"><msup id="S4.p2.2.m2.1.1" xref="S4.p2.2.m2.1.1.cmml"><mi id="S4.p2.2.m2.1.1.2" xref="S4.p2.2.m2.1.1.2.cmml">β</mi><mi id="S4.p2.2.m2.1.1.3" xref="S4.p2.2.m2.1.1.3.cmml">i</mi></msup><annotation-xml encoding="MathML-Content" id="S4.p2.2.m2.1b"><apply id="S4.p2.2.m2.1.1.cmml" xref="S4.p2.2.m2.1.1"><csymbol cd="ambiguous" id="S4.p2.2.m2.1.1.1.cmml" xref="S4.p2.2.m2.1.1">superscript</csymbol><ci id="S4.p2.2.m2.1.1.2.cmml" xref="S4.p2.2.m2.1.1.2">𝛽</ci><ci id="S4.p2.2.m2.1.1.3.cmml" xref="S4.p2.2.m2.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p2.2.m2.1c">\beta^{i}</annotation><annotation encoding="application/x-llamapun" id="S4.p2.2.m2.1d">italic_β start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT</annotation></semantics></math> are some functions of the space coordinate <math alttext="x^{i}" class="ltx_Math" display="inline" id="S4.p2.3.m3.1"><semantics id="S4.p2.3.m3.1a"><msup id="S4.p2.3.m3.1.1" xref="S4.p2.3.m3.1.1.cmml"><mi id="S4.p2.3.m3.1.1.2" xref="S4.p2.3.m3.1.1.2.cmml">x</mi><mi id="S4.p2.3.m3.1.1.3" xref="S4.p2.3.m3.1.1.3.cmml">i</mi></msup><annotation-xml encoding="MathML-Content" id="S4.p2.3.m3.1b"><apply id="S4.p2.3.m3.1.1.cmml" xref="S4.p2.3.m3.1.1"><csymbol cd="ambiguous" id="S4.p2.3.m3.1.1.1.cmml" xref="S4.p2.3.m3.1.1">superscript</csymbol><ci id="S4.p2.3.m3.1.1.2.cmml" xref="S4.p2.3.m3.1.1.2">𝑥</ci><ci id="S4.p2.3.m3.1.1.3.cmml" xref="S4.p2.3.m3.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p2.3.m3.1c">x^{i}</annotation><annotation encoding="application/x-llamapun" id="S4.p2.3.m3.1d">italic_x start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT</annotation></semantics></math> only and <math alttext="\gamma_{ij}" class="ltx_Math" display="inline" id="S4.p2.4.m4.1"><semantics id="S4.p2.4.m4.1a"><msub id="S4.p2.4.m4.1.1" xref="S4.p2.4.m4.1.1.cmml"><mi id="S4.p2.4.m4.1.1.2" xref="S4.p2.4.m4.1.1.2.cmml">γ</mi><mrow id="S4.p2.4.m4.1.1.3" xref="S4.p2.4.m4.1.1.3.cmml"><mi id="S4.p2.4.m4.1.1.3.2" xref="S4.p2.4.m4.1.1.3.2.cmml">i</mi><mo id="S4.p2.4.m4.1.1.3.1" xref="S4.p2.4.m4.1.1.3.1.cmml">⁢</mo><mi id="S4.p2.4.m4.1.1.3.3" xref="S4.p2.4.m4.1.1.3.3.cmml">j</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="S4.p2.4.m4.1b"><apply id="S4.p2.4.m4.1.1.cmml" xref="S4.p2.4.m4.1.1"><csymbol cd="ambiguous" id="S4.p2.4.m4.1.1.1.cmml" xref="S4.p2.4.m4.1.1">subscript</csymbol><ci id="S4.p2.4.m4.1.1.2.cmml" xref="S4.p2.4.m4.1.1.2">𝛾</ci><apply id="S4.p2.4.m4.1.1.3.cmml" xref="S4.p2.4.m4.1.1.3"><times id="S4.p2.4.m4.1.1.3.1.cmml" xref="S4.p2.4.m4.1.1.3.1"></times><ci id="S4.p2.4.m4.1.1.3.2.cmml" xref="S4.p2.4.m4.1.1.3.2">𝑖</ci><ci id="S4.p2.4.m4.1.1.3.3.cmml" xref="S4.p2.4.m4.1.1.3.3">𝑗</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p2.4.m4.1c">\gamma_{ij}</annotation><annotation encoding="application/x-llamapun" id="S4.p2.4.m4.1d">italic_γ start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT</annotation></semantics></math> are the components of a space metric. This form is known as <math alttext="3\!+\!1" class="ltx_Math" display="inline" id="S4.p2.5.m5.1"><semantics id="S4.p2.5.m5.1a"><mrow id="S4.p2.5.m5.1.1" xref="S4.p2.5.m5.1.1.cmml"><mn id="S4.p2.5.m5.1.1.2" xref="S4.p2.5.m5.1.1.2.cmml">3</mn><mo id="S4.p2.5.m5.1.1.1" lspace="0.052em" rspace="0.052em" xref="S4.p2.5.m5.1.1.1.cmml">+</mo><mn id="S4.p2.5.m5.1.1.3" xref="S4.p2.5.m5.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.p2.5.m5.1b"><apply id="S4.p2.5.m5.1.1.cmml" xref="S4.p2.5.m5.1.1"><plus id="S4.p2.5.m5.1.1.1.cmml" xref="S4.p2.5.m5.1.1.1"></plus><cn id="S4.p2.5.m5.1.1.2.cmml" type="integer" xref="S4.p2.5.m5.1.1.2">3</cn><cn id="S4.p2.5.m5.1.1.3.cmml" type="integer" xref="S4.p2.5.m5.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p2.5.m5.1c">3\!+\!1</annotation><annotation encoding="application/x-llamapun" id="S4.p2.5.m5.1d">3 + 1</annotation></semantics></math> decomposition (three space- and one time-coordinates) or ADM-decomposition and</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="S3.EGx56"> <tbody id="S4.E58"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_left_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle G_{\mu\nu}(x)" class="ltx_Math" display="inline" id="S4.E58.m1.1"><semantics id="S4.E58.m1.1a"><mrow id="S4.E58.m1.1.2" xref="S4.E58.m1.1.2.cmml"><msub id="S4.E58.m1.1.2.2" xref="S4.E58.m1.1.2.2.cmml"><mi id="S4.E58.m1.1.2.2.2" xref="S4.E58.m1.1.2.2.2.cmml">G</mi><mrow id="S4.E58.m1.1.2.2.3" xref="S4.E58.m1.1.2.2.3.cmml"><mi id="S4.E58.m1.1.2.2.3.2" xref="S4.E58.m1.1.2.2.3.2.cmml">μ</mi><mo id="S4.E58.m1.1.2.2.3.1" xref="S4.E58.m1.1.2.2.3.1.cmml">⁢</mo><mi id="S4.E58.m1.1.2.2.3.3" xref="S4.E58.m1.1.2.2.3.3.cmml">ν</mi></mrow></msub><mo id="S4.E58.m1.1.2.1" xref="S4.E58.m1.1.2.1.cmml">⁢</mo><mrow id="S4.E58.m1.1.2.3.2" xref="S4.E58.m1.1.2.cmml"><mo id="S4.E58.m1.1.2.3.2.1" stretchy="false" xref="S4.E58.m1.1.2.cmml">(</mo><mi id="S4.E58.m1.1.1" xref="S4.E58.m1.1.1.cmml">x</mi><mo id="S4.E58.m1.1.2.3.2.2" stretchy="false" xref="S4.E58.m1.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.E58.m1.1b"><apply id="S4.E58.m1.1.2.cmml" xref="S4.E58.m1.1.2"><times id="S4.E58.m1.1.2.1.cmml" xref="S4.E58.m1.1.2.1"></times><apply id="S4.E58.m1.1.2.2.cmml" xref="S4.E58.m1.1.2.2"><csymbol cd="ambiguous" id="S4.E58.m1.1.2.2.1.cmml" xref="S4.E58.m1.1.2.2">subscript</csymbol><ci id="S4.E58.m1.1.2.2.2.cmml" xref="S4.E58.m1.1.2.2.2">𝐺</ci><apply id="S4.E58.m1.1.2.2.3.cmml" xref="S4.E58.m1.1.2.2.3"><times id="S4.E58.m1.1.2.2.3.1.cmml" xref="S4.E58.m1.1.2.2.3.1"></times><ci id="S4.E58.m1.1.2.2.3.2.cmml" xref="S4.E58.m1.1.2.2.3.2">𝜇</ci><ci id="S4.E58.m1.1.2.2.3.3.cmml" xref="S4.E58.m1.1.2.2.3.3">𝜈</ci></apply></apply><ci id="S4.E58.m1.1.1.cmml" xref="S4.E58.m1.1.1">𝑥</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.E58.m1.1c">\displaystyle G_{\mu\nu}(x)</annotation><annotation encoding="application/x-llamapun" id="S4.E58.m1.1d">italic_G start_POSTSUBSCRIPT italic_μ italic_ν end_POSTSUBSCRIPT ( italic_x )</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle=\left(\begin{array}[]{cc}G_{00}&G_{0j}\\ G_{i0}&G_{ij}\end{array}\right)" class="ltx_Math" display="inline" id="S4.E58.m2.1"><semantics id="S4.E58.m2.1a"><mrow id="S4.E58.m2.1.2" xref="S4.E58.m2.1.2.cmml"><mi id="S4.E58.m2.1.2.2" xref="S4.E58.m2.1.2.2.cmml"></mi><mo id="S4.E58.m2.1.2.1" xref="S4.E58.m2.1.2.1.cmml">=</mo><mrow id="S4.E58.m2.1.2.3.2" xref="S4.E58.m2.1.1.cmml"><mo id="S4.E58.m2.1.2.3.2.1" xref="S4.E58.m2.1.1.cmml">(</mo><mtable columnspacing="5pt" id="S4.E58.m2.1.1" rowspacing="0pt" xref="S4.E58.m2.1.1.cmml"><mtr id="S4.E58.m2.1.1a" xref="S4.E58.m2.1.1.cmml"><mtd id="S4.E58.m2.1.1b" xref="S4.E58.m2.1.1.cmml"><msub id="S4.E56.1.1" xref="S4.E56.1.1.cmml"><mi 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id="S4.E60.2.1.3.2.cmml" xref="S4.E60.2.1.3.2">𝑖</ci><ci id="S4.E60.2.1.3.3.cmml" xref="S4.E60.2.1.3.3">𝑗</ci></apply></apply></matrixrow></matrix></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.E61.m1.2c">\displaystyle=\left(\begin{array}[]{cc}-\alpha^{2}+\gamma_{ij}\beta^{i}\beta^{% j}&\gamma_{ij}\beta^{i}\\ \gamma_{ij}\beta^{j}&\gamma_{ij}\end{array}\right),</annotation><annotation encoding="application/x-llamapun" id="S4.E61.m1.2d">= ( start_ARRAY start_ROW start_CELL - italic_α start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + italic_γ start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT italic_β start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT italic_β start_POSTSUPERSCRIPT italic_j end_POSTSUPERSCRIPT end_CELL start_CELL italic_γ start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT italic_β start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT end_CELL end_ROW start_ROW start_CELL italic_γ start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT italic_β start_POSTSUPERSCRIPT italic_j end_POSTSUPERSCRIPT end_CELL start_CELL italic_γ start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT end_CELL end_ROW end_ARRAY ) ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_left_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(61)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S4.p2.8">where <math alttext="G_{\mu\nu}(x)" class="ltx_Math" display="inline" id="S4.p2.6.m1.1"><semantics id="S4.p2.6.m1.1a"><mrow id="S4.p2.6.m1.1.2" xref="S4.p2.6.m1.1.2.cmml"><msub id="S4.p2.6.m1.1.2.2" xref="S4.p2.6.m1.1.2.2.cmml"><mi id="S4.p2.6.m1.1.2.2.2" xref="S4.p2.6.m1.1.2.2.2.cmml">G</mi><mrow id="S4.p2.6.m1.1.2.2.3" xref="S4.p2.6.m1.1.2.2.3.cmml"><mi id="S4.p2.6.m1.1.2.2.3.2" xref="S4.p2.6.m1.1.2.2.3.2.cmml">μ</mi><mo id="S4.p2.6.m1.1.2.2.3.1" xref="S4.p2.6.m1.1.2.2.3.1.cmml">⁢</mo><mi id="S4.p2.6.m1.1.2.2.3.3" xref="S4.p2.6.m1.1.2.2.3.3.cmml">ν</mi></mrow></msub><mo id="S4.p2.6.m1.1.2.1" xref="S4.p2.6.m1.1.2.1.cmml">⁢</mo><mrow id="S4.p2.6.m1.1.2.3.2" xref="S4.p2.6.m1.1.2.cmml"><mo id="S4.p2.6.m1.1.2.3.2.1" stretchy="false" xref="S4.p2.6.m1.1.2.cmml">(</mo><mi id="S4.p2.6.m1.1.1" xref="S4.p2.6.m1.1.1.cmml">x</mi><mo id="S4.p2.6.m1.1.2.3.2.2" stretchy="false" xref="S4.p2.6.m1.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.p2.6.m1.1b"><apply id="S4.p2.6.m1.1.2.cmml" xref="S4.p2.6.m1.1.2"><times id="S4.p2.6.m1.1.2.1.cmml" xref="S4.p2.6.m1.1.2.1"></times><apply id="S4.p2.6.m1.1.2.2.cmml" xref="S4.p2.6.m1.1.2.2"><csymbol cd="ambiguous" id="S4.p2.6.m1.1.2.2.1.cmml" xref="S4.p2.6.m1.1.2.2">subscript</csymbol><ci id="S4.p2.6.m1.1.2.2.2.cmml" xref="S4.p2.6.m1.1.2.2.2">𝐺</ci><apply id="S4.p2.6.m1.1.2.2.3.cmml" xref="S4.p2.6.m1.1.2.2.3"><times id="S4.p2.6.m1.1.2.2.3.1.cmml" xref="S4.p2.6.m1.1.2.2.3.1"></times><ci id="S4.p2.6.m1.1.2.2.3.2.cmml" xref="S4.p2.6.m1.1.2.2.3.2">𝜇</ci><ci id="S4.p2.6.m1.1.2.2.3.3.cmml" xref="S4.p2.6.m1.1.2.2.3.3">𝜈</ci></apply></apply><ci id="S4.p2.6.m1.1.1.cmml" xref="S4.p2.6.m1.1.1">𝑥</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p2.6.m1.1c">G_{\mu\nu}(x)</annotation><annotation encoding="application/x-llamapun" id="S4.p2.6.m1.1d">italic_G start_POSTSUBSCRIPT italic_μ italic_ν end_POSTSUBSCRIPT ( italic_x )</annotation></semantics></math> are the components of the metric <math alttext="G" class="ltx_Math" display="inline" id="S4.p2.7.m2.1"><semantics id="S4.p2.7.m2.1a"><mi id="S4.p2.7.m2.1.1" xref="S4.p2.7.m2.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S4.p2.7.m2.1b"><ci id="S4.p2.7.m2.1.1.cmml" xref="S4.p2.7.m2.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.p2.7.m2.1c">G</annotation><annotation encoding="application/x-llamapun" id="S4.p2.7.m2.1d">italic_G</annotation></semantics></math>. The associated Lagrangian for the affine parameter <math alttext="\tau" class="ltx_Math" display="inline" id="S4.p2.8.m3.1"><semantics id="S4.p2.8.m3.1a"><mi id="S4.p2.8.m3.1.1" xref="S4.p2.8.m3.1.1.cmml">τ</mi><annotation-xml encoding="MathML-Content" id="S4.p2.8.m3.1b"><ci id="S4.p2.8.m3.1.1.cmml" xref="S4.p2.8.m3.1.1">𝜏</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.p2.8.m3.1c">\tau</annotation><annotation encoding="application/x-llamapun" id="S4.p2.8.m3.1d">italic_τ</annotation></semantics></math> is</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="S3.EGx57"> <tbody id="S4.E62"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_left_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle L\left(x,\frac{dx}{d\tau}\right)=\frac{1}{2}G_{\mu\nu}(x)\frac{% dx^{\mu}}{d\tau}\frac{dx^{\nu}}{d\tau}." class="ltx_Math" display="inline" id="S4.E62.m1.4"><semantics 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end_ARG italic_G start_POSTSUBSCRIPT italic_μ italic_ν end_POSTSUBSCRIPT ( italic_x ) divide start_ARG italic_d italic_x start_POSTSUPERSCRIPT italic_μ end_POSTSUPERSCRIPT end_ARG start_ARG italic_d italic_τ end_ARG divide start_ARG italic_d italic_x start_POSTSUPERSCRIPT italic_ν end_POSTSUPERSCRIPT end_ARG start_ARG italic_d italic_τ end_ARG .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_left_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(62)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S4.p2.26">Since the canonical momenta are</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="S3.EGx58"> <tbody id="S4.E63"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_left_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle p_{\mu}:=\frac{\partial 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id="S4.E63.m1.3c">\displaystyle p_{\mu}:=\frac{\partial L}{\partial(dx^{\mu}/d\tau)}=G_{\mu\nu}(% x)\frac{dx^{\nu}}{d\tau},</annotation><annotation encoding="application/x-llamapun" id="S4.E63.m1.3d">italic_p start_POSTSUBSCRIPT italic_μ end_POSTSUBSCRIPT := divide start_ARG ∂ italic_L end_ARG start_ARG ∂ ( italic_d italic_x start_POSTSUPERSCRIPT italic_μ end_POSTSUPERSCRIPT / italic_d italic_τ ) end_ARG = italic_G start_POSTSUBSCRIPT italic_μ italic_ν end_POSTSUBSCRIPT ( italic_x ) divide start_ARG italic_d italic_x start_POSTSUPERSCRIPT italic_ν end_POSTSUPERSCRIPT end_ARG start_ARG italic_d italic_τ end_ARG ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_left_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(63)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S4.p2.27">it follows that</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" 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xref="S4.E64.m1.3.3.1.1.6.5.3.3">𝜏</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.E64.m1.3c">\displaystyle\omega^{2}:=G^{\mu\nu}(x)p_{\mu}p_{\nu}=G_{\mu\nu}(x)\frac{dx^{% \mu}}{d\tau}\frac{dx^{\nu}}{d\tau}.</annotation><annotation encoding="application/x-llamapun" id="S4.E64.m1.3d">italic_ω start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT := italic_G start_POSTSUPERSCRIPT italic_μ italic_ν end_POSTSUPERSCRIPT ( italic_x ) italic_p start_POSTSUBSCRIPT italic_μ end_POSTSUBSCRIPT italic_p start_POSTSUBSCRIPT italic_ν end_POSTSUBSCRIPT = italic_G start_POSTSUBSCRIPT italic_μ italic_ν end_POSTSUBSCRIPT ( italic_x ) divide start_ARG italic_d italic_x start_POSTSUPERSCRIPT italic_μ end_POSTSUPERSCRIPT end_ARG start_ARG italic_d italic_τ end_ARG divide start_ARG italic_d italic_x start_POSTSUPERSCRIPT italic_ν end_POSTSUPERSCRIPT end_ARG start_ARG italic_d italic_τ end_ARG .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_left_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(64)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S4.p2.9">Here the components of the inverse metric <math alttext="G^{-1}" class="ltx_Math" display="inline" id="S4.p2.9.m1.1"><semantics id="S4.p2.9.m1.1a"><msup id="S4.p2.9.m1.1.1" xref="S4.p2.9.m1.1.1.cmml"><mi id="S4.p2.9.m1.1.1.2" xref="S4.p2.9.m1.1.1.2.cmml">G</mi><mrow id="S4.p2.9.m1.1.1.3" xref="S4.p2.9.m1.1.1.3.cmml"><mo id="S4.p2.9.m1.1.1.3a" xref="S4.p2.9.m1.1.1.3.cmml">−</mo><mn id="S4.p2.9.m1.1.1.3.2" xref="S4.p2.9.m1.1.1.3.2.cmml">1</mn></mrow></msup><annotation-xml encoding="MathML-Content" id="S4.p2.9.m1.1b"><apply id="S4.p2.9.m1.1.1.cmml" xref="S4.p2.9.m1.1.1"><csymbol cd="ambiguous" id="S4.p2.9.m1.1.1.1.cmml" xref="S4.p2.9.m1.1.1">superscript</csymbol><ci id="S4.p2.9.m1.1.1.2.cmml" xref="S4.p2.9.m1.1.1.2">𝐺</ci><apply id="S4.p2.9.m1.1.1.3.cmml" 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end_CELL end_ROW start_ROW start_CELL divide start_ARG italic_β start_POSTSUPERSCRIPT italic_j end_POSTSUPERSCRIPT end_ARG start_ARG italic_α start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG end_CELL start_CELL italic_γ start_POSTSUPERSCRIPT italic_i italic_j end_POSTSUPERSCRIPT - divide start_ARG italic_β start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT italic_β start_POSTSUPERSCRIPT italic_j end_POSTSUPERSCRIPT end_ARG start_ARG italic_α start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG end_CELL end_ROW end_ARRAY ) ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_left_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(67)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S4.p2.15">where <math alttext="\gamma^{ij}" class="ltx_Math" display="inline" id="S4.p2.10.m1.1"><semantics id="S4.p2.10.m1.1a"><msup id="S4.p2.10.m1.1.1" xref="S4.p2.10.m1.1.1.cmml"><mi id="S4.p2.10.m1.1.1.2" xref="S4.p2.10.m1.1.1.2.cmml">γ</mi><mrow id="S4.p2.10.m1.1.1.3" xref="S4.p2.10.m1.1.1.3.cmml"><mi id="S4.p2.10.m1.1.1.3.2" xref="S4.p2.10.m1.1.1.3.2.cmml">i</mi><mo id="S4.p2.10.m1.1.1.3.1" xref="S4.p2.10.m1.1.1.3.1.cmml">⁢</mo><mi id="S4.p2.10.m1.1.1.3.3" xref="S4.p2.10.m1.1.1.3.3.cmml">j</mi></mrow></msup><annotation-xml encoding="MathML-Content" id="S4.p2.10.m1.1b"><apply id="S4.p2.10.m1.1.1.cmml" xref="S4.p2.10.m1.1.1"><csymbol cd="ambiguous" id="S4.p2.10.m1.1.1.1.cmml" xref="S4.p2.10.m1.1.1">superscript</csymbol><ci id="S4.p2.10.m1.1.1.2.cmml" xref="S4.p2.10.m1.1.1.2">𝛾</ci><apply id="S4.p2.10.m1.1.1.3.cmml" xref="S4.p2.10.m1.1.1.3"><times id="S4.p2.10.m1.1.1.3.1.cmml" xref="S4.p2.10.m1.1.1.3.1"></times><ci id="S4.p2.10.m1.1.1.3.2.cmml" xref="S4.p2.10.m1.1.1.3.2">𝑖</ci><ci id="S4.p2.10.m1.1.1.3.3.cmml" xref="S4.p2.10.m1.1.1.3.3">𝑗</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p2.10.m1.1c">\gamma^{ij}</annotation><annotation encoding="application/x-llamapun" id="S4.p2.10.m1.1d">italic_γ start_POSTSUPERSCRIPT italic_i italic_j end_POSTSUPERSCRIPT</annotation></semantics></math> are the inverse matrix elements for <math alttext="\gamma_{ij}" class="ltx_Math" display="inline" id="S4.p2.11.m2.1"><semantics id="S4.p2.11.m2.1a"><msub id="S4.p2.11.m2.1.1" xref="S4.p2.11.m2.1.1.cmml"><mi id="S4.p2.11.m2.1.1.2" xref="S4.p2.11.m2.1.1.2.cmml">γ</mi><mrow id="S4.p2.11.m2.1.1.3" xref="S4.p2.11.m2.1.1.3.cmml"><mi id="S4.p2.11.m2.1.1.3.2" xref="S4.p2.11.m2.1.1.3.2.cmml">i</mi><mo id="S4.p2.11.m2.1.1.3.1" xref="S4.p2.11.m2.1.1.3.1.cmml">⁢</mo><mi id="S4.p2.11.m2.1.1.3.3" xref="S4.p2.11.m2.1.1.3.3.cmml">j</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="S4.p2.11.m2.1b"><apply id="S4.p2.11.m2.1.1.cmml" xref="S4.p2.11.m2.1.1"><csymbol cd="ambiguous" id="S4.p2.11.m2.1.1.1.cmml" xref="S4.p2.11.m2.1.1">subscript</csymbol><ci id="S4.p2.11.m2.1.1.2.cmml" xref="S4.p2.11.m2.1.1.2">𝛾</ci><apply id="S4.p2.11.m2.1.1.3.cmml" xref="S4.p2.11.m2.1.1.3"><times id="S4.p2.11.m2.1.1.3.1.cmml" xref="S4.p2.11.m2.1.1.3.1"></times><ci id="S4.p2.11.m2.1.1.3.2.cmml" xref="S4.p2.11.m2.1.1.3.2">𝑖</ci><ci id="S4.p2.11.m2.1.1.3.3.cmml" xref="S4.p2.11.m2.1.1.3.3">𝑗</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p2.11.m2.1c">\gamma_{ij}</annotation><annotation encoding="application/x-llamapun" id="S4.p2.11.m2.1d">italic_γ start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT</annotation></semantics></math>. Since <math alttext="G^{\mu\nu}(x)" class="ltx_Math" display="inline" id="S4.p2.12.m3.1"><semantics id="S4.p2.12.m3.1a"><mrow id="S4.p2.12.m3.1.2" xref="S4.p2.12.m3.1.2.cmml"><msup id="S4.p2.12.m3.1.2.2" xref="S4.p2.12.m3.1.2.2.cmml"><mi id="S4.p2.12.m3.1.2.2.2" xref="S4.p2.12.m3.1.2.2.2.cmml">G</mi><mrow id="S4.p2.12.m3.1.2.2.3" xref="S4.p2.12.m3.1.2.2.3.cmml"><mi id="S4.p2.12.m3.1.2.2.3.2" xref="S4.p2.12.m3.1.2.2.3.2.cmml">μ</mi><mo id="S4.p2.12.m3.1.2.2.3.1" xref="S4.p2.12.m3.1.2.2.3.1.cmml">⁢</mo><mi id="S4.p2.12.m3.1.2.2.3.3" xref="S4.p2.12.m3.1.2.2.3.3.cmml">ν</mi></mrow></msup><mo id="S4.p2.12.m3.1.2.1" xref="S4.p2.12.m3.1.2.1.cmml">⁢</mo><mrow id="S4.p2.12.m3.1.2.3.2" xref="S4.p2.12.m3.1.2.cmml"><mo id="S4.p2.12.m3.1.2.3.2.1" stretchy="false" xref="S4.p2.12.m3.1.2.cmml">(</mo><mi id="S4.p2.12.m3.1.1" xref="S4.p2.12.m3.1.1.cmml">x</mi><mo id="S4.p2.12.m3.1.2.3.2.2" stretchy="false" xref="S4.p2.12.m3.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.p2.12.m3.1b"><apply id="S4.p2.12.m3.1.2.cmml" xref="S4.p2.12.m3.1.2"><times id="S4.p2.12.m3.1.2.1.cmml" xref="S4.p2.12.m3.1.2.1"></times><apply id="S4.p2.12.m3.1.2.2.cmml" xref="S4.p2.12.m3.1.2.2"><csymbol cd="ambiguous" id="S4.p2.12.m3.1.2.2.1.cmml" xref="S4.p2.12.m3.1.2.2">superscript</csymbol><ci id="S4.p2.12.m3.1.2.2.2.cmml" xref="S4.p2.12.m3.1.2.2.2">𝐺</ci><apply id="S4.p2.12.m3.1.2.2.3.cmml" xref="S4.p2.12.m3.1.2.2.3"><times id="S4.p2.12.m3.1.2.2.3.1.cmml" xref="S4.p2.12.m3.1.2.2.3.1"></times><ci id="S4.p2.12.m3.1.2.2.3.2.cmml" xref="S4.p2.12.m3.1.2.2.3.2">𝜇</ci><ci id="S4.p2.12.m3.1.2.2.3.3.cmml" xref="S4.p2.12.m3.1.2.2.3.3">𝜈</ci></apply></apply><ci id="S4.p2.12.m3.1.1.cmml" xref="S4.p2.12.m3.1.1">𝑥</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p2.12.m3.1c">G^{\mu\nu}(x)</annotation><annotation encoding="application/x-llamapun" id="S4.p2.12.m3.1d">italic_G start_POSTSUPERSCRIPT italic_μ italic_ν end_POSTSUPERSCRIPT ( italic_x )</annotation></semantics></math> are independent of the affine parameter <math alttext="\tau" class="ltx_Math" display="inline" id="S4.p2.13.m4.1"><semantics id="S4.p2.13.m4.1a"><mi id="S4.p2.13.m4.1.1" xref="S4.p2.13.m4.1.1.cmml">τ</mi><annotation-xml encoding="MathML-Content" id="S4.p2.13.m4.1b"><ci id="S4.p2.13.m4.1.1.cmml" xref="S4.p2.13.m4.1.1">𝜏</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.p2.13.m4.1c">\tau</annotation><annotation encoding="application/x-llamapun" id="S4.p2.13.m4.1d">italic_τ</annotation></semantics></math>, the value <math alttext="\omega^{2}" class="ltx_Math" display="inline" id="S4.p2.14.m5.1"><semantics id="S4.p2.14.m5.1a"><msup id="S4.p2.14.m5.1.1" xref="S4.p2.14.m5.1.1.cmml"><mi id="S4.p2.14.m5.1.1.2" xref="S4.p2.14.m5.1.1.2.cmml">ω</mi><mn id="S4.p2.14.m5.1.1.3" xref="S4.p2.14.m5.1.1.3.cmml">2</mn></msup><annotation-xml encoding="MathML-Content" id="S4.p2.14.m5.1b"><apply id="S4.p2.14.m5.1.1.cmml" xref="S4.p2.14.m5.1.1"><csymbol cd="ambiguous" id="S4.p2.14.m5.1.1.1.cmml" xref="S4.p2.14.m5.1.1">superscript</csymbol><ci id="S4.p2.14.m5.1.1.2.cmml" xref="S4.p2.14.m5.1.1.2">𝜔</ci><cn id="S4.p2.14.m5.1.1.3.cmml" type="integer" xref="S4.p2.14.m5.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p2.14.m5.1c">\omega^{2}</annotation><annotation encoding="application/x-llamapun" id="S4.p2.14.m5.1d">italic_ω start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT</annotation></semantics></math> in (<a class="ltx_ref" href="https://arxiv.org/html/2406.11224v2#S4.E64" title="In 4 The motions of a light-like particle in a pseudo Riemann space ‣ A Hamiltonian approach to the gradient-flow equations in information geometry"><span class="ltx_text ltx_ref_tag">64</span></a>) is a constant. For <math alttext="\omega^{2}<0,\omega^{2}=0,\omega^{2}>0" class="ltx_Math" display="inline" id="S4.p2.15.m6.3"><semantics id="S4.p2.15.m6.3a"><mrow id="S4.p2.15.m6.3.3.2" xref="S4.p2.15.m6.3.3.3.cmml"><mrow id="S4.p2.15.m6.2.2.1.1" xref="S4.p2.15.m6.2.2.1.1.cmml"><msup id="S4.p2.15.m6.2.2.1.1.2" xref="S4.p2.15.m6.2.2.1.1.2.cmml"><mi id="S4.p2.15.m6.2.2.1.1.2.2" xref="S4.p2.15.m6.2.2.1.1.2.2.cmml">ω</mi><mn id="S4.p2.15.m6.2.2.1.1.2.3" xref="S4.p2.15.m6.2.2.1.1.2.3.cmml">2</mn></msup><mo id="S4.p2.15.m6.2.2.1.1.1" xref="S4.p2.15.m6.2.2.1.1.1.cmml"><</mo><mn id="S4.p2.15.m6.1.1" xref="S4.p2.15.m6.1.1.cmml">0</mn></mrow><mo id="S4.p2.15.m6.3.3.2.3" xref="S4.p2.15.m6.3.3.3a.cmml">,</mo><mrow id="S4.p2.15.m6.3.3.2.2.2" xref="S4.p2.15.m6.3.3.2.2.3.cmml"><mrow id="S4.p2.15.m6.3.3.2.2.1.1" xref="S4.p2.15.m6.3.3.2.2.1.1.cmml"><msup id="S4.p2.15.m6.3.3.2.2.1.1.2" xref="S4.p2.15.m6.3.3.2.2.1.1.2.cmml"><mi id="S4.p2.15.m6.3.3.2.2.1.1.2.2" xref="S4.p2.15.m6.3.3.2.2.1.1.2.2.cmml">ω</mi><mn id="S4.p2.15.m6.3.3.2.2.1.1.2.3" xref="S4.p2.15.m6.3.3.2.2.1.1.2.3.cmml">2</mn></msup><mo id="S4.p2.15.m6.3.3.2.2.1.1.1" xref="S4.p2.15.m6.3.3.2.2.1.1.1.cmml">=</mo><mn id="S4.p2.15.m6.3.3.2.2.1.1.3" xref="S4.p2.15.m6.3.3.2.2.1.1.3.cmml">0</mn></mrow><mo id="S4.p2.15.m6.3.3.2.2.2.3" xref="S4.p2.15.m6.3.3.2.2.3a.cmml">,</mo><mrow id="S4.p2.15.m6.3.3.2.2.2.2" xref="S4.p2.15.m6.3.3.2.2.2.2.cmml"><msup id="S4.p2.15.m6.3.3.2.2.2.2.2" xref="S4.p2.15.m6.3.3.2.2.2.2.2.cmml"><mi id="S4.p2.15.m6.3.3.2.2.2.2.2.2" xref="S4.p2.15.m6.3.3.2.2.2.2.2.2.cmml">ω</mi><mn id="S4.p2.15.m6.3.3.2.2.2.2.2.3" xref="S4.p2.15.m6.3.3.2.2.2.2.2.3.cmml">2</mn></msup><mo id="S4.p2.15.m6.3.3.2.2.2.2.1" xref="S4.p2.15.m6.3.3.2.2.2.2.1.cmml">></mo><mn id="S4.p2.15.m6.3.3.2.2.2.2.3" xref="S4.p2.15.m6.3.3.2.2.2.2.3.cmml">0</mn></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.p2.15.m6.3b"><apply id="S4.p2.15.m6.3.3.3.cmml" xref="S4.p2.15.m6.3.3.2"><csymbol cd="ambiguous" id="S4.p2.15.m6.3.3.3a.cmml" xref="S4.p2.15.m6.3.3.2.3">formulae-sequence</csymbol><apply id="S4.p2.15.m6.2.2.1.1.cmml" xref="S4.p2.15.m6.2.2.1.1"><lt id="S4.p2.15.m6.2.2.1.1.1.cmml" xref="S4.p2.15.m6.2.2.1.1.1"></lt><apply id="S4.p2.15.m6.2.2.1.1.2.cmml" xref="S4.p2.15.m6.2.2.1.1.2"><csymbol cd="ambiguous" id="S4.p2.15.m6.2.2.1.1.2.1.cmml" xref="S4.p2.15.m6.2.2.1.1.2">superscript</csymbol><ci id="S4.p2.15.m6.2.2.1.1.2.2.cmml" xref="S4.p2.15.m6.2.2.1.1.2.2">𝜔</ci><cn id="S4.p2.15.m6.2.2.1.1.2.3.cmml" type="integer" xref="S4.p2.15.m6.2.2.1.1.2.3">2</cn></apply><cn id="S4.p2.15.m6.1.1.cmml" type="integer" xref="S4.p2.15.m6.1.1">0</cn></apply><apply id="S4.p2.15.m6.3.3.2.2.3.cmml" xref="S4.p2.15.m6.3.3.2.2.2"><csymbol cd="ambiguous" id="S4.p2.15.m6.3.3.2.2.3a.cmml" xref="S4.p2.15.m6.3.3.2.2.2.3">formulae-sequence</csymbol><apply id="S4.p2.15.m6.3.3.2.2.1.1.cmml" xref="S4.p2.15.m6.3.3.2.2.1.1"><eq id="S4.p2.15.m6.3.3.2.2.1.1.1.cmml" xref="S4.p2.15.m6.3.3.2.2.1.1.1"></eq><apply id="S4.p2.15.m6.3.3.2.2.1.1.2.cmml" xref="S4.p2.15.m6.3.3.2.2.1.1.2"><csymbol cd="ambiguous" id="S4.p2.15.m6.3.3.2.2.1.1.2.1.cmml" xref="S4.p2.15.m6.3.3.2.2.1.1.2">superscript</csymbol><ci id="S4.p2.15.m6.3.3.2.2.1.1.2.2.cmml" xref="S4.p2.15.m6.3.3.2.2.1.1.2.2">𝜔</ci><cn id="S4.p2.15.m6.3.3.2.2.1.1.2.3.cmml" type="integer" xref="S4.p2.15.m6.3.3.2.2.1.1.2.3">2</cn></apply><cn id="S4.p2.15.m6.3.3.2.2.1.1.3.cmml" type="integer" xref="S4.p2.15.m6.3.3.2.2.1.1.3">0</cn></apply><apply id="S4.p2.15.m6.3.3.2.2.2.2.cmml" xref="S4.p2.15.m6.3.3.2.2.2.2"><gt id="S4.p2.15.m6.3.3.2.2.2.2.1.cmml" xref="S4.p2.15.m6.3.3.2.2.2.2.1"></gt><apply id="S4.p2.15.m6.3.3.2.2.2.2.2.cmml" xref="S4.p2.15.m6.3.3.2.2.2.2.2"><csymbol cd="ambiguous" id="S4.p2.15.m6.3.3.2.2.2.2.2.1.cmml" xref="S4.p2.15.m6.3.3.2.2.2.2.2">superscript</csymbol><ci id="S4.p2.15.m6.3.3.2.2.2.2.2.2.cmml" xref="S4.p2.15.m6.3.3.2.2.2.2.2.2">𝜔</ci><cn id="S4.p2.15.m6.3.3.2.2.2.2.2.3.cmml" type="integer" xref="S4.p2.15.m6.3.3.2.2.2.2.2.3">2</cn></apply><cn id="S4.p2.15.m6.3.3.2.2.2.2.3.cmml" type="integer" xref="S4.p2.15.m6.3.3.2.2.2.2.3">0</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p2.15.m6.3c">\omega^{2}<0,\omega^{2}=0,\omega^{2}>0</annotation><annotation encoding="application/x-llamapun" id="S4.p2.15.m6.3d">italic_ω start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT < 0 , italic_ω start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT = 0 , italic_ω start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT > 0</annotation></semantics></math>, the metric is said to be time-like, null (or light-like), or space-like, respectively. Since we would like to consider the motions of a light-like particle, we focus on the null case</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="S3.EGx61"> <tbody id="S4.Ex13"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_left_padleft"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle 0=G^{\mu\nu}(x)p_{\mu}p_{\nu}" class="ltx_Math" display="inline" id="S4.Ex13.m1.1"><semantics id="S4.Ex13.m1.1a"><mrow id="S4.Ex13.m1.1.2" xref="S4.Ex13.m1.1.2.cmml"><mn id="S4.Ex13.m1.1.2.2" xref="S4.Ex13.m1.1.2.2.cmml">0</mn><mo id="S4.Ex13.m1.1.2.1" xref="S4.Ex13.m1.1.2.1.cmml">=</mo><mrow id="S4.Ex13.m1.1.2.3" xref="S4.Ex13.m1.1.2.3.cmml"><msup id="S4.Ex13.m1.1.2.3.2" xref="S4.Ex13.m1.1.2.3.2.cmml"><mi id="S4.Ex13.m1.1.2.3.2.2" xref="S4.Ex13.m1.1.2.3.2.2.cmml">G</mi><mrow id="S4.Ex13.m1.1.2.3.2.3" xref="S4.Ex13.m1.1.2.3.2.3.cmml"><mi id="S4.Ex13.m1.1.2.3.2.3.2" xref="S4.Ex13.m1.1.2.3.2.3.2.cmml">μ</mi><mo id="S4.Ex13.m1.1.2.3.2.3.1" xref="S4.Ex13.m1.1.2.3.2.3.1.cmml">⁢</mo><mi id="S4.Ex13.m1.1.2.3.2.3.3" xref="S4.Ex13.m1.1.2.3.2.3.3.cmml">ν</mi></mrow></msup><mo id="S4.Ex13.m1.1.2.3.1" xref="S4.Ex13.m1.1.2.3.1.cmml">⁢</mo><mrow id="S4.Ex13.m1.1.2.3.3.2" xref="S4.Ex13.m1.1.2.3.cmml"><mo id="S4.Ex13.m1.1.2.3.3.2.1" stretchy="false" xref="S4.Ex13.m1.1.2.3.cmml">(</mo><mi id="S4.Ex13.m1.1.1" xref="S4.Ex13.m1.1.1.cmml">x</mi><mo id="S4.Ex13.m1.1.2.3.3.2.2" stretchy="false" xref="S4.Ex13.m1.1.2.3.cmml">)</mo></mrow><mo id="S4.Ex13.m1.1.2.3.1a" xref="S4.Ex13.m1.1.2.3.1.cmml">⁢</mo><msub id="S4.Ex13.m1.1.2.3.4" xref="S4.Ex13.m1.1.2.3.4.cmml"><mi id="S4.Ex13.m1.1.2.3.4.2" xref="S4.Ex13.m1.1.2.3.4.2.cmml">p</mi><mi id="S4.Ex13.m1.1.2.3.4.3" xref="S4.Ex13.m1.1.2.3.4.3.cmml">μ</mi></msub><mo id="S4.Ex13.m1.1.2.3.1b" xref="S4.Ex13.m1.1.2.3.1.cmml">⁢</mo><msub id="S4.Ex13.m1.1.2.3.5" xref="S4.Ex13.m1.1.2.3.5.cmml"><mi id="S4.Ex13.m1.1.2.3.5.2" xref="S4.Ex13.m1.1.2.3.5.2.cmml">p</mi><mi id="S4.Ex13.m1.1.2.3.5.3" xref="S4.Ex13.m1.1.2.3.5.3.cmml">ν</mi></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Ex13.m1.1b"><apply id="S4.Ex13.m1.1.2.cmml" xref="S4.Ex13.m1.1.2"><eq id="S4.Ex13.m1.1.2.1.cmml" xref="S4.Ex13.m1.1.2.1"></eq><cn id="S4.Ex13.m1.1.2.2.cmml" type="integer" xref="S4.Ex13.m1.1.2.2">0</cn><apply id="S4.Ex13.m1.1.2.3.cmml" xref="S4.Ex13.m1.1.2.3"><times id="S4.Ex13.m1.1.2.3.1.cmml" xref="S4.Ex13.m1.1.2.3.1"></times><apply id="S4.Ex13.m1.1.2.3.2.cmml" xref="S4.Ex13.m1.1.2.3.2"><csymbol cd="ambiguous" id="S4.Ex13.m1.1.2.3.2.1.cmml" xref="S4.Ex13.m1.1.2.3.2">superscript</csymbol><ci id="S4.Ex13.m1.1.2.3.2.2.cmml" xref="S4.Ex13.m1.1.2.3.2.2">𝐺</ci><apply id="S4.Ex13.m1.1.2.3.2.3.cmml" xref="S4.Ex13.m1.1.2.3.2.3"><times id="S4.Ex13.m1.1.2.3.2.3.1.cmml" xref="S4.Ex13.m1.1.2.3.2.3.1"></times><ci id="S4.Ex13.m1.1.2.3.2.3.2.cmml" xref="S4.Ex13.m1.1.2.3.2.3.2">𝜇</ci><ci id="S4.Ex13.m1.1.2.3.2.3.3.cmml" xref="S4.Ex13.m1.1.2.3.2.3.3">𝜈</ci></apply></apply><ci id="S4.Ex13.m1.1.1.cmml" xref="S4.Ex13.m1.1.1">𝑥</ci><apply id="S4.Ex13.m1.1.2.3.4.cmml" xref="S4.Ex13.m1.1.2.3.4"><csymbol cd="ambiguous" id="S4.Ex13.m1.1.2.3.4.1.cmml" xref="S4.Ex13.m1.1.2.3.4">subscript</csymbol><ci id="S4.Ex13.m1.1.2.3.4.2.cmml" xref="S4.Ex13.m1.1.2.3.4.2">𝑝</ci><ci id="S4.Ex13.m1.1.2.3.4.3.cmml" xref="S4.Ex13.m1.1.2.3.4.3">𝜇</ci></apply><apply id="S4.Ex13.m1.1.2.3.5.cmml" xref="S4.Ex13.m1.1.2.3.5"><csymbol cd="ambiguous" id="S4.Ex13.m1.1.2.3.5.1.cmml" xref="S4.Ex13.m1.1.2.3.5">subscript</csymbol><ci id="S4.Ex13.m1.1.2.3.5.2.cmml" xref="S4.Ex13.m1.1.2.3.5.2">𝑝</ci><ci id="S4.Ex13.m1.1.2.3.5.3.cmml" xref="S4.Ex13.m1.1.2.3.5.3">𝜈</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Ex13.m1.1c">\displaystyle 0=G^{\mu\nu}(x)p_{\mu}p_{\nu}</annotation><annotation encoding="application/x-llamapun" id="S4.Ex13.m1.1d">0 = italic_G start_POSTSUPERSCRIPT italic_μ italic_ν 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end_POSTSUPERSCRIPT italic_β start_POSTSUPERSCRIPT italic_j end_POSTSUPERSCRIPT ) italic_p start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_p start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT } ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_left_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(68)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S4.p2.17">where <math alttext="1/\alpha^{2}" class="ltx_Math" display="inline" id="S4.p2.16.m1.1"><semantics id="S4.p2.16.m1.1a"><mrow id="S4.p2.16.m1.1.1" xref="S4.p2.16.m1.1.1.cmml"><mn id="S4.p2.16.m1.1.1.2" xref="S4.p2.16.m1.1.1.2.cmml">1</mn><mo id="S4.p2.16.m1.1.1.1" xref="S4.p2.16.m1.1.1.1.cmml">/</mo><msup id="S4.p2.16.m1.1.1.3" xref="S4.p2.16.m1.1.1.3.cmml"><mi id="S4.p2.16.m1.1.1.3.2" xref="S4.p2.16.m1.1.1.3.2.cmml">α</mi><mn id="S4.p2.16.m1.1.1.3.3" 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Solving (<a class="ltx_ref" href="https://arxiv.org/html/2406.11224v2#S4.E68" title="In 4 The motions of a light-like particle in a pseudo Riemann space ‣ A Hamiltonian approach to the gradient-flow equations in information geometry"><span class="ltx_text ltx_ref_tag">68</span></a>) for <math alttext="p_{0}" class="ltx_Math" display="inline" id="S4.p2.17.m2.1"><semantics id="S4.p2.17.m2.1a"><msub id="S4.p2.17.m2.1.1" xref="S4.p2.17.m2.1.1.cmml"><mi id="S4.p2.17.m2.1.1.2" xref="S4.p2.17.m2.1.1.2.cmml">p</mi><mn id="S4.p2.17.m2.1.1.3" xref="S4.p2.17.m2.1.1.3.cmml">0</mn></msub><annotation-xml encoding="MathML-Content" id="S4.p2.17.m2.1b"><apply id="S4.p2.17.m2.1.1.cmml" xref="S4.p2.17.m2.1.1"><csymbol cd="ambiguous" id="S4.p2.17.m2.1.1.1.cmml" xref="S4.p2.17.m2.1.1">subscript</csymbol><ci id="S4.p2.17.m2.1.1.2.cmml" xref="S4.p2.17.m2.1.1.2">𝑝</ci><cn id="S4.p2.17.m2.1.1.3.cmml" type="integer" xref="S4.p2.17.m2.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p2.17.m2.1c">p_{0}</annotation><annotation encoding="application/x-llamapun" id="S4.p2.17.m2.1d">italic_p start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math> leads to</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="S3.EGx62"> <tbody id="S4.E69"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_left_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle p_{0}=\beta^{i}p_{i}\pm\sqrt{\alpha^{2}\gamma^{ij}p_{i}p_{j}}." class="ltx_Math" display="inline" id="S4.E69.m1.1"><semantics id="S4.E69.m1.1a"><mrow id="S4.E69.m1.1.1.1" xref="S4.E69.m1.1.1.1.1.cmml"><mrow id="S4.E69.m1.1.1.1.1" xref="S4.E69.m1.1.1.1.1.cmml"><msub id="S4.E69.m1.1.1.1.1.2" xref="S4.E69.m1.1.1.1.1.2.cmml"><mi id="S4.E69.m1.1.1.1.1.2.2" xref="S4.E69.m1.1.1.1.1.2.2.cmml">p</mi><mn id="S4.E69.m1.1.1.1.1.2.3" xref="S4.E69.m1.1.1.1.1.2.3.cmml">0</mn></msub><mo id="S4.E69.m1.1.1.1.1.1" xref="S4.E69.m1.1.1.1.1.1.cmml">=</mo><mrow id="S4.E69.m1.1.1.1.1.3" xref="S4.E69.m1.1.1.1.1.3.cmml"><mrow id="S4.E69.m1.1.1.1.1.3.2" xref="S4.E69.m1.1.1.1.1.3.2.cmml"><msup id="S4.E69.m1.1.1.1.1.3.2.2" xref="S4.E69.m1.1.1.1.1.3.2.2.cmml"><mi id="S4.E69.m1.1.1.1.1.3.2.2.2" xref="S4.E69.m1.1.1.1.1.3.2.2.2.cmml">β</mi><mi id="S4.E69.m1.1.1.1.1.3.2.2.3" xref="S4.E69.m1.1.1.1.1.3.2.2.3.cmml">i</mi></msup><mo id="S4.E69.m1.1.1.1.1.3.2.1" xref="S4.E69.m1.1.1.1.1.3.2.1.cmml">⁢</mo><msub id="S4.E69.m1.1.1.1.1.3.2.3" xref="S4.E69.m1.1.1.1.1.3.2.3.cmml"><mi id="S4.E69.m1.1.1.1.1.3.2.3.2" xref="S4.E69.m1.1.1.1.1.3.2.3.2.cmml">p</mi><mi id="S4.E69.m1.1.1.1.1.3.2.3.3" xref="S4.E69.m1.1.1.1.1.3.2.3.3.cmml">i</mi></msub></mrow><mo id="S4.E69.m1.1.1.1.1.3.1" xref="S4.E69.m1.1.1.1.1.3.1.cmml">±</mo><msqrt id="S4.E69.m1.1.1.1.1.3.3" xref="S4.E69.m1.1.1.1.1.3.3.cmml"><mrow id="S4.E69.m1.1.1.1.1.3.3.2" xref="S4.E69.m1.1.1.1.1.3.3.2.cmml"><msup id="S4.E69.m1.1.1.1.1.3.3.2.2" xref="S4.E69.m1.1.1.1.1.3.3.2.2.cmml"><mi id="S4.E69.m1.1.1.1.1.3.3.2.2.2" xref="S4.E69.m1.1.1.1.1.3.3.2.2.2.cmml">α</mi><mn id="S4.E69.m1.1.1.1.1.3.3.2.2.3" xref="S4.E69.m1.1.1.1.1.3.3.2.2.3.cmml">2</mn></msup><mo id="S4.E69.m1.1.1.1.1.3.3.2.1" xref="S4.E69.m1.1.1.1.1.3.3.2.1.cmml">⁢</mo><msup id="S4.E69.m1.1.1.1.1.3.3.2.3" xref="S4.E69.m1.1.1.1.1.3.3.2.3.cmml"><mi id="S4.E69.m1.1.1.1.1.3.3.2.3.2" xref="S4.E69.m1.1.1.1.1.3.3.2.3.2.cmml">γ</mi><mrow id="S4.E69.m1.1.1.1.1.3.3.2.3.3" xref="S4.E69.m1.1.1.1.1.3.3.2.3.3.cmml"><mi id="S4.E69.m1.1.1.1.1.3.3.2.3.3.2" xref="S4.E69.m1.1.1.1.1.3.3.2.3.3.2.cmml">i</mi><mo id="S4.E69.m1.1.1.1.1.3.3.2.3.3.1" xref="S4.E69.m1.1.1.1.1.3.3.2.3.3.1.cmml">⁢</mo><mi id="S4.E69.m1.1.1.1.1.3.3.2.3.3.3" xref="S4.E69.m1.1.1.1.1.3.3.2.3.3.3.cmml">j</mi></mrow></msup><mo id="S4.E69.m1.1.1.1.1.3.3.2.1a" xref="S4.E69.m1.1.1.1.1.3.3.2.1.cmml">⁢</mo><msub id="S4.E69.m1.1.1.1.1.3.3.2.4" xref="S4.E69.m1.1.1.1.1.3.3.2.4.cmml"><mi id="S4.E69.m1.1.1.1.1.3.3.2.4.2" xref="S4.E69.m1.1.1.1.1.3.3.2.4.2.cmml">p</mi><mi id="S4.E69.m1.1.1.1.1.3.3.2.4.3" xref="S4.E69.m1.1.1.1.1.3.3.2.4.3.cmml">i</mi></msub><mo id="S4.E69.m1.1.1.1.1.3.3.2.1b" xref="S4.E69.m1.1.1.1.1.3.3.2.1.cmml">⁢</mo><msub id="S4.E69.m1.1.1.1.1.3.3.2.5" xref="S4.E69.m1.1.1.1.1.3.3.2.5.cmml"><mi id="S4.E69.m1.1.1.1.1.3.3.2.5.2" xref="S4.E69.m1.1.1.1.1.3.3.2.5.2.cmml">p</mi><mi id="S4.E69.m1.1.1.1.1.3.3.2.5.3" xref="S4.E69.m1.1.1.1.1.3.3.2.5.3.cmml">j</mi></msub></mrow></msqrt></mrow></mrow><mo id="S4.E69.m1.1.1.1.2" lspace="0em" xref="S4.E69.m1.1.1.1.1.cmml">.</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.E69.m1.1b"><apply id="S4.E69.m1.1.1.1.1.cmml" xref="S4.E69.m1.1.1.1"><eq id="S4.E69.m1.1.1.1.1.1.cmml" xref="S4.E69.m1.1.1.1.1.1"></eq><apply id="S4.E69.m1.1.1.1.1.2.cmml" xref="S4.E69.m1.1.1.1.1.2"><csymbol cd="ambiguous" id="S4.E69.m1.1.1.1.1.2.1.cmml" xref="S4.E69.m1.1.1.1.1.2">subscript</csymbol><ci id="S4.E69.m1.1.1.1.1.2.2.cmml" xref="S4.E69.m1.1.1.1.1.2.2">𝑝</ci><cn 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xref="S4.E69.m1.1.1.1.1.3.3.2.3">superscript</csymbol><ci id="S4.E69.m1.1.1.1.1.3.3.2.3.2.cmml" xref="S4.E69.m1.1.1.1.1.3.3.2.3.2">𝛾</ci><apply id="S4.E69.m1.1.1.1.1.3.3.2.3.3.cmml" xref="S4.E69.m1.1.1.1.1.3.3.2.3.3"><times id="S4.E69.m1.1.1.1.1.3.3.2.3.3.1.cmml" xref="S4.E69.m1.1.1.1.1.3.3.2.3.3.1"></times><ci id="S4.E69.m1.1.1.1.1.3.3.2.3.3.2.cmml" xref="S4.E69.m1.1.1.1.1.3.3.2.3.3.2">𝑖</ci><ci id="S4.E69.m1.1.1.1.1.3.3.2.3.3.3.cmml" xref="S4.E69.m1.1.1.1.1.3.3.2.3.3.3">𝑗</ci></apply></apply><apply id="S4.E69.m1.1.1.1.1.3.3.2.4.cmml" xref="S4.E69.m1.1.1.1.1.3.3.2.4"><csymbol cd="ambiguous" id="S4.E69.m1.1.1.1.1.3.3.2.4.1.cmml" xref="S4.E69.m1.1.1.1.1.3.3.2.4">subscript</csymbol><ci id="S4.E69.m1.1.1.1.1.3.3.2.4.2.cmml" xref="S4.E69.m1.1.1.1.1.3.3.2.4.2">𝑝</ci><ci id="S4.E69.m1.1.1.1.1.3.3.2.4.3.cmml" xref="S4.E69.m1.1.1.1.1.3.3.2.4.3">𝑖</ci></apply><apply id="S4.E69.m1.1.1.1.1.3.3.2.5.cmml" xref="S4.E69.m1.1.1.1.1.3.3.2.5"><csymbol cd="ambiguous" id="S4.E69.m1.1.1.1.1.3.3.2.5.1.cmml" xref="S4.E69.m1.1.1.1.1.3.3.2.5">subscript</csymbol><ci id="S4.E69.m1.1.1.1.1.3.3.2.5.2.cmml" xref="S4.E69.m1.1.1.1.1.3.3.2.5.2">𝑝</ci><ci id="S4.E69.m1.1.1.1.1.3.3.2.5.3.cmml" xref="S4.E69.m1.1.1.1.1.3.3.2.5.3">𝑗</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.E69.m1.1c">\displaystyle p_{0}=\beta^{i}p_{i}\pm\sqrt{\alpha^{2}\gamma^{ij}p_{i}p_{j}}.</annotation><annotation encoding="application/x-llamapun" id="S4.E69.m1.1d">italic_p start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT = italic_β start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT italic_p start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ± square-root start_ARG italic_α start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_γ start_POSTSUPERSCRIPT italic_i italic_j end_POSTSUPERSCRIPT italic_p start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_p start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT end_ARG .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_left_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(69)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S4.p2.28">We then express (<a class="ltx_ref" href="https://arxiv.org/html/2406.11224v2#S4.E68" title="In 4 The motions of a light-like particle in a pseudo Riemann space ‣ A Hamiltonian approach to the gradient-flow equations in information geometry"><span class="ltx_text ltx_ref_tag">68</span></a>) as</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="S3.EGx63"> <tbody id="S4.E70"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_left_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\frac{1}{\alpha^{2}}\left\{H^{+}(x,p)H^{-}(x,p)\right\}=0," class="ltx_Math" display="inline" id="S4.E70.m1.5"><semantics id="S4.E70.m1.5a"><mrow id="S4.E70.m1.5.5.1" xref="S4.E70.m1.5.5.1.1.cmml"><mrow id="S4.E70.m1.5.5.1.1" xref="S4.E70.m1.5.5.1.1.cmml"><mrow id="S4.E70.m1.5.5.1.1.1" xref="S4.E70.m1.5.5.1.1.1.cmml"><mstyle displaystyle="true" id="S4.E70.m1.5.5.1.1.1.3" xref="S4.E70.m1.5.5.1.1.1.3.cmml"><mfrac id="S4.E70.m1.5.5.1.1.1.3a" xref="S4.E70.m1.5.5.1.1.1.3.cmml"><mn id="S4.E70.m1.5.5.1.1.1.3.2" xref="S4.E70.m1.5.5.1.1.1.3.2.cmml">1</mn><msup id="S4.E70.m1.5.5.1.1.1.3.3" xref="S4.E70.m1.5.5.1.1.1.3.3.cmml"><mi id="S4.E70.m1.5.5.1.1.1.3.3.2" xref="S4.E70.m1.5.5.1.1.1.3.3.2.cmml">α</mi><mn id="S4.E70.m1.5.5.1.1.1.3.3.3" xref="S4.E70.m1.5.5.1.1.1.3.3.3.cmml">2</mn></msup></mfrac></mstyle><mo id="S4.E70.m1.5.5.1.1.1.2" xref="S4.E70.m1.5.5.1.1.1.2.cmml">⁢</mo><mrow id="S4.E70.m1.5.5.1.1.1.1.1" xref="S4.E70.m1.5.5.1.1.1.1.2.cmml"><mo id="S4.E70.m1.5.5.1.1.1.1.1.2" xref="S4.E70.m1.5.5.1.1.1.1.2.cmml">{</mo><mrow id="S4.E70.m1.5.5.1.1.1.1.1.1" xref="S4.E70.m1.5.5.1.1.1.1.1.1.cmml"><msup id="S4.E70.m1.5.5.1.1.1.1.1.1.2" xref="S4.E70.m1.5.5.1.1.1.1.1.1.2.cmml"><mi id="S4.E70.m1.5.5.1.1.1.1.1.1.2.2" xref="S4.E70.m1.5.5.1.1.1.1.1.1.2.2.cmml">H</mi><mo id="S4.E70.m1.5.5.1.1.1.1.1.1.2.3" xref="S4.E70.m1.5.5.1.1.1.1.1.1.2.3.cmml">+</mo></msup><mo id="S4.E70.m1.5.5.1.1.1.1.1.1.1" xref="S4.E70.m1.5.5.1.1.1.1.1.1.1.cmml">⁢</mo><mrow id="S4.E70.m1.5.5.1.1.1.1.1.1.3.2" xref="S4.E70.m1.5.5.1.1.1.1.1.1.3.1.cmml"><mo id="S4.E70.m1.5.5.1.1.1.1.1.1.3.2.1" stretchy="false" xref="S4.E70.m1.5.5.1.1.1.1.1.1.3.1.cmml">(</mo><mi id="S4.E70.m1.1.1" xref="S4.E70.m1.1.1.cmml">x</mi><mo id="S4.E70.m1.5.5.1.1.1.1.1.1.3.2.2" xref="S4.E70.m1.5.5.1.1.1.1.1.1.3.1.cmml">,</mo><mi id="S4.E70.m1.2.2" xref="S4.E70.m1.2.2.cmml">p</mi><mo id="S4.E70.m1.5.5.1.1.1.1.1.1.3.2.3" stretchy="false" xref="S4.E70.m1.5.5.1.1.1.1.1.1.3.1.cmml">)</mo></mrow><mo id="S4.E70.m1.5.5.1.1.1.1.1.1.1a" xref="S4.E70.m1.5.5.1.1.1.1.1.1.1.cmml">⁢</mo><msup id="S4.E70.m1.5.5.1.1.1.1.1.1.4" xref="S4.E70.m1.5.5.1.1.1.1.1.1.4.cmml"><mi 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italic_x , italic_p ) italic_H start_POSTSUPERSCRIPT - end_POSTSUPERSCRIPT ( italic_x , italic_p ) } = 0 ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_left_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(70)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S4.p2.29">where</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="S3.EGx64"> <tbody id="S4.E71"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_left_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle H^{\pm}(x,p):=p_{0}-\beta^{i}p_{i}\pm\sqrt{\alpha^{2}\gamma^{ij}% p_{i}p_{j}}." class="ltx_Math" display="inline" id="S4.E71.m1.3"><semantics id="S4.E71.m1.3a"><mrow id="S4.E71.m1.3.3.1" xref="S4.E71.m1.3.3.1.1.cmml"><mrow id="S4.E71.m1.3.3.1.1" xref="S4.E71.m1.3.3.1.1.cmml"><mrow 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end_POSTSUPERSCRIPT italic_p start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ± square-root start_ARG italic_α start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_γ start_POSTSUPERSCRIPT italic_i italic_j end_POSTSUPERSCRIPT italic_p start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_p start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT end_ARG .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_left_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(71)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S4.p2.19">Thus the null Hamiltonian for a light-like particle is either <math alttext="H^{+}(x,p)=0" class="ltx_Math" display="inline" id="S4.p2.18.m1.2"><semantics id="S4.p2.18.m1.2a"><mrow id="S4.p2.18.m1.2.3" xref="S4.p2.18.m1.2.3.cmml"><mrow id="S4.p2.18.m1.2.3.2" xref="S4.p2.18.m1.2.3.2.cmml"><msup id="S4.p2.18.m1.2.3.2.2" xref="S4.p2.18.m1.2.3.2.2.cmml"><mi 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= 0</annotation></semantics></math> or <math alttext="H^{-}(x,p)=0" class="ltx_Math" display="inline" id="S4.p2.19.m2.2"><semantics id="S4.p2.19.m2.2a"><mrow id="S4.p2.19.m2.2.3" xref="S4.p2.19.m2.2.3.cmml"><mrow id="S4.p2.19.m2.2.3.2" xref="S4.p2.19.m2.2.3.2.cmml"><msup id="S4.p2.19.m2.2.3.2.2" xref="S4.p2.19.m2.2.3.2.2.cmml"><mi id="S4.p2.19.m2.2.3.2.2.2" xref="S4.p2.19.m2.2.3.2.2.2.cmml">H</mi><mo id="S4.p2.19.m2.2.3.2.2.3" xref="S4.p2.19.m2.2.3.2.2.3.cmml">−</mo></msup><mo id="S4.p2.19.m2.2.3.2.1" xref="S4.p2.19.m2.2.3.2.1.cmml">⁢</mo><mrow id="S4.p2.19.m2.2.3.2.3.2" xref="S4.p2.19.m2.2.3.2.3.1.cmml"><mo id="S4.p2.19.m2.2.3.2.3.2.1" stretchy="false" xref="S4.p2.19.m2.2.3.2.3.1.cmml">(</mo><mi id="S4.p2.19.m2.1.1" xref="S4.p2.19.m2.1.1.cmml">x</mi><mo id="S4.p2.19.m2.2.3.2.3.2.2" xref="S4.p2.19.m2.2.3.2.3.1.cmml">,</mo><mi id="S4.p2.19.m2.2.2" xref="S4.p2.19.m2.2.2.cmml">p</mi><mo id="S4.p2.19.m2.2.3.2.3.2.3" stretchy="false" xref="S4.p2.19.m2.2.3.2.3.1.cmml">)</mo></mrow></mrow><mo id="S4.p2.19.m2.2.3.1" xref="S4.p2.19.m2.2.3.1.cmml">=</mo><mn id="S4.p2.19.m2.2.3.3" xref="S4.p2.19.m2.2.3.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.p2.19.m2.2b"><apply id="S4.p2.19.m2.2.3.cmml" xref="S4.p2.19.m2.2.3"><eq id="S4.p2.19.m2.2.3.1.cmml" xref="S4.p2.19.m2.2.3.1"></eq><apply id="S4.p2.19.m2.2.3.2.cmml" xref="S4.p2.19.m2.2.3.2"><times id="S4.p2.19.m2.2.3.2.1.cmml" xref="S4.p2.19.m2.2.3.2.1"></times><apply id="S4.p2.19.m2.2.3.2.2.cmml" xref="S4.p2.19.m2.2.3.2.2"><csymbol cd="ambiguous" id="S4.p2.19.m2.2.3.2.2.1.cmml" xref="S4.p2.19.m2.2.3.2.2">superscript</csymbol><ci id="S4.p2.19.m2.2.3.2.2.2.cmml" xref="S4.p2.19.m2.2.3.2.2.2">𝐻</ci><minus id="S4.p2.19.m2.2.3.2.2.3.cmml" xref="S4.p2.19.m2.2.3.2.2.3"></minus></apply><interval closure="open" id="S4.p2.19.m2.2.3.2.3.1.cmml" xref="S4.p2.19.m2.2.3.2.3.2"><ci id="S4.p2.19.m2.1.1.cmml" xref="S4.p2.19.m2.1.1">𝑥</ci><ci id="S4.p2.19.m2.2.2.cmml" xref="S4.p2.19.m2.2.2">𝑝</ci></interval></apply><cn id="S4.p2.19.m2.2.3.3.cmml" type="integer" xref="S4.p2.19.m2.2.3.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p2.19.m2.2c">H^{-}(x,p)=0</annotation><annotation encoding="application/x-llamapun" id="S4.p2.19.m2.2d">italic_H start_POSTSUPERSCRIPT - end_POSTSUPERSCRIPT ( italic_x , italic_p ) = 0</annotation></semantics></math>. Then the corresponding Hamilton equations of motion are</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="S3.EGx65"> <tbody id="S4.Ex14"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_left_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\frac{dx^{0}}{d\tau}=H^{\mp}\frac{\partial H^{\pm}}{\partial p_{0% }}=H^{\mp},\quad\frac{dx^{i}}{d\tau}=H^{\mp}\frac{\partial H^{\pm}}{\partial p% _{i}}," class="ltx_Math" display="inline" id="S4.Ex14.m1.1"><semantics id="S4.Ex14.m1.1a"><mrow id="S4.Ex14.m1.1.1.1"><mrow id="S4.Ex14.m1.1.1.1.1.2" xref="S4.Ex14.m1.1.1.1.1.3.cmml"><mrow id="S4.Ex14.m1.1.1.1.1.1.1" xref="S4.Ex14.m1.1.1.1.1.1.1.cmml"><mstyle displaystyle="true" id="S4.Ex14.m1.1.1.1.1.1.1.2" xref="S4.Ex14.m1.1.1.1.1.1.1.2.cmml"><mfrac id="S4.Ex14.m1.1.1.1.1.1.1.2a" xref="S4.Ex14.m1.1.1.1.1.1.1.2.cmml"><mrow id="S4.Ex14.m1.1.1.1.1.1.1.2.2" xref="S4.Ex14.m1.1.1.1.1.1.1.2.2.cmml"><mi id="S4.Ex14.m1.1.1.1.1.1.1.2.2.2" xref="S4.Ex14.m1.1.1.1.1.1.1.2.2.2.cmml">d</mi><mo id="S4.Ex14.m1.1.1.1.1.1.1.2.2.1" xref="S4.Ex14.m1.1.1.1.1.1.1.2.2.1.cmml">⁢</mo><msup id="S4.Ex14.m1.1.1.1.1.1.1.2.2.3" xref="S4.Ex14.m1.1.1.1.1.1.1.2.2.3.cmml"><mi id="S4.Ex14.m1.1.1.1.1.1.1.2.2.3.2" xref="S4.Ex14.m1.1.1.1.1.1.1.2.2.3.2.cmml">x</mi><mn id="S4.Ex14.m1.1.1.1.1.1.1.2.2.3.3" xref="S4.Ex14.m1.1.1.1.1.1.1.2.2.3.3.cmml">0</mn></msup></mrow><mrow id="S4.Ex14.m1.1.1.1.1.1.1.2.3" xref="S4.Ex14.m1.1.1.1.1.1.1.2.3.cmml"><mi id="S4.Ex14.m1.1.1.1.1.1.1.2.3.2" xref="S4.Ex14.m1.1.1.1.1.1.1.2.3.2.cmml">d</mi><mo id="S4.Ex14.m1.1.1.1.1.1.1.2.3.1" xref="S4.Ex14.m1.1.1.1.1.1.1.2.3.1.cmml">⁢</mo><mi id="S4.Ex14.m1.1.1.1.1.1.1.2.3.3" xref="S4.Ex14.m1.1.1.1.1.1.1.2.3.3.cmml">τ</mi></mrow></mfrac></mstyle><mo id="S4.Ex14.m1.1.1.1.1.1.1.3" xref="S4.Ex14.m1.1.1.1.1.1.1.3.cmml">=</mo><mrow id="S4.Ex14.m1.1.1.1.1.1.1.4" xref="S4.Ex14.m1.1.1.1.1.1.1.4.cmml"><msup id="S4.Ex14.m1.1.1.1.1.1.1.4.2" xref="S4.Ex14.m1.1.1.1.1.1.1.4.2.cmml"><mi id="S4.Ex14.m1.1.1.1.1.1.1.4.2.2" xref="S4.Ex14.m1.1.1.1.1.1.1.4.2.2.cmml">H</mi><mo id="S4.Ex14.m1.1.1.1.1.1.1.4.2.3" xref="S4.Ex14.m1.1.1.1.1.1.1.4.2.3.cmml">∓</mo></msup><mo id="S4.Ex14.m1.1.1.1.1.1.1.4.1" xref="S4.Ex14.m1.1.1.1.1.1.1.4.1.cmml">⁢</mo><mstyle displaystyle="true" id="S4.Ex14.m1.1.1.1.1.1.1.4.3" xref="S4.Ex14.m1.1.1.1.1.1.1.4.3.cmml"><mfrac id="S4.Ex14.m1.1.1.1.1.1.1.4.3a" xref="S4.Ex14.m1.1.1.1.1.1.1.4.3.cmml"><mrow id="S4.Ex14.m1.1.1.1.1.1.1.4.3.2" xref="S4.Ex14.m1.1.1.1.1.1.1.4.3.2.cmml"><mo id="S4.Ex14.m1.1.1.1.1.1.1.4.3.2.1" rspace="0em" xref="S4.Ex14.m1.1.1.1.1.1.1.4.3.2.1.cmml">∂</mo><msup id="S4.Ex14.m1.1.1.1.1.1.1.4.3.2.2" xref="S4.Ex14.m1.1.1.1.1.1.1.4.3.2.2.cmml"><mi id="S4.Ex14.m1.1.1.1.1.1.1.4.3.2.2.2" xref="S4.Ex14.m1.1.1.1.1.1.1.4.3.2.2.2.cmml">H</mi><mo id="S4.Ex14.m1.1.1.1.1.1.1.4.3.2.2.3" xref="S4.Ex14.m1.1.1.1.1.1.1.4.3.2.2.3.cmml">±</mo></msup></mrow><mrow id="S4.Ex14.m1.1.1.1.1.1.1.4.3.3" xref="S4.Ex14.m1.1.1.1.1.1.1.4.3.3.cmml"><mo id="S4.Ex14.m1.1.1.1.1.1.1.4.3.3.1" rspace="0em" xref="S4.Ex14.m1.1.1.1.1.1.1.4.3.3.1.cmml">∂</mo><msub id="S4.Ex14.m1.1.1.1.1.1.1.4.3.3.2" xref="S4.Ex14.m1.1.1.1.1.1.1.4.3.3.2.cmml"><mi id="S4.Ex14.m1.1.1.1.1.1.1.4.3.3.2.2" xref="S4.Ex14.m1.1.1.1.1.1.1.4.3.3.2.2.cmml">p</mi><mn id="S4.Ex14.m1.1.1.1.1.1.1.4.3.3.2.3" xref="S4.Ex14.m1.1.1.1.1.1.1.4.3.3.2.3.cmml">0</mn></msub></mrow></mfrac></mstyle></mrow><mo id="S4.Ex14.m1.1.1.1.1.1.1.5" xref="S4.Ex14.m1.1.1.1.1.1.1.5.cmml">=</mo><msup id="S4.Ex14.m1.1.1.1.1.1.1.6" xref="S4.Ex14.m1.1.1.1.1.1.1.6.cmml"><mi id="S4.Ex14.m1.1.1.1.1.1.1.6.2" xref="S4.Ex14.m1.1.1.1.1.1.1.6.2.cmml">H</mi><mo id="S4.Ex14.m1.1.1.1.1.1.1.6.3" xref="S4.Ex14.m1.1.1.1.1.1.1.6.3.cmml">∓</mo></msup></mrow><mo id="S4.Ex14.m1.1.1.1.1.2.3" rspace="1.167em" xref="S4.Ex14.m1.1.1.1.1.3a.cmml">,</mo><mrow id="S4.Ex14.m1.1.1.1.1.2.2" xref="S4.Ex14.m1.1.1.1.1.2.2.cmml"><mstyle displaystyle="true" id="S4.Ex14.m1.1.1.1.1.2.2.2" xref="S4.Ex14.m1.1.1.1.1.2.2.2.cmml"><mfrac id="S4.Ex14.m1.1.1.1.1.2.2.2a" xref="S4.Ex14.m1.1.1.1.1.2.2.2.cmml"><mrow id="S4.Ex14.m1.1.1.1.1.2.2.2.2" xref="S4.Ex14.m1.1.1.1.1.2.2.2.2.cmml"><mi id="S4.Ex14.m1.1.1.1.1.2.2.2.2.2" xref="S4.Ex14.m1.1.1.1.1.2.2.2.2.2.cmml">d</mi><mo id="S4.Ex14.m1.1.1.1.1.2.2.2.2.1" xref="S4.Ex14.m1.1.1.1.1.2.2.2.2.1.cmml">⁢</mo><msup id="S4.Ex14.m1.1.1.1.1.2.2.2.2.3" xref="S4.Ex14.m1.1.1.1.1.2.2.2.2.3.cmml"><mi id="S4.Ex14.m1.1.1.1.1.2.2.2.2.3.2" xref="S4.Ex14.m1.1.1.1.1.2.2.2.2.3.2.cmml">x</mi><mi id="S4.Ex14.m1.1.1.1.1.2.2.2.2.3.3" xref="S4.Ex14.m1.1.1.1.1.2.2.2.2.3.3.cmml">i</mi></msup></mrow><mrow id="S4.Ex14.m1.1.1.1.1.2.2.2.3" xref="S4.Ex14.m1.1.1.1.1.2.2.2.3.cmml"><mi id="S4.Ex14.m1.1.1.1.1.2.2.2.3.2" xref="S4.Ex14.m1.1.1.1.1.2.2.2.3.2.cmml">d</mi><mo id="S4.Ex14.m1.1.1.1.1.2.2.2.3.1" xref="S4.Ex14.m1.1.1.1.1.2.2.2.3.1.cmml">⁢</mo><mi id="S4.Ex14.m1.1.1.1.1.2.2.2.3.3" xref="S4.Ex14.m1.1.1.1.1.2.2.2.3.3.cmml">τ</mi></mrow></mfrac></mstyle><mo 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xref="S4.Ex14.m1.1.1.1.1.2.2.3.3.2.2.2.cmml">H</mi><mo id="S4.Ex14.m1.1.1.1.1.2.2.3.3.2.2.3" xref="S4.Ex14.m1.1.1.1.1.2.2.3.3.2.2.3.cmml">±</mo></msup></mrow><mrow id="S4.Ex14.m1.1.1.1.1.2.2.3.3.3" xref="S4.Ex14.m1.1.1.1.1.2.2.3.3.3.cmml"><mo id="S4.Ex14.m1.1.1.1.1.2.2.3.3.3.1" rspace="0em" xref="S4.Ex14.m1.1.1.1.1.2.2.3.3.3.1.cmml">∂</mo><msub id="S4.Ex14.m1.1.1.1.1.2.2.3.3.3.2" xref="S4.Ex14.m1.1.1.1.1.2.2.3.3.3.2.cmml"><mi id="S4.Ex14.m1.1.1.1.1.2.2.3.3.3.2.2" xref="S4.Ex14.m1.1.1.1.1.2.2.3.3.3.2.2.cmml">p</mi><mi id="S4.Ex14.m1.1.1.1.1.2.2.3.3.3.2.3" xref="S4.Ex14.m1.1.1.1.1.2.2.3.3.3.2.3.cmml">i</mi></msub></mrow></mfrac></mstyle></mrow></mrow></mrow><mo id="S4.Ex14.m1.1.1.1.2">,</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.Ex14.m1.1b"><apply id="S4.Ex14.m1.1.1.1.1.3.cmml" xref="S4.Ex14.m1.1.1.1.1.2"><csymbol cd="ambiguous" id="S4.Ex14.m1.1.1.1.1.3a.cmml" xref="S4.Ex14.m1.1.1.1.1.2.3">formulae-sequence</csymbol><apply id="S4.Ex14.m1.1.1.1.1.1.1.cmml" 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encoding="application/x-llamapun" id="S4.Ex14.m1.1d">divide start_ARG italic_d italic_x start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT end_ARG start_ARG italic_d italic_τ end_ARG = italic_H start_POSTSUPERSCRIPT ∓ end_POSTSUPERSCRIPT divide start_ARG ∂ italic_H start_POSTSUPERSCRIPT ± end_POSTSUPERSCRIPT end_ARG start_ARG ∂ italic_p start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_ARG = italic_H start_POSTSUPERSCRIPT ∓ end_POSTSUPERSCRIPT , divide start_ARG italic_d italic_x start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT end_ARG start_ARG italic_d italic_τ end_ARG = italic_H start_POSTSUPERSCRIPT ∓ end_POSTSUPERSCRIPT divide start_ARG ∂ italic_H start_POSTSUPERSCRIPT ± end_POSTSUPERSCRIPT end_ARG start_ARG ∂ italic_p start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_ARG ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_left_padright"></td> </tr></tbody> <tbody id="S4.Ex15"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell 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id="S4.Ex15.m1.1c">\displaystyle\frac{dp_{0}}{d\tau}=-H^{\mp}\frac{\partial H^{\pm}}{\partial x^{% 0}}=0,\quad\frac{dp^{i}}{d\tau}=-H^{\mp}\frac{\partial H^{\pm}}{\partial x^{i}},</annotation><annotation encoding="application/x-llamapun" id="S4.Ex15.m1.1d">divide start_ARG italic_d italic_p start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_ARG start_ARG italic_d italic_τ end_ARG = - italic_H start_POSTSUPERSCRIPT ∓ end_POSTSUPERSCRIPT divide start_ARG ∂ italic_H start_POSTSUPERSCRIPT ± end_POSTSUPERSCRIPT end_ARG start_ARG ∂ italic_x start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT end_ARG = 0 , divide start_ARG italic_d italic_p start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT end_ARG start_ARG italic_d italic_τ end_ARG = - italic_H start_POSTSUPERSCRIPT ∓ end_POSTSUPERSCRIPT divide start_ARG ∂ italic_H start_POSTSUPERSCRIPT ± end_POSTSUPERSCRIPT end_ARG start_ARG ∂ italic_x start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT end_ARG ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_left_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S4.p2.23">where the upper- (lower-) case in the superscript refers to the choice <math alttext="H^{+}=0" class="ltx_Math" display="inline" id="S4.p2.20.m1.1"><semantics id="S4.p2.20.m1.1a"><mrow id="S4.p2.20.m1.1.1" xref="S4.p2.20.m1.1.1.cmml"><msup id="S4.p2.20.m1.1.1.2" xref="S4.p2.20.m1.1.1.2.cmml"><mi id="S4.p2.20.m1.1.1.2.2" xref="S4.p2.20.m1.1.1.2.2.cmml">H</mi><mo id="S4.p2.20.m1.1.1.2.3" xref="S4.p2.20.m1.1.1.2.3.cmml">+</mo></msup><mo id="S4.p2.20.m1.1.1.1" xref="S4.p2.20.m1.1.1.1.cmml">=</mo><mn id="S4.p2.20.m1.1.1.3" xref="S4.p2.20.m1.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.p2.20.m1.1b"><apply id="S4.p2.20.m1.1.1.cmml" xref="S4.p2.20.m1.1.1"><eq id="S4.p2.20.m1.1.1.1.cmml" xref="S4.p2.20.m1.1.1.1"></eq><apply id="S4.p2.20.m1.1.1.2.cmml" xref="S4.p2.20.m1.1.1.2"><csymbol cd="ambiguous" id="S4.p2.20.m1.1.1.2.1.cmml" xref="S4.p2.20.m1.1.1.2">superscript</csymbol><ci id="S4.p2.20.m1.1.1.2.2.cmml" xref="S4.p2.20.m1.1.1.2.2">𝐻</ci><plus id="S4.p2.20.m1.1.1.2.3.cmml" xref="S4.p2.20.m1.1.1.2.3"></plus></apply><cn id="S4.p2.20.m1.1.1.3.cmml" type="integer" xref="S4.p2.20.m1.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p2.20.m1.1c">H^{+}=0</annotation><annotation encoding="application/x-llamapun" id="S4.p2.20.m1.1d">italic_H start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT = 0</annotation></semantics></math> ( <math alttext="H^{-}=0" class="ltx_Math" display="inline" id="S4.p2.21.m2.1"><semantics id="S4.p2.21.m2.1a"><mrow id="S4.p2.21.m2.1.1" xref="S4.p2.21.m2.1.1.cmml"><msup id="S4.p2.21.m2.1.1.2" xref="S4.p2.21.m2.1.1.2.cmml"><mi id="S4.p2.21.m2.1.1.2.2" xref="S4.p2.21.m2.1.1.2.2.cmml">H</mi><mo id="S4.p2.21.m2.1.1.2.3" xref="S4.p2.21.m2.1.1.2.3.cmml">−</mo></msup><mo id="S4.p2.21.m2.1.1.1" xref="S4.p2.21.m2.1.1.1.cmml">=</mo><mn id="S4.p2.21.m2.1.1.3" xref="S4.p2.21.m2.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.p2.21.m2.1b"><apply id="S4.p2.21.m2.1.1.cmml" xref="S4.p2.21.m2.1.1"><eq id="S4.p2.21.m2.1.1.1.cmml" xref="S4.p2.21.m2.1.1.1"></eq><apply id="S4.p2.21.m2.1.1.2.cmml" xref="S4.p2.21.m2.1.1.2"><csymbol cd="ambiguous" id="S4.p2.21.m2.1.1.2.1.cmml" xref="S4.p2.21.m2.1.1.2">superscript</csymbol><ci id="S4.p2.21.m2.1.1.2.2.cmml" xref="S4.p2.21.m2.1.1.2.2">𝐻</ci><minus id="S4.p2.21.m2.1.1.2.3.cmml" xref="S4.p2.21.m2.1.1.2.3"></minus></apply><cn id="S4.p2.21.m2.1.1.3.cmml" type="integer" xref="S4.p2.21.m2.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p2.21.m2.1c">H^{-}=0</annotation><annotation encoding="application/x-llamapun" id="S4.p2.21.m2.1d">italic_H start_POSTSUPERSCRIPT - end_POSTSUPERSCRIPT = 0</annotation></semantics></math>). By using these relations we can eliminate the affine parameter <math alttext="\tau" class="ltx_Math" display="inline" id="S4.p2.22.m3.1"><semantics id="S4.p2.22.m3.1a"><mi id="S4.p2.22.m3.1.1" xref="S4.p2.22.m3.1.1.cmml">τ</mi><annotation-xml encoding="MathML-Content" id="S4.p2.22.m3.1b"><ci id="S4.p2.22.m3.1.1.cmml" xref="S4.p2.22.m3.1.1">𝜏</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.p2.22.m3.1c">\tau</annotation><annotation encoding="application/x-llamapun" id="S4.p2.22.m3.1d">italic_τ</annotation></semantics></math> and obtain the equations of motion with respect to <math alttext="x^{0}" class="ltx_Math" display="inline" id="S4.p2.23.m4.1"><semantics id="S4.p2.23.m4.1a"><msup id="S4.p2.23.m4.1.1" xref="S4.p2.23.m4.1.1.cmml"><mi id="S4.p2.23.m4.1.1.2" xref="S4.p2.23.m4.1.1.2.cmml">x</mi><mn id="S4.p2.23.m4.1.1.3" xref="S4.p2.23.m4.1.1.3.cmml">0</mn></msup><annotation-xml encoding="MathML-Content" id="S4.p2.23.m4.1b"><apply id="S4.p2.23.m4.1.1.cmml" 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id="S4.E72.1.m1.1d">= divide start_ARG ∂ end_ARG start_ARG ∂ italic_p start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_ARG { - italic_β start_POSTSUPERSCRIPT italic_j end_POSTSUPERSCRIPT italic_p start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ± square-root start_ARG italic_α start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_γ start_POSTSUPERSCRIPT italic_j italic_k end_POSTSUPERSCRIPT italic_p start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT italic_p start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_ARG } ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_left_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(72a)</span></td> </tr> <tr class="ltx_equation ltx_eqn_row ltx_align_baseline" id="S4.E72.x2"> <td class="ltx_eqn_cell ltx_eqn_left_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\frac{dp_{i}}{dx^{0}}" 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xref="S4.E72.x2.m1.1.1.3.2">𝑑</ci><apply id="S4.E72.x2.m1.1.1.3.3.cmml" xref="S4.E72.x2.m1.1.1.3.3"><csymbol cd="ambiguous" id="S4.E72.x2.m1.1.1.3.3.1.cmml" xref="S4.E72.x2.m1.1.1.3.3">superscript</csymbol><ci id="S4.E72.x2.m1.1.1.3.3.2.cmml" xref="S4.E72.x2.m1.1.1.3.3.2">𝑥</ci><cn id="S4.E72.x2.m1.1.1.3.3.3.cmml" type="integer" xref="S4.E72.x2.m1.1.1.3.3.3">0</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.E72.x2.m1.1c">\displaystyle\frac{dp_{i}}{dx^{0}}</annotation><annotation encoding="application/x-llamapun" id="S4.E72.x2.m1.1d">divide start_ARG italic_d italic_p start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_ARG start_ARG italic_d italic_x start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT end_ARG</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle=\frac{\frac{dp_{i}}{d\tau}}{\frac{dx^{0}}{d\tau}}=-\frac{% \partial H^{\pm}}{\partial x^{i}}" class="ltx_Math" display="inline" 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encoding="application/x-llamapun" id="S4.E72.x2.m2.1d">= divide start_ARG divide start_ARG italic_d italic_p start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_ARG start_ARG italic_d italic_τ end_ARG end_ARG start_ARG divide start_ARG italic_d italic_x start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT end_ARG start_ARG italic_d italic_τ end_ARG end_ARG = - divide start_ARG ∂ italic_H start_POSTSUPERSCRIPT ± end_POSTSUPERSCRIPT end_ARG start_ARG ∂ italic_x start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT end_ARG</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_left_padright"></td> </tr> <tr class="ltx_equation ltx_eqn_row ltx_align_baseline" id="S4.E72.2"> <td class="ltx_eqn_cell ltx_eqn_left_padleft"></td> <td class="ltx_td ltx_eqn_cell"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle=-\frac{\partial}{\partial x^{i}}\left\{-\beta^{j}p_{j}\pm\sqrt{% \alpha^{2}\gamma^{jk}p_{j}p_{k}}\right\}." class="ltx_Math" display="inline" 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encoding="application/x-llamapun" id="S4.E72.2.m1.1d">= - divide start_ARG ∂ end_ARG start_ARG ∂ italic_x start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT end_ARG { - italic_β start_POSTSUPERSCRIPT italic_j end_POSTSUPERSCRIPT italic_p start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ± square-root start_ARG italic_α start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_γ start_POSTSUPERSCRIPT italic_j italic_k end_POSTSUPERSCRIPT italic_p start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT italic_p start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_ARG } .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_left_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(72b)</span></td> </tr> </tbody> </table> <p class="ltx_p" id="S4.p2.30">Consequently the expression in the curly brackets acts as the Hamilton function</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" 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id="S4.E73.m1.3.3.1.1.3.3.2.4.3.cmml" xref="S4.E73.m1.3.3.1.1.3.3.2.4.3">𝑗</ci></apply><apply id="S4.E73.m1.3.3.1.1.3.3.2.5.cmml" xref="S4.E73.m1.3.3.1.1.3.3.2.5"><csymbol cd="ambiguous" id="S4.E73.m1.3.3.1.1.3.3.2.5.1.cmml" xref="S4.E73.m1.3.3.1.1.3.3.2.5">subscript</csymbol><ci id="S4.E73.m1.3.3.1.1.3.3.2.5.2.cmml" xref="S4.E73.m1.3.3.1.1.3.3.2.5.2">𝑝</ci><ci id="S4.E73.m1.3.3.1.1.3.3.2.5.3.cmml" xref="S4.E73.m1.3.3.1.1.3.3.2.5.3">𝑘</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.E73.m1.3c">\displaystyle\mathcal{H}^{\pm}(x,p):=-\beta^{j}p_{j}\pm\sqrt{\alpha^{2}\gamma^% {jk}p_{j}p_{k}},</annotation><annotation encoding="application/x-llamapun" id="S4.E73.m1.3d">caligraphic_H start_POSTSUPERSCRIPT ± end_POSTSUPERSCRIPT ( italic_x , italic_p ) := - italic_β start_POSTSUPERSCRIPT italic_j end_POSTSUPERSCRIPT italic_p start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ± square-root start_ARG italic_α start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_γ start_POSTSUPERSCRIPT italic_j italic_k end_POSTSUPERSCRIPT italic_p start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT italic_p start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_ARG ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_left_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(73)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S4.p2.24">which describes the motions of a null particle with respect to the parameter <math alttext="x^{0}" class="ltx_Math" display="inline" id="S4.p2.24.m1.1"><semantics id="S4.p2.24.m1.1a"><msup id="S4.p2.24.m1.1.1" xref="S4.p2.24.m1.1.1.cmml"><mi id="S4.p2.24.m1.1.1.2" xref="S4.p2.24.m1.1.1.2.cmml">x</mi><mn id="S4.p2.24.m1.1.1.3" xref="S4.p2.24.m1.1.1.3.cmml">0</mn></msup><annotation-xml encoding="MathML-Content" id="S4.p2.24.m1.1b"><apply id="S4.p2.24.m1.1.1.cmml" xref="S4.p2.24.m1.1.1"><csymbol cd="ambiguous" id="S4.p2.24.m1.1.1.1.cmml" xref="S4.p2.24.m1.1.1">superscript</csymbol><ci id="S4.p2.24.m1.1.1.2.cmml" xref="S4.p2.24.m1.1.1.2">𝑥</ci><cn id="S4.p2.24.m1.1.1.3.cmml" type="integer" xref="S4.p2.24.m1.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p2.24.m1.1c">x^{0}</annotation><annotation encoding="application/x-llamapun" id="S4.p2.24.m1.1d">italic_x start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT</annotation></semantics></math>.</p> </div> <section class="ltx_subsection" id="S4.SS1"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">4.1 </span>Relation to the Randers-Finsler Lagrangian</h3> <div class="ltx_para" id="S4.SS1.p1"> <p class="ltx_p" id="S4.SS1.p1.3">Here we explain the relationship between the Hamiltonian (<a class="ltx_ref" href="https://arxiv.org/html/2406.11224v2#S4.E73" title="In 4 The motions of a light-like particle in a pseudo Riemann space ‣ A Hamiltonian approach to the gradient-flow equations in information geometry"><span class="ltx_text ltx_ref_tag">73</span></a>) and the RF Lagrangian (<a class="ltx_ref" href="https://arxiv.org/html/2406.11224v2#S2.E35" title="In 2.3 Randers-Finsler deformation of the gradient-flow equations ‣ 2 Information Geometry and Gradient-Flow Equations ‣ A Hamiltonian approach to the gradient-flow equations in information geometry"><span class="ltx_text ltx_ref_tag">35</span></a>). It is well known that the Legendre transformation maps a Hamiltonian to a Lagrangian. However, since <math alttext="\mathcal{H}^{\pm}(x,p)" class="ltx_Math" display="inline" id="S4.SS1.p1.1.m1.2"><semantics id="S4.SS1.p1.1.m1.2a"><mrow id="S4.SS1.p1.1.m1.2.3" xref="S4.SS1.p1.1.m1.2.3.cmml"><msup id="S4.SS1.p1.1.m1.2.3.2" xref="S4.SS1.p1.1.m1.2.3.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.SS1.p1.1.m1.2.3.2.2" xref="S4.SS1.p1.1.m1.2.3.2.2.cmml">ℋ</mi><mo id="S4.SS1.p1.1.m1.2.3.2.3" xref="S4.SS1.p1.1.m1.2.3.2.3.cmml">±</mo></msup><mo id="S4.SS1.p1.1.m1.2.3.1" xref="S4.SS1.p1.1.m1.2.3.1.cmml">⁢</mo><mrow id="S4.SS1.p1.1.m1.2.3.3.2" xref="S4.SS1.p1.1.m1.2.3.3.1.cmml"><mo id="S4.SS1.p1.1.m1.2.3.3.2.1" stretchy="false" xref="S4.SS1.p1.1.m1.2.3.3.1.cmml">(</mo><mi id="S4.SS1.p1.1.m1.1.1" xref="S4.SS1.p1.1.m1.1.1.cmml">x</mi><mo id="S4.SS1.p1.1.m1.2.3.3.2.2" xref="S4.SS1.p1.1.m1.2.3.3.1.cmml">,</mo><mi id="S4.SS1.p1.1.m1.2.2" xref="S4.SS1.p1.1.m1.2.2.cmml">p</mi><mo id="S4.SS1.p1.1.m1.2.3.3.2.3" stretchy="false" xref="S4.SS1.p1.1.m1.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p1.1.m1.2b"><apply id="S4.SS1.p1.1.m1.2.3.cmml" xref="S4.SS1.p1.1.m1.2.3"><times id="S4.SS1.p1.1.m1.2.3.1.cmml" xref="S4.SS1.p1.1.m1.2.3.1"></times><apply id="S4.SS1.p1.1.m1.2.3.2.cmml" xref="S4.SS1.p1.1.m1.2.3.2"><csymbol cd="ambiguous" id="S4.SS1.p1.1.m1.2.3.2.1.cmml" xref="S4.SS1.p1.1.m1.2.3.2">superscript</csymbol><ci id="S4.SS1.p1.1.m1.2.3.2.2.cmml" xref="S4.SS1.p1.1.m1.2.3.2.2">ℋ</ci><csymbol cd="latexml" id="S4.SS1.p1.1.m1.2.3.2.3.cmml" xref="S4.SS1.p1.1.m1.2.3.2.3">plus-or-minus</csymbol></apply><interval closure="open" id="S4.SS1.p1.1.m1.2.3.3.1.cmml" xref="S4.SS1.p1.1.m1.2.3.3.2"><ci id="S4.SS1.p1.1.m1.1.1.cmml" xref="S4.SS1.p1.1.m1.1.1">𝑥</ci><ci id="S4.SS1.p1.1.m1.2.2.cmml" xref="S4.SS1.p1.1.m1.2.2">𝑝</ci></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p1.1.m1.2c">\mathcal{H}^{\pm}(x,p)</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p1.1.m1.2d">caligraphic_H start_POSTSUPERSCRIPT ± end_POSTSUPERSCRIPT ( italic_x , italic_p )</annotation></semantics></math> is homogeneous of first order in momenta <math alttext="p" class="ltx_Math" display="inline" id="S4.SS1.p1.2.m2.1"><semantics id="S4.SS1.p1.2.m2.1a"><mi id="S4.SS1.p1.2.m2.1.1" xref="S4.SS1.p1.2.m2.1.1.cmml">p</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.p1.2.m2.1b"><ci id="S4.SS1.p1.2.m2.1.1.cmml" xref="S4.SS1.p1.2.m2.1.1">𝑝</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p1.2.m2.1c">p</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p1.2.m2.1d">italic_p</annotation></semantics></math>, the associated Lagrangian would vanish. A useful method <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2406.11224v2#bib.bib24" title="">24</a>]</cite> is introduce a Hamiltonian, say <math alttext="\mathcal{G}(x,p)" class="ltx_Math" display="inline" id="S4.SS1.p1.3.m3.2"><semantics id="S4.SS1.p1.3.m3.2a"><mrow id="S4.SS1.p1.3.m3.2.3" xref="S4.SS1.p1.3.m3.2.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.SS1.p1.3.m3.2.3.2" xref="S4.SS1.p1.3.m3.2.3.2.cmml">𝒢</mi><mo id="S4.SS1.p1.3.m3.2.3.1" xref="S4.SS1.p1.3.m3.2.3.1.cmml">⁢</mo><mrow id="S4.SS1.p1.3.m3.2.3.3.2" xref="S4.SS1.p1.3.m3.2.3.3.1.cmml"><mo id="S4.SS1.p1.3.m3.2.3.3.2.1" stretchy="false" xref="S4.SS1.p1.3.m3.2.3.3.1.cmml">(</mo><mi id="S4.SS1.p1.3.m3.1.1" xref="S4.SS1.p1.3.m3.1.1.cmml">x</mi><mo id="S4.SS1.p1.3.m3.2.3.3.2.2" xref="S4.SS1.p1.3.m3.2.3.3.1.cmml">,</mo><mi id="S4.SS1.p1.3.m3.2.2" xref="S4.SS1.p1.3.m3.2.2.cmml">p</mi><mo id="S4.SS1.p1.3.m3.2.3.3.2.3" stretchy="false" xref="S4.SS1.p1.3.m3.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p1.3.m3.2b"><apply id="S4.SS1.p1.3.m3.2.3.cmml" xref="S4.SS1.p1.3.m3.2.3"><times id="S4.SS1.p1.3.m3.2.3.1.cmml" xref="S4.SS1.p1.3.m3.2.3.1"></times><ci id="S4.SS1.p1.3.m3.2.3.2.cmml" xref="S4.SS1.p1.3.m3.2.3.2">𝒢</ci><interval closure="open" id="S4.SS1.p1.3.m3.2.3.3.1.cmml" xref="S4.SS1.p1.3.m3.2.3.3.2"><ci id="S4.SS1.p1.3.m3.1.1.cmml" xref="S4.SS1.p1.3.m3.1.1">𝑥</ci><ci id="S4.SS1.p1.3.m3.2.2.cmml" xref="S4.SS1.p1.3.m3.2.2">𝑝</ci></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p1.3.m3.2c">\mathcal{G}(x,p)</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p1.3.m3.2d">caligraphic_G ( italic_x , italic_p )</annotation></semantics></math>, as</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="S3.EGx68"> <tbody id="S4.E74"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_left_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\mathcal{G}(x,p)=\frac{1}{2}\left(\mathcal{H}^{+}\right)^{2}," class="ltx_Math" display="inline" id="S4.E74.m1.3"><semantics id="S4.E74.m1.3a"><mrow id="S4.E74.m1.3.3.1" xref="S4.E74.m1.3.3.1.1.cmml"><mrow id="S4.E74.m1.3.3.1.1" xref="S4.E74.m1.3.3.1.1.cmml"><mrow id="S4.E74.m1.3.3.1.1.3" xref="S4.E74.m1.3.3.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.E74.m1.3.3.1.1.3.2" xref="S4.E74.m1.3.3.1.1.3.2.cmml">𝒢</mi><mo id="S4.E74.m1.3.3.1.1.3.1" xref="S4.E74.m1.3.3.1.1.3.1.cmml">⁢</mo><mrow id="S4.E74.m1.3.3.1.1.3.3.2" xref="S4.E74.m1.3.3.1.1.3.3.1.cmml"><mo id="S4.E74.m1.3.3.1.1.3.3.2.1" stretchy="false" xref="S4.E74.m1.3.3.1.1.3.3.1.cmml">(</mo><mi id="S4.E74.m1.1.1" xref="S4.E74.m1.1.1.cmml">x</mi><mo id="S4.E74.m1.3.3.1.1.3.3.2.2" xref="S4.E74.m1.3.3.1.1.3.3.1.cmml">,</mo><mi id="S4.E74.m1.2.2" xref="S4.E74.m1.2.2.cmml">p</mi><mo id="S4.E74.m1.3.3.1.1.3.3.2.3" stretchy="false" xref="S4.E74.m1.3.3.1.1.3.3.1.cmml">)</mo></mrow></mrow><mo id="S4.E74.m1.3.3.1.1.2" xref="S4.E74.m1.3.3.1.1.2.cmml">=</mo><mrow id="S4.E74.m1.3.3.1.1.1" xref="S4.E74.m1.3.3.1.1.1.cmml"><mstyle displaystyle="true" id="S4.E74.m1.3.3.1.1.1.3" xref="S4.E74.m1.3.3.1.1.1.3.cmml"><mfrac id="S4.E74.m1.3.3.1.1.1.3a" xref="S4.E74.m1.3.3.1.1.1.3.cmml"><mn id="S4.E74.m1.3.3.1.1.1.3.2" xref="S4.E74.m1.3.3.1.1.1.3.2.cmml">1</mn><mn id="S4.E74.m1.3.3.1.1.1.3.3" xref="S4.E74.m1.3.3.1.1.1.3.3.cmml">2</mn></mfrac></mstyle><mo id="S4.E74.m1.3.3.1.1.1.2" xref="S4.E74.m1.3.3.1.1.1.2.cmml">⁢</mo><msup id="S4.E74.m1.3.3.1.1.1.1" xref="S4.E74.m1.3.3.1.1.1.1.cmml"><mrow id="S4.E74.m1.3.3.1.1.1.1.1.1" xref="S4.E74.m1.3.3.1.1.1.1.1.1.1.cmml"><mo id="S4.E74.m1.3.3.1.1.1.1.1.1.2" xref="S4.E74.m1.3.3.1.1.1.1.1.1.1.cmml">(</mo><msup id="S4.E74.m1.3.3.1.1.1.1.1.1.1" xref="S4.E74.m1.3.3.1.1.1.1.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.E74.m1.3.3.1.1.1.1.1.1.1.2" xref="S4.E74.m1.3.3.1.1.1.1.1.1.1.2.cmml">ℋ</mi><mo id="S4.E74.m1.3.3.1.1.1.1.1.1.1.3" xref="S4.E74.m1.3.3.1.1.1.1.1.1.1.3.cmml">+</mo></msup><mo id="S4.E74.m1.3.3.1.1.1.1.1.1.3" xref="S4.E74.m1.3.3.1.1.1.1.1.1.1.cmml">)</mo></mrow><mn id="S4.E74.m1.3.3.1.1.1.1.3" xref="S4.E74.m1.3.3.1.1.1.1.3.cmml">2</mn></msup></mrow></mrow><mo id="S4.E74.m1.3.3.1.2" xref="S4.E74.m1.3.3.1.1.cmml">,</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.E74.m1.3b"><apply id="S4.E74.m1.3.3.1.1.cmml" xref="S4.E74.m1.3.3.1"><eq id="S4.E74.m1.3.3.1.1.2.cmml" xref="S4.E74.m1.3.3.1.1.2"></eq><apply id="S4.E74.m1.3.3.1.1.3.cmml" xref="S4.E74.m1.3.3.1.1.3"><times id="S4.E74.m1.3.3.1.1.3.1.cmml" xref="S4.E74.m1.3.3.1.1.3.1"></times><ci id="S4.E74.m1.3.3.1.1.3.2.cmml" xref="S4.E74.m1.3.3.1.1.3.2">𝒢</ci><interval closure="open" id="S4.E74.m1.3.3.1.1.3.3.1.cmml" xref="S4.E74.m1.3.3.1.1.3.3.2"><ci id="S4.E74.m1.1.1.cmml" xref="S4.E74.m1.1.1">𝑥</ci><ci id="S4.E74.m1.2.2.cmml" xref="S4.E74.m1.2.2">𝑝</ci></interval></apply><apply id="S4.E74.m1.3.3.1.1.1.cmml" xref="S4.E74.m1.3.3.1.1.1"><times id="S4.E74.m1.3.3.1.1.1.2.cmml" xref="S4.E74.m1.3.3.1.1.1.2"></times><apply id="S4.E74.m1.3.3.1.1.1.3.cmml" xref="S4.E74.m1.3.3.1.1.1.3"><divide id="S4.E74.m1.3.3.1.1.1.3.1.cmml" xref="S4.E74.m1.3.3.1.1.1.3"></divide><cn id="S4.E74.m1.3.3.1.1.1.3.2.cmml" type="integer" xref="S4.E74.m1.3.3.1.1.1.3.2">1</cn><cn id="S4.E74.m1.3.3.1.1.1.3.3.cmml" type="integer" xref="S4.E74.m1.3.3.1.1.1.3.3">2</cn></apply><apply id="S4.E74.m1.3.3.1.1.1.1.cmml" xref="S4.E74.m1.3.3.1.1.1.1"><csymbol cd="ambiguous" id="S4.E74.m1.3.3.1.1.1.1.2.cmml" xref="S4.E74.m1.3.3.1.1.1.1">superscript</csymbol><apply id="S4.E74.m1.3.3.1.1.1.1.1.1.1.cmml" xref="S4.E74.m1.3.3.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S4.E74.m1.3.3.1.1.1.1.1.1.1.1.cmml" xref="S4.E74.m1.3.3.1.1.1.1.1.1">superscript</csymbol><ci id="S4.E74.m1.3.3.1.1.1.1.1.1.1.2.cmml" xref="S4.E74.m1.3.3.1.1.1.1.1.1.1.2">ℋ</ci><plus id="S4.E74.m1.3.3.1.1.1.1.1.1.1.3.cmml" xref="S4.E74.m1.3.3.1.1.1.1.1.1.1.3"></plus></apply><cn id="S4.E74.m1.3.3.1.1.1.1.3.cmml" type="integer" xref="S4.E74.m1.3.3.1.1.1.1.3">2</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.E74.m1.3c">\displaystyle\mathcal{G}(x,p)=\frac{1}{2}\left(\mathcal{H}^{+}\right)^{2},</annotation><annotation encoding="application/x-llamapun" id="S4.E74.m1.3d">caligraphic_G ( italic_x , italic_p ) = divide start_ARG 1 end_ARG start_ARG 2 end_ARG ( caligraphic_H start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_left_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(74)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S4.SS1.p1.5">which is homogeneous of second order in momenta <math alttext="p" class="ltx_Math" display="inline" id="S4.SS1.p1.4.m1.1"><semantics id="S4.SS1.p1.4.m1.1a"><mi id="S4.SS1.p1.4.m1.1.1" xref="S4.SS1.p1.4.m1.1.1.cmml">p</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.p1.4.m1.1b"><ci id="S4.SS1.p1.4.m1.1.1.cmml" xref="S4.SS1.p1.4.m1.1.1">𝑝</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p1.4.m1.1c">p</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p1.4.m1.1d">italic_p</annotation></semantics></math>. Then the Legendre transformation of <math alttext="\mathcal{G}" class="ltx_Math" display="inline" id="S4.SS1.p1.5.m2.1"><semantics id="S4.SS1.p1.5.m2.1a"><mi class="ltx_font_mathcaligraphic" id="S4.SS1.p1.5.m2.1.1" xref="S4.SS1.p1.5.m2.1.1.cmml">𝒢</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.p1.5.m2.1b"><ci id="S4.SS1.p1.5.m2.1.1.cmml" xref="S4.SS1.p1.5.m2.1.1">𝒢</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p1.5.m2.1c">\mathcal{G}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p1.5.m2.1d">caligraphic_G</annotation></semantics></math> leads to the Lagrangian</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="S3.EGx69"> <tbody id="S4.E75"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_left_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle L=\frac{1}{2}\mathcal{F}^{2}," class="ltx_Math" display="inline" id="S4.E75.m1.1"><semantics id="S4.E75.m1.1a"><mrow id="S4.E75.m1.1.1.1" xref="S4.E75.m1.1.1.1.1.cmml"><mrow id="S4.E75.m1.1.1.1.1" xref="S4.E75.m1.1.1.1.1.cmml"><mi id="S4.E75.m1.1.1.1.1.2" xref="S4.E75.m1.1.1.1.1.2.cmml">L</mi><mo id="S4.E75.m1.1.1.1.1.1" xref="S4.E75.m1.1.1.1.1.1.cmml">=</mo><mrow id="S4.E75.m1.1.1.1.1.3" xref="S4.E75.m1.1.1.1.1.3.cmml"><mstyle displaystyle="true" id="S4.E75.m1.1.1.1.1.3.2" xref="S4.E75.m1.1.1.1.1.3.2.cmml"><mfrac id="S4.E75.m1.1.1.1.1.3.2a" xref="S4.E75.m1.1.1.1.1.3.2.cmml"><mn id="S4.E75.m1.1.1.1.1.3.2.2" xref="S4.E75.m1.1.1.1.1.3.2.2.cmml">1</mn><mn id="S4.E75.m1.1.1.1.1.3.2.3" xref="S4.E75.m1.1.1.1.1.3.2.3.cmml">2</mn></mfrac></mstyle><mo id="S4.E75.m1.1.1.1.1.3.1" xref="S4.E75.m1.1.1.1.1.3.1.cmml">⁢</mo><msup id="S4.E75.m1.1.1.1.1.3.3" xref="S4.E75.m1.1.1.1.1.3.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.E75.m1.1.1.1.1.3.3.2" xref="S4.E75.m1.1.1.1.1.3.3.2.cmml">ℱ</mi><mn id="S4.E75.m1.1.1.1.1.3.3.3" xref="S4.E75.m1.1.1.1.1.3.3.3.cmml">2</mn></msup></mrow></mrow><mo id="S4.E75.m1.1.1.1.2" xref="S4.E75.m1.1.1.1.1.cmml">,</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.E75.m1.1b"><apply id="S4.E75.m1.1.1.1.1.cmml" xref="S4.E75.m1.1.1.1"><eq id="S4.E75.m1.1.1.1.1.1.cmml" xref="S4.E75.m1.1.1.1.1.1"></eq><ci id="S4.E75.m1.1.1.1.1.2.cmml" xref="S4.E75.m1.1.1.1.1.2">𝐿</ci><apply id="S4.E75.m1.1.1.1.1.3.cmml" xref="S4.E75.m1.1.1.1.1.3"><times id="S4.E75.m1.1.1.1.1.3.1.cmml" xref="S4.E75.m1.1.1.1.1.3.1"></times><apply id="S4.E75.m1.1.1.1.1.3.2.cmml" xref="S4.E75.m1.1.1.1.1.3.2"><divide id="S4.E75.m1.1.1.1.1.3.2.1.cmml" xref="S4.E75.m1.1.1.1.1.3.2"></divide><cn id="S4.E75.m1.1.1.1.1.3.2.2.cmml" type="integer" xref="S4.E75.m1.1.1.1.1.3.2.2">1</cn><cn id="S4.E75.m1.1.1.1.1.3.2.3.cmml" type="integer" xref="S4.E75.m1.1.1.1.1.3.2.3">2</cn></apply><apply id="S4.E75.m1.1.1.1.1.3.3.cmml" xref="S4.E75.m1.1.1.1.1.3.3"><csymbol cd="ambiguous" id="S4.E75.m1.1.1.1.1.3.3.1.cmml" xref="S4.E75.m1.1.1.1.1.3.3">superscript</csymbol><ci id="S4.E75.m1.1.1.1.1.3.3.2.cmml" xref="S4.E75.m1.1.1.1.1.3.3.2">ℱ</ci><cn id="S4.E75.m1.1.1.1.1.3.3.3.cmml" type="integer" xref="S4.E75.m1.1.1.1.1.3.3.3">2</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.E75.m1.1c">\displaystyle L=\frac{1}{2}\mathcal{F}^{2},</annotation><annotation encoding="application/x-llamapun" id="S4.E75.m1.1d">italic_L = divide start_ARG 1 end_ARG start_ARG 2 end_ARG caligraphic_F start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_left_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(75)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S4.SS1.p1.9">where <math alttext="\mathcal{F}=\mathcal{F}(x,v)" class="ltx_Math" display="inline" id="S4.SS1.p1.6.m1.2"><semantics id="S4.SS1.p1.6.m1.2a"><mrow id="S4.SS1.p1.6.m1.2.3" xref="S4.SS1.p1.6.m1.2.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.SS1.p1.6.m1.2.3.2" xref="S4.SS1.p1.6.m1.2.3.2.cmml">ℱ</mi><mo id="S4.SS1.p1.6.m1.2.3.1" xref="S4.SS1.p1.6.m1.2.3.1.cmml">=</mo><mrow id="S4.SS1.p1.6.m1.2.3.3" xref="S4.SS1.p1.6.m1.2.3.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.SS1.p1.6.m1.2.3.3.2" xref="S4.SS1.p1.6.m1.2.3.3.2.cmml">ℱ</mi><mo id="S4.SS1.p1.6.m1.2.3.3.1" xref="S4.SS1.p1.6.m1.2.3.3.1.cmml">⁢</mo><mrow id="S4.SS1.p1.6.m1.2.3.3.3.2" xref="S4.SS1.p1.6.m1.2.3.3.3.1.cmml"><mo id="S4.SS1.p1.6.m1.2.3.3.3.2.1" stretchy="false" xref="S4.SS1.p1.6.m1.2.3.3.3.1.cmml">(</mo><mi id="S4.SS1.p1.6.m1.1.1" xref="S4.SS1.p1.6.m1.1.1.cmml">x</mi><mo id="S4.SS1.p1.6.m1.2.3.3.3.2.2" xref="S4.SS1.p1.6.m1.2.3.3.3.1.cmml">,</mo><mi id="S4.SS1.p1.6.m1.2.2" xref="S4.SS1.p1.6.m1.2.2.cmml">v</mi><mo id="S4.SS1.p1.6.m1.2.3.3.3.2.3" stretchy="false" xref="S4.SS1.p1.6.m1.2.3.3.3.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p1.6.m1.2b"><apply id="S4.SS1.p1.6.m1.2.3.cmml" xref="S4.SS1.p1.6.m1.2.3"><eq id="S4.SS1.p1.6.m1.2.3.1.cmml" xref="S4.SS1.p1.6.m1.2.3.1"></eq><ci id="S4.SS1.p1.6.m1.2.3.2.cmml" xref="S4.SS1.p1.6.m1.2.3.2">ℱ</ci><apply id="S4.SS1.p1.6.m1.2.3.3.cmml" xref="S4.SS1.p1.6.m1.2.3.3"><times id="S4.SS1.p1.6.m1.2.3.3.1.cmml" xref="S4.SS1.p1.6.m1.2.3.3.1"></times><ci id="S4.SS1.p1.6.m1.2.3.3.2.cmml" xref="S4.SS1.p1.6.m1.2.3.3.2">ℱ</ci><interval closure="open" id="S4.SS1.p1.6.m1.2.3.3.3.1.cmml" xref="S4.SS1.p1.6.m1.2.3.3.3.2"><ci id="S4.SS1.p1.6.m1.1.1.cmml" xref="S4.SS1.p1.6.m1.1.1">𝑥</ci><ci id="S4.SS1.p1.6.m1.2.2.cmml" xref="S4.SS1.p1.6.m1.2.2">𝑣</ci></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p1.6.m1.2c">\mathcal{F}=\mathcal{F}(x,v)</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p1.6.m1.2d">caligraphic_F = caligraphic_F ( italic_x , italic_v )</annotation></semantics></math> is a homogeneous function of first order in velocities <math alttext="v^{i}" class="ltx_Math" display="inline" id="S4.SS1.p1.7.m2.1"><semantics id="S4.SS1.p1.7.m2.1a"><msup id="S4.SS1.p1.7.m2.1.1" xref="S4.SS1.p1.7.m2.1.1.cmml"><mi id="S4.SS1.p1.7.m2.1.1.2" xref="S4.SS1.p1.7.m2.1.1.2.cmml">v</mi><mi id="S4.SS1.p1.7.m2.1.1.3" xref="S4.SS1.p1.7.m2.1.1.3.cmml">i</mi></msup><annotation-xml encoding="MathML-Content" id="S4.SS1.p1.7.m2.1b"><apply id="S4.SS1.p1.7.m2.1.1.cmml" xref="S4.SS1.p1.7.m2.1.1"><csymbol cd="ambiguous" id="S4.SS1.p1.7.m2.1.1.1.cmml" xref="S4.SS1.p1.7.m2.1.1">superscript</csymbol><ci id="S4.SS1.p1.7.m2.1.1.2.cmml" xref="S4.SS1.p1.7.m2.1.1.2">𝑣</ci><ci id="S4.SS1.p1.7.m2.1.1.3.cmml" xref="S4.SS1.p1.7.m2.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p1.7.m2.1c">v^{i}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p1.7.m2.1d">italic_v start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT</annotation></semantics></math>, i.e., <math alttext="\mathcal{F}=v^{i}\partial\mathcal{F}/\partial v^{i}" class="ltx_Math" display="inline" id="S4.SS1.p1.8.m3.1"><semantics id="S4.SS1.p1.8.m3.1a"><mrow id="S4.SS1.p1.8.m3.1.1" xref="S4.SS1.p1.8.m3.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.SS1.p1.8.m3.1.1.2" xref="S4.SS1.p1.8.m3.1.1.2.cmml">ℱ</mi><mo id="S4.SS1.p1.8.m3.1.1.1" xref="S4.SS1.p1.8.m3.1.1.1.cmml">=</mo><mrow id="S4.SS1.p1.8.m3.1.1.3" xref="S4.SS1.p1.8.m3.1.1.3.cmml"><msup id="S4.SS1.p1.8.m3.1.1.3.2" xref="S4.SS1.p1.8.m3.1.1.3.2.cmml"><mi id="S4.SS1.p1.8.m3.1.1.3.2.2" xref="S4.SS1.p1.8.m3.1.1.3.2.2.cmml">v</mi><mi id="S4.SS1.p1.8.m3.1.1.3.2.3" xref="S4.SS1.p1.8.m3.1.1.3.2.3.cmml">i</mi></msup><mo id="S4.SS1.p1.8.m3.1.1.3.1" lspace="0em" xref="S4.SS1.p1.8.m3.1.1.3.1.cmml">⁢</mo><mrow id="S4.SS1.p1.8.m3.1.1.3.3" xref="S4.SS1.p1.8.m3.1.1.3.3.cmml"><mo id="S4.SS1.p1.8.m3.1.1.3.3.1" rspace="0em" xref="S4.SS1.p1.8.m3.1.1.3.3.1.cmml">∂</mo><mrow id="S4.SS1.p1.8.m3.1.1.3.3.2" xref="S4.SS1.p1.8.m3.1.1.3.3.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.SS1.p1.8.m3.1.1.3.3.2.2" xref="S4.SS1.p1.8.m3.1.1.3.3.2.2.cmml">ℱ</mi><mo id="S4.SS1.p1.8.m3.1.1.3.3.2.1" xref="S4.SS1.p1.8.m3.1.1.3.3.2.1.cmml">/</mo><mrow id="S4.SS1.p1.8.m3.1.1.3.3.2.3" xref="S4.SS1.p1.8.m3.1.1.3.3.2.3.cmml"><mo id="S4.SS1.p1.8.m3.1.1.3.3.2.3.1" lspace="0em" rspace="0em" xref="S4.SS1.p1.8.m3.1.1.3.3.2.3.1.cmml">∂</mo><msup id="S4.SS1.p1.8.m3.1.1.3.3.2.3.2" xref="S4.SS1.p1.8.m3.1.1.3.3.2.3.2.cmml"><mi id="S4.SS1.p1.8.m3.1.1.3.3.2.3.2.2" xref="S4.SS1.p1.8.m3.1.1.3.3.2.3.2.2.cmml">v</mi><mi id="S4.SS1.p1.8.m3.1.1.3.3.2.3.2.3" xref="S4.SS1.p1.8.m3.1.1.3.3.2.3.2.3.cmml">i</mi></msup></mrow></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p1.8.m3.1b"><apply id="S4.SS1.p1.8.m3.1.1.cmml" xref="S4.SS1.p1.8.m3.1.1"><eq id="S4.SS1.p1.8.m3.1.1.1.cmml" xref="S4.SS1.p1.8.m3.1.1.1"></eq><ci id="S4.SS1.p1.8.m3.1.1.2.cmml" xref="S4.SS1.p1.8.m3.1.1.2">ℱ</ci><apply id="S4.SS1.p1.8.m3.1.1.3.cmml" xref="S4.SS1.p1.8.m3.1.1.3"><times id="S4.SS1.p1.8.m3.1.1.3.1.cmml" xref="S4.SS1.p1.8.m3.1.1.3.1"></times><apply id="S4.SS1.p1.8.m3.1.1.3.2.cmml" xref="S4.SS1.p1.8.m3.1.1.3.2"><csymbol cd="ambiguous" id="S4.SS1.p1.8.m3.1.1.3.2.1.cmml" xref="S4.SS1.p1.8.m3.1.1.3.2">superscript</csymbol><ci id="S4.SS1.p1.8.m3.1.1.3.2.2.cmml" xref="S4.SS1.p1.8.m3.1.1.3.2.2">𝑣</ci><ci id="S4.SS1.p1.8.m3.1.1.3.2.3.cmml" xref="S4.SS1.p1.8.m3.1.1.3.2.3">𝑖</ci></apply><apply id="S4.SS1.p1.8.m3.1.1.3.3.cmml" xref="S4.SS1.p1.8.m3.1.1.3.3"><partialdiff id="S4.SS1.p1.8.m3.1.1.3.3.1.cmml" xref="S4.SS1.p1.8.m3.1.1.3.3.1"></partialdiff><apply id="S4.SS1.p1.8.m3.1.1.3.3.2.cmml" xref="S4.SS1.p1.8.m3.1.1.3.3.2"><divide id="S4.SS1.p1.8.m3.1.1.3.3.2.1.cmml" xref="S4.SS1.p1.8.m3.1.1.3.3.2.1"></divide><ci id="S4.SS1.p1.8.m3.1.1.3.3.2.2.cmml" xref="S4.SS1.p1.8.m3.1.1.3.3.2.2">ℱ</ci><apply id="S4.SS1.p1.8.m3.1.1.3.3.2.3.cmml" xref="S4.SS1.p1.8.m3.1.1.3.3.2.3"><partialdiff id="S4.SS1.p1.8.m3.1.1.3.3.2.3.1.cmml" xref="S4.SS1.p1.8.m3.1.1.3.3.2.3.1"></partialdiff><apply id="S4.SS1.p1.8.m3.1.1.3.3.2.3.2.cmml" xref="S4.SS1.p1.8.m3.1.1.3.3.2.3.2"><csymbol cd="ambiguous" id="S4.SS1.p1.8.m3.1.1.3.3.2.3.2.1.cmml" xref="S4.SS1.p1.8.m3.1.1.3.3.2.3.2">superscript</csymbol><ci id="S4.SS1.p1.8.m3.1.1.3.3.2.3.2.2.cmml" xref="S4.SS1.p1.8.m3.1.1.3.3.2.3.2.2">𝑣</ci><ci id="S4.SS1.p1.8.m3.1.1.3.3.2.3.2.3.cmml" xref="S4.SS1.p1.8.m3.1.1.3.3.2.3.2.3">𝑖</ci></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p1.8.m3.1c">\mathcal{F}=v^{i}\partial\mathcal{F}/\partial v^{i}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p1.8.m3.1d">caligraphic_F = italic_v start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT ∂ caligraphic_F / ∂ italic_v start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT</annotation></semantics></math>. Since <math alttext="p_{i}=\partial L/\partial v^{i}" class="ltx_Math" display="inline" id="S4.SS1.p1.9.m4.1"><semantics id="S4.SS1.p1.9.m4.1a"><mrow id="S4.SS1.p1.9.m4.1.1" xref="S4.SS1.p1.9.m4.1.1.cmml"><msub id="S4.SS1.p1.9.m4.1.1.2" xref="S4.SS1.p1.9.m4.1.1.2.cmml"><mi id="S4.SS1.p1.9.m4.1.1.2.2" xref="S4.SS1.p1.9.m4.1.1.2.2.cmml">p</mi><mi id="S4.SS1.p1.9.m4.1.1.2.3" xref="S4.SS1.p1.9.m4.1.1.2.3.cmml">i</mi></msub><mo id="S4.SS1.p1.9.m4.1.1.1" rspace="0.1389em" xref="S4.SS1.p1.9.m4.1.1.1.cmml">=</mo><mrow id="S4.SS1.p1.9.m4.1.1.3" xref="S4.SS1.p1.9.m4.1.1.3.cmml"><mo id="S4.SS1.p1.9.m4.1.1.3.1" lspace="0.1389em" rspace="0em" xref="S4.SS1.p1.9.m4.1.1.3.1.cmml">∂</mo><mrow id="S4.SS1.p1.9.m4.1.1.3.2" xref="S4.SS1.p1.9.m4.1.1.3.2.cmml"><mi id="S4.SS1.p1.9.m4.1.1.3.2.2" xref="S4.SS1.p1.9.m4.1.1.3.2.2.cmml">L</mi><mo id="S4.SS1.p1.9.m4.1.1.3.2.1" xref="S4.SS1.p1.9.m4.1.1.3.2.1.cmml">/</mo><mrow id="S4.SS1.p1.9.m4.1.1.3.2.3" xref="S4.SS1.p1.9.m4.1.1.3.2.3.cmml"><mo id="S4.SS1.p1.9.m4.1.1.3.2.3.1" lspace="0em" rspace="0em" xref="S4.SS1.p1.9.m4.1.1.3.2.3.1.cmml">∂</mo><msup id="S4.SS1.p1.9.m4.1.1.3.2.3.2" xref="S4.SS1.p1.9.m4.1.1.3.2.3.2.cmml"><mi id="S4.SS1.p1.9.m4.1.1.3.2.3.2.2" xref="S4.SS1.p1.9.m4.1.1.3.2.3.2.2.cmml">v</mi><mi id="S4.SS1.p1.9.m4.1.1.3.2.3.2.3" xref="S4.SS1.p1.9.m4.1.1.3.2.3.2.3.cmml">i</mi></msup></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p1.9.m4.1b"><apply id="S4.SS1.p1.9.m4.1.1.cmml" xref="S4.SS1.p1.9.m4.1.1"><eq id="S4.SS1.p1.9.m4.1.1.1.cmml" xref="S4.SS1.p1.9.m4.1.1.1"></eq><apply id="S4.SS1.p1.9.m4.1.1.2.cmml" xref="S4.SS1.p1.9.m4.1.1.2"><csymbol cd="ambiguous" id="S4.SS1.p1.9.m4.1.1.2.1.cmml" xref="S4.SS1.p1.9.m4.1.1.2">subscript</csymbol><ci id="S4.SS1.p1.9.m4.1.1.2.2.cmml" xref="S4.SS1.p1.9.m4.1.1.2.2">𝑝</ci><ci id="S4.SS1.p1.9.m4.1.1.2.3.cmml" xref="S4.SS1.p1.9.m4.1.1.2.3">𝑖</ci></apply><apply id="S4.SS1.p1.9.m4.1.1.3.cmml" xref="S4.SS1.p1.9.m4.1.1.3"><partialdiff id="S4.SS1.p1.9.m4.1.1.3.1.cmml" xref="S4.SS1.p1.9.m4.1.1.3.1"></partialdiff><apply id="S4.SS1.p1.9.m4.1.1.3.2.cmml" xref="S4.SS1.p1.9.m4.1.1.3.2"><divide id="S4.SS1.p1.9.m4.1.1.3.2.1.cmml" xref="S4.SS1.p1.9.m4.1.1.3.2.1"></divide><ci id="S4.SS1.p1.9.m4.1.1.3.2.2.cmml" xref="S4.SS1.p1.9.m4.1.1.3.2.2">𝐿</ci><apply id="S4.SS1.p1.9.m4.1.1.3.2.3.cmml" xref="S4.SS1.p1.9.m4.1.1.3.2.3"><partialdiff id="S4.SS1.p1.9.m4.1.1.3.2.3.1.cmml" xref="S4.SS1.p1.9.m4.1.1.3.2.3.1"></partialdiff><apply id="S4.SS1.p1.9.m4.1.1.3.2.3.2.cmml" xref="S4.SS1.p1.9.m4.1.1.3.2.3.2"><csymbol cd="ambiguous" id="S4.SS1.p1.9.m4.1.1.3.2.3.2.1.cmml" xref="S4.SS1.p1.9.m4.1.1.3.2.3.2">superscript</csymbol><ci id="S4.SS1.p1.9.m4.1.1.3.2.3.2.2.cmml" xref="S4.SS1.p1.9.m4.1.1.3.2.3.2.2">𝑣</ci><ci id="S4.SS1.p1.9.m4.1.1.3.2.3.2.3.cmml" xref="S4.SS1.p1.9.m4.1.1.3.2.3.2.3">𝑖</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p1.9.m4.1c">p_{i}=\partial L/\partial v^{i}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p1.9.m4.1d">italic_p start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = ∂ italic_L / ∂ italic_v start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT</annotation></semantics></math> and</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="S3.EGx70"> <tbody id="S4.E76"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_left_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle v^{i}p_{i}=v^{i}\frac{\partial L}{\partial v^{i}}=\mathcal{F}v^{% i}\frac{\partial\mathcal{F}}{\partial v_{i}}=\mathcal{F}^{2}," class="ltx_Math" display="inline" id="S4.E76.m1.1"><semantics id="S4.E76.m1.1a"><mrow id="S4.E76.m1.1.1.1" xref="S4.E76.m1.1.1.1.1.cmml"><mrow id="S4.E76.m1.1.1.1.1" xref="S4.E76.m1.1.1.1.1.cmml"><mrow id="S4.E76.m1.1.1.1.1.2" xref="S4.E76.m1.1.1.1.1.2.cmml"><msup id="S4.E76.m1.1.1.1.1.2.2" xref="S4.E76.m1.1.1.1.1.2.2.cmml"><mi id="S4.E76.m1.1.1.1.1.2.2.2" xref="S4.E76.m1.1.1.1.1.2.2.2.cmml">v</mi><mi id="S4.E76.m1.1.1.1.1.2.2.3" xref="S4.E76.m1.1.1.1.1.2.2.3.cmml">i</mi></msup><mo id="S4.E76.m1.1.1.1.1.2.1" xref="S4.E76.m1.1.1.1.1.2.1.cmml">⁢</mo><msub id="S4.E76.m1.1.1.1.1.2.3" xref="S4.E76.m1.1.1.1.1.2.3.cmml"><mi id="S4.E76.m1.1.1.1.1.2.3.2" xref="S4.E76.m1.1.1.1.1.2.3.2.cmml">p</mi><mi id="S4.E76.m1.1.1.1.1.2.3.3" xref="S4.E76.m1.1.1.1.1.2.3.3.cmml">i</mi></msub></mrow><mo id="S4.E76.m1.1.1.1.1.3" xref="S4.E76.m1.1.1.1.1.3.cmml">=</mo><mrow id="S4.E76.m1.1.1.1.1.4" xref="S4.E76.m1.1.1.1.1.4.cmml"><msup id="S4.E76.m1.1.1.1.1.4.2" xref="S4.E76.m1.1.1.1.1.4.2.cmml"><mi id="S4.E76.m1.1.1.1.1.4.2.2" xref="S4.E76.m1.1.1.1.1.4.2.2.cmml">v</mi><mi id="S4.E76.m1.1.1.1.1.4.2.3" xref="S4.E76.m1.1.1.1.1.4.2.3.cmml">i</mi></msup><mo id="S4.E76.m1.1.1.1.1.4.1" xref="S4.E76.m1.1.1.1.1.4.1.cmml">⁢</mo><mstyle displaystyle="true" id="S4.E76.m1.1.1.1.1.4.3" xref="S4.E76.m1.1.1.1.1.4.3.cmml"><mfrac id="S4.E76.m1.1.1.1.1.4.3a" xref="S4.E76.m1.1.1.1.1.4.3.cmml"><mrow id="S4.E76.m1.1.1.1.1.4.3.2" xref="S4.E76.m1.1.1.1.1.4.3.2.cmml"><mo id="S4.E76.m1.1.1.1.1.4.3.2.1" rspace="0em" xref="S4.E76.m1.1.1.1.1.4.3.2.1.cmml">∂</mo><mi id="S4.E76.m1.1.1.1.1.4.3.2.2" xref="S4.E76.m1.1.1.1.1.4.3.2.2.cmml">L</mi></mrow><mrow id="S4.E76.m1.1.1.1.1.4.3.3" xref="S4.E76.m1.1.1.1.1.4.3.3.cmml"><mo id="S4.E76.m1.1.1.1.1.4.3.3.1" rspace="0em" xref="S4.E76.m1.1.1.1.1.4.3.3.1.cmml">∂</mo><msup id="S4.E76.m1.1.1.1.1.4.3.3.2" xref="S4.E76.m1.1.1.1.1.4.3.3.2.cmml"><mi id="S4.E76.m1.1.1.1.1.4.3.3.2.2" xref="S4.E76.m1.1.1.1.1.4.3.3.2.2.cmml">v</mi><mi id="S4.E76.m1.1.1.1.1.4.3.3.2.3" xref="S4.E76.m1.1.1.1.1.4.3.3.2.3.cmml">i</mi></msup></mrow></mfrac></mstyle></mrow><mo id="S4.E76.m1.1.1.1.1.5" xref="S4.E76.m1.1.1.1.1.5.cmml">=</mo><mrow id="S4.E76.m1.1.1.1.1.6" xref="S4.E76.m1.1.1.1.1.6.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.E76.m1.1.1.1.1.6.2" xref="S4.E76.m1.1.1.1.1.6.2.cmml">ℱ</mi><mo id="S4.E76.m1.1.1.1.1.6.1" xref="S4.E76.m1.1.1.1.1.6.1.cmml">⁢</mo><msup id="S4.E76.m1.1.1.1.1.6.3" xref="S4.E76.m1.1.1.1.1.6.3.cmml"><mi id="S4.E76.m1.1.1.1.1.6.3.2" xref="S4.E76.m1.1.1.1.1.6.3.2.cmml">v</mi><mi id="S4.E76.m1.1.1.1.1.6.3.3" xref="S4.E76.m1.1.1.1.1.6.3.3.cmml">i</mi></msup><mo id="S4.E76.m1.1.1.1.1.6.1a" xref="S4.E76.m1.1.1.1.1.6.1.cmml">⁢</mo><mstyle displaystyle="true" id="S4.E76.m1.1.1.1.1.6.4" xref="S4.E76.m1.1.1.1.1.6.4.cmml"><mfrac id="S4.E76.m1.1.1.1.1.6.4a" xref="S4.E76.m1.1.1.1.1.6.4.cmml"><mrow id="S4.E76.m1.1.1.1.1.6.4.2" xref="S4.E76.m1.1.1.1.1.6.4.2.cmml"><mo id="S4.E76.m1.1.1.1.1.6.4.2.1" rspace="0em" xref="S4.E76.m1.1.1.1.1.6.4.2.1.cmml">∂</mo><mi class="ltx_font_mathcaligraphic" id="S4.E76.m1.1.1.1.1.6.4.2.2" xref="S4.E76.m1.1.1.1.1.6.4.2.2.cmml">ℱ</mi></mrow><mrow id="S4.E76.m1.1.1.1.1.6.4.3" xref="S4.E76.m1.1.1.1.1.6.4.3.cmml"><mo id="S4.E76.m1.1.1.1.1.6.4.3.1" rspace="0em" xref="S4.E76.m1.1.1.1.1.6.4.3.1.cmml">∂</mo><msub 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encoding="application/x-llamapun" id="S4.E76.m1.1d">italic_v start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT italic_p start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = italic_v start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT divide start_ARG ∂ italic_L end_ARG start_ARG ∂ italic_v start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT end_ARG = caligraphic_F italic_v start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT divide start_ARG ∂ caligraphic_F end_ARG start_ARG ∂ italic_v start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_ARG = caligraphic_F start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_left_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(76)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S4.SS1.p1.11">one readily find that</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="S3.EGx71"> <tbody id="S4.E77"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_left_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\mathcal{G}=p_{i}v^{i}-L=\mathcal{F}^{2}-\frac{1}{2}\mathcal{F}^{% 2}=L." class="ltx_Math" display="inline" id="S4.E77.m1.1"><semantics id="S4.E77.m1.1a"><mrow id="S4.E77.m1.1.1.1" xref="S4.E77.m1.1.1.1.1.cmml"><mrow id="S4.E77.m1.1.1.1.1" xref="S4.E77.m1.1.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.E77.m1.1.1.1.1.2" xref="S4.E77.m1.1.1.1.1.2.cmml">𝒢</mi><mo id="S4.E77.m1.1.1.1.1.3" xref="S4.E77.m1.1.1.1.1.3.cmml">=</mo><mrow id="S4.E77.m1.1.1.1.1.4" xref="S4.E77.m1.1.1.1.1.4.cmml"><mrow id="S4.E77.m1.1.1.1.1.4.2" xref="S4.E77.m1.1.1.1.1.4.2.cmml"><msub id="S4.E77.m1.1.1.1.1.4.2.2" xref="S4.E77.m1.1.1.1.1.4.2.2.cmml"><mi id="S4.E77.m1.1.1.1.1.4.2.2.2" xref="S4.E77.m1.1.1.1.1.4.2.2.2.cmml">p</mi><mi id="S4.E77.m1.1.1.1.1.4.2.2.3" xref="S4.E77.m1.1.1.1.1.4.2.2.3.cmml">i</mi></msub><mo id="S4.E77.m1.1.1.1.1.4.2.1" xref="S4.E77.m1.1.1.1.1.4.2.1.cmml">⁢</mo><msup id="S4.E77.m1.1.1.1.1.4.2.3" xref="S4.E77.m1.1.1.1.1.4.2.3.cmml"><mi id="S4.E77.m1.1.1.1.1.4.2.3.2" xref="S4.E77.m1.1.1.1.1.4.2.3.2.cmml">v</mi><mi id="S4.E77.m1.1.1.1.1.4.2.3.3" xref="S4.E77.m1.1.1.1.1.4.2.3.3.cmml">i</mi></msup></mrow><mo id="S4.E77.m1.1.1.1.1.4.1" xref="S4.E77.m1.1.1.1.1.4.1.cmml">−</mo><mi id="S4.E77.m1.1.1.1.1.4.3" xref="S4.E77.m1.1.1.1.1.4.3.cmml">L</mi></mrow><mo id="S4.E77.m1.1.1.1.1.5" xref="S4.E77.m1.1.1.1.1.5.cmml">=</mo><mrow id="S4.E77.m1.1.1.1.1.6" xref="S4.E77.m1.1.1.1.1.6.cmml"><msup id="S4.E77.m1.1.1.1.1.6.2" xref="S4.E77.m1.1.1.1.1.6.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.E77.m1.1.1.1.1.6.2.2" xref="S4.E77.m1.1.1.1.1.6.2.2.cmml">ℱ</mi><mn id="S4.E77.m1.1.1.1.1.6.2.3" xref="S4.E77.m1.1.1.1.1.6.2.3.cmml">2</mn></msup><mo id="S4.E77.m1.1.1.1.1.6.1" xref="S4.E77.m1.1.1.1.1.6.1.cmml">−</mo><mrow id="S4.E77.m1.1.1.1.1.6.3" xref="S4.E77.m1.1.1.1.1.6.3.cmml"><mstyle displaystyle="true" id="S4.E77.m1.1.1.1.1.6.3.2" xref="S4.E77.m1.1.1.1.1.6.3.2.cmml"><mfrac id="S4.E77.m1.1.1.1.1.6.3.2a" xref="S4.E77.m1.1.1.1.1.6.3.2.cmml"><mn id="S4.E77.m1.1.1.1.1.6.3.2.2" xref="S4.E77.m1.1.1.1.1.6.3.2.2.cmml">1</mn><mn id="S4.E77.m1.1.1.1.1.6.3.2.3" xref="S4.E77.m1.1.1.1.1.6.3.2.3.cmml">2</mn></mfrac></mstyle><mo id="S4.E77.m1.1.1.1.1.6.3.1" xref="S4.E77.m1.1.1.1.1.6.3.1.cmml">⁢</mo><msup id="S4.E77.m1.1.1.1.1.6.3.3" xref="S4.E77.m1.1.1.1.1.6.3.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.E77.m1.1.1.1.1.6.3.3.2" xref="S4.E77.m1.1.1.1.1.6.3.3.2.cmml">ℱ</mi><mn id="S4.E77.m1.1.1.1.1.6.3.3.3" xref="S4.E77.m1.1.1.1.1.6.3.3.3.cmml">2</mn></msup></mrow></mrow><mo id="S4.E77.m1.1.1.1.1.7" xref="S4.E77.m1.1.1.1.1.7.cmml">=</mo><mi id="S4.E77.m1.1.1.1.1.8" xref="S4.E77.m1.1.1.1.1.8.cmml">L</mi></mrow><mo id="S4.E77.m1.1.1.1.2" lspace="0em" xref="S4.E77.m1.1.1.1.1.cmml">.</mo></mrow><annotation-xml 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id="S4.E77.m1.1.1.1.1.6.3.3.3.cmml" type="integer" xref="S4.E77.m1.1.1.1.1.6.3.3.3">2</cn></apply></apply></apply></apply><apply id="S4.E77.m1.1.1.1.1e.cmml" xref="S4.E77.m1.1.1.1"><eq id="S4.E77.m1.1.1.1.1.7.cmml" xref="S4.E77.m1.1.1.1.1.7"></eq><share href="https://arxiv.org/html/2406.11224v2#S4.E77.m1.1.1.1.1.6.cmml" id="S4.E77.m1.1.1.1.1f.cmml" xref="S4.E77.m1.1.1.1"></share><ci id="S4.E77.m1.1.1.1.1.8.cmml" xref="S4.E77.m1.1.1.1.1.8">𝐿</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.E77.m1.1c">\displaystyle\mathcal{G}=p_{i}v^{i}-L=\mathcal{F}^{2}-\frac{1}{2}\mathcal{F}^{% 2}=L.</annotation><annotation encoding="application/x-llamapun" id="S4.E77.m1.1d">caligraphic_G = italic_p start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_v start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT - italic_L = caligraphic_F start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT - divide start_ARG 1 end_ARG start_ARG 2 end_ARG caligraphic_F start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT = italic_L .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_left_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(77)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S4.SS1.p1.10">As a result we see that <math alttext="\mathcal{H}^{+}(x,p(v))=\mathcal{F}(x,v(p))" class="ltx_Math" display="inline" id="S4.SS1.p1.10.m1.6"><semantics id="S4.SS1.p1.10.m1.6a"><mrow id="S4.SS1.p1.10.m1.6.6" xref="S4.SS1.p1.10.m1.6.6.cmml"><mrow id="S4.SS1.p1.10.m1.5.5.1" xref="S4.SS1.p1.10.m1.5.5.1.cmml"><msup id="S4.SS1.p1.10.m1.5.5.1.3" xref="S4.SS1.p1.10.m1.5.5.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.SS1.p1.10.m1.5.5.1.3.2" xref="S4.SS1.p1.10.m1.5.5.1.3.2.cmml">ℋ</mi><mo id="S4.SS1.p1.10.m1.5.5.1.3.3" xref="S4.SS1.p1.10.m1.5.5.1.3.3.cmml">+</mo></msup><mo id="S4.SS1.p1.10.m1.5.5.1.2" xref="S4.SS1.p1.10.m1.5.5.1.2.cmml">⁢</mo><mrow id="S4.SS1.p1.10.m1.5.5.1.1.1" xref="S4.SS1.p1.10.m1.5.5.1.1.2.cmml"><mo id="S4.SS1.p1.10.m1.5.5.1.1.1.2" stretchy="false" xref="S4.SS1.p1.10.m1.5.5.1.1.2.cmml">(</mo><mi id="S4.SS1.p1.10.m1.2.2" xref="S4.SS1.p1.10.m1.2.2.cmml">x</mi><mo id="S4.SS1.p1.10.m1.5.5.1.1.1.3" xref="S4.SS1.p1.10.m1.5.5.1.1.2.cmml">,</mo><mrow id="S4.SS1.p1.10.m1.5.5.1.1.1.1" xref="S4.SS1.p1.10.m1.5.5.1.1.1.1.cmml"><mi id="S4.SS1.p1.10.m1.5.5.1.1.1.1.2" xref="S4.SS1.p1.10.m1.5.5.1.1.1.1.2.cmml">p</mi><mo id="S4.SS1.p1.10.m1.5.5.1.1.1.1.1" xref="S4.SS1.p1.10.m1.5.5.1.1.1.1.1.cmml">⁢</mo><mrow id="S4.SS1.p1.10.m1.5.5.1.1.1.1.3.2" xref="S4.SS1.p1.10.m1.5.5.1.1.1.1.cmml"><mo id="S4.SS1.p1.10.m1.5.5.1.1.1.1.3.2.1" stretchy="false" xref="S4.SS1.p1.10.m1.5.5.1.1.1.1.cmml">(</mo><mi id="S4.SS1.p1.10.m1.1.1" xref="S4.SS1.p1.10.m1.1.1.cmml">v</mi><mo id="S4.SS1.p1.10.m1.5.5.1.1.1.1.3.2.2" stretchy="false" xref="S4.SS1.p1.10.m1.5.5.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.SS1.p1.10.m1.5.5.1.1.1.4" stretchy="false" xref="S4.SS1.p1.10.m1.5.5.1.1.2.cmml">)</mo></mrow></mrow><mo id="S4.SS1.p1.10.m1.6.6.3" xref="S4.SS1.p1.10.m1.6.6.3.cmml">=</mo><mrow id="S4.SS1.p1.10.m1.6.6.2" xref="S4.SS1.p1.10.m1.6.6.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.SS1.p1.10.m1.6.6.2.3" xref="S4.SS1.p1.10.m1.6.6.2.3.cmml">ℱ</mi><mo id="S4.SS1.p1.10.m1.6.6.2.2" xref="S4.SS1.p1.10.m1.6.6.2.2.cmml">⁢</mo><mrow id="S4.SS1.p1.10.m1.6.6.2.1.1" xref="S4.SS1.p1.10.m1.6.6.2.1.2.cmml"><mo id="S4.SS1.p1.10.m1.6.6.2.1.1.2" stretchy="false" xref="S4.SS1.p1.10.m1.6.6.2.1.2.cmml">(</mo><mi id="S4.SS1.p1.10.m1.4.4" xref="S4.SS1.p1.10.m1.4.4.cmml">x</mi><mo id="S4.SS1.p1.10.m1.6.6.2.1.1.3" xref="S4.SS1.p1.10.m1.6.6.2.1.2.cmml">,</mo><mrow id="S4.SS1.p1.10.m1.6.6.2.1.1.1" xref="S4.SS1.p1.10.m1.6.6.2.1.1.1.cmml"><mi id="S4.SS1.p1.10.m1.6.6.2.1.1.1.2" xref="S4.SS1.p1.10.m1.6.6.2.1.1.1.2.cmml">v</mi><mo id="S4.SS1.p1.10.m1.6.6.2.1.1.1.1" xref="S4.SS1.p1.10.m1.6.6.2.1.1.1.1.cmml">⁢</mo><mrow id="S4.SS1.p1.10.m1.6.6.2.1.1.1.3.2" xref="S4.SS1.p1.10.m1.6.6.2.1.1.1.cmml"><mo id="S4.SS1.p1.10.m1.6.6.2.1.1.1.3.2.1" stretchy="false" xref="S4.SS1.p1.10.m1.6.6.2.1.1.1.cmml">(</mo><mi id="S4.SS1.p1.10.m1.3.3" xref="S4.SS1.p1.10.m1.3.3.cmml">p</mi><mo id="S4.SS1.p1.10.m1.6.6.2.1.1.1.3.2.2" stretchy="false" xref="S4.SS1.p1.10.m1.6.6.2.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.SS1.p1.10.m1.6.6.2.1.1.4" stretchy="false" xref="S4.SS1.p1.10.m1.6.6.2.1.2.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p1.10.m1.6b"><apply id="S4.SS1.p1.10.m1.6.6.cmml" xref="S4.SS1.p1.10.m1.6.6"><eq id="S4.SS1.p1.10.m1.6.6.3.cmml" xref="S4.SS1.p1.10.m1.6.6.3"></eq><apply id="S4.SS1.p1.10.m1.5.5.1.cmml" xref="S4.SS1.p1.10.m1.5.5.1"><times id="S4.SS1.p1.10.m1.5.5.1.2.cmml" xref="S4.SS1.p1.10.m1.5.5.1.2"></times><apply id="S4.SS1.p1.10.m1.5.5.1.3.cmml" xref="S4.SS1.p1.10.m1.5.5.1.3"><csymbol cd="ambiguous" id="S4.SS1.p1.10.m1.5.5.1.3.1.cmml" xref="S4.SS1.p1.10.m1.5.5.1.3">superscript</csymbol><ci id="S4.SS1.p1.10.m1.5.5.1.3.2.cmml" xref="S4.SS1.p1.10.m1.5.5.1.3.2">ℋ</ci><plus id="S4.SS1.p1.10.m1.5.5.1.3.3.cmml" xref="S4.SS1.p1.10.m1.5.5.1.3.3"></plus></apply><interval closure="open" id="S4.SS1.p1.10.m1.5.5.1.1.2.cmml" xref="S4.SS1.p1.10.m1.5.5.1.1.1"><ci id="S4.SS1.p1.10.m1.2.2.cmml" xref="S4.SS1.p1.10.m1.2.2">𝑥</ci><apply id="S4.SS1.p1.10.m1.5.5.1.1.1.1.cmml" xref="S4.SS1.p1.10.m1.5.5.1.1.1.1"><times id="S4.SS1.p1.10.m1.5.5.1.1.1.1.1.cmml" xref="S4.SS1.p1.10.m1.5.5.1.1.1.1.1"></times><ci id="S4.SS1.p1.10.m1.5.5.1.1.1.1.2.cmml" xref="S4.SS1.p1.10.m1.5.5.1.1.1.1.2">𝑝</ci><ci id="S4.SS1.p1.10.m1.1.1.cmml" xref="S4.SS1.p1.10.m1.1.1">𝑣</ci></apply></interval></apply><apply id="S4.SS1.p1.10.m1.6.6.2.cmml" xref="S4.SS1.p1.10.m1.6.6.2"><times id="S4.SS1.p1.10.m1.6.6.2.2.cmml" xref="S4.SS1.p1.10.m1.6.6.2.2"></times><ci id="S4.SS1.p1.10.m1.6.6.2.3.cmml" xref="S4.SS1.p1.10.m1.6.6.2.3">ℱ</ci><interval closure="open" id="S4.SS1.p1.10.m1.6.6.2.1.2.cmml" xref="S4.SS1.p1.10.m1.6.6.2.1.1"><ci id="S4.SS1.p1.10.m1.4.4.cmml" xref="S4.SS1.p1.10.m1.4.4">𝑥</ci><apply id="S4.SS1.p1.10.m1.6.6.2.1.1.1.cmml" xref="S4.SS1.p1.10.m1.6.6.2.1.1.1"><times id="S4.SS1.p1.10.m1.6.6.2.1.1.1.1.cmml" xref="S4.SS1.p1.10.m1.6.6.2.1.1.1.1"></times><ci id="S4.SS1.p1.10.m1.6.6.2.1.1.1.2.cmml" xref="S4.SS1.p1.10.m1.6.6.2.1.1.1.2">𝑣</ci><ci id="S4.SS1.p1.10.m1.3.3.cmml" xref="S4.SS1.p1.10.m1.3.3">𝑝</ci></apply></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p1.10.m1.6c">\mathcal{H}^{+}(x,p(v))=\mathcal{F}(x,v(p))</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p1.10.m1.6d">caligraphic_H start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT ( italic_x , italic_p ( italic_v ) ) = caligraphic_F ( italic_x , italic_v ( italic_p ) )</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S4.SS1.p2"> <p class="ltx_p" id="S4.SS1.p2.2">Next the expression of <math alttext="\mathcal{F}(x,v)" class="ltx_Math" display="inline" id="S4.SS1.p2.1.m1.2"><semantics id="S4.SS1.p2.1.m1.2a"><mrow id="S4.SS1.p2.1.m1.2.3" xref="S4.SS1.p2.1.m1.2.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.SS1.p2.1.m1.2.3.2" xref="S4.SS1.p2.1.m1.2.3.2.cmml">ℱ</mi><mo id="S4.SS1.p2.1.m1.2.3.1" xref="S4.SS1.p2.1.m1.2.3.1.cmml">⁢</mo><mrow id="S4.SS1.p2.1.m1.2.3.3.2" xref="S4.SS1.p2.1.m1.2.3.3.1.cmml"><mo id="S4.SS1.p2.1.m1.2.3.3.2.1" stretchy="false" xref="S4.SS1.p2.1.m1.2.3.3.1.cmml">(</mo><mi id="S4.SS1.p2.1.m1.1.1" xref="S4.SS1.p2.1.m1.1.1.cmml">x</mi><mo id="S4.SS1.p2.1.m1.2.3.3.2.2" xref="S4.SS1.p2.1.m1.2.3.3.1.cmml">,</mo><mi id="S4.SS1.p2.1.m1.2.2" xref="S4.SS1.p2.1.m1.2.2.cmml">v</mi><mo id="S4.SS1.p2.1.m1.2.3.3.2.3" stretchy="false" xref="S4.SS1.p2.1.m1.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p2.1.m1.2b"><apply id="S4.SS1.p2.1.m1.2.3.cmml" xref="S4.SS1.p2.1.m1.2.3"><times id="S4.SS1.p2.1.m1.2.3.1.cmml" xref="S4.SS1.p2.1.m1.2.3.1"></times><ci id="S4.SS1.p2.1.m1.2.3.2.cmml" xref="S4.SS1.p2.1.m1.2.3.2">ℱ</ci><interval closure="open" id="S4.SS1.p2.1.m1.2.3.3.1.cmml" xref="S4.SS1.p2.1.m1.2.3.3.2"><ci id="S4.SS1.p2.1.m1.1.1.cmml" xref="S4.SS1.p2.1.m1.1.1">𝑥</ci><ci id="S4.SS1.p2.1.m1.2.2.cmml" xref="S4.SS1.p2.1.m1.2.2">𝑣</ci></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p2.1.m1.2c">\mathcal{F}(x,v)</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p2.1.m1.2d">caligraphic_F ( italic_x , italic_v )</annotation></semantics></math> is obtained as follows. From Hamilton’s equations of motion for <math alttext="\mathcal{G}" class="ltx_Math" display="inline" id="S4.SS1.p2.2.m2.1"><semantics id="S4.SS1.p2.2.m2.1a"><mi class="ltx_font_mathcaligraphic" id="S4.SS1.p2.2.m2.1.1" xref="S4.SS1.p2.2.m2.1.1.cmml">𝒢</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.p2.2.m2.1b"><ci id="S4.SS1.p2.2.m2.1.1.cmml" xref="S4.SS1.p2.2.m2.1.1">𝒢</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p2.2.m2.1c">\mathcal{G}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p2.2.m2.1d">caligraphic_G</annotation></semantics></math>, we have</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="S3.EGx72"> <tbody id="S4.Ex3"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_left_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle v^{i}" class="ltx_Math" display="inline" id="S4.Ex3.m1.1"><semantics id="S4.Ex3.m1.1a"><msup id="S4.Ex3.m1.1.1" xref="S4.Ex3.m1.1.1.cmml"><mi id="S4.Ex3.m1.1.1.2" xref="S4.Ex3.m1.1.1.2.cmml">v</mi><mi id="S4.Ex3.m1.1.1.3" xref="S4.Ex3.m1.1.1.3.cmml">i</mi></msup><annotation-xml encoding="MathML-Content" id="S4.Ex3.m1.1b"><apply id="S4.Ex3.m1.1.1.cmml" xref="S4.Ex3.m1.1.1"><csymbol cd="ambiguous" id="S4.Ex3.m1.1.1.1.cmml" xref="S4.Ex3.m1.1.1">superscript</csymbol><ci id="S4.Ex3.m1.1.1.2.cmml" xref="S4.Ex3.m1.1.1.2">𝑣</ci><ci id="S4.Ex3.m1.1.1.3.cmml" xref="S4.Ex3.m1.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Ex3.m1.1c">\displaystyle v^{i}</annotation><annotation encoding="application/x-llamapun" id="S4.Ex3.m1.1d">italic_v start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle:=\frac{dx^{i}}{dx^{0}}=\frac{\partial\mathcal{G}}{\partial p_{i}% }=\mathcal{H}^{+}\,\frac{\partial\mathcal{H}^{+}}{\partial p_{i}}=\mathcal{H}^% {+}\,(-\beta^{i}+\nu^{i})," class="ltx_Math" display="inline" id="S4.Ex3.m2.1"><semantics id="S4.Ex3.m2.1a"><mrow id="S4.Ex3.m2.1.1.1" xref="S4.Ex3.m2.1.1.1.1.cmml"><mrow id="S4.Ex3.m2.1.1.1.1" xref="S4.Ex3.m2.1.1.1.1.cmml"><mi id="S4.Ex3.m2.1.1.1.1.3" xref="S4.Ex3.m2.1.1.1.1.3.cmml"></mi><mo id="S4.Ex3.m2.1.1.1.1.4" lspace="0.278em" rspace="0.278em" xref="S4.Ex3.m2.1.1.1.1.4.cmml">:=</mo><mstyle displaystyle="true" id="S4.Ex3.m2.1.1.1.1.5" xref="S4.Ex3.m2.1.1.1.1.5.cmml"><mfrac id="S4.Ex3.m2.1.1.1.1.5a" xref="S4.Ex3.m2.1.1.1.1.5.cmml"><mrow id="S4.Ex3.m2.1.1.1.1.5.2" xref="S4.Ex3.m2.1.1.1.1.5.2.cmml"><mi id="S4.Ex3.m2.1.1.1.1.5.2.2" xref="S4.Ex3.m2.1.1.1.1.5.2.2.cmml">d</mi><mo id="S4.Ex3.m2.1.1.1.1.5.2.1" xref="S4.Ex3.m2.1.1.1.1.5.2.1.cmml">⁢</mo><msup id="S4.Ex3.m2.1.1.1.1.5.2.3" xref="S4.Ex3.m2.1.1.1.1.5.2.3.cmml"><mi id="S4.Ex3.m2.1.1.1.1.5.2.3.2" xref="S4.Ex3.m2.1.1.1.1.5.2.3.2.cmml">x</mi><mi id="S4.Ex3.m2.1.1.1.1.5.2.3.3" 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id="S4.Ex3.m2.1d">:= divide start_ARG italic_d italic_x start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT end_ARG start_ARG italic_d italic_x start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT end_ARG = divide start_ARG ∂ caligraphic_G end_ARG start_ARG ∂ italic_p start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_ARG = caligraphic_H start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT divide start_ARG ∂ caligraphic_H start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT end_ARG start_ARG ∂ italic_p start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_ARG = caligraphic_H start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT ( - italic_β start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT + italic_ν start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT ) ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_left_padright"></td> </tr></tbody> <tbody id="S4.E78"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_left_padleft"></td> <td class="ltx_td ltx_eqn_cell"></td> <td 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xref="S4.E78.m1.1.1.1.1.3.2.3.2">𝑝</ci><ci id="S4.E78.m1.1.1.1.1.3.2.3.3.cmml" xref="S4.E78.m1.1.1.1.1.3.2.3.3">𝑗</ci></apply></apply><apply id="S4.E78.m1.1.1.1.1.3.3.cmml" xref="S4.E78.m1.1.1.1.1.3.3"><root id="S4.E78.m1.1.1.1.1.3.3a.cmml" xref="S4.E78.m1.1.1.1.1.3.3"></root><apply id="S4.E78.m1.1.1.1.1.3.3.2.cmml" xref="S4.E78.m1.1.1.1.1.3.3.2"><times id="S4.E78.m1.1.1.1.1.3.3.2.1.cmml" xref="S4.E78.m1.1.1.1.1.3.3.2.1"></times><apply id="S4.E78.m1.1.1.1.1.3.3.2.2.cmml" xref="S4.E78.m1.1.1.1.1.3.3.2.2"><csymbol cd="ambiguous" id="S4.E78.m1.1.1.1.1.3.3.2.2.1.cmml" xref="S4.E78.m1.1.1.1.1.3.3.2.2">superscript</csymbol><apply id="S4.E78.m1.1.1.1.1.3.3.2.2.2.cmml" xref="S4.E78.m1.1.1.1.1.3.3.2.2.2"><ci id="S4.E78.m1.1.1.1.1.3.3.2.2.2.1.cmml" xref="S4.E78.m1.1.1.1.1.3.3.2.2.2.1">~</ci><ci id="S4.E78.m1.1.1.1.1.3.3.2.2.2.2.cmml" xref="S4.E78.m1.1.1.1.1.3.3.2.2.2.2">𝛾</ci></apply><apply id="S4.E78.m1.1.1.1.1.3.3.2.2.3.cmml" xref="S4.E78.m1.1.1.1.1.3.3.2.2.3"><times id="S4.E78.m1.1.1.1.1.3.3.2.2.3.1.cmml" xref="S4.E78.m1.1.1.1.1.3.3.2.2.3.1"></times><ci id="S4.E78.m1.1.1.1.1.3.3.2.2.3.2.cmml" xref="S4.E78.m1.1.1.1.1.3.3.2.2.3.2">𝑘</ci><ci id="S4.E78.m1.1.1.1.1.3.3.2.2.3.3.cmml" xref="S4.E78.m1.1.1.1.1.3.3.2.2.3.3">ℓ</ci></apply></apply><apply id="S4.E78.m1.1.1.1.1.3.3.2.3.cmml" xref="S4.E78.m1.1.1.1.1.3.3.2.3"><csymbol cd="ambiguous" id="S4.E78.m1.1.1.1.1.3.3.2.3.1.cmml" xref="S4.E78.m1.1.1.1.1.3.3.2.3">subscript</csymbol><ci id="S4.E78.m1.1.1.1.1.3.3.2.3.2.cmml" xref="S4.E78.m1.1.1.1.1.3.3.2.3.2">𝑝</ci><ci id="S4.E78.m1.1.1.1.1.3.3.2.3.3.cmml" xref="S4.E78.m1.1.1.1.1.3.3.2.3.3">𝑘</ci></apply><apply id="S4.E78.m1.1.1.1.1.3.3.2.4.cmml" xref="S4.E78.m1.1.1.1.1.3.3.2.4"><csymbol cd="ambiguous" id="S4.E78.m1.1.1.1.1.3.3.2.4.1.cmml" xref="S4.E78.m1.1.1.1.1.3.3.2.4">subscript</csymbol><ci id="S4.E78.m1.1.1.1.1.3.3.2.4.2.cmml" xref="S4.E78.m1.1.1.1.1.3.3.2.4.2">𝑝</ci><ci id="S4.E78.m1.1.1.1.1.3.3.2.4.3.cmml" xref="S4.E78.m1.1.1.1.1.3.3.2.4.3">ℓ</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.E78.m1.1c">\displaystyle\textrm{ with }\nu^{i}=\frac{\tilde{\gamma}^{ij}p_{j}}{\sqrt{% \tilde{\gamma}^{k\ell}p_{k}p_{\ell}}},</annotation><annotation encoding="application/x-llamapun" id="S4.E78.m1.1d">with italic_ν start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT = divide start_ARG over~ start_ARG italic_γ end_ARG start_POSTSUPERSCRIPT italic_i italic_j end_POSTSUPERSCRIPT italic_p start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT end_ARG start_ARG square-root start_ARG over~ start_ARG italic_γ end_ARG start_POSTSUPERSCRIPT italic_k roman_ℓ end_POSTSUPERSCRIPT italic_p start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT italic_p start_POSTSUBSCRIPT roman_ℓ end_POSTSUBSCRIPT end_ARG end_ARG ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_left_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(78)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S4.SS1.p2.5">where <math alttext="\tilde{\gamma}^{ij}:=\alpha^{2}\gamma^{ij}" class="ltx_Math" display="inline" id="S4.SS1.p2.3.m1.1"><semantics id="S4.SS1.p2.3.m1.1a"><mrow id="S4.SS1.p2.3.m1.1.1" xref="S4.SS1.p2.3.m1.1.1.cmml"><msup id="S4.SS1.p2.3.m1.1.1.2" xref="S4.SS1.p2.3.m1.1.1.2.cmml"><mover accent="true" id="S4.SS1.p2.3.m1.1.1.2.2" xref="S4.SS1.p2.3.m1.1.1.2.2.cmml"><mi id="S4.SS1.p2.3.m1.1.1.2.2.2" xref="S4.SS1.p2.3.m1.1.1.2.2.2.cmml">γ</mi><mo id="S4.SS1.p2.3.m1.1.1.2.2.1" xref="S4.SS1.p2.3.m1.1.1.2.2.1.cmml">~</mo></mover><mrow id="S4.SS1.p2.3.m1.1.1.2.3" xref="S4.SS1.p2.3.m1.1.1.2.3.cmml"><mi id="S4.SS1.p2.3.m1.1.1.2.3.2" xref="S4.SS1.p2.3.m1.1.1.2.3.2.cmml">i</mi><mo id="S4.SS1.p2.3.m1.1.1.2.3.1" xref="S4.SS1.p2.3.m1.1.1.2.3.1.cmml">⁢</mo><mi id="S4.SS1.p2.3.m1.1.1.2.3.3" xref="S4.SS1.p2.3.m1.1.1.2.3.3.cmml">j</mi></mrow></msup><mo id="S4.SS1.p2.3.m1.1.1.1" lspace="0.278em" rspace="0.278em" xref="S4.SS1.p2.3.m1.1.1.1.cmml">:=</mo><mrow id="S4.SS1.p2.3.m1.1.1.3" xref="S4.SS1.p2.3.m1.1.1.3.cmml"><msup id="S4.SS1.p2.3.m1.1.1.3.2" xref="S4.SS1.p2.3.m1.1.1.3.2.cmml"><mi id="S4.SS1.p2.3.m1.1.1.3.2.2" xref="S4.SS1.p2.3.m1.1.1.3.2.2.cmml">α</mi><mn id="S4.SS1.p2.3.m1.1.1.3.2.3" xref="S4.SS1.p2.3.m1.1.1.3.2.3.cmml">2</mn></msup><mo id="S4.SS1.p2.3.m1.1.1.3.1" xref="S4.SS1.p2.3.m1.1.1.3.1.cmml">⁢</mo><msup id="S4.SS1.p2.3.m1.1.1.3.3" xref="S4.SS1.p2.3.m1.1.1.3.3.cmml"><mi id="S4.SS1.p2.3.m1.1.1.3.3.2" xref="S4.SS1.p2.3.m1.1.1.3.3.2.cmml">γ</mi><mrow id="S4.SS1.p2.3.m1.1.1.3.3.3" xref="S4.SS1.p2.3.m1.1.1.3.3.3.cmml"><mi id="S4.SS1.p2.3.m1.1.1.3.3.3.2" xref="S4.SS1.p2.3.m1.1.1.3.3.3.2.cmml">i</mi><mo id="S4.SS1.p2.3.m1.1.1.3.3.3.1" xref="S4.SS1.p2.3.m1.1.1.3.3.3.1.cmml">⁢</mo><mi id="S4.SS1.p2.3.m1.1.1.3.3.3.3" xref="S4.SS1.p2.3.m1.1.1.3.3.3.3.cmml">j</mi></mrow></msup></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p2.3.m1.1b"><apply id="S4.SS1.p2.3.m1.1.1.cmml" xref="S4.SS1.p2.3.m1.1.1"><csymbol cd="latexml" id="S4.SS1.p2.3.m1.1.1.1.cmml" xref="S4.SS1.p2.3.m1.1.1.1">assign</csymbol><apply id="S4.SS1.p2.3.m1.1.1.2.cmml" xref="S4.SS1.p2.3.m1.1.1.2"><csymbol cd="ambiguous" id="S4.SS1.p2.3.m1.1.1.2.1.cmml" xref="S4.SS1.p2.3.m1.1.1.2">superscript</csymbol><apply id="S4.SS1.p2.3.m1.1.1.2.2.cmml" xref="S4.SS1.p2.3.m1.1.1.2.2"><ci id="S4.SS1.p2.3.m1.1.1.2.2.1.cmml" xref="S4.SS1.p2.3.m1.1.1.2.2.1">~</ci><ci id="S4.SS1.p2.3.m1.1.1.2.2.2.cmml" xref="S4.SS1.p2.3.m1.1.1.2.2.2">𝛾</ci></apply><apply id="S4.SS1.p2.3.m1.1.1.2.3.cmml" xref="S4.SS1.p2.3.m1.1.1.2.3"><times id="S4.SS1.p2.3.m1.1.1.2.3.1.cmml" xref="S4.SS1.p2.3.m1.1.1.2.3.1"></times><ci id="S4.SS1.p2.3.m1.1.1.2.3.2.cmml" xref="S4.SS1.p2.3.m1.1.1.2.3.2">𝑖</ci><ci id="S4.SS1.p2.3.m1.1.1.2.3.3.cmml" xref="S4.SS1.p2.3.m1.1.1.2.3.3">𝑗</ci></apply></apply><apply id="S4.SS1.p2.3.m1.1.1.3.cmml" xref="S4.SS1.p2.3.m1.1.1.3"><times id="S4.SS1.p2.3.m1.1.1.3.1.cmml" xref="S4.SS1.p2.3.m1.1.1.3.1"></times><apply id="S4.SS1.p2.3.m1.1.1.3.2.cmml" xref="S4.SS1.p2.3.m1.1.1.3.2"><csymbol cd="ambiguous" id="S4.SS1.p2.3.m1.1.1.3.2.1.cmml" xref="S4.SS1.p2.3.m1.1.1.3.2">superscript</csymbol><ci id="S4.SS1.p2.3.m1.1.1.3.2.2.cmml" xref="S4.SS1.p2.3.m1.1.1.3.2.2">𝛼</ci><cn id="S4.SS1.p2.3.m1.1.1.3.2.3.cmml" type="integer" xref="S4.SS1.p2.3.m1.1.1.3.2.3">2</cn></apply><apply id="S4.SS1.p2.3.m1.1.1.3.3.cmml" xref="S4.SS1.p2.3.m1.1.1.3.3"><csymbol cd="ambiguous" id="S4.SS1.p2.3.m1.1.1.3.3.1.cmml" xref="S4.SS1.p2.3.m1.1.1.3.3">superscript</csymbol><ci id="S4.SS1.p2.3.m1.1.1.3.3.2.cmml" xref="S4.SS1.p2.3.m1.1.1.3.3.2">𝛾</ci><apply id="S4.SS1.p2.3.m1.1.1.3.3.3.cmml" xref="S4.SS1.p2.3.m1.1.1.3.3.3"><times id="S4.SS1.p2.3.m1.1.1.3.3.3.1.cmml" xref="S4.SS1.p2.3.m1.1.1.3.3.3.1"></times><ci id="S4.SS1.p2.3.m1.1.1.3.3.3.2.cmml" xref="S4.SS1.p2.3.m1.1.1.3.3.3.2">𝑖</ci><ci id="S4.SS1.p2.3.m1.1.1.3.3.3.3.cmml" xref="S4.SS1.p2.3.m1.1.1.3.3.3.3">𝑗</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p2.3.m1.1c">\tilde{\gamma}^{ij}:=\alpha^{2}\gamma^{ij}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p2.3.m1.1d">over~ start_ARG italic_γ end_ARG start_POSTSUPERSCRIPT italic_i italic_j end_POSTSUPERSCRIPT := italic_α start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_γ start_POSTSUPERSCRIPT italic_i italic_j end_POSTSUPERSCRIPT</annotation></semantics></math>. Since <math alttext="\tilde{\gamma}_{ij}\nu^{i}\nu^{j}=1" class="ltx_Math" display="inline" id="S4.SS1.p2.4.m2.1"><semantics id="S4.SS1.p2.4.m2.1a"><mrow id="S4.SS1.p2.4.m2.1.1" xref="S4.SS1.p2.4.m2.1.1.cmml"><mrow id="S4.SS1.p2.4.m2.1.1.2" xref="S4.SS1.p2.4.m2.1.1.2.cmml"><msub id="S4.SS1.p2.4.m2.1.1.2.2" xref="S4.SS1.p2.4.m2.1.1.2.2.cmml"><mover accent="true" id="S4.SS1.p2.4.m2.1.1.2.2.2" xref="S4.SS1.p2.4.m2.1.1.2.2.2.cmml"><mi id="S4.SS1.p2.4.m2.1.1.2.2.2.2" xref="S4.SS1.p2.4.m2.1.1.2.2.2.2.cmml">γ</mi><mo id="S4.SS1.p2.4.m2.1.1.2.2.2.1" xref="S4.SS1.p2.4.m2.1.1.2.2.2.1.cmml">~</mo></mover><mrow id="S4.SS1.p2.4.m2.1.1.2.2.3" xref="S4.SS1.p2.4.m2.1.1.2.2.3.cmml"><mi id="S4.SS1.p2.4.m2.1.1.2.2.3.2" xref="S4.SS1.p2.4.m2.1.1.2.2.3.2.cmml">i</mi><mo id="S4.SS1.p2.4.m2.1.1.2.2.3.1" xref="S4.SS1.p2.4.m2.1.1.2.2.3.1.cmml">⁢</mo><mi id="S4.SS1.p2.4.m2.1.1.2.2.3.3" xref="S4.SS1.p2.4.m2.1.1.2.2.3.3.cmml">j</mi></mrow></msub><mo id="S4.SS1.p2.4.m2.1.1.2.1" xref="S4.SS1.p2.4.m2.1.1.2.1.cmml">⁢</mo><msup id="S4.SS1.p2.4.m2.1.1.2.3" xref="S4.SS1.p2.4.m2.1.1.2.3.cmml"><mi id="S4.SS1.p2.4.m2.1.1.2.3.2" xref="S4.SS1.p2.4.m2.1.1.2.3.2.cmml">ν</mi><mi id="S4.SS1.p2.4.m2.1.1.2.3.3" xref="S4.SS1.p2.4.m2.1.1.2.3.3.cmml">i</mi></msup><mo id="S4.SS1.p2.4.m2.1.1.2.1a" xref="S4.SS1.p2.4.m2.1.1.2.1.cmml">⁢</mo><msup id="S4.SS1.p2.4.m2.1.1.2.4" xref="S4.SS1.p2.4.m2.1.1.2.4.cmml"><mi id="S4.SS1.p2.4.m2.1.1.2.4.2" xref="S4.SS1.p2.4.m2.1.1.2.4.2.cmml">ν</mi><mi id="S4.SS1.p2.4.m2.1.1.2.4.3" xref="S4.SS1.p2.4.m2.1.1.2.4.3.cmml">j</mi></msup></mrow><mo id="S4.SS1.p2.4.m2.1.1.1" xref="S4.SS1.p2.4.m2.1.1.1.cmml">=</mo><mn id="S4.SS1.p2.4.m2.1.1.3" xref="S4.SS1.p2.4.m2.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p2.4.m2.1b"><apply id="S4.SS1.p2.4.m2.1.1.cmml" xref="S4.SS1.p2.4.m2.1.1"><eq id="S4.SS1.p2.4.m2.1.1.1.cmml" xref="S4.SS1.p2.4.m2.1.1.1"></eq><apply id="S4.SS1.p2.4.m2.1.1.2.cmml" xref="S4.SS1.p2.4.m2.1.1.2"><times id="S4.SS1.p2.4.m2.1.1.2.1.cmml" xref="S4.SS1.p2.4.m2.1.1.2.1"></times><apply id="S4.SS1.p2.4.m2.1.1.2.2.cmml" xref="S4.SS1.p2.4.m2.1.1.2.2"><csymbol cd="ambiguous" id="S4.SS1.p2.4.m2.1.1.2.2.1.cmml" xref="S4.SS1.p2.4.m2.1.1.2.2">subscript</csymbol><apply id="S4.SS1.p2.4.m2.1.1.2.2.2.cmml" xref="S4.SS1.p2.4.m2.1.1.2.2.2"><ci id="S4.SS1.p2.4.m2.1.1.2.2.2.1.cmml" xref="S4.SS1.p2.4.m2.1.1.2.2.2.1">~</ci><ci id="S4.SS1.p2.4.m2.1.1.2.2.2.2.cmml" xref="S4.SS1.p2.4.m2.1.1.2.2.2.2">𝛾</ci></apply><apply id="S4.SS1.p2.4.m2.1.1.2.2.3.cmml" xref="S4.SS1.p2.4.m2.1.1.2.2.3"><times id="S4.SS1.p2.4.m2.1.1.2.2.3.1.cmml" xref="S4.SS1.p2.4.m2.1.1.2.2.3.1"></times><ci id="S4.SS1.p2.4.m2.1.1.2.2.3.2.cmml" xref="S4.SS1.p2.4.m2.1.1.2.2.3.2">𝑖</ci><ci id="S4.SS1.p2.4.m2.1.1.2.2.3.3.cmml" xref="S4.SS1.p2.4.m2.1.1.2.2.3.3">𝑗</ci></apply></apply><apply id="S4.SS1.p2.4.m2.1.1.2.3.cmml" xref="S4.SS1.p2.4.m2.1.1.2.3"><csymbol cd="ambiguous" id="S4.SS1.p2.4.m2.1.1.2.3.1.cmml" xref="S4.SS1.p2.4.m2.1.1.2.3">superscript</csymbol><ci id="S4.SS1.p2.4.m2.1.1.2.3.2.cmml" xref="S4.SS1.p2.4.m2.1.1.2.3.2">𝜈</ci><ci id="S4.SS1.p2.4.m2.1.1.2.3.3.cmml" xref="S4.SS1.p2.4.m2.1.1.2.3.3">𝑖</ci></apply><apply id="S4.SS1.p2.4.m2.1.1.2.4.cmml" xref="S4.SS1.p2.4.m2.1.1.2.4"><csymbol cd="ambiguous" id="S4.SS1.p2.4.m2.1.1.2.4.1.cmml" xref="S4.SS1.p2.4.m2.1.1.2.4">superscript</csymbol><ci id="S4.SS1.p2.4.m2.1.1.2.4.2.cmml" xref="S4.SS1.p2.4.m2.1.1.2.4.2">𝜈</ci><ci id="S4.SS1.p2.4.m2.1.1.2.4.3.cmml" xref="S4.SS1.p2.4.m2.1.1.2.4.3">𝑗</ci></apply></apply><cn id="S4.SS1.p2.4.m2.1.1.3.cmml" type="integer" xref="S4.SS1.p2.4.m2.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p2.4.m2.1c">\tilde{\gamma}_{ij}\nu^{i}\nu^{j}=1</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p2.4.m2.1d">over~ start_ARG italic_γ end_ARG start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT italic_ν start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT italic_ν start_POSTSUPERSCRIPT italic_j end_POSTSUPERSCRIPT = 1</annotation></semantics></math> and <math alttext="\nu^{i}=(v^{i}/\mathcal{H}^{+})+\beta^{i}=(v^{i}/\mathcal{F})+\beta^{i}" class="ltx_Math" display="inline" id="S4.SS1.p2.5.m3.2"><semantics id="S4.SS1.p2.5.m3.2a"><mrow id="S4.SS1.p2.5.m3.2.2" xref="S4.SS1.p2.5.m3.2.2.cmml"><msup id="S4.SS1.p2.5.m3.2.2.4" xref="S4.SS1.p2.5.m3.2.2.4.cmml"><mi id="S4.SS1.p2.5.m3.2.2.4.2" xref="S4.SS1.p2.5.m3.2.2.4.2.cmml">ν</mi><mi id="S4.SS1.p2.5.m3.2.2.4.3" xref="S4.SS1.p2.5.m3.2.2.4.3.cmml">i</mi></msup><mo id="S4.SS1.p2.5.m3.2.2.5" xref="S4.SS1.p2.5.m3.2.2.5.cmml">=</mo><mrow id="S4.SS1.p2.5.m3.1.1.1" xref="S4.SS1.p2.5.m3.1.1.1.cmml"><mrow id="S4.SS1.p2.5.m3.1.1.1.1.1" xref="S4.SS1.p2.5.m3.1.1.1.1.1.1.cmml"><mo id="S4.SS1.p2.5.m3.1.1.1.1.1.2" stretchy="false" xref="S4.SS1.p2.5.m3.1.1.1.1.1.1.cmml">(</mo><mrow id="S4.SS1.p2.5.m3.1.1.1.1.1.1" xref="S4.SS1.p2.5.m3.1.1.1.1.1.1.cmml"><msup id="S4.SS1.p2.5.m3.1.1.1.1.1.1.2" xref="S4.SS1.p2.5.m3.1.1.1.1.1.1.2.cmml"><mi id="S4.SS1.p2.5.m3.1.1.1.1.1.1.2.2" xref="S4.SS1.p2.5.m3.1.1.1.1.1.1.2.2.cmml">v</mi><mi id="S4.SS1.p2.5.m3.1.1.1.1.1.1.2.3" xref="S4.SS1.p2.5.m3.1.1.1.1.1.1.2.3.cmml">i</mi></msup><mo id="S4.SS1.p2.5.m3.1.1.1.1.1.1.1" xref="S4.SS1.p2.5.m3.1.1.1.1.1.1.1.cmml">/</mo><msup id="S4.SS1.p2.5.m3.1.1.1.1.1.1.3" xref="S4.SS1.p2.5.m3.1.1.1.1.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.SS1.p2.5.m3.1.1.1.1.1.1.3.2" xref="S4.SS1.p2.5.m3.1.1.1.1.1.1.3.2.cmml">ℋ</mi><mo id="S4.SS1.p2.5.m3.1.1.1.1.1.1.3.3" xref="S4.SS1.p2.5.m3.1.1.1.1.1.1.3.3.cmml">+</mo></msup></mrow><mo id="S4.SS1.p2.5.m3.1.1.1.1.1.3" stretchy="false" xref="S4.SS1.p2.5.m3.1.1.1.1.1.1.cmml">)</mo></mrow><mo id="S4.SS1.p2.5.m3.1.1.1.2" xref="S4.SS1.p2.5.m3.1.1.1.2.cmml">+</mo><msup id="S4.SS1.p2.5.m3.1.1.1.3" xref="S4.SS1.p2.5.m3.1.1.1.3.cmml"><mi id="S4.SS1.p2.5.m3.1.1.1.3.2" xref="S4.SS1.p2.5.m3.1.1.1.3.2.cmml">β</mi><mi id="S4.SS1.p2.5.m3.1.1.1.3.3" xref="S4.SS1.p2.5.m3.1.1.1.3.3.cmml">i</mi></msup></mrow><mo id="S4.SS1.p2.5.m3.2.2.6" xref="S4.SS1.p2.5.m3.2.2.6.cmml">=</mo><mrow id="S4.SS1.p2.5.m3.2.2.2" xref="S4.SS1.p2.5.m3.2.2.2.cmml"><mrow id="S4.SS1.p2.5.m3.2.2.2.1.1" xref="S4.SS1.p2.5.m3.2.2.2.1.1.1.cmml"><mo id="S4.SS1.p2.5.m3.2.2.2.1.1.2" stretchy="false" xref="S4.SS1.p2.5.m3.2.2.2.1.1.1.cmml">(</mo><mrow id="S4.SS1.p2.5.m3.2.2.2.1.1.1" xref="S4.SS1.p2.5.m3.2.2.2.1.1.1.cmml"><msup id="S4.SS1.p2.5.m3.2.2.2.1.1.1.2" xref="S4.SS1.p2.5.m3.2.2.2.1.1.1.2.cmml"><mi id="S4.SS1.p2.5.m3.2.2.2.1.1.1.2.2" xref="S4.SS1.p2.5.m3.2.2.2.1.1.1.2.2.cmml">v</mi><mi id="S4.SS1.p2.5.m3.2.2.2.1.1.1.2.3" xref="S4.SS1.p2.5.m3.2.2.2.1.1.1.2.3.cmml">i</mi></msup><mo id="S4.SS1.p2.5.m3.2.2.2.1.1.1.1" xref="S4.SS1.p2.5.m3.2.2.2.1.1.1.1.cmml">/</mo><mi class="ltx_font_mathcaligraphic" id="S4.SS1.p2.5.m3.2.2.2.1.1.1.3" xref="S4.SS1.p2.5.m3.2.2.2.1.1.1.3.cmml">ℱ</mi></mrow><mo id="S4.SS1.p2.5.m3.2.2.2.1.1.3" stretchy="false" xref="S4.SS1.p2.5.m3.2.2.2.1.1.1.cmml">)</mo></mrow><mo id="S4.SS1.p2.5.m3.2.2.2.2" xref="S4.SS1.p2.5.m3.2.2.2.2.cmml">+</mo><msup id="S4.SS1.p2.5.m3.2.2.2.3" xref="S4.SS1.p2.5.m3.2.2.2.3.cmml"><mi id="S4.SS1.p2.5.m3.2.2.2.3.2" xref="S4.SS1.p2.5.m3.2.2.2.3.2.cmml">β</mi><mi id="S4.SS1.p2.5.m3.2.2.2.3.3" xref="S4.SS1.p2.5.m3.2.2.2.3.3.cmml">i</mi></msup></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p2.5.m3.2b"><apply id="S4.SS1.p2.5.m3.2.2.cmml" xref="S4.SS1.p2.5.m3.2.2"><and id="S4.SS1.p2.5.m3.2.2a.cmml" xref="S4.SS1.p2.5.m3.2.2"></and><apply id="S4.SS1.p2.5.m3.2.2b.cmml" xref="S4.SS1.p2.5.m3.2.2"><eq id="S4.SS1.p2.5.m3.2.2.5.cmml" xref="S4.SS1.p2.5.m3.2.2.5"></eq><apply id="S4.SS1.p2.5.m3.2.2.4.cmml" xref="S4.SS1.p2.5.m3.2.2.4"><csymbol cd="ambiguous" id="S4.SS1.p2.5.m3.2.2.4.1.cmml" xref="S4.SS1.p2.5.m3.2.2.4">superscript</csymbol><ci id="S4.SS1.p2.5.m3.2.2.4.2.cmml" xref="S4.SS1.p2.5.m3.2.2.4.2">𝜈</ci><ci id="S4.SS1.p2.5.m3.2.2.4.3.cmml" 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xref="S4.SS1.p2.5.m3.2.2.2.1.1"><divide id="S4.SS1.p2.5.m3.2.2.2.1.1.1.1.cmml" xref="S4.SS1.p2.5.m3.2.2.2.1.1.1.1"></divide><apply id="S4.SS1.p2.5.m3.2.2.2.1.1.1.2.cmml" xref="S4.SS1.p2.5.m3.2.2.2.1.1.1.2"><csymbol cd="ambiguous" id="S4.SS1.p2.5.m3.2.2.2.1.1.1.2.1.cmml" xref="S4.SS1.p2.5.m3.2.2.2.1.1.1.2">superscript</csymbol><ci id="S4.SS1.p2.5.m3.2.2.2.1.1.1.2.2.cmml" xref="S4.SS1.p2.5.m3.2.2.2.1.1.1.2.2">𝑣</ci><ci id="S4.SS1.p2.5.m3.2.2.2.1.1.1.2.3.cmml" xref="S4.SS1.p2.5.m3.2.2.2.1.1.1.2.3">𝑖</ci></apply><ci id="S4.SS1.p2.5.m3.2.2.2.1.1.1.3.cmml" xref="S4.SS1.p2.5.m3.2.2.2.1.1.1.3">ℱ</ci></apply><apply id="S4.SS1.p2.5.m3.2.2.2.3.cmml" xref="S4.SS1.p2.5.m3.2.2.2.3"><csymbol cd="ambiguous" id="S4.SS1.p2.5.m3.2.2.2.3.1.cmml" xref="S4.SS1.p2.5.m3.2.2.2.3">superscript</csymbol><ci id="S4.SS1.p2.5.m3.2.2.2.3.2.cmml" xref="S4.SS1.p2.5.m3.2.2.2.3.2">𝛽</ci><ci id="S4.SS1.p2.5.m3.2.2.2.3.3.cmml" xref="S4.SS1.p2.5.m3.2.2.2.3.3">𝑖</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p2.5.m3.2c">\nu^{i}=(v^{i}/\mathcal{H}^{+})+\beta^{i}=(v^{i}/\mathcal{F})+\beta^{i}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p2.5.m3.2d">italic_ν start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT = ( italic_v start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT / caligraphic_H start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT ) + italic_β start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT = ( italic_v start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT / caligraphic_F ) + italic_β start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT</annotation></semantics></math>, we have</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="S3.EGx73"> <tbody id="S4.E79"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_left_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle 1=\tilde{\gamma}_{ij}\left(\frac{v^{i}}{\mathcal{F}}+\beta^{i}% \right)\left(\frac{v^{j}}{\mathcal{F}}+\beta^{j}\right)." class="ltx_Math" display="inline" id="S4.E79.m1.1"><semantics id="S4.E79.m1.1a"><mrow id="S4.E79.m1.1.1.1" xref="S4.E79.m1.1.1.1.1.cmml"><mrow id="S4.E79.m1.1.1.1.1" xref="S4.E79.m1.1.1.1.1.cmml"><mn id="S4.E79.m1.1.1.1.1.4" xref="S4.E79.m1.1.1.1.1.4.cmml">1</mn><mo id="S4.E79.m1.1.1.1.1.3" xref="S4.E79.m1.1.1.1.1.3.cmml">=</mo><mrow id="S4.E79.m1.1.1.1.1.2" xref="S4.E79.m1.1.1.1.1.2.cmml"><msub id="S4.E79.m1.1.1.1.1.2.4" xref="S4.E79.m1.1.1.1.1.2.4.cmml"><mover accent="true" id="S4.E79.m1.1.1.1.1.2.4.2" xref="S4.E79.m1.1.1.1.1.2.4.2.cmml"><mi id="S4.E79.m1.1.1.1.1.2.4.2.2" xref="S4.E79.m1.1.1.1.1.2.4.2.2.cmml">γ</mi><mo id="S4.E79.m1.1.1.1.1.2.4.2.1" xref="S4.E79.m1.1.1.1.1.2.4.2.1.cmml">~</mo></mover><mrow id="S4.E79.m1.1.1.1.1.2.4.3" xref="S4.E79.m1.1.1.1.1.2.4.3.cmml"><mi id="S4.E79.m1.1.1.1.1.2.4.3.2" xref="S4.E79.m1.1.1.1.1.2.4.3.2.cmml">i</mi><mo id="S4.E79.m1.1.1.1.1.2.4.3.1" xref="S4.E79.m1.1.1.1.1.2.4.3.1.cmml">⁢</mo><mi id="S4.E79.m1.1.1.1.1.2.4.3.3" xref="S4.E79.m1.1.1.1.1.2.4.3.3.cmml">j</mi></mrow></msub><mo id="S4.E79.m1.1.1.1.1.2.3" xref="S4.E79.m1.1.1.1.1.2.3.cmml">⁢</mo><mrow id="S4.E79.m1.1.1.1.1.1.1.1" xref="S4.E79.m1.1.1.1.1.1.1.1.1.cmml"><mo id="S4.E79.m1.1.1.1.1.1.1.1.2" xref="S4.E79.m1.1.1.1.1.1.1.1.1.cmml">(</mo><mrow id="S4.E79.m1.1.1.1.1.1.1.1.1" xref="S4.E79.m1.1.1.1.1.1.1.1.1.cmml"><mstyle displaystyle="true" id="S4.E79.m1.1.1.1.1.1.1.1.1.2" xref="S4.E79.m1.1.1.1.1.1.1.1.1.2.cmml"><mfrac id="S4.E79.m1.1.1.1.1.1.1.1.1.2a" xref="S4.E79.m1.1.1.1.1.1.1.1.1.2.cmml"><msup id="S4.E79.m1.1.1.1.1.1.1.1.1.2.2" xref="S4.E79.m1.1.1.1.1.1.1.1.1.2.2.cmml"><mi id="S4.E79.m1.1.1.1.1.1.1.1.1.2.2.2" xref="S4.E79.m1.1.1.1.1.1.1.1.1.2.2.2.cmml">v</mi><mi id="S4.E79.m1.1.1.1.1.1.1.1.1.2.2.3" xref="S4.E79.m1.1.1.1.1.1.1.1.1.2.2.3.cmml">i</mi></msup><mi class="ltx_font_mathcaligraphic" id="S4.E79.m1.1.1.1.1.1.1.1.1.2.3" xref="S4.E79.m1.1.1.1.1.1.1.1.1.2.3.cmml">ℱ</mi></mfrac></mstyle><mo id="S4.E79.m1.1.1.1.1.1.1.1.1.1" xref="S4.E79.m1.1.1.1.1.1.1.1.1.1.cmml">+</mo><msup id="S4.E79.m1.1.1.1.1.1.1.1.1.3" xref="S4.E79.m1.1.1.1.1.1.1.1.1.3.cmml"><mi id="S4.E79.m1.1.1.1.1.1.1.1.1.3.2" xref="S4.E79.m1.1.1.1.1.1.1.1.1.3.2.cmml">β</mi><mi id="S4.E79.m1.1.1.1.1.1.1.1.1.3.3" xref="S4.E79.m1.1.1.1.1.1.1.1.1.3.3.cmml">i</mi></msup></mrow><mo id="S4.E79.m1.1.1.1.1.1.1.1.3" xref="S4.E79.m1.1.1.1.1.1.1.1.1.cmml">)</mo></mrow><mo id="S4.E79.m1.1.1.1.1.2.3a" xref="S4.E79.m1.1.1.1.1.2.3.cmml">⁢</mo><mrow id="S4.E79.m1.1.1.1.1.2.2.1" xref="S4.E79.m1.1.1.1.1.2.2.1.1.cmml"><mo id="S4.E79.m1.1.1.1.1.2.2.1.2" xref="S4.E79.m1.1.1.1.1.2.2.1.1.cmml">(</mo><mrow id="S4.E79.m1.1.1.1.1.2.2.1.1" xref="S4.E79.m1.1.1.1.1.2.2.1.1.cmml"><mstyle displaystyle="true" id="S4.E79.m1.1.1.1.1.2.2.1.1.2" xref="S4.E79.m1.1.1.1.1.2.2.1.1.2.cmml"><mfrac id="S4.E79.m1.1.1.1.1.2.2.1.1.2a" xref="S4.E79.m1.1.1.1.1.2.2.1.1.2.cmml"><msup id="S4.E79.m1.1.1.1.1.2.2.1.1.2.2" xref="S4.E79.m1.1.1.1.1.2.2.1.1.2.2.cmml"><mi id="S4.E79.m1.1.1.1.1.2.2.1.1.2.2.2" xref="S4.E79.m1.1.1.1.1.2.2.1.1.2.2.2.cmml">v</mi><mi id="S4.E79.m1.1.1.1.1.2.2.1.1.2.2.3" xref="S4.E79.m1.1.1.1.1.2.2.1.1.2.2.3.cmml">j</mi></msup><mi class="ltx_font_mathcaligraphic" id="S4.E79.m1.1.1.1.1.2.2.1.1.2.3" xref="S4.E79.m1.1.1.1.1.2.2.1.1.2.3.cmml">ℱ</mi></mfrac></mstyle><mo id="S4.E79.m1.1.1.1.1.2.2.1.1.1" xref="S4.E79.m1.1.1.1.1.2.2.1.1.1.cmml">+</mo><msup id="S4.E79.m1.1.1.1.1.2.2.1.1.3" xref="S4.E79.m1.1.1.1.1.2.2.1.1.3.cmml"><mi id="S4.E79.m1.1.1.1.1.2.2.1.1.3.2" xref="S4.E79.m1.1.1.1.1.2.2.1.1.3.2.cmml">β</mi><mi id="S4.E79.m1.1.1.1.1.2.2.1.1.3.3" xref="S4.E79.m1.1.1.1.1.2.2.1.1.3.3.cmml">j</mi></msup></mrow><mo id="S4.E79.m1.1.1.1.1.2.2.1.3" xref="S4.E79.m1.1.1.1.1.2.2.1.1.cmml">)</mo></mrow></mrow></mrow><mo id="S4.E79.m1.1.1.1.2" lspace="0em" xref="S4.E79.m1.1.1.1.1.cmml">.</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.E79.m1.1b"><apply id="S4.E79.m1.1.1.1.1.cmml" xref="S4.E79.m1.1.1.1"><eq id="S4.E79.m1.1.1.1.1.3.cmml" xref="S4.E79.m1.1.1.1.1.3"></eq><cn id="S4.E79.m1.1.1.1.1.4.cmml" type="integer" xref="S4.E79.m1.1.1.1.1.4">1</cn><apply id="S4.E79.m1.1.1.1.1.2.cmml" xref="S4.E79.m1.1.1.1.1.2"><times id="S4.E79.m1.1.1.1.1.2.3.cmml" xref="S4.E79.m1.1.1.1.1.2.3"></times><apply id="S4.E79.m1.1.1.1.1.2.4.cmml" xref="S4.E79.m1.1.1.1.1.2.4"><csymbol cd="ambiguous" id="S4.E79.m1.1.1.1.1.2.4.1.cmml" xref="S4.E79.m1.1.1.1.1.2.4">subscript</csymbol><apply id="S4.E79.m1.1.1.1.1.2.4.2.cmml" xref="S4.E79.m1.1.1.1.1.2.4.2"><ci id="S4.E79.m1.1.1.1.1.2.4.2.1.cmml" xref="S4.E79.m1.1.1.1.1.2.4.2.1">~</ci><ci id="S4.E79.m1.1.1.1.1.2.4.2.2.cmml" xref="S4.E79.m1.1.1.1.1.2.4.2.2">𝛾</ci></apply><apply id="S4.E79.m1.1.1.1.1.2.4.3.cmml" xref="S4.E79.m1.1.1.1.1.2.4.3"><times id="S4.E79.m1.1.1.1.1.2.4.3.1.cmml" xref="S4.E79.m1.1.1.1.1.2.4.3.1"></times><ci id="S4.E79.m1.1.1.1.1.2.4.3.2.cmml" xref="S4.E79.m1.1.1.1.1.2.4.3.2">𝑖</ci><ci id="S4.E79.m1.1.1.1.1.2.4.3.3.cmml" xref="S4.E79.m1.1.1.1.1.2.4.3.3">𝑗</ci></apply></apply><apply id="S4.E79.m1.1.1.1.1.1.1.1.1.cmml" xref="S4.E79.m1.1.1.1.1.1.1.1"><plus id="S4.E79.m1.1.1.1.1.1.1.1.1.1.cmml" xref="S4.E79.m1.1.1.1.1.1.1.1.1.1"></plus><apply id="S4.E79.m1.1.1.1.1.1.1.1.1.2.cmml" xref="S4.E79.m1.1.1.1.1.1.1.1.1.2"><divide id="S4.E79.m1.1.1.1.1.1.1.1.1.2.1.cmml" xref="S4.E79.m1.1.1.1.1.1.1.1.1.2"></divide><apply id="S4.E79.m1.1.1.1.1.1.1.1.1.2.2.cmml" xref="S4.E79.m1.1.1.1.1.1.1.1.1.2.2"><csymbol cd="ambiguous" id="S4.E79.m1.1.1.1.1.1.1.1.1.2.2.1.cmml" xref="S4.E79.m1.1.1.1.1.1.1.1.1.2.2">superscript</csymbol><ci id="S4.E79.m1.1.1.1.1.1.1.1.1.2.2.2.cmml" xref="S4.E79.m1.1.1.1.1.1.1.1.1.2.2.2">𝑣</ci><ci id="S4.E79.m1.1.1.1.1.1.1.1.1.2.2.3.cmml" xref="S4.E79.m1.1.1.1.1.1.1.1.1.2.2.3">𝑖</ci></apply><ci id="S4.E79.m1.1.1.1.1.1.1.1.1.2.3.cmml" xref="S4.E79.m1.1.1.1.1.1.1.1.1.2.3">ℱ</ci></apply><apply id="S4.E79.m1.1.1.1.1.1.1.1.1.3.cmml" xref="S4.E79.m1.1.1.1.1.1.1.1.1.3"><csymbol cd="ambiguous" id="S4.E79.m1.1.1.1.1.1.1.1.1.3.1.cmml" xref="S4.E79.m1.1.1.1.1.1.1.1.1.3">superscript</csymbol><ci id="S4.E79.m1.1.1.1.1.1.1.1.1.3.2.cmml" xref="S4.E79.m1.1.1.1.1.1.1.1.1.3.2">𝛽</ci><ci id="S4.E79.m1.1.1.1.1.1.1.1.1.3.3.cmml" xref="S4.E79.m1.1.1.1.1.1.1.1.1.3.3">𝑖</ci></apply></apply><apply id="S4.E79.m1.1.1.1.1.2.2.1.1.cmml" xref="S4.E79.m1.1.1.1.1.2.2.1"><plus id="S4.E79.m1.1.1.1.1.2.2.1.1.1.cmml" xref="S4.E79.m1.1.1.1.1.2.2.1.1.1"></plus><apply id="S4.E79.m1.1.1.1.1.2.2.1.1.2.cmml" xref="S4.E79.m1.1.1.1.1.2.2.1.1.2"><divide id="S4.E79.m1.1.1.1.1.2.2.1.1.2.1.cmml" xref="S4.E79.m1.1.1.1.1.2.2.1.1.2"></divide><apply id="S4.E79.m1.1.1.1.1.2.2.1.1.2.2.cmml" xref="S4.E79.m1.1.1.1.1.2.2.1.1.2.2"><csymbol cd="ambiguous" id="S4.E79.m1.1.1.1.1.2.2.1.1.2.2.1.cmml" xref="S4.E79.m1.1.1.1.1.2.2.1.1.2.2">superscript</csymbol><ci id="S4.E79.m1.1.1.1.1.2.2.1.1.2.2.2.cmml" xref="S4.E79.m1.1.1.1.1.2.2.1.1.2.2.2">𝑣</ci><ci id="S4.E79.m1.1.1.1.1.2.2.1.1.2.2.3.cmml" xref="S4.E79.m1.1.1.1.1.2.2.1.1.2.2.3">𝑗</ci></apply><ci id="S4.E79.m1.1.1.1.1.2.2.1.1.2.3.cmml" xref="S4.E79.m1.1.1.1.1.2.2.1.1.2.3">ℱ</ci></apply><apply id="S4.E79.m1.1.1.1.1.2.2.1.1.3.cmml" xref="S4.E79.m1.1.1.1.1.2.2.1.1.3"><csymbol cd="ambiguous" id="S4.E79.m1.1.1.1.1.2.2.1.1.3.1.cmml" xref="S4.E79.m1.1.1.1.1.2.2.1.1.3">superscript</csymbol><ci id="S4.E79.m1.1.1.1.1.2.2.1.1.3.2.cmml" xref="S4.E79.m1.1.1.1.1.2.2.1.1.3.2">𝛽</ci><ci id="S4.E79.m1.1.1.1.1.2.2.1.1.3.3.cmml" xref="S4.E79.m1.1.1.1.1.2.2.1.1.3.3">𝑗</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.E79.m1.1c">\displaystyle 1=\tilde{\gamma}_{ij}\left(\frac{v^{i}}{\mathcal{F}}+\beta^{i}% \right)\left(\frac{v^{j}}{\mathcal{F}}+\beta^{j}\right).</annotation><annotation encoding="application/x-llamapun" id="S4.E79.m1.1d">1 = over~ start_ARG italic_γ end_ARG start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT ( divide start_ARG italic_v start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT end_ARG start_ARG caligraphic_F end_ARG + italic_β start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT ) ( divide start_ARG italic_v start_POSTSUPERSCRIPT italic_j end_POSTSUPERSCRIPT end_ARG start_ARG caligraphic_F end_ARG + italic_β start_POSTSUPERSCRIPT italic_j end_POSTSUPERSCRIPT ) .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_left_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(79)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S4.SS1.p2.8">Solving for <math alttext="\mathcal{F}" class="ltx_Math" display="inline" id="S4.SS1.p2.6.m1.1"><semantics id="S4.SS1.p2.6.m1.1a"><mi class="ltx_font_mathcaligraphic" id="S4.SS1.p2.6.m1.1.1" xref="S4.SS1.p2.6.m1.1.1.cmml">ℱ</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.p2.6.m1.1b"><ci id="S4.SS1.p2.6.m1.1.1.cmml" xref="S4.SS1.p2.6.m1.1.1">ℱ</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p2.6.m1.1c">\mathcal{F}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p2.6.m1.1d">caligraphic_F</annotation></semantics></math> and introduce the transformed metric <math alttext="a_{ij}" class="ltx_Math" display="inline" id="S4.SS1.p2.7.m2.1"><semantics id="S4.SS1.p2.7.m2.1a"><msub id="S4.SS1.p2.7.m2.1.1" xref="S4.SS1.p2.7.m2.1.1.cmml"><mi id="S4.SS1.p2.7.m2.1.1.2" xref="S4.SS1.p2.7.m2.1.1.2.cmml">a</mi><mrow id="S4.SS1.p2.7.m2.1.1.3" xref="S4.SS1.p2.7.m2.1.1.3.cmml"><mi id="S4.SS1.p2.7.m2.1.1.3.2" xref="S4.SS1.p2.7.m2.1.1.3.2.cmml">i</mi><mo id="S4.SS1.p2.7.m2.1.1.3.1" xref="S4.SS1.p2.7.m2.1.1.3.1.cmml">⁢</mo><mi id="S4.SS1.p2.7.m2.1.1.3.3" xref="S4.SS1.p2.7.m2.1.1.3.3.cmml">j</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="S4.SS1.p2.7.m2.1b"><apply id="S4.SS1.p2.7.m2.1.1.cmml" xref="S4.SS1.p2.7.m2.1.1"><csymbol cd="ambiguous" id="S4.SS1.p2.7.m2.1.1.1.cmml" xref="S4.SS1.p2.7.m2.1.1">subscript</csymbol><ci id="S4.SS1.p2.7.m2.1.1.2.cmml" xref="S4.SS1.p2.7.m2.1.1.2">𝑎</ci><apply id="S4.SS1.p2.7.m2.1.1.3.cmml" xref="S4.SS1.p2.7.m2.1.1.3"><times id="S4.SS1.p2.7.m2.1.1.3.1.cmml" xref="S4.SS1.p2.7.m2.1.1.3.1"></times><ci id="S4.SS1.p2.7.m2.1.1.3.2.cmml" xref="S4.SS1.p2.7.m2.1.1.3.2">𝑖</ci><ci id="S4.SS1.p2.7.m2.1.1.3.3.cmml" xref="S4.SS1.p2.7.m2.1.1.3.3">𝑗</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p2.7.m2.1c">a_{ij}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p2.7.m2.1d">italic_a start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="b_{i}" class="ltx_Math" display="inline" id="S4.SS1.p2.8.m3.1"><semantics id="S4.SS1.p2.8.m3.1a"><msub id="S4.SS1.p2.8.m3.1.1" xref="S4.SS1.p2.8.m3.1.1.cmml"><mi id="S4.SS1.p2.8.m3.1.1.2" xref="S4.SS1.p2.8.m3.1.1.2.cmml">b</mi><mi id="S4.SS1.p2.8.m3.1.1.3" xref="S4.SS1.p2.8.m3.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS1.p2.8.m3.1b"><apply id="S4.SS1.p2.8.m3.1.1.cmml" xref="S4.SS1.p2.8.m3.1.1"><csymbol cd="ambiguous" id="S4.SS1.p2.8.m3.1.1.1.cmml" xref="S4.SS1.p2.8.m3.1.1">subscript</csymbol><ci id="S4.SS1.p2.8.m3.1.1.2.cmml" xref="S4.SS1.p2.8.m3.1.1.2">𝑏</ci><ci id="S4.SS1.p2.8.m3.1.1.3.cmml" xref="S4.SS1.p2.8.m3.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p2.8.m3.1c">b_{i}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p2.8.m3.1d">italic_b start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> by</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="S3.EGx74"> <tbody id="S4.Ex4"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_left_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle a_{ij}" class="ltx_Math" display="inline" id="S4.Ex4.m1.1"><semantics id="S4.Ex4.m1.1a"><msub id="S4.Ex4.m1.1.1" xref="S4.Ex4.m1.1.1.cmml"><mi id="S4.Ex4.m1.1.1.2" xref="S4.Ex4.m1.1.1.2.cmml">a</mi><mrow id="S4.Ex4.m1.1.1.3" xref="S4.Ex4.m1.1.1.3.cmml"><mi id="S4.Ex4.m1.1.1.3.2" xref="S4.Ex4.m1.1.1.3.2.cmml">i</mi><mo id="S4.Ex4.m1.1.1.3.1" xref="S4.Ex4.m1.1.1.3.1.cmml">⁢</mo><mi id="S4.Ex4.m1.1.1.3.3" xref="S4.Ex4.m1.1.1.3.3.cmml">j</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="S4.Ex4.m1.1b"><apply id="S4.Ex4.m1.1.1.cmml" xref="S4.Ex4.m1.1.1"><csymbol cd="ambiguous" id="S4.Ex4.m1.1.1.1.cmml" xref="S4.Ex4.m1.1.1">subscript</csymbol><ci id="S4.Ex4.m1.1.1.2.cmml" xref="S4.Ex4.m1.1.1.2">𝑎</ci><apply id="S4.Ex4.m1.1.1.3.cmml" xref="S4.Ex4.m1.1.1.3"><times id="S4.Ex4.m1.1.1.3.1.cmml" xref="S4.Ex4.m1.1.1.3.1"></times><ci id="S4.Ex4.m1.1.1.3.2.cmml" xref="S4.Ex4.m1.1.1.3.2">𝑖</ci><ci id="S4.Ex4.m1.1.1.3.3.cmml" xref="S4.Ex4.m1.1.1.3.3">𝑗</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Ex4.m1.1c">\displaystyle a_{ij}</annotation><annotation encoding="application/x-llamapun" id="S4.Ex4.m1.1d">italic_a start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle=\frac{\xi\tilde{\gamma}_{ij}+\beta_{i}\beta_{j}}{\xi^{2}}," class="ltx_Math" display="inline" id="S4.Ex4.m2.1"><semantics id="S4.Ex4.m2.1a"><mrow id="S4.Ex4.m2.1.1.1" xref="S4.Ex4.m2.1.1.1.1.cmml"><mrow id="S4.Ex4.m2.1.1.1.1" xref="S4.Ex4.m2.1.1.1.1.cmml"><mi id="S4.Ex4.m2.1.1.1.1.2" xref="S4.Ex4.m2.1.1.1.1.2.cmml"></mi><mo id="S4.Ex4.m2.1.1.1.1.1" xref="S4.Ex4.m2.1.1.1.1.1.cmml">=</mo><mstyle displaystyle="true" id="S4.Ex4.m2.1.1.1.1.3" xref="S4.Ex4.m2.1.1.1.1.3.cmml"><mfrac id="S4.Ex4.m2.1.1.1.1.3a" xref="S4.Ex4.m2.1.1.1.1.3.cmml"><mrow id="S4.Ex4.m2.1.1.1.1.3.2" 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xref="S4.Ex4.m2.1.1.1.1.3.3.3">2</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Ex4.m2.1c">\displaystyle=\frac{\xi\tilde{\gamma}_{ij}+\beta_{i}\beta_{j}}{\xi^{2}},</annotation><annotation encoding="application/x-llamapun" id="S4.Ex4.m2.1d">= divide start_ARG italic_ξ over~ start_ARG italic_γ end_ARG start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT + italic_β start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_β start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT end_ARG start_ARG italic_ξ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG ,</annotation></semantics></math></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\quad b_{i}" class="ltx_Math" display="inline" id="S4.Ex4.m3.1"><semantics id="S4.Ex4.m3.1a"><msub id="S4.Ex4.m3.1.1" xref="S4.Ex4.m3.1.1.cmml"><mi id="S4.Ex4.m3.1.1.2" xref="S4.Ex4.m3.1.1.2.cmml">b</mi><mi id="S4.Ex4.m3.1.1.3" xref="S4.Ex4.m3.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S4.Ex4.m3.1b"><apply id="S4.Ex4.m3.1.1.cmml" xref="S4.Ex4.m3.1.1"><csymbol cd="ambiguous" id="S4.Ex4.m3.1.1.1.cmml" xref="S4.Ex4.m3.1.1">subscript</csymbol><ci id="S4.Ex4.m3.1.1.2.cmml" xref="S4.Ex4.m3.1.1.2">𝑏</ci><ci id="S4.Ex4.m3.1.1.3.cmml" xref="S4.Ex4.m3.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Ex4.m3.1c">\displaystyle\quad b_{i}</annotation><annotation encoding="application/x-llamapun" id="S4.Ex4.m3.1d">italic_b start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle=\frac{\beta_{i}}{\xi}," class="ltx_Math" display="inline" id="S4.Ex4.m4.1"><semantics id="S4.Ex4.m4.1a"><mrow id="S4.Ex4.m4.1.1.1" xref="S4.Ex4.m4.1.1.1.1.cmml"><mrow id="S4.Ex4.m4.1.1.1.1" xref="S4.Ex4.m4.1.1.1.1.cmml"><mi id="S4.Ex4.m4.1.1.1.1.2" xref="S4.Ex4.m4.1.1.1.1.2.cmml"></mi><mo id="S4.Ex4.m4.1.1.1.1.1" xref="S4.Ex4.m4.1.1.1.1.1.cmml">=</mo><mstyle displaystyle="true" id="S4.Ex4.m4.1.1.1.1.3" xref="S4.Ex4.m4.1.1.1.1.3.cmml"><mfrac id="S4.Ex4.m4.1.1.1.1.3a" xref="S4.Ex4.m4.1.1.1.1.3.cmml"><msub id="S4.Ex4.m4.1.1.1.1.3.2" xref="S4.Ex4.m4.1.1.1.1.3.2.cmml"><mi id="S4.Ex4.m4.1.1.1.1.3.2.2" xref="S4.Ex4.m4.1.1.1.1.3.2.2.cmml">β</mi><mi id="S4.Ex4.m4.1.1.1.1.3.2.3" xref="S4.Ex4.m4.1.1.1.1.3.2.3.cmml">i</mi></msub><mi id="S4.Ex4.m4.1.1.1.1.3.3" xref="S4.Ex4.m4.1.1.1.1.3.3.cmml">ξ</mi></mfrac></mstyle></mrow><mo id="S4.Ex4.m4.1.1.1.2" xref="S4.Ex4.m4.1.1.1.1.cmml">,</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.Ex4.m4.1b"><apply id="S4.Ex4.m4.1.1.1.1.cmml" xref="S4.Ex4.m4.1.1.1"><eq id="S4.Ex4.m4.1.1.1.1.1.cmml" xref="S4.Ex4.m4.1.1.1.1.1"></eq><csymbol cd="latexml" id="S4.Ex4.m4.1.1.1.1.2.cmml" xref="S4.Ex4.m4.1.1.1.1.2">absent</csymbol><apply id="S4.Ex4.m4.1.1.1.1.3.cmml" xref="S4.Ex4.m4.1.1.1.1.3"><divide id="S4.Ex4.m4.1.1.1.1.3.1.cmml" xref="S4.Ex4.m4.1.1.1.1.3"></divide><apply id="S4.Ex4.m4.1.1.1.1.3.2.cmml" xref="S4.Ex4.m4.1.1.1.1.3.2"><csymbol cd="ambiguous" id="S4.Ex4.m4.1.1.1.1.3.2.1.cmml" xref="S4.Ex4.m4.1.1.1.1.3.2">subscript</csymbol><ci id="S4.Ex4.m4.1.1.1.1.3.2.2.cmml" xref="S4.Ex4.m4.1.1.1.1.3.2.2">𝛽</ci><ci id="S4.Ex4.m4.1.1.1.1.3.2.3.cmml" xref="S4.Ex4.m4.1.1.1.1.3.2.3">𝑖</ci></apply><ci id="S4.Ex4.m4.1.1.1.1.3.3.cmml" xref="S4.Ex4.m4.1.1.1.1.3.3">𝜉</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Ex4.m4.1c">\displaystyle=\frac{\beta_{i}}{\xi},</annotation><annotation encoding="application/x-llamapun" id="S4.Ex4.m4.1d">= divide start_ARG italic_β start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_ARG start_ARG italic_ξ end_ARG ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_left_padright"></td> </tr></tbody> <tbody id="S4.E80"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_left_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\xi" class="ltx_Math" display="inline" id="S4.E80.m1.1"><semantics id="S4.E80.m1.1a"><mi id="S4.E80.m1.1.1" xref="S4.E80.m1.1.1.cmml">ξ</mi><annotation-xml encoding="MathML-Content" id="S4.E80.m1.1b"><ci id="S4.E80.m1.1.1.cmml" xref="S4.E80.m1.1.1">𝜉</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.E80.m1.1c">\displaystyle\xi</annotation><annotation encoding="application/x-llamapun" id="S4.E80.m1.1d">italic_ξ</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle:=1-\tilde{\gamma}_{ij}\beta^{i}\beta^{j}," class="ltx_Math" display="inline" id="S4.E80.m2.1"><semantics id="S4.E80.m2.1a"><mrow id="S4.E80.m2.1.1.1" xref="S4.E80.m2.1.1.1.1.cmml"><mrow id="S4.E80.m2.1.1.1.1" xref="S4.E80.m2.1.1.1.1.cmml"><mi id="S4.E80.m2.1.1.1.1.2" xref="S4.E80.m2.1.1.1.1.2.cmml"></mi><mo id="S4.E80.m2.1.1.1.1.1" lspace="0.278em" rspace="0.278em" 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encoding="application/x-tex" id="S4.E80.m2.1c">\displaystyle:=1-\tilde{\gamma}_{ij}\beta^{i}\beta^{j},</annotation><annotation encoding="application/x-llamapun" id="S4.E80.m2.1d">:= 1 - over~ start_ARG italic_γ end_ARG start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT italic_β start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT italic_β start_POSTSUPERSCRIPT italic_j end_POSTSUPERSCRIPT ,</annotation></semantics></math></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\quad\beta_{i}" class="ltx_Math" display="inline" id="S4.E80.m3.1"><semantics id="S4.E80.m3.1a"><msub id="S4.E80.m3.1.1" xref="S4.E80.m3.1.1.cmml"><mi id="S4.E80.m3.1.1.2" xref="S4.E80.m3.1.1.2.cmml">β</mi><mi id="S4.E80.m3.1.1.3" xref="S4.E80.m3.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S4.E80.m3.1b"><apply id="S4.E80.m3.1.1.cmml" xref="S4.E80.m3.1.1"><csymbol cd="ambiguous" id="S4.E80.m3.1.1.1.cmml" xref="S4.E80.m3.1.1">subscript</csymbol><ci id="S4.E80.m3.1.1.2.cmml" xref="S4.E80.m3.1.1.2">𝛽</ci><ci id="S4.E80.m3.1.1.3.cmml" xref="S4.E80.m3.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.E80.m3.1c">\displaystyle\quad\beta_{i}</annotation><annotation encoding="application/x-llamapun" id="S4.E80.m3.1d">italic_β start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle=\tilde{\gamma}_{ij}\beta^{j}," class="ltx_Math" display="inline" id="S4.E80.m4.1"><semantics id="S4.E80.m4.1a"><mrow id="S4.E80.m4.1.1.1" xref="S4.E80.m4.1.1.1.1.cmml"><mrow id="S4.E80.m4.1.1.1.1" xref="S4.E80.m4.1.1.1.1.cmml"><mi id="S4.E80.m4.1.1.1.1.2" xref="S4.E80.m4.1.1.1.1.2.cmml"></mi><mo id="S4.E80.m4.1.1.1.1.1" xref="S4.E80.m4.1.1.1.1.1.cmml">=</mo><mrow id="S4.E80.m4.1.1.1.1.3" xref="S4.E80.m4.1.1.1.1.3.cmml"><msub id="S4.E80.m4.1.1.1.1.3.2" xref="S4.E80.m4.1.1.1.1.3.2.cmml"><mover accent="true" 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rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(80)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S4.SS1.p2.12">we obtain</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="S3.EGx75"> <tbody id="S4.E81"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_left_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\mathcal{F}(x,v)=\sqrt{a_{ij}(x)v^{i}v^{j}}+b_{i}(x)v^{i}," class="ltx_Math" display="inline" id="S4.E81.m1.5"><semantics id="S4.E81.m1.5a"><mrow id="S4.E81.m1.5.5.1" xref="S4.E81.m1.5.5.1.1.cmml"><mrow id="S4.E81.m1.5.5.1.1" xref="S4.E81.m1.5.5.1.1.cmml"><mrow id="S4.E81.m1.5.5.1.1.2" xref="S4.E81.m1.5.5.1.1.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.E81.m1.5.5.1.1.2.2" xref="S4.E81.m1.5.5.1.1.2.2.cmml">ℱ</mi><mo id="S4.E81.m1.5.5.1.1.2.1" xref="S4.E81.m1.5.5.1.1.2.1.cmml">⁢</mo><mrow id="S4.E81.m1.5.5.1.1.2.3.2" 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xref="S4.E81.m1.5.5.1.1.3.2.4"><csymbol cd="ambiguous" id="S4.E81.m1.5.5.1.1.3.2.4.1.cmml" xref="S4.E81.m1.5.5.1.1.3.2.4">superscript</csymbol><ci id="S4.E81.m1.5.5.1.1.3.2.4.2.cmml" xref="S4.E81.m1.5.5.1.1.3.2.4.2">𝑣</ci><ci id="S4.E81.m1.5.5.1.1.3.2.4.3.cmml" xref="S4.E81.m1.5.5.1.1.3.2.4.3">𝑖</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.E81.m1.5c">\displaystyle\mathcal{F}(x,v)=\sqrt{a_{ij}(x)v^{i}v^{j}}+b_{i}(x)v^{i},</annotation><annotation encoding="application/x-llamapun" id="S4.E81.m1.5d">caligraphic_F ( italic_x , italic_v ) = square-root start_ARG italic_a start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT ( italic_x ) italic_v start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT italic_v start_POSTSUPERSCRIPT italic_j end_POSTSUPERSCRIPT end_ARG + italic_b start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_x ) italic_v start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_left_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(81)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S4.SS1.p2.11">which is the RF Lagrangian obtained from a Randers function <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2406.11224v2#bib.bib12" title="">12</a>]</cite>. 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encoding="application/x-tex" id="S4.SS1.p2.10.m2.1c">v^{i}=d\theta^{i}/dt</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p2.10.m2.1d">italic_v start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT = italic_d italic_θ start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT / italic_d italic_t</annotation></semantics></math>, this <math alttext="\mathcal{F}" class="ltx_Math" display="inline" id="S4.SS1.p2.11.m3.1"><semantics id="S4.SS1.p2.11.m3.1a"><mi class="ltx_font_mathcaligraphic" id="S4.SS1.p2.11.m3.1.1" xref="S4.SS1.p2.11.m3.1.1.cmml">ℱ</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.p2.11.m3.1b"><ci id="S4.SS1.p2.11.m3.1.1.cmml" xref="S4.SS1.p2.11.m3.1.1">ℱ</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p2.11.m3.1c">\mathcal{F}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p2.11.m3.1d">caligraphic_F</annotation></semantics></math> becomes the RF Lagrangian (<a class="ltx_ref" href="https://arxiv.org/html/2406.11224v2#S2.E35" title="In 2.3 Randers-Finsler deformation of the gradient-flow equations ‣ 2 Information Geometry and Gradient-Flow Equations ‣ A Hamiltonian approach to the gradient-flow equations in information geometry"><span class="ltx_text ltx_ref_tag">35</span></a>).</p> </div> </section> <section class="ltx_subsection" id="S4.SS2"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">4.2 </span>Applications to the gradient-flow equations</h3> <div class="ltx_para" id="S4.SS2.p1"> <p class="ltx_p" id="S4.SS2.p1.5">Firstly, we consider the gradient-flow equations (<a class="ltx_ref" href="https://arxiv.org/html/2406.11224v2#S2.E12" title="In 2.2 Gradient-Flow Equations ‣ 2 Information Geometry and Gradient-Flow Equations ‣ A Hamiltonian approach to the gradient-flow equations in information geometry"><span class="ltx_text ltx_ref_tag">12</span></a>) for the <math alttext="\theta" class="ltx_Math" display="inline" id="S4.SS2.p1.1.m1.1"><semantics id="S4.SS2.p1.1.m1.1a"><mi id="S4.SS2.p1.1.m1.1.1" xref="S4.SS2.p1.1.m1.1.1.cmml">θ</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.p1.1.m1.1b"><ci id="S4.SS2.p1.1.m1.1.1.cmml" xref="S4.SS2.p1.1.m1.1.1">𝜃</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p1.1.m1.1c">\theta</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p1.1.m1.1d">italic_θ</annotation></semantics></math>-potential function <math alttext="\Psi(\theta)" class="ltx_Math" display="inline" id="S4.SS2.p1.2.m2.1"><semantics id="S4.SS2.p1.2.m2.1a"><mrow id="S4.SS2.p1.2.m2.1.2" xref="S4.SS2.p1.2.m2.1.2.cmml"><mi id="S4.SS2.p1.2.m2.1.2.2" mathvariant="normal" xref="S4.SS2.p1.2.m2.1.2.2.cmml">Ψ</mi><mo id="S4.SS2.p1.2.m2.1.2.1" xref="S4.SS2.p1.2.m2.1.2.1.cmml">⁢</mo><mrow id="S4.SS2.p1.2.m2.1.2.3.2" xref="S4.SS2.p1.2.m2.1.2.cmml"><mo id="S4.SS2.p1.2.m2.1.2.3.2.1" stretchy="false" xref="S4.SS2.p1.2.m2.1.2.cmml">(</mo><mi id="S4.SS2.p1.2.m2.1.1" xref="S4.SS2.p1.2.m2.1.1.cmml">θ</mi><mo id="S4.SS2.p1.2.m2.1.2.3.2.2" stretchy="false" xref="S4.SS2.p1.2.m2.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p1.2.m2.1b"><apply id="S4.SS2.p1.2.m2.1.2.cmml" xref="S4.SS2.p1.2.m2.1.2"><times id="S4.SS2.p1.2.m2.1.2.1.cmml" xref="S4.SS2.p1.2.m2.1.2.1"></times><ci id="S4.SS2.p1.2.m2.1.2.2.cmml" xref="S4.SS2.p1.2.m2.1.2.2">Ψ</ci><ci id="S4.SS2.p1.2.m2.1.1.cmml" xref="S4.SS2.p1.2.m2.1.1">𝜃</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p1.2.m2.1c">\Psi(\theta)</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p1.2.m2.1d">roman_Ψ ( italic_θ )</annotation></semantics></math> in the case of <math alttext="\theta_{\rm r}^{i}=\eta_{i}^{\rm r}=0" class="ltx_Math" display="inline" id="S4.SS2.p1.3.m3.1"><semantics id="S4.SS2.p1.3.m3.1a"><mrow id="S4.SS2.p1.3.m3.1.1" xref="S4.SS2.p1.3.m3.1.1.cmml"><msubsup id="S4.SS2.p1.3.m3.1.1.2" xref="S4.SS2.p1.3.m3.1.1.2.cmml"><mi id="S4.SS2.p1.3.m3.1.1.2.2.2" xref="S4.SS2.p1.3.m3.1.1.2.2.2.cmml">θ</mi><mi id="S4.SS2.p1.3.m3.1.1.2.2.3" mathvariant="normal" xref="S4.SS2.p1.3.m3.1.1.2.2.3.cmml">r</mi><mi id="S4.SS2.p1.3.m3.1.1.2.3" xref="S4.SS2.p1.3.m3.1.1.2.3.cmml">i</mi></msubsup><mo id="S4.SS2.p1.3.m3.1.1.3" xref="S4.SS2.p1.3.m3.1.1.3.cmml">=</mo><msubsup id="S4.SS2.p1.3.m3.1.1.4" xref="S4.SS2.p1.3.m3.1.1.4.cmml"><mi id="S4.SS2.p1.3.m3.1.1.4.2.2" xref="S4.SS2.p1.3.m3.1.1.4.2.2.cmml">η</mi><mi id="S4.SS2.p1.3.m3.1.1.4.2.3" xref="S4.SS2.p1.3.m3.1.1.4.2.3.cmml">i</mi><mi id="S4.SS2.p1.3.m3.1.1.4.3" mathvariant="normal" xref="S4.SS2.p1.3.m3.1.1.4.3.cmml">r</mi></msubsup><mo id="S4.SS2.p1.3.m3.1.1.5" xref="S4.SS2.p1.3.m3.1.1.5.cmml">=</mo><mn id="S4.SS2.p1.3.m3.1.1.6" xref="S4.SS2.p1.3.m3.1.1.6.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p1.3.m3.1b"><apply id="S4.SS2.p1.3.m3.1.1.cmml" xref="S4.SS2.p1.3.m3.1.1"><and id="S4.SS2.p1.3.m3.1.1a.cmml" xref="S4.SS2.p1.3.m3.1.1"></and><apply id="S4.SS2.p1.3.m3.1.1b.cmml" xref="S4.SS2.p1.3.m3.1.1"><eq id="S4.SS2.p1.3.m3.1.1.3.cmml" xref="S4.SS2.p1.3.m3.1.1.3"></eq><apply id="S4.SS2.p1.3.m3.1.1.2.cmml" xref="S4.SS2.p1.3.m3.1.1.2"><csymbol cd="ambiguous" id="S4.SS2.p1.3.m3.1.1.2.1.cmml" xref="S4.SS2.p1.3.m3.1.1.2">superscript</csymbol><apply id="S4.SS2.p1.3.m3.1.1.2.2.cmml" xref="S4.SS2.p1.3.m3.1.1.2"><csymbol cd="ambiguous" id="S4.SS2.p1.3.m3.1.1.2.2.1.cmml" xref="S4.SS2.p1.3.m3.1.1.2">subscript</csymbol><ci id="S4.SS2.p1.3.m3.1.1.2.2.2.cmml" xref="S4.SS2.p1.3.m3.1.1.2.2.2">𝜃</ci><ci id="S4.SS2.p1.3.m3.1.1.2.2.3.cmml" xref="S4.SS2.p1.3.m3.1.1.2.2.3">r</ci></apply><ci id="S4.SS2.p1.3.m3.1.1.2.3.cmml" xref="S4.SS2.p1.3.m3.1.1.2.3">𝑖</ci></apply><apply id="S4.SS2.p1.3.m3.1.1.4.cmml" xref="S4.SS2.p1.3.m3.1.1.4"><csymbol cd="ambiguous" id="S4.SS2.p1.3.m3.1.1.4.1.cmml" xref="S4.SS2.p1.3.m3.1.1.4">superscript</csymbol><apply id="S4.SS2.p1.3.m3.1.1.4.2.cmml" xref="S4.SS2.p1.3.m3.1.1.4"><csymbol cd="ambiguous" id="S4.SS2.p1.3.m3.1.1.4.2.1.cmml" xref="S4.SS2.p1.3.m3.1.1.4">subscript</csymbol><ci id="S4.SS2.p1.3.m3.1.1.4.2.2.cmml" xref="S4.SS2.p1.3.m3.1.1.4.2.2">𝜂</ci><ci id="S4.SS2.p1.3.m3.1.1.4.2.3.cmml" xref="S4.SS2.p1.3.m3.1.1.4.2.3">𝑖</ci></apply><ci id="S4.SS2.p1.3.m3.1.1.4.3.cmml" xref="S4.SS2.p1.3.m3.1.1.4.3">r</ci></apply></apply><apply id="S4.SS2.p1.3.m3.1.1c.cmml" xref="S4.SS2.p1.3.m3.1.1"><eq id="S4.SS2.p1.3.m3.1.1.5.cmml" xref="S4.SS2.p1.3.m3.1.1.5"></eq><share href="https://arxiv.org/html/2406.11224v2#S4.SS2.p1.3.m3.1.1.4.cmml" id="S4.SS2.p1.3.m3.1.1d.cmml" xref="S4.SS2.p1.3.m3.1.1"></share><cn id="S4.SS2.p1.3.m3.1.1.6.cmml" type="integer" xref="S4.SS2.p1.3.m3.1.1.6">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p1.3.m3.1c">\theta_{\rm r}^{i}=\eta_{i}^{\rm r}=0</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p1.3.m3.1d">italic_θ start_POSTSUBSCRIPT roman_r end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT = italic_η start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT roman_r end_POSTSUPERSCRIPT = 0</annotation></semantics></math>. Taking the derivative of <math alttext="\Psi(\theta)" class="ltx_Math" display="inline" id="S4.SS2.p1.4.m4.1"><semantics id="S4.SS2.p1.4.m4.1a"><mrow id="S4.SS2.p1.4.m4.1.2" xref="S4.SS2.p1.4.m4.1.2.cmml"><mi id="S4.SS2.p1.4.m4.1.2.2" mathvariant="normal" xref="S4.SS2.p1.4.m4.1.2.2.cmml">Ψ</mi><mo id="S4.SS2.p1.4.m4.1.2.1" xref="S4.SS2.p1.4.m4.1.2.1.cmml">⁢</mo><mrow id="S4.SS2.p1.4.m4.1.2.3.2" xref="S4.SS2.p1.4.m4.1.2.cmml"><mo id="S4.SS2.p1.4.m4.1.2.3.2.1" stretchy="false" xref="S4.SS2.p1.4.m4.1.2.cmml">(</mo><mi id="S4.SS2.p1.4.m4.1.1" xref="S4.SS2.p1.4.m4.1.1.cmml">θ</mi><mo id="S4.SS2.p1.4.m4.1.2.3.2.2" stretchy="false" xref="S4.SS2.p1.4.m4.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p1.4.m4.1b"><apply id="S4.SS2.p1.4.m4.1.2.cmml" xref="S4.SS2.p1.4.m4.1.2"><times id="S4.SS2.p1.4.m4.1.2.1.cmml" xref="S4.SS2.p1.4.m4.1.2.1"></times><ci id="S4.SS2.p1.4.m4.1.2.2.cmml" xref="S4.SS2.p1.4.m4.1.2.2">Ψ</ci><ci id="S4.SS2.p1.4.m4.1.1.cmml" xref="S4.SS2.p1.4.m4.1.1">𝜃</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p1.4.m4.1c">\Psi(\theta)</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p1.4.m4.1d">roman_Ψ ( italic_θ )</annotation></semantics></math> with respect to <math alttext="t" class="ltx_Math" display="inline" id="S4.SS2.p1.5.m5.1"><semantics id="S4.SS2.p1.5.m5.1a"><mi id="S4.SS2.p1.5.m5.1.1" xref="S4.SS2.p1.5.m5.1.1.cmml">t</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.p1.5.m5.1b"><ci id="S4.SS2.p1.5.m5.1.1.cmml" xref="S4.SS2.p1.5.m5.1.1">𝑡</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p1.5.m5.1c">t</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p1.5.m5.1d">italic_t</annotation></semantics></math> and using (<a class="ltx_ref" href="https://arxiv.org/html/2406.11224v2#S2.E2" title="In 2.1 Information Geometry ‣ 2 Information Geometry and Gradient-Flow Equations ‣ A Hamiltonian approach to the gradient-flow equations in information geometry"><span class="ltx_text ltx_ref_tag">2</span></a>), we have</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="S3.EGx76"> <tbody id="S4.E82"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_left_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\frac{d\Psi(\theta)}{dt}=\frac{\partial\Psi(\theta)}{\partial% \theta^{i}}\frac{d\theta^{i}}{dt}=\eta_{i}\frac{d\theta^{i}}{dt}=g^{ij}(\theta% )\eta_{i}\eta_{j}," class="ltx_Math" display="inline" id="S4.E82.m1.4"><semantics id="S4.E82.m1.4a"><mrow id="S4.E82.m1.4.4.1" xref="S4.E82.m1.4.4.1.1.cmml"><mrow id="S4.E82.m1.4.4.1.1" xref="S4.E82.m1.4.4.1.1.cmml"><mstyle displaystyle="true" id="S4.E82.m1.1.1" xref="S4.E82.m1.1.1.cmml"><mfrac id="S4.E82.m1.1.1a" xref="S4.E82.m1.1.1.cmml"><mrow id="S4.E82.m1.1.1.1" xref="S4.E82.m1.1.1.1.cmml"><mi id="S4.E82.m1.1.1.1.3" xref="S4.E82.m1.1.1.1.3.cmml">d</mi><mo id="S4.E82.m1.1.1.1.2" xref="S4.E82.m1.1.1.1.2.cmml">⁢</mo><mi id="S4.E82.m1.1.1.1.4" mathvariant="normal" xref="S4.E82.m1.1.1.1.4.cmml">Ψ</mi><mo id="S4.E82.m1.1.1.1.2a" xref="S4.E82.m1.1.1.1.2.cmml">⁢</mo><mrow id="S4.E82.m1.1.1.1.5.2" xref="S4.E82.m1.1.1.1.cmml"><mo id="S4.E82.m1.1.1.1.5.2.1" stretchy="false" xref="S4.E82.m1.1.1.1.cmml">(</mo><mi id="S4.E82.m1.1.1.1.1" xref="S4.E82.m1.1.1.1.1.cmml">θ</mi><mo id="S4.E82.m1.1.1.1.5.2.2" stretchy="false" xref="S4.E82.m1.1.1.1.cmml">)</mo></mrow></mrow><mrow id="S4.E82.m1.1.1.3" xref="S4.E82.m1.1.1.3.cmml"><mi id="S4.E82.m1.1.1.3.2" xref="S4.E82.m1.1.1.3.2.cmml">d</mi><mo id="S4.E82.m1.1.1.3.1" xref="S4.E82.m1.1.1.3.1.cmml">⁢</mo><mi id="S4.E82.m1.1.1.3.3" xref="S4.E82.m1.1.1.3.3.cmml">t</mi></mrow></mfrac></mstyle><mo id="S4.E82.m1.4.4.1.1.2" xref="S4.E82.m1.4.4.1.1.2.cmml">=</mo><mrow id="S4.E82.m1.4.4.1.1.3" xref="S4.E82.m1.4.4.1.1.3.cmml"><mstyle displaystyle="true" id="S4.E82.m1.2.2" xref="S4.E82.m1.2.2.cmml"><mfrac id="S4.E82.m1.2.2a" xref="S4.E82.m1.2.2.cmml"><mrow id="S4.E82.m1.2.2.1" xref="S4.E82.m1.2.2.1.cmml"><mo id="S4.E82.m1.2.2.1.2" rspace="0em" xref="S4.E82.m1.2.2.1.2.cmml">∂</mo><mrow id="S4.E82.m1.2.2.1.3" xref="S4.E82.m1.2.2.1.3.cmml"><mi id="S4.E82.m1.2.2.1.3.2" mathvariant="normal" xref="S4.E82.m1.2.2.1.3.2.cmml">Ψ</mi><mo id="S4.E82.m1.2.2.1.3.1" xref="S4.E82.m1.2.2.1.3.1.cmml">⁢</mo><mrow id="S4.E82.m1.2.2.1.3.3.2" xref="S4.E82.m1.2.2.1.3.cmml"><mo id="S4.E82.m1.2.2.1.3.3.2.1" stretchy="false" xref="S4.E82.m1.2.2.1.3.cmml">(</mo><mi id="S4.E82.m1.2.2.1.1" xref="S4.E82.m1.2.2.1.1.cmml">θ</mi><mo id="S4.E82.m1.2.2.1.3.3.2.2" stretchy="false" xref="S4.E82.m1.2.2.1.3.cmml">)</mo></mrow></mrow></mrow><mrow id="S4.E82.m1.2.2.3" xref="S4.E82.m1.2.2.3.cmml"><mo id="S4.E82.m1.2.2.3.1" rspace="0em" xref="S4.E82.m1.2.2.3.1.cmml">∂</mo><msup id="S4.E82.m1.2.2.3.2" xref="S4.E82.m1.2.2.3.2.cmml"><mi id="S4.E82.m1.2.2.3.2.2" xref="S4.E82.m1.2.2.3.2.2.cmml">θ</mi><mi id="S4.E82.m1.2.2.3.2.3" 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xref="S4.E82.m1.4.4.1.1.7.5">subscript</csymbol><ci id="S4.E82.m1.4.4.1.1.7.5.2.cmml" xref="S4.E82.m1.4.4.1.1.7.5.2">𝜂</ci><ci id="S4.E82.m1.4.4.1.1.7.5.3.cmml" xref="S4.E82.m1.4.4.1.1.7.5.3">𝑗</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.E82.m1.4c">\displaystyle\frac{d\Psi(\theta)}{dt}=\frac{\partial\Psi(\theta)}{\partial% \theta^{i}}\frac{d\theta^{i}}{dt}=\eta_{i}\frac{d\theta^{i}}{dt}=g^{ij}(\theta% )\eta_{i}\eta_{j},</annotation><annotation encoding="application/x-llamapun" id="S4.E82.m1.4d">divide start_ARG italic_d roman_Ψ ( italic_θ ) end_ARG start_ARG italic_d italic_t end_ARG = divide start_ARG ∂ roman_Ψ ( italic_θ ) end_ARG start_ARG ∂ italic_θ start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT end_ARG divide start_ARG italic_d italic_θ start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT end_ARG start_ARG italic_d italic_t end_ARG = italic_η start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT divide start_ARG italic_d italic_θ start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT end_ARG start_ARG italic_d italic_t end_ARG = italic_g start_POSTSUPERSCRIPT italic_i italic_j end_POSTSUPERSCRIPT ( italic_θ ) italic_η start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_η start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_left_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(82)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S4.SS2.p1.7">where in the last step we used (<a class="ltx_ref" href="https://arxiv.org/html/2406.11224v2#S2.E12" title="In 2.2 Gradient-Flow Equations ‣ 2 Information Geometry and Gradient-Flow Equations ‣ A Hamiltonian approach to the gradient-flow equations in information geometry"><span class="ltx_text ltx_ref_tag">12</span></a>) with <math alttext="\eta^{\rm r}_{j}=0" class="ltx_Math" display="inline" id="S4.SS2.p1.6.m1.1"><semantics id="S4.SS2.p1.6.m1.1a"><mrow id="S4.SS2.p1.6.m1.1.1" xref="S4.SS2.p1.6.m1.1.1.cmml"><msubsup id="S4.SS2.p1.6.m1.1.1.2" xref="S4.SS2.p1.6.m1.1.1.2.cmml"><mi id="S4.SS2.p1.6.m1.1.1.2.2.2" xref="S4.SS2.p1.6.m1.1.1.2.2.2.cmml">η</mi><mi id="S4.SS2.p1.6.m1.1.1.2.3" xref="S4.SS2.p1.6.m1.1.1.2.3.cmml">j</mi><mi id="S4.SS2.p1.6.m1.1.1.2.2.3" mathvariant="normal" xref="S4.SS2.p1.6.m1.1.1.2.2.3.cmml">r</mi></msubsup><mo id="S4.SS2.p1.6.m1.1.1.1" xref="S4.SS2.p1.6.m1.1.1.1.cmml">=</mo><mn id="S4.SS2.p1.6.m1.1.1.3" xref="S4.SS2.p1.6.m1.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p1.6.m1.1b"><apply id="S4.SS2.p1.6.m1.1.1.cmml" xref="S4.SS2.p1.6.m1.1.1"><eq id="S4.SS2.p1.6.m1.1.1.1.cmml" xref="S4.SS2.p1.6.m1.1.1.1"></eq><apply id="S4.SS2.p1.6.m1.1.1.2.cmml" xref="S4.SS2.p1.6.m1.1.1.2"><csymbol cd="ambiguous" id="S4.SS2.p1.6.m1.1.1.2.1.cmml" xref="S4.SS2.p1.6.m1.1.1.2">subscript</csymbol><apply id="S4.SS2.p1.6.m1.1.1.2.2.cmml" xref="S4.SS2.p1.6.m1.1.1.2"><csymbol cd="ambiguous" id="S4.SS2.p1.6.m1.1.1.2.2.1.cmml" xref="S4.SS2.p1.6.m1.1.1.2">superscript</csymbol><ci id="S4.SS2.p1.6.m1.1.1.2.2.2.cmml" xref="S4.SS2.p1.6.m1.1.1.2.2.2">𝜂</ci><ci id="S4.SS2.p1.6.m1.1.1.2.2.3.cmml" xref="S4.SS2.p1.6.m1.1.1.2.2.3">r</ci></apply><ci id="S4.SS2.p1.6.m1.1.1.2.3.cmml" xref="S4.SS2.p1.6.m1.1.1.2.3">𝑗</ci></apply><cn id="S4.SS2.p1.6.m1.1.1.3.cmml" type="integer" xref="S4.SS2.p1.6.m1.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p1.6.m1.1c">\eta^{\rm r}_{j}=0</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p1.6.m1.1d">italic_η start_POSTSUPERSCRIPT roman_r end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT = 0</annotation></semantics></math>. Now we introduce the quantity <math alttext="\eta^{2}(\theta)" class="ltx_Math" display="inline" id="S4.SS2.p1.7.m2.1"><semantics id="S4.SS2.p1.7.m2.1a"><mrow id="S4.SS2.p1.7.m2.1.2" xref="S4.SS2.p1.7.m2.1.2.cmml"><msup id="S4.SS2.p1.7.m2.1.2.2" xref="S4.SS2.p1.7.m2.1.2.2.cmml"><mi id="S4.SS2.p1.7.m2.1.2.2.2" xref="S4.SS2.p1.7.m2.1.2.2.2.cmml">η</mi><mn id="S4.SS2.p1.7.m2.1.2.2.3" xref="S4.SS2.p1.7.m2.1.2.2.3.cmml">2</mn></msup><mo id="S4.SS2.p1.7.m2.1.2.1" xref="S4.SS2.p1.7.m2.1.2.1.cmml">⁢</mo><mrow id="S4.SS2.p1.7.m2.1.2.3.2" xref="S4.SS2.p1.7.m2.1.2.cmml"><mo id="S4.SS2.p1.7.m2.1.2.3.2.1" stretchy="false" xref="S4.SS2.p1.7.m2.1.2.cmml">(</mo><mi id="S4.SS2.p1.7.m2.1.1" xref="S4.SS2.p1.7.m2.1.1.cmml">θ</mi><mo id="S4.SS2.p1.7.m2.1.2.3.2.2" stretchy="false" xref="S4.SS2.p1.7.m2.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p1.7.m2.1b"><apply id="S4.SS2.p1.7.m2.1.2.cmml" xref="S4.SS2.p1.7.m2.1.2"><times id="S4.SS2.p1.7.m2.1.2.1.cmml" xref="S4.SS2.p1.7.m2.1.2.1"></times><apply id="S4.SS2.p1.7.m2.1.2.2.cmml" xref="S4.SS2.p1.7.m2.1.2.2"><csymbol cd="ambiguous" id="S4.SS2.p1.7.m2.1.2.2.1.cmml" xref="S4.SS2.p1.7.m2.1.2.2">superscript</csymbol><ci id="S4.SS2.p1.7.m2.1.2.2.2.cmml" xref="S4.SS2.p1.7.m2.1.2.2.2">𝜂</ci><cn id="S4.SS2.p1.7.m2.1.2.2.3.cmml" type="integer" xref="S4.SS2.p1.7.m2.1.2.2.3">2</cn></apply><ci id="S4.SS2.p1.7.m2.1.1.cmml" xref="S4.SS2.p1.7.m2.1.1">𝜃</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p1.7.m2.1c">\eta^{2}(\theta)</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p1.7.m2.1d">italic_η start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( italic_θ )</annotation></semantics></math> defined in (<a class="ltx_ref" href="https://arxiv.org/html/2406.11224v2#S2.E21.1" title="In 21 ‣ 2.2 Gradient-Flow Equations ‣ 2 Information Geometry and Gradient-Flow Equations ‣ A Hamiltonian approach to the gradient-flow equations in information geometry"><span class="ltx_text ltx_ref_tag">21a</span></a>), and rewrite the above relation as</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="S3.EGx77"> <tbody id="S4.E83"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_left_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle 0=\eta^{2}(\theta)\Big{(}\tilde{g}^{ij}(\theta)\eta_{i}(\theta)% \eta_{j}(\theta)-1\Big{)}," class="ltx_Math" display="inline" id="S4.E83.m1.5"><semantics id="S4.E83.m1.5a"><mrow id="S4.E83.m1.5.5.1" xref="S4.E83.m1.5.5.1.1.cmml"><mrow id="S4.E83.m1.5.5.1.1" xref="S4.E83.m1.5.5.1.1.cmml"><mn id="S4.E83.m1.5.5.1.1.3" xref="S4.E83.m1.5.5.1.1.3.cmml">0</mn><mo id="S4.E83.m1.5.5.1.1.2" xref="S4.E83.m1.5.5.1.1.2.cmml">=</mo><mrow id="S4.E83.m1.5.5.1.1.1" xref="S4.E83.m1.5.5.1.1.1.cmml"><msup id="S4.E83.m1.5.5.1.1.1.3" xref="S4.E83.m1.5.5.1.1.1.3.cmml"><mi id="S4.E83.m1.5.5.1.1.1.3.2" xref="S4.E83.m1.5.5.1.1.1.3.2.cmml">η</mi><mn 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italic_j end_POSTSUPERSCRIPT ( italic_θ ) := divide start_ARG 1 end_ARG start_ARG italic_η start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( italic_θ ) end_ARG italic_g start_POSTSUPERSCRIPT italic_i italic_j end_POSTSUPERSCRIPT ( italic_θ ) .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_left_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(84)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S4.SS2.p1.10">We can regard the null relation (<a class="ltx_ref" href="https://arxiv.org/html/2406.11224v2#S4.E83" title="In 4.2 Applications to the gradient-flow equations ‣ 4 The motions of a light-like particle in a pseudo Riemann space ‣ A Hamiltonian approach to the gradient-flow equations in information geometry"><span class="ltx_text ltx_ref_tag">83</span></a>) as (<a class="ltx_ref" href="https://arxiv.org/html/2406.11224v2#S4.E68" title="In 4 The motions of a light-like particle in a pseudo Riemann space ‣ A Hamiltonian approach to the gradient-flow equations in information geometry"><span class="ltx_text ltx_ref_tag">68</span></a>) by setting <math alttext="p_{i}=\eta_{i},p_{0}=-1,\alpha^{2}=1/\eta^{2}(\theta),\beta^{i}=0" class="ltx_Math" display="inline" id="S4.SS2.p1.8.m1.3"><semantics id="S4.SS2.p1.8.m1.3a"><mrow id="S4.SS2.p1.8.m1.3.3.2" xref="S4.SS2.p1.8.m1.3.3.3.cmml"><mrow id="S4.SS2.p1.8.m1.2.2.1.1" xref="S4.SS2.p1.8.m1.2.2.1.1.cmml"><msub id="S4.SS2.p1.8.m1.2.2.1.1.2" xref="S4.SS2.p1.8.m1.2.2.1.1.2.cmml"><mi id="S4.SS2.p1.8.m1.2.2.1.1.2.2" xref="S4.SS2.p1.8.m1.2.2.1.1.2.2.cmml">p</mi><mi id="S4.SS2.p1.8.m1.2.2.1.1.2.3" xref="S4.SS2.p1.8.m1.2.2.1.1.2.3.cmml">i</mi></msub><mo id="S4.SS2.p1.8.m1.2.2.1.1.1" xref="S4.SS2.p1.8.m1.2.2.1.1.1.cmml">=</mo><msub id="S4.SS2.p1.8.m1.2.2.1.1.3" xref="S4.SS2.p1.8.m1.2.2.1.1.3.cmml"><mi id="S4.SS2.p1.8.m1.2.2.1.1.3.2" xref="S4.SS2.p1.8.m1.2.2.1.1.3.2.cmml">η</mi><mi 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id="S4.SS2.p1.8.m1.3.3.2.2.2.2.2.2.2.1.cmml" xref="S4.SS2.p1.8.m1.3.3.2.2.2.2.2.2.2">superscript</csymbol><ci id="S4.SS2.p1.8.m1.3.3.2.2.2.2.2.2.2.2.cmml" xref="S4.SS2.p1.8.m1.3.3.2.2.2.2.2.2.2.2">𝛽</ci><ci id="S4.SS2.p1.8.m1.3.3.2.2.2.2.2.2.2.3.cmml" xref="S4.SS2.p1.8.m1.3.3.2.2.2.2.2.2.2.3">𝑖</ci></apply><cn id="S4.SS2.p1.8.m1.3.3.2.2.2.2.2.2.3.cmml" type="integer" xref="S4.SS2.p1.8.m1.3.3.2.2.2.2.2.2.3">0</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p1.8.m1.3c">p_{i}=\eta_{i},p_{0}=-1,\alpha^{2}=1/\eta^{2}(\theta),\beta^{i}=0</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p1.8.m1.3d">italic_p start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = italic_η start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT , italic_p start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT = - 1 , italic_α start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT = 1 / italic_η start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( italic_θ ) , italic_β start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT = 0</annotation></semantics></math>, and <math alttext="\gamma^{ij}=g^{ij}(\theta)" class="ltx_Math" display="inline" id="S4.SS2.p1.9.m2.1"><semantics id="S4.SS2.p1.9.m2.1a"><mrow id="S4.SS2.p1.9.m2.1.2" xref="S4.SS2.p1.9.m2.1.2.cmml"><msup id="S4.SS2.p1.9.m2.1.2.2" xref="S4.SS2.p1.9.m2.1.2.2.cmml"><mi id="S4.SS2.p1.9.m2.1.2.2.2" xref="S4.SS2.p1.9.m2.1.2.2.2.cmml">γ</mi><mrow id="S4.SS2.p1.9.m2.1.2.2.3" xref="S4.SS2.p1.9.m2.1.2.2.3.cmml"><mi id="S4.SS2.p1.9.m2.1.2.2.3.2" xref="S4.SS2.p1.9.m2.1.2.2.3.2.cmml">i</mi><mo id="S4.SS2.p1.9.m2.1.2.2.3.1" xref="S4.SS2.p1.9.m2.1.2.2.3.1.cmml">⁢</mo><mi id="S4.SS2.p1.9.m2.1.2.2.3.3" xref="S4.SS2.p1.9.m2.1.2.2.3.3.cmml">j</mi></mrow></msup><mo id="S4.SS2.p1.9.m2.1.2.1" xref="S4.SS2.p1.9.m2.1.2.1.cmml">=</mo><mrow id="S4.SS2.p1.9.m2.1.2.3" xref="S4.SS2.p1.9.m2.1.2.3.cmml"><msup id="S4.SS2.p1.9.m2.1.2.3.2" xref="S4.SS2.p1.9.m2.1.2.3.2.cmml"><mi id="S4.SS2.p1.9.m2.1.2.3.2.2" xref="S4.SS2.p1.9.m2.1.2.3.2.2.cmml">g</mi><mrow id="S4.SS2.p1.9.m2.1.2.3.2.3" xref="S4.SS2.p1.9.m2.1.2.3.2.3.cmml"><mi id="S4.SS2.p1.9.m2.1.2.3.2.3.2" xref="S4.SS2.p1.9.m2.1.2.3.2.3.2.cmml">i</mi><mo id="S4.SS2.p1.9.m2.1.2.3.2.3.1" xref="S4.SS2.p1.9.m2.1.2.3.2.3.1.cmml">⁢</mo><mi id="S4.SS2.p1.9.m2.1.2.3.2.3.3" xref="S4.SS2.p1.9.m2.1.2.3.2.3.3.cmml">j</mi></mrow></msup><mo id="S4.SS2.p1.9.m2.1.2.3.1" xref="S4.SS2.p1.9.m2.1.2.3.1.cmml">⁢</mo><mrow id="S4.SS2.p1.9.m2.1.2.3.3.2" xref="S4.SS2.p1.9.m2.1.2.3.cmml"><mo id="S4.SS2.p1.9.m2.1.2.3.3.2.1" stretchy="false" xref="S4.SS2.p1.9.m2.1.2.3.cmml">(</mo><mi id="S4.SS2.p1.9.m2.1.1" xref="S4.SS2.p1.9.m2.1.1.cmml">θ</mi><mo id="S4.SS2.p1.9.m2.1.2.3.3.2.2" stretchy="false" xref="S4.SS2.p1.9.m2.1.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p1.9.m2.1b"><apply id="S4.SS2.p1.9.m2.1.2.cmml" xref="S4.SS2.p1.9.m2.1.2"><eq id="S4.SS2.p1.9.m2.1.2.1.cmml" xref="S4.SS2.p1.9.m2.1.2.1"></eq><apply id="S4.SS2.p1.9.m2.1.2.2.cmml" xref="S4.SS2.p1.9.m2.1.2.2"><csymbol cd="ambiguous" id="S4.SS2.p1.9.m2.1.2.2.1.cmml" xref="S4.SS2.p1.9.m2.1.2.2">superscript</csymbol><ci id="S4.SS2.p1.9.m2.1.2.2.2.cmml" xref="S4.SS2.p1.9.m2.1.2.2.2">𝛾</ci><apply id="S4.SS2.p1.9.m2.1.2.2.3.cmml" xref="S4.SS2.p1.9.m2.1.2.2.3"><times id="S4.SS2.p1.9.m2.1.2.2.3.1.cmml" xref="S4.SS2.p1.9.m2.1.2.2.3.1"></times><ci id="S4.SS2.p1.9.m2.1.2.2.3.2.cmml" xref="S4.SS2.p1.9.m2.1.2.2.3.2">𝑖</ci><ci id="S4.SS2.p1.9.m2.1.2.2.3.3.cmml" xref="S4.SS2.p1.9.m2.1.2.2.3.3">𝑗</ci></apply></apply><apply id="S4.SS2.p1.9.m2.1.2.3.cmml" xref="S4.SS2.p1.9.m2.1.2.3"><times id="S4.SS2.p1.9.m2.1.2.3.1.cmml" xref="S4.SS2.p1.9.m2.1.2.3.1"></times><apply id="S4.SS2.p1.9.m2.1.2.3.2.cmml" xref="S4.SS2.p1.9.m2.1.2.3.2"><csymbol cd="ambiguous" id="S4.SS2.p1.9.m2.1.2.3.2.1.cmml" xref="S4.SS2.p1.9.m2.1.2.3.2">superscript</csymbol><ci id="S4.SS2.p1.9.m2.1.2.3.2.2.cmml" xref="S4.SS2.p1.9.m2.1.2.3.2.2">𝑔</ci><apply id="S4.SS2.p1.9.m2.1.2.3.2.3.cmml" xref="S4.SS2.p1.9.m2.1.2.3.2.3"><times id="S4.SS2.p1.9.m2.1.2.3.2.3.1.cmml" xref="S4.SS2.p1.9.m2.1.2.3.2.3.1"></times><ci id="S4.SS2.p1.9.m2.1.2.3.2.3.2.cmml" xref="S4.SS2.p1.9.m2.1.2.3.2.3.2">𝑖</ci><ci id="S4.SS2.p1.9.m2.1.2.3.2.3.3.cmml" xref="S4.SS2.p1.9.m2.1.2.3.2.3.3">𝑗</ci></apply></apply><ci id="S4.SS2.p1.9.m2.1.1.cmml" xref="S4.SS2.p1.9.m2.1.1">𝜃</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p1.9.m2.1c">\gamma^{ij}=g^{ij}(\theta)</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p1.9.m2.1d">italic_γ start_POSTSUPERSCRIPT italic_i italic_j end_POSTSUPERSCRIPT = italic_g start_POSTSUPERSCRIPT italic_i italic_j end_POSTSUPERSCRIPT ( italic_θ )</annotation></semantics></math>. Then the corresponding Hamiltonian <math alttext="\mathcal{H}^{+}" class="ltx_Math" display="inline" id="S4.SS2.p1.10.m3.1"><semantics id="S4.SS2.p1.10.m3.1a"><msup id="S4.SS2.p1.10.m3.1.1" xref="S4.SS2.p1.10.m3.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.SS2.p1.10.m3.1.1.2" xref="S4.SS2.p1.10.m3.1.1.2.cmml">ℋ</mi><mo id="S4.SS2.p1.10.m3.1.1.3" xref="S4.SS2.p1.10.m3.1.1.3.cmml">+</mo></msup><annotation-xml encoding="MathML-Content" id="S4.SS2.p1.10.m3.1b"><apply id="S4.SS2.p1.10.m3.1.1.cmml" xref="S4.SS2.p1.10.m3.1.1"><csymbol cd="ambiguous" id="S4.SS2.p1.10.m3.1.1.1.cmml" xref="S4.SS2.p1.10.m3.1.1">superscript</csymbol><ci id="S4.SS2.p1.10.m3.1.1.2.cmml" xref="S4.SS2.p1.10.m3.1.1.2">ℋ</ci><plus id="S4.SS2.p1.10.m3.1.1.3.cmml" xref="S4.SS2.p1.10.m3.1.1.3"></plus></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p1.10.m3.1c">\mathcal{H}^{+}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p1.10.m3.1d">caligraphic_H start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT</annotation></semantics></math> in (<a class="ltx_ref" href="https://arxiv.org/html/2406.11224v2#S4.E73" title="In 4 The motions of a light-like particle in a pseudo Riemann space ‣ A Hamiltonian approach to the gradient-flow equations in information geometry"><span class="ltx_text ltx_ref_tag">73</span></a>) becomes</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="S3.EGx79"> <tbody id="S4.E85"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_left_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\mathcal{H}^{+}(\theta,\eta)=\sqrt{\tilde{g}^{ij}(\theta)\,\eta_{% i}\eta_{j}}=\sqrt{\frac{g^{ij}(\theta)}{\eta^{2}(\theta)}\,\eta_{i}\eta_{j}}," class="ltx_Math" display="inline" id="S4.E85.m1.6"><semantics id="S4.E85.m1.6a"><mrow id="S4.E85.m1.6.6.1" xref="S4.E85.m1.6.6.1.1.cmml"><mrow id="S4.E85.m1.6.6.1.1" xref="S4.E85.m1.6.6.1.1.cmml"><mrow 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xref="S4.E85.m1.3.3.2.4">subscript</csymbol><ci id="S4.E85.m1.3.3.2.4.2.cmml" xref="S4.E85.m1.3.3.2.4.2">𝜂</ci><ci id="S4.E85.m1.3.3.2.4.3.cmml" xref="S4.E85.m1.3.3.2.4.3">𝑖</ci></apply><apply id="S4.E85.m1.3.3.2.5.cmml" xref="S4.E85.m1.3.3.2.5"><csymbol cd="ambiguous" id="S4.E85.m1.3.3.2.5.1.cmml" xref="S4.E85.m1.3.3.2.5">subscript</csymbol><ci id="S4.E85.m1.3.3.2.5.2.cmml" xref="S4.E85.m1.3.3.2.5.2">𝜂</ci><ci id="S4.E85.m1.3.3.2.5.3.cmml" xref="S4.E85.m1.3.3.2.5.3">𝑗</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.E85.m1.6c">\displaystyle\mathcal{H}^{+}(\theta,\eta)=\sqrt{\tilde{g}^{ij}(\theta)\,\eta_{% i}\eta_{j}}=\sqrt{\frac{g^{ij}(\theta)}{\eta^{2}(\theta)}\,\eta_{i}\eta_{j}},</annotation><annotation encoding="application/x-llamapun" id="S4.E85.m1.6d">caligraphic_H start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT ( italic_θ , italic_η ) = square-root start_ARG over~ start_ARG italic_g end_ARG start_POSTSUPERSCRIPT italic_i italic_j end_POSTSUPERSCRIPT ( italic_θ ) italic_η start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_η start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT end_ARG = square-root start_ARG divide start_ARG italic_g start_POSTSUPERSCRIPT italic_i italic_j end_POSTSUPERSCRIPT ( italic_θ ) end_ARG start_ARG italic_η start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( italic_θ ) end_ARG italic_η start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_η start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT end_ARG ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_left_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(85)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S4.SS2.p1.12">which describes the gradient-flows in IG as a light-like particle motion in the pseudo-Riemann space with <math alttext="G^{\mu\nu}(x)=\eta^{2}(\theta)\,{\rm diag.}(-1,\tilde{g}^{ij}(\theta))" 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xref="S4.SS2.p1.11.m1.5.5.2.2.2.2.2.3.3">𝑗</ci></apply></apply><ci id="S4.SS2.p1.11.m1.3.3.cmml" xref="S4.SS2.p1.11.m1.3.3">𝜃</ci></apply></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p1.11.m1.5c">G^{\mu\nu}(x)=\eta^{2}(\theta)\,{\rm diag.}(-1,\tilde{g}^{ij}(\theta))</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p1.11.m1.5d">italic_G start_POSTSUPERSCRIPT italic_μ italic_ν end_POSTSUPERSCRIPT ( italic_x ) = italic_η start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( italic_θ ) roman_diag . ( - 1 , over~ start_ARG italic_g end_ARG start_POSTSUPERSCRIPT italic_i italic_j end_POSTSUPERSCRIPT ( italic_θ ) )</annotation></semantics></math>. The conformal factor <math alttext="1/\alpha^{2}=\eta^{2}(\theta)" class="ltx_Math" display="inline" id="S4.SS2.p1.12.m2.1"><semantics id="S4.SS2.p1.12.m2.1a"><mrow id="S4.SS2.p1.12.m2.1.2" xref="S4.SS2.p1.12.m2.1.2.cmml"><mrow id="S4.SS2.p1.12.m2.1.2.2" xref="S4.SS2.p1.12.m2.1.2.2.cmml"><mn id="S4.SS2.p1.12.m2.1.2.2.2" xref="S4.SS2.p1.12.m2.1.2.2.2.cmml">1</mn><mo id="S4.SS2.p1.12.m2.1.2.2.1" xref="S4.SS2.p1.12.m2.1.2.2.1.cmml">/</mo><msup id="S4.SS2.p1.12.m2.1.2.2.3" xref="S4.SS2.p1.12.m2.1.2.2.3.cmml"><mi id="S4.SS2.p1.12.m2.1.2.2.3.2" xref="S4.SS2.p1.12.m2.1.2.2.3.2.cmml">α</mi><mn id="S4.SS2.p1.12.m2.1.2.2.3.3" xref="S4.SS2.p1.12.m2.1.2.2.3.3.cmml">2</mn></msup></mrow><mo id="S4.SS2.p1.12.m2.1.2.1" xref="S4.SS2.p1.12.m2.1.2.1.cmml">=</mo><mrow id="S4.SS2.p1.12.m2.1.2.3" xref="S4.SS2.p1.12.m2.1.2.3.cmml"><msup id="S4.SS2.p1.12.m2.1.2.3.2" xref="S4.SS2.p1.12.m2.1.2.3.2.cmml"><mi id="S4.SS2.p1.12.m2.1.2.3.2.2" xref="S4.SS2.p1.12.m2.1.2.3.2.2.cmml">η</mi><mn id="S4.SS2.p1.12.m2.1.2.3.2.3" xref="S4.SS2.p1.12.m2.1.2.3.2.3.cmml">2</mn></msup><mo id="S4.SS2.p1.12.m2.1.2.3.1" xref="S4.SS2.p1.12.m2.1.2.3.1.cmml">⁢</mo><mrow id="S4.SS2.p1.12.m2.1.2.3.3.2" xref="S4.SS2.p1.12.m2.1.2.3.cmml"><mo id="S4.SS2.p1.12.m2.1.2.3.3.2.1" stretchy="false" xref="S4.SS2.p1.12.m2.1.2.3.cmml">(</mo><mi id="S4.SS2.p1.12.m2.1.1" xref="S4.SS2.p1.12.m2.1.1.cmml">θ</mi><mo id="S4.SS2.p1.12.m2.1.2.3.3.2.2" stretchy="false" xref="S4.SS2.p1.12.m2.1.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p1.12.m2.1b"><apply id="S4.SS2.p1.12.m2.1.2.cmml" xref="S4.SS2.p1.12.m2.1.2"><eq id="S4.SS2.p1.12.m2.1.2.1.cmml" xref="S4.SS2.p1.12.m2.1.2.1"></eq><apply id="S4.SS2.p1.12.m2.1.2.2.cmml" xref="S4.SS2.p1.12.m2.1.2.2"><divide id="S4.SS2.p1.12.m2.1.2.2.1.cmml" xref="S4.SS2.p1.12.m2.1.2.2.1"></divide><cn id="S4.SS2.p1.12.m2.1.2.2.2.cmml" type="integer" xref="S4.SS2.p1.12.m2.1.2.2.2">1</cn><apply id="S4.SS2.p1.12.m2.1.2.2.3.cmml" xref="S4.SS2.p1.12.m2.1.2.2.3"><csymbol cd="ambiguous" id="S4.SS2.p1.12.m2.1.2.2.3.1.cmml" xref="S4.SS2.p1.12.m2.1.2.2.3">superscript</csymbol><ci id="S4.SS2.p1.12.m2.1.2.2.3.2.cmml" xref="S4.SS2.p1.12.m2.1.2.2.3.2">𝛼</ci><cn id="S4.SS2.p1.12.m2.1.2.2.3.3.cmml" type="integer" xref="S4.SS2.p1.12.m2.1.2.2.3.3">2</cn></apply></apply><apply id="S4.SS2.p1.12.m2.1.2.3.cmml" xref="S4.SS2.p1.12.m2.1.2.3"><times id="S4.SS2.p1.12.m2.1.2.3.1.cmml" xref="S4.SS2.p1.12.m2.1.2.3.1"></times><apply id="S4.SS2.p1.12.m2.1.2.3.2.cmml" xref="S4.SS2.p1.12.m2.1.2.3.2"><csymbol cd="ambiguous" id="S4.SS2.p1.12.m2.1.2.3.2.1.cmml" xref="S4.SS2.p1.12.m2.1.2.3.2">superscript</csymbol><ci id="S4.SS2.p1.12.m2.1.2.3.2.2.cmml" xref="S4.SS2.p1.12.m2.1.2.3.2.2">𝜂</ci><cn id="S4.SS2.p1.12.m2.1.2.3.2.3.cmml" type="integer" xref="S4.SS2.p1.12.m2.1.2.3.2.3">2</cn></apply><ci id="S4.SS2.p1.12.m2.1.1.cmml" xref="S4.SS2.p1.12.m2.1.1">𝜃</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p1.12.m2.1c">1/\alpha^{2}=\eta^{2}(\theta)</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p1.12.m2.1d">1 / italic_α start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT = italic_η start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( italic_θ )</annotation></semantics></math> does not affect the light-like geodesic but change their parametrization <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2406.11224v2#bib.bib25" title="">25</a>]</cite>, which is explained in Appendix <a class="ltx_ref" href="https://arxiv.org/html/2406.11224v2#A2" title="Appendix B Conformal rescaling as reparametrization ‣ A Hamiltonian approach to the gradient-flow equations in information geometry"><span class="ltx_text ltx_ref_tag">B</span></a>. The corresponding Hamilton’s equations of motion are</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="S3.EGx80"> <tbody id="S4.Ex5"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_left_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\frac{d\theta^{i}}{dx^{0}}=\frac{\partial\mathcal{H}^{+}}{% \partial\eta_{i}}=\frac{1}{\eta^{2}(\theta)}\,g^{ij}(\theta)\eta_{j}," class="ltx_Math" display="inline" id="S4.Ex5.m2.3"><semantics id="S4.Ex5.m2.3a"><mrow id="S4.Ex5.m2.3.3.1" xref="S4.Ex5.m2.3.3.1.1.cmml"><mrow id="S4.Ex5.m2.3.3.1.1" xref="S4.Ex5.m2.3.3.1.1.cmml"><mstyle displaystyle="true" id="S4.Ex5.m2.3.3.1.1.2" xref="S4.Ex5.m2.3.3.1.1.2.cmml"><mfrac id="S4.Ex5.m2.3.3.1.1.2a" xref="S4.Ex5.m2.3.3.1.1.2.cmml"><mrow id="S4.Ex5.m2.3.3.1.1.2.2" xref="S4.Ex5.m2.3.3.1.1.2.2.cmml"><mi id="S4.Ex5.m2.3.3.1.1.2.2.2" 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{\partial\theta^{i}}\eta_{j}\eta_{k}+\frac{\partial\eta^{2}(\theta)}{\partial% \theta^{i}}\right).</annotation><annotation encoding="application/x-llamapun" id="S4.E86.m1.4d">= divide start_ARG 1 end_ARG start_ARG 2 italic_η start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( italic_θ ) end_ARG ( - divide start_ARG ∂ italic_g start_POSTSUPERSCRIPT italic_j italic_k end_POSTSUPERSCRIPT ( italic_θ ) end_ARG start_ARG ∂ italic_θ start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT end_ARG italic_η start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT italic_η start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT + divide start_ARG ∂ italic_η start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( italic_θ ) end_ARG start_ARG ∂ italic_θ start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT end_ARG ) .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_left_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(86)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S4.SS2.p1.14">We can change the parametrization from <math alttext="x^{0}" class="ltx_Math" display="inline" id="S4.SS2.p1.13.m1.1"><semantics id="S4.SS2.p1.13.m1.1a"><msup id="S4.SS2.p1.13.m1.1.1" xref="S4.SS2.p1.13.m1.1.1.cmml"><mi id="S4.SS2.p1.13.m1.1.1.2" xref="S4.SS2.p1.13.m1.1.1.2.cmml">x</mi><mn id="S4.SS2.p1.13.m1.1.1.3" xref="S4.SS2.p1.13.m1.1.1.3.cmml">0</mn></msup><annotation-xml encoding="MathML-Content" id="S4.SS2.p1.13.m1.1b"><apply id="S4.SS2.p1.13.m1.1.1.cmml" xref="S4.SS2.p1.13.m1.1.1"><csymbol cd="ambiguous" id="S4.SS2.p1.13.m1.1.1.1.cmml" xref="S4.SS2.p1.13.m1.1.1">superscript</csymbol><ci id="S4.SS2.p1.13.m1.1.1.2.cmml" xref="S4.SS2.p1.13.m1.1.1.2">𝑥</ci><cn id="S4.SS2.p1.13.m1.1.1.3.cmml" type="integer" xref="S4.SS2.p1.13.m1.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p1.13.m1.1c">x^{0}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p1.13.m1.1d">italic_x start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT</annotation></semantics></math> to <math alttext="t" class="ltx_Math" display="inline" id="S4.SS2.p1.14.m2.1"><semantics id="S4.SS2.p1.14.m2.1a"><mi id="S4.SS2.p1.14.m2.1.1" xref="S4.SS2.p1.14.m2.1.1.cmml">t</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.p1.14.m2.1b"><ci id="S4.SS2.p1.14.m2.1.1.cmml" xref="S4.SS2.p1.14.m2.1.1">𝑡</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p1.14.m2.1c">t</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p1.14.m2.1d">italic_t</annotation></semantics></math> according to</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="S3.EGx81"> <tbody id="S4.E87"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_left_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle dx^{0}=\eta^{2}(\theta)dt." class="ltx_Math" display="inline" id="S4.E87.m1.2"><semantics id="S4.E87.m1.2a"><mrow id="S4.E87.m1.2.2.1" xref="S4.E87.m1.2.2.1.1.cmml"><mrow id="S4.E87.m1.2.2.1.1" xref="S4.E87.m1.2.2.1.1.cmml"><mrow id="S4.E87.m1.2.2.1.1.2" xref="S4.E87.m1.2.2.1.1.2.cmml"><mi id="S4.E87.m1.2.2.1.1.2.2" xref="S4.E87.m1.2.2.1.1.2.2.cmml">d</mi><mo id="S4.E87.m1.2.2.1.1.2.1" xref="S4.E87.m1.2.2.1.1.2.1.cmml">⁢</mo><msup id="S4.E87.m1.2.2.1.1.2.3" xref="S4.E87.m1.2.2.1.1.2.3.cmml"><mi id="S4.E87.m1.2.2.1.1.2.3.2" xref="S4.E87.m1.2.2.1.1.2.3.2.cmml">x</mi><mn id="S4.E87.m1.2.2.1.1.2.3.3" xref="S4.E87.m1.2.2.1.1.2.3.3.cmml">0</mn></msup></mrow><mo id="S4.E87.m1.2.2.1.1.1" xref="S4.E87.m1.2.2.1.1.1.cmml">=</mo><mrow id="S4.E87.m1.2.2.1.1.3" xref="S4.E87.m1.2.2.1.1.3.cmml"><msup id="S4.E87.m1.2.2.1.1.3.2" xref="S4.E87.m1.2.2.1.1.3.2.cmml"><mi id="S4.E87.m1.2.2.1.1.3.2.2" xref="S4.E87.m1.2.2.1.1.3.2.2.cmml">η</mi><mn id="S4.E87.m1.2.2.1.1.3.2.3" xref="S4.E87.m1.2.2.1.1.3.2.3.cmml">2</mn></msup><mo id="S4.E87.m1.2.2.1.1.3.1" xref="S4.E87.m1.2.2.1.1.3.1.cmml">⁢</mo><mrow id="S4.E87.m1.2.2.1.1.3.3.2" xref="S4.E87.m1.2.2.1.1.3.cmml"><mo id="S4.E87.m1.2.2.1.1.3.3.2.1" stretchy="false" xref="S4.E87.m1.2.2.1.1.3.cmml">(</mo><mi id="S4.E87.m1.1.1" xref="S4.E87.m1.1.1.cmml">θ</mi><mo id="S4.E87.m1.2.2.1.1.3.3.2.2" stretchy="false" xref="S4.E87.m1.2.2.1.1.3.cmml">)</mo></mrow><mo id="S4.E87.m1.2.2.1.1.3.1a" xref="S4.E87.m1.2.2.1.1.3.1.cmml">⁢</mo><mi id="S4.E87.m1.2.2.1.1.3.4" xref="S4.E87.m1.2.2.1.1.3.4.cmml">d</mi><mo id="S4.E87.m1.2.2.1.1.3.1b" xref="S4.E87.m1.2.2.1.1.3.1.cmml">⁢</mo><mi id="S4.E87.m1.2.2.1.1.3.5" xref="S4.E87.m1.2.2.1.1.3.5.cmml">t</mi></mrow></mrow><mo id="S4.E87.m1.2.2.1.2" lspace="0em" xref="S4.E87.m1.2.2.1.1.cmml">.</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.E87.m1.2b"><apply id="S4.E87.m1.2.2.1.1.cmml" xref="S4.E87.m1.2.2.1"><eq id="S4.E87.m1.2.2.1.1.1.cmml" xref="S4.E87.m1.2.2.1.1.1"></eq><apply id="S4.E87.m1.2.2.1.1.2.cmml" xref="S4.E87.m1.2.2.1.1.2"><times id="S4.E87.m1.2.2.1.1.2.1.cmml" xref="S4.E87.m1.2.2.1.1.2.1"></times><ci id="S4.E87.m1.2.2.1.1.2.2.cmml" xref="S4.E87.m1.2.2.1.1.2.2">𝑑</ci><apply id="S4.E87.m1.2.2.1.1.2.3.cmml" xref="S4.E87.m1.2.2.1.1.2.3"><csymbol cd="ambiguous" id="S4.E87.m1.2.2.1.1.2.3.1.cmml" xref="S4.E87.m1.2.2.1.1.2.3">superscript</csymbol><ci id="S4.E87.m1.2.2.1.1.2.3.2.cmml" xref="S4.E87.m1.2.2.1.1.2.3.2">𝑥</ci><cn id="S4.E87.m1.2.2.1.1.2.3.3.cmml" type="integer" xref="S4.E87.m1.2.2.1.1.2.3.3">0</cn></apply></apply><apply id="S4.E87.m1.2.2.1.1.3.cmml" xref="S4.E87.m1.2.2.1.1.3"><times id="S4.E87.m1.2.2.1.1.3.1.cmml" xref="S4.E87.m1.2.2.1.1.3.1"></times><apply id="S4.E87.m1.2.2.1.1.3.2.cmml" xref="S4.E87.m1.2.2.1.1.3.2"><csymbol cd="ambiguous" id="S4.E87.m1.2.2.1.1.3.2.1.cmml" xref="S4.E87.m1.2.2.1.1.3.2">superscript</csymbol><ci id="S4.E87.m1.2.2.1.1.3.2.2.cmml" xref="S4.E87.m1.2.2.1.1.3.2.2">𝜂</ci><cn id="S4.E87.m1.2.2.1.1.3.2.3.cmml" type="integer" xref="S4.E87.m1.2.2.1.1.3.2.3">2</cn></apply><ci id="S4.E87.m1.1.1.cmml" xref="S4.E87.m1.1.1">𝜃</ci><ci id="S4.E87.m1.2.2.1.1.3.4.cmml" xref="S4.E87.m1.2.2.1.1.3.4">𝑑</ci><ci id="S4.E87.m1.2.2.1.1.3.5.cmml" xref="S4.E87.m1.2.2.1.1.3.5">𝑡</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.E87.m1.2c">\displaystyle dx^{0}=\eta^{2}(\theta)dt.</annotation><annotation encoding="application/x-llamapun" id="S4.E87.m1.2d">italic_d italic_x start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT = italic_η start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( italic_θ ) italic_d italic_t .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_left_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(87)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S4.SS2.p1.16">This maps the equations (<a class="ltx_ref" href="https://arxiv.org/html/2406.11224v2#S4.Ex5" title="4.2 Applications to the gradient-flow equations ‣ 4 The motions of a light-like particle in a pseudo Riemann space ‣ A Hamiltonian approach to the gradient-flow equations in information geometry"><span class="ltx_text ltx_ref_tag">4.2</span></a>) to those for the Hamiltonian (<a class="ltx_ref" href="https://arxiv.org/html/2406.11224v2#S2.E20.1" title="In 20 ‣ 2.2 Gradient-Flow Equations ‣ 2 Information Geometry and Gradient-Flow Equations ‣ A Hamiltonian approach to the gradient-flow equations in information geometry"><span class="ltx_text ltx_ref_tag">20a</span></a>). In this way we have rederived the Hamiltonian (<a class="ltx_ref" href="https://arxiv.org/html/2406.11224v2#S2.E20.1" title="In 20 ‣ 2.2 Gradient-Flow Equations ‣ 2 Information Geometry and Gradient-Flow Equations ‣ A Hamiltonian approach to the gradient-flow equations in information geometry"><span class="ltx_text ltx_ref_tag">20a</span></a>) describing the gradient-flows in IG. The other Hamiltonian (<a class="ltx_ref" href="https://arxiv.org/html/2406.11224v2#S2.E20.2" title="In 20 ‣ 2.2 Gradient-Flow Equations ‣ 2 Information Geometry and Gradient-Flow Equations ‣ A Hamiltonian approach to the gradient-flow equations in information geometry"><span class="ltx_text ltx_ref_tag">20b</span></a>) can be obtained in a similar way.</p> </div> <div class="ltx_para" id="S4.SS2.p2"> <p class="ltx_p" id="S4.SS2.p2.7">Secondly, we consider the RF deformed gradient-flow equations (<a class="ltx_ref" href="https://arxiv.org/html/2406.11224v2#S2.E28" title="In 2.3 Randers-Finsler deformation of the gradient-flow equations ‣ 2 Information Geometry and Gradient-Flow Equations ‣ A Hamiltonian approach to the gradient-flow equations in information geometry"><span class="ltx_text ltx_ref_tag">28</span></a>). Arranging the relation (<a class="ltx_ref" href="https://arxiv.org/html/2406.11224v2#S2.E30" title="In 2.3 Randers-Finsler deformation of the gradient-flow equations ‣ 2 Information Geometry and Gradient-Flow Equations ‣ A Hamiltonian approach to the gradient-flow equations in information geometry"><span class="ltx_text ltx_ref_tag">30</span></a>) leads to</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="S3.EGx82"> <tbody id="S4.E88"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_left_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle 0=g^{ij}(\theta)\eta_{i}\eta_{j}\!-\!2A^{i}(\theta)\eta_{i}\!-\!% (\chi^{2}(\theta)\!-\!A^{2}(\theta))," class="ltx_Math" display="inline" id="S4.E88.m1.5"><semantics id="S4.E88.m1.5a"><mrow id="S4.E88.m1.5.5.1" xref="S4.E88.m1.5.5.1.1.cmml"><mrow id="S4.E88.m1.5.5.1.1" xref="S4.E88.m1.5.5.1.1.cmml"><mn id="S4.E88.m1.5.5.1.1.3" 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(\chi^{2}(\theta)\!-\!A^{2}(\theta)),</annotation><annotation encoding="application/x-llamapun" id="S4.E88.m1.5d">0 = italic_g start_POSTSUPERSCRIPT italic_i italic_j end_POSTSUPERSCRIPT ( italic_θ ) italic_η start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_η start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT - 2 italic_A start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT ( italic_θ ) italic_η start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT - ( italic_χ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( italic_θ ) - italic_A start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( italic_θ ) ) ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_left_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(88)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S4.SS2.p2.2">where we used <math alttext="A^{i}(\theta)=g^{ij}(\theta)A_{j}(\theta)" class="ltx_Math" display="inline" 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id="S4.SS2.p2.1.m1.3.4.3.2.3.3.cmml" xref="S4.SS2.p2.1.m1.3.4.3.2.3.3">𝑗</ci></apply></apply><ci id="S4.SS2.p2.1.m1.2.2.cmml" xref="S4.SS2.p2.1.m1.2.2">𝜃</ci><apply id="S4.SS2.p2.1.m1.3.4.3.4.cmml" xref="S4.SS2.p2.1.m1.3.4.3.4"><csymbol cd="ambiguous" id="S4.SS2.p2.1.m1.3.4.3.4.1.cmml" xref="S4.SS2.p2.1.m1.3.4.3.4">subscript</csymbol><ci id="S4.SS2.p2.1.m1.3.4.3.4.2.cmml" xref="S4.SS2.p2.1.m1.3.4.3.4.2">𝐴</ci><ci id="S4.SS2.p2.1.m1.3.4.3.4.3.cmml" xref="S4.SS2.p2.1.m1.3.4.3.4.3">𝑗</ci></apply><ci id="S4.SS2.p2.1.m1.3.3.cmml" xref="S4.SS2.p2.1.m1.3.3">𝜃</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p2.1.m1.3c">A^{i}(\theta)=g^{ij}(\theta)A_{j}(\theta)</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p2.1.m1.3d">italic_A start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT ( italic_θ ) = italic_g start_POSTSUPERSCRIPT italic_i italic_j end_POSTSUPERSCRIPT ( italic_θ ) italic_A start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ( italic_θ )</annotation></semantics></math> and <math alttext="A^{2}(\theta):=g^{ij}(\theta)A_{i}(\theta)A_{j}(\theta)" class="ltx_Math" display="inline" id="S4.SS2.p2.2.m2.4"><semantics id="S4.SS2.p2.2.m2.4a"><mrow id="S4.SS2.p2.2.m2.4.5" xref="S4.SS2.p2.2.m2.4.5.cmml"><mrow id="S4.SS2.p2.2.m2.4.5.2" xref="S4.SS2.p2.2.m2.4.5.2.cmml"><msup id="S4.SS2.p2.2.m2.4.5.2.2" xref="S4.SS2.p2.2.m2.4.5.2.2.cmml"><mi id="S4.SS2.p2.2.m2.4.5.2.2.2" xref="S4.SS2.p2.2.m2.4.5.2.2.2.cmml">A</mi><mn id="S4.SS2.p2.2.m2.4.5.2.2.3" xref="S4.SS2.p2.2.m2.4.5.2.2.3.cmml">2</mn></msup><mo id="S4.SS2.p2.2.m2.4.5.2.1" xref="S4.SS2.p2.2.m2.4.5.2.1.cmml">⁢</mo><mrow id="S4.SS2.p2.2.m2.4.5.2.3.2" xref="S4.SS2.p2.2.m2.4.5.2.cmml"><mo id="S4.SS2.p2.2.m2.4.5.2.3.2.1" stretchy="false" xref="S4.SS2.p2.2.m2.4.5.2.cmml">(</mo><mi id="S4.SS2.p2.2.m2.1.1" xref="S4.SS2.p2.2.m2.1.1.cmml">θ</mi><mo id="S4.SS2.p2.2.m2.4.5.2.3.2.2" rspace="0.278em" stretchy="false" xref="S4.SS2.p2.2.m2.4.5.2.cmml">)</mo></mrow></mrow><mo 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xref="S4.SS2.p2.2.m2.4.5.3.2.3.3">𝑗</ci></apply></apply><ci id="S4.SS2.p2.2.m2.2.2.cmml" xref="S4.SS2.p2.2.m2.2.2">𝜃</ci><apply id="S4.SS2.p2.2.m2.4.5.3.4.cmml" xref="S4.SS2.p2.2.m2.4.5.3.4"><csymbol cd="ambiguous" id="S4.SS2.p2.2.m2.4.5.3.4.1.cmml" xref="S4.SS2.p2.2.m2.4.5.3.4">subscript</csymbol><ci id="S4.SS2.p2.2.m2.4.5.3.4.2.cmml" xref="S4.SS2.p2.2.m2.4.5.3.4.2">𝐴</ci><ci id="S4.SS2.p2.2.m2.4.5.3.4.3.cmml" xref="S4.SS2.p2.2.m2.4.5.3.4.3">𝑖</ci></apply><ci id="S4.SS2.p2.2.m2.3.3.cmml" xref="S4.SS2.p2.2.m2.3.3">𝜃</ci><apply id="S4.SS2.p2.2.m2.4.5.3.6.cmml" xref="S4.SS2.p2.2.m2.4.5.3.6"><csymbol cd="ambiguous" id="S4.SS2.p2.2.m2.4.5.3.6.1.cmml" xref="S4.SS2.p2.2.m2.4.5.3.6">subscript</csymbol><ci id="S4.SS2.p2.2.m2.4.5.3.6.2.cmml" xref="S4.SS2.p2.2.m2.4.5.3.6.2">𝐴</ci><ci id="S4.SS2.p2.2.m2.4.5.3.6.3.cmml" xref="S4.SS2.p2.2.m2.4.5.3.6.3">𝑗</ci></apply><ci id="S4.SS2.p2.2.m2.4.4.cmml" xref="S4.SS2.p2.2.m2.4.4">𝜃</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p2.2.m2.4c">A^{2}(\theta):=g^{ij}(\theta)A_{i}(\theta)A_{j}(\theta)</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p2.2.m2.4d">italic_A start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( italic_θ ) := italic_g start_POSTSUPERSCRIPT italic_i italic_j end_POSTSUPERSCRIPT ( italic_θ ) italic_A start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_θ ) italic_A start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ( italic_θ )</annotation></semantics></math>. By setting</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="S3.EGx83"> <tbody id="S4.E89"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_left_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle g^{ij}(\theta)=\xi\chi^{2}(\theta)(\tilde{\gamma}^{ij}-\beta^{i}% \beta^{j}),\;\;\;\beta^{i}=\frac{A^{i}(\theta)}{\xi\chi^{2}(\theta)}," class="ltx_Math" display="inline" id="S4.E89.m1.5"><semantics id="S4.E89.m1.5a"><mrow id="S4.E89.m1.5.5.1"><mrow id="S4.E89.m1.5.5.1.1.2" xref="S4.E89.m1.5.5.1.1.3.cmml"><mrow id="S4.E89.m1.5.5.1.1.1.1" xref="S4.E89.m1.5.5.1.1.1.1.cmml"><mrow id="S4.E89.m1.5.5.1.1.1.1.3" xref="S4.E89.m1.5.5.1.1.1.1.3.cmml"><msup id="S4.E89.m1.5.5.1.1.1.1.3.2" xref="S4.E89.m1.5.5.1.1.1.1.3.2.cmml"><mi id="S4.E89.m1.5.5.1.1.1.1.3.2.2" xref="S4.E89.m1.5.5.1.1.1.1.3.2.2.cmml">g</mi><mrow id="S4.E89.m1.5.5.1.1.1.1.3.2.3" xref="S4.E89.m1.5.5.1.1.1.1.3.2.3.cmml"><mi 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g^{ij}(\theta)=\xi\chi^{2}(\theta)(\tilde{\gamma}^{ij}-\beta^{i}% \beta^{j}),\;\;\;\beta^{i}=\frac{A^{i}(\theta)}{\xi\chi^{2}(\theta)},</annotation><annotation encoding="application/x-llamapun" id="S4.E89.m1.5d">italic_g start_POSTSUPERSCRIPT italic_i italic_j end_POSTSUPERSCRIPT ( italic_θ ) = italic_ξ italic_χ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( italic_θ ) ( over~ start_ARG italic_γ end_ARG start_POSTSUPERSCRIPT italic_i italic_j end_POSTSUPERSCRIPT - italic_β start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT italic_β start_POSTSUPERSCRIPT italic_j end_POSTSUPERSCRIPT ) , italic_β start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT = divide start_ARG italic_A start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT ( italic_θ ) end_ARG start_ARG italic_ξ italic_χ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( italic_θ ) end_ARG ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_left_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(89)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S4.SS2.p2.3">and using <math alttext="\xi=1-A^{2}(\theta)/\chi^{2}(\theta)" class="ltx_Math" display="inline" id="S4.SS2.p2.3.m1.2"><semantics id="S4.SS2.p2.3.m1.2a"><mrow id="S4.SS2.p2.3.m1.2.3" xref="S4.SS2.p2.3.m1.2.3.cmml"><mi id="S4.SS2.p2.3.m1.2.3.2" xref="S4.SS2.p2.3.m1.2.3.2.cmml">ξ</mi><mo id="S4.SS2.p2.3.m1.2.3.1" xref="S4.SS2.p2.3.m1.2.3.1.cmml">=</mo><mrow id="S4.SS2.p2.3.m1.2.3.3" xref="S4.SS2.p2.3.m1.2.3.3.cmml"><mn id="S4.SS2.p2.3.m1.2.3.3.2" xref="S4.SS2.p2.3.m1.2.3.3.2.cmml">1</mn><mo id="S4.SS2.p2.3.m1.2.3.3.1" xref="S4.SS2.p2.3.m1.2.3.3.1.cmml">−</mo><mrow id="S4.SS2.p2.3.m1.2.3.3.3" xref="S4.SS2.p2.3.m1.2.3.3.3.cmml"><mrow id="S4.SS2.p2.3.m1.2.3.3.3.2" xref="S4.SS2.p2.3.m1.2.3.3.3.2.cmml"><mrow id="S4.SS2.p2.3.m1.2.3.3.3.2.2" xref="S4.SS2.p2.3.m1.2.3.3.3.2.2.cmml"><msup id="S4.SS2.p2.3.m1.2.3.3.3.2.2.2" 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xref="S4.SS2.p2.3.m1.2.3.3.3.2.3.3.cmml">2</mn></msup></mrow><mo id="S4.SS2.p2.3.m1.2.3.3.3.1" xref="S4.SS2.p2.3.m1.2.3.3.3.1.cmml">⁢</mo><mrow id="S4.SS2.p2.3.m1.2.3.3.3.3.2" xref="S4.SS2.p2.3.m1.2.3.3.3.cmml"><mo id="S4.SS2.p2.3.m1.2.3.3.3.3.2.1" stretchy="false" xref="S4.SS2.p2.3.m1.2.3.3.3.cmml">(</mo><mi id="S4.SS2.p2.3.m1.2.2" xref="S4.SS2.p2.3.m1.2.2.cmml">θ</mi><mo id="S4.SS2.p2.3.m1.2.3.3.3.3.2.2" stretchy="false" xref="S4.SS2.p2.3.m1.2.3.3.3.cmml">)</mo></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p2.3.m1.2b"><apply id="S4.SS2.p2.3.m1.2.3.cmml" xref="S4.SS2.p2.3.m1.2.3"><eq id="S4.SS2.p2.3.m1.2.3.1.cmml" xref="S4.SS2.p2.3.m1.2.3.1"></eq><ci id="S4.SS2.p2.3.m1.2.3.2.cmml" xref="S4.SS2.p2.3.m1.2.3.2">𝜉</ci><apply id="S4.SS2.p2.3.m1.2.3.3.cmml" xref="S4.SS2.p2.3.m1.2.3.3"><minus id="S4.SS2.p2.3.m1.2.3.3.1.cmml" xref="S4.SS2.p2.3.m1.2.3.3.1"></minus><cn id="S4.SS2.p2.3.m1.2.3.3.2.cmml" type="integer" xref="S4.SS2.p2.3.m1.2.3.3.2">1</cn><apply id="S4.SS2.p2.3.m1.2.3.3.3.cmml" xref="S4.SS2.p2.3.m1.2.3.3.3"><times id="S4.SS2.p2.3.m1.2.3.3.3.1.cmml" xref="S4.SS2.p2.3.m1.2.3.3.3.1"></times><apply id="S4.SS2.p2.3.m1.2.3.3.3.2.cmml" xref="S4.SS2.p2.3.m1.2.3.3.3.2"><divide id="S4.SS2.p2.3.m1.2.3.3.3.2.1.cmml" xref="S4.SS2.p2.3.m1.2.3.3.3.2.1"></divide><apply id="S4.SS2.p2.3.m1.2.3.3.3.2.2.cmml" xref="S4.SS2.p2.3.m1.2.3.3.3.2.2"><times id="S4.SS2.p2.3.m1.2.3.3.3.2.2.1.cmml" xref="S4.SS2.p2.3.m1.2.3.3.3.2.2.1"></times><apply id="S4.SS2.p2.3.m1.2.3.3.3.2.2.2.cmml" xref="S4.SS2.p2.3.m1.2.3.3.3.2.2.2"><csymbol cd="ambiguous" id="S4.SS2.p2.3.m1.2.3.3.3.2.2.2.1.cmml" xref="S4.SS2.p2.3.m1.2.3.3.3.2.2.2">superscript</csymbol><ci id="S4.SS2.p2.3.m1.2.3.3.3.2.2.2.2.cmml" xref="S4.SS2.p2.3.m1.2.3.3.3.2.2.2.2">𝐴</ci><cn id="S4.SS2.p2.3.m1.2.3.3.3.2.2.2.3.cmml" type="integer" xref="S4.SS2.p2.3.m1.2.3.3.3.2.2.2.3">2</cn></apply><ci id="S4.SS2.p2.3.m1.1.1.cmml" xref="S4.SS2.p2.3.m1.1.1">𝜃</ci></apply><apply id="S4.SS2.p2.3.m1.2.3.3.3.2.3.cmml" xref="S4.SS2.p2.3.m1.2.3.3.3.2.3"><csymbol cd="ambiguous" id="S4.SS2.p2.3.m1.2.3.3.3.2.3.1.cmml" xref="S4.SS2.p2.3.m1.2.3.3.3.2.3">superscript</csymbol><ci id="S4.SS2.p2.3.m1.2.3.3.3.2.3.2.cmml" xref="S4.SS2.p2.3.m1.2.3.3.3.2.3.2">𝜒</ci><cn id="S4.SS2.p2.3.m1.2.3.3.3.2.3.3.cmml" type="integer" xref="S4.SS2.p2.3.m1.2.3.3.3.2.3.3">2</cn></apply></apply><ci id="S4.SS2.p2.3.m1.2.2.cmml" xref="S4.SS2.p2.3.m1.2.2">𝜃</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p2.3.m1.2c">\xi=1-A^{2}(\theta)/\chi^{2}(\theta)</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p2.3.m1.2d">italic_ξ = 1 - italic_A start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( italic_θ ) / italic_χ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( italic_θ )</annotation></semantics></math>, the above relation (<a class="ltx_ref" href="https://arxiv.org/html/2406.11224v2#S4.E88" title="In 4.2 Applications to the gradient-flow equations ‣ 4 The motions of a light-like particle in a pseudo Riemann space ‣ A Hamiltonian approach to the gradient-flow equations in information geometry"><span class="ltx_text ltx_ref_tag">88</span></a>) is rewritten as</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="S3.EGx84"> <tbody id="S4.E90"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_left_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle 0=\xi\chi^{2}(\theta)\left[(\tilde{\gamma}^{ij}-\beta^{i}\beta^{% j})\eta_{i}\eta_{j}-2\beta^{i}\eta_{i}-1\right]." class="ltx_Math" display="inline" id="S4.E90.m1.2"><semantics id="S4.E90.m1.2a"><mrow id="S4.E90.m1.2.2.1" xref="S4.E90.m1.2.2.1.1.cmml"><mrow id="S4.E90.m1.2.2.1.1" xref="S4.E90.m1.2.2.1.1.cmml"><mn id="S4.E90.m1.2.2.1.1.3" xref="S4.E90.m1.2.2.1.1.3.cmml">0</mn><mo id="S4.E90.m1.2.2.1.1.2" xref="S4.E90.m1.2.2.1.1.2.cmml">=</mo><mrow id="S4.E90.m1.2.2.1.1.1" xref="S4.E90.m1.2.2.1.1.1.cmml"><mi id="S4.E90.m1.2.2.1.1.1.3" xref="S4.E90.m1.2.2.1.1.1.3.cmml">ξ</mi><mo id="S4.E90.m1.2.2.1.1.1.2" xref="S4.E90.m1.2.2.1.1.1.2.cmml">⁢</mo><msup id="S4.E90.m1.2.2.1.1.1.4" xref="S4.E90.m1.2.2.1.1.1.4.cmml"><mi id="S4.E90.m1.2.2.1.1.1.4.2" xref="S4.E90.m1.2.2.1.1.1.4.2.cmml">χ</mi><mn id="S4.E90.m1.2.2.1.1.1.4.3" xref="S4.E90.m1.2.2.1.1.1.4.3.cmml">2</mn></msup><mo id="S4.E90.m1.2.2.1.1.1.2a" xref="S4.E90.m1.2.2.1.1.1.2.cmml">⁢</mo><mrow id="S4.E90.m1.2.2.1.1.1.5.2" xref="S4.E90.m1.2.2.1.1.1.cmml"><mo id="S4.E90.m1.2.2.1.1.1.5.2.1" stretchy="false" xref="S4.E90.m1.2.2.1.1.1.cmml">(</mo><mi id="S4.E90.m1.1.1" xref="S4.E90.m1.1.1.cmml">θ</mi><mo id="S4.E90.m1.2.2.1.1.1.5.2.2" stretchy="false" xref="S4.E90.m1.2.2.1.1.1.cmml">)</mo></mrow><mo id="S4.E90.m1.2.2.1.1.1.2b" xref="S4.E90.m1.2.2.1.1.1.2.cmml">⁢</mo><mrow id="S4.E90.m1.2.2.1.1.1.1.1" xref="S4.E90.m1.2.2.1.1.1.1.2.cmml"><mo id="S4.E90.m1.2.2.1.1.1.1.1.2" xref="S4.E90.m1.2.2.1.1.1.1.2.1.cmml">[</mo><mrow id="S4.E90.m1.2.2.1.1.1.1.1.1" xref="S4.E90.m1.2.2.1.1.1.1.1.1.cmml"><mrow id="S4.E90.m1.2.2.1.1.1.1.1.1.1" xref="S4.E90.m1.2.2.1.1.1.1.1.1.1.cmml"><mrow id="S4.E90.m1.2.2.1.1.1.1.1.1.1.1.1" xref="S4.E90.m1.2.2.1.1.1.1.1.1.1.1.1.1.cmml"><mo id="S4.E90.m1.2.2.1.1.1.1.1.1.1.1.1.2" stretchy="false" xref="S4.E90.m1.2.2.1.1.1.1.1.1.1.1.1.1.cmml">(</mo><mrow id="S4.E90.m1.2.2.1.1.1.1.1.1.1.1.1.1" xref="S4.E90.m1.2.2.1.1.1.1.1.1.1.1.1.1.cmml"><msup id="S4.E90.m1.2.2.1.1.1.1.1.1.1.1.1.1.2" xref="S4.E90.m1.2.2.1.1.1.1.1.1.1.1.1.1.2.cmml"><mover accent="true" id="S4.E90.m1.2.2.1.1.1.1.1.1.1.1.1.1.2.2" xref="S4.E90.m1.2.2.1.1.1.1.1.1.1.1.1.1.2.2.cmml"><mi id="S4.E90.m1.2.2.1.1.1.1.1.1.1.1.1.1.2.2.2" xref="S4.E90.m1.2.2.1.1.1.1.1.1.1.1.1.1.2.2.2.cmml">γ</mi><mo id="S4.E90.m1.2.2.1.1.1.1.1.1.1.1.1.1.2.2.1" xref="S4.E90.m1.2.2.1.1.1.1.1.1.1.1.1.1.2.2.1.cmml">~</mo></mover><mrow id="S4.E90.m1.2.2.1.1.1.1.1.1.1.1.1.1.2.3" xref="S4.E90.m1.2.2.1.1.1.1.1.1.1.1.1.1.2.3.cmml"><mi id="S4.E90.m1.2.2.1.1.1.1.1.1.1.1.1.1.2.3.2" xref="S4.E90.m1.2.2.1.1.1.1.1.1.1.1.1.1.2.3.2.cmml">i</mi><mo id="S4.E90.m1.2.2.1.1.1.1.1.1.1.1.1.1.2.3.1" xref="S4.E90.m1.2.2.1.1.1.1.1.1.1.1.1.1.2.3.1.cmml">⁢</mo><mi id="S4.E90.m1.2.2.1.1.1.1.1.1.1.1.1.1.2.3.3" xref="S4.E90.m1.2.2.1.1.1.1.1.1.1.1.1.1.2.3.3.cmml">j</mi></mrow></msup><mo id="S4.E90.m1.2.2.1.1.1.1.1.1.1.1.1.1.1" xref="S4.E90.m1.2.2.1.1.1.1.1.1.1.1.1.1.1.cmml">−</mo><mrow id="S4.E90.m1.2.2.1.1.1.1.1.1.1.1.1.1.3" xref="S4.E90.m1.2.2.1.1.1.1.1.1.1.1.1.1.3.cmml"><msup id="S4.E90.m1.2.2.1.1.1.1.1.1.1.1.1.1.3.2" xref="S4.E90.m1.2.2.1.1.1.1.1.1.1.1.1.1.3.2.cmml"><mi id="S4.E90.m1.2.2.1.1.1.1.1.1.1.1.1.1.3.2.2" xref="S4.E90.m1.2.2.1.1.1.1.1.1.1.1.1.1.3.2.2.cmml">β</mi><mi id="S4.E90.m1.2.2.1.1.1.1.1.1.1.1.1.1.3.2.3" xref="S4.E90.m1.2.2.1.1.1.1.1.1.1.1.1.1.3.2.3.cmml">i</mi></msup><mo id="S4.E90.m1.2.2.1.1.1.1.1.1.1.1.1.1.3.1" xref="S4.E90.m1.2.2.1.1.1.1.1.1.1.1.1.1.3.1.cmml">⁢</mo><msup id="S4.E90.m1.2.2.1.1.1.1.1.1.1.1.1.1.3.3" 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end_POSTSUPERSCRIPT ) italic_η start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_η start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT - 2 italic_β start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT italic_η start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT - 1 ] .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_left_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(90)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S4.SS2.p2.6">We again regard this null relation as (<a class="ltx_ref" href="https://arxiv.org/html/2406.11224v2#S4.E68" title="In 4 The motions of a light-like particle in a pseudo Riemann space ‣ A Hamiltonian approach to the gradient-flow equations in information geometry"><span class="ltx_text ltx_ref_tag">68</span></a>) by setting <math alttext="p_{i}=\eta_{i},p_{0}=-1" class="ltx_Math" display="inline" id="S4.SS2.p2.4.m1.2"><semantics 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xref="S4.SS2.p2.4.m1.2.2.2.2.2.2.cmml">p</mi><mn id="S4.SS2.p2.4.m1.2.2.2.2.2.3" xref="S4.SS2.p2.4.m1.2.2.2.2.2.3.cmml">0</mn></msub><mo id="S4.SS2.p2.4.m1.2.2.2.2.1" xref="S4.SS2.p2.4.m1.2.2.2.2.1.cmml">=</mo><mrow id="S4.SS2.p2.4.m1.2.2.2.2.3" xref="S4.SS2.p2.4.m1.2.2.2.2.3.cmml"><mo id="S4.SS2.p2.4.m1.2.2.2.2.3a" xref="S4.SS2.p2.4.m1.2.2.2.2.3.cmml">−</mo><mn id="S4.SS2.p2.4.m1.2.2.2.2.3.2" xref="S4.SS2.p2.4.m1.2.2.2.2.3.2.cmml">1</mn></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p2.4.m1.2b"><apply id="S4.SS2.p2.4.m1.2.2.3.cmml" xref="S4.SS2.p2.4.m1.2.2.2"><csymbol cd="ambiguous" id="S4.SS2.p2.4.m1.2.2.3a.cmml" xref="S4.SS2.p2.4.m1.2.2.2.3">formulae-sequence</csymbol><apply id="S4.SS2.p2.4.m1.1.1.1.1.cmml" xref="S4.SS2.p2.4.m1.1.1.1.1"><eq id="S4.SS2.p2.4.m1.1.1.1.1.1.cmml" xref="S4.SS2.p2.4.m1.1.1.1.1.1"></eq><apply id="S4.SS2.p2.4.m1.1.1.1.1.2.cmml" xref="S4.SS2.p2.4.m1.1.1.1.1.2"><csymbol cd="ambiguous" id="S4.SS2.p2.4.m1.1.1.1.1.2.1.cmml" xref="S4.SS2.p2.4.m1.1.1.1.1.2">subscript</csymbol><ci id="S4.SS2.p2.4.m1.1.1.1.1.2.2.cmml" xref="S4.SS2.p2.4.m1.1.1.1.1.2.2">𝑝</ci><ci id="S4.SS2.p2.4.m1.1.1.1.1.2.3.cmml" xref="S4.SS2.p2.4.m1.1.1.1.1.2.3">𝑖</ci></apply><apply id="S4.SS2.p2.4.m1.1.1.1.1.3.cmml" xref="S4.SS2.p2.4.m1.1.1.1.1.3"><csymbol cd="ambiguous" id="S4.SS2.p2.4.m1.1.1.1.1.3.1.cmml" xref="S4.SS2.p2.4.m1.1.1.1.1.3">subscript</csymbol><ci id="S4.SS2.p2.4.m1.1.1.1.1.3.2.cmml" xref="S4.SS2.p2.4.m1.1.1.1.1.3.2">𝜂</ci><ci id="S4.SS2.p2.4.m1.1.1.1.1.3.3.cmml" xref="S4.SS2.p2.4.m1.1.1.1.1.3.3">𝑖</ci></apply></apply><apply id="S4.SS2.p2.4.m1.2.2.2.2.cmml" xref="S4.SS2.p2.4.m1.2.2.2.2"><eq id="S4.SS2.p2.4.m1.2.2.2.2.1.cmml" xref="S4.SS2.p2.4.m1.2.2.2.2.1"></eq><apply id="S4.SS2.p2.4.m1.2.2.2.2.2.cmml" xref="S4.SS2.p2.4.m1.2.2.2.2.2"><csymbol cd="ambiguous" id="S4.SS2.p2.4.m1.2.2.2.2.2.1.cmml" xref="S4.SS2.p2.4.m1.2.2.2.2.2">subscript</csymbol><ci id="S4.SS2.p2.4.m1.2.2.2.2.2.2.cmml" xref="S4.SS2.p2.4.m1.2.2.2.2.2.2">𝑝</ci><cn id="S4.SS2.p2.4.m1.2.2.2.2.2.3.cmml" type="integer" xref="S4.SS2.p2.4.m1.2.2.2.2.2.3">0</cn></apply><apply id="S4.SS2.p2.4.m1.2.2.2.2.3.cmml" xref="S4.SS2.p2.4.m1.2.2.2.2.3"><minus id="S4.SS2.p2.4.m1.2.2.2.2.3.1.cmml" xref="S4.SS2.p2.4.m1.2.2.2.2.3"></minus><cn id="S4.SS2.p2.4.m1.2.2.2.2.3.2.cmml" type="integer" xref="S4.SS2.p2.4.m1.2.2.2.2.3.2">1</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p2.4.m1.2c">p_{i}=\eta_{i},p_{0}=-1</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p2.4.m1.2d">italic_p start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = italic_η start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT , italic_p start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT = - 1</annotation></semantics></math> and <math alttext="1/\alpha^{2}=\xi\chi^{2}(\theta)" class="ltx_Math" display="inline" id="S4.SS2.p2.5.m2.1"><semantics id="S4.SS2.p2.5.m2.1a"><mrow id="S4.SS2.p2.5.m2.1.2" xref="S4.SS2.p2.5.m2.1.2.cmml"><mrow id="S4.SS2.p2.5.m2.1.2.2" xref="S4.SS2.p2.5.m2.1.2.2.cmml"><mn id="S4.SS2.p2.5.m2.1.2.2.2" xref="S4.SS2.p2.5.m2.1.2.2.2.cmml">1</mn><mo id="S4.SS2.p2.5.m2.1.2.2.1" xref="S4.SS2.p2.5.m2.1.2.2.1.cmml">/</mo><msup id="S4.SS2.p2.5.m2.1.2.2.3" xref="S4.SS2.p2.5.m2.1.2.2.3.cmml"><mi id="S4.SS2.p2.5.m2.1.2.2.3.2" xref="S4.SS2.p2.5.m2.1.2.2.3.2.cmml">α</mi><mn id="S4.SS2.p2.5.m2.1.2.2.3.3" xref="S4.SS2.p2.5.m2.1.2.2.3.3.cmml">2</mn></msup></mrow><mo id="S4.SS2.p2.5.m2.1.2.1" xref="S4.SS2.p2.5.m2.1.2.1.cmml">=</mo><mrow id="S4.SS2.p2.5.m2.1.2.3" xref="S4.SS2.p2.5.m2.1.2.3.cmml"><mi id="S4.SS2.p2.5.m2.1.2.3.2" xref="S4.SS2.p2.5.m2.1.2.3.2.cmml">ξ</mi><mo id="S4.SS2.p2.5.m2.1.2.3.1" xref="S4.SS2.p2.5.m2.1.2.3.1.cmml">⁢</mo><msup id="S4.SS2.p2.5.m2.1.2.3.3" xref="S4.SS2.p2.5.m2.1.2.3.3.cmml"><mi id="S4.SS2.p2.5.m2.1.2.3.3.2" xref="S4.SS2.p2.5.m2.1.2.3.3.2.cmml">χ</mi><mn id="S4.SS2.p2.5.m2.1.2.3.3.3" xref="S4.SS2.p2.5.m2.1.2.3.3.3.cmml">2</mn></msup><mo id="S4.SS2.p2.5.m2.1.2.3.1a" xref="S4.SS2.p2.5.m2.1.2.3.1.cmml">⁢</mo><mrow id="S4.SS2.p2.5.m2.1.2.3.4.2" xref="S4.SS2.p2.5.m2.1.2.3.cmml"><mo id="S4.SS2.p2.5.m2.1.2.3.4.2.1" stretchy="false" xref="S4.SS2.p2.5.m2.1.2.3.cmml">(</mo><mi id="S4.SS2.p2.5.m2.1.1" xref="S4.SS2.p2.5.m2.1.1.cmml">θ</mi><mo id="S4.SS2.p2.5.m2.1.2.3.4.2.2" stretchy="false" xref="S4.SS2.p2.5.m2.1.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p2.5.m2.1b"><apply id="S4.SS2.p2.5.m2.1.2.cmml" xref="S4.SS2.p2.5.m2.1.2"><eq id="S4.SS2.p2.5.m2.1.2.1.cmml" xref="S4.SS2.p2.5.m2.1.2.1"></eq><apply id="S4.SS2.p2.5.m2.1.2.2.cmml" xref="S4.SS2.p2.5.m2.1.2.2"><divide id="S4.SS2.p2.5.m2.1.2.2.1.cmml" xref="S4.SS2.p2.5.m2.1.2.2.1"></divide><cn id="S4.SS2.p2.5.m2.1.2.2.2.cmml" type="integer" xref="S4.SS2.p2.5.m2.1.2.2.2">1</cn><apply id="S4.SS2.p2.5.m2.1.2.2.3.cmml" xref="S4.SS2.p2.5.m2.1.2.2.3"><csymbol cd="ambiguous" id="S4.SS2.p2.5.m2.1.2.2.3.1.cmml" xref="S4.SS2.p2.5.m2.1.2.2.3">superscript</csymbol><ci id="S4.SS2.p2.5.m2.1.2.2.3.2.cmml" xref="S4.SS2.p2.5.m2.1.2.2.3.2">𝛼</ci><cn id="S4.SS2.p2.5.m2.1.2.2.3.3.cmml" type="integer" xref="S4.SS2.p2.5.m2.1.2.2.3.3">2</cn></apply></apply><apply id="S4.SS2.p2.5.m2.1.2.3.cmml" xref="S4.SS2.p2.5.m2.1.2.3"><times id="S4.SS2.p2.5.m2.1.2.3.1.cmml" xref="S4.SS2.p2.5.m2.1.2.3.1"></times><ci id="S4.SS2.p2.5.m2.1.2.3.2.cmml" xref="S4.SS2.p2.5.m2.1.2.3.2">𝜉</ci><apply id="S4.SS2.p2.5.m2.1.2.3.3.cmml" xref="S4.SS2.p2.5.m2.1.2.3.3"><csymbol cd="ambiguous" id="S4.SS2.p2.5.m2.1.2.3.3.1.cmml" xref="S4.SS2.p2.5.m2.1.2.3.3">superscript</csymbol><ci id="S4.SS2.p2.5.m2.1.2.3.3.2.cmml" xref="S4.SS2.p2.5.m2.1.2.3.3.2">𝜒</ci><cn id="S4.SS2.p2.5.m2.1.2.3.3.3.cmml" type="integer" xref="S4.SS2.p2.5.m2.1.2.3.3.3">2</cn></apply><ci id="S4.SS2.p2.5.m2.1.1.cmml" xref="S4.SS2.p2.5.m2.1.1">𝜃</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p2.5.m2.1c">1/\alpha^{2}=\xi\chi^{2}(\theta)</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p2.5.m2.1d">1 / italic_α start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT = italic_ξ italic_χ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( italic_θ )</annotation></semantics></math>. Then the corresponding Hamiltonian <math alttext="\mathcal{H}^{+}" class="ltx_Math" display="inline" id="S4.SS2.p2.6.m3.1"><semantics id="S4.SS2.p2.6.m3.1a"><msup id="S4.SS2.p2.6.m3.1.1" xref="S4.SS2.p2.6.m3.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.SS2.p2.6.m3.1.1.2" xref="S4.SS2.p2.6.m3.1.1.2.cmml">ℋ</mi><mo id="S4.SS2.p2.6.m3.1.1.3" xref="S4.SS2.p2.6.m3.1.1.3.cmml">+</mo></msup><annotation-xml encoding="MathML-Content" id="S4.SS2.p2.6.m3.1b"><apply id="S4.SS2.p2.6.m3.1.1.cmml" xref="S4.SS2.p2.6.m3.1.1"><csymbol cd="ambiguous" id="S4.SS2.p2.6.m3.1.1.1.cmml" xref="S4.SS2.p2.6.m3.1.1">superscript</csymbol><ci id="S4.SS2.p2.6.m3.1.1.2.cmml" xref="S4.SS2.p2.6.m3.1.1.2">ℋ</ci><plus id="S4.SS2.p2.6.m3.1.1.3.cmml" xref="S4.SS2.p2.6.m3.1.1.3"></plus></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p2.6.m3.1c">\mathcal{H}^{+}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p2.6.m3.1d">caligraphic_H start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT</annotation></semantics></math> in (<a class="ltx_ref" href="https://arxiv.org/html/2406.11224v2#S4.E73" title="In 4 The motions of a light-like particle in a pseudo Riemann space ‣ A Hamiltonian approach to the gradient-flow equations in information geometry"><span class="ltx_text ltx_ref_tag">73</span></a>) becomes</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="S3.EGx85"> <tbody id="S4.Ex7"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_left_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\mathcal{H}^{+}" class="ltx_Math" display="inline" id="S4.Ex7.m1.1"><semantics id="S4.Ex7.m1.1a"><msup id="S4.Ex7.m1.1.1" xref="S4.Ex7.m1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.Ex7.m1.1.1.2" xref="S4.Ex7.m1.1.1.2.cmml">ℋ</mi><mo id="S4.Ex7.m1.1.1.3" xref="S4.Ex7.m1.1.1.3.cmml">+</mo></msup><annotation-xml encoding="MathML-Content" id="S4.Ex7.m1.1b"><apply id="S4.Ex7.m1.1.1.cmml" xref="S4.Ex7.m1.1.1"><csymbol cd="ambiguous" id="S4.Ex7.m1.1.1.1.cmml" xref="S4.Ex7.m1.1.1">superscript</csymbol><ci id="S4.Ex7.m1.1.1.2.cmml" xref="S4.Ex7.m1.1.1.2">ℋ</ci><plus id="S4.Ex7.m1.1.1.3.cmml" xref="S4.Ex7.m1.1.1.3"></plus></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Ex7.m1.1c">\displaystyle\mathcal{H}^{+}</annotation><annotation encoding="application/x-llamapun" id="S4.Ex7.m1.1d">caligraphic_H start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle(\theta,\eta)=-\frac{A^{i}(\theta)}{\xi\chi^{2}(\theta)}\eta_{i}" class="ltx_Math" display="inline" id="S4.Ex7.m2.4"><semantics id="S4.Ex7.m2.4a"><mrow id="S4.Ex7.m2.4.5" xref="S4.Ex7.m2.4.5.cmml"><mrow id="S4.Ex7.m2.4.5.2.2" xref="S4.Ex7.m2.4.5.2.1.cmml"><mo id="S4.Ex7.m2.4.5.2.2.1" stretchy="false" xref="S4.Ex7.m2.4.5.2.1.cmml">(</mo><mi id="S4.Ex7.m2.3.3" 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xref="S4.Ex7.m2.2.2.2.4.2">𝜒</ci><cn id="S4.Ex7.m2.2.2.2.4.3.cmml" type="integer" xref="S4.Ex7.m2.2.2.2.4.3">2</cn></apply><ci id="S4.Ex7.m2.2.2.2.1.cmml" xref="S4.Ex7.m2.2.2.2.1">𝜃</ci></apply></apply><apply id="S4.Ex7.m2.4.5.3.2.2.cmml" xref="S4.Ex7.m2.4.5.3.2.2"><csymbol cd="ambiguous" id="S4.Ex7.m2.4.5.3.2.2.1.cmml" xref="S4.Ex7.m2.4.5.3.2.2">subscript</csymbol><ci id="S4.Ex7.m2.4.5.3.2.2.2.cmml" xref="S4.Ex7.m2.4.5.3.2.2.2">𝜂</ci><ci id="S4.Ex7.m2.4.5.3.2.2.3.cmml" xref="S4.Ex7.m2.4.5.3.2.2.3">𝑖</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Ex7.m2.4c">\displaystyle(\theta,\eta)=-\frac{A^{i}(\theta)}{\xi\chi^{2}(\theta)}\eta_{i}</annotation><annotation encoding="application/x-llamapun" id="S4.Ex7.m2.4d">( italic_θ , italic_η ) = - divide start_ARG italic_A start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT ( italic_θ ) end_ARG start_ARG italic_ξ italic_χ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( italic_θ ) end_ARG italic_η 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italic_ξ italic_χ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( italic_θ ) end_ARG ) italic_η start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_η start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT end_ARG .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_left_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(91)</span></td> </tr></tbody> </table> </div> <div class="ltx_para" id="S4.SS2.p3"> <p class="ltx_p" id="S4.SS2.p3.5">As we mentioned in a few lines after (<a class="ltx_ref" href="https://arxiv.org/html/2406.11224v2#S2.E28" title="In 2.3 Randers-Finsler deformation of the gradient-flow equations ‣ 2 Information Geometry and Gradient-Flow Equations ‣ A Hamiltonian approach to the gradient-flow equations in information geometry"><span class="ltx_text ltx_ref_tag">28</span></a>), the gradient-flow equations (<a class="ltx_ref" href="https://arxiv.org/html/2406.11224v2#S2.E12" title="In 2.2 Gradient-Flow Equations ‣ 2 Information Geometry and Gradient-Flow Equations ‣ A Hamiltonian approach to the gradient-flow equations in information geometry"><span class="ltx_text ltx_ref_tag">12</span></a>) correspond to the RF deformed equations (<a class="ltx_ref" href="https://arxiv.org/html/2406.11224v2#S2.E28" title="In 2.3 Randers-Finsler deformation of the gradient-flow equations ‣ 2 Information Geometry and Gradient-Flow Equations ‣ A Hamiltonian approach to the gradient-flow equations in information geometry"><span class="ltx_text ltx_ref_tag">28</span></a>) when <math alttext="A_{j}(\theta)=\eta^{\rm r}_{j}" class="ltx_Math" display="inline" id="S4.SS2.p3.1.m1.1"><semantics id="S4.SS2.p3.1.m1.1a"><mrow id="S4.SS2.p3.1.m1.1.2" xref="S4.SS2.p3.1.m1.1.2.cmml"><mrow id="S4.SS2.p3.1.m1.1.2.2" xref="S4.SS2.p3.1.m1.1.2.2.cmml"><msub id="S4.SS2.p3.1.m1.1.2.2.2" xref="S4.SS2.p3.1.m1.1.2.2.2.cmml"><mi id="S4.SS2.p3.1.m1.1.2.2.2.2" xref="S4.SS2.p3.1.m1.1.2.2.2.2.cmml">A</mi><mi id="S4.SS2.p3.1.m1.1.2.2.2.3" xref="S4.SS2.p3.1.m1.1.2.2.2.3.cmml">j</mi></msub><mo id="S4.SS2.p3.1.m1.1.2.2.1" xref="S4.SS2.p3.1.m1.1.2.2.1.cmml">⁢</mo><mrow id="S4.SS2.p3.1.m1.1.2.2.3.2" xref="S4.SS2.p3.1.m1.1.2.2.cmml"><mo id="S4.SS2.p3.1.m1.1.2.2.3.2.1" stretchy="false" xref="S4.SS2.p3.1.m1.1.2.2.cmml">(</mo><mi id="S4.SS2.p3.1.m1.1.1" xref="S4.SS2.p3.1.m1.1.1.cmml">θ</mi><mo id="S4.SS2.p3.1.m1.1.2.2.3.2.2" stretchy="false" xref="S4.SS2.p3.1.m1.1.2.2.cmml">)</mo></mrow></mrow><mo id="S4.SS2.p3.1.m1.1.2.1" xref="S4.SS2.p3.1.m1.1.2.1.cmml">=</mo><msubsup id="S4.SS2.p3.1.m1.1.2.3" xref="S4.SS2.p3.1.m1.1.2.3.cmml"><mi id="S4.SS2.p3.1.m1.1.2.3.2.2" xref="S4.SS2.p3.1.m1.1.2.3.2.2.cmml">η</mi><mi id="S4.SS2.p3.1.m1.1.2.3.3" xref="S4.SS2.p3.1.m1.1.2.3.3.cmml">j</mi><mi id="S4.SS2.p3.1.m1.1.2.3.2.3" mathvariant="normal" xref="S4.SS2.p3.1.m1.1.2.3.2.3.cmml">r</mi></msubsup></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p3.1.m1.1b"><apply id="S4.SS2.p3.1.m1.1.2.cmml" xref="S4.SS2.p3.1.m1.1.2"><eq id="S4.SS2.p3.1.m1.1.2.1.cmml" xref="S4.SS2.p3.1.m1.1.2.1"></eq><apply id="S4.SS2.p3.1.m1.1.2.2.cmml" xref="S4.SS2.p3.1.m1.1.2.2"><times id="S4.SS2.p3.1.m1.1.2.2.1.cmml" xref="S4.SS2.p3.1.m1.1.2.2.1"></times><apply id="S4.SS2.p3.1.m1.1.2.2.2.cmml" xref="S4.SS2.p3.1.m1.1.2.2.2"><csymbol cd="ambiguous" id="S4.SS2.p3.1.m1.1.2.2.2.1.cmml" xref="S4.SS2.p3.1.m1.1.2.2.2">subscript</csymbol><ci id="S4.SS2.p3.1.m1.1.2.2.2.2.cmml" xref="S4.SS2.p3.1.m1.1.2.2.2.2">𝐴</ci><ci id="S4.SS2.p3.1.m1.1.2.2.2.3.cmml" xref="S4.SS2.p3.1.m1.1.2.2.2.3">𝑗</ci></apply><ci id="S4.SS2.p3.1.m1.1.1.cmml" xref="S4.SS2.p3.1.m1.1.1">𝜃</ci></apply><apply id="S4.SS2.p3.1.m1.1.2.3.cmml" xref="S4.SS2.p3.1.m1.1.2.3"><csymbol cd="ambiguous" id="S4.SS2.p3.1.m1.1.2.3.1.cmml" xref="S4.SS2.p3.1.m1.1.2.3">subscript</csymbol><apply id="S4.SS2.p3.1.m1.1.2.3.2.cmml" xref="S4.SS2.p3.1.m1.1.2.3"><csymbol cd="ambiguous" id="S4.SS2.p3.1.m1.1.2.3.2.1.cmml" xref="S4.SS2.p3.1.m1.1.2.3">superscript</csymbol><ci id="S4.SS2.p3.1.m1.1.2.3.2.2.cmml" xref="S4.SS2.p3.1.m1.1.2.3.2.2">𝜂</ci><ci id="S4.SS2.p3.1.m1.1.2.3.2.3.cmml" xref="S4.SS2.p3.1.m1.1.2.3.2.3">r</ci></apply><ci id="S4.SS2.p3.1.m1.1.2.3.3.cmml" xref="S4.SS2.p3.1.m1.1.2.3.3">𝑗</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p3.1.m1.1c">A_{j}(\theta)=\eta^{\rm r}_{j}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p3.1.m1.1d">italic_A start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ( italic_θ ) = italic_η start_POSTSUPERSCRIPT roman_r end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT</annotation></semantics></math>. Consequently, the corresponding Hamiltonian for (<a class="ltx_ref" href="https://arxiv.org/html/2406.11224v2#S2.E12" title="In 2.2 Gradient-Flow Equations ‣ 2 Information Geometry and Gradient-Flow Equations ‣ A Hamiltonian approach to the gradient-flow equations in information geometry"><span class="ltx_text ltx_ref_tag">12</span></a>) is obtained by replacing <math alttext="A^{i}(\theta)=g^{ik}(\theta)A_{k}(\theta)" class="ltx_Math" display="inline" id="S4.SS2.p3.2.m2.3"><semantics id="S4.SS2.p3.2.m2.3a"><mrow id="S4.SS2.p3.2.m2.3.4" xref="S4.SS2.p3.2.m2.3.4.cmml"><mrow id="S4.SS2.p3.2.m2.3.4.2" xref="S4.SS2.p3.2.m2.3.4.2.cmml"><msup id="S4.SS2.p3.2.m2.3.4.2.2" xref="S4.SS2.p3.2.m2.3.4.2.2.cmml"><mi id="S4.SS2.p3.2.m2.3.4.2.2.2" xref="S4.SS2.p3.2.m2.3.4.2.2.2.cmml">A</mi><mi id="S4.SS2.p3.2.m2.3.4.2.2.3" xref="S4.SS2.p3.2.m2.3.4.2.2.3.cmml">i</mi></msup><mo id="S4.SS2.p3.2.m2.3.4.2.1" xref="S4.SS2.p3.2.m2.3.4.2.1.cmml">⁢</mo><mrow id="S4.SS2.p3.2.m2.3.4.2.3.2" 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xref="S4.SS2.p3.2.m2.3.4.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p3.2.m2.3b"><apply id="S4.SS2.p3.2.m2.3.4.cmml" xref="S4.SS2.p3.2.m2.3.4"><eq id="S4.SS2.p3.2.m2.3.4.1.cmml" xref="S4.SS2.p3.2.m2.3.4.1"></eq><apply id="S4.SS2.p3.2.m2.3.4.2.cmml" xref="S4.SS2.p3.2.m2.3.4.2"><times id="S4.SS2.p3.2.m2.3.4.2.1.cmml" xref="S4.SS2.p3.2.m2.3.4.2.1"></times><apply id="S4.SS2.p3.2.m2.3.4.2.2.cmml" xref="S4.SS2.p3.2.m2.3.4.2.2"><csymbol cd="ambiguous" id="S4.SS2.p3.2.m2.3.4.2.2.1.cmml" xref="S4.SS2.p3.2.m2.3.4.2.2">superscript</csymbol><ci id="S4.SS2.p3.2.m2.3.4.2.2.2.cmml" xref="S4.SS2.p3.2.m2.3.4.2.2.2">𝐴</ci><ci id="S4.SS2.p3.2.m2.3.4.2.2.3.cmml" xref="S4.SS2.p3.2.m2.3.4.2.2.3">𝑖</ci></apply><ci id="S4.SS2.p3.2.m2.1.1.cmml" xref="S4.SS2.p3.2.m2.1.1">𝜃</ci></apply><apply id="S4.SS2.p3.2.m2.3.4.3.cmml" xref="S4.SS2.p3.2.m2.3.4.3"><times id="S4.SS2.p3.2.m2.3.4.3.1.cmml" xref="S4.SS2.p3.2.m2.3.4.3.1"></times><apply id="S4.SS2.p3.2.m2.3.4.3.2.cmml" 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xref="S4.SS2.p3.2.m2.3.3">𝜃</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p3.2.m2.3c">A^{i}(\theta)=g^{ik}(\theta)A_{k}(\theta)</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p3.2.m2.3d">italic_A start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT ( italic_θ ) = italic_g start_POSTSUPERSCRIPT italic_i italic_k end_POSTSUPERSCRIPT ( italic_θ ) italic_A start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ( italic_θ )</annotation></semantics></math> with <math alttext="g^{ik}(\theta)\eta^{\rm r}_{k}" class="ltx_Math" display="inline" id="S4.SS2.p3.3.m3.1"><semantics id="S4.SS2.p3.3.m3.1a"><mrow id="S4.SS2.p3.3.m3.1.2" xref="S4.SS2.p3.3.m3.1.2.cmml"><msup id="S4.SS2.p3.3.m3.1.2.2" xref="S4.SS2.p3.3.m3.1.2.2.cmml"><mi id="S4.SS2.p3.3.m3.1.2.2.2" xref="S4.SS2.p3.3.m3.1.2.2.2.cmml">g</mi><mrow id="S4.SS2.p3.3.m3.1.2.2.3" xref="S4.SS2.p3.3.m3.1.2.2.3.cmml"><mi id="S4.SS2.p3.3.m3.1.2.2.3.2" xref="S4.SS2.p3.3.m3.1.2.2.3.2.cmml">i</mi><mo 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xref="S4.SS2.p3.5.m5.2.2.1.1.1.1.1.1">subscript</csymbol><ci id="S4.SS2.p3.5.m5.2.2.1.1.1.1.1.1.1.2.cmml" xref="S4.SS2.p3.5.m5.2.2.1.1.1.1.1.1.1.2">𝜂</ci><ci id="S4.SS2.p3.5.m5.2.2.1.1.1.1.1.1.1.3.cmml" xref="S4.SS2.p3.5.m5.2.2.1.1.1.1.1.1.1.3">𝑟</ci></apply><cn id="S4.SS2.p3.5.m5.2.2.1.1.1.1.3.cmml" type="integer" xref="S4.SS2.p3.5.m5.2.2.1.1.1.1.3">2</cn></apply><apply id="S4.SS2.p3.5.m5.2.2.1.1.1.3.cmml" xref="S4.SS2.p3.5.m5.2.2.1.1.1.3"><csymbol cd="ambiguous" id="S4.SS2.p3.5.m5.2.2.1.1.1.3.1.cmml" xref="S4.SS2.p3.5.m5.2.2.1.1.1.3">superscript</csymbol><ci id="S4.SS2.p3.5.m5.2.2.1.1.1.3.2.cmml" xref="S4.SS2.p3.5.m5.2.2.1.1.1.3.2">𝜒</ci><cn id="S4.SS2.p3.5.m5.2.2.1.1.1.3.3.cmml" type="integer" xref="S4.SS2.p3.5.m5.2.2.1.1.1.3.3">2</cn></apply></apply><ci id="S4.SS2.p3.5.m5.1.1.cmml" xref="S4.SS2.p3.5.m5.1.1">𝜃</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p3.5.m5.2c">\xi=1-(\eta_{r})^{2}/\chi^{2}(\theta)</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p3.5.m5.2d">italic_ξ = 1 - ( italic_η start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / italic_χ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( italic_θ )</annotation></semantics></math> in (<a class="ltx_ref" href="https://arxiv.org/html/2406.11224v2#S4.E91" title="In 4.2 Applications to the gradient-flow equations ‣ 4 The motions of a light-like particle in a pseudo Riemann space ‣ A Hamiltonian approach to the gradient-flow equations in information geometry"><span class="ltx_text ltx_ref_tag">91</span></a>).</p> </div> </section> <section class="ltx_subsection" id="S4.SS3"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">4.3 </span>Gaussian model</h3> <div class="ltx_para" id="S4.SS3.p1"> <p class="ltx_p" id="S4.SS3.p1.1">As a concrete example of the gradient-flows, we here consider the Gaussian, or Normal <math alttext="N(\mu,\sigma^{2})" class="ltx_Math" display="inline" id="S4.SS3.p1.1.m1.2"><semantics id="S4.SS3.p1.1.m1.2a"><mrow id="S4.SS3.p1.1.m1.2.2" xref="S4.SS3.p1.1.m1.2.2.cmml"><mi id="S4.SS3.p1.1.m1.2.2.3" xref="S4.SS3.p1.1.m1.2.2.3.cmml">N</mi><mo id="S4.SS3.p1.1.m1.2.2.2" xref="S4.SS3.p1.1.m1.2.2.2.cmml">⁢</mo><mrow id="S4.SS3.p1.1.m1.2.2.1.1" xref="S4.SS3.p1.1.m1.2.2.1.2.cmml"><mo id="S4.SS3.p1.1.m1.2.2.1.1.2" stretchy="false" xref="S4.SS3.p1.1.m1.2.2.1.2.cmml">(</mo><mi id="S4.SS3.p1.1.m1.1.1" xref="S4.SS3.p1.1.m1.1.1.cmml">μ</mi><mo id="S4.SS3.p1.1.m1.2.2.1.1.3" xref="S4.SS3.p1.1.m1.2.2.1.2.cmml">,</mo><msup id="S4.SS3.p1.1.m1.2.2.1.1.1" xref="S4.SS3.p1.1.m1.2.2.1.1.1.cmml"><mi id="S4.SS3.p1.1.m1.2.2.1.1.1.2" xref="S4.SS3.p1.1.m1.2.2.1.1.1.2.cmml">σ</mi><mn id="S4.SS3.p1.1.m1.2.2.1.1.1.3" xref="S4.SS3.p1.1.m1.2.2.1.1.1.3.cmml">2</mn></msup><mo id="S4.SS3.p1.1.m1.2.2.1.1.4" stretchy="false" xref="S4.SS3.p1.1.m1.2.2.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.p1.1.m1.2b"><apply 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type="integer" xref="S4.E92.m1.1.1.1.3">2</cn></apply><apply id="S4.E92.m1.1.1.3.cmml" xref="S4.E92.m1.1.1.3"><times id="S4.E92.m1.1.1.3.1.cmml" xref="S4.E92.m1.1.1.3.1"></times><cn id="S4.E92.m1.1.1.3.2.cmml" type="integer" xref="S4.E92.m1.1.1.3.2">2</cn><apply id="S4.E92.m1.1.1.3.3.cmml" xref="S4.E92.m1.1.1.3.3"><csymbol cd="ambiguous" id="S4.E92.m1.1.1.3.3.1.cmml" xref="S4.E92.m1.1.1.3.3">superscript</csymbol><ci id="S4.E92.m1.1.1.3.3.2.cmml" xref="S4.E92.m1.1.1.3.3.2">𝜎</ci><cn id="S4.E92.m1.1.1.3.3.3.cmml" type="integer" xref="S4.E92.m1.1.1.3.3.3">2</cn></apply></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.E92.m1.6c">\displaystyle p_{\rm G}(x;\mu,\sigma)=\frac{1}{\sqrt{2\pi\,\sigma^{2}}}\,\exp% \left[-\frac{(x-\mu)^{2}}{2\sigma^{2}}\right].</annotation><annotation encoding="application/x-llamapun" id="S4.E92.m1.6d">italic_p start_POSTSUBSCRIPT roman_G end_POSTSUBSCRIPT ( italic_x ; italic_μ , italic_σ ) = divide start_ARG 1 end_ARG start_ARG square-root start_ARG 2 italic_π italic_σ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG end_ARG roman_exp [ - divide start_ARG ( italic_x - italic_μ ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG 2 italic_σ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG ] .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_left_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(92)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S4.SS3.p1.5">Here <math alttext="\mu" class="ltx_Math" display="inline" id="S4.SS3.p1.2.m1.1"><semantics id="S4.SS3.p1.2.m1.1a"><mi id="S4.SS3.p1.2.m1.1.1" xref="S4.SS3.p1.2.m1.1.1.cmml">μ</mi><annotation-xml encoding="MathML-Content" id="S4.SS3.p1.2.m1.1b"><ci id="S4.SS3.p1.2.m1.1.1.cmml" xref="S4.SS3.p1.2.m1.1.1">𝜇</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p1.2.m1.1c">\mu</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p1.2.m1.1d">italic_μ</annotation></semantics></math> denotes the mean and <math alttext="\sigma^{2}" class="ltx_Math" display="inline" id="S4.SS3.p1.3.m2.1"><semantics id="S4.SS3.p1.3.m2.1a"><msup id="S4.SS3.p1.3.m2.1.1" xref="S4.SS3.p1.3.m2.1.1.cmml"><mi id="S4.SS3.p1.3.m2.1.1.2" xref="S4.SS3.p1.3.m2.1.1.2.cmml">σ</mi><mn id="S4.SS3.p1.3.m2.1.1.3" xref="S4.SS3.p1.3.m2.1.1.3.cmml">2</mn></msup><annotation-xml encoding="MathML-Content" id="S4.SS3.p1.3.m2.1b"><apply id="S4.SS3.p1.3.m2.1.1.cmml" xref="S4.SS3.p1.3.m2.1.1"><csymbol cd="ambiguous" id="S4.SS3.p1.3.m2.1.1.1.cmml" xref="S4.SS3.p1.3.m2.1.1">superscript</csymbol><ci id="S4.SS3.p1.3.m2.1.1.2.cmml" xref="S4.SS3.p1.3.m2.1.1.2">𝜎</ci><cn id="S4.SS3.p1.3.m2.1.1.3.cmml" type="integer" xref="S4.SS3.p1.3.m2.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p1.3.m2.1c">\sigma^{2}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p1.3.m2.1d">italic_σ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT</annotation></semantics></math> is the variance. It is known that the natural <math alttext="\theta" class="ltx_Math" display="inline" id="S4.SS3.p1.4.m3.1"><semantics id="S4.SS3.p1.4.m3.1a"><mi id="S4.SS3.p1.4.m3.1.1" xref="S4.SS3.p1.4.m3.1.1.cmml">θ</mi><annotation-xml encoding="MathML-Content" id="S4.SS3.p1.4.m3.1b"><ci id="S4.SS3.p1.4.m3.1.1.cmml" xref="S4.SS3.p1.4.m3.1.1">𝜃</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p1.4.m3.1c">\theta</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p1.4.m3.1d">italic_θ</annotation></semantics></math>-coordinates and <math alttext="\eta" class="ltx_Math" display="inline" id="S4.SS3.p1.5.m4.1"><semantics id="S4.SS3.p1.5.m4.1a"><mi id="S4.SS3.p1.5.m4.1.1" xref="S4.SS3.p1.5.m4.1.1.cmml">η</mi><annotation-xml encoding="MathML-Content" id="S4.SS3.p1.5.m4.1b"><ci id="S4.SS3.p1.5.m4.1.1.cmml" xref="S4.SS3.p1.5.m4.1.1">𝜂</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p1.5.m4.1c">\eta</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p1.5.m4.1d">italic_η</annotation></semantics></math>-coordinates <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2406.11224v2#bib.bib1" title="">1</a>]</cite> are <span class="ltx_note ltx_role_footnote" id="footnote3"><sup class="ltx_note_mark">3</sup><span class="ltx_note_outer"><span class="ltx_note_content"><sup class="ltx_note_mark">3</sup><span class="ltx_tag ltx_tag_note">3</span>Do not confuse the superscript in <math alttext="\theta" class="ltx_Math" display="inline" id="footnote3.m1.1"><semantics id="footnote3.m1.1b"><mi id="footnote3.m1.1.1" xref="footnote3.m1.1.1.cmml">θ</mi><annotation-xml encoding="MathML-Content" id="footnote3.m1.1c"><ci id="footnote3.m1.1.1.cmml" xref="footnote3.m1.1.1">𝜃</ci></annotation-xml><annotation encoding="application/x-tex" id="footnote3.m1.1d">\theta</annotation><annotation encoding="application/x-llamapun" id="footnote3.m1.1e">italic_θ</annotation></semantics></math> variables with exponents.</span></span></span></p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="S3.EGx87"> <tbody id="S4.Ex8"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_left_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\theta^{1}" class="ltx_Math" display="inline" id="S4.Ex8.m1.1"><semantics id="S4.Ex8.m1.1a"><msup id="S4.Ex8.m1.1.1" xref="S4.Ex8.m1.1.1.cmml"><mi id="S4.Ex8.m1.1.1.2" xref="S4.Ex8.m1.1.1.2.cmml">θ</mi><mn id="S4.Ex8.m1.1.1.3" xref="S4.Ex8.m1.1.1.3.cmml">1</mn></msup><annotation-xml encoding="MathML-Content" id="S4.Ex8.m1.1b"><apply id="S4.Ex8.m1.1.1.cmml" xref="S4.Ex8.m1.1.1"><csymbol cd="ambiguous" id="S4.Ex8.m1.1.1.1.cmml" xref="S4.Ex8.m1.1.1">superscript</csymbol><ci id="S4.Ex8.m1.1.1.2.cmml" xref="S4.Ex8.m1.1.1.2">𝜃</ci><cn id="S4.Ex8.m1.1.1.3.cmml" type="integer" xref="S4.Ex8.m1.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Ex8.m1.1c">\displaystyle\theta^{1}</annotation><annotation encoding="application/x-llamapun" id="S4.Ex8.m1.1d">italic_θ start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle=\frac{\mu}{\sigma^{2}},\quad\theta^{2}=-\frac{1}{2\sigma^{2}}," class="ltx_Math" display="inline" id="S4.Ex8.m2.1"><semantics id="S4.Ex8.m2.1a"><mrow id="S4.Ex8.m2.1.1.1"><mrow id="S4.Ex8.m2.1.1.1.1.2" xref="S4.Ex8.m2.1.1.1.1.3.cmml"><mrow id="S4.Ex8.m2.1.1.1.1.1.1" xref="S4.Ex8.m2.1.1.1.1.1.1.cmml"><mi id="S4.Ex8.m2.1.1.1.1.1.1.2" xref="S4.Ex8.m2.1.1.1.1.1.1.2.cmml"></mi><mo id="S4.Ex8.m2.1.1.1.1.1.1.1" xref="S4.Ex8.m2.1.1.1.1.1.1.1.cmml">=</mo><mstyle displaystyle="true" id="S4.Ex8.m2.1.1.1.1.1.1.3" xref="S4.Ex8.m2.1.1.1.1.1.1.3.cmml"><mfrac id="S4.Ex8.m2.1.1.1.1.1.1.3a" xref="S4.Ex8.m2.1.1.1.1.1.1.3.cmml"><mi 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xref="S4.Ex8.m2.1.1.1.1.2.2.3.2.3"><times id="S4.Ex8.m2.1.1.1.1.2.2.3.2.3.1.cmml" xref="S4.Ex8.m2.1.1.1.1.2.2.3.2.3.1"></times><cn id="S4.Ex8.m2.1.1.1.1.2.2.3.2.3.2.cmml" type="integer" xref="S4.Ex8.m2.1.1.1.1.2.2.3.2.3.2">2</cn><apply id="S4.Ex8.m2.1.1.1.1.2.2.3.2.3.3.cmml" xref="S4.Ex8.m2.1.1.1.1.2.2.3.2.3.3"><csymbol cd="ambiguous" id="S4.Ex8.m2.1.1.1.1.2.2.3.2.3.3.1.cmml" xref="S4.Ex8.m2.1.1.1.1.2.2.3.2.3.3">superscript</csymbol><ci id="S4.Ex8.m2.1.1.1.1.2.2.3.2.3.3.2.cmml" xref="S4.Ex8.m2.1.1.1.1.2.2.3.2.3.3.2">𝜎</ci><cn id="S4.Ex8.m2.1.1.1.1.2.2.3.2.3.3.3.cmml" type="integer" xref="S4.Ex8.m2.1.1.1.1.2.2.3.2.3.3.3">2</cn></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Ex8.m2.1c">\displaystyle=\frac{\mu}{\sigma^{2}},\quad\theta^{2}=-\frac{1}{2\sigma^{2}},</annotation><annotation encoding="application/x-llamapun" id="S4.Ex8.m2.1d">= divide start_ARG italic_μ end_ARG start_ARG italic_σ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG , italic_θ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT = - divide start_ARG 1 end_ARG start_ARG 2 italic_σ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_left_padright"></td> </tr></tbody> <tbody id="S4.E93"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_left_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\eta_{1}" class="ltx_Math" display="inline" id="S4.E93.m1.1"><semantics id="S4.E93.m1.1a"><msub id="S4.E93.m1.1.1" xref="S4.E93.m1.1.1.cmml"><mi id="S4.E93.m1.1.1.2" xref="S4.E93.m1.1.1.2.cmml">η</mi><mn id="S4.E93.m1.1.1.3" xref="S4.E93.m1.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S4.E93.m1.1b"><apply id="S4.E93.m1.1.1.cmml" xref="S4.E93.m1.1.1"><csymbol cd="ambiguous" id="S4.E93.m1.1.1.1.cmml" xref="S4.E93.m1.1.1">subscript</csymbol><ci id="S4.E93.m1.1.1.2.cmml" xref="S4.E93.m1.1.1.2">𝜂</ci><cn id="S4.E93.m1.1.1.3.cmml" type="integer" xref="S4.E93.m1.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.E93.m1.1c">\displaystyle\eta_{1}</annotation><annotation encoding="application/x-llamapun" id="S4.E93.m1.1d">italic_η start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle=\mu,\quad\eta_{2}=\mu^{2}+\sigma^{2}." class="ltx_Math" display="inline" id="S4.E93.m2.1"><semantics id="S4.E93.m2.1a"><mrow id="S4.E93.m2.1.1.1"><mrow id="S4.E93.m2.1.1.1.1.2" xref="S4.E93.m2.1.1.1.1.3.cmml"><mrow id="S4.E93.m2.1.1.1.1.1.1" xref="S4.E93.m2.1.1.1.1.1.1.cmml"><mi id="S4.E93.m2.1.1.1.1.1.1.2" xref="S4.E93.m2.1.1.1.1.1.1.2.cmml"></mi><mo id="S4.E93.m2.1.1.1.1.1.1.1" xref="S4.E93.m2.1.1.1.1.1.1.1.cmml">=</mo><mi id="S4.E93.m2.1.1.1.1.1.1.3" xref="S4.E93.m2.1.1.1.1.1.1.3.cmml">μ</mi></mrow><mo id="S4.E93.m2.1.1.1.1.2.3" rspace="1.167em" xref="S4.E93.m2.1.1.1.1.3a.cmml">,</mo><mrow id="S4.E93.m2.1.1.1.1.2.2" xref="S4.E93.m2.1.1.1.1.2.2.cmml"><msub id="S4.E93.m2.1.1.1.1.2.2.2" xref="S4.E93.m2.1.1.1.1.2.2.2.cmml"><mi id="S4.E93.m2.1.1.1.1.2.2.2.2" xref="S4.E93.m2.1.1.1.1.2.2.2.2.cmml">η</mi><mn id="S4.E93.m2.1.1.1.1.2.2.2.3" xref="S4.E93.m2.1.1.1.1.2.2.2.3.cmml">2</mn></msub><mo id="S4.E93.m2.1.1.1.1.2.2.1" xref="S4.E93.m2.1.1.1.1.2.2.1.cmml">=</mo><mrow id="S4.E93.m2.1.1.1.1.2.2.3" xref="S4.E93.m2.1.1.1.1.2.2.3.cmml"><msup id="S4.E93.m2.1.1.1.1.2.2.3.2" xref="S4.E93.m2.1.1.1.1.2.2.3.2.cmml"><mi id="S4.E93.m2.1.1.1.1.2.2.3.2.2" xref="S4.E93.m2.1.1.1.1.2.2.3.2.2.cmml">μ</mi><mn id="S4.E93.m2.1.1.1.1.2.2.3.2.3" xref="S4.E93.m2.1.1.1.1.2.2.3.2.3.cmml">2</mn></msup><mo id="S4.E93.m2.1.1.1.1.2.2.3.1" xref="S4.E93.m2.1.1.1.1.2.2.3.1.cmml">+</mo><msup id="S4.E93.m2.1.1.1.1.2.2.3.3" xref="S4.E93.m2.1.1.1.1.2.2.3.3.cmml"><mi id="S4.E93.m2.1.1.1.1.2.2.3.3.2" xref="S4.E93.m2.1.1.1.1.2.2.3.3.2.cmml">σ</mi><mn 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xref="S4.E93.m2.1.1.1.1.2.2.3.3.2">𝜎</ci><cn id="S4.E93.m2.1.1.1.1.2.2.3.3.3.cmml" type="integer" xref="S4.E93.m2.1.1.1.1.2.2.3.3.3">2</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.E93.m2.1c">\displaystyle=\mu,\quad\eta_{2}=\mu^{2}+\sigma^{2}.</annotation><annotation encoding="application/x-llamapun" id="S4.E93.m2.1d">= italic_μ , italic_η start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT = italic_μ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + italic_σ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_left_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(93)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S4.SS3.p1.7">The components <math alttext="g_{ij}(\eta)" class="ltx_Math" display="inline" id="S4.SS3.p1.6.m1.1"><semantics id="S4.SS3.p1.6.m1.1a"><mrow 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xref="S4.SS3.p1.6.m1.1.2"><times id="S4.SS3.p1.6.m1.1.2.1.cmml" xref="S4.SS3.p1.6.m1.1.2.1"></times><apply id="S4.SS3.p1.6.m1.1.2.2.cmml" xref="S4.SS3.p1.6.m1.1.2.2"><csymbol cd="ambiguous" id="S4.SS3.p1.6.m1.1.2.2.1.cmml" xref="S4.SS3.p1.6.m1.1.2.2">subscript</csymbol><ci id="S4.SS3.p1.6.m1.1.2.2.2.cmml" xref="S4.SS3.p1.6.m1.1.2.2.2">𝑔</ci><apply id="S4.SS3.p1.6.m1.1.2.2.3.cmml" xref="S4.SS3.p1.6.m1.1.2.2.3"><times id="S4.SS3.p1.6.m1.1.2.2.3.1.cmml" xref="S4.SS3.p1.6.m1.1.2.2.3.1"></times><ci id="S4.SS3.p1.6.m1.1.2.2.3.2.cmml" xref="S4.SS3.p1.6.m1.1.2.2.3.2">𝑖</ci><ci id="S4.SS3.p1.6.m1.1.2.2.3.3.cmml" xref="S4.SS3.p1.6.m1.1.2.2.3.3">𝑗</ci></apply></apply><ci id="S4.SS3.p1.6.m1.1.1.cmml" xref="S4.SS3.p1.6.m1.1.1">𝜂</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p1.6.m1.1c">g_{ij}(\eta)</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p1.6.m1.1d">italic_g start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT ( italic_η )</annotation></semantics></math> of the metric tensor <math alttext="g(\eta)" class="ltx_Math" display="inline" id="S4.SS3.p1.7.m2.1"><semantics id="S4.SS3.p1.7.m2.1a"><mrow id="S4.SS3.p1.7.m2.1.2" xref="S4.SS3.p1.7.m2.1.2.cmml"><mi id="S4.SS3.p1.7.m2.1.2.2" xref="S4.SS3.p1.7.m2.1.2.2.cmml">g</mi><mo id="S4.SS3.p1.7.m2.1.2.1" xref="S4.SS3.p1.7.m2.1.2.1.cmml">⁢</mo><mrow id="S4.SS3.p1.7.m2.1.2.3.2" xref="S4.SS3.p1.7.m2.1.2.cmml"><mo id="S4.SS3.p1.7.m2.1.2.3.2.1" stretchy="false" xref="S4.SS3.p1.7.m2.1.2.cmml">(</mo><mi id="S4.SS3.p1.7.m2.1.1" xref="S4.SS3.p1.7.m2.1.1.cmml">η</mi><mo id="S4.SS3.p1.7.m2.1.2.3.2.2" stretchy="false" xref="S4.SS3.p1.7.m2.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.p1.7.m2.1b"><apply id="S4.SS3.p1.7.m2.1.2.cmml" xref="S4.SS3.p1.7.m2.1.2"><times id="S4.SS3.p1.7.m2.1.2.1.cmml" xref="S4.SS3.p1.7.m2.1.2.1"></times><ci id="S4.SS3.p1.7.m2.1.2.2.cmml" xref="S4.SS3.p1.7.m2.1.2.2">𝑔</ci><ci id="S4.SS3.p1.7.m2.1.1.cmml" 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id="S4.E96.m1.1.2.2.3.2.cmml" xref="S4.E96.m1.1.2.2.3.2">𝑖</ci><ci id="S4.E96.m1.1.2.2.3.3.cmml" xref="S4.E96.m1.1.2.2.3.3">𝑗</ci></apply></apply><ci id="S4.E96.m1.1.1.cmml" xref="S4.E96.m1.1.1">𝜂</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.E96.m1.1c">\displaystyle g_{ij}(\eta)</annotation><annotation encoding="application/x-llamapun" id="S4.E96.m1.1d">italic_g start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT ( italic_η )</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle=2\big{(}\eta_{2}-(\eta_{1})^{2}\big{)}\left(\begin{array}[]{cc}% \frac{1}{2}&\eta_{1}\\[4.30554pt] \eta_{1}&(\eta_{1})^{2}+\eta_{2}\end{array}\right)" class="ltx_Math" display="inline" id="S4.E96.m2.2"><semantics id="S4.E96.m2.2a"><mrow id="S4.E96.m2.2.2" xref="S4.E96.m2.2.2.cmml"><mi id="S4.E96.m2.2.2.3" xref="S4.E96.m2.2.2.3.cmml"></mi><mo id="S4.E96.m2.2.2.2" xref="S4.E96.m2.2.2.2.cmml">=</mo><mrow 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start_ROW start_CELL italic_η start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_CELL start_CELL ( italic_η start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + italic_η start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_CELL end_ROW end_ARRAY )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_left_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(96)</span></td> </tr></tbody> <tbody id="S4.E99"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_left_padleft"></td> <td class="ltx_td ltx_eqn_cell"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle=2\sigma^{2}\left(\begin{array}[]{cc}\frac{1}{2}&\mu\\[4.30554pt] \mu&2\mu^{2}+\sigma^{2}\end{array}\right)." class="ltx_Math" display="inline" id="S4.E99.m1.2"><semantics id="S4.E99.m1.2a"><mrow id="S4.E99.m1.2.2.1" 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id="S4.E98.2.1.2.2.cmml" type="integer" xref="S4.E98.2.1.2.2">2</cn><apply id="S4.E98.2.1.2.3.cmml" xref="S4.E98.2.1.2.3"><csymbol cd="ambiguous" id="S4.E98.2.1.2.3.1.cmml" xref="S4.E98.2.1.2.3">superscript</csymbol><ci id="S4.E98.2.1.2.3.2.cmml" xref="S4.E98.2.1.2.3.2">𝜇</ci><cn id="S4.E98.2.1.2.3.3.cmml" type="integer" xref="S4.E98.2.1.2.3.3">2</cn></apply></apply><apply id="S4.E98.2.1.3.cmml" xref="S4.E98.2.1.3"><csymbol cd="ambiguous" id="S4.E98.2.1.3.1.cmml" xref="S4.E98.2.1.3">superscript</csymbol><ci id="S4.E98.2.1.3.2.cmml" xref="S4.E98.2.1.3.2">𝜎</ci><cn id="S4.E98.2.1.3.3.cmml" type="integer" xref="S4.E98.2.1.3.3">2</cn></apply></apply></matrixrow></matrix></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.E99.m1.2c">\displaystyle=2\sigma^{2}\left(\begin{array}[]{cc}\frac{1}{2}&\mu\\[4.30554pt] \mu&2\mu^{2}+\sigma^{2}\end{array}\right).</annotation><annotation encoding="application/x-llamapun" id="S4.E99.m1.2d">= 2 italic_σ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( start_ARRAY start_ROW start_CELL divide start_ARG 1 end_ARG start_ARG 2 end_ARG end_CELL start_CELL italic_μ end_CELL end_ROW start_ROW start_CELL italic_μ end_CELL start_CELL 2 italic_μ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + italic_σ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_CELL end_ROW end_ARRAY ) .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_left_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(99)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S4.SS3.p1.23">The linear differential equation (<a class="ltx_ref" href="https://arxiv.org/html/2406.11224v2#S2.E16" title="In 2.2 Gradient-Flow Equations ‣ 2 Information Geometry and Gradient-Flow Equations ‣ A Hamiltonian approach to the gradient-flow equations in information geometry"><span class="ltx_text ltx_ref_tag">16</span></a>) of the gradient-flow equations (<a class="ltx_ref" href="https://arxiv.org/html/2406.11224v2#S2.E15" title="In 2.2 Gradient-Flow Equations ‣ 2 Information Geometry and Gradient-Flow Equations ‣ A Hamiltonian approach to the gradient-flow equations in information geometry"><span class="ltx_text ltx_ref_tag">15</span></a>) are</p> <table class="ltx_equationgroup ltx_eqn_table" id="S4.E100"> <tbody> <tr class="ltx_eqn_row" id="S3.EGx89"><td class="ltx_eqn_cell" colspan="5"></td></tr> <tr class="ltx_equation ltx_eqn_row ltx_align_baseline" id="S4.E100.1"> <td class="ltx_eqn_cell ltx_eqn_left_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\frac{d}{dt}\theta^{1}" class="ltx_Math" display="inline" id="S4.E100.1.m1.1"><semantics id="S4.E100.1.m1.1a"><mrow id="S4.E100.1.m1.1.1" xref="S4.E100.1.m1.1.1.cmml"><mstyle displaystyle="true" id="S4.E100.1.m1.1.1.2" xref="S4.E100.1.m1.1.1.2.cmml"><mfrac id="S4.E100.1.m1.1.1.2a" xref="S4.E100.1.m1.1.1.2.cmml"><mi id="S4.E100.1.m1.1.1.2.2" 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xref="S4.E100.1.m1.1.1.2.2">𝑑</ci><apply id="S4.E100.1.m1.1.1.2.3.cmml" xref="S4.E100.1.m1.1.1.2.3"><times id="S4.E100.1.m1.1.1.2.3.1.cmml" xref="S4.E100.1.m1.1.1.2.3.1"></times><ci id="S4.E100.1.m1.1.1.2.3.2.cmml" xref="S4.E100.1.m1.1.1.2.3.2">𝑑</ci><ci id="S4.E100.1.m1.1.1.2.3.3.cmml" xref="S4.E100.1.m1.1.1.2.3.3">𝑡</ci></apply></apply><apply id="S4.E100.1.m1.1.1.3.cmml" xref="S4.E100.1.m1.1.1.3"><csymbol cd="ambiguous" id="S4.E100.1.m1.1.1.3.1.cmml" xref="S4.E100.1.m1.1.1.3">superscript</csymbol><ci id="S4.E100.1.m1.1.1.3.2.cmml" xref="S4.E100.1.m1.1.1.3.2">𝜃</ci><cn id="S4.E100.1.m1.1.1.3.3.cmml" type="integer" xref="S4.E100.1.m1.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.E100.1.m1.1c">\displaystyle\frac{d}{dt}\theta^{1}</annotation><annotation encoding="application/x-llamapun" id="S4.E100.1.m1.1d">divide start_ARG italic_d end_ARG start_ARG italic_d italic_t end_ARG italic_θ start_POSTSUPERSCRIPT 1 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xref="S4.E100.1.m2.1.1.1.1.6.3.3.2.2">𝜎</ci><ci id="S4.E100.1.m2.1.1.1.1.6.3.3.2.3.cmml" xref="S4.E100.1.m2.1.1.1.1.6.3.3.2.3">r</ci></apply><cn id="S4.E100.1.m2.1.1.1.1.6.3.3.3.cmml" type="integer" xref="S4.E100.1.m2.1.1.1.1.6.3.3.3">2</cn></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.E100.1.m2.1c">\displaystyle=\frac{1}{\sigma^{2}}\frac{d\mu}{dt}-\frac{2\mu}{\sigma^{3}}\frac% {d\sigma}{dt}=-\frac{\mu}{\sigma^{2}}+\frac{\mu_{\rm r}}{\sigma_{\rm r}^{2}},</annotation><annotation encoding="application/x-llamapun" id="S4.E100.1.m2.1d">= divide start_ARG 1 end_ARG start_ARG italic_σ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG divide start_ARG italic_d italic_μ end_ARG start_ARG italic_d italic_t end_ARG - divide start_ARG 2 italic_μ end_ARG start_ARG italic_σ start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT end_ARG divide start_ARG italic_d italic_σ end_ARG start_ARG italic_d italic_t end_ARG = - divide start_ARG italic_μ end_ARG start_ARG italic_σ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG + divide start_ARG italic_μ start_POSTSUBSCRIPT roman_r end_POSTSUBSCRIPT end_ARG start_ARG italic_σ start_POSTSUBSCRIPT roman_r end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_left_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(100a)</span></td> </tr> <tr class="ltx_equation ltx_eqn_row ltx_align_baseline" id="S4.E100.2"> <td class="ltx_eqn_cell ltx_eqn_left_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\frac{d}{dt}\theta^{2}" class="ltx_Math" display="inline" id="S4.E100.2.m1.1"><semantics id="S4.E100.2.m1.1a"><mrow id="S4.E100.2.m1.1.1" xref="S4.E100.2.m1.1.1.cmml"><mstyle displaystyle="true" id="S4.E100.2.m1.1.1.2" xref="S4.E100.2.m1.1.1.2.cmml"><mfrac 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end_ARG start_ARG italic_d italic_t end_ARG italic_θ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle=\frac{1}{\sigma^{3}}\frac{d\sigma}{dt}=\frac{1}{2\sigma^{2}}-% \frac{1}{2\sigma_{\rm r}^{2}}," class="ltx_Math" display="inline" id="S4.E100.2.m2.1"><semantics id="S4.E100.2.m2.1a"><mrow id="S4.E100.2.m2.1.1.1" xref="S4.E100.2.m2.1.1.1.1.cmml"><mrow id="S4.E100.2.m2.1.1.1.1" xref="S4.E100.2.m2.1.1.1.1.cmml"><mi id="S4.E100.2.m2.1.1.1.1.2" xref="S4.E100.2.m2.1.1.1.1.2.cmml"></mi><mo id="S4.E100.2.m2.1.1.1.1.3" xref="S4.E100.2.m2.1.1.1.1.3.cmml">=</mo><mrow id="S4.E100.2.m2.1.1.1.1.4" xref="S4.E100.2.m2.1.1.1.1.4.cmml"><mstyle displaystyle="true" id="S4.E100.2.m2.1.1.1.1.4.2" xref="S4.E100.2.m2.1.1.1.1.4.2.cmml"><mfrac id="S4.E100.2.m2.1.1.1.1.4.2a" xref="S4.E100.2.m2.1.1.1.1.4.2.cmml"><mn id="S4.E100.2.m2.1.1.1.1.4.2.2" xref="S4.E100.2.m2.1.1.1.1.4.2.2.cmml">1</mn><msup 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id="S4.E100.2.m2.1.1.1.1.6.3.3.3.3.cmml" type="integer" xref="S4.E100.2.m2.1.1.1.1.6.3.3.3.3">2</cn></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.E100.2.m2.1c">\displaystyle=\frac{1}{\sigma^{3}}\frac{d\sigma}{dt}=\frac{1}{2\sigma^{2}}-% \frac{1}{2\sigma_{\rm r}^{2}},</annotation><annotation encoding="application/x-llamapun" id="S4.E100.2.m2.1d">= divide start_ARG 1 end_ARG start_ARG italic_σ start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT end_ARG divide start_ARG italic_d italic_σ end_ARG start_ARG italic_d italic_t end_ARG = divide start_ARG 1 end_ARG start_ARG 2 italic_σ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG - divide start_ARG 1 end_ARG start_ARG 2 italic_σ start_POSTSUBSCRIPT roman_r end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_left_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(100b)</span></td> </tr> </tbody> </table> <p class="ltx_p" id="S4.SS3.p1.14">where the set of <math alttext="\mu_{\rm r}" class="ltx_Math" display="inline" id="S4.SS3.p1.8.m1.1"><semantics id="S4.SS3.p1.8.m1.1a"><msub id="S4.SS3.p1.8.m1.1.1" xref="S4.SS3.p1.8.m1.1.1.cmml"><mi id="S4.SS3.p1.8.m1.1.1.2" xref="S4.SS3.p1.8.m1.1.1.2.cmml">μ</mi><mi id="S4.SS3.p1.8.m1.1.1.3" mathvariant="normal" xref="S4.SS3.p1.8.m1.1.1.3.cmml">r</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS3.p1.8.m1.1b"><apply id="S4.SS3.p1.8.m1.1.1.cmml" xref="S4.SS3.p1.8.m1.1.1"><csymbol cd="ambiguous" id="S4.SS3.p1.8.m1.1.1.1.cmml" xref="S4.SS3.p1.8.m1.1.1">subscript</csymbol><ci id="S4.SS3.p1.8.m1.1.1.2.cmml" xref="S4.SS3.p1.8.m1.1.1.2">𝜇</ci><ci id="S4.SS3.p1.8.m1.1.1.3.cmml" xref="S4.SS3.p1.8.m1.1.1.3">r</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p1.8.m1.1c">\mu_{\rm r}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p1.8.m1.1d">italic_μ start_POSTSUBSCRIPT roman_r end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="\sigma_{\rm r}" class="ltx_Math" display="inline" id="S4.SS3.p1.9.m2.1"><semantics id="S4.SS3.p1.9.m2.1a"><msub id="S4.SS3.p1.9.m2.1.1" xref="S4.SS3.p1.9.m2.1.1.cmml"><mi id="S4.SS3.p1.9.m2.1.1.2" xref="S4.SS3.p1.9.m2.1.1.2.cmml">σ</mi><mi id="S4.SS3.p1.9.m2.1.1.3" mathvariant="normal" xref="S4.SS3.p1.9.m2.1.1.3.cmml">r</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS3.p1.9.m2.1b"><apply id="S4.SS3.p1.9.m2.1.1.cmml" xref="S4.SS3.p1.9.m2.1.1"><csymbol cd="ambiguous" id="S4.SS3.p1.9.m2.1.1.1.cmml" xref="S4.SS3.p1.9.m2.1.1">subscript</csymbol><ci id="S4.SS3.p1.9.m2.1.1.2.cmml" xref="S4.SS3.p1.9.m2.1.1.2">𝜎</ci><ci id="S4.SS3.p1.9.m2.1.1.3.cmml" xref="S4.SS3.p1.9.m2.1.1.3">r</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p1.9.m2.1c">\sigma_{\rm r}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p1.9.m2.1d">italic_σ start_POSTSUBSCRIPT roman_r end_POSTSUBSCRIPT</annotation></semantics></math> specify the reference state, whose <math alttext="\theta" class="ltx_Math" display="inline" id="S4.SS3.p1.10.m3.1"><semantics id="S4.SS3.p1.10.m3.1a"><mi id="S4.SS3.p1.10.m3.1.1" xref="S4.SS3.p1.10.m3.1.1.cmml">θ</mi><annotation-xml encoding="MathML-Content" id="S4.SS3.p1.10.m3.1b"><ci id="S4.SS3.p1.10.m3.1.1.cmml" xref="S4.SS3.p1.10.m3.1.1">𝜃</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p1.10.m3.1c">\theta</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p1.10.m3.1d">italic_θ</annotation></semantics></math>-coordinates are <math alttext="\theta_{\rm r}^{1}=\mu_{\rm r}/\sigma_{\rm r}^{2}" class="ltx_Math" display="inline" id="S4.SS3.p1.11.m4.1"><semantics id="S4.SS3.p1.11.m4.1a"><mrow id="S4.SS3.p1.11.m4.1.1" xref="S4.SS3.p1.11.m4.1.1.cmml"><msubsup 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mathvariant="normal" xref="S4.SS3.p1.11.m4.1.1.3.3.2.3.cmml">r</mi><mn id="S4.SS3.p1.11.m4.1.1.3.3.3" xref="S4.SS3.p1.11.m4.1.1.3.3.3.cmml">2</mn></msubsup></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.p1.11.m4.1b"><apply id="S4.SS3.p1.11.m4.1.1.cmml" xref="S4.SS3.p1.11.m4.1.1"><eq id="S4.SS3.p1.11.m4.1.1.1.cmml" xref="S4.SS3.p1.11.m4.1.1.1"></eq><apply id="S4.SS3.p1.11.m4.1.1.2.cmml" xref="S4.SS3.p1.11.m4.1.1.2"><csymbol cd="ambiguous" id="S4.SS3.p1.11.m4.1.1.2.1.cmml" xref="S4.SS3.p1.11.m4.1.1.2">superscript</csymbol><apply id="S4.SS3.p1.11.m4.1.1.2.2.cmml" xref="S4.SS3.p1.11.m4.1.1.2"><csymbol cd="ambiguous" id="S4.SS3.p1.11.m4.1.1.2.2.1.cmml" xref="S4.SS3.p1.11.m4.1.1.2">subscript</csymbol><ci id="S4.SS3.p1.11.m4.1.1.2.2.2.cmml" xref="S4.SS3.p1.11.m4.1.1.2.2.2">𝜃</ci><ci id="S4.SS3.p1.11.m4.1.1.2.2.3.cmml" xref="S4.SS3.p1.11.m4.1.1.2.2.3">r</ci></apply><cn id="S4.SS3.p1.11.m4.1.1.2.3.cmml" type="integer" xref="S4.SS3.p1.11.m4.1.1.2.3">1</cn></apply><apply id="S4.SS3.p1.11.m4.1.1.3.cmml" xref="S4.SS3.p1.11.m4.1.1.3"><divide id="S4.SS3.p1.11.m4.1.1.3.1.cmml" xref="S4.SS3.p1.11.m4.1.1.3.1"></divide><apply id="S4.SS3.p1.11.m4.1.1.3.2.cmml" xref="S4.SS3.p1.11.m4.1.1.3.2"><csymbol cd="ambiguous" id="S4.SS3.p1.11.m4.1.1.3.2.1.cmml" xref="S4.SS3.p1.11.m4.1.1.3.2">subscript</csymbol><ci id="S4.SS3.p1.11.m4.1.1.3.2.2.cmml" xref="S4.SS3.p1.11.m4.1.1.3.2.2">𝜇</ci><ci id="S4.SS3.p1.11.m4.1.1.3.2.3.cmml" xref="S4.SS3.p1.11.m4.1.1.3.2.3">r</ci></apply><apply id="S4.SS3.p1.11.m4.1.1.3.3.cmml" xref="S4.SS3.p1.11.m4.1.1.3.3"><csymbol cd="ambiguous" id="S4.SS3.p1.11.m4.1.1.3.3.1.cmml" xref="S4.SS3.p1.11.m4.1.1.3.3">superscript</csymbol><apply id="S4.SS3.p1.11.m4.1.1.3.3.2.cmml" xref="S4.SS3.p1.11.m4.1.1.3.3"><csymbol cd="ambiguous" id="S4.SS3.p1.11.m4.1.1.3.3.2.1.cmml" xref="S4.SS3.p1.11.m4.1.1.3.3">subscript</csymbol><ci id="S4.SS3.p1.11.m4.1.1.3.3.2.2.cmml" xref="S4.SS3.p1.11.m4.1.1.3.3.2.2">𝜎</ci><ci id="S4.SS3.p1.11.m4.1.1.3.3.2.3.cmml" xref="S4.SS3.p1.11.m4.1.1.3.3.2.3">r</ci></apply><cn id="S4.SS3.p1.11.m4.1.1.3.3.3.cmml" type="integer" xref="S4.SS3.p1.11.m4.1.1.3.3.3">2</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p1.11.m4.1c">\theta_{\rm r}^{1}=\mu_{\rm r}/\sigma_{\rm r}^{2}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p1.11.m4.1d">italic_θ start_POSTSUBSCRIPT roman_r end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT = italic_μ start_POSTSUBSCRIPT roman_r end_POSTSUBSCRIPT / italic_σ start_POSTSUBSCRIPT roman_r end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT</annotation></semantics></math> and <math alttext="\theta_{\rm r}^{2}=-1/(2\sigma_{\rm r}^{2})" class="ltx_Math" display="inline" id="S4.SS3.p1.12.m5.1"><semantics id="S4.SS3.p1.12.m5.1a"><mrow id="S4.SS3.p1.12.m5.1.1" xref="S4.SS3.p1.12.m5.1.1.cmml"><msubsup id="S4.SS3.p1.12.m5.1.1.3" xref="S4.SS3.p1.12.m5.1.1.3.cmml"><mi id="S4.SS3.p1.12.m5.1.1.3.2.2" xref="S4.SS3.p1.12.m5.1.1.3.2.2.cmml">θ</mi><mi id="S4.SS3.p1.12.m5.1.1.3.2.3" mathvariant="normal" xref="S4.SS3.p1.12.m5.1.1.3.2.3.cmml">r</mi><mn id="S4.SS3.p1.12.m5.1.1.3.3" xref="S4.SS3.p1.12.m5.1.1.3.3.cmml">2</mn></msubsup><mo id="S4.SS3.p1.12.m5.1.1.2" xref="S4.SS3.p1.12.m5.1.1.2.cmml">=</mo><mrow id="S4.SS3.p1.12.m5.1.1.1" xref="S4.SS3.p1.12.m5.1.1.1.cmml"><mo id="S4.SS3.p1.12.m5.1.1.1a" xref="S4.SS3.p1.12.m5.1.1.1.cmml">−</mo><mrow id="S4.SS3.p1.12.m5.1.1.1.1" xref="S4.SS3.p1.12.m5.1.1.1.1.cmml"><mn id="S4.SS3.p1.12.m5.1.1.1.1.3" xref="S4.SS3.p1.12.m5.1.1.1.1.3.cmml">1</mn><mo id="S4.SS3.p1.12.m5.1.1.1.1.2" xref="S4.SS3.p1.12.m5.1.1.1.1.2.cmml">/</mo><mrow id="S4.SS3.p1.12.m5.1.1.1.1.1.1" xref="S4.SS3.p1.12.m5.1.1.1.1.1.1.1.cmml"><mo id="S4.SS3.p1.12.m5.1.1.1.1.1.1.2" stretchy="false" xref="S4.SS3.p1.12.m5.1.1.1.1.1.1.1.cmml">(</mo><mrow id="S4.SS3.p1.12.m5.1.1.1.1.1.1.1" xref="S4.SS3.p1.12.m5.1.1.1.1.1.1.1.cmml"><mn id="S4.SS3.p1.12.m5.1.1.1.1.1.1.1.2" 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id="S4.SS3.p1.12.m5.1.1.3.1.cmml" xref="S4.SS3.p1.12.m5.1.1.3">superscript</csymbol><apply id="S4.SS3.p1.12.m5.1.1.3.2.cmml" xref="S4.SS3.p1.12.m5.1.1.3"><csymbol cd="ambiguous" id="S4.SS3.p1.12.m5.1.1.3.2.1.cmml" xref="S4.SS3.p1.12.m5.1.1.3">subscript</csymbol><ci id="S4.SS3.p1.12.m5.1.1.3.2.2.cmml" xref="S4.SS3.p1.12.m5.1.1.3.2.2">𝜃</ci><ci id="S4.SS3.p1.12.m5.1.1.3.2.3.cmml" xref="S4.SS3.p1.12.m5.1.1.3.2.3">r</ci></apply><cn id="S4.SS3.p1.12.m5.1.1.3.3.cmml" type="integer" xref="S4.SS3.p1.12.m5.1.1.3.3">2</cn></apply><apply id="S4.SS3.p1.12.m5.1.1.1.cmml" xref="S4.SS3.p1.12.m5.1.1.1"><minus id="S4.SS3.p1.12.m5.1.1.1.2.cmml" xref="S4.SS3.p1.12.m5.1.1.1"></minus><apply id="S4.SS3.p1.12.m5.1.1.1.1.cmml" xref="S4.SS3.p1.12.m5.1.1.1.1"><divide id="S4.SS3.p1.12.m5.1.1.1.1.2.cmml" xref="S4.SS3.p1.12.m5.1.1.1.1.2"></divide><cn id="S4.SS3.p1.12.m5.1.1.1.1.3.cmml" type="integer" xref="S4.SS3.p1.12.m5.1.1.1.1.3">1</cn><apply id="S4.SS3.p1.12.m5.1.1.1.1.1.1.1.cmml" xref="S4.SS3.p1.12.m5.1.1.1.1.1.1"><times id="S4.SS3.p1.12.m5.1.1.1.1.1.1.1.1.cmml" xref="S4.SS3.p1.12.m5.1.1.1.1.1.1.1.1"></times><cn id="S4.SS3.p1.12.m5.1.1.1.1.1.1.1.2.cmml" type="integer" xref="S4.SS3.p1.12.m5.1.1.1.1.1.1.1.2">2</cn><apply id="S4.SS3.p1.12.m5.1.1.1.1.1.1.1.3.cmml" xref="S4.SS3.p1.12.m5.1.1.1.1.1.1.1.3"><csymbol cd="ambiguous" id="S4.SS3.p1.12.m5.1.1.1.1.1.1.1.3.1.cmml" xref="S4.SS3.p1.12.m5.1.1.1.1.1.1.1.3">superscript</csymbol><apply id="S4.SS3.p1.12.m5.1.1.1.1.1.1.1.3.2.cmml" xref="S4.SS3.p1.12.m5.1.1.1.1.1.1.1.3"><csymbol cd="ambiguous" id="S4.SS3.p1.12.m5.1.1.1.1.1.1.1.3.2.1.cmml" xref="S4.SS3.p1.12.m5.1.1.1.1.1.1.1.3">subscript</csymbol><ci id="S4.SS3.p1.12.m5.1.1.1.1.1.1.1.3.2.2.cmml" xref="S4.SS3.p1.12.m5.1.1.1.1.1.1.1.3.2.2">𝜎</ci><ci id="S4.SS3.p1.12.m5.1.1.1.1.1.1.1.3.2.3.cmml" xref="S4.SS3.p1.12.m5.1.1.1.1.1.1.1.3.2.3">r</ci></apply><cn id="S4.SS3.p1.12.m5.1.1.1.1.1.1.1.3.3.cmml" type="integer" xref="S4.SS3.p1.12.m5.1.1.1.1.1.1.1.3.3">2</cn></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p1.12.m5.1c">\theta_{\rm r}^{2}=-1/(2\sigma_{\rm r}^{2})</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p1.12.m5.1d">italic_θ start_POSTSUBSCRIPT roman_r end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT = - 1 / ( 2 italic_σ start_POSTSUBSCRIPT roman_r end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT )</annotation></semantics></math>. From (<a class="ltx_ref" href="https://arxiv.org/html/2406.11224v2#S4.E100" title="In 4.3 Gaussian model ‣ 4 The motions of a light-like particle in a pseudo Riemann space ‣ A Hamiltonian approach to the gradient-flow equations in information geometry"><span class="ltx_text ltx_ref_tag">100</span></a>), we obtain the differential equations for <math alttext="\mu(t)" class="ltx_Math" display="inline" id="S4.SS3.p1.13.m6.1"><semantics id="S4.SS3.p1.13.m6.1a"><mrow id="S4.SS3.p1.13.m6.1.2" xref="S4.SS3.p1.13.m6.1.2.cmml"><mi id="S4.SS3.p1.13.m6.1.2.2" xref="S4.SS3.p1.13.m6.1.2.2.cmml">μ</mi><mo id="S4.SS3.p1.13.m6.1.2.1" xref="S4.SS3.p1.13.m6.1.2.1.cmml">⁢</mo><mrow id="S4.SS3.p1.13.m6.1.2.3.2" xref="S4.SS3.p1.13.m6.1.2.cmml"><mo id="S4.SS3.p1.13.m6.1.2.3.2.1" stretchy="false" xref="S4.SS3.p1.13.m6.1.2.cmml">(</mo><mi id="S4.SS3.p1.13.m6.1.1" xref="S4.SS3.p1.13.m6.1.1.cmml">t</mi><mo id="S4.SS3.p1.13.m6.1.2.3.2.2" stretchy="false" xref="S4.SS3.p1.13.m6.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.p1.13.m6.1b"><apply id="S4.SS3.p1.13.m6.1.2.cmml" xref="S4.SS3.p1.13.m6.1.2"><times id="S4.SS3.p1.13.m6.1.2.1.cmml" xref="S4.SS3.p1.13.m6.1.2.1"></times><ci id="S4.SS3.p1.13.m6.1.2.2.cmml" xref="S4.SS3.p1.13.m6.1.2.2">𝜇</ci><ci id="S4.SS3.p1.13.m6.1.1.cmml" xref="S4.SS3.p1.13.m6.1.1">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p1.13.m6.1c">\mu(t)</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p1.13.m6.1d">italic_μ ( italic_t )</annotation></semantics></math> and <math alttext="\sigma(t)" class="ltx_Math" display="inline" id="S4.SS3.p1.14.m7.1"><semantics id="S4.SS3.p1.14.m7.1a"><mrow id="S4.SS3.p1.14.m7.1.2" xref="S4.SS3.p1.14.m7.1.2.cmml"><mi id="S4.SS3.p1.14.m7.1.2.2" xref="S4.SS3.p1.14.m7.1.2.2.cmml">σ</mi><mo id="S4.SS3.p1.14.m7.1.2.1" xref="S4.SS3.p1.14.m7.1.2.1.cmml">⁢</mo><mrow id="S4.SS3.p1.14.m7.1.2.3.2" xref="S4.SS3.p1.14.m7.1.2.cmml"><mo id="S4.SS3.p1.14.m7.1.2.3.2.1" stretchy="false" xref="S4.SS3.p1.14.m7.1.2.cmml">(</mo><mi id="S4.SS3.p1.14.m7.1.1" xref="S4.SS3.p1.14.m7.1.1.cmml">t</mi><mo id="S4.SS3.p1.14.m7.1.2.3.2.2" stretchy="false" xref="S4.SS3.p1.14.m7.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.p1.14.m7.1b"><apply id="S4.SS3.p1.14.m7.1.2.cmml" xref="S4.SS3.p1.14.m7.1.2"><times id="S4.SS3.p1.14.m7.1.2.1.cmml" xref="S4.SS3.p1.14.m7.1.2.1"></times><ci id="S4.SS3.p1.14.m7.1.2.2.cmml" xref="S4.SS3.p1.14.m7.1.2.2">𝜎</ci><ci id="S4.SS3.p1.14.m7.1.1.cmml" xref="S4.SS3.p1.14.m7.1.1">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p1.14.m7.1c">\sigma(t)</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p1.14.m7.1d">italic_σ ( italic_t )</annotation></semantics></math> as</p> <table class="ltx_equationgroup ltx_eqn_table" id="S4.E101"> <tbody> <tr class="ltx_eqn_row" id="S3.EGx90"><td class="ltx_eqn_cell" colspan="5"></td></tr> <tr class="ltx_equation ltx_eqn_row ltx_align_baseline" id="S4.E101.1"> <td class="ltx_eqn_cell ltx_eqn_left_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\frac{d}{dt}\mu(t)" class="ltx_Math" display="inline" id="S4.E101.1.m1.1"><semantics id="S4.E101.1.m1.1a"><mrow id="S4.E101.1.m1.1.2" xref="S4.E101.1.m1.1.2.cmml"><mstyle displaystyle="true" id="S4.E101.1.m1.1.2.2" xref="S4.E101.1.m1.1.2.2.cmml"><mfrac id="S4.E101.1.m1.1.2.2a" xref="S4.E101.1.m1.1.2.2.cmml"><mi id="S4.E101.1.m1.1.2.2.2" xref="S4.E101.1.m1.1.2.2.2.cmml">d</mi><mrow id="S4.E101.1.m1.1.2.2.3" xref="S4.E101.1.m1.1.2.2.3.cmml"><mi id="S4.E101.1.m1.1.2.2.3.2" xref="S4.E101.1.m1.1.2.2.3.2.cmml">d</mi><mo id="S4.E101.1.m1.1.2.2.3.1" xref="S4.E101.1.m1.1.2.2.3.1.cmml">⁢</mo><mi id="S4.E101.1.m1.1.2.2.3.3" xref="S4.E101.1.m1.1.2.2.3.3.cmml">t</mi></mrow></mfrac></mstyle><mo id="S4.E101.1.m1.1.2.1" xref="S4.E101.1.m1.1.2.1.cmml">⁢</mo><mi id="S4.E101.1.m1.1.2.3" xref="S4.E101.1.m1.1.2.3.cmml">μ</mi><mo id="S4.E101.1.m1.1.2.1a" xref="S4.E101.1.m1.1.2.1.cmml">⁢</mo><mrow id="S4.E101.1.m1.1.2.4.2" xref="S4.E101.1.m1.1.2.cmml"><mo id="S4.E101.1.m1.1.2.4.2.1" stretchy="false" xref="S4.E101.1.m1.1.2.cmml">(</mo><mi id="S4.E101.1.m1.1.1" xref="S4.E101.1.m1.1.1.cmml">t</mi><mo id="S4.E101.1.m1.1.2.4.2.2" stretchy="false" xref="S4.E101.1.m1.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.E101.1.m1.1b"><apply id="S4.E101.1.m1.1.2.cmml" xref="S4.E101.1.m1.1.2"><times id="S4.E101.1.m1.1.2.1.cmml" xref="S4.E101.1.m1.1.2.1"></times><apply id="S4.E101.1.m1.1.2.2.cmml" xref="S4.E101.1.m1.1.2.2"><divide id="S4.E101.1.m1.1.2.2.1.cmml" xref="S4.E101.1.m1.1.2.2"></divide><ci id="S4.E101.1.m1.1.2.2.2.cmml" xref="S4.E101.1.m1.1.2.2.2">𝑑</ci><apply id="S4.E101.1.m1.1.2.2.3.cmml" xref="S4.E101.1.m1.1.2.2.3"><times id="S4.E101.1.m1.1.2.2.3.1.cmml" xref="S4.E101.1.m1.1.2.2.3.1"></times><ci id="S4.E101.1.m1.1.2.2.3.2.cmml" xref="S4.E101.1.m1.1.2.2.3.2">𝑑</ci><ci id="S4.E101.1.m1.1.2.2.3.3.cmml" xref="S4.E101.1.m1.1.2.2.3.3">𝑡</ci></apply></apply><ci id="S4.E101.1.m1.1.2.3.cmml" xref="S4.E101.1.m1.1.2.3">𝜇</ci><ci id="S4.E101.1.m1.1.1.cmml" xref="S4.E101.1.m1.1.1">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.E101.1.m1.1c">\displaystyle\frac{d}{dt}\mu(t)</annotation><annotation encoding="application/x-llamapun" id="S4.E101.1.m1.1d">divide start_ARG italic_d end_ARG start_ARG italic_d italic_t end_ARG italic_μ ( italic_t )</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle=\frac{\sigma^{2}}{\sigma_{\rm r}^{2}}(\mu_{\rm r}-\mu(t))," class="ltx_Math" display="inline" id="S4.E101.1.m2.2"><semantics id="S4.E101.1.m2.2a"><mrow id="S4.E101.1.m2.2.2.1" xref="S4.E101.1.m2.2.2.1.1.cmml"><mrow id="S4.E101.1.m2.2.2.1.1" xref="S4.E101.1.m2.2.2.1.1.cmml"><mi id="S4.E101.1.m2.2.2.1.1.3" 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id="S4.E101.1.m2.2.2.1.1.1.2" xref="S4.E101.1.m2.2.2.1.1.1.2.cmml">⁢</mo><mrow id="S4.E101.1.m2.2.2.1.1.1.1.1" xref="S4.E101.1.m2.2.2.1.1.1.1.1.1.cmml"><mo id="S4.E101.1.m2.2.2.1.1.1.1.1.2" stretchy="false" xref="S4.E101.1.m2.2.2.1.1.1.1.1.1.cmml">(</mo><mrow id="S4.E101.1.m2.2.2.1.1.1.1.1.1" xref="S4.E101.1.m2.2.2.1.1.1.1.1.1.cmml"><msub id="S4.E101.1.m2.2.2.1.1.1.1.1.1.2" xref="S4.E101.1.m2.2.2.1.1.1.1.1.1.2.cmml"><mi id="S4.E101.1.m2.2.2.1.1.1.1.1.1.2.2" xref="S4.E101.1.m2.2.2.1.1.1.1.1.1.2.2.cmml">μ</mi><mi id="S4.E101.1.m2.2.2.1.1.1.1.1.1.2.3" mathvariant="normal" xref="S4.E101.1.m2.2.2.1.1.1.1.1.1.2.3.cmml">r</mi></msub><mo id="S4.E101.1.m2.2.2.1.1.1.1.1.1.1" xref="S4.E101.1.m2.2.2.1.1.1.1.1.1.1.cmml">−</mo><mrow id="S4.E101.1.m2.2.2.1.1.1.1.1.1.3" xref="S4.E101.1.m2.2.2.1.1.1.1.1.1.3.cmml"><mi id="S4.E101.1.m2.2.2.1.1.1.1.1.1.3.2" xref="S4.E101.1.m2.2.2.1.1.1.1.1.1.3.2.cmml">μ</mi><mo id="S4.E101.1.m2.2.2.1.1.1.1.1.1.3.1" xref="S4.E101.1.m2.2.2.1.1.1.1.1.1.3.1.cmml">⁢</mo><mrow 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id="S4.E101.1.m2.2.2.1.1.1.1.1.1.3.1.cmml" xref="S4.E101.1.m2.2.2.1.1.1.1.1.1.3.1"></times><ci id="S4.E101.1.m2.2.2.1.1.1.1.1.1.3.2.cmml" xref="S4.E101.1.m2.2.2.1.1.1.1.1.1.3.2">𝜇</ci><ci id="S4.E101.1.m2.1.1.cmml" xref="S4.E101.1.m2.1.1">𝑡</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.E101.1.m2.2c">\displaystyle=\frac{\sigma^{2}}{\sigma_{\rm r}^{2}}(\mu_{\rm r}-\mu(t)),</annotation><annotation encoding="application/x-llamapun" id="S4.E101.1.m2.2d">= divide start_ARG italic_σ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG italic_σ start_POSTSUBSCRIPT roman_r end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG ( italic_μ start_POSTSUBSCRIPT roman_r end_POSTSUBSCRIPT - italic_μ ( italic_t ) ) ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_left_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(101a)</span></td> </tr> <tr class="ltx_equation ltx_eqn_row ltx_align_baseline" id="S4.E101.2"> <td class="ltx_eqn_cell ltx_eqn_left_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\frac{d}{dt}\sigma(t)" class="ltx_Math" display="inline" id="S4.E101.2.m1.1"><semantics id="S4.E101.2.m1.1a"><mrow id="S4.E101.2.m1.1.2" xref="S4.E101.2.m1.1.2.cmml"><mstyle displaystyle="true" id="S4.E101.2.m1.1.2.2" xref="S4.E101.2.m1.1.2.2.cmml"><mfrac id="S4.E101.2.m1.1.2.2a" xref="S4.E101.2.m1.1.2.2.cmml"><mi id="S4.E101.2.m1.1.2.2.2" xref="S4.E101.2.m1.1.2.2.2.cmml">d</mi><mrow id="S4.E101.2.m1.1.2.2.3" xref="S4.E101.2.m1.1.2.2.3.cmml"><mi id="S4.E101.2.m1.1.2.2.3.2" xref="S4.E101.2.m1.1.2.2.3.2.cmml">d</mi><mo id="S4.E101.2.m1.1.2.2.3.1" xref="S4.E101.2.m1.1.2.2.3.1.cmml">⁢</mo><mi id="S4.E101.2.m1.1.2.2.3.3" xref="S4.E101.2.m1.1.2.2.3.3.cmml">t</mi></mrow></mfrac></mstyle><mo id="S4.E101.2.m1.1.2.1" xref="S4.E101.2.m1.1.2.1.cmml">⁢</mo><mi id="S4.E101.2.m1.1.2.3" xref="S4.E101.2.m1.1.2.3.cmml">σ</mi><mo id="S4.E101.2.m1.1.2.1a" xref="S4.E101.2.m1.1.2.1.cmml">⁢</mo><mrow id="S4.E101.2.m1.1.2.4.2" xref="S4.E101.2.m1.1.2.cmml"><mo id="S4.E101.2.m1.1.2.4.2.1" stretchy="false" xref="S4.E101.2.m1.1.2.cmml">(</mo><mi id="S4.E101.2.m1.1.1" xref="S4.E101.2.m1.1.1.cmml">t</mi><mo id="S4.E101.2.m1.1.2.4.2.2" stretchy="false" xref="S4.E101.2.m1.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.E101.2.m1.1b"><apply id="S4.E101.2.m1.1.2.cmml" xref="S4.E101.2.m1.1.2"><times id="S4.E101.2.m1.1.2.1.cmml" xref="S4.E101.2.m1.1.2.1"></times><apply id="S4.E101.2.m1.1.2.2.cmml" xref="S4.E101.2.m1.1.2.2"><divide id="S4.E101.2.m1.1.2.2.1.cmml" xref="S4.E101.2.m1.1.2.2"></divide><ci id="S4.E101.2.m1.1.2.2.2.cmml" xref="S4.E101.2.m1.1.2.2.2">𝑑</ci><apply id="S4.E101.2.m1.1.2.2.3.cmml" xref="S4.E101.2.m1.1.2.2.3"><times id="S4.E101.2.m1.1.2.2.3.1.cmml" xref="S4.E101.2.m1.1.2.2.3.1"></times><ci id="S4.E101.2.m1.1.2.2.3.2.cmml" xref="S4.E101.2.m1.1.2.2.3.2">𝑑</ci><ci id="S4.E101.2.m1.1.2.2.3.3.cmml" xref="S4.E101.2.m1.1.2.2.3.3">𝑡</ci></apply></apply><ci id="S4.E101.2.m1.1.2.3.cmml" xref="S4.E101.2.m1.1.2.3">𝜎</ci><ci id="S4.E101.2.m1.1.1.cmml" xref="S4.E101.2.m1.1.1">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.E101.2.m1.1c">\displaystyle\frac{d}{dt}\sigma(t)</annotation><annotation encoding="application/x-llamapun" id="S4.E101.2.m1.1d">divide start_ARG italic_d end_ARG start_ARG italic_d italic_t end_ARG italic_σ ( italic_t )</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle=\frac{1}{2}\left(\sigma(t)-\frac{\sigma^{3}(t)}{\sigma_{\rm r}^{% 2}}\right)," class="ltx_Math" display="inline" id="S4.E101.2.m2.3"><semantics id="S4.E101.2.m2.3a"><mrow id="S4.E101.2.m2.3.3.1" xref="S4.E101.2.m2.3.3.1.1.cmml"><mrow id="S4.E101.2.m2.3.3.1.1" xref="S4.E101.2.m2.3.3.1.1.cmml"><mi id="S4.E101.2.m2.3.3.1.1.3" xref="S4.E101.2.m2.3.3.1.1.3.cmml"></mi><mo id="S4.E101.2.m2.3.3.1.1.2" xref="S4.E101.2.m2.3.3.1.1.2.cmml">=</mo><mrow id="S4.E101.2.m2.3.3.1.1.1" xref="S4.E101.2.m2.3.3.1.1.1.cmml"><mstyle displaystyle="true" id="S4.E101.2.m2.3.3.1.1.1.3" xref="S4.E101.2.m2.3.3.1.1.1.3.cmml"><mfrac id="S4.E101.2.m2.3.3.1.1.1.3a" xref="S4.E101.2.m2.3.3.1.1.1.3.cmml"><mn id="S4.E101.2.m2.3.3.1.1.1.3.2" xref="S4.E101.2.m2.3.3.1.1.1.3.2.cmml">1</mn><mn id="S4.E101.2.m2.3.3.1.1.1.3.3" xref="S4.E101.2.m2.3.3.1.1.1.3.3.cmml">2</mn></mfrac></mstyle><mo id="S4.E101.2.m2.3.3.1.1.1.2" xref="S4.E101.2.m2.3.3.1.1.1.2.cmml">⁢</mo><mrow id="S4.E101.2.m2.3.3.1.1.1.1.1" xref="S4.E101.2.m2.3.3.1.1.1.1.1.1.cmml"><mo id="S4.E101.2.m2.3.3.1.1.1.1.1.2" xref="S4.E101.2.m2.3.3.1.1.1.1.1.1.cmml">(</mo><mrow id="S4.E101.2.m2.3.3.1.1.1.1.1.1" xref="S4.E101.2.m2.3.3.1.1.1.1.1.1.cmml"><mrow id="S4.E101.2.m2.3.3.1.1.1.1.1.1.2" xref="S4.E101.2.m2.3.3.1.1.1.1.1.1.2.cmml"><mi id="S4.E101.2.m2.3.3.1.1.1.1.1.1.2.2" xref="S4.E101.2.m2.3.3.1.1.1.1.1.1.2.2.cmml">σ</mi><mo id="S4.E101.2.m2.3.3.1.1.1.1.1.1.2.1" xref="S4.E101.2.m2.3.3.1.1.1.1.1.1.2.1.cmml">⁢</mo><mrow id="S4.E101.2.m2.3.3.1.1.1.1.1.1.2.3.2" xref="S4.E101.2.m2.3.3.1.1.1.1.1.1.2.cmml"><mo id="S4.E101.2.m2.3.3.1.1.1.1.1.1.2.3.2.1" stretchy="false" xref="S4.E101.2.m2.3.3.1.1.1.1.1.1.2.cmml">(</mo><mi id="S4.E101.2.m2.2.2" xref="S4.E101.2.m2.2.2.cmml">t</mi><mo id="S4.E101.2.m2.3.3.1.1.1.1.1.1.2.3.2.2" stretchy="false" xref="S4.E101.2.m2.3.3.1.1.1.1.1.1.2.cmml">)</mo></mrow></mrow><mo id="S4.E101.2.m2.3.3.1.1.1.1.1.1.1" xref="S4.E101.2.m2.3.3.1.1.1.1.1.1.1.cmml">−</mo><mstyle displaystyle="true" id="S4.E101.2.m2.1.1" xref="S4.E101.2.m2.1.1.cmml"><mfrac id="S4.E101.2.m2.1.1a" xref="S4.E101.2.m2.1.1.cmml"><mrow id="S4.E101.2.m2.1.1.1" xref="S4.E101.2.m2.1.1.1.cmml"><msup id="S4.E101.2.m2.1.1.1.3" xref="S4.E101.2.m2.1.1.1.3.cmml"><mi id="S4.E101.2.m2.1.1.1.3.2" xref="S4.E101.2.m2.1.1.1.3.2.cmml">σ</mi><mn id="S4.E101.2.m2.1.1.1.3.3" xref="S4.E101.2.m2.1.1.1.3.3.cmml">3</mn></msup><mo id="S4.E101.2.m2.1.1.1.2" xref="S4.E101.2.m2.1.1.1.2.cmml">⁢</mo><mrow id="S4.E101.2.m2.1.1.1.4.2" xref="S4.E101.2.m2.1.1.1.cmml"><mo id="S4.E101.2.m2.1.1.1.4.2.1" stretchy="false" xref="S4.E101.2.m2.1.1.1.cmml">(</mo><mi id="S4.E101.2.m2.1.1.1.1" xref="S4.E101.2.m2.1.1.1.1.cmml">t</mi><mo id="S4.E101.2.m2.1.1.1.4.2.2" stretchy="false" xref="S4.E101.2.m2.1.1.1.cmml">)</mo></mrow></mrow><msubsup id="S4.E101.2.m2.1.1.3" xref="S4.E101.2.m2.1.1.3.cmml"><mi id="S4.E101.2.m2.1.1.3.2.2" xref="S4.E101.2.m2.1.1.3.2.2.cmml">σ</mi><mi id="S4.E101.2.m2.1.1.3.2.3" mathvariant="normal" xref="S4.E101.2.m2.1.1.3.2.3.cmml">r</mi><mn id="S4.E101.2.m2.1.1.3.3" xref="S4.E101.2.m2.1.1.3.3.cmml">2</mn></msubsup></mfrac></mstyle></mrow><mo id="S4.E101.2.m2.3.3.1.1.1.1.1.3" xref="S4.E101.2.m2.3.3.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><mo id="S4.E101.2.m2.3.3.1.2" xref="S4.E101.2.m2.3.3.1.1.cmml">,</mo></mrow><annotation-xml 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xref="S4.E101.2.m2.1.1.1.1">𝑡</ci></apply><apply id="S4.E101.2.m2.1.1.3.cmml" xref="S4.E101.2.m2.1.1.3"><csymbol cd="ambiguous" id="S4.E101.2.m2.1.1.3.1.cmml" xref="S4.E101.2.m2.1.1.3">superscript</csymbol><apply id="S4.E101.2.m2.1.1.3.2.cmml" xref="S4.E101.2.m2.1.1.3"><csymbol cd="ambiguous" id="S4.E101.2.m2.1.1.3.2.1.cmml" xref="S4.E101.2.m2.1.1.3">subscript</csymbol><ci id="S4.E101.2.m2.1.1.3.2.2.cmml" xref="S4.E101.2.m2.1.1.3.2.2">𝜎</ci><ci id="S4.E101.2.m2.1.1.3.2.3.cmml" xref="S4.E101.2.m2.1.1.3.2.3">r</ci></apply><cn id="S4.E101.2.m2.1.1.3.3.cmml" type="integer" xref="S4.E101.2.m2.1.1.3.3">2</cn></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.E101.2.m2.3c">\displaystyle=\frac{1}{2}\left(\sigma(t)-\frac{\sigma^{3}(t)}{\sigma_{\rm r}^{% 2}}\right),</annotation><annotation encoding="application/x-llamapun" id="S4.E101.2.m2.3d">= divide start_ARG 1 end_ARG start_ARG 2 end_ARG ( italic_σ ( italic_t ) - divide start_ARG italic_σ start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ( italic_t ) end_ARG start_ARG italic_σ start_POSTSUBSCRIPT roman_r end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG ) ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_left_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(101b)</span></td> </tr> </tbody> </table> <p class="ltx_p" id="S4.SS3.p1.24">and the solutions are</p> <table class="ltx_equationgroup ltx_eqn_table" id="S4.E102"> <tbody> <tr class="ltx_eqn_row" id="S3.EGx91"><td class="ltx_eqn_cell" colspan="5"></td></tr> <tr class="ltx_equation ltx_eqn_row ltx_align_baseline" id="S4.E102.1"> <td class="ltx_eqn_cell ltx_eqn_left_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\mu(t)" class="ltx_Math" display="inline" id="S4.E102.1.m1.1"><semantics id="S4.E102.1.m1.1a"><mrow id="S4.E102.1.m1.1.2" xref="S4.E102.1.m1.1.2.cmml"><mi id="S4.E102.1.m1.1.2.2" xref="S4.E102.1.m1.1.2.2.cmml">μ</mi><mo id="S4.E102.1.m1.1.2.1" xref="S4.E102.1.m1.1.2.1.cmml">⁢</mo><mrow id="S4.E102.1.m1.1.2.3.2" xref="S4.E102.1.m1.1.2.cmml"><mo id="S4.E102.1.m1.1.2.3.2.1" stretchy="false" xref="S4.E102.1.m1.1.2.cmml">(</mo><mi id="S4.E102.1.m1.1.1" xref="S4.E102.1.m1.1.1.cmml">t</mi><mo id="S4.E102.1.m1.1.2.3.2.2" stretchy="false" xref="S4.E102.1.m1.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.E102.1.m1.1b"><apply id="S4.E102.1.m1.1.2.cmml" xref="S4.E102.1.m1.1.2"><times id="S4.E102.1.m1.1.2.1.cmml" xref="S4.E102.1.m1.1.2.1"></times><ci id="S4.E102.1.m1.1.2.2.cmml" xref="S4.E102.1.m1.1.2.2">𝜇</ci><ci id="S4.E102.1.m1.1.1.cmml" xref="S4.E102.1.m1.1.1">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.E102.1.m1.1c">\displaystyle\mu(t)</annotation><annotation encoding="application/x-llamapun" id="S4.E102.1.m1.1d">italic_μ ( italic_t )</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle=\frac{\mu_{0}\sigma_{\rm r}^{2}+\mu_{\rm r}\sigma_{0}^{2}(\exp(t% )-1)}{\sigma_{\rm r}^{2}+\sigma_{0}^{2}(\exp(t)-1)}," class="ltx_Math" display="inline" id="S4.E102.1.m2.7"><semantics id="S4.E102.1.m2.7a"><mrow id="S4.E102.1.m2.7.7.1" xref="S4.E102.1.m2.7.7.1.1.cmml"><mrow id="S4.E102.1.m2.7.7.1.1" xref="S4.E102.1.m2.7.7.1.1.cmml"><mi id="S4.E102.1.m2.7.7.1.1.2" xref="S4.E102.1.m2.7.7.1.1.2.cmml"></mi><mo id="S4.E102.1.m2.7.7.1.1.1" xref="S4.E102.1.m2.7.7.1.1.1.cmml">=</mo><mstyle displaystyle="true" id="S4.E102.1.m2.6.6" xref="S4.E102.1.m2.6.6.cmml"><mfrac id="S4.E102.1.m2.6.6a" xref="S4.E102.1.m2.6.6.cmml"><mrow id="S4.E102.1.m2.3.3.3" xref="S4.E102.1.m2.3.3.3.cmml"><mrow id="S4.E102.1.m2.3.3.3.5" xref="S4.E102.1.m2.3.3.3.5.cmml"><msub id="S4.E102.1.m2.3.3.3.5.2" xref="S4.E102.1.m2.3.3.3.5.2.cmml"><mi id="S4.E102.1.m2.3.3.3.5.2.2" xref="S4.E102.1.m2.3.3.3.5.2.2.cmml">μ</mi><mn id="S4.E102.1.m2.3.3.3.5.2.3" xref="S4.E102.1.m2.3.3.3.5.2.3.cmml">0</mn></msub><mo id="S4.E102.1.m2.3.3.3.5.1" xref="S4.E102.1.m2.3.3.3.5.1.cmml">⁢</mo><msubsup id="S4.E102.1.m2.3.3.3.5.3" xref="S4.E102.1.m2.3.3.3.5.3.cmml"><mi id="S4.E102.1.m2.3.3.3.5.3.2.2" xref="S4.E102.1.m2.3.3.3.5.3.2.2.cmml">σ</mi><mi id="S4.E102.1.m2.3.3.3.5.3.2.3" mathvariant="normal" xref="S4.E102.1.m2.3.3.3.5.3.2.3.cmml">r</mi><mn id="S4.E102.1.m2.3.3.3.5.3.3" xref="S4.E102.1.m2.3.3.3.5.3.3.cmml">2</mn></msubsup></mrow><mo id="S4.E102.1.m2.3.3.3.4" xref="S4.E102.1.m2.3.3.3.4.cmml">+</mo><mrow id="S4.E102.1.m2.3.3.3.3" xref="S4.E102.1.m2.3.3.3.3.cmml"><msub id="S4.E102.1.m2.3.3.3.3.3" xref="S4.E102.1.m2.3.3.3.3.3.cmml"><mi id="S4.E102.1.m2.3.3.3.3.3.2" xref="S4.E102.1.m2.3.3.3.3.3.2.cmml">μ</mi><mi id="S4.E102.1.m2.3.3.3.3.3.3" mathvariant="normal" xref="S4.E102.1.m2.3.3.3.3.3.3.cmml">r</mi></msub><mo id="S4.E102.1.m2.3.3.3.3.2" 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start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( roman_exp ( italic_t ) - 1 ) end_ARG start_ARG italic_σ start_POSTSUBSCRIPT roman_r end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( roman_exp ( italic_t ) - 1 ) end_ARG ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_left_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(102a)</span></td> </tr> <tr class="ltx_equation ltx_eqn_row ltx_align_baseline" id="S4.E102.2"> <td class="ltx_eqn_cell ltx_eqn_left_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\sigma(t)" class="ltx_Math" display="inline" id="S4.E102.2.m1.1"><semantics id="S4.E102.2.m1.1a"><mrow id="S4.E102.2.m1.1.2" xref="S4.E102.2.m1.1.2.cmml"><mi id="S4.E102.2.m1.1.2.2" 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xref="S4.E102.2.m2.2.2.2.2"><divide id="S4.E102.2.m2.2.2.2.2.1.cmml" xref="S4.E102.2.m2.2.2.2.2"></divide><ci id="S4.E102.2.m2.2.2.2.2.2.cmml" xref="S4.E102.2.m2.2.2.2.2.2">𝑡</ci><cn id="S4.E102.2.m2.2.2.2.2.3.cmml" type="integer" xref="S4.E102.2.m2.2.2.2.2.3">2</cn></apply></apply></apply><apply id="S4.E102.2.m2.5.5.5.cmml" xref="S4.E102.2.m2.5.5.5"><root id="S4.E102.2.m2.5.5.5a.cmml" xref="S4.E102.2.m2.5.5.5"></root><apply id="S4.E102.2.m2.5.5.5.3.3.cmml" xref="S4.E102.2.m2.5.5.5.3.3"><plus id="S4.E102.2.m2.5.5.5.3.3.4.cmml" xref="S4.E102.2.m2.5.5.5.3.3.4"></plus><apply id="S4.E102.2.m2.5.5.5.3.3.5.cmml" xref="S4.E102.2.m2.5.5.5.3.3.5"><csymbol cd="ambiguous" id="S4.E102.2.m2.5.5.5.3.3.5.1.cmml" xref="S4.E102.2.m2.5.5.5.3.3.5">superscript</csymbol><apply id="S4.E102.2.m2.5.5.5.3.3.5.2.cmml" xref="S4.E102.2.m2.5.5.5.3.3.5"><csymbol cd="ambiguous" id="S4.E102.2.m2.5.5.5.3.3.5.2.1.cmml" xref="S4.E102.2.m2.5.5.5.3.3.5">subscript</csymbol><ci id="S4.E102.2.m2.5.5.5.3.3.5.2.2.cmml" xref="S4.E102.2.m2.5.5.5.3.3.5.2.2">𝜎</ci><ci id="S4.E102.2.m2.5.5.5.3.3.5.2.3.cmml" xref="S4.E102.2.m2.5.5.5.3.3.5.2.3">r</ci></apply><cn id="S4.E102.2.m2.5.5.5.3.3.5.3.cmml" type="integer" xref="S4.E102.2.m2.5.5.5.3.3.5.3">2</cn></apply><apply id="S4.E102.2.m2.5.5.5.3.3.3.cmml" xref="S4.E102.2.m2.5.5.5.3.3.3"><times id="S4.E102.2.m2.5.5.5.3.3.3.2.cmml" xref="S4.E102.2.m2.5.5.5.3.3.3.2"></times><apply id="S4.E102.2.m2.5.5.5.3.3.3.3.cmml" xref="S4.E102.2.m2.5.5.5.3.3.3.3"><csymbol cd="ambiguous" id="S4.E102.2.m2.5.5.5.3.3.3.3.1.cmml" xref="S4.E102.2.m2.5.5.5.3.3.3.3">superscript</csymbol><apply id="S4.E102.2.m2.5.5.5.3.3.3.3.2.cmml" xref="S4.E102.2.m2.5.5.5.3.3.3.3"><csymbol cd="ambiguous" id="S4.E102.2.m2.5.5.5.3.3.3.3.2.1.cmml" xref="S4.E102.2.m2.5.5.5.3.3.3.3">subscript</csymbol><ci id="S4.E102.2.m2.5.5.5.3.3.3.3.2.2.cmml" xref="S4.E102.2.m2.5.5.5.3.3.3.3.2.2">𝜎</ci><cn id="S4.E102.2.m2.5.5.5.3.3.3.3.2.3.cmml" type="integer" xref="S4.E102.2.m2.5.5.5.3.3.3.3.2.3">0</cn></apply><cn id="S4.E102.2.m2.5.5.5.3.3.3.3.3.cmml" type="integer" xref="S4.E102.2.m2.5.5.5.3.3.3.3.3">2</cn></apply><apply id="S4.E102.2.m2.5.5.5.3.3.3.1.1.1.cmml" xref="S4.E102.2.m2.5.5.5.3.3.3.1.1"><minus id="S4.E102.2.m2.5.5.5.3.3.3.1.1.1.1.cmml" xref="S4.E102.2.m2.5.5.5.3.3.3.1.1.1.1"></minus><apply id="S4.E102.2.m2.5.5.5.3.3.3.1.1.1.2.1.cmml" xref="S4.E102.2.m2.5.5.5.3.3.3.1.1.1.2.2"><exp id="S4.E102.2.m2.3.3.3.1.1.1.cmml" xref="S4.E102.2.m2.3.3.3.1.1.1"></exp><ci id="S4.E102.2.m2.4.4.4.2.2.2.cmml" xref="S4.E102.2.m2.4.4.4.2.2.2">𝑡</ci></apply><cn id="S4.E102.2.m2.5.5.5.3.3.3.1.1.1.3.cmml" type="integer" xref="S4.E102.2.m2.5.5.5.3.3.3.1.1.1.3">1</cn></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.E102.2.m2.6c">\displaystyle=\frac{\sigma_{\rm r}\sigma_{0}\exp\left(\frac{t}{2}\right)}{% \sqrt{\sigma_{\rm r}^{2}+\sigma_{0}^{2}(\exp(t)-1)}},</annotation><annotation encoding="application/x-llamapun" id="S4.E102.2.m2.6d">= divide start_ARG italic_σ start_POSTSUBSCRIPT roman_r end_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT roman_exp ( divide start_ARG italic_t end_ARG start_ARG 2 end_ARG ) end_ARG start_ARG square-root start_ARG italic_σ start_POSTSUBSCRIPT roman_r end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( roman_exp ( italic_t ) - 1 ) end_ARG end_ARG ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_left_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(102b)</span></td> </tr> </tbody> </table> <p class="ltx_p" id="S4.SS3.p1.22">where <math alttext="\mu_{0}" class="ltx_Math" display="inline" id="S4.SS3.p1.15.m1.1"><semantics id="S4.SS3.p1.15.m1.1a"><msub id="S4.SS3.p1.15.m1.1.1" xref="S4.SS3.p1.15.m1.1.1.cmml"><mi id="S4.SS3.p1.15.m1.1.1.2" xref="S4.SS3.p1.15.m1.1.1.2.cmml">μ</mi><mn id="S4.SS3.p1.15.m1.1.1.3" xref="S4.SS3.p1.15.m1.1.1.3.cmml">0</mn></msub><annotation-xml encoding="MathML-Content" id="S4.SS3.p1.15.m1.1b"><apply id="S4.SS3.p1.15.m1.1.1.cmml" xref="S4.SS3.p1.15.m1.1.1"><csymbol cd="ambiguous" id="S4.SS3.p1.15.m1.1.1.1.cmml" xref="S4.SS3.p1.15.m1.1.1">subscript</csymbol><ci id="S4.SS3.p1.15.m1.1.1.2.cmml" xref="S4.SS3.p1.15.m1.1.1.2">𝜇</ci><cn id="S4.SS3.p1.15.m1.1.1.3.cmml" type="integer" xref="S4.SS3.p1.15.m1.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p1.15.m1.1c">\mu_{0}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p1.15.m1.1d">italic_μ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="\sigma_{0}" class="ltx_Math" display="inline" id="S4.SS3.p1.16.m2.1"><semantics id="S4.SS3.p1.16.m2.1a"><msub id="S4.SS3.p1.16.m2.1.1" xref="S4.SS3.p1.16.m2.1.1.cmml"><mi id="S4.SS3.p1.16.m2.1.1.2" xref="S4.SS3.p1.16.m2.1.1.2.cmml">σ</mi><mn id="S4.SS3.p1.16.m2.1.1.3" xref="S4.SS3.p1.16.m2.1.1.3.cmml">0</mn></msub><annotation-xml encoding="MathML-Content" id="S4.SS3.p1.16.m2.1b"><apply id="S4.SS3.p1.16.m2.1.1.cmml" xref="S4.SS3.p1.16.m2.1.1"><csymbol cd="ambiguous" id="S4.SS3.p1.16.m2.1.1.1.cmml" xref="S4.SS3.p1.16.m2.1.1">subscript</csymbol><ci id="S4.SS3.p1.16.m2.1.1.2.cmml" xref="S4.SS3.p1.16.m2.1.1.2">𝜎</ci><cn id="S4.SS3.p1.16.m2.1.1.3.cmml" type="integer" xref="S4.SS3.p1.16.m2.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p1.16.m2.1c">\sigma_{0}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p1.16.m2.1d">italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math> are the initial values, i.e., <math alttext="\mu(0)=\mu_{0}" class="ltx_Math" display="inline" id="S4.SS3.p1.17.m3.1"><semantics id="S4.SS3.p1.17.m3.1a"><mrow id="S4.SS3.p1.17.m3.1.2" xref="S4.SS3.p1.17.m3.1.2.cmml"><mrow id="S4.SS3.p1.17.m3.1.2.2" xref="S4.SS3.p1.17.m3.1.2.2.cmml"><mi id="S4.SS3.p1.17.m3.1.2.2.2" xref="S4.SS3.p1.17.m3.1.2.2.2.cmml">μ</mi><mo id="S4.SS3.p1.17.m3.1.2.2.1" xref="S4.SS3.p1.17.m3.1.2.2.1.cmml">⁢</mo><mrow id="S4.SS3.p1.17.m3.1.2.2.3.2" xref="S4.SS3.p1.17.m3.1.2.2.cmml"><mo id="S4.SS3.p1.17.m3.1.2.2.3.2.1" stretchy="false" xref="S4.SS3.p1.17.m3.1.2.2.cmml">(</mo><mn id="S4.SS3.p1.17.m3.1.1" xref="S4.SS3.p1.17.m3.1.1.cmml">0</mn><mo id="S4.SS3.p1.17.m3.1.2.2.3.2.2" stretchy="false" xref="S4.SS3.p1.17.m3.1.2.2.cmml">)</mo></mrow></mrow><mo id="S4.SS3.p1.17.m3.1.2.1" xref="S4.SS3.p1.17.m3.1.2.1.cmml">=</mo><msub id="S4.SS3.p1.17.m3.1.2.3" xref="S4.SS3.p1.17.m3.1.2.3.cmml"><mi id="S4.SS3.p1.17.m3.1.2.3.2" xref="S4.SS3.p1.17.m3.1.2.3.2.cmml">μ</mi><mn id="S4.SS3.p1.17.m3.1.2.3.3" xref="S4.SS3.p1.17.m3.1.2.3.3.cmml">0</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.p1.17.m3.1b"><apply id="S4.SS3.p1.17.m3.1.2.cmml" xref="S4.SS3.p1.17.m3.1.2"><eq id="S4.SS3.p1.17.m3.1.2.1.cmml" xref="S4.SS3.p1.17.m3.1.2.1"></eq><apply id="S4.SS3.p1.17.m3.1.2.2.cmml" xref="S4.SS3.p1.17.m3.1.2.2"><times id="S4.SS3.p1.17.m3.1.2.2.1.cmml" xref="S4.SS3.p1.17.m3.1.2.2.1"></times><ci id="S4.SS3.p1.17.m3.1.2.2.2.cmml" xref="S4.SS3.p1.17.m3.1.2.2.2">𝜇</ci><cn id="S4.SS3.p1.17.m3.1.1.cmml" type="integer" xref="S4.SS3.p1.17.m3.1.1">0</cn></apply><apply id="S4.SS3.p1.17.m3.1.2.3.cmml" xref="S4.SS3.p1.17.m3.1.2.3"><csymbol cd="ambiguous" id="S4.SS3.p1.17.m3.1.2.3.1.cmml" xref="S4.SS3.p1.17.m3.1.2.3">subscript</csymbol><ci id="S4.SS3.p1.17.m3.1.2.3.2.cmml" xref="S4.SS3.p1.17.m3.1.2.3.2">𝜇</ci><cn id="S4.SS3.p1.17.m3.1.2.3.3.cmml" type="integer" xref="S4.SS3.p1.17.m3.1.2.3.3">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p1.17.m3.1c">\mu(0)=\mu_{0}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p1.17.m3.1d">italic_μ ( 0 ) = italic_μ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="\sigma(0)=\sigma_{0}" class="ltx_Math" display="inline" id="S4.SS3.p1.18.m4.1"><semantics id="S4.SS3.p1.18.m4.1a"><mrow id="S4.SS3.p1.18.m4.1.2" xref="S4.SS3.p1.18.m4.1.2.cmml"><mrow id="S4.SS3.p1.18.m4.1.2.2" xref="S4.SS3.p1.18.m4.1.2.2.cmml"><mi id="S4.SS3.p1.18.m4.1.2.2.2" xref="S4.SS3.p1.18.m4.1.2.2.2.cmml">σ</mi><mo id="S4.SS3.p1.18.m4.1.2.2.1" xref="S4.SS3.p1.18.m4.1.2.2.1.cmml">⁢</mo><mrow id="S4.SS3.p1.18.m4.1.2.2.3.2" xref="S4.SS3.p1.18.m4.1.2.2.cmml"><mo id="S4.SS3.p1.18.m4.1.2.2.3.2.1" stretchy="false" xref="S4.SS3.p1.18.m4.1.2.2.cmml">(</mo><mn id="S4.SS3.p1.18.m4.1.1" xref="S4.SS3.p1.18.m4.1.1.cmml">0</mn><mo id="S4.SS3.p1.18.m4.1.2.2.3.2.2" stretchy="false" xref="S4.SS3.p1.18.m4.1.2.2.cmml">)</mo></mrow></mrow><mo id="S4.SS3.p1.18.m4.1.2.1" xref="S4.SS3.p1.18.m4.1.2.1.cmml">=</mo><msub id="S4.SS3.p1.18.m4.1.2.3" xref="S4.SS3.p1.18.m4.1.2.3.cmml"><mi id="S4.SS3.p1.18.m4.1.2.3.2" xref="S4.SS3.p1.18.m4.1.2.3.2.cmml">σ</mi><mn id="S4.SS3.p1.18.m4.1.2.3.3" xref="S4.SS3.p1.18.m4.1.2.3.3.cmml">0</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.p1.18.m4.1b"><apply id="S4.SS3.p1.18.m4.1.2.cmml" xref="S4.SS3.p1.18.m4.1.2"><eq id="S4.SS3.p1.18.m4.1.2.1.cmml" xref="S4.SS3.p1.18.m4.1.2.1"></eq><apply id="S4.SS3.p1.18.m4.1.2.2.cmml" xref="S4.SS3.p1.18.m4.1.2.2"><times id="S4.SS3.p1.18.m4.1.2.2.1.cmml" xref="S4.SS3.p1.18.m4.1.2.2.1"></times><ci id="S4.SS3.p1.18.m4.1.2.2.2.cmml" xref="S4.SS3.p1.18.m4.1.2.2.2">𝜎</ci><cn id="S4.SS3.p1.18.m4.1.1.cmml" type="integer" xref="S4.SS3.p1.18.m4.1.1">0</cn></apply><apply id="S4.SS3.p1.18.m4.1.2.3.cmml" xref="S4.SS3.p1.18.m4.1.2.3"><csymbol cd="ambiguous" id="S4.SS3.p1.18.m4.1.2.3.1.cmml" xref="S4.SS3.p1.18.m4.1.2.3">subscript</csymbol><ci id="S4.SS3.p1.18.m4.1.2.3.2.cmml" xref="S4.SS3.p1.18.m4.1.2.3.2">𝜎</ci><cn id="S4.SS3.p1.18.m4.1.2.3.3.cmml" type="integer" xref="S4.SS3.p1.18.m4.1.2.3.3">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p1.18.m4.1c">\sigma(0)=\sigma_{0}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p1.18.m4.1d">italic_σ ( 0 ) = italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math>. Note that <math alttext="\mu_{\rm r}" class="ltx_Math" display="inline" id="S4.SS3.p1.19.m5.1"><semantics id="S4.SS3.p1.19.m5.1a"><msub id="S4.SS3.p1.19.m5.1.1" xref="S4.SS3.p1.19.m5.1.1.cmml"><mi id="S4.SS3.p1.19.m5.1.1.2" xref="S4.SS3.p1.19.m5.1.1.2.cmml">μ</mi><mi id="S4.SS3.p1.19.m5.1.1.3" mathvariant="normal" xref="S4.SS3.p1.19.m5.1.1.3.cmml">r</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS3.p1.19.m5.1b"><apply id="S4.SS3.p1.19.m5.1.1.cmml" xref="S4.SS3.p1.19.m5.1.1"><csymbol cd="ambiguous" id="S4.SS3.p1.19.m5.1.1.1.cmml" xref="S4.SS3.p1.19.m5.1.1">subscript</csymbol><ci id="S4.SS3.p1.19.m5.1.1.2.cmml" xref="S4.SS3.p1.19.m5.1.1.2">𝜇</ci><ci id="S4.SS3.p1.19.m5.1.1.3.cmml" xref="S4.SS3.p1.19.m5.1.1.3">r</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p1.19.m5.1c">\mu_{\rm r}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p1.19.m5.1d">italic_μ start_POSTSUBSCRIPT roman_r end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="\sigma_{\rm r}" class="ltx_Math" display="inline" id="S4.SS3.p1.20.m6.1"><semantics id="S4.SS3.p1.20.m6.1a"><msub id="S4.SS3.p1.20.m6.1.1" xref="S4.SS3.p1.20.m6.1.1.cmml"><mi id="S4.SS3.p1.20.m6.1.1.2" xref="S4.SS3.p1.20.m6.1.1.2.cmml">σ</mi><mi id="S4.SS3.p1.20.m6.1.1.3" mathvariant="normal" xref="S4.SS3.p1.20.m6.1.1.3.cmml">r</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS3.p1.20.m6.1b"><apply id="S4.SS3.p1.20.m6.1.1.cmml" xref="S4.SS3.p1.20.m6.1.1"><csymbol cd="ambiguous" id="S4.SS3.p1.20.m6.1.1.1.cmml" xref="S4.SS3.p1.20.m6.1.1">subscript</csymbol><ci id="S4.SS3.p1.20.m6.1.1.2.cmml" xref="S4.SS3.p1.20.m6.1.1.2">𝜎</ci><ci id="S4.SS3.p1.20.m6.1.1.3.cmml" xref="S4.SS3.p1.20.m6.1.1.3">r</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p1.20.m6.1c">\sigma_{\rm r}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p1.20.m6.1d">italic_σ start_POSTSUBSCRIPT roman_r end_POSTSUBSCRIPT</annotation></semantics></math> are the final values, i.e., <math alttext="\lim_{t\to\infty}\mu(t)=\mu_{\rm r}" class="ltx_Math" display="inline" id="S4.SS3.p1.21.m7.1"><semantics id="S4.SS3.p1.21.m7.1a"><mrow id="S4.SS3.p1.21.m7.1.2" xref="S4.SS3.p1.21.m7.1.2.cmml"><mrow id="S4.SS3.p1.21.m7.1.2.2" xref="S4.SS3.p1.21.m7.1.2.2.cmml"><msub id="S4.SS3.p1.21.m7.1.2.2.1" xref="S4.SS3.p1.21.m7.1.2.2.1.cmml"><mo id="S4.SS3.p1.21.m7.1.2.2.1.2" xref="S4.SS3.p1.21.m7.1.2.2.1.2.cmml">lim</mo><mrow id="S4.SS3.p1.21.m7.1.2.2.1.3" xref="S4.SS3.p1.21.m7.1.2.2.1.3.cmml"><mi id="S4.SS3.p1.21.m7.1.2.2.1.3.2" xref="S4.SS3.p1.21.m7.1.2.2.1.3.2.cmml">t</mi><mo id="S4.SS3.p1.21.m7.1.2.2.1.3.1" stretchy="false" xref="S4.SS3.p1.21.m7.1.2.2.1.3.1.cmml">→</mo><mi id="S4.SS3.p1.21.m7.1.2.2.1.3.3" mathvariant="normal" xref="S4.SS3.p1.21.m7.1.2.2.1.3.3.cmml">∞</mi></mrow></msub><mrow id="S4.SS3.p1.21.m7.1.2.2.2" xref="S4.SS3.p1.21.m7.1.2.2.2.cmml"><mi id="S4.SS3.p1.21.m7.1.2.2.2.2" xref="S4.SS3.p1.21.m7.1.2.2.2.2.cmml">μ</mi><mo id="S4.SS3.p1.21.m7.1.2.2.2.1" xref="S4.SS3.p1.21.m7.1.2.2.2.1.cmml">⁢</mo><mrow id="S4.SS3.p1.21.m7.1.2.2.2.3.2" xref="S4.SS3.p1.21.m7.1.2.2.2.cmml"><mo id="S4.SS3.p1.21.m7.1.2.2.2.3.2.1" stretchy="false" xref="S4.SS3.p1.21.m7.1.2.2.2.cmml">(</mo><mi id="S4.SS3.p1.21.m7.1.1" xref="S4.SS3.p1.21.m7.1.1.cmml">t</mi><mo id="S4.SS3.p1.21.m7.1.2.2.2.3.2.2" stretchy="false" xref="S4.SS3.p1.21.m7.1.2.2.2.cmml">)</mo></mrow></mrow></mrow><mo id="S4.SS3.p1.21.m7.1.2.1" xref="S4.SS3.p1.21.m7.1.2.1.cmml">=</mo><msub id="S4.SS3.p1.21.m7.1.2.3" xref="S4.SS3.p1.21.m7.1.2.3.cmml"><mi id="S4.SS3.p1.21.m7.1.2.3.2" xref="S4.SS3.p1.21.m7.1.2.3.2.cmml">μ</mi><mi id="S4.SS3.p1.21.m7.1.2.3.3" mathvariant="normal" xref="S4.SS3.p1.21.m7.1.2.3.3.cmml">r</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.p1.21.m7.1b"><apply id="S4.SS3.p1.21.m7.1.2.cmml" xref="S4.SS3.p1.21.m7.1.2"><eq id="S4.SS3.p1.21.m7.1.2.1.cmml" 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xref="S4.SS3.p1.21.m7.1.1">𝑡</ci></apply></apply><apply id="S4.SS3.p1.21.m7.1.2.3.cmml" xref="S4.SS3.p1.21.m7.1.2.3"><csymbol cd="ambiguous" id="S4.SS3.p1.21.m7.1.2.3.1.cmml" xref="S4.SS3.p1.21.m7.1.2.3">subscript</csymbol><ci id="S4.SS3.p1.21.m7.1.2.3.2.cmml" xref="S4.SS3.p1.21.m7.1.2.3.2">𝜇</ci><ci id="S4.SS3.p1.21.m7.1.2.3.3.cmml" xref="S4.SS3.p1.21.m7.1.2.3.3">r</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p1.21.m7.1c">\lim_{t\to\infty}\mu(t)=\mu_{\rm r}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p1.21.m7.1d">roman_lim start_POSTSUBSCRIPT italic_t → ∞ end_POSTSUBSCRIPT italic_μ ( italic_t ) = italic_μ start_POSTSUBSCRIPT roman_r end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="\lim_{t\to\infty}\sigma(t)=\sigma_{\rm r}" class="ltx_Math" display="inline" id="S4.SS3.p1.22.m8.1"><semantics id="S4.SS3.p1.22.m8.1a"><mrow id="S4.SS3.p1.22.m8.1.2" xref="S4.SS3.p1.22.m8.1.2.cmml"><mrow id="S4.SS3.p1.22.m8.1.2.2" xref="S4.SS3.p1.22.m8.1.2.2.cmml"><msub id="S4.SS3.p1.22.m8.1.2.2.1" xref="S4.SS3.p1.22.m8.1.2.2.1.cmml"><mo id="S4.SS3.p1.22.m8.1.2.2.1.2" xref="S4.SS3.p1.22.m8.1.2.2.1.2.cmml">lim</mo><mrow id="S4.SS3.p1.22.m8.1.2.2.1.3" xref="S4.SS3.p1.22.m8.1.2.2.1.3.cmml"><mi id="S4.SS3.p1.22.m8.1.2.2.1.3.2" xref="S4.SS3.p1.22.m8.1.2.2.1.3.2.cmml">t</mi><mo id="S4.SS3.p1.22.m8.1.2.2.1.3.1" stretchy="false" xref="S4.SS3.p1.22.m8.1.2.2.1.3.1.cmml">→</mo><mi id="S4.SS3.p1.22.m8.1.2.2.1.3.3" mathvariant="normal" xref="S4.SS3.p1.22.m8.1.2.2.1.3.3.cmml">∞</mi></mrow></msub><mrow id="S4.SS3.p1.22.m8.1.2.2.2" xref="S4.SS3.p1.22.m8.1.2.2.2.cmml"><mi id="S4.SS3.p1.22.m8.1.2.2.2.2" xref="S4.SS3.p1.22.m8.1.2.2.2.2.cmml">σ</mi><mo id="S4.SS3.p1.22.m8.1.2.2.2.1" xref="S4.SS3.p1.22.m8.1.2.2.2.1.cmml">⁢</mo><mrow id="S4.SS3.p1.22.m8.1.2.2.2.3.2" xref="S4.SS3.p1.22.m8.1.2.2.2.cmml"><mo id="S4.SS3.p1.22.m8.1.2.2.2.3.2.1" stretchy="false" xref="S4.SS3.p1.22.m8.1.2.2.2.cmml">(</mo><mi id="S4.SS3.p1.22.m8.1.1" xref="S4.SS3.p1.22.m8.1.1.cmml">t</mi><mo id="S4.SS3.p1.22.m8.1.2.2.2.3.2.2" stretchy="false" xref="S4.SS3.p1.22.m8.1.2.2.2.cmml">)</mo></mrow></mrow></mrow><mo id="S4.SS3.p1.22.m8.1.2.1" xref="S4.SS3.p1.22.m8.1.2.1.cmml">=</mo><msub id="S4.SS3.p1.22.m8.1.2.3" xref="S4.SS3.p1.22.m8.1.2.3.cmml"><mi id="S4.SS3.p1.22.m8.1.2.3.2" xref="S4.SS3.p1.22.m8.1.2.3.2.cmml">σ</mi><mi id="S4.SS3.p1.22.m8.1.2.3.3" mathvariant="normal" xref="S4.SS3.p1.22.m8.1.2.3.3.cmml">r</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.p1.22.m8.1b"><apply id="S4.SS3.p1.22.m8.1.2.cmml" xref="S4.SS3.p1.22.m8.1.2"><eq id="S4.SS3.p1.22.m8.1.2.1.cmml" xref="S4.SS3.p1.22.m8.1.2.1"></eq><apply id="S4.SS3.p1.22.m8.1.2.2.cmml" xref="S4.SS3.p1.22.m8.1.2.2"><apply id="S4.SS3.p1.22.m8.1.2.2.1.cmml" xref="S4.SS3.p1.22.m8.1.2.2.1"><csymbol cd="ambiguous" id="S4.SS3.p1.22.m8.1.2.2.1.1.cmml" xref="S4.SS3.p1.22.m8.1.2.2.1">subscript</csymbol><limit id="S4.SS3.p1.22.m8.1.2.2.1.2.cmml" 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xref="S4.SS3.p1.22.m8.1.2.3.3">r</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p1.22.m8.1c">\lim_{t\to\infty}\sigma(t)=\sigma_{\rm r}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p1.22.m8.1d">roman_lim start_POSTSUBSCRIPT italic_t → ∞ end_POSTSUBSCRIPT italic_σ ( italic_t ) = italic_σ start_POSTSUBSCRIPT roman_r end_POSTSUBSCRIPT</annotation></semantics></math>.</p> </div> <figure class="ltx_figure" id="S4.F1"><img alt="Refer to caption" class="ltx_graphics ltx_centering ltx_img_landscape" height="268" id="S4.F1.g1" src="x1.png" width="381"/> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure">Figure 1: </span>The gradient-flows of the equations (<a class="ltx_ref" href="https://arxiv.org/html/2406.11224v2#S4.E101" title="In 4.3 Gaussian model ‣ 4 The motions of a light-like particle in a pseudo Riemann space ‣ A Hamiltonian approach to the gradient-flow equations in information geometry"><span class="ltx_text ltx_ref_tag">101</span></a>) with <math alttext="\mu_{\rm r}=1.2,\sigma_{\rm r}=0.8" class="ltx_Math" display="inline" id="S4.F1.2.m1.2"><semantics id="S4.F1.2.m1.2b"><mrow id="S4.F1.2.m1.2.2.2" xref="S4.F1.2.m1.2.2.3.cmml"><mrow id="S4.F1.2.m1.1.1.1.1" xref="S4.F1.2.m1.1.1.1.1.cmml"><msub id="S4.F1.2.m1.1.1.1.1.2" xref="S4.F1.2.m1.1.1.1.1.2.cmml"><mi id="S4.F1.2.m1.1.1.1.1.2.2" xref="S4.F1.2.m1.1.1.1.1.2.2.cmml">μ</mi><mi id="S4.F1.2.m1.1.1.1.1.2.3" mathvariant="normal" xref="S4.F1.2.m1.1.1.1.1.2.3.cmml">r</mi></msub><mo id="S4.F1.2.m1.1.1.1.1.1" xref="S4.F1.2.m1.1.1.1.1.1.cmml">=</mo><mn id="S4.F1.2.m1.1.1.1.1.3" xref="S4.F1.2.m1.1.1.1.1.3.cmml">1.2</mn></mrow><mo id="S4.F1.2.m1.2.2.2.3" xref="S4.F1.2.m1.2.2.3a.cmml">,</mo><mrow id="S4.F1.2.m1.2.2.2.2" xref="S4.F1.2.m1.2.2.2.2.cmml"><msub id="S4.F1.2.m1.2.2.2.2.2" xref="S4.F1.2.m1.2.2.2.2.2.cmml"><mi id="S4.F1.2.m1.2.2.2.2.2.2" xref="S4.F1.2.m1.2.2.2.2.2.2.cmml">σ</mi><mi id="S4.F1.2.m1.2.2.2.2.2.3" mathvariant="normal" xref="S4.F1.2.m1.2.2.2.2.2.3.cmml">r</mi></msub><mo id="S4.F1.2.m1.2.2.2.2.1" xref="S4.F1.2.m1.2.2.2.2.1.cmml">=</mo><mn id="S4.F1.2.m1.2.2.2.2.3" xref="S4.F1.2.m1.2.2.2.2.3.cmml">0.8</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.F1.2.m1.2c"><apply id="S4.F1.2.m1.2.2.3.cmml" xref="S4.F1.2.m1.2.2.2"><csymbol cd="ambiguous" id="S4.F1.2.m1.2.2.3a.cmml" xref="S4.F1.2.m1.2.2.2.3">formulae-sequence</csymbol><apply id="S4.F1.2.m1.1.1.1.1.cmml" xref="S4.F1.2.m1.1.1.1.1"><eq id="S4.F1.2.m1.1.1.1.1.1.cmml" xref="S4.F1.2.m1.1.1.1.1.1"></eq><apply id="S4.F1.2.m1.1.1.1.1.2.cmml" xref="S4.F1.2.m1.1.1.1.1.2"><csymbol cd="ambiguous" id="S4.F1.2.m1.1.1.1.1.2.1.cmml" xref="S4.F1.2.m1.1.1.1.1.2">subscript</csymbol><ci id="S4.F1.2.m1.1.1.1.1.2.2.cmml" xref="S4.F1.2.m1.1.1.1.1.2.2">𝜇</ci><ci id="S4.F1.2.m1.1.1.1.1.2.3.cmml" xref="S4.F1.2.m1.1.1.1.1.2.3">r</ci></apply><cn id="S4.F1.2.m1.1.1.1.1.3.cmml" type="float" xref="S4.F1.2.m1.1.1.1.1.3">1.2</cn></apply><apply id="S4.F1.2.m1.2.2.2.2.cmml" xref="S4.F1.2.m1.2.2.2.2"><eq id="S4.F1.2.m1.2.2.2.2.1.cmml" xref="S4.F1.2.m1.2.2.2.2.1"></eq><apply id="S4.F1.2.m1.2.2.2.2.2.cmml" xref="S4.F1.2.m1.2.2.2.2.2"><csymbol cd="ambiguous" id="S4.F1.2.m1.2.2.2.2.2.1.cmml" xref="S4.F1.2.m1.2.2.2.2.2">subscript</csymbol><ci id="S4.F1.2.m1.2.2.2.2.2.2.cmml" xref="S4.F1.2.m1.2.2.2.2.2.2">𝜎</ci><ci id="S4.F1.2.m1.2.2.2.2.2.3.cmml" xref="S4.F1.2.m1.2.2.2.2.2.3">r</ci></apply><cn id="S4.F1.2.m1.2.2.2.2.3.cmml" type="float" xref="S4.F1.2.m1.2.2.2.2.3">0.8</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.F1.2.m1.2d">\mu_{\rm r}=1.2,\sigma_{\rm r}=0.8</annotation><annotation encoding="application/x-llamapun" id="S4.F1.2.m1.2e">italic_μ start_POSTSUBSCRIPT roman_r end_POSTSUBSCRIPT = 1.2 , italic_σ start_POSTSUBSCRIPT roman_r end_POSTSUBSCRIPT = 0.8</annotation></semantics></math>. </figcaption> </figure> <div class="ltx_para" id="S4.SS3.p2"> <p class="ltx_p" id="S4.SS3.p2.4">Fig. <a class="ltx_ref" href="https://arxiv.org/html/2406.11224v2#S4.F1" title="Figure 1 ‣ 4.3 Gaussian model ‣ 4 The motions of a light-like particle in a pseudo Riemann space ‣ A Hamiltonian approach to the gradient-flow equations in information geometry"><span class="ltx_text ltx_ref_tag">1</span></a> shows the gradient-flows of (<a class="ltx_ref" href="https://arxiv.org/html/2406.11224v2#S4.E101" title="In 4.3 Gaussian model ‣ 4 The motions of a light-like particle in a pseudo Riemann space ‣ A Hamiltonian approach to the gradient-flow equations in information geometry"><span class="ltx_text ltx_ref_tag">101</span></a>) in <math alttext="(\mu,\sigma)" class="ltx_Math" display="inline" id="S4.SS3.p2.1.m1.2"><semantics id="S4.SS3.p2.1.m1.2a"><mrow id="S4.SS3.p2.1.m1.2.3.2" xref="S4.SS3.p2.1.m1.2.3.1.cmml"><mo id="S4.SS3.p2.1.m1.2.3.2.1" stretchy="false" xref="S4.SS3.p2.1.m1.2.3.1.cmml">(</mo><mi id="S4.SS3.p2.1.m1.1.1" xref="S4.SS3.p2.1.m1.1.1.cmml">μ</mi><mo id="S4.SS3.p2.1.m1.2.3.2.2" xref="S4.SS3.p2.1.m1.2.3.1.cmml">,</mo><mi id="S4.SS3.p2.1.m1.2.2" xref="S4.SS3.p2.1.m1.2.2.cmml">σ</mi><mo id="S4.SS3.p2.1.m1.2.3.2.3" stretchy="false" xref="S4.SS3.p2.1.m1.2.3.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.p2.1.m1.2b"><interval closure="open" id="S4.SS3.p2.1.m1.2.3.1.cmml" xref="S4.SS3.p2.1.m1.2.3.2"><ci id="S4.SS3.p2.1.m1.1.1.cmml" xref="S4.SS3.p2.1.m1.1.1">𝜇</ci><ci id="S4.SS3.p2.1.m1.2.2.cmml" xref="S4.SS3.p2.1.m1.2.2">𝜎</ci></interval></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p2.1.m1.2c">(\mu,\sigma)</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p2.1.m1.2d">( italic_μ , italic_σ )</annotation></semantics></math>-space. The reference state is specified by <math alttext="\mu_{\rm r}=1.2,\sigma_{\rm r}=0.8" class="ltx_Math" display="inline" id="S4.SS3.p2.2.m2.2"><semantics id="S4.SS3.p2.2.m2.2a"><mrow id="S4.SS3.p2.2.m2.2.2.2" xref="S4.SS3.p2.2.m2.2.2.3.cmml"><mrow id="S4.SS3.p2.2.m2.1.1.1.1" xref="S4.SS3.p2.2.m2.1.1.1.1.cmml"><msub id="S4.SS3.p2.2.m2.1.1.1.1.2" xref="S4.SS3.p2.2.m2.1.1.1.1.2.cmml"><mi id="S4.SS3.p2.2.m2.1.1.1.1.2.2" xref="S4.SS3.p2.2.m2.1.1.1.1.2.2.cmml">μ</mi><mi id="S4.SS3.p2.2.m2.1.1.1.1.2.3" mathvariant="normal" xref="S4.SS3.p2.2.m2.1.1.1.1.2.3.cmml">r</mi></msub><mo id="S4.SS3.p2.2.m2.1.1.1.1.1" xref="S4.SS3.p2.2.m2.1.1.1.1.1.cmml">=</mo><mn id="S4.SS3.p2.2.m2.1.1.1.1.3" xref="S4.SS3.p2.2.m2.1.1.1.1.3.cmml">1.2</mn></mrow><mo id="S4.SS3.p2.2.m2.2.2.2.3" xref="S4.SS3.p2.2.m2.2.2.3a.cmml">,</mo><mrow id="S4.SS3.p2.2.m2.2.2.2.2" xref="S4.SS3.p2.2.m2.2.2.2.2.cmml"><msub id="S4.SS3.p2.2.m2.2.2.2.2.2" xref="S4.SS3.p2.2.m2.2.2.2.2.2.cmml"><mi id="S4.SS3.p2.2.m2.2.2.2.2.2.2" xref="S4.SS3.p2.2.m2.2.2.2.2.2.2.cmml">σ</mi><mi id="S4.SS3.p2.2.m2.2.2.2.2.2.3" mathvariant="normal" xref="S4.SS3.p2.2.m2.2.2.2.2.2.3.cmml">r</mi></msub><mo id="S4.SS3.p2.2.m2.2.2.2.2.1" xref="S4.SS3.p2.2.m2.2.2.2.2.1.cmml">=</mo><mn id="S4.SS3.p2.2.m2.2.2.2.2.3" xref="S4.SS3.p2.2.m2.2.2.2.2.3.cmml">0.8</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.p2.2.m2.2b"><apply id="S4.SS3.p2.2.m2.2.2.3.cmml" xref="S4.SS3.p2.2.m2.2.2.2"><csymbol cd="ambiguous" id="S4.SS3.p2.2.m2.2.2.3a.cmml" xref="S4.SS3.p2.2.m2.2.2.2.3">formulae-sequence</csymbol><apply id="S4.SS3.p2.2.m2.1.1.1.1.cmml" xref="S4.SS3.p2.2.m2.1.1.1.1"><eq id="S4.SS3.p2.2.m2.1.1.1.1.1.cmml" xref="S4.SS3.p2.2.m2.1.1.1.1.1"></eq><apply id="S4.SS3.p2.2.m2.1.1.1.1.2.cmml" xref="S4.SS3.p2.2.m2.1.1.1.1.2"><csymbol cd="ambiguous" id="S4.SS3.p2.2.m2.1.1.1.1.2.1.cmml" xref="S4.SS3.p2.2.m2.1.1.1.1.2">subscript</csymbol><ci id="S4.SS3.p2.2.m2.1.1.1.1.2.2.cmml" xref="S4.SS3.p2.2.m2.1.1.1.1.2.2">𝜇</ci><ci id="S4.SS3.p2.2.m2.1.1.1.1.2.3.cmml" xref="S4.SS3.p2.2.m2.1.1.1.1.2.3">r</ci></apply><cn id="S4.SS3.p2.2.m2.1.1.1.1.3.cmml" type="float" xref="S4.SS3.p2.2.m2.1.1.1.1.3">1.2</cn></apply><apply id="S4.SS3.p2.2.m2.2.2.2.2.cmml" xref="S4.SS3.p2.2.m2.2.2.2.2"><eq id="S4.SS3.p2.2.m2.2.2.2.2.1.cmml" xref="S4.SS3.p2.2.m2.2.2.2.2.1"></eq><apply id="S4.SS3.p2.2.m2.2.2.2.2.2.cmml" xref="S4.SS3.p2.2.m2.2.2.2.2.2"><csymbol cd="ambiguous" id="S4.SS3.p2.2.m2.2.2.2.2.2.1.cmml" xref="S4.SS3.p2.2.m2.2.2.2.2.2">subscript</csymbol><ci id="S4.SS3.p2.2.m2.2.2.2.2.2.2.cmml" xref="S4.SS3.p2.2.m2.2.2.2.2.2.2">𝜎</ci><ci id="S4.SS3.p2.2.m2.2.2.2.2.2.3.cmml" xref="S4.SS3.p2.2.m2.2.2.2.2.2.3">r</ci></apply><cn id="S4.SS3.p2.2.m2.2.2.2.2.3.cmml" type="float" xref="S4.SS3.p2.2.m2.2.2.2.2.3">0.8</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p2.2.m2.2c">\mu_{\rm r}=1.2,\sigma_{\rm r}=0.8</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p2.2.m2.2d">italic_μ start_POSTSUBSCRIPT roman_r end_POSTSUBSCRIPT = 1.2 , italic_σ start_POSTSUBSCRIPT roman_r end_POSTSUBSCRIPT = 0.8</annotation></semantics></math>, at which <math alttext="\eta=\eta_{\rm r}" class="ltx_Math" display="inline" id="S4.SS3.p2.3.m3.1"><semantics id="S4.SS3.p2.3.m3.1a"><mrow id="S4.SS3.p2.3.m3.1.1" xref="S4.SS3.p2.3.m3.1.1.cmml"><mi id="S4.SS3.p2.3.m3.1.1.2" xref="S4.SS3.p2.3.m3.1.1.2.cmml">η</mi><mo id="S4.SS3.p2.3.m3.1.1.1" xref="S4.SS3.p2.3.m3.1.1.1.cmml">=</mo><msub id="S4.SS3.p2.3.m3.1.1.3" xref="S4.SS3.p2.3.m3.1.1.3.cmml"><mi id="S4.SS3.p2.3.m3.1.1.3.2" xref="S4.SS3.p2.3.m3.1.1.3.2.cmml">η</mi><mi id="S4.SS3.p2.3.m3.1.1.3.3" mathvariant="normal" xref="S4.SS3.p2.3.m3.1.1.3.3.cmml">r</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.p2.3.m3.1b"><apply id="S4.SS3.p2.3.m3.1.1.cmml" xref="S4.SS3.p2.3.m3.1.1"><eq id="S4.SS3.p2.3.m3.1.1.1.cmml" xref="S4.SS3.p2.3.m3.1.1.1"></eq><ci id="S4.SS3.p2.3.m3.1.1.2.cmml" xref="S4.SS3.p2.3.m3.1.1.2">𝜂</ci><apply id="S4.SS3.p2.3.m3.1.1.3.cmml" xref="S4.SS3.p2.3.m3.1.1.3"><csymbol cd="ambiguous" id="S4.SS3.p2.3.m3.1.1.3.1.cmml" xref="S4.SS3.p2.3.m3.1.1.3">subscript</csymbol><ci id="S4.SS3.p2.3.m3.1.1.3.2.cmml" xref="S4.SS3.p2.3.m3.1.1.3.2">𝜂</ci><ci id="S4.SS3.p2.3.m3.1.1.3.3.cmml" xref="S4.SS3.p2.3.m3.1.1.3.3">r</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p2.3.m3.1c">\eta=\eta_{\rm r}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p2.3.m3.1d">italic_η = italic_η start_POSTSUBSCRIPT roman_r end_POSTSUBSCRIPT</annotation></semantics></math> and the divergence <math alttext="D(\eta,\eta_{\rm r})" class="ltx_Math" display="inline" id="S4.SS3.p2.4.m4.2"><semantics id="S4.SS3.p2.4.m4.2a"><mrow id="S4.SS3.p2.4.m4.2.2" xref="S4.SS3.p2.4.m4.2.2.cmml"><mi id="S4.SS3.p2.4.m4.2.2.3" xref="S4.SS3.p2.4.m4.2.2.3.cmml">D</mi><mo id="S4.SS3.p2.4.m4.2.2.2" xref="S4.SS3.p2.4.m4.2.2.2.cmml">⁢</mo><mrow id="S4.SS3.p2.4.m4.2.2.1.1" xref="S4.SS3.p2.4.m4.2.2.1.2.cmml"><mo id="S4.SS3.p2.4.m4.2.2.1.1.2" stretchy="false" xref="S4.SS3.p2.4.m4.2.2.1.2.cmml">(</mo><mi id="S4.SS3.p2.4.m4.1.1" xref="S4.SS3.p2.4.m4.1.1.cmml">η</mi><mo id="S4.SS3.p2.4.m4.2.2.1.1.3" xref="S4.SS3.p2.4.m4.2.2.1.2.cmml">,</mo><msub id="S4.SS3.p2.4.m4.2.2.1.1.1" xref="S4.SS3.p2.4.m4.2.2.1.1.1.cmml"><mi id="S4.SS3.p2.4.m4.2.2.1.1.1.2" xref="S4.SS3.p2.4.m4.2.2.1.1.1.2.cmml">η</mi><mi id="S4.SS3.p2.4.m4.2.2.1.1.1.3" mathvariant="normal" xref="S4.SS3.p2.4.m4.2.2.1.1.1.3.cmml">r</mi></msub><mo id="S4.SS3.p2.4.m4.2.2.1.1.4" stretchy="false" xref="S4.SS3.p2.4.m4.2.2.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.p2.4.m4.2b"><apply id="S4.SS3.p2.4.m4.2.2.cmml" xref="S4.SS3.p2.4.m4.2.2"><times id="S4.SS3.p2.4.m4.2.2.2.cmml" xref="S4.SS3.p2.4.m4.2.2.2"></times><ci id="S4.SS3.p2.4.m4.2.2.3.cmml" xref="S4.SS3.p2.4.m4.2.2.3">𝐷</ci><interval closure="open" id="S4.SS3.p2.4.m4.2.2.1.2.cmml" xref="S4.SS3.p2.4.m4.2.2.1.1"><ci id="S4.SS3.p2.4.m4.1.1.cmml" xref="S4.SS3.p2.4.m4.1.1">𝜂</ci><apply id="S4.SS3.p2.4.m4.2.2.1.1.1.cmml" xref="S4.SS3.p2.4.m4.2.2.1.1.1"><csymbol cd="ambiguous" id="S4.SS3.p2.4.m4.2.2.1.1.1.1.cmml" xref="S4.SS3.p2.4.m4.2.2.1.1.1">subscript</csymbol><ci id="S4.SS3.p2.4.m4.2.2.1.1.1.2.cmml" xref="S4.SS3.p2.4.m4.2.2.1.1.1.2">𝜂</ci><ci id="S4.SS3.p2.4.m4.2.2.1.1.1.3.cmml" xref="S4.SS3.p2.4.m4.2.2.1.1.1.3">r</ci></apply></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p2.4.m4.2c">D(\eta,\eta_{\rm r})</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p2.4.m4.2d">italic_D ( italic_η , italic_η start_POSTSUBSCRIPT roman_r end_POSTSUBSCRIPT )</annotation></semantics></math> vanishes.</p> </div> <div class="ltx_para" id="S4.SS3.p3"> <p class="ltx_p" id="S4.SS3.p3.3">From (<a class="ltx_ref" href="https://arxiv.org/html/2406.11224v2#S4.E101" title="In 4.3 Gaussian model ‣ 4 The motions of a light-like particle in a pseudo Riemann space ‣ A Hamiltonian approach to the gradient-flow equations in information geometry"><span class="ltx_text ltx_ref_tag">101</span></a>), we also obtain the explicit relations <math alttext="dt" class="ltx_Math" display="inline" id="S4.SS3.p3.1.m1.1"><semantics id="S4.SS3.p3.1.m1.1a"><mrow id="S4.SS3.p3.1.m1.1.1" xref="S4.SS3.p3.1.m1.1.1.cmml"><mi id="S4.SS3.p3.1.m1.1.1.2" xref="S4.SS3.p3.1.m1.1.1.2.cmml">d</mi><mo id="S4.SS3.p3.1.m1.1.1.1" xref="S4.SS3.p3.1.m1.1.1.1.cmml">⁢</mo><mi id="S4.SS3.p3.1.m1.1.1.3" xref="S4.SS3.p3.1.m1.1.1.3.cmml">t</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.p3.1.m1.1b"><apply id="S4.SS3.p3.1.m1.1.1.cmml" xref="S4.SS3.p3.1.m1.1.1"><times id="S4.SS3.p3.1.m1.1.1.1.cmml" xref="S4.SS3.p3.1.m1.1.1.1"></times><ci id="S4.SS3.p3.1.m1.1.1.2.cmml" xref="S4.SS3.p3.1.m1.1.1.2">𝑑</ci><ci id="S4.SS3.p3.1.m1.1.1.3.cmml" xref="S4.SS3.p3.1.m1.1.1.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p3.1.m1.1c">dt</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p3.1.m1.1d">italic_d italic_t</annotation></semantics></math>, <math alttext="d\mu" class="ltx_Math" display="inline" id="S4.SS3.p3.2.m2.1"><semantics id="S4.SS3.p3.2.m2.1a"><mrow id="S4.SS3.p3.2.m2.1.1" xref="S4.SS3.p3.2.m2.1.1.cmml"><mi id="S4.SS3.p3.2.m2.1.1.2" xref="S4.SS3.p3.2.m2.1.1.2.cmml">d</mi><mo id="S4.SS3.p3.2.m2.1.1.1" xref="S4.SS3.p3.2.m2.1.1.1.cmml">⁢</mo><mi id="S4.SS3.p3.2.m2.1.1.3" xref="S4.SS3.p3.2.m2.1.1.3.cmml">μ</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.p3.2.m2.1b"><apply id="S4.SS3.p3.2.m2.1.1.cmml" xref="S4.SS3.p3.2.m2.1.1"><times id="S4.SS3.p3.2.m2.1.1.1.cmml" xref="S4.SS3.p3.2.m2.1.1.1"></times><ci id="S4.SS3.p3.2.m2.1.1.2.cmml" xref="S4.SS3.p3.2.m2.1.1.2">𝑑</ci><ci id="S4.SS3.p3.2.m2.1.1.3.cmml" xref="S4.SS3.p3.2.m2.1.1.3">𝜇</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p3.2.m2.1c">d\mu</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p3.2.m2.1d">italic_d italic_μ</annotation></semantics></math> and <math alttext="d\sigma" class="ltx_Math" display="inline" id="S4.SS3.p3.3.m3.1"><semantics id="S4.SS3.p3.3.m3.1a"><mrow id="S4.SS3.p3.3.m3.1.1" xref="S4.SS3.p3.3.m3.1.1.cmml"><mi id="S4.SS3.p3.3.m3.1.1.2" xref="S4.SS3.p3.3.m3.1.1.2.cmml">d</mi><mo id="S4.SS3.p3.3.m3.1.1.1" xref="S4.SS3.p3.3.m3.1.1.1.cmml">⁢</mo><mi id="S4.SS3.p3.3.m3.1.1.3" xref="S4.SS3.p3.3.m3.1.1.3.cmml">σ</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.p3.3.m3.1b"><apply id="S4.SS3.p3.3.m3.1.1.cmml" xref="S4.SS3.p3.3.m3.1.1"><times id="S4.SS3.p3.3.m3.1.1.1.cmml" xref="S4.SS3.p3.3.m3.1.1.1"></times><ci id="S4.SS3.p3.3.m3.1.1.2.cmml" xref="S4.SS3.p3.3.m3.1.1.2">𝑑</ci><ci id="S4.SS3.p3.3.m3.1.1.3.cmml" xref="S4.SS3.p3.3.m3.1.1.3">𝜎</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p3.3.m3.1c">d\sigma</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p3.3.m3.1d">italic_d italic_σ</annotation></semantics></math> along these gradient-flows as</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="S3.EGx92"> <tbody id="S4.E103"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_left_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle dt=\frac{\sigma_{\rm r}^{2}}{\sigma^{2}}\frac{d\mu}{\mu_{\rm r}-% \mu}=\frac{2d\sigma}{\sigma\left(1-\frac{\sigma^{2}}{\sigma_{\rm r}^{2}}\right% )}." class="ltx_Math" display="inline" id="S4.E103.m1.2"><semantics id="S4.E103.m1.2a"><mrow id="S4.E103.m1.2.2.1" xref="S4.E103.m1.2.2.1.1.cmml"><mrow id="S4.E103.m1.2.2.1.1" xref="S4.E103.m1.2.2.1.1.cmml"><mrow id="S4.E103.m1.2.2.1.1.2" xref="S4.E103.m1.2.2.1.1.2.cmml"><mi id="S4.E103.m1.2.2.1.1.2.2" xref="S4.E103.m1.2.2.1.1.2.2.cmml">d</mi><mo id="S4.E103.m1.2.2.1.1.2.1" xref="S4.E103.m1.2.2.1.1.2.1.cmml">⁢</mo><mi id="S4.E103.m1.2.2.1.1.2.3" xref="S4.E103.m1.2.2.1.1.2.3.cmml">t</mi></mrow><mo id="S4.E103.m1.2.2.1.1.3" xref="S4.E103.m1.2.2.1.1.3.cmml">=</mo><mrow id="S4.E103.m1.2.2.1.1.4" xref="S4.E103.m1.2.2.1.1.4.cmml"><mstyle displaystyle="true" id="S4.E103.m1.2.2.1.1.4.2" xref="S4.E103.m1.2.2.1.1.4.2.cmml"><mfrac id="S4.E103.m1.2.2.1.1.4.2a" xref="S4.E103.m1.2.2.1.1.4.2.cmml"><msubsup id="S4.E103.m1.2.2.1.1.4.2.2" xref="S4.E103.m1.2.2.1.1.4.2.2.cmml"><mi id="S4.E103.m1.2.2.1.1.4.2.2.2.2" xref="S4.E103.m1.2.2.1.1.4.2.2.2.2.cmml">σ</mi><mi id="S4.E103.m1.2.2.1.1.4.2.2.2.3" mathvariant="normal" xref="S4.E103.m1.2.2.1.1.4.2.2.2.3.cmml">r</mi><mn id="S4.E103.m1.2.2.1.1.4.2.2.3" xref="S4.E103.m1.2.2.1.1.4.2.2.3.cmml">2</mn></msubsup><msup id="S4.E103.m1.2.2.1.1.4.2.3" xref="S4.E103.m1.2.2.1.1.4.2.3.cmml"><mi id="S4.E103.m1.2.2.1.1.4.2.3.2" xref="S4.E103.m1.2.2.1.1.4.2.3.2.cmml">σ</mi><mn id="S4.E103.m1.2.2.1.1.4.2.3.3" xref="S4.E103.m1.2.2.1.1.4.2.3.3.cmml">2</mn></msup></mfrac></mstyle><mo id="S4.E103.m1.2.2.1.1.4.1" xref="S4.E103.m1.2.2.1.1.4.1.cmml">⁢</mo><mstyle displaystyle="true" id="S4.E103.m1.2.2.1.1.4.3" xref="S4.E103.m1.2.2.1.1.4.3.cmml"><mfrac id="S4.E103.m1.2.2.1.1.4.3a" xref="S4.E103.m1.2.2.1.1.4.3.cmml"><mrow id="S4.E103.m1.2.2.1.1.4.3.2" xref="S4.E103.m1.2.2.1.1.4.3.2.cmml"><mi id="S4.E103.m1.2.2.1.1.4.3.2.2" xref="S4.E103.m1.2.2.1.1.4.3.2.2.cmml">d</mi><mo id="S4.E103.m1.2.2.1.1.4.3.2.1" xref="S4.E103.m1.2.2.1.1.4.3.2.1.cmml">⁢</mo><mi id="S4.E103.m1.2.2.1.1.4.3.2.3" xref="S4.E103.m1.2.2.1.1.4.3.2.3.cmml">μ</mi></mrow><mrow id="S4.E103.m1.2.2.1.1.4.3.3" xref="S4.E103.m1.2.2.1.1.4.3.3.cmml"><msub id="S4.E103.m1.2.2.1.1.4.3.3.2" xref="S4.E103.m1.2.2.1.1.4.3.3.2.cmml"><mi id="S4.E103.m1.2.2.1.1.4.3.3.2.2" xref="S4.E103.m1.2.2.1.1.4.3.3.2.2.cmml">μ</mi><mi id="S4.E103.m1.2.2.1.1.4.3.3.2.3" mathvariant="normal" xref="S4.E103.m1.2.2.1.1.4.3.3.2.3.cmml">r</mi></msub><mo id="S4.E103.m1.2.2.1.1.4.3.3.1" xref="S4.E103.m1.2.2.1.1.4.3.3.1.cmml">−</mo><mi id="S4.E103.m1.2.2.1.1.4.3.3.3" 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\mu}=\frac{2d\sigma}{\sigma\left(1-\frac{\sigma^{2}}{\sigma_{\rm r}^{2}}\right% )}.</annotation><annotation encoding="application/x-llamapun" id="S4.E103.m1.2d">italic_d italic_t = divide start_ARG italic_σ start_POSTSUBSCRIPT roman_r end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG italic_σ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG divide start_ARG italic_d italic_μ end_ARG start_ARG italic_μ start_POSTSUBSCRIPT roman_r end_POSTSUBSCRIPT - italic_μ end_ARG = divide start_ARG 2 italic_d italic_σ end_ARG start_ARG italic_σ ( 1 - divide start_ARG italic_σ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG italic_σ start_POSTSUBSCRIPT roman_r end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG ) end_ARG .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_left_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(103)</span></td> </tr></tbody> </table> </div> </section> </section> <section class="ltx_section" id="S5"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">5 </span>Conclusions and perspectives</h2> <div class="ltx_para" id="S5.p1"> <p class="ltx_p" id="S5.p1.1">We have related the motions of a light-like particle in a curved space to the gradient-flows in IG. Based on the point-particle viewpoint, we have rederived the Hamiltonians (<a class="ltx_ref" href="https://arxiv.org/html/2406.11224v2#S2.E20" title="In 2.2 Gradient-Flow Equations ‣ 2 Information Geometry and Gradient-Flow Equations ‣ A Hamiltonian approach to the gradient-flow equations in information geometry"><span class="ltx_text ltx_ref_tag">20</span></a>) in our previous works <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2406.11224v2#bib.bib10" title="">10</a>, <a class="ltx_ref" href="https://arxiv.org/html/2406.11224v2#bib.bib13" title="">13</a>]</cite>. In addition, it is shown that the complete integrability of Pfaffian systems for the Poincaré-Cartan one-form supports this type of Hamiltonian (<a class="ltx_ref" href="https://arxiv.org/html/2406.11224v2#S2.E20" title="In 2.2 Gradient-Flow Equations ‣ 2 Information Geometry and Gradient-Flow Equations ‣ A Hamiltonian approach to the gradient-flow equations in information geometry"><span class="ltx_text ltx_ref_tag">20</span></a>).</p> </div> <div class="ltx_para" id="S5.p2"> <p class="ltx_p" id="S5.p2.8">As mentioned in Introduction, our studies on the gradient-flow equations in IG have some relations to different fields such as analytical mechanics, geometric optics, thermodynamics, general relativity, cosmology and so on. 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ltx_align_right">(104)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S5.p2.7">where <math alttext="X_{i}" class="ltx_Math" display="inline" id="S5.p2.1.m1.1"><semantics id="S5.p2.1.m1.1a"><msub id="S5.p2.1.m1.1.1" xref="S5.p2.1.m1.1.1.cmml"><mi id="S5.p2.1.m1.1.1.2" xref="S5.p2.1.m1.1.1.2.cmml">X</mi><mi id="S5.p2.1.m1.1.1.3" xref="S5.p2.1.m1.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S5.p2.1.m1.1b"><apply id="S5.p2.1.m1.1.1.cmml" xref="S5.p2.1.m1.1.1"><csymbol cd="ambiguous" id="S5.p2.1.m1.1.1.1.cmml" xref="S5.p2.1.m1.1.1">subscript</csymbol><ci id="S5.p2.1.m1.1.1.2.cmml" xref="S5.p2.1.m1.1.1.2">𝑋</ci><ci id="S5.p2.1.m1.1.1.3.cmml" xref="S5.p2.1.m1.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.p2.1.m1.1c">X_{i}</annotation><annotation encoding="application/x-llamapun" id="S5.p2.1.m1.1d">italic_X start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> denotes an extensive variable, <math alttext="S(X)" class="ltx_Math" display="inline" id="S5.p2.2.m2.1"><semantics id="S5.p2.2.m2.1a"><mrow id="S5.p2.2.m2.1.2" xref="S5.p2.2.m2.1.2.cmml"><mi id="S5.p2.2.m2.1.2.2" xref="S5.p2.2.m2.1.2.2.cmml">S</mi><mo id="S5.p2.2.m2.1.2.1" xref="S5.p2.2.m2.1.2.1.cmml">⁢</mo><mrow id="S5.p2.2.m2.1.2.3.2" xref="S5.p2.2.m2.1.2.cmml"><mo id="S5.p2.2.m2.1.2.3.2.1" stretchy="false" xref="S5.p2.2.m2.1.2.cmml">(</mo><mi id="S5.p2.2.m2.1.1" xref="S5.p2.2.m2.1.1.cmml">X</mi><mo id="S5.p2.2.m2.1.2.3.2.2" stretchy="false" xref="S5.p2.2.m2.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.p2.2.m2.1b"><apply id="S5.p2.2.m2.1.2.cmml" xref="S5.p2.2.m2.1.2"><times id="S5.p2.2.m2.1.2.1.cmml" xref="S5.p2.2.m2.1.2.1"></times><ci id="S5.p2.2.m2.1.2.2.cmml" xref="S5.p2.2.m2.1.2.2">𝑆</ci><ci id="S5.p2.2.m2.1.1.cmml" xref="S5.p2.2.m2.1.1">𝑋</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.p2.2.m2.1c">S(X)</annotation><annotation encoding="application/x-llamapun" id="S5.p2.2.m2.1d">italic_S ( italic_X )</annotation></semantics></math> is the thermodynamic entropy, and <math alttext="L_{ij}(X)" class="ltx_Math" display="inline" id="S5.p2.3.m3.1"><semantics id="S5.p2.3.m3.1a"><mrow id="S5.p2.3.m3.1.2" xref="S5.p2.3.m3.1.2.cmml"><msub id="S5.p2.3.m3.1.2.2" xref="S5.p2.3.m3.1.2.2.cmml"><mi id="S5.p2.3.m3.1.2.2.2" xref="S5.p2.3.m3.1.2.2.2.cmml">L</mi><mrow id="S5.p2.3.m3.1.2.2.3" xref="S5.p2.3.m3.1.2.2.3.cmml"><mi id="S5.p2.3.m3.1.2.2.3.2" xref="S5.p2.3.m3.1.2.2.3.2.cmml">i</mi><mo id="S5.p2.3.m3.1.2.2.3.1" xref="S5.p2.3.m3.1.2.2.3.1.cmml">⁢</mo><mi id="S5.p2.3.m3.1.2.2.3.3" xref="S5.p2.3.m3.1.2.2.3.3.cmml">j</mi></mrow></msub><mo id="S5.p2.3.m3.1.2.1" xref="S5.p2.3.m3.1.2.1.cmml">⁢</mo><mrow id="S5.p2.3.m3.1.2.3.2" xref="S5.p2.3.m3.1.2.cmml"><mo id="S5.p2.3.m3.1.2.3.2.1" stretchy="false" xref="S5.p2.3.m3.1.2.cmml">(</mo><mi id="S5.p2.3.m3.1.1" xref="S5.p2.3.m3.1.1.cmml">X</mi><mo id="S5.p2.3.m3.1.2.3.2.2" 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start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT ( italic_X )</annotation></semantics></math> are the components of Onsager’s phenomenological matrix. Note that (<a class="ltx_ref" href="https://arxiv.org/html/2406.11224v2#S5.E104" title="In 5 Conclusions and perspectives ‣ A Hamiltonian approach to the gradient-flow equations in information geometry"><span class="ltx_text ltx_ref_tag">104</span></a>) can be regarded as the gradient-flow equations (<a class="ltx_ref" href="https://arxiv.org/html/2406.11224v2#S2.E15" title="In 2.2 Gradient-Flow Equations ‣ 2 Information Geometry and Gradient-Flow Equations ‣ A Hamiltonian approach to the gradient-flow equations in information geometry"><span class="ltx_text ltx_ref_tag">15</span></a>) if we make the correspondence that <math alttext="X_{i}\leftrightarrow\eta_{i},L_{ij}\leftrightarrow g_{ij}(\eta)" class="ltx_Math" display="inline" id="S5.p2.4.m4.3"><semantics id="S5.p2.4.m4.3a"><mrow id="S5.p2.4.m4.3.3" xref="S5.p2.4.m4.3.3.cmml"><msub id="S5.p2.4.m4.3.3.4" xref="S5.p2.4.m4.3.3.4.cmml"><mi id="S5.p2.4.m4.3.3.4.2" xref="S5.p2.4.m4.3.3.4.2.cmml">X</mi><mi id="S5.p2.4.m4.3.3.4.3" 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id="S5.p2.4.m4.3.3.7.2.2.cmml" xref="S5.p2.4.m4.3.3.7.2.2">𝑔</ci><apply id="S5.p2.4.m4.3.3.7.2.3.cmml" xref="S5.p2.4.m4.3.3.7.2.3"><times id="S5.p2.4.m4.3.3.7.2.3.1.cmml" xref="S5.p2.4.m4.3.3.7.2.3.1"></times><ci id="S5.p2.4.m4.3.3.7.2.3.2.cmml" xref="S5.p2.4.m4.3.3.7.2.3.2">𝑖</ci><ci id="S5.p2.4.m4.3.3.7.2.3.3.cmml" xref="S5.p2.4.m4.3.3.7.2.3.3">𝑗</ci></apply></apply><ci id="S5.p2.4.m4.1.1.cmml" xref="S5.p2.4.m4.1.1">𝜂</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.p2.4.m4.3c">X_{i}\leftrightarrow\eta_{i},L_{ij}\leftrightarrow g_{ij}(\eta)</annotation><annotation encoding="application/x-llamapun" id="S5.p2.4.m4.3d">italic_X start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ↔ italic_η start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT , italic_L start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT ↔ italic_g start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT ( italic_η )</annotation></semantics></math>, and <math alttext="S(X)\leftrightarrow-\Psi^{\star}(\eta)" class="ltx_Math" display="inline" id="S5.p2.5.m5.2"><semantics id="S5.p2.5.m5.2a"><mrow id="S5.p2.5.m5.2.3" xref="S5.p2.5.m5.2.3.cmml"><mrow id="S5.p2.5.m5.2.3.2" xref="S5.p2.5.m5.2.3.2.cmml"><mi id="S5.p2.5.m5.2.3.2.2" xref="S5.p2.5.m5.2.3.2.2.cmml">S</mi><mo id="S5.p2.5.m5.2.3.2.1" xref="S5.p2.5.m5.2.3.2.1.cmml">⁢</mo><mrow id="S5.p2.5.m5.2.3.2.3.2" xref="S5.p2.5.m5.2.3.2.cmml"><mo id="S5.p2.5.m5.2.3.2.3.2.1" stretchy="false" xref="S5.p2.5.m5.2.3.2.cmml">(</mo><mi id="S5.p2.5.m5.1.1" xref="S5.p2.5.m5.1.1.cmml">X</mi><mo id="S5.p2.5.m5.2.3.2.3.2.2" stretchy="false" xref="S5.p2.5.m5.2.3.2.cmml">)</mo></mrow></mrow><mo id="S5.p2.5.m5.2.3.1" stretchy="false" xref="S5.p2.5.m5.2.3.1.cmml">↔</mo><mrow id="S5.p2.5.m5.2.3.3" xref="S5.p2.5.m5.2.3.3.cmml"><mo id="S5.p2.5.m5.2.3.3a" xref="S5.p2.5.m5.2.3.3.cmml">−</mo><mrow id="S5.p2.5.m5.2.3.3.2" xref="S5.p2.5.m5.2.3.3.2.cmml"><msup id="S5.p2.5.m5.2.3.3.2.2" xref="S5.p2.5.m5.2.3.3.2.2.cmml"><mi id="S5.p2.5.m5.2.3.3.2.2.2" mathvariant="normal" xref="S5.p2.5.m5.2.3.3.2.2.2.cmml">Ψ</mi><mo id="S5.p2.5.m5.2.3.3.2.2.3" xref="S5.p2.5.m5.2.3.3.2.2.3.cmml">⋆</mo></msup><mo id="S5.p2.5.m5.2.3.3.2.1" xref="S5.p2.5.m5.2.3.3.2.1.cmml">⁢</mo><mrow id="S5.p2.5.m5.2.3.3.2.3.2" xref="S5.p2.5.m5.2.3.3.2.cmml"><mo id="S5.p2.5.m5.2.3.3.2.3.2.1" stretchy="false" xref="S5.p2.5.m5.2.3.3.2.cmml">(</mo><mi id="S5.p2.5.m5.2.2" xref="S5.p2.5.m5.2.2.cmml">η</mi><mo id="S5.p2.5.m5.2.3.3.2.3.2.2" stretchy="false" xref="S5.p2.5.m5.2.3.3.2.cmml">)</mo></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.p2.5.m5.2b"><apply id="S5.p2.5.m5.2.3.cmml" xref="S5.p2.5.m5.2.3"><ci id="S5.p2.5.m5.2.3.1.cmml" xref="S5.p2.5.m5.2.3.1">↔</ci><apply id="S5.p2.5.m5.2.3.2.cmml" xref="S5.p2.5.m5.2.3.2"><times id="S5.p2.5.m5.2.3.2.1.cmml" xref="S5.p2.5.m5.2.3.2.1"></times><ci id="S5.p2.5.m5.2.3.2.2.cmml" xref="S5.p2.5.m5.2.3.2.2">𝑆</ci><ci id="S5.p2.5.m5.1.1.cmml" xref="S5.p2.5.m5.1.1">𝑋</ci></apply><apply id="S5.p2.5.m5.2.3.3.cmml" xref="S5.p2.5.m5.2.3.3"><minus id="S5.p2.5.m5.2.3.3.1.cmml" xref="S5.p2.5.m5.2.3.3"></minus><apply id="S5.p2.5.m5.2.3.3.2.cmml" xref="S5.p2.5.m5.2.3.3.2"><times id="S5.p2.5.m5.2.3.3.2.1.cmml" xref="S5.p2.5.m5.2.3.3.2.1"></times><apply id="S5.p2.5.m5.2.3.3.2.2.cmml" xref="S5.p2.5.m5.2.3.3.2.2"><csymbol cd="ambiguous" id="S5.p2.5.m5.2.3.3.2.2.1.cmml" xref="S5.p2.5.m5.2.3.3.2.2">superscript</csymbol><ci id="S5.p2.5.m5.2.3.3.2.2.2.cmml" xref="S5.p2.5.m5.2.3.3.2.2.2">Ψ</ci><ci id="S5.p2.5.m5.2.3.3.2.2.3.cmml" xref="S5.p2.5.m5.2.3.3.2.2.3">⋆</ci></apply><ci id="S5.p2.5.m5.2.2.cmml" xref="S5.p2.5.m5.2.2">𝜂</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.p2.5.m5.2c">S(X)\leftrightarrow-\Psi^{\star}(\eta)</annotation><annotation encoding="application/x-llamapun" id="S5.p2.5.m5.2d">italic_S ( italic_X ) ↔ - roman_Ψ start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT ( italic_η )</annotation></semantics></math>. In this correspondence, famous Onsager’s reciprocal relations <math alttext="L_{ij}=L_{jk}" class="ltx_Math" display="inline" id="S5.p2.6.m6.1"><semantics id="S5.p2.6.m6.1a"><mrow id="S5.p2.6.m6.1.1" xref="S5.p2.6.m6.1.1.cmml"><msub id="S5.p2.6.m6.1.1.2" xref="S5.p2.6.m6.1.1.2.cmml"><mi id="S5.p2.6.m6.1.1.2.2" xref="S5.p2.6.m6.1.1.2.2.cmml">L</mi><mrow id="S5.p2.6.m6.1.1.2.3" xref="S5.p2.6.m6.1.1.2.3.cmml"><mi id="S5.p2.6.m6.1.1.2.3.2" xref="S5.p2.6.m6.1.1.2.3.2.cmml">i</mi><mo id="S5.p2.6.m6.1.1.2.3.1" xref="S5.p2.6.m6.1.1.2.3.1.cmml">⁢</mo><mi id="S5.p2.6.m6.1.1.2.3.3" xref="S5.p2.6.m6.1.1.2.3.3.cmml">j</mi></mrow></msub><mo id="S5.p2.6.m6.1.1.1" xref="S5.p2.6.m6.1.1.1.cmml">=</mo><msub id="S5.p2.6.m6.1.1.3" xref="S5.p2.6.m6.1.1.3.cmml"><mi id="S5.p2.6.m6.1.1.3.2" xref="S5.p2.6.m6.1.1.3.2.cmml">L</mi><mrow id="S5.p2.6.m6.1.1.3.3" xref="S5.p2.6.m6.1.1.3.3.cmml"><mi id="S5.p2.6.m6.1.1.3.3.2" xref="S5.p2.6.m6.1.1.3.3.2.cmml">j</mi><mo id="S5.p2.6.m6.1.1.3.3.1" xref="S5.p2.6.m6.1.1.3.3.1.cmml">⁢</mo><mi id="S5.p2.6.m6.1.1.3.3.3" xref="S5.p2.6.m6.1.1.3.3.3.cmml">k</mi></mrow></msub></mrow><annotation-xml encoding="MathML-Content" id="S5.p2.6.m6.1b"><apply id="S5.p2.6.m6.1.1.cmml" xref="S5.p2.6.m6.1.1"><eq id="S5.p2.6.m6.1.1.1.cmml" xref="S5.p2.6.m6.1.1.1"></eq><apply id="S5.p2.6.m6.1.1.2.cmml" xref="S5.p2.6.m6.1.1.2"><csymbol cd="ambiguous" id="S5.p2.6.m6.1.1.2.1.cmml" xref="S5.p2.6.m6.1.1.2">subscript</csymbol><ci id="S5.p2.6.m6.1.1.2.2.cmml" xref="S5.p2.6.m6.1.1.2.2">𝐿</ci><apply id="S5.p2.6.m6.1.1.2.3.cmml" xref="S5.p2.6.m6.1.1.2.3"><times id="S5.p2.6.m6.1.1.2.3.1.cmml" xref="S5.p2.6.m6.1.1.2.3.1"></times><ci id="S5.p2.6.m6.1.1.2.3.2.cmml" xref="S5.p2.6.m6.1.1.2.3.2">𝑖</ci><ci id="S5.p2.6.m6.1.1.2.3.3.cmml" xref="S5.p2.6.m6.1.1.2.3.3">𝑗</ci></apply></apply><apply id="S5.p2.6.m6.1.1.3.cmml" xref="S5.p2.6.m6.1.1.3"><csymbol cd="ambiguous" id="S5.p2.6.m6.1.1.3.1.cmml" xref="S5.p2.6.m6.1.1.3">subscript</csymbol><ci id="S5.p2.6.m6.1.1.3.2.cmml" xref="S5.p2.6.m6.1.1.3.2">𝐿</ci><apply id="S5.p2.6.m6.1.1.3.3.cmml" xref="S5.p2.6.m6.1.1.3.3"><times id="S5.p2.6.m6.1.1.3.3.1.cmml" xref="S5.p2.6.m6.1.1.3.3.1"></times><ci id="S5.p2.6.m6.1.1.3.3.2.cmml" xref="S5.p2.6.m6.1.1.3.3.2">𝑗</ci><ci id="S5.p2.6.m6.1.1.3.3.3.cmml" xref="S5.p2.6.m6.1.1.3.3.3">𝑘</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.p2.6.m6.1c">L_{ij}=L_{jk}</annotation><annotation encoding="application/x-llamapun" id="S5.p2.6.m6.1d">italic_L start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT = italic_L start_POSTSUBSCRIPT italic_j italic_k end_POSTSUBSCRIPT</annotation></semantics></math> can be understood in IG as the symmetry of the metric <math alttext="g_{ij}(\eta)=\partial\eta_{i}/\partial\theta^{j}=\partial\eta_{j}/\partial% \theta^{i}=g_{ji}(\eta)" class="ltx_Math" display="inline" id="S5.p2.7.m7.2"><semantics id="S5.p2.7.m7.2a"><mrow id="S5.p2.7.m7.2.3" xref="S5.p2.7.m7.2.3.cmml"><mrow id="S5.p2.7.m7.2.3.2" 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italic_i end_POSTSUBSCRIPT ( italic_η )</annotation></semantics></math>, which is due to integrability. Recently, Katagiri <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2406.11224v2#bib.bib26" title="">26</a>]</cite> extended the constitutive relations of Onsager’s non-equilibrium thermodynamics by considering a thermodynamic force as a gauge fixing. It is noticed that some relations in Ref. <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2406.11224v2#bib.bib26" title="">26</a>]</cite> can be regarded as the RF deformed gradient-flow equations in IG.</p> </div> <div class="ltx_para" id="S5.p3"> <p class="ltx_p" id="S5.p3.1">In regard to general relativity and cosmology, Ref. <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2406.11224v2#bib.bib11" title="">11</a>]</cite> described the dynamical evolutions of flat metrics for Kerr and Reissner-Nordström black holes, and Ref. <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2406.11224v2#bib.bib13" title="">13</a>]</cite> shows that the significance of Weyl integrable geometry in IG. Moreover Gibbons et al. <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2406.11224v2#bib.bib24" title="">24</a>]</cite> showed the triality among the Zermelo navigation problem, the geodesic flow on a RF function, and optical metric of one dimension higher stationary spacetime. The Zermelo/Randers/spacetime triality allows us to translate one of the three viewpoints to any of the other two viewpoints, resulting in significant simplifications or complications. Since the stationary metric (<a class="ltx_ref" href="https://arxiv.org/html/2406.11224v2#S4.E55" title="In 4 The motions of a light-like particle in a pseudo Riemann space ‣ A Hamiltonian approach to the gradient-flow equations in information geometry"><span class="ltx_text ltx_ref_tag">55</span></a>) is equivalent to the Zermelo form Eq. (31) in Ref. <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2406.11224v2#bib.bib24" title="">24</a>]</cite> as shown in Appendix <a class="ltx_ref" href="https://arxiv.org/html/2406.11224v2#A3" title="Appendix C Zermelo form ‣ A Hamiltonian approach to the gradient-flow equations in information geometry"><span class="ltx_text ltx_ref_tag">C</span></a>, it will be interesting to study the gradient-flows in IG from these viewpoints.</p> </div> <div class="ltx_para" id="S5.p4"> <p class="ltx_p" id="S5.p4.1">Furthermore, in regard to the approaches to the gradient-flows in IG based on analytical mechanics, Pistone et al. <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2406.11224v2#bib.bib7" title="">7</a>, <a class="ltx_ref" href="https://arxiv.org/html/2406.11224v2#bib.bib8" title="">8</a>]</cite> have been developed their Lagrangian and Hamiltonian formalism. It is intriguing to study the relation between their method and this work. Finally we believe that it is worthwhile to further explore the information geometric studies on the gradient-flow equations from some different perspectives in the fields of physics.</p> </div> <div class="ltx_para" id="S5.p5"> <span class="ltx_ERROR undefined" id="S5.p5.1">\bmhead</span> <p class="ltx_p" id="S5.p5.2">Acknowledgements</p> </div> <div class="ltx_para" id="S5.p6"> <p class="ltx_p" id="S5.p6.1">The first named author (T.W.) was supported by Japan Society for the Promotion of Science (JSPS) Grants-in-Aid for Scientific Research (KAKENHI) Grant Number JP22K03431. We thank the anonymous referees for their valuable comments.</p> </div> <section class="ltx_subsection" id="S5.SSx1"> <h3 class="ltx_title ltx_title_subsection">Declarations</h3> </section> <section class="ltx_subsection" id="S5.SSx2"> <h3 class="ltx_title ltx_title_subsection">Conflict of interest</h3> <div class="ltx_para" id="S5.SSx2.p1"> <p class="ltx_p" id="S5.SSx2.p1.1">The authors have no relevant financial or non-financial interests to disclose.</p> </div> </section> <section class="ltx_subsection" id="S5.SSx3"> <h3 class="ltx_title ltx_title_subsection">Author contribution</h3> <div class="ltx_para" id="S5.SSx3.p1"> <p class="ltx_p" id="S5.SSx3.p1.1">Tatsuaki Wada: Conceptualization, Methodology, Validation, Writing–original draft, review and editing. Antonio Maria Scarfone: Conceptualization, Validation, Writing–review and editing.</p> </div> </section> <section class="ltx_subsection" id="S5.SSx4"> <h3 class="ltx_title ltx_title_subsection">Data availability statement</h3> <div class="ltx_para" id="S5.SSx4.p1"> <p class="ltx_p" id="S5.SSx4.p1.1">No associated data.</p> </div> </section> </section> <section class="ltx_appendix" id="A1"> <h2 class="ltx_title ltx_title_appendix"> <span class="ltx_tag ltx_tag_appendix">Appendix A </span>The proof of (<a class="ltx_ref" href="https://arxiv.org/html/2406.11224v2#S3.E38" title="In 3 Complete integrability and geodesic Hamiltonian ‣ A Hamiltonian approach to the gradient-flow equations in information geometry"><span class="ltx_text ltx_ref_tag">38</span></a>)</h2> <div class="ltx_para" id="A1.p1"> <p class="ltx_p" id="A1.p1.1">In the extended phase space of <math alttext="(x,p,t)\in T^{\star}\mathcal{M}\times\mathbb{R}" class="ltx_Math" display="inline" id="A1.p1.1.m1.3"><semantics id="A1.p1.1.m1.3a"><mrow id="A1.p1.1.m1.3.4" xref="A1.p1.1.m1.3.4.cmml"><mrow id="A1.p1.1.m1.3.4.2.2" xref="A1.p1.1.m1.3.4.2.1.cmml"><mo id="A1.p1.1.m1.3.4.2.2.1" stretchy="false" xref="A1.p1.1.m1.3.4.2.1.cmml">(</mo><mi id="A1.p1.1.m1.1.1" xref="A1.p1.1.m1.1.1.cmml">x</mi><mo id="A1.p1.1.m1.3.4.2.2.2" xref="A1.p1.1.m1.3.4.2.1.cmml">,</mo><mi id="A1.p1.1.m1.2.2" xref="A1.p1.1.m1.2.2.cmml">p</mi><mo id="A1.p1.1.m1.3.4.2.2.3" xref="A1.p1.1.m1.3.4.2.1.cmml">,</mo><mi id="A1.p1.1.m1.3.3" xref="A1.p1.1.m1.3.3.cmml">t</mi><mo id="A1.p1.1.m1.3.4.2.2.4" stretchy="false" xref="A1.p1.1.m1.3.4.2.1.cmml">)</mo></mrow><mo id="A1.p1.1.m1.3.4.1" xref="A1.p1.1.m1.3.4.1.cmml">∈</mo><mrow id="A1.p1.1.m1.3.4.3" xref="A1.p1.1.m1.3.4.3.cmml"><mrow id="A1.p1.1.m1.3.4.3.2" xref="A1.p1.1.m1.3.4.3.2.cmml"><msup id="A1.p1.1.m1.3.4.3.2.2" xref="A1.p1.1.m1.3.4.3.2.2.cmml"><mi id="A1.p1.1.m1.3.4.3.2.2.2" xref="A1.p1.1.m1.3.4.3.2.2.2.cmml">T</mi><mo id="A1.p1.1.m1.3.4.3.2.2.3" xref="A1.p1.1.m1.3.4.3.2.2.3.cmml">⋆</mo></msup><mo id="A1.p1.1.m1.3.4.3.2.1" xref="A1.p1.1.m1.3.4.3.2.1.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="A1.p1.1.m1.3.4.3.2.3" xref="A1.p1.1.m1.3.4.3.2.3.cmml">ℳ</mi></mrow><mo id="A1.p1.1.m1.3.4.3.1" lspace="0.222em" rspace="0.222em" xref="A1.p1.1.m1.3.4.3.1.cmml">×</mo><mi id="A1.p1.1.m1.3.4.3.3" xref="A1.p1.1.m1.3.4.3.3.cmml">ℝ</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="A1.p1.1.m1.3b"><apply id="A1.p1.1.m1.3.4.cmml" xref="A1.p1.1.m1.3.4"><in id="A1.p1.1.m1.3.4.1.cmml" xref="A1.p1.1.m1.3.4.1"></in><vector id="A1.p1.1.m1.3.4.2.1.cmml" xref="A1.p1.1.m1.3.4.2.2"><ci id="A1.p1.1.m1.1.1.cmml" xref="A1.p1.1.m1.1.1">𝑥</ci><ci id="A1.p1.1.m1.2.2.cmml" xref="A1.p1.1.m1.2.2">𝑝</ci><ci id="A1.p1.1.m1.3.3.cmml" xref="A1.p1.1.m1.3.3">𝑡</ci></vector><apply id="A1.p1.1.m1.3.4.3.cmml" xref="A1.p1.1.m1.3.4.3"><times id="A1.p1.1.m1.3.4.3.1.cmml" xref="A1.p1.1.m1.3.4.3.1"></times><apply id="A1.p1.1.m1.3.4.3.2.cmml" xref="A1.p1.1.m1.3.4.3.2"><times id="A1.p1.1.m1.3.4.3.2.1.cmml" xref="A1.p1.1.m1.3.4.3.2.1"></times><apply id="A1.p1.1.m1.3.4.3.2.2.cmml" xref="A1.p1.1.m1.3.4.3.2.2"><csymbol cd="ambiguous" id="A1.p1.1.m1.3.4.3.2.2.1.cmml" xref="A1.p1.1.m1.3.4.3.2.2">superscript</csymbol><ci id="A1.p1.1.m1.3.4.3.2.2.2.cmml" xref="A1.p1.1.m1.3.4.3.2.2.2">𝑇</ci><ci id="A1.p1.1.m1.3.4.3.2.2.3.cmml" xref="A1.p1.1.m1.3.4.3.2.2.3">⋆</ci></apply><ci id="A1.p1.1.m1.3.4.3.2.3.cmml" xref="A1.p1.1.m1.3.4.3.2.3">ℳ</ci></apply><ci id="A1.p1.1.m1.3.4.3.3.cmml" xref="A1.p1.1.m1.3.4.3.3">ℝ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.p1.1.m1.3c">(x,p,t)\in T^{\star}\mathcal{M}\times\mathbb{R}</annotation><annotation encoding="application/x-llamapun" id="A1.p1.1.m1.3d">( italic_x , italic_p , italic_t ) ∈ italic_T start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT caligraphic_M × blackboard_R</annotation></semantics></math>, the canonical equation of motion can be expressed as</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="S3.EGx94"> <tbody id="A1.E105"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_left_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\iota(X)\,d\omega_{\rm PC}=0," class="ltx_Math" display="inline" id="A1.E105.m1.2"><semantics id="A1.E105.m1.2a"><mrow id="A1.E105.m1.2.2.1" xref="A1.E105.m1.2.2.1.1.cmml"><mrow id="A1.E105.m1.2.2.1.1" xref="A1.E105.m1.2.2.1.1.cmml"><mrow id="A1.E105.m1.2.2.1.1.2" xref="A1.E105.m1.2.2.1.1.2.cmml"><mi id="A1.E105.m1.2.2.1.1.2.2" xref="A1.E105.m1.2.2.1.1.2.2.cmml">ι</mi><mo id="A1.E105.m1.2.2.1.1.2.1" xref="A1.E105.m1.2.2.1.1.2.1.cmml">⁢</mo><mrow id="A1.E105.m1.2.2.1.1.2.3.2" xref="A1.E105.m1.2.2.1.1.2.cmml"><mo id="A1.E105.m1.2.2.1.1.2.3.2.1" stretchy="false" xref="A1.E105.m1.2.2.1.1.2.cmml">(</mo><mi id="A1.E105.m1.1.1" xref="A1.E105.m1.1.1.cmml">X</mi><mo id="A1.E105.m1.2.2.1.1.2.3.2.2" stretchy="false" xref="A1.E105.m1.2.2.1.1.2.cmml">)</mo></mrow><mo id="A1.E105.m1.2.2.1.1.2.1a" lspace="0.170em" xref="A1.E105.m1.2.2.1.1.2.1.cmml">⁢</mo><mi id="A1.E105.m1.2.2.1.1.2.4" xref="A1.E105.m1.2.2.1.1.2.4.cmml">d</mi><mo id="A1.E105.m1.2.2.1.1.2.1b" xref="A1.E105.m1.2.2.1.1.2.1.cmml">⁢</mo><msub id="A1.E105.m1.2.2.1.1.2.5" xref="A1.E105.m1.2.2.1.1.2.5.cmml"><mi id="A1.E105.m1.2.2.1.1.2.5.2" xref="A1.E105.m1.2.2.1.1.2.5.2.cmml">ω</mi><mi id="A1.E105.m1.2.2.1.1.2.5.3" xref="A1.E105.m1.2.2.1.1.2.5.3.cmml">PC</mi></msub></mrow><mo id="A1.E105.m1.2.2.1.1.1" xref="A1.E105.m1.2.2.1.1.1.cmml">=</mo><mn id="A1.E105.m1.2.2.1.1.3" xref="A1.E105.m1.2.2.1.1.3.cmml">0</mn></mrow><mo id="A1.E105.m1.2.2.1.2" xref="A1.E105.m1.2.2.1.1.cmml">,</mo></mrow><annotation-xml encoding="MathML-Content" id="A1.E105.m1.2b"><apply id="A1.E105.m1.2.2.1.1.cmml" xref="A1.E105.m1.2.2.1"><eq id="A1.E105.m1.2.2.1.1.1.cmml" xref="A1.E105.m1.2.2.1.1.1"></eq><apply id="A1.E105.m1.2.2.1.1.2.cmml" xref="A1.E105.m1.2.2.1.1.2"><times id="A1.E105.m1.2.2.1.1.2.1.cmml" xref="A1.E105.m1.2.2.1.1.2.1"></times><ci id="A1.E105.m1.2.2.1.1.2.2.cmml" xref="A1.E105.m1.2.2.1.1.2.2">𝜄</ci><ci id="A1.E105.m1.1.1.cmml" xref="A1.E105.m1.1.1">𝑋</ci><ci id="A1.E105.m1.2.2.1.1.2.4.cmml" xref="A1.E105.m1.2.2.1.1.2.4">𝑑</ci><apply id="A1.E105.m1.2.2.1.1.2.5.cmml" xref="A1.E105.m1.2.2.1.1.2.5"><csymbol cd="ambiguous" id="A1.E105.m1.2.2.1.1.2.5.1.cmml" xref="A1.E105.m1.2.2.1.1.2.5">subscript</csymbol><ci id="A1.E105.m1.2.2.1.1.2.5.2.cmml" xref="A1.E105.m1.2.2.1.1.2.5.2">𝜔</ci><ci id="A1.E105.m1.2.2.1.1.2.5.3.cmml" xref="A1.E105.m1.2.2.1.1.2.5.3">PC</ci></apply></apply><cn id="A1.E105.m1.2.2.1.1.3.cmml" type="integer" xref="A1.E105.m1.2.2.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.E105.m1.2c">\displaystyle\iota(X)\,d\omega_{\rm PC}=0,</annotation><annotation encoding="application/x-llamapun" id="A1.E105.m1.2d">italic_ι ( italic_X ) italic_d italic_ω start_POSTSUBSCRIPT roman_PC end_POSTSUBSCRIPT = 0 ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_left_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(105)</span></td> </tr></tbody> </table> <p class="ltx_p" id="A1.p1.3">where <math alttext="X" class="ltx_Math" display="inline" id="A1.p1.2.m1.1"><semantics id="A1.p1.2.m1.1a"><mi id="A1.p1.2.m1.1.1" xref="A1.p1.2.m1.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="A1.p1.2.m1.1b"><ci id="A1.p1.2.m1.1.1.cmml" xref="A1.p1.2.m1.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="A1.p1.2.m1.1c">X</annotation><annotation encoding="application/x-llamapun" id="A1.p1.2.m1.1d">italic_X</annotation></semantics></math> is a vector field in the extended phase space and <math alttext="\iota(X)" class="ltx_Math" display="inline" id="A1.p1.3.m2.1"><semantics id="A1.p1.3.m2.1a"><mrow id="A1.p1.3.m2.1.2" xref="A1.p1.3.m2.1.2.cmml"><mi id="A1.p1.3.m2.1.2.2" xref="A1.p1.3.m2.1.2.2.cmml">ι</mi><mo id="A1.p1.3.m2.1.2.1" xref="A1.p1.3.m2.1.2.1.cmml">⁢</mo><mrow id="A1.p1.3.m2.1.2.3.2" xref="A1.p1.3.m2.1.2.cmml"><mo id="A1.p1.3.m2.1.2.3.2.1" stretchy="false" xref="A1.p1.3.m2.1.2.cmml">(</mo><mi id="A1.p1.3.m2.1.1" xref="A1.p1.3.m2.1.1.cmml">X</mi><mo id="A1.p1.3.m2.1.2.3.2.2" stretchy="false" xref="A1.p1.3.m2.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A1.p1.3.m2.1b"><apply id="A1.p1.3.m2.1.2.cmml" xref="A1.p1.3.m2.1.2"><times id="A1.p1.3.m2.1.2.1.cmml" xref="A1.p1.3.m2.1.2.1"></times><ci id="A1.p1.3.m2.1.2.2.cmml" xref="A1.p1.3.m2.1.2.2">𝜄</ci><ci id="A1.p1.3.m2.1.1.cmml" xref="A1.p1.3.m2.1.1">𝑋</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.p1.3.m2.1c">\iota(X)</annotation><annotation encoding="application/x-llamapun" id="A1.p1.3.m2.1d">italic_ι ( italic_X )</annotation></semantics></math> denote the interior product. More concretely we have</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="S3.EGx95"> <tbody id="A1.Ex1"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_left_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle-\iota\!\left(\frac{\partial}{\partial x^{i}}\right)\!d\omega_{% \rm PC}" class="ltx_Math" display="inline" id="A1.Ex1.m1.1"><semantics id="A1.Ex1.m1.1a"><mrow id="A1.Ex1.m1.1.2" xref="A1.Ex1.m1.1.2.cmml"><mo id="A1.Ex1.m1.1.2a" xref="A1.Ex1.m1.1.2.cmml">−</mo><mrow id="A1.Ex1.m1.1.2.2" xref="A1.Ex1.m1.1.2.2.cmml"><mpadded width="0.184em"><mi id="A1.Ex1.m1.1.2.2.2" xref="A1.Ex1.m1.1.2.2.2.cmml">ι</mi></mpadded><mo id="A1.Ex1.m1.1.2.2.1" xref="A1.Ex1.m1.1.2.2.1.cmml">⁢</mo><mrow id="A1.Ex1.m1.1.2.2.3.2" xref="A1.Ex1.m1.1.1.cmml"><mo id="A1.Ex1.m1.1.2.2.3.2.1" xref="A1.Ex1.m1.1.1.cmml">(</mo><mstyle displaystyle="true" id="A1.Ex1.m1.1.1" xref="A1.Ex1.m1.1.1.cmml"><mfrac 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xref="A1.Ex1.m1.1.1.3.2.3">𝑖</ci></apply></apply></apply><ci id="A1.Ex1.m1.1.2.2.4.cmml" xref="A1.Ex1.m1.1.2.2.4">𝑑</ci><apply id="A1.Ex1.m1.1.2.2.5.cmml" xref="A1.Ex1.m1.1.2.2.5"><csymbol cd="ambiguous" id="A1.Ex1.m1.1.2.2.5.1.cmml" xref="A1.Ex1.m1.1.2.2.5">subscript</csymbol><ci id="A1.Ex1.m1.1.2.2.5.2.cmml" xref="A1.Ex1.m1.1.2.2.5.2">𝜔</ci><ci id="A1.Ex1.m1.1.2.2.5.3.cmml" xref="A1.Ex1.m1.1.2.2.5.3">PC</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.Ex1.m1.1c">\displaystyle-\iota\!\left(\frac{\partial}{\partial x^{i}}\right)\!d\omega_{% \rm PC}</annotation><annotation encoding="application/x-llamapun" id="A1.Ex1.m1.1d">- italic_ι ( divide start_ARG ∂ end_ARG start_ARG ∂ italic_x start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT end_ARG ) italic_d italic_ω start_POSTSUBSCRIPT roman_PC end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle=dp_{i}+\frac{\partial 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id="A1.Ex1.m2.1.1.1.1.5.cmml" xref="A1.Ex1.m2.1.1.1.1.5"></eq><share href="https://arxiv.org/html/2406.11224v2#A1.Ex1.m2.1.1.1.1.4.cmml" id="A1.Ex1.m2.1.1.1.1d.cmml" xref="A1.Ex1.m2.1.1.1"></share><cn id="A1.Ex1.m2.1.1.1.1.6.cmml" type="integer" xref="A1.Ex1.m2.1.1.1.1.6">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.Ex1.m2.1c">\displaystyle=dp_{i}+\frac{\partial H}{\partial x^{i}}dt=0,</annotation><annotation encoding="application/x-llamapun" id="A1.Ex1.m2.1d">= italic_d italic_p start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT + divide start_ARG ∂ italic_H end_ARG start_ARG ∂ italic_x start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT end_ARG italic_d italic_t = 0 ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_left_padright"></td> </tr></tbody> <tbody id="A1.E106"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_left_padleft"></td> <td class="ltx_td ltx_align_right 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p_{i}}\right)\!d\omega_{\rm PC}</annotation><annotation encoding="application/x-llamapun" id="A1.E106.m1.1d">italic_ι ( divide start_ARG ∂ end_ARG start_ARG ∂ italic_p start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_ARG ) italic_d italic_ω start_POSTSUBSCRIPT roman_PC end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle=dx^{i}-\frac{\partial H}{\partial p_{i}}dt=0," class="ltx_Math" display="inline" id="A1.E106.m2.1"><semantics id="A1.E106.m2.1a"><mrow id="A1.E106.m2.1.1.1" xref="A1.E106.m2.1.1.1.1.cmml"><mrow id="A1.E106.m2.1.1.1.1" xref="A1.E106.m2.1.1.1.1.cmml"><mi id="A1.E106.m2.1.1.1.1.2" xref="A1.E106.m2.1.1.1.1.2.cmml"></mi><mo id="A1.E106.m2.1.1.1.1.3" xref="A1.E106.m2.1.1.1.1.3.cmml">=</mo><mrow id="A1.E106.m2.1.1.1.1.4" xref="A1.E106.m2.1.1.1.1.4.cmml"><mrow id="A1.E106.m2.1.1.1.1.4.2" xref="A1.E106.m2.1.1.1.1.4.2.cmml"><mi id="A1.E106.m2.1.1.1.1.4.2.2" 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cd="ambiguous" id="A1.E106.m2.1.1.1.1.4.3.2.3.2.1.cmml" xref="A1.E106.m2.1.1.1.1.4.3.2.3.2">subscript</csymbol><ci id="A1.E106.m2.1.1.1.1.4.3.2.3.2.2.cmml" xref="A1.E106.m2.1.1.1.1.4.3.2.3.2.2">𝑝</ci><ci id="A1.E106.m2.1.1.1.1.4.3.2.3.2.3.cmml" xref="A1.E106.m2.1.1.1.1.4.3.2.3.2.3">𝑖</ci></apply></apply></apply><ci id="A1.E106.m2.1.1.1.1.4.3.3.cmml" xref="A1.E106.m2.1.1.1.1.4.3.3">𝑑</ci><ci id="A1.E106.m2.1.1.1.1.4.3.4.cmml" xref="A1.E106.m2.1.1.1.1.4.3.4">𝑡</ci></apply></apply></apply><apply id="A1.E106.m2.1.1.1.1c.cmml" xref="A1.E106.m2.1.1.1"><eq id="A1.E106.m2.1.1.1.1.5.cmml" xref="A1.E106.m2.1.1.1.1.5"></eq><share href="https://arxiv.org/html/2406.11224v2#A1.E106.m2.1.1.1.1.4.cmml" id="A1.E106.m2.1.1.1.1d.cmml" xref="A1.E106.m2.1.1.1"></share><cn id="A1.E106.m2.1.1.1.1.6.cmml" type="integer" xref="A1.E106.m2.1.1.1.1.6">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.E106.m2.1c">\displaystyle=dx^{i}-\frac{\partial H}{\partial p_{i}}dt=0,</annotation><annotation encoding="application/x-llamapun" id="A1.E106.m2.1d">= italic_d italic_x start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT - divide start_ARG ∂ italic_H end_ARG start_ARG ∂ italic_p start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_ARG italic_d italic_t = 0 ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_left_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(106)</span></td> </tr></tbody> </table> <p class="ltx_p" id="A1.p1.4">as Hamilton’s equations of motion.</p> </div> <div class="ltx_para" id="A1.p2"> <p class="ltx_p" id="A1.p2.3">Now we consider the complete integrability of the Pfaffian equation (<a class="ltx_ref" href="https://arxiv.org/html/2406.11224v2#S3.E37" title="In 3 Complete integrability and geodesic Hamiltonian ‣ A Hamiltonian approach to the gradient-flow equations in information geometry"><span class="ltx_text ltx_ref_tag">37</span></a>). Recall that a differential form <math alttext="\alpha" class="ltx_Math" display="inline" id="A1.p2.1.m1.1"><semantics id="A1.p2.1.m1.1a"><mi id="A1.p2.1.m1.1.1" xref="A1.p2.1.m1.1.1.cmml">α</mi><annotation-xml encoding="MathML-Content" id="A1.p2.1.m1.1b"><ci id="A1.p2.1.m1.1.1.cmml" xref="A1.p2.1.m1.1.1">𝛼</ci></annotation-xml><annotation encoding="application/x-tex" id="A1.p2.1.m1.1c">\alpha</annotation><annotation encoding="application/x-llamapun" id="A1.p2.1.m1.1d">italic_α</annotation></semantics></math> is <span class="ltx_text ltx_font_italic" id="A1.p2.3.1">closed</span> if its exterior derivative is zero (<math alttext="d\alpha=0" class="ltx_Math" display="inline" id="A1.p2.2.m2.1"><semantics id="A1.p2.2.m2.1a"><mrow id="A1.p2.2.m2.1.1" xref="A1.p2.2.m2.1.1.cmml"><mrow id="A1.p2.2.m2.1.1.2" xref="A1.p2.2.m2.1.1.2.cmml"><mi id="A1.p2.2.m2.1.1.2.2" xref="A1.p2.2.m2.1.1.2.2.cmml">d</mi><mo id="A1.p2.2.m2.1.1.2.1" xref="A1.p2.2.m2.1.1.2.1.cmml">⁢</mo><mi id="A1.p2.2.m2.1.1.2.3" xref="A1.p2.2.m2.1.1.2.3.cmml">α</mi></mrow><mo id="A1.p2.2.m2.1.1.1" xref="A1.p2.2.m2.1.1.1.cmml">=</mo><mn id="A1.p2.2.m2.1.1.3" xref="A1.p2.2.m2.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="A1.p2.2.m2.1b"><apply id="A1.p2.2.m2.1.1.cmml" xref="A1.p2.2.m2.1.1"><eq id="A1.p2.2.m2.1.1.1.cmml" xref="A1.p2.2.m2.1.1.1"></eq><apply id="A1.p2.2.m2.1.1.2.cmml" xref="A1.p2.2.m2.1.1.2"><times id="A1.p2.2.m2.1.1.2.1.cmml" xref="A1.p2.2.m2.1.1.2.1"></times><ci id="A1.p2.2.m2.1.1.2.2.cmml" xref="A1.p2.2.m2.1.1.2.2">𝑑</ci><ci id="A1.p2.2.m2.1.1.2.3.cmml" xref="A1.p2.2.m2.1.1.2.3">𝛼</ci></apply><cn id="A1.p2.2.m2.1.1.3.cmml" type="integer" xref="A1.p2.2.m2.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.p2.2.m2.1c">d\alpha=0</annotation><annotation encoding="application/x-llamapun" id="A1.p2.2.m2.1d">italic_d italic_α = 0</annotation></semantics></math>), and is <span class="ltx_text ltx_font_italic" id="A1.p2.3.2">exact</span> if it is the exterior deriative of another differential form (<math alttext="\alpha=d\beta" class="ltx_Math" display="inline" id="A1.p2.3.m3.1"><semantics id="A1.p2.3.m3.1a"><mrow id="A1.p2.3.m3.1.1" xref="A1.p2.3.m3.1.1.cmml"><mi id="A1.p2.3.m3.1.1.2" xref="A1.p2.3.m3.1.1.2.cmml">α</mi><mo id="A1.p2.3.m3.1.1.1" xref="A1.p2.3.m3.1.1.1.cmml">=</mo><mrow id="A1.p2.3.m3.1.1.3" xref="A1.p2.3.m3.1.1.3.cmml"><mi id="A1.p2.3.m3.1.1.3.2" xref="A1.p2.3.m3.1.1.3.2.cmml">d</mi><mo id="A1.p2.3.m3.1.1.3.1" xref="A1.p2.3.m3.1.1.3.1.cmml">⁢</mo><mi id="A1.p2.3.m3.1.1.3.3" xref="A1.p2.3.m3.1.1.3.3.cmml">β</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="A1.p2.3.m3.1b"><apply id="A1.p2.3.m3.1.1.cmml" xref="A1.p2.3.m3.1.1"><eq id="A1.p2.3.m3.1.1.1.cmml" xref="A1.p2.3.m3.1.1.1"></eq><ci id="A1.p2.3.m3.1.1.2.cmml" xref="A1.p2.3.m3.1.1.2">𝛼</ci><apply id="A1.p2.3.m3.1.1.3.cmml" xref="A1.p2.3.m3.1.1.3"><times id="A1.p2.3.m3.1.1.3.1.cmml" xref="A1.p2.3.m3.1.1.3.1"></times><ci id="A1.p2.3.m3.1.1.3.2.cmml" xref="A1.p2.3.m3.1.1.3.2">𝑑</ci><ci id="A1.p2.3.m3.1.1.3.3.cmml" xref="A1.p2.3.m3.1.1.3.3">𝛽</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.p2.3.m3.1c">\alpha=d\beta</annotation><annotation encoding="application/x-llamapun" id="A1.p2.3.m3.1d">italic_α = italic_d italic_β</annotation></semantics></math>). From (<a class="ltx_ref" href="https://arxiv.org/html/2406.11224v2#S3.E36" title="In 3 Complete integrability and geodesic Hamiltonian ‣ A Hamiltonian approach to the gradient-flow equations in information geometry"><span class="ltx_text ltx_ref_tag">36</span></a>) we have</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="S3.EGx96"> <tbody id="A1.E107"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_left_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle d\omega_{\rm PC}=dp_{j}\wedge dx^{j}-dH(x,p,t)\wedge dt." class="ltx_Math" display="inline" id="A1.E107.m1.4"><semantics id="A1.E107.m1.4a"><mrow id="A1.E107.m1.4.4.1" xref="A1.E107.m1.4.4.1.1.cmml"><mrow id="A1.E107.m1.4.4.1.1" xref="A1.E107.m1.4.4.1.1.cmml"><mrow id="A1.E107.m1.4.4.1.1.2" xref="A1.E107.m1.4.4.1.1.2.cmml"><mi id="A1.E107.m1.4.4.1.1.2.2" xref="A1.E107.m1.4.4.1.1.2.2.cmml">d</mi><mo id="A1.E107.m1.4.4.1.1.2.1" xref="A1.E107.m1.4.4.1.1.2.1.cmml">⁢</mo><msub id="A1.E107.m1.4.4.1.1.2.3" xref="A1.E107.m1.4.4.1.1.2.3.cmml"><mi id="A1.E107.m1.4.4.1.1.2.3.2" xref="A1.E107.m1.4.4.1.1.2.3.2.cmml">ω</mi><mi id="A1.E107.m1.4.4.1.1.2.3.3" xref="A1.E107.m1.4.4.1.1.2.3.3.cmml">PC</mi></msub></mrow><mo id="A1.E107.m1.4.4.1.1.1" xref="A1.E107.m1.4.4.1.1.1.cmml">=</mo><mrow id="A1.E107.m1.4.4.1.1.3" xref="A1.E107.m1.4.4.1.1.3.cmml"><mrow id="A1.E107.m1.4.4.1.1.3.2" xref="A1.E107.m1.4.4.1.1.3.2.cmml"><mrow id="A1.E107.m1.4.4.1.1.3.2.2" xref="A1.E107.m1.4.4.1.1.3.2.2.cmml"><mrow id="A1.E107.m1.4.4.1.1.3.2.2.2" xref="A1.E107.m1.4.4.1.1.3.2.2.2.cmml"><mi id="A1.E107.m1.4.4.1.1.3.2.2.2.2" xref="A1.E107.m1.4.4.1.1.3.2.2.2.2.cmml">d</mi><mo id="A1.E107.m1.4.4.1.1.3.2.2.2.1" xref="A1.E107.m1.4.4.1.1.3.2.2.2.1.cmml">⁢</mo><msub id="A1.E107.m1.4.4.1.1.3.2.2.2.3" xref="A1.E107.m1.4.4.1.1.3.2.2.2.3.cmml"><mi id="A1.E107.m1.4.4.1.1.3.2.2.2.3.2" xref="A1.E107.m1.4.4.1.1.3.2.2.2.3.2.cmml">p</mi><mi 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xref="A1.E107.m1.4.4.1.1.3.3.cmml"><mi id="A1.E107.m1.4.4.1.1.3.3.2" xref="A1.E107.m1.4.4.1.1.3.3.2.cmml">d</mi><mo id="A1.E107.m1.4.4.1.1.3.3.1" xref="A1.E107.m1.4.4.1.1.3.3.1.cmml">⁢</mo><mi id="A1.E107.m1.4.4.1.1.3.3.3" xref="A1.E107.m1.4.4.1.1.3.3.3.cmml">t</mi></mrow></mrow></mrow><mo id="A1.E107.m1.4.4.1.2" lspace="0em" xref="A1.E107.m1.4.4.1.1.cmml">.</mo></mrow><annotation-xml encoding="MathML-Content" id="A1.E107.m1.4b"><apply id="A1.E107.m1.4.4.1.1.cmml" xref="A1.E107.m1.4.4.1"><eq id="A1.E107.m1.4.4.1.1.1.cmml" xref="A1.E107.m1.4.4.1.1.1"></eq><apply id="A1.E107.m1.4.4.1.1.2.cmml" xref="A1.E107.m1.4.4.1.1.2"><times id="A1.E107.m1.4.4.1.1.2.1.cmml" xref="A1.E107.m1.4.4.1.1.2.1"></times><ci id="A1.E107.m1.4.4.1.1.2.2.cmml" xref="A1.E107.m1.4.4.1.1.2.2">𝑑</ci><apply id="A1.E107.m1.4.4.1.1.2.3.cmml" xref="A1.E107.m1.4.4.1.1.2.3"><csymbol cd="ambiguous" id="A1.E107.m1.4.4.1.1.2.3.1.cmml" xref="A1.E107.m1.4.4.1.1.2.3">subscript</csymbol><ci id="A1.E107.m1.4.4.1.1.2.3.2.cmml" 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id="A1.E107.m1.4.4.1.1.3.2.3.1.cmml" xref="A1.E107.m1.4.4.1.1.3.2.3.1"></times><ci id="A1.E107.m1.4.4.1.1.3.2.3.2.cmml" xref="A1.E107.m1.4.4.1.1.3.2.3.2">𝑑</ci><ci id="A1.E107.m1.4.4.1.1.3.2.3.3.cmml" xref="A1.E107.m1.4.4.1.1.3.2.3.3">𝐻</ci><vector id="A1.E107.m1.4.4.1.1.3.2.3.4.1.cmml" xref="A1.E107.m1.4.4.1.1.3.2.3.4.2"><ci id="A1.E107.m1.1.1.cmml" xref="A1.E107.m1.1.1">𝑥</ci><ci id="A1.E107.m1.2.2.cmml" xref="A1.E107.m1.2.2">𝑝</ci><ci id="A1.E107.m1.3.3.cmml" xref="A1.E107.m1.3.3">𝑡</ci></vector></apply></apply><apply id="A1.E107.m1.4.4.1.1.3.3.cmml" xref="A1.E107.m1.4.4.1.1.3.3"><times id="A1.E107.m1.4.4.1.1.3.3.1.cmml" xref="A1.E107.m1.4.4.1.1.3.3.1"></times><ci id="A1.E107.m1.4.4.1.1.3.3.2.cmml" xref="A1.E107.m1.4.4.1.1.3.3.2">𝑑</ci><ci id="A1.E107.m1.4.4.1.1.3.3.3.cmml" xref="A1.E107.m1.4.4.1.1.3.3.3">𝑡</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.E107.m1.4c">\displaystyle d\omega_{\rm PC}=dp_{j}\wedge dx^{j}-dH(x,p,t)\wedge dt.</annotation><annotation encoding="application/x-llamapun" id="A1.E107.m1.4d">italic_d italic_ω start_POSTSUBSCRIPT roman_PC end_POSTSUBSCRIPT = italic_d italic_p start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ∧ italic_d italic_x start_POSTSUPERSCRIPT italic_j end_POSTSUPERSCRIPT - italic_d italic_H ( italic_x , italic_p , italic_t ) ∧ italic_d italic_t .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_left_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(107)</span></td> </tr></tbody> </table> <p class="ltx_p" id="A1.p2.5">By using (<a class="ltx_ref" href="https://arxiv.org/html/2406.11224v2#A1.E106" title="In Appendix A The proof of (38) ‣ A Hamiltonian approach to the gradient-flow equations in information geometry"><span class="ltx_text ltx_ref_tag">106</span></a>) we have</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="S3.EGx97"> <tbody id="A1.Ex2"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_left_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle d" class="ltx_Math" display="inline" id="A1.Ex2.m1.1"><semantics id="A1.Ex2.m1.1a"><mi id="A1.Ex2.m1.1.1" xref="A1.Ex2.m1.1.1.cmml">d</mi><annotation-xml encoding="MathML-Content" id="A1.Ex2.m1.1b"><ci id="A1.Ex2.m1.1.1.cmml" xref="A1.Ex2.m1.1.1">𝑑</ci></annotation-xml><annotation encoding="application/x-tex" id="A1.Ex2.m1.1c">\displaystyle d</annotation><annotation encoding="application/x-llamapun" id="A1.Ex2.m1.1d">italic_d</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle H\wedge dt=\left(dx^{i}\frac{\partial H}{\partial x^{i}}+dp_{i}% \frac{\partial H}{\partial p_{i}}+dt\frac{\partial H}{\partial t}\right)\wedge dt" class="ltx_Math" display="inline" id="A1.Ex2.m2.1"><semantics id="A1.Ex2.m2.1a"><mrow id="A1.Ex2.m2.1.1" 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xref="A1.Ex2.m2.1.1.1.3"><times id="A1.Ex2.m2.1.1.1.3.1.cmml" xref="A1.Ex2.m2.1.1.1.3.1"></times><ci id="A1.Ex2.m2.1.1.1.3.2.cmml" xref="A1.Ex2.m2.1.1.1.3.2">𝑑</ci><ci id="A1.Ex2.m2.1.1.1.3.3.cmml" xref="A1.Ex2.m2.1.1.1.3.3">𝑡</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.Ex2.m2.1c">\displaystyle H\wedge dt=\left(dx^{i}\frac{\partial H}{\partial x^{i}}+dp_{i}% \frac{\partial H}{\partial p_{i}}+dt\frac{\partial H}{\partial t}\right)\wedge dt</annotation><annotation encoding="application/x-llamapun" id="A1.Ex2.m2.1d">italic_H ∧ italic_d italic_t = ( italic_d italic_x start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT divide start_ARG ∂ italic_H end_ARG start_ARG ∂ italic_x start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT end_ARG + italic_d italic_p start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT divide start_ARG ∂ italic_H end_ARG start_ARG ∂ italic_p start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_ARG + italic_d italic_t divide start_ARG ∂ italic_H end_ARG start_ARG ∂ italic_t end_ARG ) ∧ italic_d italic_t</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_left_padright"></td> </tr></tbody> <tbody id="A1.Ex3"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_left_padleft"></td> <td class="ltx_td ltx_eqn_cell"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle=dx^{i}\wedge\frac{\partial H}{\partial x^{i}}dt+dp_{i}\wedge% \frac{\partial H}{\partial p_{i}}dt" class="ltx_Math" display="inline" id="A1.Ex3.m1.1"><semantics id="A1.Ex3.m1.1a"><mrow id="A1.Ex3.m1.1.1" xref="A1.Ex3.m1.1.1.cmml"><mi id="A1.Ex3.m1.1.1.2" xref="A1.Ex3.m1.1.1.2.cmml"></mi><mo id="A1.Ex3.m1.1.1.1" xref="A1.Ex3.m1.1.1.1.cmml">=</mo><mrow id="A1.Ex3.m1.1.1.3" xref="A1.Ex3.m1.1.1.3.cmml"><mrow id="A1.Ex3.m1.1.1.3.2" xref="A1.Ex3.m1.1.1.3.2.cmml"><mrow id="A1.Ex3.m1.1.1.3.2.2" xref="A1.Ex3.m1.1.1.3.2.2.cmml"><mrow id="A1.Ex3.m1.1.1.3.2.2.2" 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dx^{i}.</annotation><annotation encoding="application/x-llamapun" id="A1.E108.m1.1d">start_OPFUNCTION overEOM start_ARG = end_ARG end_OPFUNCTION italic_d italic_x start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT ∧ ( - italic_d italic_p start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) + italic_d italic_p start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ∧ italic_d italic_x start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT = 2 italic_d italic_p start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ∧ italic_d italic_x start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_left_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(108)</span></td> </tr></tbody> </table> <p class="ltx_p" id="A1.p2.4">then (<a class="ltx_ref" href="https://arxiv.org/html/2406.11224v2#A1.E107" title="In Appendix A The proof of (38) ‣ A Hamiltonian approach to the gradient-flow equations in information geometry"><span class="ltx_text ltx_ref_tag">107</span></a>) becomes <math alttext="d\omega_{\rm PC}\operatorname{\overset{\textrm{ \tiny EOM}}{=}}-dp_{j}\wedge dx% ^{j}" class="ltx_Math" display="inline" id="A1.p2.4.m1.1"><semantics id="A1.p2.4.m1.1a"><mrow id="A1.p2.4.m1.1.1" xref="A1.p2.4.m1.1.1.cmml"><mrow id="A1.p2.4.m1.1.1.2" xref="A1.p2.4.m1.1.1.2.cmml"><mrow id="A1.p2.4.m1.1.1.2.2" xref="A1.p2.4.m1.1.1.2.2.cmml"><mi id="A1.p2.4.m1.1.1.2.2.2" xref="A1.p2.4.m1.1.1.2.2.2.cmml">d</mi><mo id="A1.p2.4.m1.1.1.2.2.1" xref="A1.p2.4.m1.1.1.2.2.1.cmml">⁢</mo><msub id="A1.p2.4.m1.1.1.2.2.3" xref="A1.p2.4.m1.1.1.2.2.3.cmml"><mi id="A1.p2.4.m1.1.1.2.2.3.2" xref="A1.p2.4.m1.1.1.2.2.3.2.cmml">ω</mi><mi id="A1.p2.4.m1.1.1.2.2.3.3" xref="A1.p2.4.m1.1.1.2.2.3.3.cmml">PC</mi></msub><mo id="A1.p2.4.m1.1.1.2.2.1a" xref="A1.p2.4.m1.1.1.2.2.1.cmml">⁢</mo><mover accent="true" id="A1.p2.4.m1.1.1.2.2.4" xref="A1.p2.4.m1.1.1.2.2.4.cmml"><mo id="A1.p2.4.m1.1.1.2.2.4.2" rspace="0em" xref="A1.p2.4.m1.1.1.2.2.4.2.cmml">=</mo><mrow id="A1.p2.4.m1.1.1.2.2.4.1" xref="A1.p2.4.m1.1.1.2.2.4.1c.cmml"><mtext id="A1.p2.4.m1.1.1.2.2.4.1a" xref="A1.p2.4.m1.1.1.2.2.4.1c.cmml"> </mtext><mtext id="A1.p2.4.m1.1.1.2.2.4.1b" mathsize="50%" xref="A1.p2.4.m1.1.1.2.2.4.1c.cmml">EOM</mtext></mrow></mover></mrow><mo id="A1.p2.4.m1.1.1.2.1" lspace="0em" xref="A1.p2.4.m1.1.1.2.1.cmml">−</mo><mrow id="A1.p2.4.m1.1.1.2.3" xref="A1.p2.4.m1.1.1.2.3.cmml"><mi id="A1.p2.4.m1.1.1.2.3.2" xref="A1.p2.4.m1.1.1.2.3.2.cmml">d</mi><mo id="A1.p2.4.m1.1.1.2.3.1" xref="A1.p2.4.m1.1.1.2.3.1.cmml">⁢</mo><msub id="A1.p2.4.m1.1.1.2.3.3" xref="A1.p2.4.m1.1.1.2.3.3.cmml"><mi id="A1.p2.4.m1.1.1.2.3.3.2" xref="A1.p2.4.m1.1.1.2.3.3.2.cmml">p</mi><mi id="A1.p2.4.m1.1.1.2.3.3.3" xref="A1.p2.4.m1.1.1.2.3.3.3.cmml">j</mi></msub></mrow></mrow><mo id="A1.p2.4.m1.1.1.1" xref="A1.p2.4.m1.1.1.1.cmml">∧</mo><mrow id="A1.p2.4.m1.1.1.3" xref="A1.p2.4.m1.1.1.3.cmml"><mi id="A1.p2.4.m1.1.1.3.2" xref="A1.p2.4.m1.1.1.3.2.cmml">d</mi><mo id="A1.p2.4.m1.1.1.3.1" xref="A1.p2.4.m1.1.1.3.1.cmml">⁢</mo><msup id="A1.p2.4.m1.1.1.3.3" xref="A1.p2.4.m1.1.1.3.3.cmml"><mi id="A1.p2.4.m1.1.1.3.3.2" xref="A1.p2.4.m1.1.1.3.3.2.cmml">x</mi><mi id="A1.p2.4.m1.1.1.3.3.3" xref="A1.p2.4.m1.1.1.3.3.3.cmml">j</mi></msup></mrow></mrow><annotation-xml encoding="MathML-Content" id="A1.p2.4.m1.1b"><apply id="A1.p2.4.m1.1.1.cmml" xref="A1.p2.4.m1.1.1"><and id="A1.p2.4.m1.1.1.1.cmml" xref="A1.p2.4.m1.1.1.1"></and><apply id="A1.p2.4.m1.1.1.2.cmml" xref="A1.p2.4.m1.1.1.2"><minus id="A1.p2.4.m1.1.1.2.1.cmml" xref="A1.p2.4.m1.1.1.2.1"></minus><apply id="A1.p2.4.m1.1.1.2.2.cmml" xref="A1.p2.4.m1.1.1.2.2"><times id="A1.p2.4.m1.1.1.2.2.1.cmml" xref="A1.p2.4.m1.1.1.2.2.1"></times><ci id="A1.p2.4.m1.1.1.2.2.2.cmml" xref="A1.p2.4.m1.1.1.2.2.2">𝑑</ci><apply id="A1.p2.4.m1.1.1.2.2.3.cmml" xref="A1.p2.4.m1.1.1.2.2.3"><csymbol cd="ambiguous" id="A1.p2.4.m1.1.1.2.2.3.1.cmml" xref="A1.p2.4.m1.1.1.2.2.3">subscript</csymbol><ci id="A1.p2.4.m1.1.1.2.2.3.2.cmml" xref="A1.p2.4.m1.1.1.2.2.3.2">𝜔</ci><ci id="A1.p2.4.m1.1.1.2.2.3.3.cmml" xref="A1.p2.4.m1.1.1.2.2.3.3">PC</ci></apply><apply id="A1.p2.4.m1.1.1.2.2.4.cmml" xref="A1.p2.4.m1.1.1.2.2.4"><ci id="A1.p2.4.m1.1.1.2.2.4.1c.cmml" xref="A1.p2.4.m1.1.1.2.2.4.1"><mrow id="A1.p2.4.m1.1.1.2.2.4.1.cmml" xref="A1.p2.4.m1.1.1.2.2.4.1"><mtext id="A1.p2.4.m1.1.1.2.2.4.1a.cmml" xref="A1.p2.4.m1.1.1.2.2.4.1"> </mtext><mtext id="A1.p2.4.m1.1.1.2.2.4.1b.cmml" mathsize="50%" xref="A1.p2.4.m1.1.1.2.2.4.1">EOM</mtext></mrow></ci><eq id="A1.p2.4.m1.1.1.2.2.4.2.cmml" xref="A1.p2.4.m1.1.1.2.2.4.2"></eq></apply></apply><apply id="A1.p2.4.m1.1.1.2.3.cmml" xref="A1.p2.4.m1.1.1.2.3"><times id="A1.p2.4.m1.1.1.2.3.1.cmml" xref="A1.p2.4.m1.1.1.2.3.1"></times><ci id="A1.p2.4.m1.1.1.2.3.2.cmml" xref="A1.p2.4.m1.1.1.2.3.2">𝑑</ci><apply id="A1.p2.4.m1.1.1.2.3.3.cmml" xref="A1.p2.4.m1.1.1.2.3.3"><csymbol cd="ambiguous" id="A1.p2.4.m1.1.1.2.3.3.1.cmml" xref="A1.p2.4.m1.1.1.2.3.3">subscript</csymbol><ci id="A1.p2.4.m1.1.1.2.3.3.2.cmml" xref="A1.p2.4.m1.1.1.2.3.3.2">𝑝</ci><ci id="A1.p2.4.m1.1.1.2.3.3.3.cmml" xref="A1.p2.4.m1.1.1.2.3.3.3">𝑗</ci></apply></apply></apply><apply id="A1.p2.4.m1.1.1.3.cmml" xref="A1.p2.4.m1.1.1.3"><times id="A1.p2.4.m1.1.1.3.1.cmml" xref="A1.p2.4.m1.1.1.3.1"></times><ci id="A1.p2.4.m1.1.1.3.2.cmml" xref="A1.p2.4.m1.1.1.3.2">𝑑</ci><apply id="A1.p2.4.m1.1.1.3.3.cmml" xref="A1.p2.4.m1.1.1.3.3"><csymbol cd="ambiguous" id="A1.p2.4.m1.1.1.3.3.1.cmml" xref="A1.p2.4.m1.1.1.3.3">superscript</csymbol><ci id="A1.p2.4.m1.1.1.3.3.2.cmml" xref="A1.p2.4.m1.1.1.3.3.2">𝑥</ci><ci id="A1.p2.4.m1.1.1.3.3.3.cmml" xref="A1.p2.4.m1.1.1.3.3.3">𝑗</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.p2.4.m1.1c">d\omega_{\rm PC}\operatorname{\overset{\textrm{ \tiny EOM}}{=}}-dp_{j}\wedge dx% ^{j}</annotation><annotation encoding="application/x-llamapun" id="A1.p2.4.m1.1d">italic_d italic_ω start_POSTSUBSCRIPT roman_PC end_POSTSUBSCRIPT start_OPFUNCTION over EOM start_ARG = end_ARG end_OPFUNCTION - italic_d italic_p start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ∧ italic_d italic_x start_POSTSUPERSCRIPT italic_j end_POSTSUPERSCRIPT</annotation></semantics></math>. It follows that</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="S3.EGx98"> <tbody id="A1.Ex4"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_left_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle d\omega_{\rm PC}" class="ltx_Math" display="inline" id="A1.Ex4.m1.1"><semantics id="A1.Ex4.m1.1a"><mrow id="A1.Ex4.m1.1.1" xref="A1.Ex4.m1.1.1.cmml"><mi id="A1.Ex4.m1.1.1.2" xref="A1.Ex4.m1.1.1.2.cmml">d</mi><mo id="A1.Ex4.m1.1.1.1" xref="A1.Ex4.m1.1.1.1.cmml">⁢</mo><msub id="A1.Ex4.m1.1.1.3" xref="A1.Ex4.m1.1.1.3.cmml"><mi id="A1.Ex4.m1.1.1.3.2" xref="A1.Ex4.m1.1.1.3.2.cmml">ω</mi><mi id="A1.Ex4.m1.1.1.3.3" xref="A1.Ex4.m1.1.1.3.3.cmml">PC</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="A1.Ex4.m1.1b"><apply id="A1.Ex4.m1.1.1.cmml" xref="A1.Ex4.m1.1.1"><times id="A1.Ex4.m1.1.1.1.cmml" xref="A1.Ex4.m1.1.1.1"></times><ci id="A1.Ex4.m1.1.1.2.cmml" xref="A1.Ex4.m1.1.1.2">𝑑</ci><apply id="A1.Ex4.m1.1.1.3.cmml" xref="A1.Ex4.m1.1.1.3"><csymbol cd="ambiguous" id="A1.Ex4.m1.1.1.3.1.cmml" xref="A1.Ex4.m1.1.1.3">subscript</csymbol><ci id="A1.Ex4.m1.1.1.3.2.cmml" xref="A1.Ex4.m1.1.1.3.2">𝜔</ci><ci id="A1.Ex4.m1.1.1.3.3.cmml" xref="A1.Ex4.m1.1.1.3.3">PC</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.Ex4.m1.1c">\displaystyle d\omega_{\rm PC}</annotation><annotation encoding="application/x-llamapun" id="A1.Ex4.m1.1d">italic_d italic_ω start_POSTSUBSCRIPT roman_PC end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\wedge\omega_{\rm PC}\operatorname{\overset{\textrm{ \tiny EOM}}{% =}}Hdp_{i}\wedge dx^{i}\wedge dt-p_{j}dx^{j}\wedge dp_{i}\wedge dx^{i}" class="ltx_Math" display="inline" id="A1.Ex4.m2.1"><semantics id="A1.Ex4.m2.1a"><mrow id="A1.Ex4.m2.1.1" xref="A1.Ex4.m2.1.1.cmml"><mrow id="A1.Ex4.m2.1.1.2" xref="A1.Ex4.m2.1.1.2.cmml"><mrow id="A1.Ex4.m2.1.1.2.2" xref="A1.Ex4.m2.1.1.2.2.cmml"><mrow id="A1.Ex4.m2.1.1.2.2.2" xref="A1.Ex4.m2.1.1.2.2.2.cmml"><mo id="A1.Ex4.m2.1.1.2.2.2a" rspace="0em" xref="A1.Ex4.m2.1.1.2.2.2.cmml">∧</mo><mrow id="A1.Ex4.m2.1.1.2.2.2.2" xref="A1.Ex4.m2.1.1.2.2.2.2.cmml"><msub id="A1.Ex4.m2.1.1.2.2.2.2.2" xref="A1.Ex4.m2.1.1.2.2.2.2.2.cmml"><mi id="A1.Ex4.m2.1.1.2.2.2.2.2.2" xref="A1.Ex4.m2.1.1.2.2.2.2.2.2.cmml">ω</mi><mi id="A1.Ex4.m2.1.1.2.2.2.2.2.3" xref="A1.Ex4.m2.1.1.2.2.2.2.2.3.cmml">PC</mi></msub><mo id="A1.Ex4.m2.1.1.2.2.2.2.1" xref="A1.Ex4.m2.1.1.2.2.2.2.1.cmml">⁢</mo><mrow id="A1.Ex4.m2.1.1.2.2.2.2.3" xref="A1.Ex4.m2.1.1.2.2.2.2.3.cmml"><mover accent="true" id="A1.Ex4.m2.1.1.2.2.2.2.3.1" xref="A1.Ex4.m2.1.1.2.2.2.2.3.1.cmml"><mo id="A1.Ex4.m2.1.1.2.2.2.2.3.1.2" xref="A1.Ex4.m2.1.1.2.2.2.2.3.1.2.cmml">=</mo><mrow id="A1.Ex4.m2.1.1.2.2.2.2.3.1.1" xref="A1.Ex4.m2.1.1.2.2.2.2.3.1.1c.cmml"><mtext id="A1.Ex4.m2.1.1.2.2.2.2.3.1.1a" xref="A1.Ex4.m2.1.1.2.2.2.2.3.1.1c.cmml"> </mtext><mtext id="A1.Ex4.m2.1.1.2.2.2.2.3.1.1b" mathsize="50%" xref="A1.Ex4.m2.1.1.2.2.2.2.3.1.1c.cmml">EOM</mtext></mrow></mover><mrow id="A1.Ex4.m2.1.1.2.2.2.2.3.2" xref="A1.Ex4.m2.1.1.2.2.2.2.3.2.cmml"><mi id="A1.Ex4.m2.1.1.2.2.2.2.3.2.2" xref="A1.Ex4.m2.1.1.2.2.2.2.3.2.2.cmml">H</mi><mo id="A1.Ex4.m2.1.1.2.2.2.2.3.2.1" xref="A1.Ex4.m2.1.1.2.2.2.2.3.2.1.cmml">⁢</mo><mi id="A1.Ex4.m2.1.1.2.2.2.2.3.2.3" xref="A1.Ex4.m2.1.1.2.2.2.2.3.2.3.cmml">d</mi><mo id="A1.Ex4.m2.1.1.2.2.2.2.3.2.1a" xref="A1.Ex4.m2.1.1.2.2.2.2.3.2.1.cmml">⁢</mo><msub id="A1.Ex4.m2.1.1.2.2.2.2.3.2.4" xref="A1.Ex4.m2.1.1.2.2.2.2.3.2.4.cmml"><mi id="A1.Ex4.m2.1.1.2.2.2.2.3.2.4.2" xref="A1.Ex4.m2.1.1.2.2.2.2.3.2.4.2.cmml">p</mi><mi id="A1.Ex4.m2.1.1.2.2.2.2.3.2.4.3" xref="A1.Ex4.m2.1.1.2.2.2.2.3.2.4.3.cmml">i</mi></msub></mrow></mrow></mrow></mrow><mo id="A1.Ex4.m2.1.1.2.2.1" xref="A1.Ex4.m2.1.1.2.2.1.cmml">∧</mo><mrow id="A1.Ex4.m2.1.1.2.2.3" 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over EOM start_ARG = end_ARG end_OPFUNCTION italic_H italic_d italic_p start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ∧ italic_d italic_x start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT ∧ italic_d italic_t - italic_p start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT italic_d italic_x start_POSTSUPERSCRIPT italic_j end_POSTSUPERSCRIPT ∧ italic_d italic_p start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ∧ italic_d italic_x start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_left_padright"></td> </tr></tbody> <tbody id="A1.Ex5"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_left_padleft"></td> <td class="ltx_td ltx_eqn_cell"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\operatorname{\overset{\textrm{ \tiny EOM}}{=}}Hdp_{i}\wedge dx^{% i}\wedge dt-p_{j}\frac{\partial H}{\partial p_{j}}dt\wedge dp_{i}\wedge dx^{i}" class="ltx_Math" display="inline" 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id="A1.E109.m1.1c">\displaystyle=\left(H-p_{j}\frac{\partial H}{\partial p_{j}}\right)dp_{i}% \wedge dx^{i}\wedge dt,</annotation><annotation encoding="application/x-llamapun" id="A1.E109.m1.1d">= ( italic_H - italic_p start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT divide start_ARG ∂ italic_H end_ARG start_ARG ∂ italic_p start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT end_ARG ) italic_d italic_p start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ∧ italic_d italic_x start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT ∧ italic_d italic_t ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_left_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(109)</span></td> </tr></tbody> </table> <p class="ltx_p" id="A1.p2.6">where we used (<a class="ltx_ref" href="https://arxiv.org/html/2406.11224v2#A1.E106" title="In Appendix A The proof of (38) ‣ A Hamiltonian approach to the gradient-flow equations in information geometry"><span class="ltx_text ltx_ref_tag">106</span></a>).</p> </div> </section> <section class="ltx_appendix" id="A2"> <h2 class="ltx_title ltx_title_appendix"> <span class="ltx_tag ltx_tag_appendix">Appendix B </span>Conformal rescaling as reparametrization</h2> <div class="ltx_para" id="A2.p1"> <p class="ltx_p" id="A2.p1.1">Here we briefly explain the relation between the conformal rescaling and reparametrization of null geodesics <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2406.11224v2#bib.bib25" title="">25</a>]</cite>. 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the null geodesic Hamiltonian</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="S3.EGx99"> <tbody id="A2.E110"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_left_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\mathcal{H}=\frac{1}{2}g^{\mu\nu}(x)p_{\mu}p_{\nu}=0." class="ltx_Math" display="inline" id="A2.E110.m1.2"><semantics id="A2.E110.m1.2a"><mrow id="A2.E110.m1.2.2.1" xref="A2.E110.m1.2.2.1.1.cmml"><mrow id="A2.E110.m1.2.2.1.1" xref="A2.E110.m1.2.2.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="A2.E110.m1.2.2.1.1.2" xref="A2.E110.m1.2.2.1.1.2.cmml">ℋ</mi><mo id="A2.E110.m1.2.2.1.1.3" xref="A2.E110.m1.2.2.1.1.3.cmml">=</mo><mrow id="A2.E110.m1.2.2.1.1.4" xref="A2.E110.m1.2.2.1.1.4.cmml"><mstyle displaystyle="true" id="A2.E110.m1.2.2.1.1.4.2" xref="A2.E110.m1.2.2.1.1.4.2.cmml"><mfrac id="A2.E110.m1.2.2.1.1.4.2a" xref="A2.E110.m1.2.2.1.1.4.2.cmml"><mn 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end_POSTSUBSCRIPT italic_p start_POSTSUBSCRIPT italic_ν end_POSTSUBSCRIPT = 0 .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_left_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(110)</span></td> </tr></tbody> </table> <p class="ltx_p" id="A2.p1.9">Its null geodesics are the solutions of Hamilton’s equations of motion,</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="S3.EGx100"> <tbody id="A2.E111"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_left_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\frac{dx^{\mu}}{dt}" class="ltx_Math" display="inline" id="A2.E111.m1.1"><semantics id="A2.E111.m1.1a"><mstyle displaystyle="true" id="A2.E111.m1.1.1" xref="A2.E111.m1.1.1.cmml"><mfrac id="A2.E111.m1.1.1a" xref="A2.E111.m1.1.1.cmml"><mrow id="A2.E111.m1.1.1.2" 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xref="A2.E111.m2.2.2.1.1.6.2.3.2">𝜇</ci><ci id="A2.E111.m2.2.2.1.1.6.2.3.3.cmml" xref="A2.E111.m2.2.2.1.1.6.2.3.3">𝜈</ci></apply></apply><ci id="A2.E111.m2.1.1.cmml" xref="A2.E111.m2.1.1">𝑥</ci><apply id="A2.E111.m2.2.2.1.1.6.4.cmml" xref="A2.E111.m2.2.2.1.1.6.4"><csymbol cd="ambiguous" id="A2.E111.m2.2.2.1.1.6.4.1.cmml" xref="A2.E111.m2.2.2.1.1.6.4">subscript</csymbol><ci id="A2.E111.m2.2.2.1.1.6.4.2.cmml" xref="A2.E111.m2.2.2.1.1.6.4.2">𝑝</ci><ci id="A2.E111.m2.2.2.1.1.6.4.3.cmml" xref="A2.E111.m2.2.2.1.1.6.4.3">𝜈</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.E111.m2.2c">\displaystyle=\frac{\partial\mathcal{H}}{\partial p_{\mu}}=g^{\mu\nu}(x)p_{\nu},</annotation><annotation encoding="application/x-llamapun" id="A2.E111.m2.2d">= divide start_ARG ∂ caligraphic_H end_ARG start_ARG ∂ italic_p start_POSTSUBSCRIPT italic_μ end_POSTSUBSCRIPT end_ARG = italic_g start_POSTSUPERSCRIPT italic_μ italic_ν end_POSTSUPERSCRIPT ( italic_x ) italic_p start_POSTSUBSCRIPT italic_ν end_POSTSUBSCRIPT ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_left_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(111)</span></td> </tr></tbody> <tbody id="A2.E112"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_left_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\frac{dp_{\mu}}{dt}" class="ltx_Math" display="inline" id="A2.E112.m1.1"><semantics id="A2.E112.m1.1a"><mstyle displaystyle="true" id="A2.E112.m1.1.1" xref="A2.E112.m1.1.1.cmml"><mfrac id="A2.E112.m1.1.1a" xref="A2.E112.m1.1.1.cmml"><mrow id="A2.E112.m1.1.1.2" xref="A2.E112.m1.1.1.2.cmml"><mi id="A2.E112.m1.1.1.2.2" xref="A2.E112.m1.1.1.2.2.cmml">d</mi><mo id="A2.E112.m1.1.1.2.1" xref="A2.E112.m1.1.1.2.1.cmml">⁢</mo><msub id="A2.E112.m1.1.1.2.3" xref="A2.E112.m1.1.1.2.3.cmml"><mi id="A2.E112.m1.1.1.2.3.2" xref="A2.E112.m1.1.1.2.3.2.cmml">p</mi><mi id="A2.E112.m1.1.1.2.3.3" xref="A2.E112.m1.1.1.2.3.3.cmml">μ</mi></msub></mrow><mrow id="A2.E112.m1.1.1.3" xref="A2.E112.m1.1.1.3.cmml"><mi id="A2.E112.m1.1.1.3.2" xref="A2.E112.m1.1.1.3.2.cmml">d</mi><mo id="A2.E112.m1.1.1.3.1" xref="A2.E112.m1.1.1.3.1.cmml">⁢</mo><mi id="A2.E112.m1.1.1.3.3" xref="A2.E112.m1.1.1.3.3.cmml">t</mi></mrow></mfrac></mstyle><annotation-xml encoding="MathML-Content" id="A2.E112.m1.1b"><apply id="A2.E112.m1.1.1.cmml" xref="A2.E112.m1.1.1"><divide id="A2.E112.m1.1.1.1.cmml" xref="A2.E112.m1.1.1"></divide><apply id="A2.E112.m1.1.1.2.cmml" xref="A2.E112.m1.1.1.2"><times id="A2.E112.m1.1.1.2.1.cmml" xref="A2.E112.m1.1.1.2.1"></times><ci id="A2.E112.m1.1.1.2.2.cmml" xref="A2.E112.m1.1.1.2.2">𝑑</ci><apply id="A2.E112.m1.1.1.2.3.cmml" xref="A2.E112.m1.1.1.2.3"><csymbol cd="ambiguous" id="A2.E112.m1.1.1.2.3.1.cmml" xref="A2.E112.m1.1.1.2.3">subscript</csymbol><ci id="A2.E112.m1.1.1.2.3.2.cmml" xref="A2.E112.m1.1.1.2.3.2">𝑝</ci><ci id="A2.E112.m1.1.1.2.3.3.cmml" xref="A2.E112.m1.1.1.2.3.3">𝜇</ci></apply></apply><apply id="A2.E112.m1.1.1.3.cmml" xref="A2.E112.m1.1.1.3"><times id="A2.E112.m1.1.1.3.1.cmml" xref="A2.E112.m1.1.1.3.1"></times><ci id="A2.E112.m1.1.1.3.2.cmml" xref="A2.E112.m1.1.1.3.2">𝑑</ci><ci id="A2.E112.m1.1.1.3.3.cmml" xref="A2.E112.m1.1.1.3.3">𝑡</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.E112.m1.1c">\displaystyle\frac{dp_{\mu}}{dt}</annotation><annotation encoding="application/x-llamapun" id="A2.E112.m1.1d">divide start_ARG italic_d italic_p start_POSTSUBSCRIPT italic_μ end_POSTSUBSCRIPT end_ARG start_ARG italic_d italic_t end_ARG</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle=-\frac{\partial\mathcal{H}}{\partial x^{\mu}}=-\frac{1}{2}\frac{% \partial g^{\nu\rho}(x)}{\partial x^{\mu}}p_{\nu}p_{\rho}," class="ltx_Math" 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,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_left_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(112)</span></td> </tr></tbody> </table> <p class="ltx_p" id="A2.p1.2">where <math alttext="t" class="ltx_Math" display="inline" id="A2.p1.2.m1.1"><semantics id="A2.p1.2.m1.1a"><mi id="A2.p1.2.m1.1.1" xref="A2.p1.2.m1.1.1.cmml">t</mi><annotation-xml encoding="MathML-Content" id="A2.p1.2.m1.1b"><ci id="A2.p1.2.m1.1.1.cmml" xref="A2.p1.2.m1.1.1">𝑡</ci></annotation-xml><annotation encoding="application/x-tex" id="A2.p1.2.m1.1c">t</annotation><annotation encoding="application/x-llamapun" id="A2.p1.2.m1.1d">italic_t</annotation></semantics></math> is the evolution parameter. Now consider the rescaled metric</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="S3.EGx101"> <tbody id="A2.E113"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_left_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\tilde{g}_{\mu\nu}(x)=\Omega^{2}(x)g_{\mu\nu}(x)," class="ltx_Math" display="inline" id="A2.E113.m1.4"><semantics id="A2.E113.m1.4a"><mrow id="A2.E113.m1.4.4.1" xref="A2.E113.m1.4.4.1.1.cmml"><mrow id="A2.E113.m1.4.4.1.1" xref="A2.E113.m1.4.4.1.1.cmml"><mrow id="A2.E113.m1.4.4.1.1.2" xref="A2.E113.m1.4.4.1.1.2.cmml"><msub id="A2.E113.m1.4.4.1.1.2.2" xref="A2.E113.m1.4.4.1.1.2.2.cmml"><mover accent="true" id="A2.E113.m1.4.4.1.1.2.2.2" xref="A2.E113.m1.4.4.1.1.2.2.2.cmml"><mi id="A2.E113.m1.4.4.1.1.2.2.2.2" xref="A2.E113.m1.4.4.1.1.2.2.2.2.cmml">g</mi><mo id="A2.E113.m1.4.4.1.1.2.2.2.1" xref="A2.E113.m1.4.4.1.1.2.2.2.1.cmml">~</mo></mover><mrow id="A2.E113.m1.4.4.1.1.2.2.3" 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encoding="application/x-tex" id="A2.E113.m1.4c">\displaystyle\tilde{g}_{\mu\nu}(x)=\Omega^{2}(x)g_{\mu\nu}(x),</annotation><annotation encoding="application/x-llamapun" id="A2.E113.m1.4d">over~ start_ARG italic_g end_ARG start_POSTSUBSCRIPT italic_μ italic_ν end_POSTSUBSCRIPT ( italic_x ) = roman_Ω start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( italic_x ) italic_g start_POSTSUBSCRIPT italic_μ italic_ν end_POSTSUBSCRIPT ( italic_x ) ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_left_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(113)</span></td> </tr></tbody> </table> <p class="ltx_p" id="A2.p1.10">and the associated new Hamiltonian</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="S3.EGx102"> <tbody id="A2.E114"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_left_padleft"></td> <td class="ltx_td 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end_POSTSUPERSCRIPT ( italic_x ) end_ARG start_ARG 2 roman_Ω start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( italic_x ) end_ARG italic_p start_POSTSUBSCRIPT italic_μ end_POSTSUBSCRIPT italic_p start_POSTSUBSCRIPT italic_ν end_POSTSUBSCRIPT = 0 ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_left_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(114)</span></td> </tr></tbody> </table> <p class="ltx_p" id="A2.p1.3">where <math alttext="\Omega^{2}(x)>0" class="ltx_Math" display="inline" id="A2.p1.3.m1.1"><semantics id="A2.p1.3.m1.1a"><mrow id="A2.p1.3.m1.1.2" xref="A2.p1.3.m1.1.2.cmml"><mrow id="A2.p1.3.m1.1.2.2" xref="A2.p1.3.m1.1.2.2.cmml"><msup id="A2.p1.3.m1.1.2.2.2" xref="A2.p1.3.m1.1.2.2.2.cmml"><mi id="A2.p1.3.m1.1.2.2.2.2" mathvariant="normal" xref="A2.p1.3.m1.1.2.2.2.2.cmml">Ω</mi><mn id="A2.p1.3.m1.1.2.2.2.3" 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The new equations of motion are</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="S3.EGx103"> <tbody id="A2.Ex6"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_left_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\frac{dx^{\mu}}{d\tilde{t}}" class="ltx_Math" display="inline" id="A2.Ex6.m1.1"><semantics id="A2.Ex6.m1.1a"><mstyle displaystyle="true" id="A2.Ex6.m1.1.1" xref="A2.Ex6.m1.1.1.cmml"><mfrac id="A2.Ex6.m1.1.1a" xref="A2.Ex6.m1.1.1.cmml"><mrow id="A2.Ex6.m1.1.1.2" xref="A2.Ex6.m1.1.1.2.cmml"><mi id="A2.Ex6.m1.1.1.2.2" xref="A2.Ex6.m1.1.1.2.2.cmml">d</mi><mo id="A2.Ex6.m1.1.1.2.1" xref="A2.Ex6.m1.1.1.2.1.cmml">⁢</mo><msup id="A2.Ex6.m1.1.1.2.3" xref="A2.Ex6.m1.1.1.2.3.cmml"><mi id="A2.Ex6.m1.1.1.2.3.2" xref="A2.Ex6.m1.1.1.2.3.2.cmml">x</mi><mi id="A2.Ex6.m1.1.1.2.3.3" xref="A2.Ex6.m1.1.1.2.3.3.cmml">μ</mi></msup></mrow><mrow id="A2.Ex6.m1.1.1.3" xref="A2.Ex6.m1.1.1.3.cmml"><mi 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xref="A2.Ex6.m1.1.1.2.3.3">𝜇</ci></apply></apply><apply id="A2.Ex6.m1.1.1.3.cmml" xref="A2.Ex6.m1.1.1.3"><times id="A2.Ex6.m1.1.1.3.1.cmml" xref="A2.Ex6.m1.1.1.3.1"></times><ci id="A2.Ex6.m1.1.1.3.2.cmml" xref="A2.Ex6.m1.1.1.3.2">𝑑</ci><apply id="A2.Ex6.m1.1.1.3.3.cmml" xref="A2.Ex6.m1.1.1.3.3"><ci id="A2.Ex6.m1.1.1.3.3.1.cmml" xref="A2.Ex6.m1.1.1.3.3.1">~</ci><ci id="A2.Ex6.m1.1.1.3.3.2.cmml" xref="A2.Ex6.m1.1.1.3.3.2">𝑡</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.Ex6.m1.1c">\displaystyle\frac{dx^{\mu}}{d\tilde{t}}</annotation><annotation encoding="application/x-llamapun" id="A2.Ex6.m1.1d">divide start_ARG italic_d italic_x start_POSTSUPERSCRIPT italic_μ end_POSTSUPERSCRIPT end_ARG start_ARG italic_d over~ start_ARG italic_t end_ARG end_ARG</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle=\frac{\partial\tilde{\mathcal{H}}}{\partial p_{\mu}}=\frac{1}{% 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encoding="application/x-llamapun" id="A2.Ex6.m2.4d">= divide start_ARG ∂ over~ start_ARG caligraphic_H end_ARG end_ARG start_ARG ∂ italic_p start_POSTSUBSCRIPT italic_μ end_POSTSUBSCRIPT end_ARG = divide start_ARG 1 end_ARG start_ARG roman_Ω start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( italic_x ) end_ARG italic_g start_POSTSUPERSCRIPT italic_μ italic_ν end_POSTSUPERSCRIPT ( italic_x ) italic_p start_POSTSUBSCRIPT italic_ν end_POSTSUBSCRIPT = divide start_ARG 1 end_ARG start_ARG roman_Ω start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( italic_x ) end_ARG divide start_ARG italic_d italic_x start_POSTSUPERSCRIPT italic_μ end_POSTSUPERSCRIPT end_ARG start_ARG italic_d italic_t end_ARG ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_left_padright"></td> </tr></tbody> <tbody id="A2.Ex7"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_left_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\frac{dp_{\mu}}{d\tilde{t}}" class="ltx_Math" display="inline" id="A2.Ex7.m1.1"><semantics id="A2.Ex7.m1.1a"><mstyle displaystyle="true" id="A2.Ex7.m1.1.1" xref="A2.Ex7.m1.1.1.cmml"><mfrac id="A2.Ex7.m1.1.1a" xref="A2.Ex7.m1.1.1.cmml"><mrow id="A2.Ex7.m1.1.1.2" xref="A2.Ex7.m1.1.1.2.cmml"><mi id="A2.Ex7.m1.1.1.2.2" xref="A2.Ex7.m1.1.1.2.2.cmml">d</mi><mo id="A2.Ex7.m1.1.1.2.1" xref="A2.Ex7.m1.1.1.2.1.cmml">⁢</mo><msub id="A2.Ex7.m1.1.1.2.3" xref="A2.Ex7.m1.1.1.2.3.cmml"><mi id="A2.Ex7.m1.1.1.2.3.2" xref="A2.Ex7.m1.1.1.2.3.2.cmml">p</mi><mi id="A2.Ex7.m1.1.1.2.3.3" xref="A2.Ex7.m1.1.1.2.3.3.cmml">μ</mi></msub></mrow><mrow id="A2.Ex7.m1.1.1.3" xref="A2.Ex7.m1.1.1.3.cmml"><mi id="A2.Ex7.m1.1.1.3.2" xref="A2.Ex7.m1.1.1.3.2.cmml">d</mi><mo id="A2.Ex7.m1.1.1.3.1" xref="A2.Ex7.m1.1.1.3.1.cmml">⁢</mo><mover accent="true" id="A2.Ex7.m1.1.1.3.3" xref="A2.Ex7.m1.1.1.3.3.cmml"><mi id="A2.Ex7.m1.1.1.3.3.2" xref="A2.Ex7.m1.1.1.3.3.2.cmml">t</mi><mo id="A2.Ex7.m1.1.1.3.3.1" xref="A2.Ex7.m1.1.1.3.3.1.cmml">~</mo></mover></mrow></mfrac></mstyle><annotation-xml encoding="MathML-Content" id="A2.Ex7.m1.1b"><apply id="A2.Ex7.m1.1.1.cmml" xref="A2.Ex7.m1.1.1"><divide id="A2.Ex7.m1.1.1.1.cmml" xref="A2.Ex7.m1.1.1"></divide><apply id="A2.Ex7.m1.1.1.2.cmml" xref="A2.Ex7.m1.1.1.2"><times id="A2.Ex7.m1.1.1.2.1.cmml" xref="A2.Ex7.m1.1.1.2.1"></times><ci id="A2.Ex7.m1.1.1.2.2.cmml" xref="A2.Ex7.m1.1.1.2.2">𝑑</ci><apply id="A2.Ex7.m1.1.1.2.3.cmml" xref="A2.Ex7.m1.1.1.2.3"><csymbol cd="ambiguous" id="A2.Ex7.m1.1.1.2.3.1.cmml" xref="A2.Ex7.m1.1.1.2.3">subscript</csymbol><ci id="A2.Ex7.m1.1.1.2.3.2.cmml" xref="A2.Ex7.m1.1.1.2.3.2">𝑝</ci><ci id="A2.Ex7.m1.1.1.2.3.3.cmml" xref="A2.Ex7.m1.1.1.2.3.3">𝜇</ci></apply></apply><apply id="A2.Ex7.m1.1.1.3.cmml" xref="A2.Ex7.m1.1.1.3"><times id="A2.Ex7.m1.1.1.3.1.cmml" xref="A2.Ex7.m1.1.1.3.1"></times><ci id="A2.Ex7.m1.1.1.3.2.cmml" xref="A2.Ex7.m1.1.1.3.2">𝑑</ci><apply id="A2.Ex7.m1.1.1.3.3.cmml" xref="A2.Ex7.m1.1.1.3.3"><ci id="A2.Ex7.m1.1.1.3.3.1.cmml" xref="A2.Ex7.m1.1.1.3.3.1">~</ci><ci id="A2.Ex7.m1.1.1.3.3.2.cmml" xref="A2.Ex7.m1.1.1.3.3.2">𝑡</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.Ex7.m1.1c">\displaystyle\frac{dp_{\mu}}{d\tilde{t}}</annotation><annotation encoding="application/x-llamapun" id="A2.Ex7.m1.1d">divide start_ARG italic_d italic_p start_POSTSUBSCRIPT italic_μ end_POSTSUBSCRIPT end_ARG start_ARG italic_d over~ start_ARG italic_t end_ARG end_ARG</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle=-\frac{\partial\tilde{\mathcal{H}}}{\partial x^{\mu}}" class="ltx_Math" display="inline" id="A2.Ex7.m2.1"><semantics id="A2.Ex7.m2.1a"><mrow id="A2.Ex7.m2.1.1" xref="A2.Ex7.m2.1.1.cmml"><mi id="A2.Ex7.m2.1.1.2" xref="A2.Ex7.m2.1.1.2.cmml"></mi><mo id="A2.Ex7.m2.1.1.1" xref="A2.Ex7.m2.1.1.1.cmml">=</mo><mrow id="A2.Ex7.m2.1.1.3" xref="A2.Ex7.m2.1.1.3.cmml"><mo id="A2.Ex7.m2.1.1.3a" 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xref="A2.Ex7.m2.1.1.3.2.3.2.3.cmml">μ</mi></msup></mrow></mfrac></mstyle></mrow></mrow><annotation-xml encoding="MathML-Content" id="A2.Ex7.m2.1b"><apply id="A2.Ex7.m2.1.1.cmml" xref="A2.Ex7.m2.1.1"><eq id="A2.Ex7.m2.1.1.1.cmml" xref="A2.Ex7.m2.1.1.1"></eq><csymbol cd="latexml" id="A2.Ex7.m2.1.1.2.cmml" xref="A2.Ex7.m2.1.1.2">absent</csymbol><apply id="A2.Ex7.m2.1.1.3.cmml" xref="A2.Ex7.m2.1.1.3"><minus id="A2.Ex7.m2.1.1.3.1.cmml" xref="A2.Ex7.m2.1.1.3"></minus><apply id="A2.Ex7.m2.1.1.3.2.cmml" xref="A2.Ex7.m2.1.1.3.2"><divide id="A2.Ex7.m2.1.1.3.2.1.cmml" xref="A2.Ex7.m2.1.1.3.2"></divide><apply id="A2.Ex7.m2.1.1.3.2.2.cmml" xref="A2.Ex7.m2.1.1.3.2.2"><partialdiff id="A2.Ex7.m2.1.1.3.2.2.1.cmml" xref="A2.Ex7.m2.1.1.3.2.2.1"></partialdiff><apply id="A2.Ex7.m2.1.1.3.2.2.2.cmml" xref="A2.Ex7.m2.1.1.3.2.2.2"><ci id="A2.Ex7.m2.1.1.3.2.2.2.1.cmml" xref="A2.Ex7.m2.1.1.3.2.2.2.1">~</ci><ci id="A2.Ex7.m2.1.1.3.2.2.2.2.cmml" xref="A2.Ex7.m2.1.1.3.2.2.2.2">ℋ</ci></apply></apply><apply 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id="A2.Ex8"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_left_padleft"></td> <td class="ltx_td ltx_eqn_cell"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle=-\frac{1}{2\Omega^{2}(x)}\frac{\partial g^{\nu\rho}(x)}{\partial x% ^{\mu}}p_{\nu}p_{\rho}-\mathcal{H}\frac{\partial}{\partial x^{\mu}}\frac{1}{% \Omega^{2}(x)}" class="ltx_Math" display="inline" id="A2.Ex8.m1.3"><semantics id="A2.Ex8.m1.3a"><mrow id="A2.Ex8.m1.3.4" xref="A2.Ex8.m1.3.4.cmml"><mi id="A2.Ex8.m1.3.4.2" xref="A2.Ex8.m1.3.4.2.cmml"></mi><mo id="A2.Ex8.m1.3.4.1" xref="A2.Ex8.m1.3.4.1.cmml">=</mo><mrow id="A2.Ex8.m1.3.4.3" xref="A2.Ex8.m1.3.4.3.cmml"><mrow id="A2.Ex8.m1.3.4.3.2" xref="A2.Ex8.m1.3.4.3.2.cmml"><mo id="A2.Ex8.m1.3.4.3.2a" xref="A2.Ex8.m1.3.4.3.2.cmml">−</mo><mrow id="A2.Ex8.m1.3.4.3.2.2" xref="A2.Ex8.m1.3.4.3.2.2.cmml"><mstyle displaystyle="true" id="A2.Ex8.m1.1.1" xref="A2.Ex8.m1.1.1.cmml"><mfrac id="A2.Ex8.m1.1.1a" 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italic_p start_POSTSUBSCRIPT italic_ρ end_POSTSUBSCRIPT - caligraphic_H divide start_ARG ∂ end_ARG start_ARG ∂ italic_x start_POSTSUPERSCRIPT italic_μ end_POSTSUPERSCRIPT end_ARG divide start_ARG 1 end_ARG start_ARG roman_Ω start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( italic_x ) end_ARG</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_left_padright"></td> </tr></tbody> <tbody id="A2.Ex9"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_left_padleft"></td> <td class="ltx_td ltx_eqn_cell"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle=\frac{1}{\Omega^{2}(x)}\frac{dp_{\mu}}{dt}," class="ltx_Math" display="inline" id="A2.Ex9.m1.2"><semantics id="A2.Ex9.m1.2a"><mrow id="A2.Ex9.m1.2.2.1" xref="A2.Ex9.m1.2.2.1.1.cmml"><mrow id="A2.Ex9.m1.2.2.1.1" xref="A2.Ex9.m1.2.2.1.1.cmml"><mi id="A2.Ex9.m1.2.2.1.1.2" xref="A2.Ex9.m1.2.2.1.1.2.cmml"></mi><mo id="A2.Ex9.m1.2.2.1.1.1" 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xref="A2.Ex9.m1.2.2.1.1.3.2.2.3.3">𝜇</ci></apply></apply><apply id="A2.Ex9.m1.2.2.1.1.3.2.3.cmml" xref="A2.Ex9.m1.2.2.1.1.3.2.3"><times id="A2.Ex9.m1.2.2.1.1.3.2.3.1.cmml" xref="A2.Ex9.m1.2.2.1.1.3.2.3.1"></times><ci id="A2.Ex9.m1.2.2.1.1.3.2.3.2.cmml" xref="A2.Ex9.m1.2.2.1.1.3.2.3.2">𝑑</ci><ci id="A2.Ex9.m1.2.2.1.1.3.2.3.3.cmml" xref="A2.Ex9.m1.2.2.1.1.3.2.3.3">𝑡</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.Ex9.m1.2c">\displaystyle=\frac{1}{\Omega^{2}(x)}\frac{dp_{\mu}}{dt},</annotation><annotation encoding="application/x-llamapun" id="A2.Ex9.m1.2d">= divide start_ARG 1 end_ARG start_ARG roman_Ω start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( italic_x ) end_ARG divide start_ARG italic_d italic_p start_POSTSUBSCRIPT italic_μ end_POSTSUBSCRIPT end_ARG start_ARG italic_d italic_t end_ARG ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_left_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="A2.p1.6">where <math alttext="\tilde{t}" class="ltx_Math" display="inline" id="A2.p1.4.m1.1"><semantics id="A2.p1.4.m1.1a"><mover accent="true" id="A2.p1.4.m1.1.1" xref="A2.p1.4.m1.1.1.cmml"><mi id="A2.p1.4.m1.1.1.2" xref="A2.p1.4.m1.1.1.2.cmml">t</mi><mo id="A2.p1.4.m1.1.1.1" xref="A2.p1.4.m1.1.1.1.cmml">~</mo></mover><annotation-xml encoding="MathML-Content" id="A2.p1.4.m1.1b"><apply id="A2.p1.4.m1.1.1.cmml" xref="A2.p1.4.m1.1.1"><ci id="A2.p1.4.m1.1.1.1.cmml" xref="A2.p1.4.m1.1.1.1">~</ci><ci id="A2.p1.4.m1.1.1.2.cmml" xref="A2.p1.4.m1.1.1.2">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.p1.4.m1.1c">\tilde{t}</annotation><annotation encoding="application/x-llamapun" id="A2.p1.4.m1.1d">over~ start_ARG italic_t end_ARG</annotation></semantics></math> is the evolution parameter in the new <math alttext="\tilde{\mathcal{H}}" class="ltx_Math" display="inline" id="A2.p1.5.m2.1"><semantics id="A2.p1.5.m2.1a"><mover accent="true" id="A2.p1.5.m2.1.1" xref="A2.p1.5.m2.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="A2.p1.5.m2.1.1.2" xref="A2.p1.5.m2.1.1.2.cmml">ℋ</mi><mo id="A2.p1.5.m2.1.1.1" xref="A2.p1.5.m2.1.1.1.cmml">~</mo></mover><annotation-xml encoding="MathML-Content" id="A2.p1.5.m2.1b"><apply id="A2.p1.5.m2.1.1.cmml" xref="A2.p1.5.m2.1.1"><ci id="A2.p1.5.m2.1.1.1.cmml" xref="A2.p1.5.m2.1.1.1">~</ci><ci id="A2.p1.5.m2.1.1.2.cmml" xref="A2.p1.5.m2.1.1.2">ℋ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.p1.5.m2.1c">\tilde{\mathcal{H}}</annotation><annotation encoding="application/x-llamapun" id="A2.p1.5.m2.1d">over~ start_ARG caligraphic_H end_ARG</annotation></semantics></math> and we used <math alttext="\mathcal{H}=0" class="ltx_Math" display="inline" id="A2.p1.6.m3.1"><semantics id="A2.p1.6.m3.1a"><mrow id="A2.p1.6.m3.1.1" xref="A2.p1.6.m3.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="A2.p1.6.m3.1.1.2" xref="A2.p1.6.m3.1.1.2.cmml">ℋ</mi><mo id="A2.p1.6.m3.1.1.1" xref="A2.p1.6.m3.1.1.1.cmml">=</mo><mn id="A2.p1.6.m3.1.1.3" xref="A2.p1.6.m3.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="A2.p1.6.m3.1b"><apply id="A2.p1.6.m3.1.1.cmml" xref="A2.p1.6.m3.1.1"><eq id="A2.p1.6.m3.1.1.1.cmml" xref="A2.p1.6.m3.1.1.1"></eq><ci id="A2.p1.6.m3.1.1.2.cmml" xref="A2.p1.6.m3.1.1.2">ℋ</ci><cn id="A2.p1.6.m3.1.1.3.cmml" type="integer" xref="A2.p1.6.m3.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.p1.6.m3.1c">\mathcal{H}=0</annotation><annotation encoding="application/x-llamapun" id="A2.p1.6.m3.1d">caligraphic_H = 0</annotation></semantics></math> in the last step. From these relations, one sees that both Hamiltonians share the same null geodesics and changing the evolution parameter according to</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="S3.EGx104"> <tbody id="A2.E115"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_left_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle d\tilde{t}=\Omega^{2}(x)dt." class="ltx_Math" display="inline" id="A2.E115.m1.2"><semantics id="A2.E115.m1.2a"><mrow id="A2.E115.m1.2.2.1" xref="A2.E115.m1.2.2.1.1.cmml"><mrow id="A2.E115.m1.2.2.1.1" xref="A2.E115.m1.2.2.1.1.cmml"><mrow id="A2.E115.m1.2.2.1.1.2" xref="A2.E115.m1.2.2.1.1.2.cmml"><mi id="A2.E115.m1.2.2.1.1.2.2" xref="A2.E115.m1.2.2.1.1.2.2.cmml">d</mi><mo id="A2.E115.m1.2.2.1.1.2.1" xref="A2.E115.m1.2.2.1.1.2.1.cmml">⁢</mo><mover accent="true" id="A2.E115.m1.2.2.1.1.2.3" xref="A2.E115.m1.2.2.1.1.2.3.cmml"><mi id="A2.E115.m1.2.2.1.1.2.3.2" 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xref="A2.E115.m1.2.2.1.1.3"><times id="A2.E115.m1.2.2.1.1.3.1.cmml" xref="A2.E115.m1.2.2.1.1.3.1"></times><apply id="A2.E115.m1.2.2.1.1.3.2.cmml" xref="A2.E115.m1.2.2.1.1.3.2"><csymbol cd="ambiguous" id="A2.E115.m1.2.2.1.1.3.2.1.cmml" xref="A2.E115.m1.2.2.1.1.3.2">superscript</csymbol><ci id="A2.E115.m1.2.2.1.1.3.2.2.cmml" xref="A2.E115.m1.2.2.1.1.3.2.2">Ω</ci><cn id="A2.E115.m1.2.2.1.1.3.2.3.cmml" type="integer" xref="A2.E115.m1.2.2.1.1.3.2.3">2</cn></apply><ci id="A2.E115.m1.1.1.cmml" xref="A2.E115.m1.1.1">𝑥</ci><ci id="A2.E115.m1.2.2.1.1.3.4.cmml" xref="A2.E115.m1.2.2.1.1.3.4">𝑑</ci><ci id="A2.E115.m1.2.2.1.1.3.5.cmml" xref="A2.E115.m1.2.2.1.1.3.5">𝑡</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.E115.m1.2c">\displaystyle d\tilde{t}=\Omega^{2}(x)dt.</annotation><annotation encoding="application/x-llamapun" id="A2.E115.m1.2d">italic_d over~ start_ARG italic_t end_ARG = roman_Ω start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( italic_x ) italic_d italic_t .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_left_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(115)</span></td> </tr></tbody> </table> <p class="ltx_p" id="A2.p1.8">This maps the equations of motion for <math alttext="\tilde{\mathcal{H}}" class="ltx_Math" display="inline" id="A2.p1.7.m1.1"><semantics id="A2.p1.7.m1.1a"><mover accent="true" id="A2.p1.7.m1.1.1" xref="A2.p1.7.m1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="A2.p1.7.m1.1.1.2" xref="A2.p1.7.m1.1.1.2.cmml">ℋ</mi><mo id="A2.p1.7.m1.1.1.1" xref="A2.p1.7.m1.1.1.1.cmml">~</mo></mover><annotation-xml encoding="MathML-Content" id="A2.p1.7.m1.1b"><apply id="A2.p1.7.m1.1.1.cmml" xref="A2.p1.7.m1.1.1"><ci id="A2.p1.7.m1.1.1.1.cmml" xref="A2.p1.7.m1.1.1.1">~</ci><ci id="A2.p1.7.m1.1.1.2.cmml" xref="A2.p1.7.m1.1.1.2">ℋ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.p1.7.m1.1c">\tilde{\mathcal{H}}</annotation><annotation encoding="application/x-llamapun" id="A2.p1.7.m1.1d">over~ start_ARG caligraphic_H end_ARG</annotation></semantics></math> into those fo <math alttext="\mathcal{H}" class="ltx_Math" display="inline" id="A2.p1.8.m2.1"><semantics id="A2.p1.8.m2.1a"><mi class="ltx_font_mathcaligraphic" id="A2.p1.8.m2.1.1" xref="A2.p1.8.m2.1.1.cmml">ℋ</mi><annotation-xml encoding="MathML-Content" id="A2.p1.8.m2.1b"><ci id="A2.p1.8.m2.1.1.cmml" xref="A2.p1.8.m2.1.1">ℋ</ci></annotation-xml><annotation encoding="application/x-tex" id="A2.p1.8.m2.1c">\mathcal{H}</annotation><annotation encoding="application/x-llamapun" id="A2.p1.8.m2.1d">caligraphic_H</annotation></semantics></math>. In this way, the conformal rescaling (<a class="ltx_ref" href="https://arxiv.org/html/2406.11224v2#A2.E113" title="In Appendix B Conformal rescaling as reparametrization ‣ A Hamiltonian approach to the gradient-flow equations in information geometry"><span class="ltx_text ltx_ref_tag">113</span></a>) is functioning as the change of the evolution parameter (<a class="ltx_ref" href="https://arxiv.org/html/2406.11224v2#A2.E115" title="In Appendix B Conformal rescaling as reparametrization ‣ A Hamiltonian approach to the gradient-flow equations in information geometry"><span class="ltx_text ltx_ref_tag">115</span></a>).</p> </div> </section> <section class="ltx_appendix" id="A3"> <h2 class="ltx_title ltx_title_appendix"> <span class="ltx_tag ltx_tag_appendix">Appendix C </span>Zermelo form</h2> <div class="ltx_para" id="A3.p1"> <p class="ltx_p" id="A3.p1.1">The Zermelo navigation problem is a time-optimal control problem, which aims at finding the shortest-time path under the influence of a window vector <math alttext="W^{i}" class="ltx_Math" display="inline" id="A3.p1.1.m1.1"><semantics id="A3.p1.1.m1.1a"><msup id="A3.p1.1.m1.1.1" xref="A3.p1.1.m1.1.1.cmml"><mi id="A3.p1.1.m1.1.1.2" xref="A3.p1.1.m1.1.1.2.cmml">W</mi><mi id="A3.p1.1.m1.1.1.3" xref="A3.p1.1.m1.1.1.3.cmml">i</mi></msup><annotation-xml encoding="MathML-Content" id="A3.p1.1.m1.1b"><apply id="A3.p1.1.m1.1.1.cmml" xref="A3.p1.1.m1.1.1"><csymbol cd="ambiguous" id="A3.p1.1.m1.1.1.1.cmml" xref="A3.p1.1.m1.1.1">superscript</csymbol><ci id="A3.p1.1.m1.1.1.2.cmml" xref="A3.p1.1.m1.1.1.2">𝑊</ci><ci id="A3.p1.1.m1.1.1.3.cmml" xref="A3.p1.1.m1.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.p1.1.m1.1c">W^{i}</annotation><annotation encoding="application/x-llamapun" id="A3.p1.1.m1.1d">italic_W start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT</annotation></semantics></math>. We here show the explicit relations between the stationary metric (<a class="ltx_ref" href="https://arxiv.org/html/2406.11224v2#S4.E55" title="In 4 The motions of a light-like particle in a pseudo Riemann space ‣ A Hamiltonian approach to the gradient-flow equations in information geometry"><span class="ltx_text ltx_ref_tag">55</span></a>) and the Zermelo form:</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="S3.EGx105"> <tbody id="A3.Ex10"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_left_padleft"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle ds^{2}=\frac{V^{2}}{1-h_{ij}W^{i}W^{j}}" class="ltx_Math" display="inline" id="A3.Ex10.m1.1"><semantics id="A3.Ex10.m1.1a"><mrow id="A3.Ex10.m1.1.1" xref="A3.Ex10.m1.1.1.cmml"><mrow id="A3.Ex10.m1.1.1.2" xref="A3.Ex10.m1.1.1.2.cmml"><mi id="A3.Ex10.m1.1.1.2.2" xref="A3.Ex10.m1.1.1.2.2.cmml">d</mi><mo id="A3.Ex10.m1.1.1.2.1" 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xref="A3.E116.m1.1.1.1.1.1.1.1.2.2.1.1.3.2">superscript</csymbol><ci id="A3.E116.m1.1.1.1.1.1.1.1.2.2.1.1.3.2.2.cmml" xref="A3.E116.m1.1.1.1.1.1.1.1.2.2.1.1.3.2.2">𝑊</ci><ci id="A3.E116.m1.1.1.1.1.1.1.1.2.2.1.1.3.2.3.cmml" xref="A3.E116.m1.1.1.1.1.1.1.1.2.2.1.1.3.2.3">𝑗</ci></apply><ci id="A3.E116.m1.1.1.1.1.1.1.1.2.2.1.1.3.3.cmml" xref="A3.E116.m1.1.1.1.1.1.1.1.2.2.1.1.3.3">𝑑</ci><ci id="A3.E116.m1.1.1.1.1.1.1.1.2.2.1.1.3.4.cmml" xref="A3.E116.m1.1.1.1.1.1.1.1.2.2.1.1.3.4">𝑡</ci></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.E116.m1.1c">\displaystyle\times\left[-dt^{2}\!+\!h_{ij}(dx^{i}\!-\!W^{i}dt)(dx^{j}\!-\!W^{% j}dt)\right],</annotation><annotation encoding="application/x-llamapun" id="A3.E116.m1.1d">× [ - italic_d italic_t start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + italic_h start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT ( italic_d italic_x start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT - italic_W start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT italic_d italic_t ) ( italic_d italic_x start_POSTSUPERSCRIPT italic_j end_POSTSUPERSCRIPT - italic_W start_POSTSUPERSCRIPT italic_j end_POSTSUPERSCRIPT italic_d italic_t ) ] ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_left_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(116)</span></td> </tr></tbody> </table> <p class="ltx_p" id="A3.p1.2">which is Eq. (31) in Ref. <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2406.11224v2#bib.bib24" title="">24</a>]</cite>. By setting</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="S3.EGx106"> <tbody id="A3.Ex11"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_left_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle V^{2}" class="ltx_Math" display="inline" id="A3.Ex11.m1.1"><semantics id="A3.Ex11.m1.1a"><msup id="A3.Ex11.m1.1.1" xref="A3.Ex11.m1.1.1.cmml"><mi id="A3.Ex11.m1.1.1.2" xref="A3.Ex11.m1.1.1.2.cmml">V</mi><mn id="A3.Ex11.m1.1.1.3" xref="A3.Ex11.m1.1.1.3.cmml">2</mn></msup><annotation-xml encoding="MathML-Content" id="A3.Ex11.m1.1b"><apply id="A3.Ex11.m1.1.1.cmml" xref="A3.Ex11.m1.1.1"><csymbol cd="ambiguous" id="A3.Ex11.m1.1.1.1.cmml" xref="A3.Ex11.m1.1.1">superscript</csymbol><ci id="A3.Ex11.m1.1.1.2.cmml" xref="A3.Ex11.m1.1.1.2">𝑉</ci><cn id="A3.Ex11.m1.1.1.3.cmml" type="integer" xref="A3.Ex11.m1.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" 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end_POSTSUPERSCRIPT , italic_h start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT := over~ start_ARG italic_γ end_ARG start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT = divide start_ARG italic_γ start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT end_ARG start_ARG italic_α start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_left_padright"></td> </tr></tbody> <tbody id="A3.E117"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_left_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle W^{i}" class="ltx_Math" display="inline" id="A3.E117.m1.1"><semantics id="A3.E117.m1.1a"><msup id="A3.E117.m1.1.1" xref="A3.E117.m1.1.1.cmml"><mi id="A3.E117.m1.1.1.2" xref="A3.E117.m1.1.1.2.cmml">W</mi><mi id="A3.E117.m1.1.1.3" xref="A3.E117.m1.1.1.3.cmml">i</mi></msup><annotation-xml encoding="MathML-Content" id="A3.E117.m1.1b"><apply id="A3.E117.m1.1.1.cmml" xref="A3.E117.m1.1.1"><csymbol cd="ambiguous" id="A3.E117.m1.1.1.1.cmml" xref="A3.E117.m1.1.1">superscript</csymbol><ci id="A3.E117.m1.1.1.2.cmml" xref="A3.E117.m1.1.1.2">𝑊</ci><ci id="A3.E117.m1.1.1.3.cmml" xref="A3.E117.m1.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.E117.m1.1c">\displaystyle W^{i}</annotation><annotation encoding="application/x-llamapun" id="A3.E117.m1.1d">italic_W start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle:=-\beta^{i},\quad dt=dx^{0}," class="ltx_Math" display="inline" id="A3.E117.m2.1"><semantics id="A3.E117.m2.1a"><mrow id="A3.E117.m2.1.1.1"><mrow id="A3.E117.m2.1.1.1.1.2" xref="A3.E117.m2.1.1.1.1.3.cmml"><mrow id="A3.E117.m2.1.1.1.1.1.1" xref="A3.E117.m2.1.1.1.1.1.1.cmml"><mi id="A3.E117.m2.1.1.1.1.1.1.2" xref="A3.E117.m2.1.1.1.1.1.1.2.cmml"></mi><mo id="A3.E117.m2.1.1.1.1.1.1.1" lspace="0.278em" rspace="0.278em" xref="A3.E117.m2.1.1.1.1.1.1.1.cmml">:=</mo><mrow id="A3.E117.m2.1.1.1.1.1.1.3" xref="A3.E117.m2.1.1.1.1.1.1.3.cmml"><mo id="A3.E117.m2.1.1.1.1.1.1.3a" xref="A3.E117.m2.1.1.1.1.1.1.3.cmml">−</mo><msup id="A3.E117.m2.1.1.1.1.1.1.3.2" xref="A3.E117.m2.1.1.1.1.1.1.3.2.cmml"><mi id="A3.E117.m2.1.1.1.1.1.1.3.2.2" xref="A3.E117.m2.1.1.1.1.1.1.3.2.2.cmml">β</mi><mi id="A3.E117.m2.1.1.1.1.1.1.3.2.3" xref="A3.E117.m2.1.1.1.1.1.1.3.2.3.cmml">i</mi></msup></mrow></mrow><mo id="A3.E117.m2.1.1.1.1.2.3" rspace="1.167em" xref="A3.E117.m2.1.1.1.1.3a.cmml">,</mo><mrow id="A3.E117.m2.1.1.1.1.2.2" xref="A3.E117.m2.1.1.1.1.2.2.cmml"><mrow id="A3.E117.m2.1.1.1.1.2.2.2" xref="A3.E117.m2.1.1.1.1.2.2.2.cmml"><mi id="A3.E117.m2.1.1.1.1.2.2.2.2" xref="A3.E117.m2.1.1.1.1.2.2.2.2.cmml">d</mi><mo id="A3.E117.m2.1.1.1.1.2.2.2.1" xref="A3.E117.m2.1.1.1.1.2.2.2.1.cmml">⁢</mo><mi id="A3.E117.m2.1.1.1.1.2.2.2.3" xref="A3.E117.m2.1.1.1.1.2.2.2.3.cmml">t</mi></mrow><mo id="A3.E117.m2.1.1.1.1.2.2.1" xref="A3.E117.m2.1.1.1.1.2.2.1.cmml">=</mo><mrow id="A3.E117.m2.1.1.1.1.2.2.3" xref="A3.E117.m2.1.1.1.1.2.2.3.cmml"><mi id="A3.E117.m2.1.1.1.1.2.2.3.2" xref="A3.E117.m2.1.1.1.1.2.2.3.2.cmml">d</mi><mo id="A3.E117.m2.1.1.1.1.2.2.3.1" xref="A3.E117.m2.1.1.1.1.2.2.3.1.cmml">⁢</mo><msup id="A3.E117.m2.1.1.1.1.2.2.3.3" xref="A3.E117.m2.1.1.1.1.2.2.3.3.cmml"><mi id="A3.E117.m2.1.1.1.1.2.2.3.3.2" xref="A3.E117.m2.1.1.1.1.2.2.3.3.2.cmml">x</mi><mn id="A3.E117.m2.1.1.1.1.2.2.3.3.3" xref="A3.E117.m2.1.1.1.1.2.2.3.3.3.cmml">0</mn></msup></mrow></mrow></mrow><mo id="A3.E117.m2.1.1.1.2">,</mo></mrow><annotation-xml encoding="MathML-Content" id="A3.E117.m2.1b"><apply id="A3.E117.m2.1.1.1.1.3.cmml" xref="A3.E117.m2.1.1.1.1.2"><csymbol cd="ambiguous" id="A3.E117.m2.1.1.1.1.3a.cmml" xref="A3.E117.m2.1.1.1.1.2.3">formulae-sequence</csymbol><apply id="A3.E117.m2.1.1.1.1.1.1.cmml" xref="A3.E117.m2.1.1.1.1.1.1"><csymbol cd="latexml" id="A3.E117.m2.1.1.1.1.1.1.1.cmml" xref="A3.E117.m2.1.1.1.1.1.1.1">assign</csymbol><csymbol cd="latexml" id="A3.E117.m2.1.1.1.1.1.1.2.cmml" xref="A3.E117.m2.1.1.1.1.1.1.2">absent</csymbol><apply id="A3.E117.m2.1.1.1.1.1.1.3.cmml" xref="A3.E117.m2.1.1.1.1.1.1.3"><minus id="A3.E117.m2.1.1.1.1.1.1.3.1.cmml" xref="A3.E117.m2.1.1.1.1.1.1.3"></minus><apply id="A3.E117.m2.1.1.1.1.1.1.3.2.cmml" xref="A3.E117.m2.1.1.1.1.1.1.3.2"><csymbol cd="ambiguous" id="A3.E117.m2.1.1.1.1.1.1.3.2.1.cmml" xref="A3.E117.m2.1.1.1.1.1.1.3.2">superscript</csymbol><ci id="A3.E117.m2.1.1.1.1.1.1.3.2.2.cmml" xref="A3.E117.m2.1.1.1.1.1.1.3.2.2">𝛽</ci><ci id="A3.E117.m2.1.1.1.1.1.1.3.2.3.cmml" xref="A3.E117.m2.1.1.1.1.1.1.3.2.3">𝑖</ci></apply></apply></apply><apply id="A3.E117.m2.1.1.1.1.2.2.cmml" xref="A3.E117.m2.1.1.1.1.2.2"><eq id="A3.E117.m2.1.1.1.1.2.2.1.cmml" xref="A3.E117.m2.1.1.1.1.2.2.1"></eq><apply id="A3.E117.m2.1.1.1.1.2.2.2.cmml" xref="A3.E117.m2.1.1.1.1.2.2.2"><times id="A3.E117.m2.1.1.1.1.2.2.2.1.cmml" xref="A3.E117.m2.1.1.1.1.2.2.2.1"></times><ci id="A3.E117.m2.1.1.1.1.2.2.2.2.cmml" xref="A3.E117.m2.1.1.1.1.2.2.2.2">𝑑</ci><ci id="A3.E117.m2.1.1.1.1.2.2.2.3.cmml" xref="A3.E117.m2.1.1.1.1.2.2.2.3">𝑡</ci></apply><apply id="A3.E117.m2.1.1.1.1.2.2.3.cmml" xref="A3.E117.m2.1.1.1.1.2.2.3"><times id="A3.E117.m2.1.1.1.1.2.2.3.1.cmml" xref="A3.E117.m2.1.1.1.1.2.2.3.1"></times><ci id="A3.E117.m2.1.1.1.1.2.2.3.2.cmml" xref="A3.E117.m2.1.1.1.1.2.2.3.2">𝑑</ci><apply id="A3.E117.m2.1.1.1.1.2.2.3.3.cmml" xref="A3.E117.m2.1.1.1.1.2.2.3.3"><csymbol cd="ambiguous" id="A3.E117.m2.1.1.1.1.2.2.3.3.1.cmml" xref="A3.E117.m2.1.1.1.1.2.2.3.3">superscript</csymbol><ci id="A3.E117.m2.1.1.1.1.2.2.3.3.2.cmml" xref="A3.E117.m2.1.1.1.1.2.2.3.3.2">𝑥</ci><cn id="A3.E117.m2.1.1.1.1.2.2.3.3.3.cmml" type="integer" xref="A3.E117.m2.1.1.1.1.2.2.3.3.3">0</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.E117.m2.1c">\displaystyle:=-\beta^{i},\quad dt=dx^{0},</annotation><annotation encoding="application/x-llamapun" id="A3.E117.m2.1d">:= - italic_β start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT , italic_d italic_t = italic_d italic_x start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_left_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(117)</span></td> </tr></tbody> </table> <p class="ltx_p" id="A3.p1.3">we have</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="S3.EGx107"> <tbody id="A3.E118"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_left_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\frac{V^{2}}{1-h_{ij}W^{i}W^{j}}=\frac{\alpha^{2}-\gamma_{ij}% \beta^{i}\beta^{j}}{1-\frac{\gamma_{ij}}{\alpha^{2}}\beta^{i}\beta^{j}}=\alpha% ^{2}." class="ltx_Math" display="inline" id="A3.E118.m1.1"><semantics id="A3.E118.m1.1a"><mrow id="A3.E118.m1.1.1.1" xref="A3.E118.m1.1.1.1.1.cmml"><mrow id="A3.E118.m1.1.1.1.1" xref="A3.E118.m1.1.1.1.1.cmml"><mstyle displaystyle="true" id="A3.E118.m1.1.1.1.1.2" xref="A3.E118.m1.1.1.1.1.2.cmml"><mfrac id="A3.E118.m1.1.1.1.1.2a" xref="A3.E118.m1.1.1.1.1.2.cmml"><msup id="A3.E118.m1.1.1.1.1.2.2" xref="A3.E118.m1.1.1.1.1.2.2.cmml"><mi id="A3.E118.m1.1.1.1.1.2.2.2" xref="A3.E118.m1.1.1.1.1.2.2.2.cmml">V</mi><mn id="A3.E118.m1.1.1.1.1.2.2.3" xref="A3.E118.m1.1.1.1.1.2.2.3.cmml">2</mn></msup><mrow id="A3.E118.m1.1.1.1.1.2.3" xref="A3.E118.m1.1.1.1.1.2.3.cmml"><mn id="A3.E118.m1.1.1.1.1.2.3.2" xref="A3.E118.m1.1.1.1.1.2.3.2.cmml">1</mn><mo id="A3.E118.m1.1.1.1.1.2.3.1" xref="A3.E118.m1.1.1.1.1.2.3.1.cmml">−</mo><mrow id="A3.E118.m1.1.1.1.1.2.3.3" xref="A3.E118.m1.1.1.1.1.2.3.3.cmml"><msub 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xref="A3.E118.m1.1.1.1.1.2.3.3.4.cmml"><mi id="A3.E118.m1.1.1.1.1.2.3.3.4.2" xref="A3.E118.m1.1.1.1.1.2.3.3.4.2.cmml">W</mi><mi id="A3.E118.m1.1.1.1.1.2.3.3.4.3" xref="A3.E118.m1.1.1.1.1.2.3.3.4.3.cmml">j</mi></msup></mrow></mrow></mfrac></mstyle><mo id="A3.E118.m1.1.1.1.1.3" xref="A3.E118.m1.1.1.1.1.3.cmml">=</mo><mstyle displaystyle="true" id="A3.E118.m1.1.1.1.1.4" xref="A3.E118.m1.1.1.1.1.4.cmml"><mfrac id="A3.E118.m1.1.1.1.1.4a" xref="A3.E118.m1.1.1.1.1.4.cmml"><mrow id="A3.E118.m1.1.1.1.1.4.2" xref="A3.E118.m1.1.1.1.1.4.2.cmml"><msup id="A3.E118.m1.1.1.1.1.4.2.2" xref="A3.E118.m1.1.1.1.1.4.2.2.cmml"><mi id="A3.E118.m1.1.1.1.1.4.2.2.2" xref="A3.E118.m1.1.1.1.1.4.2.2.2.cmml">α</mi><mn id="A3.E118.m1.1.1.1.1.4.2.2.3" xref="A3.E118.m1.1.1.1.1.4.2.2.3.cmml">2</mn></msup><mo id="A3.E118.m1.1.1.1.1.4.2.1" xref="A3.E118.m1.1.1.1.1.4.2.1.cmml">−</mo><mrow id="A3.E118.m1.1.1.1.1.4.2.3" xref="A3.E118.m1.1.1.1.1.4.2.3.cmml"><msub id="A3.E118.m1.1.1.1.1.4.2.3.2" 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xref="A3.E118.m1.1.1.1.1.4.3.3.4.3">𝑗</ci></apply></apply></apply></apply></apply><apply id="A3.E118.m1.1.1.1.1c.cmml" xref="A3.E118.m1.1.1.1"><eq id="A3.E118.m1.1.1.1.1.5.cmml" xref="A3.E118.m1.1.1.1.1.5"></eq><share href="https://arxiv.org/html/2406.11224v2#A3.E118.m1.1.1.1.1.4.cmml" id="A3.E118.m1.1.1.1.1d.cmml" xref="A3.E118.m1.1.1.1"></share><apply id="A3.E118.m1.1.1.1.1.6.cmml" xref="A3.E118.m1.1.1.1.1.6"><csymbol cd="ambiguous" id="A3.E118.m1.1.1.1.1.6.1.cmml" xref="A3.E118.m1.1.1.1.1.6">superscript</csymbol><ci id="A3.E118.m1.1.1.1.1.6.2.cmml" xref="A3.E118.m1.1.1.1.1.6.2">𝛼</ci><cn id="A3.E118.m1.1.1.1.1.6.3.cmml" type="integer" xref="A3.E118.m1.1.1.1.1.6.3">2</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.E118.m1.1c">\displaystyle\frac{V^{2}}{1-h_{ij}W^{i}W^{j}}=\frac{\alpha^{2}-\gamma_{ij}% \beta^{i}\beta^{j}}{1-\frac{\gamma_{ij}}{\alpha^{2}}\beta^{i}\beta^{j}}=\alpha% ^{2}.</annotation><annotation encoding="application/x-llamapun" id="A3.E118.m1.1d">divide start_ARG italic_V start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG 1 - italic_h start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT italic_W start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT italic_W start_POSTSUPERSCRIPT italic_j end_POSTSUPERSCRIPT end_ARG = divide start_ARG italic_α start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT - italic_γ start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT italic_β start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT italic_β start_POSTSUPERSCRIPT italic_j end_POSTSUPERSCRIPT end_ARG start_ARG 1 - divide start_ARG italic_γ start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT end_ARG start_ARG italic_α start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG italic_β start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT italic_β start_POSTSUPERSCRIPT italic_j end_POSTSUPERSCRIPT end_ARG = italic_α start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_left_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(118)</span></td> </tr></tbody> </table> <p class="ltx_p" id="A3.p1.4">Then the Zermelo form (<a class="ltx_ref" href="https://arxiv.org/html/2406.11224v2#A3.E116" title="In Appendix C Zermelo form ‣ A Hamiltonian approach to the gradient-flow equations in information geometry"><span class="ltx_text ltx_ref_tag">116</span></a>) becomes (<a class="ltx_ref" href="https://arxiv.org/html/2406.11224v2#S4.E55" title="In 4 The motions of a light-like particle in a pseudo Riemann space ‣ A Hamiltonian approach to the gradient-flow equations in information geometry"><span class="ltx_text ltx_ref_tag">55</span></a>).</p> </div> </section> <section class="ltx_bibliography" id="bib"> <h2 class="ltx_title ltx_title_bibliography">References</h2> <ul class="ltx_biblist"> <li class="ltx_bibitem" id="bib.bib1"> <span class="ltx_tag ltx_role_refnum ltx_tag_bibitem">[1]</span> <span class="ltx_bibblock"> S-I. Amari, <span class="ltx_text ltx_font_italic" id="bib.bib1.1.1">Information geometry and its applications</span>, <span class="ltx_text ltx_font_italic" id="bib.bib1.2.2">Appl. Math. Sci.</span> <span class="ltx_text ltx_font_bold" id="bib.bib1.3.3">194</span> (Springer, Tokyo, 2016) </span> </li> <li class="ltx_bibitem" id="bib.bib2"> <span class="ltx_tag ltx_role_refnum ltx_tag_bibitem">[2]</span> <span class="ltx_bibblock"> T. Wada, A.M. Scarfone, Information geometry on the <math alttext="\kappa" class="ltx_Math" display="inline" id="bib.bib2.1.m1.1"><semantics id="bib.bib2.1.m1.1a"><mi id="bib.bib2.1.m1.1.1" xref="bib.bib2.1.m1.1.1.cmml">κ</mi><annotation-xml encoding="MathML-Content" id="bib.bib2.1.m1.1b"><ci id="bib.bib2.1.m1.1.1.cmml" xref="bib.bib2.1.m1.1.1">𝜅</ci></annotation-xml><annotation encoding="application/x-tex" id="bib.bib2.1.m1.1c">\kappa</annotation><annotation encoding="application/x-llamapun" id="bib.bib2.1.m1.1d">italic_κ</annotation></semantics></math>-thermostatistics. Entropy <span class="ltx_text ltx_font_bold" id="bib.bib2.2.1">17</span>, 1204-1217 (2015) </span> </li> <li class="ltx_bibitem" id="bib.bib3"> <span class="ltx_tag ltx_role_refnum ltx_tag_bibitem">[3]</span> <span class="ltx_bibblock"> Y. Nakamura, Gradient systems associated with probability distributions. <span class="ltx_text ltx_font_italic" id="bib.bib3.1.1">Japan J. Indust. Appl. Math.</span> <span class="ltx_text ltx_font_bold" id="bib.bib3.2.2">11</span> 21-30 (1994) </span> </li> <li class="ltx_bibitem" id="bib.bib4"> <span class="ltx_tag ltx_role_refnum ltx_tag_bibitem">[4]</span> <span class="ltx_bibblock"> A. Fujiwara, S-I. Amari, Gradient systems in view of information geometry. <span class="ltx_text ltx_font_italic" id="bib.bib4.1.1">Physica</span> D <span class="ltx_text ltx_font_bold" id="bib.bib4.2.2">80</span> 317 (1995) </span> </li> <li class="ltx_bibitem" id="bib.bib5"> <span class="ltx_tag ltx_role_refnum ltx_tag_bibitem">[5]</span> <span class="ltx_bibblock"> Malagó L. and Pistone G., Natural Gradient Flow in the Mixture Geometry of a Discrete Exponential Family. <span class="ltx_text ltx_font_italic" id="bib.bib5.1.1">Entropy</span> <span class="ltx_text ltx_font_bold" id="bib.bib5.2.2">17</span> 4215–4254 (2015) </span> </li> <li class="ltx_bibitem" id="bib.bib6"> <span class="ltx_tag ltx_role_refnum ltx_tag_bibitem">[6]</span> <span class="ltx_bibblock"> N. Boumuki, T. Noda, On gradient and Hamiltonian flows on even dimensional dually flat spaces. <span class="ltx_text ltx_font_italic" id="bib.bib6.1.1">Fundamental J. Math. and Math Sci. </span> <span class="ltx_text ltx_font_bold" id="bib.bib6.2.2">6</span> 51-66 (2016) </span> </li> <li class="ltx_bibitem" id="bib.bib7"> <span class="ltx_tag ltx_role_refnum ltx_tag_bibitem">[7]</span> <span class="ltx_bibblock"> G. Chirco, L. Malagó, G. Pistone, Lagrangian and Hamiltonian Mechanics for Probabilities on the Statistical Manifold. arXiv:2009.09431v2 </span> </li> <li class="ltx_bibitem" id="bib.bib8"> <span class="ltx_tag ltx_role_refnum ltx_tag_bibitem">[8]</span> <span class="ltx_bibblock"> G. Pistone, Lagrangian Function on the Finite State Space Statistical Bundle. <span class="ltx_text ltx_font_italic" id="bib.bib8.1.1">Entropy</span>, <span class="ltx_text ltx_font_bold" id="bib.bib8.2.2">20</span>, 139 (2018) <br class="ltx_break"/><a class="ltx_ref ltx_url ltx_font_typewriter" href="https://doi.org/10.3390/e20020139" title="">https://doi.org/10.3390/e20020139</a> </span> </li> <li class="ltx_bibitem" id="bib.bib9"> <span class="ltx_tag ltx_role_refnum ltx_tag_bibitem">[9]</span> <span class="ltx_bibblock"> T. Wada, A.M. Scarfone, H. Matsuzoe, An eikonal equation approach to thermodynamics and the gradient flows in information geometry. <span class="ltx_text ltx_font_italic" id="bib.bib9.1.1">Physica</span> A, <span class="ltx_text ltx_font_bold" id="bib.bib9.2.2">570</span> 125820 (2021) </span> </li> <li class="ltx_bibitem" id="bib.bib10"> <span class="ltx_tag ltx_role_refnum ltx_tag_bibitem">[10]</span> <span class="ltx_bibblock"> T. Wada, A.M. Scarfone, H. Matsuzoe, Huygens’ equations and the gradient-flow equations in information geometry. <span class="ltx_text ltx_font_italic" id="bib.bib10.1.1"> Int. J. Geom. Methods Mod. Phys.</span>, <span class="ltx_text ltx_font_bold" id="bib.bib10.2.2">20</span> 2450012 (2023) </span> </li> <li class="ltx_bibitem" id="bib.bib11"> <span class="ltx_tag ltx_role_refnum ltx_tag_bibitem">[11]</span> <span class="ltx_bibblock"> S. Chanda, T. Wada, Mechanics of geodesics in information geometry and black hole thermodynamics. <span class="ltx_text ltx_font_italic" id="bib.bib11.1.1"> Int. J. Geom. Methods Mod. Phys.</span>, <span class="ltx_text ltx_font_bold" id="bib.bib11.2.2"> 21</span> 2450098 (2024) </span> </li> <li class="ltx_bibitem" id="bib.bib12"> <span class="ltx_tag ltx_role_refnum ltx_tag_bibitem">[12]</span> <span class="ltx_bibblock"> G. Randers, On an asymmetrical metric in the four-space of general relativity. <span class="ltx_text ltx_font_italic" id="bib.bib12.1.1">Phys. Rev. </span> <span class="ltx_text ltx_font_bold" id="bib.bib12.2.2">59</span> 195 (1941) </span> </li> <li class="ltx_bibitem" id="bib.bib13"> <span class="ltx_tag ltx_role_refnum ltx_tag_bibitem">[13]</span> <span class="ltx_bibblock"> T. Wada, Weyl geometric approach to the gradient-flow equations in information geometry. <span class="ltx_text ltx_font_italic" id="bib.bib13.1.1">J. Geom. Symmetry Phys.</span>, <span class="ltx_text ltx_font_bold" id="bib.bib13.2.2">66</span> 59-70 (2023) </span> </li> <li class="ltx_bibitem" id="bib.bib14"> <span class="ltx_tag ltx_role_refnum ltx_tag_bibitem">[14]</span> <span class="ltx_bibblock"> M. Blau, <span class="ltx_text ltx_font_italic" id="bib.bib14.1.1">Lecture Notes on General Relativity <br class="ltx_break"/><a class="ltx_ref ltx_url ltx_font_typewriter ltx_font_upright" href="http://www.blau.itp.unibe.ch/GRLecturenotes.html" title="">http://www.blau.itp.unibe.ch/GRLecturenotes.html</a></span> </span> </li> <li class="ltx_bibitem" id="bib.bib15"> <span class="ltx_tag ltx_role_refnum ltx_tag_bibitem">[15]</span> <span class="ltx_bibblock"> Y. Gu, Natural coordinate system in curved space-time. <span class="ltx_text ltx_font_italic" id="bib.bib15.1.1">J. Geom. Symmetry Phys. </span><span class="ltx_text ltx_font_bold" id="bib.bib15.2.2">47</span> 51-62 (2018) </span> </li> <li class="ltx_bibitem" id="bib.bib16"> <span class="ltx_tag ltx_role_refnum ltx_tag_bibitem">[16]</span> <span class="ltx_bibblock"> V.P. Frolov, Generalized Fermat’s principle and action for light rays in a curved spacetime. <span class="ltx_text ltx_font_italic" id="bib.bib16.1.1">Phys. Rev. D</span> <span class="ltx_text ltx_font_bold" id="bib.bib16.2.2">88</span> 064039 (2013) </span> </li> <li class="ltx_bibitem" id="bib.bib17"> <span class="ltx_tag ltx_role_refnum ltx_tag_bibitem">[17]</span> <span class="ltx_bibblock"> E. Cartan, Chap. X, <span class="ltx_text ltx_font_italic" id="bib.bib17.1.1">Lessons on integral invariants</span> translated by D.H. Delphenich (Hermann and Co., Paris, 1922) </span> </li> <li class="ltx_bibitem" id="bib.bib18"> <span class="ltx_tag ltx_role_refnum ltx_tag_bibitem">[18]</span> <span class="ltx_bibblock"> V.I. Arnold, <span class="ltx_text ltx_font_italic" id="bib.bib18.1.1">Mathematical methods of classical mechanics</span>, 2nd edition, (Springer-Verlag, New York, 1989) </span> </li> <li class="ltx_bibitem" id="bib.bib19"> <span class="ltx_tag ltx_role_refnum ltx_tag_bibitem">[19]</span> <span class="ltx_bibblock"> D. Drosdoff, A. Widom, Snell’s law from an elementary particle viewpoint. <span class="ltx_text ltx_font_italic" id="bib.bib19.1.1">Am. J. Phys.</span> <span class="ltx_text ltx_font_bold" id="bib.bib19.2.2">73</span> 973-975 (2005) </span> </li> <li class="ltx_bibitem" id="bib.bib20"> <span class="ltx_tag ltx_role_refnum ltx_tag_bibitem">[20]</span> <span class="ltx_bibblock"> D.D. Holm, <span class="ltx_text ltx_font_italic" id="bib.bib20.1.1">Fermat’s principle and the geometric mechanics of ray optics</span>. Summer School Lectures, Fields Institute, Toronto. (2012) </span> </li> <li class="ltx_bibitem" id="bib.bib21"> <span class="ltx_tag ltx_role_refnum ltx_tag_bibitem">[21]</span> <span class="ltx_bibblock"> R. Arnowitt, S. Deser, C.W. Misner, Dynamical structure and definition of energy in general relativity. <span class="ltx_text ltx_font_italic" id="bib.bib21.1.1">Phys. Rev.</span> <span class="ltx_text ltx_font_bold" id="bib.bib21.2.2">116</span>, 1322 (1959) </span> </li> <li class="ltx_bibitem" id="bib.bib22"> <span class="ltx_tag ltx_role_refnum ltx_tag_bibitem">[22]</span> <span class="ltx_bibblock"> S.A. Caveny, M. Anderson, R.A. Matzner, Tracking black holes in numerical relativity. <span class="ltx_text ltx_font_italic" id="bib.bib22.1.1">Phys. Rev. D</span> <span class="ltx_text ltx_font_bold" id="bib.bib22.2.2">68</span> 104009 (2003) </span> </li> <li class="ltx_bibitem" id="bib.bib23"> <span class="ltx_tag ltx_role_refnum ltx_tag_bibitem">[23]</span> <span class="ltx_bibblock"> W. Belayev, Application of Lagrange mechanics for analysis of the light-like particle motion in pseudo-Riemann space. <span class="ltx_text ltx_font_italic" id="bib.bib23.1.1">Int. J. Theor. Math. Phys.</span> <span class="ltx_text ltx_font_bold" id="bib.bib23.2.2">2</span> 10-15 (2012) </span> </li> <li class="ltx_bibitem" id="bib.bib24"> <span class="ltx_tag ltx_role_refnum ltx_tag_bibitem">[24]</span> <span class="ltx_bibblock"> G.W. Gibbons, C.A.R. Herdeiro, C.M. Warnick, M.C. Werner, Stationary metrics and optical Zermelo-Randers-Finsler geometry. <span class="ltx_text ltx_font_italic" id="bib.bib24.1.1">Phys. Rev. D</span> <span class="ltx_text ltx_font_bold" id="bib.bib24.2.2">79</span> 044022 (2009) </span> </li> <li class="ltx_bibitem" id="bib.bib25"> <span class="ltx_tag ltx_role_refnum ltx_tag_bibitem">[25]</span> <span class="ltx_bibblock"> M. Cariglia, Null lifts and projective dynamics. <span class="ltx_text ltx_font_italic" id="bib.bib25.1.1">Annals of Physics</span> <span class="ltx_text ltx_font_bold" id="bib.bib25.2.2">362</span> 642-658 (2015) </span> </li> <li class="ltx_bibitem" id="bib.bib26"> <span class="ltx_tag ltx_role_refnum ltx_tag_bibitem">[26]</span> <span class="ltx_bibblock"> S. Katagiri, Non-equilibrium thermodynamics as gauge fixing. <span class="ltx_text ltx_font_italic" id="bib.bib26.1.1">Prog. Theor. Exp. Phys.</span>, <span class="ltx_text ltx_font_bold" id="bib.bib26.2.2">2018</span>, issue 9 (2018) </span> </li> </ul> </section> </article> </div> <footer class="ltx_page_footer"> <div class="ltx_page_logo">Generated on Wed Jul 17 04:25:37 2024 by <a class="ltx_LaTeXML_logo" href="http://dlmf.nist.gov/LaTeXML/"><span style="letter-spacing:-0.2em; margin-right:0.1em;">L<span class="ltx_font_smallcaps" style="position:relative; bottom:2.2pt;">a</span>T<span class="ltx_font_smallcaps" style="font-size:120%;position:relative; bottom:-0.2ex;">e</span></span><span style="font-size:90%; position:relative; bottom:-0.2ex;">XML</span><img alt="Mascot Sammy" src="data:image/png;base64,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"/></a> </div></footer> </div> </body> </html>