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<button class="button is-small is-link">Go</button> </div> </div> </form> </div> </div> <ol class="breathe-horizontal" start="1"> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2404.11997">arXiv:2404.11997</a> <span> [<a href="https://arxiv.org/pdf/2404.11997">pdf</a>, <a href="https://arxiv.org/format/2404.11997">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Differential Geometry">math.DG</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Mathematical Physics">math-ph</span> </div> </div> <p class="title is-5 mathjax"> Geodesic extensions of mechanical systems with nonholonomic constraints </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math-ph?searchtype=author&query=Belrhazi%2C+M">Malika Belrhazi</a>, <a href="/search/math-ph?searchtype=author&query=Mestdag%2C+T">Tom Mestdag</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2404.11997v2-abstract-short" style="display: inline;"> For a Lagrangian system with nonholonomic constraints, we construct extensions of the equations of motion to sets of second-order ordinary differential equations. In the case of a purely kinetic Lagrangian, we investigate the conditions under which the nonholonomic trajectories are geodesics of a Riemannian metric, while preserving the constrained Lagrangian. We interpret the algebraic and PDE con… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2404.11997v2-abstract-full').style.display = 'inline'; document.getElementById('2404.11997v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2404.11997v2-abstract-full" style="display: none;"> For a Lagrangian system with nonholonomic constraints, we construct extensions of the equations of motion to sets of second-order ordinary differential equations. In the case of a purely kinetic Lagrangian, we investigate the conditions under which the nonholonomic trajectories are geodesics of a Riemannian metric, while preserving the constrained Lagrangian. We interpret the algebraic and PDE conditions of this problem as infinitesimal versions of the relation between the nonholonomic exponential map and the Riemannian metric. We discuss the special case of a Chaplygin system with symmetries and we end the paper with a worked-out example. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2404.11997v2-abstract-full').style.display = 'none'; document.getElementById('2404.11997v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 8 February, 2025; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 18 April, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> April 2024. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">to appear in J Nonlinear Science</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 37J06; 37J60; 53Z05; 70G45; 70G65 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2303.16037">arXiv:2303.16037</a> <span> [<a href="https://arxiv.org/pdf/2303.16037">pdf</a>, <a href="https://arxiv.org/ps/2303.16037">ps</a>, <a href="https://arxiv.org/format/2303.16037">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Mathematical Physics">math-ph</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Differential Geometry">math.DG</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1088/1751-8121/ace74c">10.1088/1751-8121/ace74c <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Conditions for symmetry reduction of polysymplectic and polycosymplectic structures </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math-ph?searchtype=author&query=Andr%C3%A9s%2C+E+G">Eduardo Garc铆a-Tora帽o Andr茅s</a>, <a href="/search/math-ph?searchtype=author&query=Mestdag%2C+T">Tom Mestdag</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2303.16037v2-abstract-short" style="display: inline;"> For Hamiltonian field theories on polysymplectic manifolds with a symmetry group action and a momentum map, we explore the redundancy in a set of necessary conditions that has appeared in the literature, for a generalized version of the Marsden-Weinstein symmetry reduction theorem. Next, we prove a necessary and sufficient condition for polycosymplectic reduction. We relate polycosymplectic reduct… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2303.16037v2-abstract-full').style.display = 'inline'; document.getElementById('2303.16037v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2303.16037v2-abstract-full" style="display: none;"> For Hamiltonian field theories on polysymplectic manifolds with a symmetry group action and a momentum map, we explore the redundancy in a set of necessary conditions that has appeared in the literature, for a generalized version of the Marsden-Weinstein symmetry reduction theorem. Next, we prove a necessary and sufficient condition for polycosymplectic reduction. We relate polycosymplectic reduction in a one-to-one way to the reduction of an associated larger polysymplectic manifold. Throughout the paper, we provide examples and discuss special cases. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2303.16037v2-abstract-full').style.display = 'none'; document.getElementById('2303.16037v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 29 July, 2023; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 28 March, 2023; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> March 2023. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> J. Phys. A: Math. Theor. 56 (2023) 335202 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2205.00309">arXiv:2205.00309</a> <span> [<a href="https://arxiv.org/pdf/2205.00309">pdf</a>, <a href="https://arxiv.org/ps/2205.00309">ps</a>, <a href="https://arxiv.org/format/2205.00309">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Mathematical Physics">math-ph</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Differential Geometry">math.DG</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1088/1751-8121/ac91b3">10.1088/1751-8121/ac91b3 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Cotangent bundle reduction and Routh reduction for polysymplectic manifolds </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math-ph?searchtype=author&query=Capriotti%2C+S">Santiago Capriotti</a>, <a href="/search/math-ph?searchtype=author&query=D%C3%ADaz%2C+V+A">Viviana Alejandra D铆az</a>, <a href="/search/math-ph?searchtype=author&query=Andr%C3%A9s%2C+E+G">Eduardo Garc铆a-Tora帽o Andr茅s</a>, <a href="/search/math-ph?searchtype=author&query=Mestdag%2C+T">Tom Mestdag</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2205.00309v2-abstract-short" style="display: inline;"> We discuss Lagrangian and Hamiltonian field theories that are invariant under a symmetry group. We apply the polysymplectic reduction theorem for both types of field equations and we investigate aspects of the corresponding reconstruction process. We identify the polysymplectic structures that lie at the basis of cotangent bundle reduction and Routh reduction in this setting and we relate them by… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2205.00309v2-abstract-full').style.display = 'inline'; document.getElementById('2205.00309v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2205.00309v2-abstract-full" style="display: none;"> We discuss Lagrangian and Hamiltonian field theories that are invariant under a symmetry group. We apply the polysymplectic reduction theorem for both types of field equations and we investigate aspects of the corresponding reconstruction process. We identify the polysymplectic structures that lie at the basis of cotangent bundle reduction and Routh reduction in this setting and we relate them by means of the Routhian function and its associated Legendre transformation. We end the paper with examples that illustrate the applicability of our results. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2205.00309v2-abstract-full').style.display = 'none'; document.getElementById('2205.00309v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 5 October, 2022; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 30 April, 2022; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> May 2022. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> J. Phys. A: Math. Theor. 55 (2022) 415401 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2111.04411">arXiv:2111.04411</a> <span> [<a href="https://arxiv.org/pdf/2111.04411">pdf</a>, <a href="https://arxiv.org/format/2111.04411">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Differential Geometry">math.DG</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Mathematical Physics">math-ph</span> </div> </div> <p class="title is-5 mathjax"> Homogeneous nonlinear splittings and Finsler submersions </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math-ph?searchtype=author&query=Hajdu%2C+S">S. Hajdu</a>, <a href="/search/math-ph?searchtype=author&query=Mestdag%2C+T">T. Mestdag</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2111.04411v2-abstract-short" style="display: inline;"> A nonlinear splitting on a fibre bundle is a generalization of an Ehresmann connection. An example is given by the homogeneous nonlinear splitting of a Finsler function on the total manifold of a fibre bundle. We show how homogeneous nonlinear splittings and nonlinear lifts can be used to construct submersions between Euclidean, Minkowski and Finsler spaces. As an application we consider a semisim… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2111.04411v2-abstract-full').style.display = 'inline'; document.getElementById('2111.04411v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2111.04411v2-abstract-full" style="display: none;"> A nonlinear splitting on a fibre bundle is a generalization of an Ehresmann connection. An example is given by the homogeneous nonlinear splitting of a Finsler function on the total manifold of a fibre bundle. We show how homogeneous nonlinear splittings and nonlinear lifts can be used to construct submersions between Euclidean, Minkowski and Finsler spaces. As an application we consider a semisimple Lie algebra and use our methods to give new examples of Finsler functions on a reductive homogeneous space. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2111.04411v2-abstract-full').style.display = 'none'; document.getElementById('2111.04411v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 5 December, 2022; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 8 November, 2021; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> November 2021. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">24 pages</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 53B40; 53C60; 53C05; 53C30 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2107.14192">arXiv:2107.14192</a> <span> [<a href="https://arxiv.org/pdf/2107.14192">pdf</a>, <a href="https://arxiv.org/ps/2107.14192">ps</a>, <a href="https://arxiv.org/format/2107.14192">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Differential Geometry">math.DG</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Mathematical Physics">math-ph</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.3934/jgm.2021019">10.3934/jgm.2021019 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Compatibility aspects of the method of phase synchronization for decoupling linear second-order differential equations </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math-ph?searchtype=author&query=Sarlet%2C+W">W. Sarlet</a>, <a href="/search/math-ph?searchtype=author&query=Mestdag%2C+T">T. Mestdag</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2107.14192v1-abstract-short" style="display: inline;"> The so-called method of phase synchronization has been advocated in a number of papers as a way of decoupling a system of linear second-order differential equations by a linear transformation of coordinates and velocities. This is a rather unusual approach because velocity-dependent transformations in general do not preserve the second-order character of differential equations. Moreover, at least… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2107.14192v1-abstract-full').style.display = 'inline'; document.getElementById('2107.14192v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2107.14192v1-abstract-full" style="display: none;"> The so-called method of phase synchronization has been advocated in a number of papers as a way of decoupling a system of linear second-order differential equations by a linear transformation of coordinates and velocities. This is a rather unusual approach because velocity-dependent transformations in general do not preserve the second-order character of differential equations. Moreover, at least in the case of linear transformations, such a velocity-dependent one defines by itself a second-order system, which need not have anything to do, in principle, with the given system or its reformulation. This aspect, and the related questions of compatibility it raises, seem to have been overlooked in the existing literature. The purpose of this paper is to clarify this issue and to suggest topics for further research in conjunction with the general theory of decoupling in a differential geometric context. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2107.14192v1-abstract-full').style.display = 'none'; document.getElementById('2107.14192v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 29 July, 2021; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> July 2021. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">17 pages, to appear in J Geometric Mechanics</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Journal of Geometric Mechanics 14 (2021), 91-104 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2107.00428">arXiv:2107.00428</a> <span> [<a href="https://arxiv.org/pdf/2107.00428">pdf</a>, <a href="https://arxiv.org/ps/2107.00428">ps</a>, <a href="https://arxiv.org/format/2107.00428">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Differential Geometry">math.DG</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Mathematical Physics">math-ph</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1007/s13324-021-00622-0">10.1007/s13324-021-00622-0 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Nonlinear splittings on fibre bundles </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math-ph?searchtype=author&query=Hajd%C3%BA%2C+S">S. Hajd煤</a>, <a href="/search/math-ph?searchtype=author&query=Mestdag%2C+T">T. Mestdag</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2107.00428v2-abstract-short" style="display: inline;"> We introduce the notion of a nonlinear splitting on a fibre bundle as a generalization of an Ehresmann connection. We present its basic properties and we pay attention to the special cases of affine, homogeneous and principal nonlinear splittings. We explain where nonlinear splittings appear in the context of Lagrangian systems and Finsler geometry and we show their relation to Routh symmetry redu… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2107.00428v2-abstract-full').style.display = 'inline'; document.getElementById('2107.00428v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2107.00428v2-abstract-full" style="display: none;"> We introduce the notion of a nonlinear splitting on a fibre bundle as a generalization of an Ehresmann connection. We present its basic properties and we pay attention to the special cases of affine, homogeneous and principal nonlinear splittings. We explain where nonlinear splittings appear in the context of Lagrangian systems and Finsler geometry and we show their relation to Routh symmetry reduction, submersive second-order differential equations and unreduction. We define a curvature map for a nonlinear splitting, and we indicate where this concept appears in the context of nonholonomic systems with affine constraints and Lagrangian systems of magnetic type. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2107.00428v2-abstract-full').style.display = 'none'; document.getElementById('2107.00428v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 6 November, 2021; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 1 July, 2021; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> July 2021. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">24 pages, to appear in "Analysis and Mathematical Physics''</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 34A26; 37J15; 53C05; 70G65; 70H03 </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Analysis and Mathematical Physics 12 (2022), paper 14 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2009.00913">arXiv:2009.00913</a> <span> [<a href="https://arxiv.org/pdf/2009.00913">pdf</a>, <a href="https://arxiv.org/format/2009.00913">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Differential Geometry">math.DG</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Mathematical Physics">math-ph</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1007/s00009-021-01736-2">10.1007/s00009-021-01736-2 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Jacobi fields and conjugate points for a projective class of sprays </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math-ph?searchtype=author&query=Hajd%C3%BA%2C+S">S. Hajd煤</a>, <a href="/search/math-ph?searchtype=author&query=Mestdag%2C+T">T. Mestdag</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2009.00913v1-abstract-short" style="display: inline;"> We investigate Jacobi fields and conjugate points in the context of sprays. We first prove that the conjugate points of a spray remain preserved under a projective change. Then, we establish conditions on the projective factor so that the projectively deformed spray meets the conditions of a proposition that ensures the existence of conjugate points. We discuss our methods by means of illustrative… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2009.00913v1-abstract-full').style.display = 'inline'; document.getElementById('2009.00913v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2009.00913v1-abstract-full" style="display: none;"> We investigate Jacobi fields and conjugate points in the context of sprays. We first prove that the conjugate points of a spray remain preserved under a projective change. Then, we establish conditions on the projective factor so that the projectively deformed spray meets the conditions of a proposition that ensures the existence of conjugate points. We discuss our methods by means of illustrative examples, throughout the paper. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2009.00913v1-abstract-full').style.display = 'none'; document.getElementById('2009.00913v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 2 September, 2020; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> September 2020. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">18 pages, 3 figures</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 34A26; 53B40; 58E10 </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Mediterranean Journal of Mathematics 18 (2021), article 73 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1909.05012">arXiv:1909.05012</a> <span> [<a href="https://arxiv.org/pdf/1909.05012">pdf</a>, <a href="https://arxiv.org/format/1909.05012">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Differential Geometry">math.DG</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Mathematical Physics">math-ph</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1142/S0219887820500127">10.1142/S0219887820500127 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Conjugate points for systems of second-order ordinary differential equations </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math-ph?searchtype=author&query=Hajd%C3%BA%2C+S">S. Hajd煤</a>, <a href="/search/math-ph?searchtype=author&query=Mestdag%2C+T">T. Mestdag</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1909.05012v1-abstract-short" style="display: inline;"> We recall the notion of Jacobi fields, as it was extended to systems of second-order ordinary differential equations. Two points along a base integral curve are conjugate if there exists a non-trivial Jacobi field along that curve that vanishes on both points. Based on arguments that involve the eigendistributions of the Jacobi endomorphism, we discuss conjugate points for a certain generalization… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1909.05012v1-abstract-full').style.display = 'inline'; document.getElementById('1909.05012v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1909.05012v1-abstract-full" style="display: none;"> We recall the notion of Jacobi fields, as it was extended to systems of second-order ordinary differential equations. Two points along a base integral curve are conjugate if there exists a non-trivial Jacobi field along that curve that vanishes on both points. Based on arguments that involve the eigendistributions of the Jacobi endomorphism, we discuss conjugate points for a certain generalization (to the current setting) of locally symmetric spaces. Next, we study conjugate points along relative equilibria of Lagrangian systems with a symmetry Lie group. We end the paper with some examples and applications. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1909.05012v1-abstract-full').style.display = 'none'; document.getElementById('1909.05012v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 11 September, 2019; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> September 2019. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">23 pages, 1 figure</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 34A26; 37J15; 53C21; 53C22; 70G65 </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> International Journal of Geometric Methods in Modern Physics 17 (2020), 2050012 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1612.04638">arXiv:1612.04638</a> <span> [<a href="https://arxiv.org/pdf/1612.04638">pdf</a>, <a href="https://arxiv.org/ps/1612.04638">ps</a>, <a href="https://arxiv.org/format/1612.04638">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Mathematical Physics">math-ph</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Differential Geometry">math.DG</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1016/S0034-4877(17)30049-6">10.1016/S0034-4877(17)30049-6 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> A generalization of Szebehely's inverse problem of dynamics in dimension three </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math-ph?searchtype=author&query=Sarlet%2C+W">W. Sarlet</a>, <a href="/search/math-ph?searchtype=author&query=Mestdag%2C+T">T. Mestdag</a>, <a href="/search/math-ph?searchtype=author&query=Prince%2C+G">G. Prince</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1612.04638v1-abstract-short" style="display: inline;"> Extending a previous paper, we present a generalization in dimension 3 of the traditional Szebehely-type inverse problem. In that traditional setting, the data are curves determined as the intersection of two families of surfaces, and the problem is to find a potential V such that the Lagrangian L = T - V, where T is the standard Euclidean kinetic energy function, generates integral curves which i… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1612.04638v1-abstract-full').style.display = 'inline'; document.getElementById('1612.04638v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1612.04638v1-abstract-full" style="display: none;"> Extending a previous paper, we present a generalization in dimension 3 of the traditional Szebehely-type inverse problem. In that traditional setting, the data are curves determined as the intersection of two families of surfaces, and the problem is to find a potential V such that the Lagrangian L = T - V, where T is the standard Euclidean kinetic energy function, generates integral curves which include the given family of curves. Our more general way of posing the problem makes use of ideas of the inverse problem of the calculus of variations and essentially consists of allowing more general kinetic energy functions, with a metric which is still constant, but need not be the standard Euclidean one. In developing our generalization, we review and clarify different aspects of the existing literature on the problem and illustrate the relevance of the newly introduced additional freedom with many examples. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1612.04638v1-abstract-full').style.display = 'none'; document.getElementById('1612.04638v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 14 December, 2016; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> December 2016. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">23 pages, to appear in Rep. Math. Phys</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1606.07649">arXiv:1606.07649</a> <span> [<a href="https://arxiv.org/pdf/1606.07649">pdf</a>, <a href="https://arxiv.org/format/1606.07649">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Differential Geometry">math.DG</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Mathematical Physics">math-ph</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.3842/SIGMA.2016.115">10.3842/SIGMA.2016.115 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Un-Reduction of Systems of Second-Order Ordinary Differential Equations </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math-ph?searchtype=author&query=Andr%C3%A9s%2C+E+G">Eduardo Garc铆a-Tora帽o Andr茅s</a>, <a href="/search/math-ph?searchtype=author&query=Mestdag%2C+T">Tom Mestdag</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1606.07649v2-abstract-short" style="display: inline;"> In this paper we consider an alternative approach to "un-reduction". This is the process where one associates to a Lagrangian system on a manifold a dynamical system on a principal bundle over that manifold, in such a way that solutions project. We show that, when written in terms of second-order ordinary differential equations (SODEs), one may associate to the first system a (what we have called)… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1606.07649v2-abstract-full').style.display = 'inline'; document.getElementById('1606.07649v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1606.07649v2-abstract-full" style="display: none;"> In this paper we consider an alternative approach to "un-reduction". This is the process where one associates to a Lagrangian system on a manifold a dynamical system on a principal bundle over that manifold, in such a way that solutions project. We show that, when written in terms of second-order ordinary differential equations (SODEs), one may associate to the first system a (what we have called) "primary un-reduced SODE", and we explain how all other un-reduced SODEs relate to it. We give examples that show that the considered procedure exceeds the realm of Lagrangian systems and that relate our results to those in the literature. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1606.07649v2-abstract-full').style.display = 'none'; document.getElementById('1606.07649v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 7 December, 2016; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 24 June, 2016; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> June 2016. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 34A26; 37J15; 70H33; 70G65 </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> SIGMA 12 (2016), 115, 20 pages </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1602.01673">arXiv:1602.01673</a> <span> [<a href="https://arxiv.org/pdf/1602.01673">pdf</a>, <a href="https://arxiv.org/ps/1602.01673">ps</a>, <a href="https://arxiv.org/format/1602.01673">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Mathematical Physics">math-ph</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Dynamical Systems">math.DS</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Optimization and Control">math.OC</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1137/16M1060091">10.1137/16M1060091 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> The inverse problem of the calculus of variations and the stabilization of controlled Lagrangian systems </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math-ph?searchtype=author&query=Puiggal%C3%AD%2C+M+F">M. Farr茅 Puiggal铆</a>, <a href="/search/math-ph?searchtype=author&query=Mestdag%2C+T">T. Mestdag</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1602.01673v2-abstract-short" style="display: inline;"> We apply methods of the so-called `inverse problem of the calculus of variations' to the stabilization of an equilibrium of a class of two-dimensional controlled mechanical systems. The class is general enough to include, among others, the inverted pendulum on a cart and the inertia wheel pendulum. By making use of a condition that follows from Douglas' classification, we derive feedback controls… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1602.01673v2-abstract-full').style.display = 'inline'; document.getElementById('1602.01673v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1602.01673v2-abstract-full" style="display: none;"> We apply methods of the so-called `inverse problem of the calculus of variations' to the stabilization of an equilibrium of a class of two-dimensional controlled mechanical systems. The class is general enough to include, among others, the inverted pendulum on a cart and the inertia wheel pendulum. By making use of a condition that follows from Douglas' classification, we derive feedback controls for which the control system is variational. We then use the energy of a suitable controlled Lagrangian to provide a stability criterion for the equilibrium. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1602.01673v2-abstract-full').style.display = 'none'; document.getElementById('1602.01673v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 16 December, 2016; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 4 February, 2016; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> February 2016. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 70H03; 70Q05; 49N45 </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> SIAM J. Control Optim. 54-6 (2016), pp. 3297-3318 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1509.01946">arXiv:1509.01946</a> <span> [<a href="https://arxiv.org/pdf/1509.01946">pdf</a>, <a href="https://arxiv.org/ps/1509.01946">ps</a>, <a href="https://arxiv.org/format/1509.01946">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Differential Geometry">math.DG</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Mathematical Physics">math-ph</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1016/j.geomphys.2016.02.010">10.1016/j.geomphys.2016.02.010 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Implicit Lagrange-Routh Equations and Dirac Reduction </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math-ph?searchtype=author&query=Andr%C3%A9s%2C+E+G">Eduardo Garc铆a-Tora帽o Andr茅s</a>, <a href="/search/math-ph?searchtype=author&query=Mestdag%2C+T">Tom Mestdag</a>, <a href="/search/math-ph?searchtype=author&query=Yoshimura%2C+H">Hiroaki Yoshimura</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1509.01946v2-abstract-short" style="display: inline;"> In this paper, we make a generalization of Routh's reduction method for Lagrangian systems with symmetry to the case where not any regularity condition is imposed on the Lagrangian. First, we show how implicit Lagrange-Routh equations can be obtained from the Hamilton-Pontryagin principle, by making use of an anholonomic frame, and how these equations can be reduced. To do this, we keep the moment… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1509.01946v2-abstract-full').style.display = 'inline'; document.getElementById('1509.01946v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1509.01946v2-abstract-full" style="display: none;"> In this paper, we make a generalization of Routh's reduction method for Lagrangian systems with symmetry to the case where not any regularity condition is imposed on the Lagrangian. First, we show how implicit Lagrange-Routh equations can be obtained from the Hamilton-Pontryagin principle, by making use of an anholonomic frame, and how these equations can be reduced. To do this, we keep the momentum constraint implicit throughout and we make use of a Routhian function defined on a certain submanifold of the Pontryagin bundle. Then, we show how the reduced implicit Lagrange-Routh equations can be described in the context of dynamical systems associated to Dirac structures, in which we fully utilize a symmetry reduction procedure for implicit Hamiltonian systems with symmetry. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1509.01946v2-abstract-full').style.display = 'none'; document.getElementById('1509.01946v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 25 February, 2016; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 7 September, 2015; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> September 2015. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">21 pages, to appear in J. Geom. Phys</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 37J15; 53D20; 70G65; 70H33 </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Journal of Geometry and Physics 104 (2016) 291-304 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1504.06462">arXiv:1504.06462</a> <span> [<a href="https://arxiv.org/pdf/1504.06462">pdf</a>, <a href="https://arxiv.org/ps/1504.06462">ps</a>, <a href="https://arxiv.org/format/1504.06462">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Mathematical Physics">math-ph</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Differential Geometry">math.DG</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.3934/jgm.2015.7.395">10.3934/jgm.2015.7.395 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Symmetry reduction, integrability and reconstruction in k-symplectic field theory </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math-ph?searchtype=author&query=Bua%2C+L">L. Bua</a>, <a href="/search/math-ph?searchtype=author&query=Mestdag%2C+T">T. Mestdag</a>, <a href="/search/math-ph?searchtype=author&query=Salgado%2C+M">M. Salgado</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1504.06462v1-abstract-short" style="display: inline;"> We investigate the reduction process of a k-symplectic field theory whose Lagrangian is invariant under a symmetry group. We give explicit coordinate expressions of the resulting reduced partial differential equations, the so-called Lagrange-Poincare field equations. We discuss two issues about reconstructing a solution from a given solution of the reduced equations. The first one is an interpreta… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1504.06462v1-abstract-full').style.display = 'inline'; document.getElementById('1504.06462v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1504.06462v1-abstract-full" style="display: none;"> We investigate the reduction process of a k-symplectic field theory whose Lagrangian is invariant under a symmetry group. We give explicit coordinate expressions of the resulting reduced partial differential equations, the so-called Lagrange-Poincare field equations. We discuss two issues about reconstructing a solution from a given solution of the reduced equations. The first one is an interpretation of the integrability conditions, in terms of the curvatures of some connections. The second includes the introduction of the concept of a k-connection to provide a reconstruction method. We show that an invariant Lagrangian, under suitable regularity conditions, defines a `mechanical' k-connection. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1504.06462v1-abstract-full').style.display = 'none'; document.getElementById('1504.06462v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 24 April, 2015; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> April 2015. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">37 pages</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 37J15; 53Z05; 70S05; 70S10 </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Journal of Geometric Mechanics 7 (2015) 395-429 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1412.3526">arXiv:1412.3526</a> <span> [<a href="https://arxiv.org/pdf/1412.3526">pdf</a>, <a href="https://arxiv.org/ps/1412.3526">ps</a>, <a href="https://arxiv.org/format/1412.3526">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Differential Geometry">math.DG</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Mathematical Physics">math-ph</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1007/s00009-014-0505-z">10.1007/s00009-014-0505-z <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Finsler geodesics of Lagrangian systems through Routh reduction </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math-ph?searchtype=author&query=Mestdag%2C+T">T. Mestdag</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1412.3526v1-abstract-short" style="display: inline;"> We make use of a symmetry reduction technique called Routh reduction to show that the solutions of the Euler-Lagrange equations of a strongly convex autonomous Lagrangian which lie on a specific energy level can be thought of as geodesics of an associated Finsler function. </span> <span class="abstract-full has-text-grey-dark mathjax" id="1412.3526v1-abstract-full" style="display: none;"> We make use of a symmetry reduction technique called Routh reduction to show that the solutions of the Euler-Lagrange equations of a strongly convex autonomous Lagrangian which lie on a specific energy level can be thought of as geodesics of an associated Finsler function. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1412.3526v1-abstract-full').style.display = 'none'; document.getElementById('1412.3526v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 10 December, 2014; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> December 2014. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">Accepted for publ. in Mediterranean Journal of Mathematics</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 53C60; 70H03; 70H33 </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Mediterranean Journal of Mathematics 13 (2016) 825-839 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1402.2847">arXiv:1402.2847</a> <span> [<a href="https://arxiv.org/pdf/1402.2847">pdf</a>, <a href="https://arxiv.org/ps/1402.2847">ps</a>, <a href="https://arxiv.org/format/1402.2847">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Differential Geometry">math.DG</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Mathematical Physics">math-ph</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1088/1751-8113/47/22/225203">10.1088/1751-8113/47/22/225203 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Reduced dynamics and Lagrangian submanifolds of symplectic manifolds </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math-ph?searchtype=author&query=Andr%C3%A9s%2C+E+G">E. Garc铆a-Tora帽o Andr茅s</a>, <a href="/search/math-ph?searchtype=author&query=Guzm%C3%A1n%2C+E">E. Guzm谩n</a>, <a href="/search/math-ph?searchtype=author&query=Marrero%2C+J+C">J. C. Marrero</a>, <a href="/search/math-ph?searchtype=author&query=Mestdag%2C+T">T. Mestdag</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1402.2847v2-abstract-short" style="display: inline;"> In this paper, we will see that the symplectic creed by Weinstein "everything is a Lagrangian submanifold" also holds for Hamilton-Poincar茅 and Lagrange-Poincar茅 reduction. In fact, we show that solutions of the Hamilton-Poincar茅 equations and of the Lagrange-Poincar茅 equations are in one-to-one correspondence with distinguished curves in a Lagrangian submanifold of a symplectic manifold. For this… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1402.2847v2-abstract-full').style.display = 'inline'; document.getElementById('1402.2847v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1402.2847v2-abstract-full" style="display: none;"> In this paper, we will see that the symplectic creed by Weinstein "everything is a Lagrangian submanifold" also holds for Hamilton-Poincar茅 and Lagrange-Poincar茅 reduction. In fact, we show that solutions of the Hamilton-Poincar茅 equations and of the Lagrange-Poincar茅 equations are in one-to-one correspondence with distinguished curves in a Lagrangian submanifold of a symplectic manifold. For this purpose, we will combine the concept of a Tulczyjew triple with Marsden-Weinstein symplectic reduction. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1402.2847v2-abstract-full').style.display = 'none'; document.getElementById('1402.2847v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 19 May, 2014; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 12 February, 2014; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> February 2014. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">26 pages</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 53D05; 53D12; 70G65; 70H03; 70H05; 70H33 </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> J. Phys. A: Math. Theor. 47 (2014) 225203 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1305.3175">arXiv:1305.3175</a> <span> [<a href="https://arxiv.org/pdf/1305.3175">pdf</a>, <a href="https://arxiv.org/ps/1305.3175">ps</a>, <a href="https://arxiv.org/format/1305.3175">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Mathematical Physics">math-ph</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Differential Geometry">math.DG</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1016/S0034-4877(14)60005-7">10.1016/S0034-4877(14)60005-7 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> A generalization of Szebehely's inverse problem of dynamics </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math-ph?searchtype=author&query=Sarlet%2C+W">W. Sarlet</a>, <a href="/search/math-ph?searchtype=author&query=Mestdag%2C+T">T. Mestdag</a>, <a href="/search/math-ph?searchtype=author&query=Prince%2C+G">G. Prince</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1305.3175v1-abstract-short" style="display: inline;"> The so-called inverse problem of dynamics is about constructing a potential for a given family of curves. We observe that there is a more general way of posing the problem by making use of ideas of another inverse problem, namely the inverse problem of the calculus of variations. We critically review and clarify different aspects of the current state of the art of the problem (mainly restricted to… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1305.3175v1-abstract-full').style.display = 'inline'; document.getElementById('1305.3175v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1305.3175v1-abstract-full" style="display: none;"> The so-called inverse problem of dynamics is about constructing a potential for a given family of curves. We observe that there is a more general way of posing the problem by making use of ideas of another inverse problem, namely the inverse problem of the calculus of variations. We critically review and clarify different aspects of the current state of the art of the problem (mainly restricted to the case of planar curves), and then develop our more general approach. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1305.3175v1-abstract-full').style.display = 'none'; document.getElementById('1305.3175v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 14 May, 2013; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> May 2013. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">21 pages, to appear in Rep. Math. Phys</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Reports on Mathematical Physics 72 (2013), 65-84 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1112.0162">arXiv:1112.0162</a> <span> [<a href="https://arxiv.org/pdf/1112.0162">pdf</a>, <a href="https://arxiv.org/ps/1112.0162">ps</a>, <a href="https://arxiv.org/format/1112.0162">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Mathematical Physics">math-ph</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Dynamical Systems">math.DS</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1088/1751-8113/45/8/085208">10.1088/1751-8113/45/8/085208 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Time-dependent kinetic energy metrics for Lagrangians of electromagnetic type </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math-ph?searchtype=author&query=Sarlet%2C+W">W. Sarlet</a>, <a href="/search/math-ph?searchtype=author&query=Prince%2C+G">G. Prince</a>, <a href="/search/math-ph?searchtype=author&query=Mestdag%2C+T">T. Mestdag</a>, <a href="/search/math-ph?searchtype=author&query=Krupkova%2C+O">O. Krupkova</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1112.0162v1-abstract-short" style="display: inline;"> We extend the results obtained in a previous paper about a class of Lagrangian systems which admit alternative kinetic energy metrics to second-order mechanical systems with explicit time-dependence. The main results are that a time-dependent alternative metric will have constant eigenvalues, and will give rise to a time-dependent coordinate transformation which partially decouples the system. </span> <span class="abstract-full has-text-grey-dark mathjax" id="1112.0162v1-abstract-full" style="display: none;"> We extend the results obtained in a previous paper about a class of Lagrangian systems which admit alternative kinetic energy metrics to second-order mechanical systems with explicit time-dependence. The main results are that a time-dependent alternative metric will have constant eigenvalues, and will give rise to a time-dependent coordinate transformation which partially decouples the system. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1112.0162v1-abstract-full').style.display = 'none'; document.getElementById('1112.0162v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 1 December, 2011; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> December 2011. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 34A26; 34A55; 58E30; 70H03 </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> J. Phys. A: Math. Theor. 45 (2012) 085208 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1106.2950">arXiv:1106.2950</a> <span> [<a href="https://arxiv.org/pdf/1106.2950">pdf</a>, <a href="https://arxiv.org/ps/1106.2950">ps</a>, <a href="https://arxiv.org/format/1106.2950">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Mathematical Physics">math-ph</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Dynamical Systems">math.DS</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.3842/SIGMA.2011.109">10.3842/SIGMA.2011.109 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Routh Reduction by Stages </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math-ph?searchtype=author&query=Langerock%2C+B">Bavo Langerock</a>, <a href="/search/math-ph?searchtype=author&query=Mestdag%2C+T">Tom Mestdag</a>, <a href="/search/math-ph?searchtype=author&query=Vankerschaver%2C+J">Joris Vankerschaver</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1106.2950v2-abstract-short" style="display: inline;"> This paper deals with the Lagrangian analogue of symplectic or point reduction by stages. We develop Routh reduction as a reduction technique that preserves the Lagrangian nature of the dynamics. To do so we heavily rely on the relation between Routh reduction and cotangent symplectic reduction. The main results in this paper are: (i) we develop a class of so called magnetic Lagrangian systems and… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1106.2950v2-abstract-full').style.display = 'inline'; document.getElementById('1106.2950v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1106.2950v2-abstract-full" style="display: none;"> This paper deals with the Lagrangian analogue of symplectic or point reduction by stages. We develop Routh reduction as a reduction technique that preserves the Lagrangian nature of the dynamics. To do so we heavily rely on the relation between Routh reduction and cotangent symplectic reduction. The main results in this paper are: (i) we develop a class of so called magnetic Lagrangian systems and this class has the property that it is closed under Routh reduction; (ii) we construct a transformation relating the magnetic Lagrangian system obtained after two subsequent Routh reductions and the magnetic Lagrangian system obtained after Routh reduction w.r.t. to the full symmetry group. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1106.2950v2-abstract-full').style.display = 'none'; document.getElementById('1106.2950v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 29 November, 2011; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 15 June, 2011; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> June 2011. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> SIGMA 7 (2011), 109, 31 pages </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1105.5223">arXiv:1105.5223</a> <span> [<a href="https://arxiv.org/pdf/1105.5223">pdf</a>, <a href="https://arxiv.org/format/1105.5223">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Mathematical Physics">math-ph</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Dynamical Systems">math.DS</span> </div> </div> <p class="title is-5 mathjax"> Hamiltonization and geometric integration of nonholonomic mechanical systems </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math-ph?searchtype=author&query=Mestdag%2C+T">T. Mestdag</a>, <a href="/search/math-ph?searchtype=author&query=Bloch%2C+A+M">A. M. Bloch</a>, <a href="/search/math-ph?searchtype=author&query=Fernandez%2C+O+E">O. E. Fernandez</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1105.5223v1-abstract-short" style="display: inline;"> In this paper we study a Hamiltonization procedure for mechanical systems with velocity-depending (nonholonomic) constraints. We first rewrite the nonholonomic equations of motion as Euler-Lagrange equations, with a Lagrangian that follows from rephrasing the issue in terms of the inverse problem of Lagrangian mechanics. Second, the Legendre transformation transforms the Lagrangian in the sought-f… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1105.5223v1-abstract-full').style.display = 'inline'; document.getElementById('1105.5223v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1105.5223v1-abstract-full" style="display: none;"> In this paper we study a Hamiltonization procedure for mechanical systems with velocity-depending (nonholonomic) constraints. We first rewrite the nonholonomic equations of motion as Euler-Lagrange equations, with a Lagrangian that follows from rephrasing the issue in terms of the inverse problem of Lagrangian mechanics. Second, the Legendre transformation transforms the Lagrangian in the sought-for Hamiltonian. As an application, we compare some variational integrators for the new Lagrangians with some known nonholonomic integrators. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1105.5223v1-abstract-full').style.display = 'none'; document.getElementById('1105.5223v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 26 May, 2011; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> May 2011. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">11 pages, 19 figures</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 70-08; 70H03; 70F25 </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Proceedings 8th National Congress on Theoretical and Applied Mechanics, Brussels (Belgium), 2009, 230-236 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1103.2935">arXiv:1103.2935</a> <span> [<a href="https://arxiv.org/pdf/1103.2935">pdf</a>, <a href="https://arxiv.org/ps/1103.2935">ps</a>, <a href="https://arxiv.org/format/1103.2935">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Differential Geometry">math.DG</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Mathematical Physics">math-ph</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1016/j.difgeo.2011.08.003">10.1016/j.difgeo.2011.08.003 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Involutive distributions and dynamical systems of second-order type </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math-ph?searchtype=author&query=Mestdag%2C+T">T. Mestdag</a>, <a href="/search/math-ph?searchtype=author&query=Crampin%2C+M">M. Crampin</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1103.2935v1-abstract-short" style="display: inline;"> We investigate the existence of coordinate transformations which bring a given vector field on a manifold equipped with an involutive distribution into the form of a second-order differential equation field with parameters. We define associated connections and we give a coordinate-independent criterion for determining whether the vector field is of quadratic type. Further, we investigate the under… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1103.2935v1-abstract-full').style.display = 'inline'; document.getElementById('1103.2935v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1103.2935v1-abstract-full" style="display: none;"> We investigate the existence of coordinate transformations which bring a given vector field on a manifold equipped with an involutive distribution into the form of a second-order differential equation field with parameters. We define associated connections and we give a coordinate-independent criterion for determining whether the vector field is of quadratic type. Further, we investigate the underlying global bundle structure of the manifold under consideration, induced by the vector field and the involutive distribution. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1103.2935v1-abstract-full').style.display = 'none'; document.getElementById('1103.2935v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 15 March, 2011; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> March 2011. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">18 pages</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 34A26; 53B05; 53C13; 58A30 </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Diff Geom Appl 29 (2011) 747 - 757 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1101.3153">arXiv:1101.3153</a> <span> [<a href="https://arxiv.org/pdf/1101.3153">pdf</a>, <a href="https://arxiv.org/ps/1101.3153">ps</a>, <a href="https://arxiv.org/format/1101.3153">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Differential Geometry">math.DG</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Mathematical Physics">math-ph</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1142/S0219887811005452">10.1142/S0219887811005452 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> The Cartan form for constrained Lagrangian systems and the nonholonomic Noether theorem </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math-ph?searchtype=author&query=Crampin%2C+M">M. Crampin</a>, <a href="/search/math-ph?searchtype=author&query=Mestdag%2C+T">T. Mestdag</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1101.3153v1-abstract-short" style="display: inline;"> This paper deals with conservation laws for mechanical systems with nonholonomic constraints. It uses a Lagrangian formulation of nonholonomic systems and a Cartan form approach. We present what we believe to be the most general relations between symmetries and first integrals. We discuss the so-called nonholonomic Noether theorem in terms of our formalism, and we give applications to Riemannian s… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1101.3153v1-abstract-full').style.display = 'inline'; document.getElementById('1101.3153v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1101.3153v1-abstract-full" style="display: none;"> This paper deals with conservation laws for mechanical systems with nonholonomic constraints. It uses a Lagrangian formulation of nonholonomic systems and a Cartan form approach. We present what we believe to be the most general relations between symmetries and first integrals. We discuss the so-called nonholonomic Noether theorem in terms of our formalism, and we give applications to Riemannian submanifolds, to Lagrangians of mechanical type, and to the determination of quadratic first integrals. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1101.3153v1-abstract-full').style.display = 'none'; document.getElementById('1101.3153v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 17 January, 2011; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> January 2011. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">25 pages</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 34A26; 37J60; 70G45; 70H03; 70H33 </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Int. J. Geom. Methods Mod. Phys. 8 (2011) 897-923 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1101.2551">arXiv:1101.2551</a> <span> [<a href="https://arxiv.org/pdf/1101.2551">pdf</a>, <a href="https://arxiv.org/ps/1101.2551">ps</a>, <a href="https://arxiv.org/format/1101.2551">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Differential Geometry">math.DG</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Mathematical Physics">math-ph</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.3934/jgm.2011.3.23">10.3934/jgm.2011.3.23 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Reduction of invariant constrained systems using anholonomic frames </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math-ph?searchtype=author&query=Crampin%2C+M">M. Crampin</a>, <a href="/search/math-ph?searchtype=author&query=Mestdag%2C+T">T. Mestdag</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1101.2551v1-abstract-short" style="display: inline;"> We analyze two reduction methods for nonholonomic systems that are invariant under the action of a Lie group on the configuration space. Our approach for obtaining the reduced equations is entirely based on the observation that the dynamics can be represented by a second-order differential equations vector field and that in both cases the reduced dynamics can be described by expressing that vector… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1101.2551v1-abstract-full').style.display = 'inline'; document.getElementById('1101.2551v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1101.2551v1-abstract-full" style="display: none;"> We analyze two reduction methods for nonholonomic systems that are invariant under the action of a Lie group on the configuration space. Our approach for obtaining the reduced equations is entirely based on the observation that the dynamics can be represented by a second-order differential equations vector field and that in both cases the reduced dynamics can be described by expressing that vector field in terms of an appropriately chosen anholonomic frame. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1101.2551v1-abstract-full').style.display = 'none'; document.getElementById('1101.2551v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 13 January, 2011; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> January 2011. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">19 pages</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 34A26; 37J60; 70G45; 70H03 </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> J. Geom. Mech 3 (2011) 23 - 40 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1004.0674">arXiv:1004.0674</a> <span> [<a href="https://arxiv.org/pdf/1004.0674">pdf</a>, <a href="https://arxiv.org/ps/1004.0674">ps</a>, <a href="https://arxiv.org/format/1004.0674">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Differential Geometry">math.DG</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Mathematical Physics">math-ph</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1016/j.difgeo.2010.11.002">10.1016/j.difgeo.2010.11.002 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> The inverse problem for Lagrangian systems with certain non-conservative forces </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math-ph?searchtype=author&query=Mestdag%2C+T">T. Mestdag</a>, <a href="/search/math-ph?searchtype=author&query=Sarlet%2C+W">W. Sarlet</a>, <a href="/search/math-ph?searchtype=author&query=Crampin%2C+M">M. Crampin</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1004.0674v1-abstract-short" style="display: inline;"> We discuss two generalizations of the inverse problem of the calculus of variations, one in which a given mechanical system can be brought into the form of Lagrangian equations with non-conservative forces of a generalized Rayleigh dissipation type, the other leading to Lagrangian equations with so-called gyroscopic forces. Our approach focusses primarily on obtaining coordinate-free conditions fo… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1004.0674v1-abstract-full').style.display = 'inline'; document.getElementById('1004.0674v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1004.0674v1-abstract-full" style="display: none;"> We discuss two generalizations of the inverse problem of the calculus of variations, one in which a given mechanical system can be brought into the form of Lagrangian equations with non-conservative forces of a generalized Rayleigh dissipation type, the other leading to Lagrangian equations with so-called gyroscopic forces. Our approach focusses primarily on obtaining coordinate-free conditions for the existence of a suitable non-singular multiplier matrix, which will lead to an equivalent representation of a given system of second-order equations as one of these Lagrangian systems with non-conservative forces. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1004.0674v1-abstract-full').style.display = 'none'; document.getElementById('1004.0674v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 5 April, 2010; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> April 2010. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">28 pages</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 70H03; 70F17; 49N45 </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Diff Geom Appl 29 (2011) 55 - 72 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1003.1840">arXiv:1003.1840</a> <span> [<a href="https://arxiv.org/pdf/1003.1840">pdf</a>, <a href="https://arxiv.org/ps/1003.1840">ps</a>, <a href="https://arxiv.org/format/1003.1840">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Differential Geometry">math.DG</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Mathematical Physics">math-ph</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1002/zamm.200900327">10.1002/zamm.200900327 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> On the generalized Helmholtz conditions for Lagrangian systems with dissipative forces </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math-ph?searchtype=author&query=Crampin%2C+M">M. Crampin</a>, <a href="/search/math-ph?searchtype=author&query=Mestdag%2C+T">T. Mestdag</a>, <a href="/search/math-ph?searchtype=author&query=Sarlet%2C+W">W. Sarlet</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1003.1840v1-abstract-short" style="display: inline;"> In two recent papers necessary and sufficient conditions for a given system of second-order ordinary differential equations to be of Lagrangian form with additional dissipative forces were derived. We point out that these conditions are not independent and prove a stronger result accordingly. </span> <span class="abstract-full has-text-grey-dark mathjax" id="1003.1840v1-abstract-full" style="display: none;"> In two recent papers necessary and sufficient conditions for a given system of second-order ordinary differential equations to be of Lagrangian form with additional dissipative forces were derived. We point out that these conditions are not independent and prove a stronger result accordingly. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1003.1840v1-abstract-full').style.display = 'none'; document.getElementById('1003.1840v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 9 March, 2010; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> March 2010. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">10 pages, accepted for publ in Z. Angew. Math. Mech.</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 70H03; 70F17; 49N45 </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Z. Angew. Math. Mech. 90 (2010), 502-508 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/0909.4230">arXiv:0909.4230</a> <span> [<a href="https://arxiv.org/pdf/0909.4230">pdf</a>, <a href="https://arxiv.org/format/0909.4230">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Mathematical Physics">math-ph</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1080/14689360903360888">10.1080/14689360903360888 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Anholonomic frames in constrained dynamics </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math-ph?searchtype=author&query=Crampin%2C+M">M. Crampin</a>, <a href="/search/math-ph?searchtype=author&query=Mestdag%2C+T">T. Mestdag</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="0909.4230v1-abstract-short" style="display: inline;"> We demonstrate the usefulness of anholonomic frames in the contexts of nonholonomic and vakonomic systems. We take a consistently differential-geometric approach. As an application, we investigate the conditions under which the dynamics of the two systems will be consistent. A few illustrative examples confirm the results. </span> <span class="abstract-full has-text-grey-dark mathjax" id="0909.4230v1-abstract-full" style="display: none;"> We demonstrate the usefulness of anholonomic frames in the contexts of nonholonomic and vakonomic systems. We take a consistently differential-geometric approach. As an application, we investigate the conditions under which the dynamics of the two systems will be consistent. A few illustrative examples confirm the results. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('0909.4230v1-abstract-full').style.display = 'none'; document.getElementById('0909.4230v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 23 September, 2009; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> September 2009. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">33 pages, to appear in `Dynamical Systems. An international journal.'</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 34A26; 37J60; 70G45; 70G75; 70H03 </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Dynamical Systems: An International Journal 25 (2010) 159 - 187 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/0909.4018">arXiv:0909.4018</a> <span> [<a href="https://arxiv.org/pdf/0909.4018">pdf</a>, <a href="https://arxiv.org/format/0909.4018">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Mathematical Physics">math-ph</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1134/S1560354709060033">10.1134/S1560354709060033 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> A Generalization of Chaplygin's Reducibility Theorem </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math-ph?searchtype=author&query=Fernandez%2C+O+E">O. E. Fernandez</a>, <a href="/search/math-ph?searchtype=author&query=Mestdag%2C+T">T. Mestdag</a>, <a href="/search/math-ph?searchtype=author&query=Bloch%2C+A+M">A. M. Bloch</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="0909.4018v2-abstract-short" style="display: inline;"> In this paper we study Chaplygin's Reducibility Theorem and extend its applicability to nonholonomic systems with symmetry described by the Hamilton-Poincare-d'Alembert equations in arbitrary degrees of freedom. As special cases we extract the extension of the Theorem to nonholonomic Chaplygin systems with nonabelian symmetry groups as well as Euler-Poincare-Suslov systems in arbitrary degrees o… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('0909.4018v2-abstract-full').style.display = 'inline'; document.getElementById('0909.4018v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="0909.4018v2-abstract-full" style="display: none;"> In this paper we study Chaplygin's Reducibility Theorem and extend its applicability to nonholonomic systems with symmetry described by the Hamilton-Poincare-d'Alembert equations in arbitrary degrees of freedom. As special cases we extract the extension of the Theorem to nonholonomic Chaplygin systems with nonabelian symmetry groups as well as Euler-Poincare-Suslov systems in arbitrary degrees of freedom. In the latter case, we also extend the Hamiltonization Theorem to nonholonomic systems which do not possess an invariant measure. Lastly, we extend previous work on conditionally variational systems using the results above. We illustrate the results through various examples of well-known nonholonomic systems. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('0909.4018v2-abstract-full').style.display = 'none'; document.getElementById('0909.4018v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 10 November, 2009; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 22 September, 2009; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> September 2009. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">27 pages, 3 figures, submitted to Reg. and Chaotic Dyn</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 37J60 (Primary); 70H06; 37J15 (Secondary) </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Reg. and Chaotic Dyn. 14(6) 2009 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/0812.0437">arXiv:0812.0437</a> <span> [<a href="https://arxiv.org/pdf/0812.0437">pdf</a>, <a href="https://arxiv.org/ps/0812.0437">ps</a>, <a href="https://arxiv.org/format/0812.0437">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Mathematical Physics">math-ph</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1016/S0034-4877(09)90001-5">10.1016/S0034-4877(09)90001-5 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Hamiltonization of Nonholonomic Systems and the Inverse Problem of the Calculus of Variations </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math-ph?searchtype=author&query=Bloch%2C+A+M">A. M. Bloch</a>, <a href="/search/math-ph?searchtype=author&query=Fernandez%2C+O+E">O. E. Fernandez</a>, <a href="/search/math-ph?searchtype=author&query=Mestdag%2C+T">T. Mestdag</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="0812.0437v1-abstract-short" style="display: inline;"> We introduce a method which allows one to recover the equations of motion of a class of nonholonomic systems by finding instead an unconstrained Hamiltonian system on the full phase space, and to restrict the resulting canonical equations to an appropriate submanifold of phase space. We focus first on the Lagrangian picture of the method and deduce the corresponding Hamiltonian from the Legendre… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('0812.0437v1-abstract-full').style.display = 'inline'; document.getElementById('0812.0437v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="0812.0437v1-abstract-full" style="display: none;"> We introduce a method which allows one to recover the equations of motion of a class of nonholonomic systems by finding instead an unconstrained Hamiltonian system on the full phase space, and to restrict the resulting canonical equations to an appropriate submanifold of phase space. We focus first on the Lagrangian picture of the method and deduce the corresponding Hamiltonian from the Legendre transformation. We illustrate the method with several examples and we discuss its relationship to the Pontryagin maximum principle. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('0812.0437v1-abstract-full').style.display = 'none'; document.getElementById('0812.0437v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 2 December, 2008; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> December 2008. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">23 pages, accepted for publication in Rep. Math. Phys</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Rep. Math. Phys. 63 (2009) 225-249. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/0807.0156">arXiv:0807.0156</a> <span> [<a href="https://arxiv.org/pdf/0807.0156">pdf</a>, <a href="https://arxiv.org/ps/0807.0156">ps</a>, <a href="https://arxiv.org/format/0807.0156">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Differential Geometry">math.DG</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Mathematical Physics">math-ph</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1007/s10440-008-9274-7">10.1007/s10440-008-9274-7 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Reduction and reconstruction aspects of second-order dynamical systems with symmetry </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math-ph?searchtype=author&query=Crampin%2C+M">M. Crampin</a>, <a href="/search/math-ph?searchtype=author&query=Mestdag%2C+T">T. Mestdag</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="0807.0156v1-abstract-short" style="display: inline;"> We examine the reduction process of a system of second-order ordinary differential equations which is invariant under a Lie group action. With the aid of connection theory, we explain why the associated vector field decomposes in three parts and we show how the integral curves of the original system can be reconstructed from the reduced dynamics. An illustrative example confirms the results. </span> <span class="abstract-full has-text-grey-dark mathjax" id="0807.0156v1-abstract-full" style="display: none;"> We examine the reduction process of a system of second-order ordinary differential equations which is invariant under a Lie group action. With the aid of connection theory, we explain why the associated vector field decomposes in three parts and we show how the integral curves of the original system can be reconstructed from the reduced dynamics. An illustrative example confirms the results. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('0807.0156v1-abstract-full').style.display = 'none'; document.getElementById('0807.0156v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 1 July, 2008; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> July 2008. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">28 pages, to appear in Acta Appl. Math</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 34A26; 37J15; 53C05 </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Acta Appl. Math.105 (2009), 241-266. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/0802.1421">arXiv:0802.1421</a> <span> [<a href="https://arxiv.org/pdf/0802.1421">pdf</a>, <a href="https://arxiv.org/ps/0802.1421">ps</a>, <a href="https://arxiv.org/format/0802.1421">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Differential Geometry">math.DG</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Mathematical Physics">math-ph</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1016/j.geomphys.2008.02.008">10.1016/j.geomphys.2008.02.008 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Relative equilibria of Lagrangian systems with symmetry </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math-ph?searchtype=author&query=Crampin%2C+M">M. Crampin</a>, <a href="/search/math-ph?searchtype=author&query=Mestdag%2C+T">T. Mestdag</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="0802.1421v1-abstract-short" style="display: inline;"> We discuss the characterization of relative equilibria of Lagrangian systems with symmetry. </span> <span class="abstract-full has-text-grey-dark mathjax" id="0802.1421v1-abstract-full" style="display: none;"> We discuss the characterization of relative equilibria of Lagrangian systems with symmetry. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('0802.1421v1-abstract-full').style.display = 'none'; document.getElementById('0802.1421v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 11 February, 2008; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> February 2008. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">16 pages, to appear in J. Geom. Phys</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 34A05; 34A26; 37J15; 37J15; 70H03 </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> J. Geom. Phys. 58 (2008) 874--887. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/0802.0528">arXiv:0802.0528</a> <span> [<a href="https://arxiv.org/pdf/0802.0528">pdf</a>, <a href="https://arxiv.org/ps/0802.0528">ps</a>, <a href="https://arxiv.org/format/0802.0528">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Differential Geometry">math.DG</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Mathematical Physics">math-ph</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1063/1.2885077">10.1063/1.2885077 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Routh's procedure for non-Abelian symmetry groups </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math-ph?searchtype=author&query=Crampin%2C+M">M. Crampin</a>, <a href="/search/math-ph?searchtype=author&query=Mestdag%2C+T">T. Mestdag</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="0802.0528v1-abstract-short" style="display: inline;"> We extend Routh's reduction procedure to an arbitrary Lagrangian system (that is, one whose Lagrangian is not necessarily the difference of kinetic and potential energies) with a symmetry group which is not necessarily Abelian. To do so we analyse the restriction of the Euler-Lagrange field to a level set of momentum in velocity phase space. We present a new method of analysis based on the use o… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('0802.0528v1-abstract-full').style.display = 'inline'; document.getElementById('0802.0528v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="0802.0528v1-abstract-full" style="display: none;"> We extend Routh's reduction procedure to an arbitrary Lagrangian system (that is, one whose Lagrangian is not necessarily the difference of kinetic and potential energies) with a symmetry group which is not necessarily Abelian. To do so we analyse the restriction of the Euler-Lagrange field to a level set of momentum in velocity phase space. We present a new method of analysis based on the use of quasi-velocities. We discuss the reconstruction of solutions of the full Euler-Lagrange equations from those of the reduced equations. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('0802.0528v1-abstract-full').style.display = 'none'; document.getElementById('0802.0528v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 5 February, 2008; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> February 2008. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">30 pages, to appear in J Math Phys</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 34A26; 37J15; 53C05; 70H03 </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> J Math Phys (2008) 49, 032901. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/0802.0146">arXiv:0802.0146</a> <span> [<a href="https://arxiv.org/pdf/0802.0146">pdf</a>, <a href="https://arxiv.org/ps/0802.0146">ps</a>, <a href="https://arxiv.org/format/0802.0146">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Differential Geometry">math.DG</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Mathematical Physics">math-ph</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1088/1751-8113/41/34/344015">10.1088/1751-8113/41/34/344015 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Invariant Lagrangians, mechanical connections and the Lagrange-Poincare equations </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math-ph?searchtype=author&query=Mestdag%2C+T">T. Mestdag</a>, <a href="/search/math-ph?searchtype=author&query=Crampin%2C+M">M. Crampin</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="0802.0146v1-abstract-short" style="display: inline;"> We deal with Lagrangian systems that are invariant under the action of a symmetry group. The mechanical connection is a principal connection that is associated to Lagrangians which have a kinetic energy function that is defined by a Riemannian metric. In this paper we extend this notion to arbitrary Lagrangians. We then derive the reduced Lagrange-Poincare equations in a new fashion and we show… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('0802.0146v1-abstract-full').style.display = 'inline'; document.getElementById('0802.0146v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="0802.0146v1-abstract-full" style="display: none;"> We deal with Lagrangian systems that are invariant under the action of a symmetry group. The mechanical connection is a principal connection that is associated to Lagrangians which have a kinetic energy function that is defined by a Riemannian metric. In this paper we extend this notion to arbitrary Lagrangians. We then derive the reduced Lagrange-Poincare equations in a new fashion and we show how solutions of the Euler-Lagrange equations can be reconstructed with the help of the mechanical connection. Illustrative examples confirm the theory. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('0802.0146v1-abstract-full').style.display = 'none'; document.getElementById('0802.0146v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 1 February, 2008; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> February 2008. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">22 pages, to appear in J. Phys. A: Math. Theor., D2HFest special issue</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 34A26; 37J15; 53C05; 70H03 </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> J. Phys. A: Math. Theor. 41 (2008) 344015 (20pp) </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/0801.4735">arXiv:0801.4735</a> <span> [<a href="https://arxiv.org/pdf/0801.4735">pdf</a>, <a href="https://arxiv.org/ps/0801.4735">ps</a>, <a href="https://arxiv.org/format/0801.4735">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Differential Geometry">math.DG</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Mathematical Physics">math-ph</span> </div> </div> <p class="title is-5 mathjax"> The inverse problem for invariant Lagrangians on a Lie group </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math-ph?searchtype=author&query=Crampin%2C+M">M. Crampin</a>, <a href="/search/math-ph?searchtype=author&query=Mestdag%2C+T">T. Mestdag</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="0801.4735v1-abstract-short" style="display: inline;"> We discuss the problem of the existence of a regular invariant Lagrangian for a given system of invariant second-order differential equations on a Lie group $G$, using approaches based on the Helmholtz conditions. Although we deal with the problem directly on $TG$, our main result relies on a reduction of the system on $TG$ to a system on the Lie algebra of $G$. We conclude with some illustrativ… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('0801.4735v1-abstract-full').style.display = 'inline'; document.getElementById('0801.4735v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="0801.4735v1-abstract-full" style="display: none;"> We discuss the problem of the existence of a regular invariant Lagrangian for a given system of invariant second-order differential equations on a Lie group $G$, using approaches based on the Helmholtz conditions. Although we deal with the problem directly on $TG$, our main result relies on a reduction of the system on $TG$ to a system on the Lie algebra of $G$. We conclude with some illustrative examples. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('0801.4735v1-abstract-full').style.display = 'none'; document.getElementById('0801.4735v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 30 January, 2008; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> January 2008. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">31 pages</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 34A26; 37J15; 49N45 </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Journal of Lie Theory 18 (2008), 471-502. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/math/0410460">arXiv:math/0410460</a> <span> [<a href="https://arxiv.org/pdf/math/0410460">pdf</a>, <a href="https://arxiv.org/ps/math/0410460">ps</a>, <a href="https://arxiv.org/format/math/0410460">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Differential Geometry">math.DG</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Mathematical Physics">math-ph</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1088/0305-4470/38/5/011">10.1088/0305-4470/38/5/011 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> A Lie algebroid framework for non-holonomic systems </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math-ph?searchtype=author&query=Mestdag%2C+T">Tom Mestdag</a>, <a href="/search/math-ph?searchtype=author&query=Langerock%2C+B">Bavo Langerock</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="math/0410460v1-abstract-short" style="display: inline;"> In order to obtain a framework in which both non-holonomic mechanical systems and non-holonomic mechanical systems with symmetry can be described, we introduce in this paper the notion of a Lagrangian system on a subbundle of a Lie algebroid. </span> <span class="abstract-full has-text-grey-dark mathjax" id="math/0410460v1-abstract-full" style="display: none;"> In order to obtain a framework in which both non-holonomic mechanical systems and non-holonomic mechanical systems with symmetry can be described, we introduce in this paper the notion of a Lagrangian system on a subbundle of a Lie algebroid. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('math/0410460v1-abstract-full').style.display = 'none'; document.getElementById('math/0410460v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 21 October, 2004; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> October 2004. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">18 pages</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 17B66; 53C05; 70G45; 70H03; 70H05 </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> J. Phys. A: Math. Gen. 38 (2005), 1097-1111 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/math/0304221">arXiv:math/0304221</a> <span> [<a href="https://arxiv.org/pdf/math/0304221">pdf</a>, <a href="https://arxiv.org/ps/math/0304221">ps</a>, <a href="https://arxiv.org/format/math/0304221">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Differential Geometry">math.DG</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Mathematical Physics">math-ph</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1088/0305-4470/36/29/310">10.1088/0305-4470/36/29/310 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> The Berwald-type linearisation of generalised connections </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math-ph?searchtype=author&query=Mestdag%2C+T">Tom Mestdag</a>, <a href="/search/math-ph?searchtype=author&query=Sarlet%2C+W">Willy Sarlet</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="math/0304221v1-abstract-short" style="display: inline;"> We study the existence of a natural `linearisation' process for generalised connections on an affine bundle. It is shown that this leads to an affine generalised connection over a prolonged bundle, which is the analogue of what is called a connection of Berwald type in the standard theory of connections. Various new insights are being obtained in the fine structure of affine bundles over an anch… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('math/0304221v1-abstract-full').style.display = 'inline'; document.getElementById('math/0304221v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="math/0304221v1-abstract-full" style="display: none;"> We study the existence of a natural `linearisation' process for generalised connections on an affine bundle. It is shown that this leads to an affine generalised connection over a prolonged bundle, which is the analogue of what is called a connection of Berwald type in the standard theory of connections. Various new insights are being obtained in the fine structure of affine bundles over an anchored vector bundle and affineness of generalised connections on such bundles. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('math/0304221v1-abstract-full').style.display = 'none'; document.getElementById('math/0304221v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 16 April, 2003; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> April 2003. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">25 pages</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 53C05; 58A32 </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> J. Phys. A: Math. Gen. 36 (2003), 8049--8069 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/math/0207189">arXiv:math/0207189</a> <span> [<a href="https://arxiv.org/pdf/math/0207189">pdf</a>, <a href="https://arxiv.org/ps/math/0207189">ps</a>, <a href="https://arxiv.org/format/math/0207189">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Differential Geometry">math.DG</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Mathematical Physics">math-ph</span> </div> </div> <p class="title is-5 mathjax"> Note on generalised connections and affine bundles </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math-ph?searchtype=author&query=Mestdag%2C+T">T. Mestdag</a>, <a href="/search/math-ph?searchtype=author&query=Sarlet%2C+W">W. Sarlet</a>, <a href="/search/math-ph?searchtype=author&query=Martinez%2C+E">E. Martinez</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="math/0207189v1-abstract-short" style="display: inline;"> We develop an alternative view on the concept of connections over a vector bundle map, which consists of a horizontal lift procedure to a prolonged bundle. We further focus on prolongations to an affine bundle and introduce the concept of affineness of a generalised connection. </span> <span class="abstract-full has-text-grey-dark mathjax" id="math/0207189v1-abstract-full" style="display: none;"> We develop an alternative view on the concept of connections over a vector bundle map, which consists of a horizontal lift procedure to a prolonged bundle. We further focus on prolongations to an affine bundle and introduce the concept of affineness of a generalised connection. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('math/0207189v1-abstract-full').style.display = 'none'; document.getElementById('math/0207189v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 22 July, 2002; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> July 2002. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">17 pages</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 53C05; 58A32 </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> J. Phys. A: Math. Gen. 35 (2002), 9843--9856 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/math/0203178">arXiv:math/0203178</a> <span> [<a href="https://arxiv.org/pdf/math/0203178">pdf</a>, <a href="https://arxiv.org/ps/math/0203178">ps</a>, <a href="https://arxiv.org/format/math/0203178">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Differential Geometry">math.DG</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Mathematical Physics">math-ph</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1016/S0393-0440(02)00114-6">10.1016/S0393-0440(02)00114-6 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Lie algebroid structures and Lagrangian systems on affine bundles </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math-ph?searchtype=author&query=Martinez%2C+E">Eduardo Martinez</a>, <a href="/search/math-ph?searchtype=author&query=Mestdag%2C+T">Tom Mestdag</a>, <a href="/search/math-ph?searchtype=author&query=Sarlet%2C+W">Willy Sarlet</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="math/0203178v1-abstract-short" style="display: inline;"> As a continuation of previous papers, we study the concept of a Lie algebroid structure on an affine bundle by means of the canonical immersion of the affine bundle into its bidual. We pay particular attention to the prolongation and various lifting procedures, and to the geometrical construction of Lagrangian-type dynamics on an affine Lie algebroid. </span> <span class="abstract-full has-text-grey-dark mathjax" id="math/0203178v1-abstract-full" style="display: none;"> As a continuation of previous papers, we study the concept of a Lie algebroid structure on an affine bundle by means of the canonical immersion of the affine bundle into its bidual. We pay particular attention to the prolongation and various lifting procedures, and to the geometrical construction of Lagrangian-type dynamics on an affine Lie algebroid. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('math/0203178v1-abstract-full').style.display = 'none'; document.getElementById('math/0203178v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 18 March, 2002; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> March 2002. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">28 pages, Latex</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 17B66; 53C15; 53D17; 37J99 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/math/0201264">arXiv:math/0201264</a> <span> [<a href="https://arxiv.org/pdf/math/0201264">pdf</a>, <a href="https://arxiv.org/ps/math/0201264">ps</a>, <a href="https://arxiv.org/format/math/0201264">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Differential Geometry">math.DG</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Mathematical Physics">math-ph</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1063/1.1510958">10.1063/1.1510958 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Lie algebroid structures on a class of affine bundles </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math-ph?searchtype=author&query=Sarlet%2C+W">W. Sarlet</a>, <a href="/search/math-ph?searchtype=author&query=Mestdag%2C+T">T. Mestdag</a>, <a href="/search/math-ph?searchtype=author&query=Martinez%2C+E">E. Martinez</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="math/0201264v1-abstract-short" style="display: inline;"> We introduce the notion of a Lie algebroid structure on an affine bundle whose base manifold is fibred over the real numbers. It is argued that this is the framework which one needs for coming to a time-dependent generalization of the theory of Lagrangian systems on Lie algebroids. An extensive discussion is given of a way one can think of forms acting on sections of the affine bundle. It is fur… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('math/0201264v1-abstract-full').style.display = 'inline'; document.getElementById('math/0201264v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="math/0201264v1-abstract-full" style="display: none;"> We introduce the notion of a Lie algebroid structure on an affine bundle whose base manifold is fibred over the real numbers. It is argued that this is the framework which one needs for coming to a time-dependent generalization of the theory of Lagrangian systems on Lie algebroids. An extensive discussion is given of a way one can think of forms acting on sections of the affine bundle. It is further shown that the affine Lie algebroid structure gives rise to a coboundary operator on such forms. The concept of admissible curves and dynamical systems whose integral curves are admissible, brings an associated affine bundle into the picture, on which one can define in a natural way a prolongation of the original affine Lie algebroid structure. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('math/0201264v1-abstract-full').style.display = 'none'; document.getElementById('math/0201264v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 28 January, 2002; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> January 2002. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">28 pages</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 37J99; 17B60; 53C15 </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> J. Math. Phys. 43 (2002) 5654-5674 </p> </li> </ol> <div class="is-hidden-tablet">  <span class="help" style="display: inline-block;"><a href="https://github.com/arXiv/arxiv-search/releases">Search v0.5.6 released 2020-02-24</a>  </span> </div> </div> </main> <footer> <div class="columns is-desktop" role="navigation" aria-label="Secondary">  <div class="column"> <div class="columns"> <div class="column"> <ul class="nav-spaced"> <li><a href="https://info.arxiv.org/about">About</a></li> <li><a href="https://info.arxiv.org/help">Help</a></li> </ul> </div> <div class="column"> <ul class="nav-spaced"> <li> <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 512 512" class="icon filter-black" role="presentation"><title>contact arXiv</title><desc>Click here to contact arXiv</desc><path d="M502.3 190.8c3.9-3.1 9.7-.2 9.7 4.7V400c0 26.5-21.5 48-48 48H48c-26.5 0-48-21.5-48-48V195.6c0-5 5.7-7.8 9.7-4.7 22.4 17.4 52.1 39.5 154.1 113.6 21.1 15.4 56.7 47.8 92.2 47.6 35.7.3 72-32.8 92.3-47.6 102-74.1 131.6-96.3 154-113.7zM256 320c23.2.4 56.6-29.2 73.4-41.4 132.7-96.3 142.8-104.7 173.4-128.7 5.8-4.5 9.2-11.5 9.2-18.9v-19c0-26.5-21.5-48-48-48H48C21.5 64 0 85.5 0 112v19c0 7.4 3.4 14.3 9.2 18.9 30.6 23.9 40.7 32.4 173.4 128.7 16.8 12.2 50.2 41.8 73.4 41.4z"/></svg> <a href="https://info.arxiv.org/help/contact.html"> Contact</a> </li> <li> <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 512 512" class="icon filter-black" role="presentation"><title>subscribe to arXiv mailings</title><desc>Click here to subscribe</desc><path d="M476 3.2L12.5 270.6c-18.1 10.4-15.8 35.6 2.2 43.2L121 358.4l287.3-253.2c5.5-4.9 13.3 2.6 8.6 8.3L176 407v80.5c0 23.6 28.5 32.9 42.5 15.8L282 426l124.6 52.2c14.2 6 30.4-2.9 33-18.2l72-432C515 7.8 493.3-6.8 476 3.2z"/></svg> <a href="https://info.arxiv.org/help/subscribe"> Subscribe</a> </li> </ul> </div> </div> </div>   <div class="column"> <div class="columns"> <div class="column"> <ul class="nav-spaced"> <li><a href="https://info.arxiv.org/help/license/index.html">Copyright</a></li> <li><a href="https://info.arxiv.org/help/policies/privacy_policy.html">Privacy Policy</a></li> </ul> </div> <div class="column sorry-app-links"> <ul class="nav-spaced"> <li><a href="https://info.arxiv.org/help/web_accessibility.html">Web Accessibility Assistance</a></li> <li> <p class="help"> <a class="a11y-main-link" href="https://status.arxiv.org" target="_blank">arXiv Operational Status <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 256 512" class="icon filter-dark_grey" role="presentation"><path d="M224.3 273l-136 136c-9.4 9.4-24.6 9.4-33.9 0l-22.6-22.6c-9.4-9.4-9.4-24.6 0-33.9l96.4-96.4-96.4-96.4c-9.4-9.4-9.4-24.6 0-33.9L54.3 103c9.4-9.4 24.6-9.4 33.9 0l136 136c9.5 9.4 9.5 24.6.1 34z"/></svg></a><br> Get status notifications via <a class="is-link" href="https://subscribe.sorryapp.com/24846f03/email/new" target="_blank"><svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 512 512" class="icon filter-black" role="presentation"><path d="M502.3 190.8c3.9-3.1 9.7-.2 9.7 4.7V400c0 26.5-21.5 48-48 48H48c-26.5 0-48-21.5-48-48V195.6c0-5 5.7-7.8 9.7-4.7 22.4 17.4 52.1 39.5 154.1 113.6 21.1 15.4 56.7 47.8 92.2 47.6 35.7.3 72-32.8 92.3-47.6 102-74.1 131.6-96.3 154-113.7zM256 320c23.2.4 56.6-29.2 73.4-41.4 132.7-96.3 142.8-104.7 173.4-128.7 5.8-4.5 9.2-11.5 9.2-18.9v-19c0-26.5-21.5-48-48-48H48C21.5 64 0 85.5 0 112v19c0 7.4 3.4 14.3 9.2 18.9 30.6 23.9 40.7 32.4 173.4 128.7 16.8 12.2 50.2 41.8 73.4 41.4z"/></svg>email</a> or <a class="is-link" href="https://subscribe.sorryapp.com/24846f03/slack/new" target="_blank"><svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 448 512" class="icon filter-black" role="presentation"><path d="M94.12 315.1c0 25.9-21.16 47.06-47.06 47.06S0 341 0 315.1c0-25.9 21.16-47.06 47.06-47.06h47.06v47.06zm23.72 0c0-25.9 21.16-47.06 47.06-47.06s47.06 21.16 47.06 47.06v117.84c0 25.9-21.16 47.06-47.06 47.06s-47.06-21.16-47.06-47.06V315.1zm47.06-188.98c-25.9 0-47.06-21.16-47.06-47.06S139 32 164.9 32s47.06 21.16 47.06 47.06v47.06H164.9zm0 23.72c25.9 0 47.06 21.16 47.06 47.06s-21.16 47.06-47.06 47.06H47.06C21.16 243.96 0 222.8 0 196.9s21.16-47.06 47.06-47.06H164.9zm188.98 47.06c0-25.9 21.16-47.06 47.06-47.06 25.9 0 47.06 21.16 47.06 47.06s-21.16 47.06-47.06 47.06h-47.06V196.9zm-23.72 0c0 25.9-21.16 47.06-47.06 47.06-25.9 0-47.06-21.16-47.06-47.06V79.06c0-25.9 21.16-47.06 47.06-47.06 25.9 0 47.06 21.16 47.06 47.06V196.9zM283.1 385.88c25.9 0 47.06 21.16 47.06 47.06 0 25.9-21.16 47.06-47.06 47.06-25.9 0-47.06-21.16-47.06-47.06v-47.06h47.06zm0-23.72c-25.9 0-47.06-21.16-47.06-47.06 0-25.9 21.16-47.06 47.06-47.06h117.84c25.9 0 47.06 21.16 47.06 47.06 0 25.9-21.16 47.06-47.06 47.06H283.1z"/></svg>slack</a> </p> </li> </ul> </div> </div> </div>  </div> </footer> <script src="https://static.arxiv.org/static/base/1.0.0a5/js/member_acknowledgement.js"></script> </body> </html>

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