Hamiltonian vector field

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<div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">From Wikipedia, the free encyclopedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><p>In <a href="/wiki/Mathematics" title="Mathematics">mathematics</a> and <a href="/wiki/Physics" title="Physics">physics</a>, a <b>Hamiltonian vector field</b> on a <a href="/wiki/Symplectic_manifold" title="Symplectic manifold">symplectic manifold</a> is a <a href="/wiki/Vector_field" title="Vector field">vector field</a> defined for any <b>energy function</b> or <b>Hamiltonian</b>. Named after the physicist and mathematician <a href="/wiki/William_Rowan_Hamilton" title="William Rowan Hamilton">Sir William Rowan Hamilton</a>, a Hamiltonian vector field is a geometric manifestation of <a href="/wiki/Hamilton%27s_equations" class="mw-redirect" title="Hamilton's equations">Hamilton's equations</a> in <a href="/wiki/Classical_mechanics" title="Classical mechanics">classical mechanics</a>. The <a href="/wiki/Integral_curve" title="Integral curve">integral curves</a> of a Hamiltonian vector field represent solutions to the equations of motion in the Hamiltonian form. The <a href="/wiki/Diffeomorphism" title="Diffeomorphism">diffeomorphisms</a> of a symplectic manifold arising from the <a href="/wiki/Flow_(mathematics)" title="Flow (mathematics)">flow</a> of a Hamiltonian vector field are known as <a href="/wiki/Canonical_transformation" title="Canonical transformation">canonical transformations</a> in physics and (Hamiltonian) <a href="/wiki/Symplectomorphism" title="Symplectomorphism">symplectomorphisms</a> in mathematics.<sup id="cite_ref-FOOTNOTELee2003Chapter_18_1-0" class="reference"><a href="#cite_note-FOOTNOTELee2003Chapter_18-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p><p>Hamiltonian vector fields can be defined more generally on an arbitrary <a href="/wiki/Poisson_manifold" title="Poisson manifold">Poisson manifold</a>. The <a href="/wiki/Lie_bracket_of_vector_fields" title="Lie bracket of vector fields">Lie bracket</a> of two Hamiltonian vector fields corresponding to functions <i>f</i> and <i>g</i> on the manifold is itself a Hamiltonian vector field, with the Hamiltonian given by the <a href="/wiki/Poisson_bracket" title="Poisson bracket">Poisson bracket</a> of <i>f</i> and <i>g</i>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hamiltonian_vector_field&action=edit&section=1" title="Edit section: Definition"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Suppose that <span class="texhtml">(<i>M</i>, <i>ω</i>)</span> is a <a href="/wiki/Symplectic_manifold" title="Symplectic manifold">symplectic manifold</a>. Since the <a href="/wiki/Symplectic_form" class="mw-redirect" title="Symplectic form">symplectic form</a> <span class="texhtml"><i>ω</i></span> is nondegenerate, it sets up a <i>fiberwise-linear</i> <a href="/wiki/Isomorphism" title="Isomorphism">isomorphism</a> </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega :TM\to T^{*}M,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ω</mi> <mo>:</mo> <mi>T</mi> <mi>M</mi> <mo stretchy="false">→</mo> <msup> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mi>M</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega :TM\to T^{*}M,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6b378c126fddd1bcc3a5632c905a80fee4a0c2de" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:16.939ex; height:2.676ex;" alt="{\displaystyle \omega :TM\to T^{*}M,}"></span> </p><p>between the <a href="/wiki/Tangent_bundle" title="Tangent bundle">tangent bundle</a> <span class="texhtml"><i>TM</i></span> and the <a href="/wiki/Cotangent_bundle" title="Cotangent bundle">cotangent bundle</a> <span class="texhtml"><i>T*M</i></span>, with the inverse </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega :T^{*}M\to TM,\quad \Omega =\omega ^{-1}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Ω</mi> <mo>:</mo> <msup> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mi>M</mi> <mo stretchy="false">→</mo> <mi>T</mi> <mi>M</mi> <mo>,</mo> <mspace width="1em" /> <mi mathvariant="normal">Ω</mi> <mo>=</mo> <msup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Omega :T^{*}M\to TM,\quad \Omega =\omega ^{-1}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c37d00d607897484563491ba565c8ae1f45de670" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:29.083ex; height:3.009ex;" alt="{\displaystyle \Omega :T^{*}M\to TM,\quad \Omega =\omega ^{-1}.}"></span> </p><p>Therefore, <a href="/wiki/One-form" class="mw-redirect" title="One-form">one-forms</a> on a symplectic manifold <span class="texhtml"><i>M</i></span> may be identified with <a href="/wiki/Vector_field" title="Vector field">vector fields</a> and every <a href="/wiki/Differentiable_function" title="Differentiable function">differentiable function</a> <span class="texhtml"><i>H</i>: <i>M</i> → <b>R</b></span> determines a unique <a href="/wiki/Vector_field" title="Vector field">vector field</a> <span class="texhtml"><i>X<sub>H</sub></i></span>, called the <i>Hamiltonian vector field</i> with the <i>Hamiltonian</i> <span class="texhtml"><i>H</i></span>, by defining for every vector field <span class="texhtml"><i>Y</i></span> on <span class="texhtml"><i>M</i></span>, </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {d} H(Y)=\omega (X_{H},Y).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>H</mi> <mo stretchy="false">(</mo> <mi>Y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>ω</mi> <mo stretchy="false">(</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>H</mi> </mrow> </msub> <mo>,</mo> <mi>Y</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {d} H(Y)=\omega (X_{H},Y).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ac368ec47276a2b4ddba0a0eb1679e1df48d1a31" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.362ex; height:2.843ex;" alt="{\displaystyle \mathrm {d} H(Y)=\omega (X_{H},Y).}"></span> </p><p><b>Note</b>: Some authors define the Hamiltonian vector field with the opposite sign. One has to be mindful of varying conventions in physical and mathematical literature. </p> <div class="mw-heading mw-heading2"><h2 id="Examples">Examples</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hamiltonian_vector_field&action=edit&section=2" title="Edit section: Examples"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Suppose that <span class="texhtml"><i>M</i></span> is a <span class="texhtml">2<i>n</i></span>-dimensional symplectic manifold. Then locally, one may choose <a href="/wiki/Canonical_coordinates" title="Canonical coordinates">canonical coordinates</a> <span class="texhtml">(<i>q</i><sup>1</sup>, ..., <i>q<sup>n</sup></i>, <i>p</i><sub>1</sub>, ..., <i>p<sub>n</sub></i>)</span> on <span class="texhtml"><i>M</i></span>, in which the symplectic form is expressed as:<sup id="cite_ref-FOOTNOTELee2003Chapter_12_2-0" class="reference"><a href="#cite_note-FOOTNOTELee2003Chapter_12-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega =\sum _{i}\mathrm {d} q^{i}\wedge \mathrm {d} p_{i},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ω</mi> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> <mo>∧</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega =\sum _{i}\mathrm {d} q^{i}\wedge \mathrm {d} p_{i},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/26be08c6b0583926dfaff37719da89676bc96fcc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:17.949ex; height:5.509ex;" alt="{\displaystyle \omega =\sum _{i}\mathrm {d} q^{i}\wedge \mathrm {d} p_{i},}"></span> </p><p>where <span class="texhtml">d</span> denotes the <a href="/wiki/Exterior_derivative" title="Exterior derivative">exterior derivative</a> and <span class="texhtml">∧</span> denotes the <a href="/wiki/Exterior_product" class="mw-redirect" title="Exterior product">exterior product</a>. Then the Hamiltonian vector field with Hamiltonian <span class="texhtml"><i>H</i></span> takes the form:<sup id="cite_ref-FOOTNOTELee2003Chapter_18_1-1" class="reference"><a href="#cite_note-FOOTNOTELee2003Chapter_18-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {X} _{H}=\left({\frac {\partial H}{\partial p_{i}}},-{\frac {\partial H}{\partial q^{i}}}\right)=\Omega \,\mathrm {d} H,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">X</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>H</mi> </mrow> </msub> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>H</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>,</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>H</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mi mathvariant="normal">Ω</mi> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>H</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {X} _{H}=\left({\frac {\partial H}{\partial p_{i}}},-{\frac {\partial H}{\partial q^{i}}}\right)=\Omega \,\mathrm {d} H,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8a8e1a5ef41129d1483eae5a57be0febda0a7116" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:30.399ex; height:6.176ex;" alt="{\displaystyle \mathrm {X} _{H}=\left({\frac {\partial H}{\partial p_{i}}},-{\frac {\partial H}{\partial q^{i}}}\right)=\Omega \,\mathrm {d} H,}"></span> </p><p>where <span class="texhtml">Ω</span> is a <span class="texhtml">2<i>n</i> × 2<i>n</i></span> square matrix </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega ={\begin{bmatrix}0&I_{n}\\-I_{n}&0\\\end{bmatrix}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Ω</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Omega ={\begin{bmatrix}0&I_{n}\\-I_{n}&0\\\end{bmatrix}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d382eac5f49e429f9cacde1f8eb3f8e6c7008f24" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:17.244ex; height:6.176ex;" alt="{\displaystyle \Omega ={\begin{bmatrix}0&I_{n}\\-I_{n}&0\\\end{bmatrix}},}"></span> </p><p>and </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {d} H={\begin{bmatrix}{\frac {\partial H}{\partial q^{i}}}\\{\frac {\partial H}{\partial p_{i}}}\end{bmatrix}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>H</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>H</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>H</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </mfrac> </mrow> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {d} H={\begin{bmatrix}{\frac {\partial H}{\partial q^{i}}}\\{\frac {\partial H}{\partial p_{i}}}\end{bmatrix}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/838a20f7a8f5a09ebfc0f4625544950c916d33d1" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.838ex; width:14.181ex; height:8.843ex;" alt="{\displaystyle \mathrm {d} H={\begin{bmatrix}{\frac {\partial H}{\partial q^{i}}}\\{\frac {\partial H}{\partial p_{i}}}\end{bmatrix}}.}"></span> </p><p>The matrix <span class="texhtml">Ω</span> is frequently denoted with <span class="texhtml"><b>J</b></span>. </p><p>Suppose that <i>M</i> = <b>R</b><sup>2<i>n</i></sup> is the 2<i>n</i>-dimensional <a href="/wiki/Symplectic_vector_space" title="Symplectic vector space">symplectic vector space</a> with (global) canonical coordinates. </p> <ul><li>If <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H=p_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> <mo>=</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H=p_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0332ba74c10cea0d0d24735fb6500c7449886bfd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.131ex; height:2.509ex;" alt="{\displaystyle H=p_{i}}"></span> then <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{H}=\partial /\partial q^{i};}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>H</mi> </mrow> </msub> <mo>=</mo> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> <mo>;</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{H}=\partial /\partial q^{i};}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d280053c1b964e8a7b2f937ee4dde967dfddef82" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.039ex; height:3.176ex;" alt="{\displaystyle X_{H}=\partial /\partial q^{i};}"></span></li> <li>if <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H=q_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> <mo>=</mo> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H=q_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fa2f13827a48bbf1a8c9a8f3e712ebb8f84c537f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.999ex; height:2.509ex;" alt="{\displaystyle H=q_{i}}"></span> then <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{H}=-\partial /\partial p^{i};}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>H</mi> </mrow> </msub> <mo>=</mo> <mo>−</mo> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> <mo>;</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{H}=-\partial /\partial p^{i};}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6c5a253976fc003a1e9709559b1a174f526d5e91" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.937ex; height:3.176ex;" alt="{\displaystyle X_{H}=-\partial /\partial p^{i};}"></span></li> <li>if <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle H={\frac {1}{2}}\sum (p_{i})^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>H</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo>∑</mo> <mo stretchy="false">(</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle H={\frac {1}{2}}\sum (p_{i})^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/34ca75e9b695255fff91bb01c1c3cc3958ed3413" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:14.494ex; height:3.509ex;" alt="{\textstyle H={\frac {1}{2}}\sum (p_{i})^{2}}"></span> then <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle X_{H}=\sum p_{i}\partial /\partial q^{i};}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>H</mi> </mrow> </msub> <mo>=</mo> <mo>∑</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> <mo>;</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle X_{H}=\sum p_{i}\partial /\partial q^{i};}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/491bb956c5e876eeed56175858d9dc1c5f33e7bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.849ex; height:3.009ex;" alt="{\textstyle X_{H}=\sum p_{i}\partial /\partial q^{i};}"></span></li> <li>if <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle H={\frac {1}{2}}\sum a_{ij}q^{i}q^{j},a_{ij}=a_{ji}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>H</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo>∑</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msup> <mo>,</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle H={\frac {1}{2}}\sum a_{ij}q^{i}q^{j},a_{ij}=a_{ji}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a092dd166234d3b240eb22663d5b118aead96713" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:26.17ex; height:3.509ex;" alt="{\textstyle H={\frac {1}{2}}\sum a_{ij}q^{i}q^{j},a_{ij}=a_{ji}}"></span> then <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle X_{H}=-\sum a_{ij}q_{i}\partial /\partial p^{j}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>H</mi> </mrow> </msub> <mo>=</mo> <mo>−</mo> <mo>∑</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle X_{H}=-\sum a_{ij}q_{i}\partial /\partial p^{j}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7423caa6e9abcbdb9ed9080930f86e671aeb359e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:22.819ex; height:3.176ex;" alt="{\textstyle X_{H}=-\sum a_{ij}q_{i}\partial /\partial p^{j}.}"></span></li></ul> <div class="mw-heading mw-heading2"><h2 id="Properties">Properties</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hamiltonian_vector_field&action=edit&section=3" title="Edit section: Properties"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>The assignment <span class="texhtml"><i>f</i> ↦ <i>X<sub>f</sub></i></span> is <a href="/wiki/Linear_map" title="Linear map">linear</a>, so that the sum of two Hamiltonian functions transforms into the sum of the corresponding Hamiltonian vector fields.</li> <li>Suppose that <span class="texhtml">(<i>q</i><sup>1</sup>, ..., <i>q<sup>n</sup></i>, <i>p</i><sub>1</sub>, ..., <i>p<sub>n</sub></i>)</span> are canonical coordinates on <span class="texhtml"><i>M</i></span> (see above). Then a curve <span class="texhtml"><i>γ</i>(<i>t</i>) = (<i>q</i>(<i>t</i>),<i>p</i>(<i>t</i>))</span> is an <a href="/wiki/Integral_curve" title="Integral curve">integral curve</a> of the Hamiltonian vector field <span class="texhtml"><i>X<sub>H</sub></i></span> if and only if it is a solution of <a href="/wiki/Hamilton%27s_equations" class="mw-redirect" title="Hamilton's equations">Hamilton's equations</a>:<sup id="cite_ref-FOOTNOTELee2003Chapter_18_1-2" class="reference"><a href="#cite_note-FOOTNOTELee2003Chapter_18-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}{\dot {q}}^{i}&={\frac {\partial H}{\partial p_{i}}}\\{\dot {p}}_{i}&=-{\frac {\partial H}{\partial q^{i}}}.\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>q</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>H</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>p</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>H</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\dot {q}}^{i}&={\frac {\partial H}{\partial p_{i}}}\\{\dot {p}}_{i}&=-{\frac {\partial H}{\partial q^{i}}}.\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dbe8e152ada934af076e1315e44993d88622b78e" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.211ex; margin-bottom: -0.294ex; width:12.7ex; height:12.176ex;" alt="{\displaystyle {\begin{aligned}{\dot {q}}^{i}&={\frac {\partial H}{\partial p_{i}}}\\{\dot {p}}_{i}&=-{\frac {\partial H}{\partial q^{i}}}.\end{aligned}}}"></span></li> <li>The Hamiltonian <span class="texhtml"><i>H</i></span> is constant along the integral curves, because <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle dH,{\dot {\gamma }}\rangle =\omega (X_{H}(\gamma ),X_{H}(\gamma ))=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mi>d</mi> <mi>H</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>γ</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <mi>ω</mi> <mo stretchy="false">(</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>H</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>γ</mi> <mo stretchy="false">)</mo> <mo>,</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>H</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>γ</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle dH,{\dot {\gamma }}\rangle =\omega (X_{H}(\gamma ),X_{H}(\gamma ))=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ea8f69f650992389b86deb06418c3dba0ecc079c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:32.408ex; height:2.843ex;" alt="{\displaystyle \langle dH,{\dot {\gamma }}\rangle =\omega (X_{H}(\gamma ),X_{H}(\gamma ))=0}"></span>. That is, <span class="texhtml"><i>H</i>(<i>γ</i>(<i>t</i>))</span> is actually independent of <span class="texhtml"><i>t</i></span>. This property corresponds to the <a href="/wiki/Conservation_of_energy" title="Conservation of energy">conservation of energy</a> in <a href="/wiki/Hamiltonian_mechanics" title="Hamiltonian mechanics">Hamiltonian mechanics</a>.</li> <li>More generally, if two functions <span class="texhtml"><i>F</i></span> and <span class="texhtml"><i>H</i></span> have a zero <a href="/wiki/Poisson_bracket" title="Poisson bracket">Poisson bracket</a> (cf. below), then <span class="texhtml"><i>F</i></span> is constant along the integral curves of <span class="texhtml"><i>H</i></span>, and similarly, <span class="texhtml"><i>H</i></span> is constant along the integral curves of <span class="texhtml"><i>F</i></span>. This fact is the abstract mathematical principle behind <a href="/wiki/Noether%27s_theorem" title="Noether's theorem">Noether's theorem</a>.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>nb 1<span class="cite-bracket">]</span></a></sup></li> <li>The <a href="/wiki/Symplectic_form" class="mw-redirect" title="Symplectic form">symplectic form</a> <span class="texhtml mvar" style="font-style:italic;">ω</span> is preserved by the Hamiltonian flow. Equivalently, the <a href="/wiki/Lie_derivative" title="Lie derivative">Lie derivative</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {L}}_{X_{H}}\omega =0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>H</mi> </mrow> </msub> </mrow> </msub> <mi>ω</mi> <mo>=</mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {L}}_{X_{H}}\omega =0.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6c9ba23c9b4b9012a3f35a8de790eed6be4e4ddf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:10.899ex; height:2.843ex;" alt="{\displaystyle {\mathcal {L}}_{X_{H}}\omega =0.}"></span></li></ul> <div class="mw-heading mw-heading2"><h2 id="Poisson_bracket">Poisson bracket</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hamiltonian_vector_field&action=edit&section=4" title="Edit section: Poisson bracket"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The notion of a Hamiltonian vector field leads to a <a href="/wiki/Bilinear_form#Symmetric,_skew-symmetric_and_alternating_forms" title="Bilinear form">skew-symmetric</a> bilinear operation on the differentiable functions on a symplectic manifold <i>M</i>, the <b><a href="/wiki/Poisson_bracket" title="Poisson bracket">Poisson bracket</a></b>, defined by the formula </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{f,g\}=\omega (X_{g},X_{f})=dg(X_{f})={\mathcal {L}}_{X_{f}}g}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mi>f</mi> <mo>,</mo> <mi>g</mi> <mo fence="false" stretchy="false">}</mo> <mo>=</mo> <mi>ω</mi> <mo stretchy="false">(</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>g</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mi>d</mi> <mi>g</mi> <mo stretchy="false">(</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> </msub> </mrow> </msub> <mi>g</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{f,g\}=\omega (X_{g},X_{f})=dg(X_{f})={\mathcal {L}}_{X_{f}}g}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/02a3b190798e5c08b4349083c6608593a36d4b1a" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:37.757ex; height:3.176ex;" alt="{\displaystyle \{f,g\}=\omega (X_{g},X_{f})=dg(X_{f})={\mathcal {L}}_{X_{f}}g}"></span> </p><p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {L}}_{X}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {L}}_{X}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ce4db2d17b365a7321dbfdb8f8bc512dd911ea54" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.236ex; height:2.509ex;" alt="{\displaystyle {\mathcal {L}}_{X}}"></span> denotes the <a href="/wiki/Lie_derivative" title="Lie derivative">Lie derivative</a> along a vector field <i>X</i>. Moreover, one can check that the following identity holds:<sup id="cite_ref-FOOTNOTELee2003Chapter_18_1-3" class="reference"><a href="#cite_note-FOOTNOTELee2003Chapter_18-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{\{f,g\}}=-[X_{f},X_{g}],}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mo fence="false" stretchy="false">{</mo> <mi>f</mi> <mo>,</mo> <mi>g</mi> <mo fence="false" stretchy="false">}</mo> </mrow> </msub> <mo>=</mo> <mo>−</mo> <mo stretchy="false">[</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>g</mi> </mrow> </msub> <mo stretchy="false">]</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{\{f,g\}}=-[X_{f},X_{g}],}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/34d7e458dd64e9cd7f7b55b966875a200adef820" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:19.838ex; height:3.176ex;" alt="{\displaystyle X_{\{f,g\}}=-[X_{f},X_{g}],}"></span> </p><p>where the right hand side represents the Lie bracket of the Hamiltonian vector fields with Hamiltonians <i>f</i> and <i>g</i>. As a consequence (a proof at <a href="/wiki/Poisson_bracket" title="Poisson bracket">Poisson bracket</a>), the Poisson bracket satisfies the <a href="/wiki/Jacobi_identity" title="Jacobi identity">Jacobi identity</a>:<sup id="cite_ref-FOOTNOTELee2003Chapter_18_1-4" class="reference"><a href="#cite_note-FOOTNOTELee2003Chapter_18-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{\{f,g\},h\}+\{\{g,h\},f\}+\{\{h,f\},g\}=0,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mo fence="false" stretchy="false">{</mo> <mi>f</mi> <mo>,</mo> <mi>g</mi> <mo fence="false" stretchy="false">}</mo> <mo>,</mo> <mi>h</mi> <mo fence="false" stretchy="false">}</mo> <mo>+</mo> <mo fence="false" stretchy="false">{</mo> <mo fence="false" stretchy="false">{</mo> <mi>g</mi> <mo>,</mo> <mi>h</mi> <mo fence="false" stretchy="false">}</mo> <mo>,</mo> <mi>f</mi> <mo fence="false" stretchy="false">}</mo> <mo>+</mo> <mo fence="false" stretchy="false">{</mo> <mo fence="false" stretchy="false">{</mo> <mi>h</mi> <mo>,</mo> <mi>f</mi> <mo fence="false" stretchy="false">}</mo> <mo>,</mo> <mi>g</mi> <mo fence="false" stretchy="false">}</mo> <mo>=</mo> <mn>0</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{\{f,g\},h\}+\{\{g,h\},f\}+\{\{h,f\},g\}=0,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/488599f2cced293b3d1256031e4c1f814b08b866" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:41.942ex; height:2.843ex;" alt="{\displaystyle \{\{f,g\},h\}+\{\{g,h\},f\}+\{\{h,f\},g\}=0,}"></span> </p><p>which means that the vector space of differentiable functions on <span class="texhtml"><i>M</i></span>, endowed with the Poisson bracket, has the structure of a <a href="/wiki/Lie_algebra" title="Lie algebra">Lie algebra</a> over <span class="texhtml"><b>R</b></span>, and the assignment <span class="texhtml"><i>f</i> ↦ <i>X<sub>f</sub></i></span> is a <a href="/wiki/Lie_algebra_homomorphism" class="mw-redirect" title="Lie algebra homomorphism">Lie algebra homomorphism</a>, whose <a href="/wiki/Kernel_(linear_algebra)" title="Kernel (linear algebra)">kernel</a> consists of the locally constant functions (constant functions if <span class="texhtml"><i>M</i></span> is connected). </p> <div class="mw-heading mw-heading2"><h2 id="Remarks">Remarks</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hamiltonian_vector_field&action=edit&section=5" title="Edit section: Remarks"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text">See <a href="#CITEREFLee2003">Lee (2003</a>, Chapter 18) for a very concise statement and proof of Noether's theorem.</span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hamiltonian_vector_field&action=edit&section=6" title="Edit section: Notes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239543626"><div class="reflist reflist-columns references-column-width" style="column-width: 22em;"> <ol class="references"> <li id="cite_note-FOOTNOTELee2003Chapter_18-1"><span class="mw-cite-backlink">^ <a href="#cite_ref-FOOTNOTELee2003Chapter_18_1-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-FOOTNOTELee2003Chapter_18_1-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-FOOTNOTELee2003Chapter_18_1-2"><sup><i><b>c</b></i></sup></a> <a href="#cite_ref-FOOTNOTELee2003Chapter_18_1-3"><sup><i><b>d</b></i></sup></a> <a href="#cite_ref-FOOTNOTELee2003Chapter_18_1-4"><sup><i><b>e</b></i></sup></a></span> <span class="reference-text"><a href="#CITEREFLee2003">Lee 2003</a>, Chapter 18.</span> </li> <li id="cite_note-FOOTNOTELee2003Chapter_12-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTELee2003Chapter_12_2-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFLee2003">Lee 2003</a>, Chapter 12.</span> </li> </ol></div> <div class="mw-heading mw-heading2"><h2 id="Works_cited">Works cited</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hamiltonian_vector_field&action=edit&section=7" title="Edit section: Works cited"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239549316">.mw-parser-output .refbegin{margin-bottom:0.5em}.mw-parser-output .refbegin-hanging-indents>ul{margin-left:0}.mw-parser-output .refbegin-hanging-indents>ul>li{margin-left:0;padding-left:3.2em;text-indent:-3.2em}.mw-parser-output .refbegin-hanging-indents ul,.mw-parser-output .refbegin-hanging-indents ul li{list-style:none}@media(max-width:720px){.mw-parser-output .refbegin-hanging-indents>ul>li{padding-left:1.6em;text-indent:-1.6em}}.mw-parser-output .refbegin-columns{margin-top:0.3em}.mw-parser-output .refbegin-columns ul{margin-top:0}.mw-parser-output .refbegin-columns li{page-break-inside:avoid;break-inside:avoid-column}@media screen{.mw-parser-output .refbegin{font-size:90%}}</style><div class="refbegin" style=""> <ul><li><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFAbrahamMarsden1978" class="citation book cs1"><a href="/wiki/Ralph_Abraham_(mathematician)" title="Ralph Abraham (mathematician)">Abraham, Ralph</a>; <a href="/wiki/Jerrold_E._Marsden" title="Jerrold E. Marsden">Marsden, Jerrold E.</a> (1978). <i>Foundations of Mechanics</i>. London: Benjamin-Cummings. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-080530102-1" title="Special:BookSources/978-080530102-1"><bdi>978-080530102-1</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Foundations+of+Mechanics&rft.place=London&rft.pub=Benjamin-Cummings&rft.date=1978&rft.isbn=978-080530102-1&rft.aulast=Abraham&rft.aufirst=Ralph&rft.au=Marsden%2C+Jerrold+E.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHamiltonian+vector+field" class="Z3988"></span><i>See section 3.2</i>.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFArnol'd1997" class="citation book cs1"><a href="/wiki/Vladimir_Arnold" title="Vladimir Arnold">Arnol'd, V.I.</a> (1997). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/mathematicalmeth0000arno"><i>Mathematical Methods of Classical Mechanics</i></a></span>. Berlin etc: Springer. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-387-96890-3" title="Special:BookSources/0-387-96890-3"><bdi>0-387-96890-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Mathematical+Methods+of+Classical+Mechanics&rft.place=Berlin+etc&rft.pub=Springer&rft.date=1997&rft.isbn=0-387-96890-3&rft.aulast=Arnol%27d&rft.aufirst=V.I.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fmathematicalmeth0000arno&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHamiltonian+vector+field" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFrankel1997" class="citation book cs1">Frankel, Theodore (1997). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/geometryofphysic0000fran"><i>The Geometry of Physics</i></a></span>. Cambridge University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-521-38753-1" title="Special:BookSources/0-521-38753-1"><bdi>0-521-38753-1</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Geometry+of+Physics&rft.pub=Cambridge+University+Press&rft.date=1997&rft.isbn=0-521-38753-1&rft.aulast=Frankel&rft.aufirst=Theodore&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fgeometryofphysic0000fran&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHamiltonian+vector+field" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLee2003" class="citation cs2">Lee, J. M. (2003), <i>Introduction to Smooth manifolds</i>, Springer Graduate Texts in Mathematics, vol. 218, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-387-95448-1" title="Special:BookSources/0-387-95448-1"><bdi>0-387-95448-1</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introduction+to+Smooth+manifolds&rft.series=Springer+Graduate+Texts+in+Mathematics&rft.date=2003&rft.isbn=0-387-95448-1&rft.aulast=Lee&rft.aufirst=J.+M.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHamiltonian+vector+field" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMcDuffSalamon1998" class="citation book cs1"><a href="/wiki/Dusa_McDuff" title="Dusa McDuff">McDuff, Dusa</a>; Salamon, D. (1998). <i>Introduction to Symplectic Topology</i>. Oxford Mathematical Monographs. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-19-850451-9" title="Special:BookSources/0-19-850451-9"><bdi>0-19-850451-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introduction+to+Symplectic+Topology&rft.series=Oxford+Mathematical+Monographs&rft.date=1998&rft.isbn=0-19-850451-9&rft.aulast=McDuff&rft.aufirst=Dusa&rft.au=Salamon%2C+D.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHamiltonian+vector+field" class="Z3988"></span></li></ul> </div> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hamiltonian_vector_field&action=edit&section=8" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="https://ncatlab.org/nlab/show/Hamiltonian%2Bvector%2Bfield" class="extiw" title="nlab:Hamiltonian+vector+field">Hamiltonian vector field</a> on <a href="/wiki/NLab" title="NLab">nLab</a></li></ul>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?useformat=desktop&type=1x1&usesul3=0" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Retrieved from "<a dir="ltr" href="https://en.wikipedia.org/w/index.php?title=Hamiltonian_vector_field&oldid=1262301364">https://en.wikipedia.org/w/index.php?title=Hamiltonian_vector_field&oldid=1262301364</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Help:Category" title="Help:Category">Categories</a>: <ul><li><a href="/wiki/Category:Hamiltonian_mechanics" title="Category:Hamiltonian mechanics">Hamiltonian mechanics</a></li><li><a href="/wiki/Category:Symplectic_geometry" title="Category:Symplectic geometry">Symplectic geometry</a></li><li><a href="/wiki/Category:William_Rowan_Hamilton" title="Category:William Rowan Hamilton">William Rowan Hamilton</a></li></ul></div></div> </div> </main> </div> <div class="mw-footer-container"> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> This page was last edited on 10 December 2024, at 18:01<span class="anonymous-show"> (UTC)</span>.</li> <li id="footer-info-copyright">Text is available under the <a href="/wiki/Wikipedia:Text_of_the_Creative_Commons_Attribution-ShareAlike_4.0_International_License" title="Wikipedia:Text of the Creative Commons Attribution-ShareAlike 4.0 International License">Creative Commons Attribution-ShareAlike 4.0 License</a>; additional terms may apply. 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