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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Poisson bracket</span></span>
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</style><table class="sidebar sidebar-collapse nomobile nowraplinks cm-sidebar"><tbody><tr><td class="sidebar-pretitle">Part of a series on</td></tr><tr><th class="sidebar-title-with-pretitle cm-sidebar-title"><a href="Classical_mechanics" title="Classical mechanics">Classical mechanics</a></th></tr><tr><td class="sidebar-image"><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\textbf {F}}={\frac {d\mathbf {p} }{dt}}}">
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<annotation encoding="application/x-tex">{\displaystyle {\textbf {F}}={\frac {d\mathbf {p} }{dt}}}</annotation>
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</math></span><img src="./c2ad0a6d6780c3abc5247abd82bd8a2249d56ff3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:8.318ex; height:5.509ex;" alt="{\displaystyle {\textbf {F}}={\frac {d\mathbf {p} }{dt}}}" loading="lazy"></span><div class="sidebar-caption"><a href="Second_law_of_motion" class="mw-redirect" title="Second law of motion">Second law of motion</a></div></td></tr><tr><th class="sidebar-heading cm-sidebar-above">
<div class="hlist">
<ul><li><a href="History_of_classical_mechanics" title="History of classical mechanics">History</a></li>
<li><a href="Timeline_of_classical_mechanics" title="Timeline of classical mechanics">Timeline</a></li>
<li><a href="List_of_textbooks_on_classical_mechanics_and_quantum_mechanics" title="List of textbooks on classical mechanics and quantum mechanics">Textbooks</a></li></ul>
</div></th></tr><tr><td class="sidebar-content">
<div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)"><div class="sidebar-list-title-c">Branches</div></div><div class="sidebar-list-content mw-collapsible-content plainlist"><div class="hlist">
<ul><li><a href="Applied_mechanics" title="Applied mechanics">Applied</a></li>
<li><a href="Celestial_mechanics" title="Celestial mechanics">Celestial</a></li>
<li><a href="Continuum_mechanics" title="Continuum mechanics">Continuum</a></li>
<li><a href="Dynamics_(mechanics)" title="Dynamics (mechanics)">Dynamics</a></li>
<li><a href="Classical_field_theory" title="Classical field theory">Field theory</a></li>
<li><a href="Kinematics" title="Kinematics">Kinematics</a></li>
<li><a href="Kinetics_(physics)" title="Kinetics (physics)">Kinetics</a></li>
<li><a href="Statics" title="Statics">Statics</a></li>
<li><a href="Statistical_mechanics" title="Statistical mechanics">Statistical mechanics</a></li></ul>
</div></div></div></td>
</tr><tr><td class="sidebar-content">
<div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)"><div class="sidebar-list-title-c">Fundamentals</div></div><div class="sidebar-list-content mw-collapsible-content plainlist"><div class="hlist">
<ul><li><a href="Acceleration" title="Acceleration">Acceleration</a></li>
<li><a href="Angular_momentum" title="Angular momentum">Angular momentum</a></li>
<li><a href="Couple_(mechanics)" title="Couple (mechanics)">Couple</a></li>
<li><a href="D'Alembert's_principle" title="D'Alembert's principle">D'Alembert's principle</a></li>
<li><a href="Energy" title="Energy">Energy</a>
<ul><li><a href="Kinetic_energy#Newtonian_kinetic_energy" title="Kinetic energy">kinetic</a></li>
<li><a href="Potential_energy" title="Potential energy">potential</a></li></ul></li>
<li><a href="Force" title="Force">Force</a></li>
<li><a href="Frame_of_reference" title="Frame of reference">Frame of reference</a></li>
<li><a href="Inertial_frame_of_reference" title="Inertial frame of reference">Inertial frame of reference</a></li>
<li><a href="Impulse_(physics)" title="Impulse (physics)">Impulse</a></li>
<li><span class="nowrap"><a href="Inertia" title="Inertia">Inertia</a>&nbsp;/ <a href="Moment_of_inertia" title="Moment of inertia">Moment of inertia</a></span></li>
<li><a href="Mass" title="Mass">Mass</a></li>
<li><br><a href="Mechanical_power_(physics)" class="mw-redirect" title="Mechanical power (physics)">Mechanical power</a></li>
<li><a href="Work_(physics)" title="Work (physics)">Mechanical work</a></li>
<li><br><a href="Moment_(physics)" title="Moment (physics)">Moment</a></li>
<li><a href="Momentum" title="Momentum">Momentum</a></li>
<li><a href="Space" title="Space">Space</a></li>
<li><a href="Speed" title="Speed">Speed</a></li>
<li><a href="Time" title="Time">Time</a></li>
<li><a href="Torque" title="Torque">Torque</a></li>
<li><a href="Velocity" title="Velocity">Velocity</a></li>
<li><a href="Virtual_work" title="Virtual work">Virtual work</a></li></ul>
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<div class="sidebar-list mw-collapsible"><div class="sidebar-list-title" style="color: var(--color-base)"><div class="sidebar-list-title-c">Formulations</div></div><div class="sidebar-list-content mw-collapsible-content plainlist">
<ul><li><div style="display: inline-block; line-height: 1.2em; padding: .1em 0;"><b><a href="Newton's_laws_of_motion" title="Newton's laws of motion">Newton's laws of motion</a></b></div></li>
<li><div style="display: inline-block; line-height: 1.2em; padding: .1em 0;"><b><a href="Analytical_mechanics" title="Analytical mechanics">Analytical mechanics</a></b> <div class="plainlist"><ul><li><a href="Lagrangian_mechanics" title="Lagrangian mechanics">Lagrangian mechanics</a></li><li><a href="Hamiltonian_mechanics" title="Hamiltonian mechanics">Hamiltonian mechanics</a></li><li><a href="Routhian_mechanics" title="Routhian mechanics">Routhian mechanics</a></li><li><a href="Hamilton%E2%80%93Jacobi_equation" title="Hamilton–Jacobi equation">Hamilton–Jacobi equation</a></li><li><a href="Appell's_equation_of_motion" title="Appell's equation of motion">Appell's equation of motion</a></li><li><a href="Koopman%E2%80%93von_Neumann_classical_mechanics" title="Koopman–von Neumann classical mechanics">Koopman–von Neumann mechanics</a></li></ul></div></div></li></ul></div></div></td>
</tr><tr><td class="sidebar-content">
<div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)"><div class="sidebar-list-title-c">Core topics</div></div><div class="sidebar-list-content mw-collapsible-content plainlist"><div class="hlist">
<ul><li><a href="Damping" title="Damping">Damping</a></li>
<li><a href="Displacement_(geometry)" title="Displacement (geometry)">Displacement</a></li>
<li><a href="Equations_of_motion" title="Equations of motion">Equations of motion</a></li>
<li><a href="Euler's_laws_of_motion" title="Euler's laws of motion"><span class="wrap">Euler's laws of motion</span></a></li>
<li><a href="Fictitious_force" title="Fictitious force">Fictitious force</a></li>
<li><a href="Friction" title="Friction">Friction</a></li>
<li><a href="Harmonic_oscillator" title="Harmonic oscillator">Harmonic oscillator</a></li></ul>
</div>
<ul><li><span class="nowrap"><a href="Inertial_frame_of_reference" title="Inertial frame of reference">Inertial</a>&nbsp;/ <a href="Non-inertial_reference_frame" title="Non-inertial reference frame">Non-inertial reference frame</a></span></li></ul>
<div class="hlist">
<ul><li><a href="Motion" title="Motion">Motion</a>&nbsp;(<a href="Linear_motion" title="Linear motion">linear</a>)</li>
<li><a href="Newton's_law_of_universal_gravitation" title="Newton's law of universal gravitation"><span class="wrap">Newton's law of universal gravitation</span></a></li>
<li><a href="Newton's_laws_of_motion" title="Newton's laws of motion">Newton's laws of motion</a></li>
<li><a href="Relative_velocity" title="Relative velocity">Relative velocity</a></li>
<li><a href="Rigid_body" title="Rigid body">Rigid body</a>
<ul><li><a href="Rigid_body_dynamics" title="Rigid body dynamics">dynamics</a></li>
<li><a href="Euler's_equations_(rigid_body_dynamics)" title="Euler's equations (rigid body dynamics)">Euler's equations</a></li></ul></li>
<li><a href="Simple_harmonic_motion" title="Simple harmonic motion">Simple harmonic motion</a></li>
<li><a href="Vibration" title="Vibration">Vibration</a></li></ul>
</div></div></div></td>
</tr><tr><td class="sidebar-content">
<div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)"><div class="sidebar-list-title-c"><a href="Rotation_around_a_fixed_axis" title="Rotation around a fixed axis">Rotation</a></div></div><div class="sidebar-list-content mw-collapsible-content plainlist"><div class="hlist">
<ul><li><a href="Circular_motion" title="Circular motion">Circular motion</a></li>
<li><a href="Rotating_reference_frame" title="Rotating reference frame">Rotating reference frame</a></li>
<li><a href="Centripetal_force" title="Centripetal force">Centripetal force</a></li>
<li><a href="Centrifugal_force" title="Centrifugal force">Centrifugal force</a>
<ul><li><a href="Reactive_centrifugal_force" title="Reactive centrifugal force">reactive</a></li></ul></li>
<li><a href="Coriolis_force" title="Coriolis force">Coriolis force</a></li>
<li><a href="Pendulum_(mechanics)" title="Pendulum (mechanics)">Pendulum</a></li>
<li><a href="Tangential_speed" title="Tangential speed">Tangential speed</a></li>
<li><a href="Rotational_frequency" title="Rotational frequency">Rotational frequency</a></li></ul>
</div>
<ul><li><a href="Angular_acceleration" title="Angular acceleration">Angular acceleration</a>&nbsp;/ <a href="Angular_displacement" title="Angular displacement">displacement</a>&nbsp;/ <a href="Angular_frequency" title="Angular frequency">frequency</a>&nbsp;/ <a href="Angular_velocity" title="Angular velocity">velocity</a></li></ul></div></div></td>
</tr><tr><td class="sidebar-content">
<div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)"><div class="sidebar-list-title-c">Scientists</div></div><div class="sidebar-list-content mw-collapsible-content plainlist"><div class="hlist">
<ul><li><a href="Johannes_Kepler" title="Johannes Kepler">Kepler</a></li>
<li><a href="Galileo_Galilei" title="Galileo Galilei">Galileo</a></li>
<li><a href="Christiaan_Huygens" title="Christiaan Huygens">Huygens</a></li>
<li><a href="Isaac_Newton" title="Isaac Newton">Newton</a></li>
<li><a href="Jeremiah_Horrocks" title="Jeremiah Horrocks">Horrocks</a></li>
<li><a href="Edmond_Halley" title="Edmond Halley">Halley</a></li>
<li><a href="Pierre_Louis_Maupertuis" title="Pierre Louis Maupertuis">Maupertuis</a></li>
<li><a href="Daniel_Bernoulli" title="Daniel Bernoulli">Daniel Bernoulli</a></li>
<li><a href="Johann_Bernoulli" title="Johann Bernoulli">Johann Bernoulli</a></li>
<li><a href="Leonhard_Euler" title="Leonhard Euler">Euler</a></li>
<li><a href="Jean_le_Rond_d'Alembert" title="Jean le Rond d'Alembert">d'Alembert</a></li>
<li><a href="Alexis_Clairaut" title="Alexis Clairaut">Clairaut</a></li>
<li><a href="Joseph-Louis_Lagrange" title="Joseph-Louis Lagrange">Lagrange</a></li>
<li><a href="Pierre-Simon_Laplace" title="Pierre-Simon Laplace">Laplace</a></li>
<li><a href="Sim%C3%A9on_Denis_Poisson" title="Siméon Denis Poisson">Poisson</a></li>
<li><a href="William_Rowan_Hamilton" title="William Rowan Hamilton">Hamilton</a></li>
<li><a href="Carl_Gustav_Jacob_Jacobi" title="Carl Gustav Jacob Jacobi">Jacobi</a></li>
<li><a href="Augustin-Louis_Cauchy" title="Augustin-Louis Cauchy">Cauchy</a></li>
<li><a href="Edward_Routh" title="Edward Routh">Routh</a></li>
<li><a href="Joseph_Liouville" title="Joseph Liouville">Liouville</a></li>
<li><a href="Paul_%C3%89mile_Appell" title="Paul Émile Appell">Appell</a></li>
<li><a href="Josiah_Willard_Gibbs" title="Josiah Willard Gibbs">Gibbs</a></li>
<li><a href="Bernard_Koopman" title="Bernard Koopman">Koopman</a></li>
<li><a href="John_von_Neumann" title="John von Neumann">von Neumann</a></li></ul>
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<p>In <a href="Mathematics" title="Mathematics">mathematics</a> and <a href="Classical_mechanics" title="Classical mechanics">classical mechanics</a>, the <b>Poisson bracket</b> is an important <a href="Binary_operation" title="Binary operation">binary operation</a> in <a href="Hamiltonian_mechanics" title="Hamiltonian mechanics">Hamiltonian mechanics</a>, playing a central role in Hamilton's equations of motion, which govern the time evolution of a Hamiltonian <a href="Dynamical_system" title="Dynamical system">dynamical system</a>. The Poisson bracket also distinguishes a certain class of coordinate transformations, called <i><a href="Canonical_transformations" class="mw-redirect" title="Canonical transformations">canonical transformations</a></i>, which map <a href="Canonical_coordinates" title="Canonical coordinates">canonical coordinate systems</a> into other canonical coordinate systems. A "canonical coordinate system" consists of canonical position and momentum variables (below symbolized by <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q_{i}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>q</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle q_{i}}</annotation>
</semantics>
</math></span><img src="./2752dcbff884354069fe332b8e51eb0a70a531b6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.837ex; height:2.009ex;" alt="{\displaystyle q_{i}}" loading="lazy"></span> and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p_{i}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>p</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
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</mrow>
<annotation encoding="application/x-tex">{\displaystyle p_{i}}</annotation>
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</math></span><img src="./5bab39399bf5424f25d957cdc57c84a0622626d2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:2.059ex; height:2.009ex;" alt="{\displaystyle p_{i}}" loading="lazy"></span>, respectively) that satisfy canonical Poisson bracket relations. The set of possible canonical transformations is always very rich. For instance, it is often possible to choose the Hamiltonian itself <span class="mwe-math-element mwe-math-element-inline"><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 {H}}={\mathcal {H}}(q,p,t)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi class="MJX-tex-caligraphic" mathvariant="script">H</mi>
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</mrow>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi class="MJX-tex-caligraphic" mathvariant="script">H</mi>
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<mo stretchy="false">(</mo>
<mi>q</mi>
<mo>,</mo>
<mi>p</mi>
<mo>,</mo>
<mi>t</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\mathcal {H}}={\mathcal {H}}(q,p,t)}</annotation>
</semantics>
</math></span><img src="./fe5be22330190d67b91fcbb021eaffb6121472c0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.982ex; height:2.843ex;" alt="{\displaystyle {\mathcal {H}}={\mathcal {H}}(q,p,t)}" loading="lazy"></span> as one of the new canonical momentum coordinates.
</p><p>In a more general sense, the Poisson bracket is used to define a <a href="Poisson_algebra" title="Poisson algebra">Poisson algebra</a>, of which the algebra of functions on a <a href="Poisson_manifold" title="Poisson manifold">Poisson manifold</a> is a special case. There are other general examples, as well: it occurs in the theory of <a href="Lie_algebra" title="Lie algebra">Lie algebras</a>, where the <a href="Tensor_algebra" title="Tensor algebra">tensor algebra</a> of a Lie algebra forms a Poisson algebra; a detailed construction of how this comes about is given in the <a href="Universal_enveloping_algebra" title="Universal enveloping algebra">universal enveloping algebra</a> article. Quantum deformations of the universal enveloping algebra lead to the notion of <a href="Quantum_group" title="Quantum group">quantum groups</a>.
</p><p>All of these objects are named in honor of French mathematician <a href="Sim%C3%A9on_Denis_Poisson" title="Siméon Denis Poisson">Siméon Denis Poisson</a>. He introduced the Poisson bracket in his 1809 treatise on mechanics.<sup id="cite_ref-Poisson1809_1-0" class="reference"><a href="#cite_note-Poisson1809-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Marle2009_2-0" class="reference"><a href="#cite_note-Marle2009-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup>
</p>
<meta property="mw:PageProp/toc">
<div class="mw-heading mw-heading2"><h2 id="Properties">Properties</h2></div>
<p>Given two functions <span class="texhtml mvar" style="font-style:italic;">f</span> and <span class="texhtml mvar" style="font-style:italic;">g</span> that depend on <a href="Phase_space" title="Phase space">phase space</a> and time, their Poisson bracket <span class="mwe-math-element mwe-math-element-inline"><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\}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo fence="false" stretchy="false">{</mo>
<mi>f</mi>
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<mi>g</mi>
<mo fence="false" stretchy="false">}</mo>
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<annotation encoding="application/x-tex">{\displaystyle \{f,g\}}</annotation>
</semantics>
</math></span><img src="./342713616a282b5a224271900f5cb0464f880c1c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.753ex; height:2.843ex;" alt="{\displaystyle \{f,g\}}" loading="lazy"></span> is another function that depends on phase space and time. The following rules hold for any three functions <span class="mwe-math-element mwe-math-element-inline"><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}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>f</mi>
<mo>,</mo>
<mspace width="thinmathspace"></mspace>
<mi>g</mi>
<mo>,</mo>
<mspace width="thinmathspace"></mspace>
<mi>h</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f,\,g,\,h}</annotation>
</semantics>
</math></span><img src="./e1b1973006bcbcc0d3d1a2a302e640c5bb5e184d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.576ex; height:2.509ex;" alt="{\displaystyle f,\,g,\,h}" loading="lazy"></span> of phase space and time:
</p>
<dl><dt><a href="Anticommutativity" class="mw-redirect" title="Anticommutativity">Anticommutativity</a></dt>
<dd><span class="mwe-math-element mwe-math-element-inline"><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\}=-\{g,f\}}">
<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>
<mo>−<!-- − --></mo>
<mo fence="false" stretchy="false">{</mo>
<mi>g</mi>
<mo>,</mo>
<mi>f</mi>
<mo fence="false" stretchy="false">}</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \{f,g\}=-\{g,f\}}</annotation>
</semantics>
</math></span><img src="./eff1f1f4a7eef368856844d55a700e559463abe1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.413ex; height:2.843ex;" alt="{\displaystyle \{f,g\}=-\{g,f\}}" loading="lazy"></span></dd>
<dt><a href="Bilinearity" class="mw-redirect" title="Bilinearity">Bilinearity</a></dt>
<dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{af+bg,h\}=a\{f,h\}+b\{g,h\},}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo fence="false" stretchy="false">{</mo>
<mi>a</mi>
<mi>f</mi>
<mo>+</mo>
<mi>b</mi>
<mi>g</mi>
<mo>,</mo>
<mi>h</mi>
<mo fence="false" stretchy="false">}</mo>
<mo>=</mo>
<mi>a</mi>
<mo fence="false" stretchy="false">{</mo>
<mi>f</mi>
<mo>,</mo>
<mi>h</mi>
<mo fence="false" stretchy="false">}</mo>
<mo>+</mo>
<mi>b</mi>
<mo fence="false" stretchy="false">{</mo>
<mi>g</mi>
<mo>,</mo>
<mi>h</mi>
<mo fence="false" stretchy="false">}</mo>
<mo>,</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \{af+bg,h\}=a\{f,h\}+b\{g,h\},}</annotation>
</semantics>
</math></span><img src="./f1659d89f3606daeb9a96c2fba1d66212afef197.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:32.763ex; height:2.843ex;" alt="{\displaystyle \{af+bg,h\}=a\{f,h\}+b\{g,h\},}" loading="lazy"></span><span class="mwe-math-element mwe-math-element-inline"><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,af+bg\}=a\{h,f\}+b\{h,g\},\quad a,b\in \mathbb {R} }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo fence="false" stretchy="false">{</mo>
<mi>h</mi>
<mo>,</mo>
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<mi>f</mi>
<mo>+</mo>
<mi>b</mi>
<mi>g</mi>
<mo fence="false" stretchy="false">}</mo>
<mo>=</mo>
<mi>a</mi>
<mo fence="false" stretchy="false">{</mo>
<mi>h</mi>
<mo>,</mo>
<mi>f</mi>
<mo fence="false" stretchy="false">}</mo>
<mo>+</mo>
<mi>b</mi>
<mo fence="false" stretchy="false">{</mo>
<mi>h</mi>
<mo>,</mo>
<mi>g</mi>
<mo fence="false" stretchy="false">}</mo>
<mo>,</mo>
<mspace width="1em"></mspace>
<mi>a</mi>
<mo>,</mo>
<mi>b</mi>
<mo>∈<!-- ∈ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">R</mi>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \{h,af+bg\}=a\{h,f\}+b\{h,g\},\quad a,b\in \mathbb {R} }</annotation>
</semantics>
</math></span><img src="./691f8a28bc90902b01f1086f890427a1047472d2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:43.253ex; height:2.843ex;" alt="{\displaystyle \{h,af+bg\}=a\{h,f\}+b\{h,g\},\quad a,b\in \mathbb {R} }" loading="lazy"></span></dd>
<dt><a href="Product_rule" title="Product rule">Leibniz's rule</a></dt>
<dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{fg,h\}=\{f,h\}g+f\{g,h\}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo fence="false" stretchy="false">{</mo>
<mi>f</mi>
<mi>g</mi>
<mo>,</mo>
<mi>h</mi>
<mo fence="false" stretchy="false">}</mo>
<mo>=</mo>
<mo fence="false" stretchy="false">{</mo>
<mi>f</mi>
<mo>,</mo>
<mi>h</mi>
<mo fence="false" stretchy="false">}</mo>
<mi>g</mi>
<mo>+</mo>
<mi>f</mi>
<mo fence="false" stretchy="false">{</mo>
<mi>g</mi>
<mo>,</mo>
<mi>h</mi>
<mo fence="false" stretchy="false">}</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \{fg,h\}=\{f,h\}g+f\{g,h\}}</annotation>
</semantics>
</math></span><img src="./e2b6f742232f680ffab1030dba8f89aee85a0f5c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.216ex; height:2.843ex;" alt="{\displaystyle \{fg,h\}=\{f,h\}g+f\{g,h\}}" loading="lazy"></span></dd>
<dt><a href="Jacobi_identity" title="Jacobi identity">Jacobi identity</a></dt>
<dd><span class="mwe-math-element mwe-math-element-inline"><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>
<mi>f</mi>
<mo>,</mo>
<mo fence="false" stretchy="false">{</mo>
<mi>g</mi>
<mo>,</mo>
<mi>h</mi>
<mo fence="false" stretchy="false">}</mo>
<mo fence="false" stretchy="false">}</mo>
<mo>+</mo>
<mo fence="false" stretchy="false">{</mo>
<mi>g</mi>
<mo>,</mo>
<mo fence="false" stretchy="false">{</mo>
<mi>h</mi>
<mo>,</mo>
<mi>f</mi>
<mo fence="false" stretchy="false">}</mo>
<mo fence="false" stretchy="false">}</mo>
<mo>+</mo>
<mo fence="false" stretchy="false">{</mo>
<mi>h</mi>
<mo>,</mo>
<mo fence="false" stretchy="false">{</mo>
<mi>f</mi>
<mo>,</mo>
<mi>g</mi>
<mo fence="false" stretchy="false">}</mo>
<mo fence="false" stretchy="false">}</mo>
<mo>=</mo>
<mn>0</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \{f,\{g,h\}\}+\{g,\{h,f\}\}+\{h,\{f,g\}\}=0}</annotation>
</semantics>
</math></span><img src="./022941efe73a03dbf8e90d3ca46fa4aba6bce0bd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:41.296ex; height:2.843ex;" alt="{\displaystyle \{f,\{g,h\}\}+\{g,\{h,f\}\}+\{h,\{f,g\}\}=0}" loading="lazy"></span></dd></dl>
<p>Also, if a function <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>k</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle k}</annotation>
</semantics>
</math></span><img src="./c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy"></span> is constant over phase space (but may depend on time), then <span class="mwe-math-element mwe-math-element-inline"><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,\,k\}=0}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo fence="false" stretchy="false">{</mo>
<mi>f</mi>
<mo>,</mo>
<mspace width="thinmathspace"></mspace>
<mi>k</mi>
<mo fence="false" stretchy="false">}</mo>
<mo>=</mo>
<mn>0</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \{f,\,k\}=0}</annotation>
</semantics>
</math></span><img src="./cc30f577986578a814ca01de1017f36b41fdce67.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.497ex; height:2.843ex;" alt="{\displaystyle \{f,\,k\}=0}" loading="lazy"></span> for any <span class="mwe-math-element mwe-math-element-inline"><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}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>f</mi>
</mstyle>
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<annotation encoding="application/x-tex">{\displaystyle f}</annotation>
</semantics>
</math></span><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"></span>.
</p>
<div class="mw-heading mw-heading2"><h2 id="Definition_in_canonical_coordinates">Definition in canonical coordinates</h2></div>
<p>In <a href="Canonical_coordinates" title="Canonical coordinates">canonical coordinates</a> (also known as <a href="Darboux_coordinates" class="mw-redirect" title="Darboux coordinates">Darboux coordinates</a>) <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (q_{i},\,p_{i})}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo stretchy="false">(</mo>
<msub>
<mi>q</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
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</msub>
<mo>,</mo>
<mspace width="thinmathspace"></mspace>
<msub>
<mi>p</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle (q_{i},\,p_{i})}</annotation>
</semantics>
</math></span><img src="./27dfbd6039581e4e28bdb488ac4d5a42b87d9d37.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.036ex; height:2.843ex;" alt="{\displaystyle (q_{i},\,p_{i})}" loading="lazy"></span> on the <a href="Phase_space" title="Phase space">phase space</a>, given two functions <span class="mwe-math-element mwe-math-element-inline"><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(p_{i},\,q_{i},t)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>f</mi>
<mo stretchy="false">(</mo>
<msub>
<mi>p</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
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</msub>
<mo>,</mo>
<mspace width="thinmathspace"></mspace>
<msub>
<mi>q</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
<mo>,</mo>
<mi>t</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f(p_{i},\,q_{i},t)}</annotation>
</semantics>
</math></span><img src="./19a19a89fc0cf08c17a834e8445df162e33f4675.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.188ex; height:2.843ex;" alt="{\displaystyle f(p_{i},\,q_{i},t)}" loading="lazy"></span> and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g(p_{i},\,q_{i},t)}">
<semantics>
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<annotation encoding="application/x-tex">{\displaystyle g(p_{i},\,q_{i},t)}</annotation>
</semantics>
</math></span><img src="./ca5c55349fa472f9956fa3a4a6ade1fdff6622e8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.026ex; height:2.843ex;" alt="{\displaystyle g(p_{i},\,q_{i},t)}" loading="lazy"></span>,<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>Note 1<span class="cite-bracket">]</span></a></sup> the Poisson bracket takes the form
<span class="mwe-math-element mwe-math-element-block"><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\}=\sum _{i=1}^{N}\left({\frac {\partial f}{\partial q_{i}}}{\frac {\partial g}{\partial p_{i}}}-{\frac {\partial f}{\partial p_{i}}}{\frac {\partial g}{\partial q_{i}}}\right).}">
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<annotation encoding="application/x-tex">{\displaystyle \{f,g\}=\sum _{i=1}^{N}\left({\frac {\partial f}{\partial q_{i}}}{\frac {\partial g}{\partial p_{i}}}-{\frac {\partial f}{\partial p_{i}}}{\frac {\partial g}{\partial q_{i}}}\right).}</annotation>
</semantics>
</math></span></span>
</p><p>The Poisson brackets of the canonical coordinates are
<span class="mwe-math-element mwe-math-element-block"><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}\{q_{k},q_{l}\}&amp;=\sum _{i=1}^{N}\left({\frac {\partial q_{k}}{\partial q_{i}}}{\frac {\partial q_{l}}{\partial p_{i}}}-{\frac {\partial q_{k}}{\partial p_{i}}}{\frac {\partial q_{l}}{\partial q_{i}}}\right)=\sum _{i=1}^{N}\left(\delta _{ki}\cdot 0-0\cdot \delta _{li}\right)=0,\\\{p_{k},p_{l}\}&amp;=\sum _{i=1}^{N}\left({\frac {\partial p_{k}}{\partial q_{i}}}{\frac {\partial p_{l}}{\partial p_{i}}}-{\frac {\partial p_{k}}{\partial p_{i}}}{\frac {\partial p_{l}}{\partial q_{i}}}\right)=\sum _{i=1}^{N}\left(0\cdot \delta _{li}-\delta _{ki}\cdot 0\right)=0,\\\{q_{k},p_{l}\}&amp;=\sum _{i=1}^{N}\left({\frac {\partial q_{k}}{\partial q_{i}}}{\frac {\partial p_{l}}{\partial p_{i}}}-{\frac {\partial q_{k}}{\partial p_{i}}}{\frac {\partial p_{l}}{\partial q_{i}}}\right)=\sum _{i=1}^{N}\left(\delta _{ki}\cdot \delta _{li}-0\cdot 0\right)=\delta _{kl},\end{aligned}}}">
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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\{q_{k},q_{l}\}&amp;=\sum _{i=1}^{N}\left({\frac {\partial q_{k}}{\partial q_{i}}}{\frac {\partial q_{l}}{\partial p_{i}}}-{\frac {\partial q_{k}}{\partial p_{i}}}{\frac {\partial q_{l}}{\partial q_{i}}}\right)=\sum _{i=1}^{N}\left(\delta _{ki}\cdot 0-0\cdot \delta _{li}\right)=0,\\\{p_{k},p_{l}\}&amp;=\sum _{i=1}^{N}\left({\frac {\partial p_{k}}{\partial q_{i}}}{\frac {\partial p_{l}}{\partial p_{i}}}-{\frac {\partial p_{k}}{\partial p_{i}}}{\frac {\partial p_{l}}{\partial q_{i}}}\right)=\sum _{i=1}^{N}\left(0\cdot \delta _{li}-\delta _{ki}\cdot 0\right)=0,\\\{q_{k},p_{l}\}&amp;=\sum _{i=1}^{N}\left({\frac {\partial q_{k}}{\partial q_{i}}}{\frac {\partial p_{l}}{\partial p_{i}}}-{\frac {\partial q_{k}}{\partial p_{i}}}{\frac {\partial p_{l}}{\partial q_{i}}}\right)=\sum _{i=1}^{N}\left(\delta _{ki}\cdot \delta _{li}-0\cdot 0\right)=\delta _{kl},\end{aligned}}}</annotation>
</semantics>
</math></span></span>
where <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta _{ij}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>δ<!-- δ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
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</mrow>
<annotation encoding="application/x-tex">{\displaystyle \delta _{ij}}</annotation>
</semantics>
</math></span><img src="./fa75d04c11480d976e1396951e02cbb3c4f71568.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.51ex; height:3.009ex;" alt="{\displaystyle \delta _{ij}}" loading="lazy"></span> is the <a href="Kronecker_delta" title="Kronecker delta">Kronecker delta</a>.
</p>
<div class="mw-heading mw-heading2"><h2 id="Hamilton's_equations_of_motion">Hamilton's equations of motion</h2></div>
<p><a href="Hamilton's_equations_of_motion" class="mw-redirect" title="Hamilton's equations of motion">Hamilton's equations of motion</a> have an equivalent expression in terms of the Poisson bracket. This may be most directly demonstrated in an explicit coordinate frame. Suppose that <span class="mwe-math-element mwe-math-element-inline"><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(p,q,t)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
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</mrow>
<annotation encoding="application/x-tex">{\displaystyle f(p,q,t)}</annotation>
</semantics>
</math></span><img src="./800ae2372ac6e8a7cdedb6b66225bab8c516f685.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.234ex; height:2.843ex;" alt="{\displaystyle f(p,q,t)}" loading="lazy"></span> is a function on the solution's trajectory-manifold. Then from the multivariable <a href="Chain_rule" title="Chain rule">chain rule</a>,
<span class="mwe-math-element mwe-math-element-block"><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 {\frac {d}{dt}}f(p,q,t)={\frac {\partial f}{\partial q}}{\frac {dq}{dt}}+{\frac {\partial f}{\partial p}}{\frac {dp}{dt}}+{\frac {\partial f}{\partial t}}.}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mi>d</mi>
<mrow>
<mi>d</mi>
<mi>t</mi>
</mrow>
</mfrac>
</mrow>
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>p</mi>
<mo>,</mo>
<mi>q</mi>
<mo>,</mo>
<mi>t</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<mi>f</mi>
</mrow>
<mrow>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<mi>q</mi>
</mrow>
</mfrac>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mi>d</mi>
<mi>q</mi>
</mrow>
<mrow>
<mi>d</mi>
<mi>t</mi>
</mrow>
</mfrac>
</mrow>
<mo>+</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<mi>f</mi>
</mrow>
<mrow>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<mi>p</mi>
</mrow>
</mfrac>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mi>d</mi>
<mi>p</mi>
</mrow>
<mrow>
<mi>d</mi>
<mi>t</mi>
</mrow>
</mfrac>
</mrow>
<mo>+</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<mi>f</mi>
</mrow>
<mrow>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<mi>t</mi>
</mrow>
</mfrac>
</mrow>
<mo>.</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\frac {d}{dt}}f(p,q,t)={\frac {\partial f}{\partial q}}{\frac {dq}{dt}}+{\frac {\partial f}{\partial p}}{\frac {dp}{dt}}+{\frac {\partial f}{\partial t}}.}</annotation>
</semantics>
</math></span></span>
</p><p>Further, one may take <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p=p(t)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>p</mi>
<mo>=</mo>
<mi>p</mi>
<mo stretchy="false">(</mo>
<mi>t</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle p=p(t)}</annotation>
</semantics>
</math></span><img src="./8126309e9692644f4f2bb23f7f76211e1f80ae30.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:8.176ex; height:2.843ex;" alt="{\displaystyle p=p(t)}" loading="lazy"></span> and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q=q(t)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>q</mi>
<mo>=</mo>
<mi>q</mi>
<mo stretchy="false">(</mo>
<mi>t</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle q=q(t)}</annotation>
</semantics>
</math></span><img src="./bccff540f4b254e8998503162fe3a42f1c3bdc53.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.886ex; height:2.843ex;" alt="{\displaystyle q=q(t)}" loading="lazy"></span> to be solutions to <a href="Hamilton's_equations" class="mw-redirect" title="Hamilton's equations">Hamilton's equations</a>; that is,
<span class="mwe-math-element mwe-math-element-block"><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}{\frac {dq}{dt}}&amp;={\frac {\partial {\mathcal {H}}}{\partial p}}=\{q,{\mathcal {H}}\},\\{\frac {dp}{dt}}&amp;=-{\frac {\partial {\mathcal {H}}}{\partial q}}=\{p,{\mathcal {H}}\}.\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>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mi>d</mi>
<mi>q</mi>
</mrow>
<mrow>
<mi>d</mi>
<mi>t</mi>
</mrow>
</mfrac>
</mrow>
</mtd>
<mtd>
<mi></mi>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi class="MJX-tex-caligraphic" mathvariant="script">H</mi>
</mrow>
</mrow>
</mrow>
<mrow>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<mi>p</mi>
</mrow>
</mfrac>
</mrow>
<mo>=</mo>
<mo fence="false" stretchy="false">{</mo>
<mi>q</mi>
<mo>,</mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi class="MJX-tex-caligraphic" mathvariant="script">H</mi>
</mrow>
</mrow>
<mo fence="false" stretchy="false">}</mo>
<mo>,</mo>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mi>d</mi>
<mi>p</mi>
</mrow>
<mrow>
<mi>d</mi>
<mi>t</mi>
</mrow>
</mfrac>
</mrow>
</mtd>
<mtd>
<mi></mi>
<mo>=</mo>
<mo>−<!-- − --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi class="MJX-tex-caligraphic" mathvariant="script">H</mi>
</mrow>
</mrow>
</mrow>
<mrow>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<mi>q</mi>
</mrow>
</mfrac>
</mrow>
<mo>=</mo>
<mo fence="false" stretchy="false">{</mo>
<mi>p</mi>
<mo>,</mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi class="MJX-tex-caligraphic" mathvariant="script">H</mi>
</mrow>
</mrow>
<mo fence="false" stretchy="false">}</mo>
<mo>.</mo>
</mtd>
</mtr>
</mtable>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\frac {dq}{dt}}&amp;={\frac {\partial {\mathcal {H}}}{\partial p}}=\{q,{\mathcal {H}}\},\\{\frac {dp}{dt}}&amp;=-{\frac {\partial {\mathcal {H}}}{\partial q}}=\{p,{\mathcal {H}}\}.\end{aligned}}}</annotation>
</semantics>
</math></span></span>
</p><p>Then
<span class="mwe-math-element mwe-math-element-block"><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}{\frac {d}{dt}}f(p,q,t)&amp;={\frac {\partial f}{\partial q}}{\frac {\partial {\mathcal {H}}}{\partial p}}-{\frac {\partial f}{\partial p}}{\frac {\partial {\mathcal {H}}}{\partial q}}+{\frac {\partial f}{\partial t}}\\&amp;=\{f,{\mathcal {H}}\}+{\frac {\partial f}{\partial t}}~.\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>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mi>d</mi>
<mrow>
<mi>d</mi>
<mi>t</mi>
</mrow>
</mfrac>
</mrow>
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>p</mi>
<mo>,</mo>
<mi>q</mi>
<mo>,</mo>
<mi>t</mi>
<mo stretchy="false">)</mo>
</mtd>
<mtd>
<mi></mi>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<mi>f</mi>
</mrow>
<mrow>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<mi>q</mi>
</mrow>
</mfrac>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi class="MJX-tex-caligraphic" mathvariant="script">H</mi>
</mrow>
</mrow>
</mrow>
<mrow>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<mi>p</mi>
</mrow>
</mfrac>
</mrow>
<mo>−<!-- − --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<mi>f</mi>
</mrow>
<mrow>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<mi>p</mi>
</mrow>
</mfrac>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi class="MJX-tex-caligraphic" mathvariant="script">H</mi>
</mrow>
</mrow>
</mrow>
<mrow>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<mi>q</mi>
</mrow>
</mfrac>
</mrow>
<mo>+</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<mi>f</mi>
</mrow>
<mrow>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<mi>t</mi>
</mrow>
</mfrac>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd></mtd>
<mtd>
<mi></mi>
<mo>=</mo>
<mo fence="false" stretchy="false">{</mo>
<mi>f</mi>
<mo>,</mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi class="MJX-tex-caligraphic" mathvariant="script">H</mi>
</mrow>
</mrow>
<mo fence="false" stretchy="false">}</mo>
<mo>+</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<mi>f</mi>
</mrow>
<mrow>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<mi>t</mi>
</mrow>
</mfrac>
</mrow>
<mtext>&nbsp;</mtext>
<mo>.</mo>
</mtd>
</mtr>
</mtable>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\frac {d}{dt}}f(p,q,t)&amp;={\frac {\partial f}{\partial q}}{\frac {\partial {\mathcal {H}}}{\partial p}}-{\frac {\partial f}{\partial p}}{\frac {\partial {\mathcal {H}}}{\partial q}}+{\frac {\partial f}{\partial t}}\\&amp;=\{f,{\mathcal {H}}\}+{\frac {\partial f}{\partial t}}~.\end{aligned}}}</annotation>
</semantics>
</math></span></span>
</p><p>Thus, the time evolution of a function <span class="mwe-math-element mwe-math-element-inline"><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}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>f</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f}</annotation>
</semantics>
</math></span><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"></span> on a <a href="Symplectic_manifold" title="Symplectic manifold">symplectic manifold</a> can be given as a <a href="Flow_(mathematics)" title="Flow (mathematics)">one-parameter family</a> of <a href="Symplectomorphism" title="Symplectomorphism">symplectomorphisms</a> (i.e., <a href="Canonical_transformations" class="mw-redirect" title="Canonical transformations">canonical transformations</a>, area-preserving diffeomorphisms), with the time <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>t</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle t}</annotation>
</semantics>
</math></span><img src="./65658b7b223af9e1acc877d848888ecdb4466560.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}" loading="lazy"></span> being the parameter: Hamiltonian motion is a canonical transformation generated by the Hamiltonian. That is, Poisson brackets are preserved in it, so that <i>any time <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>t</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle t}</annotation>
</semantics>
</math></span><img src="./65658b7b223af9e1acc877d848888ecdb4466560.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}" loading="lazy"></span></i> in the solution to Hamilton's equations,
<span class="mwe-math-element mwe-math-element-block"><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 q(t)=\exp(-t\{{\mathcal {H}},\cdot \})q(0),\quad p(t)=\exp(-t\{{\mathcal {H}},\cdot \})p(0),}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>q</mi>
<mo stretchy="false">(</mo>
<mi>t</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi>exp</mi>
<mo>⁡<!-- ⁡ --></mo>
<mo stretchy="false">(</mo>
<mo>−<!-- − --></mo>
<mi>t</mi>
<mo fence="false" stretchy="false">{</mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi class="MJX-tex-caligraphic" mathvariant="script">H</mi>
</mrow>
</mrow>
<mo>,</mo>
<mo>⋅<!-- ⋅ --></mo>
<mo fence="false" stretchy="false">}</mo>
<mo stretchy="false">)</mo>
<mi>q</mi>
<mo stretchy="false">(</mo>
<mn>0</mn>
<mo stretchy="false">)</mo>
<mo>,</mo>
<mspace width="1em"></mspace>
<mi>p</mi>
<mo stretchy="false">(</mo>
<mi>t</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi>exp</mi>
<mo>⁡<!-- ⁡ --></mo>
<mo stretchy="false">(</mo>
<mo>−<!-- − --></mo>
<mi>t</mi>
<mo fence="false" stretchy="false">{</mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi class="MJX-tex-caligraphic" mathvariant="script">H</mi>
</mrow>
</mrow>
<mo>,</mo>
<mo>⋅<!-- ⋅ --></mo>
<mo fence="false" stretchy="false">}</mo>
<mo stretchy="false">)</mo>
<mi>p</mi>
<mo stretchy="false">(</mo>
<mn>0</mn>
<mo stretchy="false">)</mo>
<mo>,</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle q(t)=\exp(-t\{{\mathcal {H}},\cdot \})q(0),\quad p(t)=\exp(-t\{{\mathcal {H}},\cdot \})p(0),}</annotation>
</semantics>
</math></span></span>
can serve as the bracket coordinates. <i>Poisson brackets are <a href="Canonical_transformation" title="Canonical transformation">canonical invariants</a></i>.
</p><p>Dropping the coordinates,
<span class="mwe-math-element mwe-math-element-block"><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 {\frac {d}{dt}}f=\left({\frac {\partial }{\partial t}}-\{{\mathcal {H}},\cdot \}\right)f.}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mi>d</mi>
<mrow>
<mi>d</mi>
<mi>t</mi>
</mrow>
</mfrac>
</mrow>
<mi>f</mi>
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<mrow>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<mi>t</mi>
</mrow>
</mfrac>
</mrow>
<mo>−<!-- − --></mo>
<mo fence="false" stretchy="false">{</mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi class="MJX-tex-caligraphic" mathvariant="script">H</mi>
</mrow>
</mrow>
<mo>,</mo>
<mo>⋅<!-- ⋅ --></mo>
<mo fence="false" stretchy="false">}</mo>
</mrow>
<mo>)</mo>
</mrow>
<mi>f</mi>
<mo>.</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\frac {d}{dt}}f=\left({\frac {\partial }{\partial t}}-\{{\mathcal {H}},\cdot \}\right)f.}</annotation>
</semantics>
</math></span></span>
</p><p>The operator in the convective part of the derivative, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i{\hat {L}}=-\{{\mathcal {H}},\cdot \}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>i</mi>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>L</mi>
<mo stretchy="false">^<!-- ^ --></mo>
</mover>
</mrow>
</mrow>
<mo>=</mo>
<mo>−<!-- − --></mo>
<mo fence="false" stretchy="false">{</mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi class="MJX-tex-caligraphic" mathvariant="script">H</mi>
</mrow>
</mrow>
<mo>,</mo>
<mo>⋅<!-- ⋅ --></mo>
<mo fence="false" stretchy="false">}</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle i{\hat {L}}=-\{{\mathcal {H}},\cdot \}}</annotation>
</semantics>
</math></span><img src="./da1b25f309a76550c7cd986cb2faef5e4bf15e54.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.261ex; height:3.343ex;" alt="{\displaystyle i{\hat {L}}=-\{{\mathcal {H}},\cdot \}}" loading="lazy"></span>, is sometimes referred to as the Liouvillian (see <a href="Liouville's_theorem_(Hamiltonian)" title="Liouville's theorem (Hamiltonian)">Liouville's theorem (Hamiltonian)</a>).
</p>
<div class="mw-heading mw-heading2"><h2 id="Poisson_matrix_in_canonical_transformations">Poisson matrix in canonical transformations</h2></div>
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</style><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Canonical_transformation" title="Canonical transformation">Canonical transformation</a></div>
<p>The concept of Poisson brackets can be expanded to that of matrices by defining the Poisson matrix.
</p><p>Consider the following canonical transformation:<span class="mwe-math-element mwe-math-element-block"><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 \eta ={\begin{bmatrix}q_{1}\\\vdots \\q_{N}\\p_{1}\\\vdots \\p_{N}\\\end{bmatrix}}\quad \rightarrow \quad \varepsilon ={\begin{bmatrix}Q_{1}\\\vdots \\Q_{N}\\P_{1}\\\vdots \\P_{N}\\\end{bmatrix}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>η<!-- η --></mi>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow>
<mo>[</mo>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<msub>
<mi>q</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
</mtd>
</mtr>
<mtr>
<mtd>
<mo>⋮<!-- ⋮ --></mo>
</mtd>
</mtr>
<mtr>
<mtd>
<msub>
<mi>q</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>N</mi>
</mrow>
</msub>
</mtd>
</mtr>
<mtr>
<mtd>
<msub>
<mi>p</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
</mtd>
</mtr>
<mtr>
<mtd>
<mo>⋮<!-- ⋮ --></mo>
</mtd>
</mtr>
<mtr>
<mtd>
<msub>
<mi>p</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>N</mi>
</mrow>
</msub>
</mtd>
</mtr>
</mtable>
<mo>]</mo>
</mrow>
</mrow>
<mspace width="1em"></mspace>
<mo stretchy="false">→<!-- → --></mo>
<mspace width="1em"></mspace>
<mi>ε<!-- ε --></mi>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow>
<mo>[</mo>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<msub>
<mi>Q</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
</mtd>
</mtr>
<mtr>
<mtd>
<mo>⋮<!-- ⋮ --></mo>
</mtd>
</mtr>
<mtr>
<mtd>
<msub>
<mi>Q</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>N</mi>
</mrow>
</msub>
</mtd>
</mtr>
<mtr>
<mtd>
<msub>
<mi>P</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
</mtd>
</mtr>
<mtr>
<mtd>
<mo>⋮<!-- ⋮ --></mo>
</mtd>
</mtr>
<mtr>
<mtd>
<msub>
<mi>P</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>N</mi>
</mrow>
</msub>
</mtd>
</mtr>
</mtable>
<mo>]</mo>
</mrow>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \eta ={\begin{bmatrix}q_{1}\\\vdots \\q_{N}\\p_{1}\\\vdots \\p_{N}\\\end{bmatrix}}\quad \rightarrow \quad \varepsilon ={\begin{bmatrix}Q_{1}\\\vdots \\Q_{N}\\P_{1}\\\vdots \\P_{N}\\\end{bmatrix}}}</annotation>
</semantics>
</math></span></span>Defining <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle M:={\frac {\partial (\mathbf {Q} ,\mathbf {P} )}{\partial (\mathbf {q} ,\mathbf {p} )}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="false" scriptlevel="0">
<mi>M</mi>
<mo>:=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<mo stretchy="false">(</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">Q</mi>
</mrow>
<mo>,</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">P</mi>
</mrow>
<mo stretchy="false">)</mo>
</mrow>
<mrow>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<mo stretchy="false">(</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">q</mi>
</mrow>
<mo>,</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">p</mi>
</mrow>
<mo stretchy="false">)</mo>
</mrow>
</mfrac>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\textstyle M:={\frac {\partial (\mathbf {Q} ,\mathbf {P} )}{\partial (\mathbf {q} ,\mathbf {p} )}}}</annotation>
</semantics>
</math></span><img src="./049891c97cd0b9c16a5383ce84ee604db8b234a7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:12.404ex; height:4.843ex;" alt="{\textstyle M:={\frac {\partial (\mathbf {Q} ,\mathbf {P} )}{\partial (\mathbf {q} ,\mathbf {p} )}}}" loading="lazy"></span>, the Poisson matrix is defined as <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle {\mathcal {P}}(\varepsilon )=MJM^{T}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="false" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi class="MJX-tex-caligraphic" mathvariant="script">P</mi>
</mrow>
</mrow>
<mo stretchy="false">(</mo>
<mi>ε<!-- ε --></mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi>M</mi>
<mi>J</mi>
<msup>
<mi>M</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>T</mi>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\textstyle {\mathcal {P}}(\varepsilon )=MJM^{T}}</annotation>
</semantics>
</math></span><img src="./8047559802051ca86f12d5eaa9a3c95c62d18475.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.496ex; height:3.009ex;" alt="{\textstyle {\mathcal {P}}(\varepsilon )=MJM^{T}}" loading="lazy"></span>, where <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>J</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle J}</annotation>
</semantics>
</math></span><img src="./359e4f407b49910e02c27c2f52e87a36cd74c053.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.471ex; height:2.176ex;" alt="{\displaystyle J}" loading="lazy"></span> is the <a href="Symplectic_matrix" title="Symplectic matrix">symplectic matrix</a> under the same conventions used to order the set of coordinates. It follows from the definition that:<span class="mwe-math-element mwe-math-element-block"><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 {\mathcal {P}}_{ij}(\varepsilon )=[MJM^{T}]_{ij}=\sum _{k=1}^{N}\left({\frac {\partial \varepsilon _{i}}{\partial \eta _{k}}}{\frac {\partial \varepsilon _{j}}{\partial \eta _{N+k}}}-{\frac {\partial \varepsilon _{i}}{\partial \eta _{N+k}}}{\frac {\partial \varepsilon _{j}}{\partial \eta _{k}}}\right)=\sum _{k=1}^{N}\left({\frac {\partial \varepsilon _{i}}{\partial q_{k}}}{\frac {\partial \varepsilon _{j}}{\partial p_{k}}}-{\frac {\partial \varepsilon _{i}}{\partial p_{k}}}{\frac {\partial \varepsilon _{j}}{\partial q_{k}}}\right)=\{\varepsilon _{i},\varepsilon _{j}\}_{\eta }.}">
<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">P</mi>
</mrow>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
<mi>j</mi>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<mi>ε<!-- ε --></mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mo stretchy="false">[</mo>
<mi>M</mi>
<mi>J</mi>
<msup>
<mi>M</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>T</mi>
</mrow>
</msup>
<msub>
<mo stretchy="false">]</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
<mi>j</mi>
</mrow>
</msub>
<mo>=</mo>
<munderover>
<mo>∑<!-- ∑ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>k</mi>
<mo>=</mo>
<mn>1</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>N</mi>
</mrow>
</munderover>
<mrow>
<mo>(</mo>
<mrow>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<msub>
<mi>ε<!-- ε --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
</mrow>
<mrow>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<msub>
<mi>η<!-- η --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>k</mi>
</mrow>
</msub>
</mrow>
</mfrac>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<msub>
<mi>ε<!-- ε --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>j</mi>
</mrow>
</msub>
</mrow>
<mrow>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<msub>
<mi>η<!-- η --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>N</mi>
<mo>+</mo>
<mi>k</mi>
</mrow>
</msub>
</mrow>
</mfrac>
</mrow>
<mo>−<!-- − --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<msub>
<mi>ε<!-- ε --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
</mrow>
<mrow>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<msub>
<mi>η<!-- η --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>N</mi>
<mo>+</mo>
<mi>k</mi>
</mrow>
</msub>
</mrow>
</mfrac>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<msub>
<mi>ε<!-- ε --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>j</mi>
</mrow>
</msub>
</mrow>
<mrow>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<msub>
<mi>η<!-- η --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>k</mi>
</mrow>
</msub>
</mrow>
</mfrac>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<munderover>
<mo>∑<!-- ∑ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>k</mi>
<mo>=</mo>
<mn>1</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>N</mi>
</mrow>
</munderover>
<mrow>
<mo>(</mo>
<mrow>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<msub>
<mi>ε<!-- ε --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
</mrow>
<mrow>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<msub>
<mi>q</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>k</mi>
</mrow>
</msub>
</mrow>
</mfrac>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<msub>
<mi>ε<!-- ε --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>j</mi>
</mrow>
</msub>
</mrow>
<mrow>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<msub>
<mi>p</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>k</mi>
</mrow>
</msub>
</mrow>
</mfrac>
</mrow>
<mo>−<!-- − --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<msub>
<mi>ε<!-- ε --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
</mrow>
<mrow>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<msub>
<mi>p</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>k</mi>
</mrow>
</msub>
</mrow>
</mfrac>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<msub>
<mi>ε<!-- ε --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>j</mi>
</mrow>
</msub>
</mrow>
<mrow>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<msub>
<mi>q</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>k</mi>
</mrow>
</msub>
</mrow>
</mfrac>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mo fence="false" stretchy="false">{</mo>
<msub>
<mi>ε<!-- ε --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
<mo>,</mo>
<msub>
<mi>ε<!-- ε --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>j</mi>
</mrow>
</msub>
<msub>
<mo fence="false" stretchy="false">}</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>η<!-- η --></mi>
</mrow>
</msub>
<mo>.</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\mathcal {P}}_{ij}(\varepsilon )=[MJM^{T}]_{ij}=\sum _{k=1}^{N}\left({\frac {\partial \varepsilon _{i}}{\partial \eta _{k}}}{\frac {\partial \varepsilon _{j}}{\partial \eta _{N+k}}}-{\frac {\partial \varepsilon _{i}}{\partial \eta _{N+k}}}{\frac {\partial \varepsilon _{j}}{\partial \eta _{k}}}\right)=\sum _{k=1}^{N}\left({\frac {\partial \varepsilon _{i}}{\partial q_{k}}}{\frac {\partial \varepsilon _{j}}{\partial p_{k}}}-{\frac {\partial \varepsilon _{i}}{\partial p_{k}}}{\frac {\partial \varepsilon _{j}}{\partial q_{k}}}\right)=\{\varepsilon _{i},\varepsilon _{j}\}_{\eta }.}</annotation>
</semantics>
</math></span></span>
</p><p>The Poisson matrix satisfies the following known properties:<span class="mwe-math-element mwe-math-element-block"><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}{\mathcal {P}}^{T}&amp;=-{\mathcal {P}}\\|{\mathcal {P}}|&amp;={\frac {1}{|M|^{2}}}\\{\mathcal {P}}^{-1}(\varepsilon )&amp;=-(M^{-1})^{T}JM^{-1}=-{\mathcal {L}}(\varepsilon )\\\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">
<mi class="MJX-tex-caligraphic" mathvariant="script">P</mi>
</mrow>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>T</mi>
</mrow>
</msup>
</mtd>
<mtd>
<mi></mi>
<mo>=</mo>
<mo>−<!-- − --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi class="MJX-tex-caligraphic" mathvariant="script">P</mi>
</mrow>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi class="MJX-tex-caligraphic" mathvariant="script">P</mi>
</mrow>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
</mtd>
<mtd>
<mi></mi>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mn>1</mn>
<mrow>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
<mi>M</mi>
<msup>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
</mrow>
</mfrac>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<msup>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi class="MJX-tex-caligraphic" mathvariant="script">P</mi>
</mrow>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mo>−<!-- − --></mo>
<mn>1</mn>
</mrow>
</msup>
<mo stretchy="false">(</mo>
<mi>ε<!-- ε --></mi>
<mo stretchy="false">)</mo>
</mtd>
<mtd>
<mi></mi>
<mo>=</mo>
<mo>−<!-- − --></mo>
<mo stretchy="false">(</mo>
<msup>
<mi>M</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>−<!-- − --></mo>
<mn>1</mn>
</mrow>
</msup>
<msup>
<mo stretchy="false">)</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>T</mi>
</mrow>
</msup>
<mi>J</mi>
<msup>
<mi>M</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>−<!-- − --></mo>
<mn>1</mn>
</mrow>
</msup>
<mo>=</mo>
<mo>−<!-- − --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi class="MJX-tex-caligraphic" mathvariant="script">L</mi>
</mrow>
</mrow>
<mo stretchy="false">(</mo>
<mi>ε<!-- ε --></mi>
<mo stretchy="false">)</mo>
</mtd>
</mtr>
</mtable>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\mathcal {P}}^{T}&amp;=-{\mathcal {P}}\\|{\mathcal {P}}|&amp;={\frac {1}{|M|^{2}}}\\{\mathcal {P}}^{-1}(\varepsilon )&amp;=-(M^{-1})^{T}JM^{-1}=-{\mathcal {L}}(\varepsilon )\\\end{aligned}}}</annotation>
</semantics>
</math></span></span>
</p><p>where the <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle {\mathcal {L}}(\varepsilon )}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="false" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi class="MJX-tex-caligraphic" mathvariant="script">L</mi>
</mrow>
</mrow>
<mo stretchy="false">(</mo>
<mi>ε<!-- ε --></mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\textstyle {\mathcal {L}}(\varepsilon )}</annotation>
</semantics>
</math></span><img src="./ec3ad3480531270152db218b7754cd9e68b3521c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.497ex; height:2.843ex;" alt="{\textstyle {\mathcal {L}}(\varepsilon )}" loading="lazy"></span> is known as a Lagrange matrix and whose elements correspond to <a href="Lagrange_bracket" title="Lagrange bracket">Lagrange brackets</a>. The last identity can also be stated as the following:<span class="mwe-math-element mwe-math-element-block"><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 \sum _{k=1}^{2N}\{\eta _{i},\eta _{k}\}[\eta _{k},\eta _{j}]=-\delta _{ij}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<munderover>
<mo>∑<!-- ∑ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>k</mi>
<mo>=</mo>
<mn>1</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
<mi>N</mi>
</mrow>
</munderover>
<mo fence="false" stretchy="false">{</mo>
<msub>
<mi>η<!-- η --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
<mo>,</mo>
<msub>
<mi>η<!-- η --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>k</mi>
</mrow>
</msub>
<mo fence="false" stretchy="false">}</mo>
<mo stretchy="false">[</mo>
<msub>
<mi>η<!-- η --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>k</mi>
</mrow>
</msub>
<mo>,</mo>
<msub>
<mi>η<!-- η --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>j</mi>
</mrow>
</msub>
<mo stretchy="false">]</mo>
<mo>=</mo>
<mo>−<!-- − --></mo>
<msub>
<mi>δ<!-- δ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
<mi>j</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \sum _{k=1}^{2N}\{\eta _{i},\eta _{k}\}[\eta _{k},\eta _{j}]=-\delta _{ij}}</annotation>
</semantics>
</math></span></span>Note that the summation here involves generalized coordinates as well as generalized momentum.
</p><p>The invariance of Poisson bracket can be expressed as: <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \{\varepsilon _{i},\varepsilon _{j}\}_{\eta }=\{\varepsilon _{i},\varepsilon _{j}\}_{\varepsilon }=J_{ij}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="false" scriptlevel="0">
<mo fence="false" stretchy="false">{</mo>
<msub>
<mi>ε<!-- ε --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
<mo>,</mo>
<msub>
<mi>ε<!-- ε --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>j</mi>
</mrow>
</msub>
<msub>
<mo fence="false" stretchy="false">}</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>η<!-- η --></mi>
</mrow>
</msub>
<mo>=</mo>
<mo fence="false" stretchy="false">{</mo>
<msub>
<mi>ε<!-- ε --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
<mo>,</mo>
<msub>
<mi>ε<!-- ε --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>j</mi>
</mrow>
</msub>
<msub>
<mo fence="false" stretchy="false">}</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>ε<!-- ε --></mi>
</mrow>
</msub>
<mo>=</mo>
<msub>
<mi>J</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
<mi>j</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\textstyle \{\varepsilon _{i},\varepsilon _{j}\}_{\eta }=\{\varepsilon _{i},\varepsilon _{j}\}_{\varepsilon }=J_{ij}}</annotation>
</semantics>
</math></span><img src="./1af681232916b1fbf5749679993588220ce9c4aa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:25.492ex; height:3.009ex;" alt="{\textstyle \{\varepsilon _{i},\varepsilon _{j}\}_{\eta }=\{\varepsilon _{i},\varepsilon _{j}\}_{\varepsilon }=J_{ij}}" loading="lazy"></span>, which directly leads to the symplectic condition: <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle MJM^{T}=J}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="false" scriptlevel="0">
<mi>M</mi>
<mi>J</mi>
<msup>
<mi>M</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>T</mi>
</mrow>
</msup>
<mo>=</mo>
<mi>J</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\textstyle MJM^{T}=J}</annotation>
</semantics>
</math></span><img src="./ad1baf25fb1d989aec0e4a15933e537a9c0bec47.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:12.371ex; height:2.509ex;" alt="{\textstyle MJM^{T}=J}" loading="lazy"></span>.<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup>
</p>
<div class="mw-heading mw-heading2"><h2 id="Constants_of_motion">Constants of motion</h2></div>
<p>An <a href="Integrable_system" title="Integrable system">integrable system</a> will have <a href="Constants_of_motion" class="mw-redirect" title="Constants of motion">constants of motion</a> in addition to the energy. Such constants of motion will commute with the Hamiltonian under the Poisson bracket. Suppose some function <span class="mwe-math-element mwe-math-element-inline"><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(p,q)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>p</mi>
<mo>,</mo>
<mi>q</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f(p,q)}</annotation>
</semantics>
</math></span><img src="./f1f73ce44a1cb676deb977b8d833ad1decd3e09a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.361ex; height:2.843ex;" alt="{\displaystyle f(p,q)}" loading="lazy"></span> is a constant of motion. This implies that if <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p(t),q(t)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>p</mi>
<mo stretchy="false">(</mo>
<mi>t</mi>
<mo stretchy="false">)</mo>
<mo>,</mo>
<mi>q</mi>
<mo stretchy="false">(</mo>
<mi>t</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle p(t),q(t)}</annotation>
</semantics>
</math></span><img src="./cb47a84e6f06bb36212097929ad2cd42e9bfb781.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:8.66ex; height:2.843ex;" alt="{\displaystyle p(t),q(t)}" loading="lazy"></span> is a <a href="Trajectory" title="Trajectory">trajectory</a> or solution to <a href="Hamilton's_equations_of_motion" class="mw-redirect" title="Hamilton's equations of motion">Hamilton's equations of motion</a>, then along that trajectory:<span class="mwe-math-element mwe-math-element-block"><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 0={\frac {df}{dt}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mn>0</mn>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mi>d</mi>
<mi>f</mi>
</mrow>
<mrow>
<mi>d</mi>
<mi>t</mi>
</mrow>
</mfrac>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle 0={\frac {df}{dt}}}</annotation>
</semantics>
</math></span></span>Where, as above, the intermediate step follows by applying the equations of motion and we assume that <span class="mwe-math-element mwe-math-element-inline"><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}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>f</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f}</annotation>
</semantics>
</math></span><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"></span> does not explicitly depend on time. This equation is known as the <a href="Liouville's_theorem_(Hamiltonian)#Liouville_equations" title="Liouville's theorem (Hamiltonian)">Liouville equation</a>. The content of <a href="Liouville's_theorem_(Hamiltonian)" title="Liouville's theorem (Hamiltonian)">Liouville's theorem</a> is that the time evolution of a <a href="Measure_(mathematics)" title="Measure (mathematics)">measure</a> given by a <a href="Distribution_function_(physics)" title="Distribution function (physics)">distribution function</a> <span class="mwe-math-element mwe-math-element-inline"><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}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>f</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f}</annotation>
</semantics>
</math></span><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"></span> is given by the above equation.
</p><p>If the Poisson bracket of <span class="mwe-math-element mwe-math-element-inline"><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}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>f</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f}</annotation>
</semantics>
</math></span><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"></span> and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>g</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle g}</annotation>
</semantics>
</math></span><img src="./d3556280e66fe2c0d0140df20935a6f057381d77.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}" loading="lazy"></span> vanishes (<span class="mwe-math-element mwe-math-element-inline"><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\}=0}">
<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>
<mn>0</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \{f,g\}=0}</annotation>
</semantics>
</math></span><img src="./e3b33bdfb9f245ef4b8b2dd7aa78a4fb8ef77ed5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.014ex; height:2.843ex;" alt="{\displaystyle \{f,g\}=0}" loading="lazy"></span>), then <span class="mwe-math-element mwe-math-element-inline"><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}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>f</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f}</annotation>
</semantics>
</math></span><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"></span> and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>g</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle g}</annotation>
</semantics>
</math></span><img src="./d3556280e66fe2c0d0140df20935a6f057381d77.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}" loading="lazy"></span> are said to be <b>in involution</b>. In order for a Hamiltonian system to be <a href="Completely_integrable" class="mw-redirect" title="Completely integrable">completely integrable</a>, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>n</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle n}</annotation>
</semantics>
</math></span><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"></span> independent constants of motion must be in <a href="Distribution_(differential_geometry)#Involutive_distributions" title="Distribution (differential geometry)">mutual involution</a>, where <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>n</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle n}</annotation>
</semantics>
</math></span><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"></span> is the number of degrees of freedom.
</p><p>Furthermore, according to <b>Poisson's Theorem</b>, if two quantities <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>A</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle A}</annotation>
</semantics>
</math></span><img src="./7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy"></span> and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>B</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle B}</annotation>
</semantics>
</math></span><img src="./47136aad860d145f75f3eed3022df827cee94d7a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}" loading="lazy"></span> are explicitly time independent (<span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A(p,q),B(p,q)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>A</mi>
<mo stretchy="false">(</mo>
<mi>p</mi>
<mo>,</mo>
<mi>q</mi>
<mo stretchy="false">)</mo>
<mo>,</mo>
<mi>B</mi>
<mo stretchy="false">(</mo>
<mi>p</mi>
<mo>,</mo>
<mi>q</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle A(p,q),B(p,q)}</annotation>
</semantics>
</math></span><img src="./e9ccd197afb1b10ec5b2784f162d539941eab14e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.705ex; height:2.843ex;" alt="{\displaystyle A(p,q),B(p,q)}" loading="lazy"></span>) constants of motion, so is their Poisson bracket <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{A,\,B\}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo fence="false" stretchy="false">{</mo>
<mi>A</mi>
<mo>,</mo>
<mspace width="thinmathspace"></mspace>
<mi>B</mi>
<mo fence="false" stretchy="false">}</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \{A,\,B\}}</annotation>
</semantics>
</math></span><img src="./790fae42caa999d320d97bacb7de37bccb8fa5e2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.253ex; height:2.843ex;" alt="{\displaystyle \{A,\,B\}}" loading="lazy"></span>. This follows from the Jacobi identity (see section below). Poisson's Theorem does not always supply a useful result, however, since the number of possible constants of motion is limited (<span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2n-1}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mn>2</mn>
<mi>n</mi>
<mo>−<!-- − --></mo>
<mn>1</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle 2n-1}</annotation>
</semantics>
</math></span><img src="./fb6ce8a02613283b1e60305814a1457335b44437.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.56ex; height:2.343ex;" alt="{\displaystyle 2n-1}" loading="lazy"></span> for a system with <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>n</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle n}</annotation>
</semantics>
</math></span><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"></span> degrees of freedom), and so the result may be trivial (a constant, or a function of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>A</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle A}</annotation>
</semantics>
</math></span><img src="./7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy"></span> and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>B</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle B}</annotation>
</semantics>
</math></span><img src="./47136aad860d145f75f3eed3022df827cee94d7a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}" loading="lazy"></span>.)
</p>
<div class="mw-heading mw-heading2"><h2 id="The_Poisson_bracket_in_coordinate-free_language">The Poisson bracket in coordinate-free language</h2></div>
<p>Let <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>M</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle M}</annotation>
</semantics>
</math></span><img src="./f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy"></span> be a <a href="Symplectic_manifold" title="Symplectic manifold">symplectic manifold</a>, that is, a <a href="Manifold" title="Manifold">manifold</a> equipped with a <a href="Symplectic_form" class="mw-redirect" title="Symplectic form">symplectic form</a>: a <a href="Differential_form" title="Differential form">2-form</a> <span class="mwe-math-element mwe-math-element-inline"><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 }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>ω<!-- ω --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \omega }</annotation>
</semantics>
</math></span><img src="./48eff443f9de7a985bb94ca3bde20813ea737be8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.446ex; height:1.676ex;" alt="{\displaystyle \omega }" loading="lazy"></span> which is both <b>closed</b> (i.e., its <a href="Exterior_derivative" title="Exterior derivative">exterior derivative</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d\omega }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>d</mi>
<mi>ω<!-- ω --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle d\omega }</annotation>
</semantics>
</math></span><img src="./ad5fe5ef6e9587a089d370dc32119c60e91f1d4c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.662ex; height:2.176ex;" alt="{\displaystyle d\omega }" loading="lazy"></span> vanishes) and <b>non-degenerate</b>. For example, in the treatment above, take <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>M</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle M}</annotation>
</semantics>
</math></span><img src="./f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy"></span> to be <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{2n}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">R</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
<mi>n</mi>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{2n}}</annotation>
</semantics>
</math></span><img src="./47460f1a92774729807be11cf62b9178b5771b4a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.719ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{2n}}" loading="lazy"></span> and take
<span class="mwe-math-element mwe-math-element-block"><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 =\sum _{i=1}^{n}dq_{i}\wedge dp_{i}.}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>ω<!-- ω --></mi>
<mo>=</mo>
<munderover>
<mo>∑<!-- ∑ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
<mo>=</mo>
<mn>1</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</munderover>
<mi>d</mi>
<msub>
<mi>q</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
<mo>∧<!-- ∧ --></mo>
<mi>d</mi>
<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=1}^{n}dq_{i}\wedge dp_{i}.}</annotation>
</semantics>
</math></span></span>
</p><p>If <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \iota _{v}\omega }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>ι<!-- ι --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>v</mi>
</mrow>
</msub>
<mi>ω<!-- ω --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \iota _{v}\omega }</annotation>
</semantics>
</math></span><img src="./135214e1e88dbd2255e42e2027217c0fd6f59904.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.299ex; height:2.009ex;" alt="{\displaystyle \iota _{v}\omega }" loading="lazy"></span> is the <a href="Interior_product" title="Interior product">interior product</a> or <a href="Tensor_contraction" title="Tensor contraction">contraction</a> operation defined by <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\iota _{v}\omega )(u)=\omega (v,\,u)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo stretchy="false">(</mo>
<msub>
<mi>ι<!-- ι --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>v</mi>
</mrow>
</msub>
<mi>ω<!-- ω --></mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">(</mo>
<mi>u</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi>ω<!-- ω --></mi>
<mo stretchy="false">(</mo>
<mi>v</mi>
<mo>,</mo>
<mspace width="thinmathspace"></mspace>
<mi>u</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle (\iota _{v}\omega )(u)=\omega (v,\,u)}</annotation>
</semantics>
</math></span><img src="./3e1be627aa763d7f66a2e7e8b2776f1dfda4dfc7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.479ex; height:2.843ex;" alt="{\displaystyle (\iota _{v}\omega )(u)=\omega (v,\,u)}" loading="lazy"></span>, then non-degeneracy is equivalent to saying that for every one-form <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>α<!-- α --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation>
</semantics>
</math></span><img src="./b79333175c8b3f0840bfb4ec41b8072c83ea88d3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }" loading="lazy"></span> there is a unique vector field <span class="mwe-math-element mwe-math-element-inline"><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 _{\alpha }}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi mathvariant="normal">Ω<!-- Ω --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>α<!-- α --></mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \Omega _{\alpha }}</annotation>
</semantics>
</math></span><img src="./cb42c33a7b17ff4164064c8d31e0f636767d12f3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.962ex; height:2.509ex;" alt="{\displaystyle \Omega _{\alpha }}" loading="lazy"></span> such that <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \iota _{\Omega _{\alpha }}\omega =\alpha }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>ι<!-- ι --></mi>
<mrow class="MJX-TeXAtom-ORD">
<msub>
<mi mathvariant="normal">Ω<!-- Ω --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>α<!-- α --></mi>
</mrow>
</msub>
</mrow>
</msub>
<mi>ω<!-- ω --></mi>
<mo>=</mo>
<mi>α<!-- α --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \iota _{\Omega _{\alpha }}\omega =\alpha }</annotation>
</semantics>
</math></span><img src="./9a322a6d9207e6784bf1c99c22230a3a9cab7341.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.292ex; height:2.176ex;" alt="{\displaystyle \iota _{\Omega _{\alpha }}\omega =\alpha }" loading="lazy"></span>. Alternatively, <span class="mwe-math-element mwe-math-element-inline"><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 _{dH}=\omega ^{-1}(dH)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi mathvariant="normal">Ω<!-- Ω --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>d</mi>
<mi>H</mi>
</mrow>
</msub>
<mo>=</mo>
<msup>
<mi>ω<!-- ω --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>−<!-- − --></mo>
<mn>1</mn>
</mrow>
</msup>
<mo stretchy="false">(</mo>
<mi>d</mi>
<mi>H</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \Omega _{dH}=\omega ^{-1}(dH)}</annotation>
</semantics>
</math></span><img src="./0d8800bb44aa06ff9c8554260edc64ca460de48c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.195ex; height:3.176ex;" alt="{\displaystyle \Omega _{dH}=\omega ^{-1}(dH)}" loading="lazy"></span>. Then if <span class="mwe-math-element mwe-math-element-inline"><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}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>H</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle H}</annotation>
</semantics>
</math></span><img src="./75a9edddcca2f782014371f75dca39d7e13a9c1b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle H}" loading="lazy"></span> is a smooth function on <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>M</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle M}</annotation>
</semantics>
</math></span><img src="./f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy"></span>, the <a href="Hamiltonian_vector_field" title="Hamiltonian vector field">Hamiltonian vector field</a> <span class="mwe-math-element mwe-math-element-inline"><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}}">
<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>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle X_{H}}</annotation>
</semantics>
</math></span><img src="./d229e5e97360e167654b28a40234485f65027ac8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.616ex; height:2.509ex;" alt="{\displaystyle X_{H}}" loading="lazy"></span> can be defined to be <span class="mwe-math-element mwe-math-element-inline"><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 _{dH}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi mathvariant="normal">Ω<!-- Ω --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>d</mi>
<mi>H</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \Omega _{dH}}</annotation>
</semantics>
</math></span><img src="./0891cd6ddeab8765a1a79dc616b3cdf7b63b0ae9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.229ex; height:2.509ex;" alt="{\displaystyle \Omega _{dH}}" loading="lazy"></span>. It is easy to see that
<span class="mwe-math-element mwe-math-element-block"><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}X_{p_{i}}&amp;={\frac {\partial }{\partial q_{i}}}\\X_{q_{i}}&amp;=-{\frac {\partial }{\partial p_{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>
<msub>
<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<msub>
<mi>p</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
</mrow>
</msub>
</mtd>
<mtd>
<mi></mi>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<mrow>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<msub>
<mi>q</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
</mrow>
</mfrac>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<msub>
<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<msub>
<mi>q</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
</mrow>
</msub>
</mtd>
<mtd>
<mi></mi>
<mo>=</mo>
<mo>−<!-- − --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<mrow>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<msub>
<mi>p</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
</mrow>
</mfrac>
</mrow>
<mo>.</mo>
</mtd>
</mtr>
</mtable>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}X_{p_{i}}&amp;={\frac {\partial }{\partial q_{i}}}\\X_{q_{i}}&amp;=-{\frac {\partial }{\partial p_{i}}}.\end{aligned}}}</annotation>
</semantics>
</math></span></span>
</p><p>The <b>Poisson bracket</b> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ \{\cdot ,\,\cdot \}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mtext>&nbsp;</mtext>
<mo fence="false" stretchy="false">{</mo>
<mo>⋅<!-- ⋅ --></mo>
<mo>,</mo>
<mspace width="thinmathspace"></mspace>
<mo>⋅<!-- ⋅ --></mo>
<mo fence="false" stretchy="false">}</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \ \{\cdot ,\,\cdot \}}</annotation>
</semantics>
</math></span><img src="./c9d755bdfb13226106697b8c6f7cfc3c0fb261bc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.62ex; height:2.843ex;" alt="{\displaystyle \ \{\cdot ,\,\cdot \}}" loading="lazy"></span> on <span class="texhtml">(<i>M</i>, <i>ω</i>)</span> is a <a href="Bilinear_map" title="Bilinear map">bilinear operation</a> on <a href="Differentiable_function" title="Differentiable function">differentiable functions</a>, defined by <span class="mwe-math-element mwe-math-element-inline"><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\}\;=\;\omega (X_{f},\,X_{g})}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo fence="false" stretchy="false">{</mo>
<mi>f</mi>
<mo>,</mo>
<mspace width="thinmathspace"></mspace>
<mi>g</mi>
<mo fence="false" stretchy="false">}</mo>
<mspace width="thickmathspace"></mspace>
<mo>=</mo>
<mspace width="thickmathspace"></mspace>
<mi>ω<!-- ω --></mi>
<mo stretchy="false">(</mo>
<msub>
<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>f</mi>
</mrow>
</msub>
<mo>,</mo>
<mspace width="thinmathspace"></mspace>
<msub>
<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>g</mi>
</mrow>
</msub>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \{f,\,g\}\;=\;\omega (X_{f},\,X_{g})}</annotation>
</semantics>
</math></span><img src="./f55ad3734e67d81ceebbf22bd30a927098395c4d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:21.212ex; height:3.009ex;" alt="{\displaystyle \{f,\,g\}\;=\;\omega (X_{f},\,X_{g})}" loading="lazy"></span>; the Poisson bracket of two functions on <span class="texhtml"><i>M</i></span> is itself a function on <span class="texhtml"><i>M</i></span>. The Poisson bracket is antisymmetric because:
<span class="mwe-math-element mwe-math-element-block"><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_{f},X_{g})=-\omega (X_{g},X_{f})=-\{g,f\}.}">
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<mstyle displaystyle="true" scriptlevel="0">
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<mo fence="false" stretchy="false">}</mo>
<mo>=</mo>
<mi>ω<!-- ω --></mi>
<mo stretchy="false">(</mo>
<msub>
<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>f</mi>
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<mi>g</mi>
</mrow>
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<mo stretchy="false">)</mo>
<mo>=</mo>
<mo>−<!-- − --></mo>
<mi>ω<!-- ω --></mi>
<mo stretchy="false">(</mo>
<msub>
<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>g</mi>
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<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>f</mi>
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<mo stretchy="false">)</mo>
<mo>=</mo>
<mo>−<!-- − --></mo>
<mo fence="false" stretchy="false">{</mo>
<mi>g</mi>
<mo>,</mo>
<mi>f</mi>
<mo fence="false" stretchy="false">}</mo>
<mo>.</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \{f,g\}=\omega (X_{f},X_{g})=-\omega (X_{g},X_{f})=-\{g,f\}.}</annotation>
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</math></span></span>
</p><p>Furthermore,
</p>
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</style><table role="presentation" class="numblk" style="margin-left: 0em;"><tbody><tr><td class="nowrap"><span class="mwe-math-element mwe-math-element-block"><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}\{f,g\}&amp;=\omega (X_{f},X_{g})=\omega (\Omega _{df},X_{g})\\&amp;=(\iota _{\Omega _{df}}\omega )(X_{g})=df(X_{g})\\&amp;=X_{g}f={\mathcal {L}}_{X_{g}}f.\end{aligned}}}">
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<mtd>
<mo fence="false" stretchy="false">{</mo>
<mi>f</mi>
<mo>,</mo>
<mi>g</mi>
<mo fence="false" stretchy="false">}</mo>
</mtd>
<mtd>
<mi></mi>
<mo>=</mo>
<mi>ω<!-- ω --></mi>
<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>
<mi>ω<!-- ω --></mi>
<mo stretchy="false">(</mo>
<msub>
<mi mathvariant="normal">Ω<!-- Ω --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>d</mi>
<mi>f</mi>
</mrow>
</msub>
<mo>,</mo>
<msub>
<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>g</mi>
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<mo stretchy="false">)</mo>
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</mtr>
<mtr>
<mtd></mtd>
<mtd>
<mi></mi>
<mo>=</mo>
<mo stretchy="false">(</mo>
<msub>
<mi>ι<!-- ι --></mi>
<mrow class="MJX-TeXAtom-ORD">
<msub>
<mi mathvariant="normal">Ω<!-- Ω --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>d</mi>
<mi>f</mi>
</mrow>
</msub>
</mrow>
</msub>
<mi>ω<!-- ω --></mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">(</mo>
<msub>
<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>g</mi>
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</msub>
<mo stretchy="false">)</mo>
<mo>=</mo>
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<mo stretchy="false">(</mo>
<msub>
<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>g</mi>
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<mo stretchy="false">)</mo>
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<mtr>
<mtd></mtd>
<mtd>
<mi></mi>
<mo>=</mo>
<msub>
<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>g</mi>
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<mi>f</mi>
<mo>=</mo>
<msub>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi class="MJX-tex-caligraphic" mathvariant="script">L</mi>
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<mrow class="MJX-TeXAtom-ORD">
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<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>g</mi>
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<mi>f</mi>
<mo>.</mo>
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</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\{f,g\}&amp;=\omega (X_{f},X_{g})=\omega (\Omega _{df},X_{g})\\&amp;=(\iota _{\Omega _{df}}\omega )(X_{g})=df(X_{g})\\&amp;=X_{g}f={\mathcal {L}}_{X_{g}}f.\end{aligned}}}</annotation>
</semantics>
</math></span></span></td> <td></td> <td class="nowrap"><span id="math_1" class="reference nourlexpansion" style="font-weight:bold;">1</span></td></tr></tbody></table>
<p>Here <span class="texhtml"><i>X<sub>g</sub>f</i></span> denotes the vector field <span class="texhtml"><i>X<sub>g</sub></i></span> applied to the function <span class="texhtml"><i>f</i></span> as a directional derivative, and <span class="mwe-math-element mwe-math-element-inline"><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_{g}}f}">
<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>g</mi>
</mrow>
</msub>
</mrow>
</msub>
<mi>f</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\mathcal {L}}_{X_{g}}f}</annotation>
</semantics>
</math></span><img src="./3bb2a5c22e9bd21e635eb1d706d6075aa4180938.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:5.28ex; height:3.009ex;" alt="{\displaystyle {\mathcal {L}}_{X_{g}}f}" loading="lazy"></span> denotes the (entirely equivalent) <a href="Lie_derivative" title="Lie derivative">Lie derivative</a> of the function <span class="texhtml"><i>f</i></span>.
</p><p>If <span class="texhtml">α</span> is an arbitrary one-form on <span class="texhtml"><i>M</i></span>, the vector field <span class="texhtml">Ω<sub>α</sub></span> generates (at least locally) a <a href="Flow_(mathematics)" title="Flow (mathematics)">flow</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi _{x}(t)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>ϕ<!-- ϕ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>x</mi>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<mi>t</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \phi _{x}(t)}</annotation>
</semantics>
</math></span><img src="./81f87a76cc3104b5779bbdbbfb02dc955632f7d2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.207ex; height:2.843ex;" alt="{\displaystyle \phi _{x}(t)}" loading="lazy"></span> satisfying the boundary condition <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi _{x}(0)=x}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>ϕ<!-- ϕ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>x</mi>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<mn>0</mn>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi>x</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \phi _{x}(0)=x}</annotation>
</semantics>
</math></span><img src="./1b635b9e034aaa7df15b73c9dba144ba3e2f402f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.958ex; height:2.843ex;" alt="{\displaystyle \phi _{x}(0)=x}" loading="lazy"></span> and the first-order differential equation
<span class="mwe-math-element mwe-math-element-block"><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 {\frac {d\phi _{x}}{dt}}=\left.\Omega _{\alpha }\right|_{\phi _{x}(t)}.}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mi>d</mi>
<msub>
<mi>ϕ<!-- ϕ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>x</mi>
</mrow>
</msub>
</mrow>
<mrow>
<mi>d</mi>
<mi>t</mi>
</mrow>
</mfrac>
</mrow>
<mo>=</mo>
<msub>
<mrow>
<mo fence="true" stretchy="true" symmetric="true"></mo>
<msub>
<mi mathvariant="normal">Ω<!-- Ω --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>α<!-- α --></mi>
</mrow>
</msub>
<mo>|</mo>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<msub>
<mi>ϕ<!-- ϕ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>x</mi>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<mi>t</mi>
<mo stretchy="false">)</mo>
</mrow>
</msub>
<mo>.</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\frac {d\phi _{x}}{dt}}=\left.\Omega _{\alpha }\right|_{\phi _{x}(t)}.}</annotation>
</semantics>
</math></span></span>
</p><p>The <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi _{x}(t)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>ϕ<!-- ϕ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>x</mi>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<mi>t</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \phi _{x}(t)}</annotation>
</semantics>
</math></span><img src="./81f87a76cc3104b5779bbdbbfb02dc955632f7d2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.207ex; height:2.843ex;" alt="{\displaystyle \phi _{x}(t)}" loading="lazy"></span> will be <a href="Symplectomorphism" title="Symplectomorphism">symplectomorphisms</a> (<a href="Canonical_transformation" title="Canonical transformation">canonical transformations</a>) for every <span class="texhtml"><i>t</i></span> as a function of <span class="texhtml"><i>x</i></span> if and only if <span class="mwe-math-element mwe-math-element-inline"><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}}_{\Omega _{\alpha }}\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 mathvariant="normal">Ω<!-- Ω --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>α<!-- α --></mi>
</mrow>
</msub>
</mrow>
</msub>
<mi>ω<!-- ω --></mi>
<mspace width="thickmathspace"></mspace>
<mo>=</mo>
<mspace width="thickmathspace"></mspace>
<mn>0</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\mathcal {L}}_{\Omega _{\alpha }}\omega \;=\;0}</annotation>
</semantics>
</math></span><img src="./9a83a1809361cb73ccc2ce06f8de29c9cd739cc7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.038ex; height:2.676ex;" alt="{\displaystyle {\mathcal {L}}_{\Omega _{\alpha }}\omega \;=\;0}" loading="lazy"></span>; when this is true, <span class="texhtml">Ω<sub>α</sub></span> is called a <a href="Symplectic_vector_field" title="Symplectic vector field">symplectic vector field</a>. Recalling <a href="Cartan's_identity" class="mw-redirect" title="Cartan's identity">Cartan's identity</a> <span class="mwe-math-element mwe-math-element-inline"><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}\omega \;=\;d(\iota _{X}\omega )\,+\,\iota _{X}d\omega }">
<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>
<mi>ω<!-- ω --></mi>
<mspace width="thickmathspace"></mspace>
<mo>=</mo>
<mspace width="thickmathspace"></mspace>
<mi>d</mi>
<mo stretchy="false">(</mo>
<msub>
<mi>ι<!-- ι --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>X</mi>
</mrow>
</msub>
<mi>ω<!-- ω --></mi>
<mo stretchy="false">)</mo>
<mspace width="thinmathspace"></mspace>
<mo>+</mo>
<mspace width="thinmathspace"></mspace>
<msub>
<mi>ι<!-- ι --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>X</mi>
</mrow>
</msub>
<mi>d</mi>
<mi>ω<!-- ω --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\mathcal {L}}_{X}\omega \;=\;d(\iota _{X}\omega )\,+\,\iota _{X}d\omega }</annotation>
</semantics>
</math></span><img src="./d3ef9f9cfd6f3ac74a6ca6e299b46815f6693a7c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.729ex; height:2.843ex;" alt="{\displaystyle {\mathcal {L}}_{X}\omega \;=\;d(\iota _{X}\omega )\,+\,\iota _{X}d\omega }" loading="lazy"></span> and <span class="texhtml"><i>d</i>ω = 0</span>, it follows that <span class="mwe-math-element mwe-math-element-inline"><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}}_{\Omega _{\alpha }}\omega \;=\;d\left(\iota _{\Omega _{\alpha }}\omega \right)\;=\;d\alpha }">
<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 mathvariant="normal">Ω<!-- Ω --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>α<!-- α --></mi>
</mrow>
</msub>
</mrow>
</msub>
<mi>ω<!-- ω --></mi>
<mspace width="thickmathspace"></mspace>
<mo>=</mo>
<mspace width="thickmathspace"></mspace>
<mi>d</mi>
<mrow>
<mo>(</mo>
<mrow>
<msub>
<mi>ι<!-- ι --></mi>
<mrow class="MJX-TeXAtom-ORD">
<msub>
<mi mathvariant="normal">Ω<!-- Ω --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>α<!-- α --></mi>
</mrow>
</msub>
</mrow>
</msub>
<mi>ω<!-- ω --></mi>
</mrow>
<mo>)</mo>
</mrow>
<mspace width="thickmathspace"></mspace>
<mo>=</mo>
<mspace width="thickmathspace"></mspace>
<mi>d</mi>
<mi>α<!-- α --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\mathcal {L}}_{\Omega _{\alpha }}\omega \;=\;d\left(\iota _{\Omega _{\alpha }}\omega \right)\;=\;d\alpha }</annotation>
</semantics>
</math></span><img src="./8be43a78cbe7f667c6a615b29d702dc16632a6c0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.086ex; height:2.843ex;" alt="{\displaystyle {\mathcal {L}}_{\Omega _{\alpha }}\omega \;=\;d\left(\iota _{\Omega _{\alpha }}\omega \right)\;=\;d\alpha }" loading="lazy"></span>. Therefore, <span class="texhtml">Ω<sub>α</sub></span> is a symplectic vector field if and only if α is a <a href="Closed_and_exact_differential_forms" title="Closed and exact differential forms">closed form</a>. Since <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d(df)\;=\;d^{2}f\;=\;0}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>d</mi>
<mo stretchy="false">(</mo>
<mi>d</mi>
<mi>f</mi>
<mo stretchy="false">)</mo>
<mspace width="thickmathspace"></mspace>
<mo>=</mo>
<mspace width="thickmathspace"></mspace>
<msup>
<mi>d</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mi>f</mi>
<mspace width="thickmathspace"></mspace>
<mo>=</mo>
<mspace width="thickmathspace"></mspace>
<mn>0</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle d(df)\;=\;d^{2}f\;=\;0}</annotation>
</semantics>
</math></span><img src="./bbee0fb98b9ab0af6cac2d022c221f8451e84040.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.011ex; height:3.176ex;" alt="{\displaystyle d(df)\;=\;d^{2}f\;=\;0}" loading="lazy"></span>, it follows that every Hamiltonian vector field <span class="texhtml"><i>X<sub>f</sub></i></span> is a symplectic vector field, and that the Hamiltonian flow consists of canonical transformations. From <b><a href="#math_1">(1)</a></b> above, under the Hamiltonian flow <span class="mwe-math-element mwe-math-element-inline"><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_{\mathcal {H}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi class="MJX-tex-caligraphic" mathvariant="script">H</mi>
</mrow>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle X_{\mathcal {H}}}</annotation>
</semantics>
</math></span><img src="./91ed9819b34aa803f3d3b79a40acc4c4bd1c2790.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.545ex; height:2.509ex;" alt="{\displaystyle X_{\mathcal {H}}}" loading="lazy"></span>,
<span class="mwe-math-element mwe-math-element-block"><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 {\frac {d}{dt}}f(\phi _{x}(t))=X_{\mathcal {H}}f=\{f,{\mathcal {H}}\}.}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mi>d</mi>
<mrow>
<mi>d</mi>
<mi>t</mi>
</mrow>
</mfrac>
</mrow>
<mi>f</mi>
<mo stretchy="false">(</mo>
<msub>
<mi>ϕ<!-- ϕ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>x</mi>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<mi>t</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">)</mo>
<mo>=</mo>
<msub>
<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi class="MJX-tex-caligraphic" mathvariant="script">H</mi>
</mrow>
</mrow>
</msub>
<mi>f</mi>
<mo>=</mo>
<mo fence="false" stretchy="false">{</mo>
<mi>f</mi>
<mo>,</mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi class="MJX-tex-caligraphic" mathvariant="script">H</mi>
</mrow>
</mrow>
<mo fence="false" stretchy="false">}</mo>
<mo>.</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\frac {d}{dt}}f(\phi _{x}(t))=X_{\mathcal {H}}f=\{f,{\mathcal {H}}\}.}</annotation>
</semantics>
</math></span></span>
</p><p>This is a fundamental result in Hamiltonian mechanics, governing the time evolution of functions defined on phase space. As noted above, when <span class="mwe-math-element mwe-math-element-inline"><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,{\mathcal {H}}\}=0}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo fence="false" stretchy="false">{</mo>
<mi>f</mi>
<mo>,</mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi class="MJX-tex-caligraphic" mathvariant="script">H</mi>
</mrow>
</mrow>
<mo fence="false" stretchy="false">}</mo>
<mo>=</mo>
<mn>0</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \{f,{\mathcal {H}}\}=0}</annotation>
</semantics>
</math></span><img src="./5fb042b4e8caaf090e158f8cbe2d807e9af524b7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.862ex; height:2.843ex;" alt="{\displaystyle \{f,{\mathcal {H}}\}=0}" loading="lazy"></span>, <span class="texhtml"><i>f</i></span> is a constant of motion of the system. In addition, in canonical coordinates (with <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{p_{i},\,p_{j}\}\;=\;\{q_{i},q_{j}\}\;=\;0}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo fence="false" stretchy="false">{</mo>
<msub>
<mi>p</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
<mo>,</mo>
<mspace width="thinmathspace"></mspace>
<msub>
<mi>p</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>j</mi>
</mrow>
</msub>
<mo fence="false" stretchy="false">}</mo>
<mspace width="thickmathspace"></mspace>
<mo>=</mo>
<mspace width="thickmathspace"></mspace>
<mo fence="false" stretchy="false">{</mo>
<msub>
<mi>q</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
<mo>,</mo>
<msub>
<mi>q</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>j</mi>
</mrow>
</msub>
<mo fence="false" stretchy="false">}</mo>
<mspace width="thickmathspace"></mspace>
<mo>=</mo>
<mspace width="thickmathspace"></mspace>
<mn>0</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \{p_{i},\,p_{j}\}\;=\;\{q_{i},q_{j}\}\;=\;0}</annotation>
</semantics>
</math></span><img src="./75ec93326306d6e97ec7b991d1ba4afacf078c11.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:24.877ex; height:3.009ex;" alt="{\displaystyle \{p_{i},\,p_{j}\}\;=\;\{q_{i},q_{j}\}\;=\;0}" loading="lazy"></span> and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{q_{i},\,p_{j}\}\;=\;\delta _{ij}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo fence="false" stretchy="false">{</mo>
<msub>
<mi>q</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
<mo>,</mo>
<mspace width="thinmathspace"></mspace>
<msub>
<mi>p</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>j</mi>
</mrow>
</msub>
<mo fence="false" stretchy="false">}</mo>
<mspace width="thickmathspace"></mspace>
<mo>=</mo>
<mspace width="thickmathspace"></mspace>
<msub>
<mi>δ<!-- δ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
<mi>j</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \{q_{i},\,p_{j}\}\;=\;\delta _{ij}}</annotation>
</semantics>
</math></span><img src="./bfaeb184778351142286fd0d6aa09a4b8bdaf5f2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:14.56ex; height:3.009ex;" alt="{\displaystyle \{q_{i},\,p_{j}\}\;=\;\delta _{ij}}" loading="lazy"></span>), Hamilton's equations for the time evolution of the system follow immediately from this formula.
</p><p>It also follows from <b><a href="#math_1">(1)</a></b> that the Poisson bracket is a <a href="Derivation_(abstract_algebra)" class="mw-redirect" title="Derivation (abstract algebra)">derivation</a>; that is, it satisfies a non-commutative version of Leibniz's <a href="Product_rule" title="Product rule">product rule</a>:
</p>
<table role="presentation" class="numblk" style="margin-left: 0em;"><tbody><tr><td class="nowrap"><span class="mwe-math-element mwe-math-element-block"><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 \{fg,h\}=f\{g,h\}+g\{f,h\},}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo fence="false" stretchy="false">{</mo>
<mi>f</mi>
<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>
<mi>g</mi>
<mo>,</mo>
<mi>h</mi>
<mo fence="false" stretchy="false">}</mo>
<mo>+</mo>
<mi>g</mi>
<mo fence="false" stretchy="false">{</mo>
<mi>f</mi>
<mo>,</mo>
<mi>h</mi>
<mo fence="false" stretchy="false">}</mo>
<mo>,</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \{fg,h\}=f\{g,h\}+g\{f,h\},}</annotation>
</semantics>
</math></span></span> and <span class="mwe-math-element mwe-math-element-block"><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,gh\}=g\{f,h\}+h\{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>
<mi>h</mi>
<mo fence="false" stretchy="false">}</mo>
<mo>=</mo>
<mi>g</mi>
<mo fence="false" stretchy="false">{</mo>
<mi>f</mi>
<mo>,</mo>
<mi>h</mi>
<mo fence="false" stretchy="false">}</mo>
<mo>+</mo>
<mi>h</mi>
<mo fence="false" stretchy="false">{</mo>
<mi>f</mi>
<mo>,</mo>
<mi>g</mi>
<mo fence="false" stretchy="false">}</mo>
<mo>.</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \{f,gh\}=g\{f,h\}+h\{f,g\}.}</annotation>
</semantics>
</math></span></span></td> <td></td> <td class="nowrap"><span id="math_2" class="reference nourlexpansion" style="font-weight:bold;">2</span></td></tr></tbody></table>
<p>The Poisson bracket is intimately connected to the <a href="Lie_bracket_of_vector_fields" title="Lie bracket of vector fields">Lie bracket</a> of the Hamiltonian vector fields. Because the Lie derivative is a derivation,
<span class="mwe-math-element mwe-math-element-block"><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 {\mathcal {L}}_{v}\iota _{u}\omega =\iota _{{\mathcal {L}}_{v}u}\omega +\iota _{u}{\mathcal {L}}_{v}\omega =\iota _{[v,u]}\omega +\iota _{u}{\mathcal {L}}_{v}\omega .}">
<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>v</mi>
</mrow>
</msub>
<msub>
<mi>ι<!-- ι --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>u</mi>
</mrow>
</msub>
<mi>ω<!-- ω --></mi>
<mo>=</mo>
<msub>
<mi>ι<!-- ι --></mi>
<mrow class="MJX-TeXAtom-ORD">
<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>v</mi>
</mrow>
</msub>
<mi>u</mi>
</mrow>
</msub>
<mi>ω<!-- ω --></mi>
<mo>+</mo>
<msub>
<mi>ι<!-- ι --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>u</mi>
</mrow>
</msub>
<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>v</mi>
</mrow>
</msub>
<mi>ω<!-- ω --></mi>
<mo>=</mo>
<msub>
<mi>ι<!-- ι --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">[</mo>
<mi>v</mi>
<mo>,</mo>
<mi>u</mi>
<mo stretchy="false">]</mo>
</mrow>
</msub>
<mi>ω<!-- ω --></mi>
<mo>+</mo>
<msub>
<mi>ι<!-- ι --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>u</mi>
</mrow>
</msub>
<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>v</mi>
</mrow>
</msub>
<mi>ω<!-- ω --></mi>
<mo>.</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\mathcal {L}}_{v}\iota _{u}\omega =\iota _{{\mathcal {L}}_{v}u}\omega +\iota _{u}{\mathcal {L}}_{v}\omega =\iota _{[v,u]}\omega +\iota _{u}{\mathcal {L}}_{v}\omega .}</annotation>
</semantics>
</math></span></span>
</p><p>Thus if <span class="texhtml"><i>v</i></span> and <span class="texhtml"><i>u</i></span> are symplectic, using <span class="mwe-math-element mwe-math-element-inline"><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}}_{v}\omega =0={\mathcal {L}}_{u}\omega }">
<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>v</mi>
</mrow>
</msub>
<mi>ω<!-- ω --></mi>
<mo>=</mo>
<mn>0</mn>
<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">
<mi>u</mi>
</mrow>
</msub>
<mi>ω<!-- ω --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\mathcal {L}}_{v}\omega =0={\mathcal {L}}_{u}\omega }</annotation>
</semantics>
</math></span><img src="./fab6d438763dcfe9dec42eb467c108e8b1e3303c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.661ex; height:2.509ex;" alt="{\displaystyle {\mathcal {L}}_{v}\omega =0={\mathcal {L}}_{u}\omega }" loading="lazy"></span>, Cartan's identity, and the fact that <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \iota _{u}\omega }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>ι<!-- ι --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>u</mi>
</mrow>
</msub>
<mi>ω<!-- ω --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \iota _{u}\omega }</annotation>
</semantics>
</math></span><img src="./9f7bf6016291c0939d2176bc3307355fe9a4d093.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.442ex; height:2.009ex;" alt="{\displaystyle \iota _{u}\omega }" loading="lazy"></span> is a closed form,
<span class="mwe-math-element mwe-math-element-block"><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 \iota _{[v,u]}\omega ={\mathcal {L}}_{v}\iota _{u}\omega =d(\iota _{v}\iota _{u}\omega )+\iota _{v}d(\iota _{u}\omega )=d(\iota _{v}\iota _{u}\omega )=d(\omega (u,v)).}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>ι<!-- ι --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">[</mo>
<mi>v</mi>
<mo>,</mo>
<mi>u</mi>
<mo stretchy="false">]</mo>
</mrow>
</msub>
<mi>ω<!-- ω --></mi>
<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">
<mi>v</mi>
</mrow>
</msub>
<msub>
<mi>ι<!-- ι --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>u</mi>
</mrow>
</msub>
<mi>ω<!-- ω --></mi>
<mo>=</mo>
<mi>d</mi>
<mo stretchy="false">(</mo>
<msub>
<mi>ι<!-- ι --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>v</mi>
</mrow>
</msub>
<msub>
<mi>ι<!-- ι --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>u</mi>
</mrow>
</msub>
<mi>ω<!-- ω --></mi>
<mo stretchy="false">)</mo>
<mo>+</mo>
<msub>
<mi>ι<!-- ι --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>v</mi>
</mrow>
</msub>
<mi>d</mi>
<mo stretchy="false">(</mo>
<msub>
<mi>ι<!-- ι --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>u</mi>
</mrow>
</msub>
<mi>ω<!-- ω --></mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi>d</mi>
<mo stretchy="false">(</mo>
<msub>
<mi>ι<!-- ι --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>v</mi>
</mrow>
</msub>
<msub>
<mi>ι<!-- ι --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>u</mi>
</mrow>
</msub>
<mi>ω<!-- ω --></mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi>d</mi>
<mo stretchy="false">(</mo>
<mi>ω<!-- ω --></mi>
<mo stretchy="false">(</mo>
<mi>u</mi>
<mo>,</mo>
<mi>v</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">)</mo>
<mo>.</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \iota _{[v,u]}\omega ={\mathcal {L}}_{v}\iota _{u}\omega =d(\iota _{v}\iota _{u}\omega )+\iota _{v}d(\iota _{u}\omega )=d(\iota _{v}\iota _{u}\omega )=d(\omega (u,v)).}</annotation>
</semantics>
</math></span></span>
</p><p>It follows that <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [v,u]=X_{\omega (u,v)}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo stretchy="false">[</mo>
<mi>v</mi>
<mo>,</mo>
<mi>u</mi>
<mo stretchy="false">]</mo>
<mo>=</mo>
<msub>
<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>ω<!-- ω --></mi>
<mo stretchy="false">(</mo>
<mi>u</mi>
<mo>,</mo>
<mi>v</mi>
<mo stretchy="false">)</mo>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle [v,u]=X_{\omega (u,v)}}</annotation>
</semantics>
</math></span><img src="./4d8f3e43d192a6a5d02c9d440fb5f54e7ace1a4d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:14.537ex; height:3.176ex;" alt="{\displaystyle [v,u]=X_{\omega (u,v)}}" loading="lazy"></span>, so that
</p>
<table role="presentation" class="numblk" style="margin-left: 0em;"><tbody><tr><td class="nowrap"><span class="mwe-math-element mwe-math-element-block"><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 [X_{f},X_{g}]=X_{\omega (X_{g},X_{f})}=-X_{\omega (X_{f},X_{g})}=-X_{\{f,g\}}.}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<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>
<msub>
<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>ω<!-- 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>
</mrow>
</msub>
<mo>=</mo>
<mo>−<!-- − --></mo>
<msub>
<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>ω<!-- ω --></mi>
<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>
</mrow>
</msub>
<mo>=</mo>
<mo>−<!-- − --></mo>
<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>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle [X_{f},X_{g}]=X_{\omega (X_{g},X_{f})}=-X_{\omega (X_{f},X_{g})}=-X_{\{f,g\}}.}</annotation>
</semantics>
</math></span></span></td> <td></td> <td class="nowrap"><span id="math_3" class="reference nourlexpansion" style="font-weight:bold;">3</span></td></tr></tbody></table>
<p>Thus, the Poisson bracket on functions corresponds to the Lie bracket of the associated Hamiltonian vector fields. We have also shown that the Lie bracket of two symplectic vector fields is a Hamiltonian vector field and hence is also symplectic. In the language of <a href="Abstract_algebra" title="Abstract algebra">abstract algebra</a>, the symplectic vector fields form a <a href="Subalgebra" title="Subalgebra">subalgebra</a> of the <a href="Lie_algebra" title="Lie algebra">Lie algebra</a> of smooth vector fields on <span class="texhtml"><i>M</i></span>, and the Hamiltonian vector fields form an <a href="Algebraic_ideal" class="mw-redirect" title="Algebraic ideal">ideal</a> of this subalgebra. The symplectic vector fields are the Lie algebra of the (infinite-dimensional) <a href="Lie_group" title="Lie group">Lie group</a> of <a href="Symplectomorphism" title="Symplectomorphism">symplectomorphisms</a> of <span class="texhtml"><i>M</i></span>.
</p><p>It is widely asserted that the <a href="Jacobi_identity" title="Jacobi identity">Jacobi identity</a> for the Poisson bracket,
<span class="mwe-math-element mwe-math-element-block"><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,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>
<mi>f</mi>
<mo>,</mo>
<mo fence="false" stretchy="false">{</mo>
<mi>g</mi>
<mo>,</mo>
<mi>h</mi>
<mo fence="false" stretchy="false">}</mo>
<mo fence="false" stretchy="false">}</mo>
<mo>+</mo>
<mo fence="false" stretchy="false">{</mo>
<mi>g</mi>
<mo>,</mo>
<mo fence="false" stretchy="false">{</mo>
<mi>h</mi>
<mo>,</mo>
<mi>f</mi>
<mo fence="false" stretchy="false">}</mo>
<mo fence="false" stretchy="false">}</mo>
<mo>+</mo>
<mo fence="false" stretchy="false">{</mo>
<mi>h</mi>
<mo>,</mo>
<mo fence="false" stretchy="false">{</mo>
<mi>f</mi>
<mo>,</mo>
<mi>g</mi>
<mo fence="false" stretchy="false">}</mo>
<mo fence="false" stretchy="false">}</mo>
<mo>=</mo>
<mn>0</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \{f,\{g,h\}\}+\{g,\{h,f\}\}+\{h,\{f,g\}\}=0}</annotation>
</semantics>
</math></span></span>
follows from the corresponding identity for the Lie bracket of vector fields, but this is true only up to a locally constant function. However, to prove the Jacobi identity for the Poisson bracket, it is <a href="Jacobi_identity#Examples" title="Jacobi identity">sufficient</a> to show that:
<span class="mwe-math-element mwe-math-element-block"><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 \operatorname {ad} _{\{g,f\}}=\operatorname {ad} _{-\{f,g\}}=[\operatorname {ad} _{f},\operatorname {ad} _{g}]}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>ad</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo fence="false" stretchy="false">{</mo>
<mi>g</mi>
<mo>,</mo>
<mi>f</mi>
<mo fence="false" stretchy="false">}</mo>
</mrow>
</msub>
<mo>=</mo>
<msub>
<mi>ad</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>−<!-- − --></mo>
<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 stretchy="false">[</mo>
<msub>
<mi>ad</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>f</mi>
</mrow>
</msub>
<mo>,</mo>
<msub>
<mi>ad</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>g</mi>
</mrow>
</msub>
<mo stretchy="false">]</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \operatorname {ad} _{\{g,f\}}=\operatorname {ad} _{-\{f,g\}}=[\operatorname {ad} _{f},\operatorname {ad} _{g}]}</annotation>
</semantics>
</math></span></span>
where the operator <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {ad} _{g}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>ad</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>g</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \operatorname {ad} _{g}}</annotation>
</semantics>
</math></span><img src="./3e71199d30b9419c35306d8247ba8c5a640cf031.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.476ex; height:2.843ex;" alt="{\displaystyle \operatorname {ad} _{g}}" loading="lazy"></span> on smooth functions on <span class="texhtml"><i>M</i></span> is defined by <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {ad} _{g}(\cdot )\;=\;\{\cdot ,\,g\}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>ad</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>g</mi>
</mrow>
</msub>
<mo>⁡<!-- ⁡ --></mo>
<mo stretchy="false">(</mo>
<mo>⋅<!-- ⋅ --></mo>
<mo stretchy="false">)</mo>
<mspace width="thickmathspace"></mspace>
<mo>=</mo>
<mspace width="thickmathspace"></mspace>
<mo fence="false" stretchy="false">{</mo>
<mo>⋅<!-- ⋅ --></mo>
<mo>,</mo>
<mspace width="thinmathspace"></mspace>
<mi>g</mi>
<mo fence="false" stretchy="false">}</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \operatorname {ad} _{g}(\cdot )\;=\;\{\cdot ,\,g\}}</annotation>
</semantics>
</math></span><img src="./177d78f12faea3fd6a0043f5dd0b0f06d3740357.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:15.83ex; height:3.009ex;" alt="{\displaystyle \operatorname {ad} _{g}(\cdot )\;=\;\{\cdot ,\,g\}}" loading="lazy"></span> and the bracket on the right-hand side is the commutator of operators, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [\operatorname {A} ,\,\operatorname {B} ]\;=\;\operatorname {A} \operatorname {B} -\operatorname {B} \operatorname {A} }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo stretchy="false">[</mo>
<mi mathvariant="normal">A</mi>
<mo>,</mo>
<mspace width="thinmathspace"></mspace>
<mi mathvariant="normal">B</mi>
<mo stretchy="false">]</mo>
<mspace width="thickmathspace"></mspace>
<mo>=</mo>
<mspace width="thickmathspace"></mspace>
<mi mathvariant="normal">A</mi>
<mo>⁡<!-- ⁡ --></mo>
<mi mathvariant="normal">B</mi>
<mo>−<!-- − --></mo>
<mi mathvariant="normal">B</mi>
<mo>⁡<!-- ⁡ --></mo>
<mi mathvariant="normal">A</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle [\operatorname {A} ,\,\operatorname {B} ]\;=\;\operatorname {A} \operatorname {B} -\operatorname {B} \operatorname {A} }</annotation>
</semantics>
</math></span><img src="./d7f97f3ece87400d2b60f4bff90b63e785f72198.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.626ex; height:2.843ex;" alt="{\displaystyle [\operatorname {A} ,\,\operatorname {B} ]\;=\;\operatorname {A} \operatorname {B} -\operatorname {B} \operatorname {A} }" loading="lazy"></span>. By <b><a href="#math_1">(1)</a></b>, the operator <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {ad} _{g}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>ad</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>g</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \operatorname {ad} _{g}}</annotation>
</semantics>
</math></span><img src="./3e71199d30b9419c35306d8247ba8c5a640cf031.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.476ex; height:2.843ex;" alt="{\displaystyle \operatorname {ad} _{g}}" loading="lazy"></span> is equal to the operator <span class="texhtml"><i>X<sub>g</sub></i></span>. The proof of the Jacobi identity follows from <b><a href="#math_3">(3)</a></b> because, up to the factor of -1, the Lie bracket of vector fields is just their commutator as differential operators.
</p><p>The <a href="Algebra_over_a_field" title="Algebra over a field">algebra</a> of smooth functions on M, together with the Poisson bracket forms a <a href="Poisson_algebra" title="Poisson algebra">Poisson algebra</a>, because it is a <a href="Lie_algebra" title="Lie algebra">Lie algebra</a> under the Poisson bracket, which additionally satisfies Leibniz's rule <b><a href="#math_2">(2)</a></b>. We have shown that every <a href="Symplectic_manifold" title="Symplectic manifold">symplectic manifold</a> is a <a href="Poisson_manifold" title="Poisson manifold">Poisson manifold</a>, that is a manifold with a "curly-bracket" operator on smooth functions such that the smooth functions form a Poisson algebra. However, not every Poisson manifold arises in this way, because Poisson manifolds allow for degeneracy which cannot arise in the symplectic case.
</p>
<div class="mw-heading mw-heading2"><h2 id="A_result_on_conjugate_momenta">A result on conjugate momenta</h2></div>
<p>Given a smooth <a href="Vector_field" title="Vector field">vector field</a> <span class="mwe-math-element mwe-math-element-inline"><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}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>X</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle X}</annotation>
</semantics>
</math></span><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"></span> on the configuration space, let <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{X}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>P</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>X</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle P_{X}}</annotation>
</semantics>
</math></span><img src="./8348dd8ce7e6f7f4778ee01fa5bdc7b828afd98c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.125ex; height:2.509ex;" alt="{\displaystyle P_{X}}" loading="lazy"></span> be its <a href="Conjugate_momentum" class="mw-redirect" title="Conjugate momentum">conjugate momentum</a>. The conjugate momentum mapping is a <a href="Lie_algebra" title="Lie algebra">Lie algebra</a> anti-homomorphism from the <a href="Lie_bracket_of_vector_fields" title="Lie bracket of vector fields">Lie bracket</a> to the Poisson bracket:
<span class="mwe-math-element mwe-math-element-block"><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 \{P_{X},P_{Y}\}=-P_{[X,Y]}.}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo fence="false" stretchy="false">{</mo>
<msub>
<mi>P</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>X</mi>
</mrow>
</msub>
<mo>,</mo>
<msub>
<mi>P</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>Y</mi>
</mrow>
</msub>
<mo fence="false" stretchy="false">}</mo>
<mo>=</mo>
<mo>−<!-- − --></mo>
<msub>
<mi>P</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">[</mo>
<mi>X</mi>
<mo>,</mo>
<mi>Y</mi>
<mo stretchy="false">]</mo>
</mrow>
</msub>
<mo>.</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \{P_{X},P_{Y}\}=-P_{[X,Y]}.}</annotation>
</semantics>
</math></span></span>
</p><p>This important result is worth a short proof. Write a vector field <span class="mwe-math-element mwe-math-element-inline"><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}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>X</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle X}</annotation>
</semantics>
</math></span><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"></span> at point <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>q</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle q}</annotation>
</semantics>
</math></span><img src="./06809d64fa7c817ffc7e323f85997f783dbdf71d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.07ex; height:2.009ex;" alt="{\displaystyle q}" loading="lazy"></span> in the <a href="Configuration_space_(physics)" title="Configuration space (physics)">configuration space</a> as
<span class="mwe-math-element mwe-math-element-block"><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 X_{q}=\sum _{i}X^{i}(q){\frac {\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>q</mi>
</mrow>
</msub>
<mo>=</mo>
<munder>
<mo>∑<!-- ∑ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</munder>
<msup>
<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msup>
<mo stretchy="false">(</mo>
<mi>q</mi>
<mo stretchy="false">)</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<mrow>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<msup>
<mi>q</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msup>
</mrow>
</mfrac>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle X_{q}=\sum _{i}X^{i}(q){\frac {\partial }{\partial q^{i}}}}</annotation>
</semantics>
</math></span></span>
where <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle {\frac {\partial }{\partial q^{i}}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="false" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<mrow>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<msup>
<mi>q</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msup>
</mrow>
</mfrac>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\textstyle {\frac {\partial }{\partial q^{i}}}}</annotation>
</semantics>
</math></span><img src="./7dcbf7bb6ad5d91bb777b67ad7f5d273b6bfc502.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:3.156ex; height:4.343ex;" alt="{\textstyle {\frac {\partial }{\partial q^{i}}}}" loading="lazy"></span> is the local coordinate frame. The conjugate momentum to <span class="mwe-math-element mwe-math-element-inline"><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}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>X</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle X}</annotation>
</semantics>
</math></span><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"></span> has the expression
<span class="mwe-math-element mwe-math-element-block"><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 P_{X}(q,p)=\sum _{i}X^{i}(q)\;p_{i}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>P</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>X</mi>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<mi>q</mi>
<mo>,</mo>
<mi>p</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<munder>
<mo>∑<!-- ∑ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</munder>
<msup>
<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msup>
<mo stretchy="false">(</mo>
<mi>q</mi>
<mo stretchy="false">)</mo>
<mspace width="thickmathspace"></mspace>
<msub>
<mi>p</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle P_{X}(q,p)=\sum _{i}X^{i}(q)\;p_{i}}</annotation>
</semantics>
</math></span></span>
where the <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p_{i}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>p</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle p_{i}}</annotation>
</semantics>
</math></span><img src="./5bab39399bf5424f25d957cdc57c84a0622626d2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:2.059ex; height:2.009ex;" alt="{\displaystyle p_{i}}" loading="lazy"></span> are the momentum functions conjugate to the coordinates. One then has, for a point <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (q,p)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo stretchy="false">(</mo>
<mi>q</mi>
<mo>,</mo>
<mi>p</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle (q,p)}</annotation>
</semantics>
</math></span><img src="./3e542f3143f252633d3d5633d5c9c9933e15dbfb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.082ex; height:2.843ex;" alt="{\displaystyle (q,p)}" loading="lazy"></span> in the <a href="Phase_space" title="Phase space">phase space</a>,
<span class="mwe-math-element mwe-math-element-block"><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}\{P_{X},P_{Y}\}(q,p)&amp;=\sum _{i}\sum _{j}\left\{X^{i}(q)\;p_{i},Y^{j}(q)\;p_{j}\right\}\\&amp;=\sum _{ij}p_{i}Y^{j}(q){\frac {\partial X^{i}}{\partial q^{j}}}-p_{j}X^{i}(q){\frac {\partial Y^{j}}{\partial q^{i}}}\\&amp;=-\sum _{i}p_{i}\;[X,Y]^{i}(q)\\&amp;=-P_{[X,Y]}(q,p).\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>
<mo fence="false" stretchy="false">{</mo>
<msub>
<mi>P</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>X</mi>
</mrow>
</msub>
<mo>,</mo>
<msub>
<mi>P</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>Y</mi>
</mrow>
</msub>
<mo fence="false" stretchy="false">}</mo>
<mo stretchy="false">(</mo>
<mi>q</mi>
<mo>,</mo>
<mi>p</mi>
<mo stretchy="false">)</mo>
</mtd>
<mtd>
<mi></mi>
<mo>=</mo>
<munder>
<mo>∑<!-- ∑ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</munder>
<munder>
<mo>∑<!-- ∑ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>j</mi>
</mrow>
</munder>
<mrow>
<mo>{</mo>
<mrow>
<msup>
<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msup>
<mo stretchy="false">(</mo>
<mi>q</mi>
<mo stretchy="false">)</mo>
<mspace width="thickmathspace"></mspace>
<msub>
<mi>p</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
<mo>,</mo>
<msup>
<mi>Y</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>j</mi>
</mrow>
</msup>
<mo stretchy="false">(</mo>
<mi>q</mi>
<mo stretchy="false">)</mo>
<mspace width="thickmathspace"></mspace>
<msub>
<mi>p</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>j</mi>
</mrow>
</msub>
</mrow>
<mo>}</mo>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd></mtd>
<mtd>
<mi></mi>
<mo>=</mo>
<munder>
<mo>∑<!-- ∑ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
<mi>j</mi>
</mrow>
</munder>
<msub>
<mi>p</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
<msup>
<mi>Y</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>j</mi>
</mrow>
</msup>
<mo stretchy="false">(</mo>
<mi>q</mi>
<mo stretchy="false">)</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<msup>
<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msup>
</mrow>
<mrow>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<msup>
<mi>q</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>j</mi>
</mrow>
</msup>
</mrow>
</mfrac>
</mrow>
<mo>−<!-- − --></mo>
<msub>
<mi>p</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>j</mi>
</mrow>
</msub>
<msup>
<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msup>
<mo stretchy="false">(</mo>
<mi>q</mi>
<mo stretchy="false">)</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<msup>
<mi>Y</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>j</mi>
</mrow>
</msup>
</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></mtd>
<mtd>
<mi></mi>
<mo>=</mo>
<mo>−<!-- − --></mo>
<munder>
<mo>∑<!-- ∑ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</munder>
<msub>
<mi>p</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
<mspace width="thickmathspace"></mspace>
<mo stretchy="false">[</mo>
<mi>X</mi>
<mo>,</mo>
<mi>Y</mi>
<msup>
<mo stretchy="false">]</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msup>
<mo stretchy="false">(</mo>
<mi>q</mi>
<mo stretchy="false">)</mo>
</mtd>
</mtr>
<mtr>
<mtd></mtd>
<mtd>
<mi></mi>
<mo>=</mo>
<mo>−<!-- − --></mo>
<msub>
<mi>P</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">[</mo>
<mi>X</mi>
<mo>,</mo>
<mi>Y</mi>
<mo stretchy="false">]</mo>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<mi>q</mi>
<mo>,</mo>
<mi>p</mi>
<mo stretchy="false">)</mo>
<mo>.</mo>
</mtd>
</mtr>
</mtable>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\{P_{X},P_{Y}\}(q,p)&amp;=\sum _{i}\sum _{j}\left\{X^{i}(q)\;p_{i},Y^{j}(q)\;p_{j}\right\}\\&amp;=\sum _{ij}p_{i}Y^{j}(q){\frac {\partial X^{i}}{\partial q^{j}}}-p_{j}X^{i}(q){\frac {\partial Y^{j}}{\partial q^{i}}}\\&amp;=-\sum _{i}p_{i}\;[X,Y]^{i}(q)\\&amp;=-P_{[X,Y]}(q,p).\end{aligned}}}</annotation>
</semantics>
</math></span></span>
</p><p>The above holds for all <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (q,p)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo stretchy="false">(</mo>
<mi>q</mi>
<mo>,</mo>
<mi>p</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle (q,p)}</annotation>
</semantics>
</math></span><img src="./3e542f3143f252633d3d5633d5c9c9933e15dbfb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.082ex; height:2.843ex;" alt="{\displaystyle (q,p)}" loading="lazy"></span>, giving the desired result.
</p>
<div class="mw-heading mw-heading2"><h2 id="Quantization">Quantization</h2></div>
<p>Poisson brackets <a href="Deformation_theory" class="mw-redirect" title="Deformation theory">deform</a> to <a href="Moyal_bracket" title="Moyal bracket">Moyal brackets</a> upon <a href="Weyl_quantization" class="mw-redirect" title="Weyl quantization">quantization</a>, that is, they generalize to a different Lie algebra, the <a href="Moyal_bracket" title="Moyal bracket">Moyal algebra</a>, or, equivalently in <a href="Hilbert_space" title="Hilbert space">Hilbert space</a>, quantum <a href="Commutator" title="Commutator">commutators</a>. The Wigner-İnönü <a href="Group_contraction" title="Group contraction">group contraction</a> of these (the classical limit, <span class="texhtml">ħ → 0</span>) yields the above Lie algebra.
</p><p>To state this more explicitly and precisely, the <a href="Universal_enveloping_algebra" title="Universal enveloping algebra">universal enveloping algebra</a> of the <a href="Heisenberg_algebra" class="mw-redirect" title="Heisenberg algebra">Heisenberg algebra</a> is the <a href="Weyl_algebra" title="Weyl algebra">Weyl algebra</a> (modulo the relation that the center be the unit). The Moyal product is then a special case of the star product on the algebra of symbols. An explicit definition of the algebra of symbols, and the star product is given in the article on the <a href="Universal_enveloping_algebra" title="Universal enveloping algebra">universal enveloping algebra</a>.
</p>
<div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2></div>
<style data-mw-deduplicate="TemplateStyles:r1184024115">
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.mw-parser-output .div-col{margin-top:0.3em;column-width:30em}.mw-parser-output .div-col-small{font-size:90%}.mw-parser-output .div-col-rules{column-rule:1px solid #aaa}.mw-parser-output .div-col dl,.mw-parser-output .div-col ol,.mw-parser-output .div-col ul{margin-top:0}.mw-parser-output .div-col li,.mw-parser-output .div-col dd{page-break-inside:avoid;break-inside:avoid-column}


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<ul><li><a href="Commutator" title="Commutator">Commutator</a></li>
<li><a href="Dirac_bracket" title="Dirac bracket">Dirac bracket</a></li>
<li><a href="Lagrange_bracket" title="Lagrange bracket">Lagrange bracket</a></li>
<li><a href="Moyal_bracket" title="Moyal bracket">Moyal bracket</a></li>
<li><a href="Peierls_bracket" title="Peierls bracket">Peierls bracket</a></li>
<li><a href="Phase_space" title="Phase space">Phase space</a></li>
<li><a href="Poisson_algebra" title="Poisson algebra">Poisson algebra</a></li>
<li><a href="Poisson_ring" title="Poisson ring">Poisson ring</a></li>
<li><a href="Poisson_superalgebra" title="Poisson superalgebra">Poisson superalgebra</a></li>
<li><a href="Poisson_superbracket" class="mw-redirect" title="Poisson superbracket">Poisson superbracket</a></li></ul>
</div>
<div class="mw-heading mw-heading2"><h2 id="Remarks">Remarks</h2></div>
<style data-mw-deduplicate="TemplateStyles:r1239543626">
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/* end https://en.wikipedia.org/ */
</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"><span class="mwe-math-element mwe-math-element-inline"><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(p_{i},\,q_{i},\,t)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>f</mi>
<mo stretchy="false">(</mo>
<msub>
<mi>p</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
<mo>,</mo>
<mspace width="thinmathspace"></mspace>
<msub>
<mi>q</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
<mo>,</mo>
<mspace width="thinmathspace"></mspace>
<mi>t</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f(p_{i},\,q_{i},\,t)}</annotation>
</semantics>
</math></span><img src="./3d09f5c1587f43bd85a789272cea23ba76841a08.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.575ex; height:2.843ex;" alt="{\displaystyle f(p_{i},\,q_{i},\,t)}" loading="lazy"></span> means <span class="mwe-math-element mwe-math-element-inline"><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}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>f</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f}</annotation>
</semantics>
</math></span><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"></span> is a function of the <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2N+1}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mn>2</mn>
<mi>N</mi>
<mo>+</mo>
<mn>1</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle 2N+1}</annotation>
</semantics>
</math></span><img src="./0d862395d02c6c80fb2ec2ebd027e13279c2c8ea.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:7.229ex; height:2.343ex;" alt="{\displaystyle 2N+1}" loading="lazy"></span> independent variables: momentum, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p_{1\dots N}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>p</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
<mo>…<!-- … --></mo>
<mi>N</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle p_{1\dots N}}</annotation>
</semantics>
</math></span><img src="./8cbbaf1342ab99c9bcd5fae626bb75c80b60b714.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:5.698ex; height:2.009ex;" alt="{\displaystyle p_{1\dots N}}" loading="lazy"></span>; position, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q_{1\dots N}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>q</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
<mo>…<!-- … --></mo>
<mi>N</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle q_{1\dots N}}</annotation>
</semantics>
</math></span><img src="./a34bd1b8252a24efc58273442392cf1b6e5f4784.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.476ex; height:2.009ex;" alt="{\displaystyle q_{1\dots N}}" loading="lazy"></span>; and time, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>t</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle t}</annotation>
</semantics>
</math></span><img src="./65658b7b223af9e1acc877d848888ecdb4466560.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}" loading="lazy"></span></span>
</li>
</ol></div></div>
<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>
<div class="reflist">
<div class="mw-references-wrap"><ol class="references">
<li id="cite_note-Poisson1809-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-Poisson1809_1-0">^</a></b></span> <span class="reference-text"><a href="#poisson1881">S. D. Poisson (1809)</a></span>
</li>
<li id="cite_note-Marle2009-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-Marle2009_2-0">^</a></b></span> <span class="reference-text"><a href="#marle2009">C. M. Marle (2009)</a></span>
</li>
<li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">
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</style><cite id="CITEREFGiacaglia1972" class="citation book cs1">Giacaglia, Giorgio E. O. (1972). <i>Perturbation methods in non-linear systems</i>. Applied mathematical sciences. New York Heidelberg: Springer. pp.&nbsp;<span class="nowrap">8–</span>9. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a>&nbsp;<bdi>978-3-540-90054-2</bdi>.</cite></span>
</li>
</ol></div></div>
<ul><li><cite id="CITEREFArnold1989" class="citation book cs1"><a href="Vladimir_Arnold" title="Vladimir Arnold">Arnold, Vladimir I.</a> (1989). <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> (2nd&nbsp;ed.). New York: Springer. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a>&nbsp;<bdi>978-0-387-96890-2</bdi>.</cite></li>
<li><cite id="CITEREFLandauLifshitz1982" class="citation book cs1"><a href="Lev_Landau" title="Lev Landau">Landau, Lev D.</a>; <a href="Evgeny_Lifshitz" title="Evgeny Lifshitz">Lifshitz, Evegeny M.</a> (1982). <i>Mechanics</i>. <a href="Course_of_Theoretical_Physics" title="Course of Theoretical Physics">Course of Theoretical Physics</a>. Vol.&nbsp;1 (3rd&nbsp;ed.). Butterworth-Heinemann. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a>&nbsp;<bdi>978-0-7506-2896-9</bdi>.</cite></li>
<li><cite id="CITEREFKarasëvMaslov1993" class="citation book cs1">Karasëv, Mikhail V.; <a href="Victor_Pavlovich_Maslov" class="mw-redirect" title="Victor Pavlovich Maslov">Maslov, Victor P.</a> (1993). <i>Nonlinear Poisson brackets, Geometry and Quantization</i>. Translations of Mathematical Monographs. Vol.&nbsp;119. Translated by Sossinsky, Alexey; Shishkova, M.A. Providence, RI: American Mathematical Society. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a>&nbsp;<bdi>978-0821887967</bdi>. <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a>&nbsp;<a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=1214142">1214142</a>.</cite></li>
<li><cite id="CITEREFMoretti2023" class="citation book cs1">Moretti, Valter (2023). <i>Analytical Mechanics, Classical, Lagrangian and Hamiltonian Mechanics, Stability Theory, Special Relativity</i>. UNITEXT. Vol.&nbsp;150. Springer. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a>&nbsp;<bdi>978-3-031-27612-5</bdi>.</cite></li>
<li><cite id="poisson1809" class="citation journal cs1">Poisson, Siméon-Denis (1809). <a rel="nofollow" class="external text" href="https://math.huji.ac.il/~piz/documents-others/SDP-1809.pdf">"Mémoire sur la variation des constantes arbitraires dans les questions de Mécanique"</a> <span class="cs1-format">(PDF)</span>. <i>Journal de l'École polytechnique, 15e cahier</i>. <b>8</b>: 266-344.</cite></li>
<li><cite id="marle2009" class="citation journal cs1"><a href="Charles-Michel_Marle" title="Charles-Michel Marle">Marle, Charles-Michel</a> (2009). "The Inception of Symplectic Geometry: the Works of Lagrange and Poisson During the Years 1808-1810". <i>Letters in Mathematical Physics</i>. <b>90</b> (<span class="nowrap">1–</span>3): 3-21. <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/0902.0685">0902.0685</a></span>. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2009LMaPh..90....3M">2009LMaPh..90....3M</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2Fs11005-009-0347-y">10.1007/s11005-009-0347-y</a>.</cite></li></ul>
<div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2></div>
<ul><li><cite class="citation cs2"><a rel="nofollow" class="external text" href="https://www.encyclopediaofmath.org/index.php?title=Poisson_brackets">"Poisson brackets"</a>, <i><a href="Encyclopedia_of_Mathematics" title="Encyclopedia of Mathematics">Encyclopedia of Mathematics</a></i>, <a href="European_Mathematical_Society" title="European Mathematical Society">EMS Press</a>, 2001 [1994]</cite></li>
<li><span class="citation mathworld" id="Reference-Mathworld-Poisson_bracket"><cite id="CITEREFWeisstein" class="citation web cs1"><a href="Eric_W._Weisstein" title="Eric W. Weisstein">Eric W. Weisstein</a>. <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/PoissonBracket.html">"Poisson bracket"</a>. <i><a href="MathWorld" title="MathWorld">MathWorld</a></i>.</cite></span></li></ul></div><!--htdig_noindex--><div><div class="zim-footer">
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