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<h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Poisson-Klammer</span></h1>
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<div id="mw-content-text" class="mw-body-content mw-content-ltr" lang="de" dir="ltr"><div class="mw-content-ltr mw-parser-output" lang="de" dir="ltr"><p>Die <b>Poisson-Klammer</b>, benannt nach <a href="Sim%C3%A9on_Denis_Poisson" title="Siméon Denis Poisson">Siméon Denis Poisson</a>, ist ein <a href="Bilineare_Abbildung" title="Bilineare Abbildung">bilinearer</a> <a href="Differentialoperator" title="Differentialoperator">Differentialoperator</a> in der kanonischen (<a href="Hamiltonsche_Mechanik" title="Hamiltonsche Mechanik">hamiltonschen</a>) <a href="Mechanik" title="Mechanik">Mechanik</a>. Sie ist ein Beispiel für eine <a href="Lie-Klammer" title="Lie-Klammer">Lie-Klammer</a>, also für eine Multiplikation in einer <a href="Lie-Algebra" title="Lie-Algebra">Lie-Algebra</a>.
</p>
<div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2></div>
<p>Die Poisson-Klammer ist definiert als
</p>
<dl><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 \left\{f,g\right\}:=\sum _{k=1}^{s}{\left({\frac {\partial f}{\partial q_{k}}}{\frac {\partial g}{\partial p_{k}}}-{\frac {\partial f}{\partial p_{k}}}{\frac {\partial g}{\partial q_{k}}}\right)}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow>
<mo>{</mo>
<mrow>
<mi>f</mi>
<mo>,</mo>
<mi>g</mi>
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<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>s</mi>
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</munderover>
<mrow class="MJX-TeXAtom-ORD">
<mrow>
<mo>(</mo>
<mrow>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<mi>f</mi>
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<mi mathvariant="normal">∂<!-- ∂ --></mi>
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<mi>q</mi>
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<mi>k</mi>
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</msub>
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</mfrac>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<mi>g</mi>
</mrow>
<mrow>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<msub>
<mi>p</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>k</mi>
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</msub>
</mrow>
</mfrac>
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<mo>−<!-- − --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<mi>f</mi>
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<mrow>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<msub>
<mi>p</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>k</mi>
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</msub>
</mrow>
</mfrac>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<mi>g</mi>
</mrow>
<mrow>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<msub>
<mi>q</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>k</mi>
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<mo>)</mo>
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</mrow>
</mstyle>
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<annotation encoding="application/x-tex">{\displaystyle \left\{f,g\right\}:=\sum _{k=1}^{s}{\left({\frac {\partial f}{\partial q_{k}}}{\frac {\partial g}{\partial p_{k}}}-{\frac {\partial f}{\partial p_{k}}}{\frac {\partial g}{\partial q_{k}}}\right)}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/155ee527a97d4e232009c92491b3d3a2bd7fe704.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:36.887ex; height:6.843ex;" alt="{\displaystyle \left\{f,g\right\}:=\sum _{k=1}^{s}{\left({\frac {\partial f}{\partial q_{k}}}{\frac {\partial g}{\partial p_{k}}}-{\frac {\partial f}{\partial p_{k}}}{\frac {\partial g}{\partial q_{k}}}\right)}}" loading="lazy"></span></dd></dl>
<p>mit
</p>
<ul><li><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="./_assets_/eb734a37dd21ce173a46342d1cc64c92/132e57acb643253e7810ee9702d9581f159a1c61.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> und <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="./_assets_/eb734a37dd21ce173a46342d1cc64c92/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> <a href="Funktion_(Mathematik)" title="Funktion (Mathematik)">Funktionen</a> der <a href="Generalisierte_Koordinate" title="Generalisierte Koordinate">generalisierten Koordinaten</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_{k}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>q</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>k</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle q_{k}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f27215e46abcad60f100434d2c8003310580af95.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.126ex; height:2.009ex;" alt="{\displaystyle q_{k}}" loading="lazy"></span> und der <a href="Kanonischer_Impuls" class="mw-redirect" title="Kanonischer Impuls">kanonisch konjugierten Impulse</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 p_{k}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>p</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>k</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle p_{k}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/01084a31964201514f3e6bd0136989e11ea6e58a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:2.348ex; height:2.009ex;" alt="{\displaystyle p_{k}}" loading="lazy"></span></li>
<li><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 s}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>s</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle s}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/01d131dfd7673938b947072a13a9744fe997e632.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:1.676ex;" alt="{\displaystyle s}" loading="lazy"></span> Anzahl der <a href="Freiheitsgrad" title="Freiheitsgrad">Freiheitsgrade</a>.</li></ul>
<p>Allgemein kann die Poisson-Klammer auch für Funktionen <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="./_assets_/eb734a37dd21ce173a46342d1cc64c92/545fd099af8541605f7ee55f08225526be88ce57.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}" loading="lazy"></span> und <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="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy"></span> definiert werden, die nicht von generalisierten Koordinaten und kanonischen Impulsen abhängen. Zur Verdeutlichung, auf welche Variablen sich die Poisson-Klammer beziehen soll, werden diese als <a href="Index_(Mathematik)" class="mw-redirect" title="Index (Mathematik)">Indizes</a> an die Klammer geschrieben:
</p>
<dl><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\}_{ab}:=\sum _{k=1}^{s}\left({\frac {\partial F}{\partial a_{k}}}{\frac {\partial G}{\partial b_{k}}}-{\frac {\partial F}{\partial b_{k}}}{\frac {\partial G}{\partial a_{k}}}\right)}">
<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>
<msub>
<mo fence="false" stretchy="false">}</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>a</mi>
<mi>b</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>s</mi>
</mrow>
</munderover>
<mrow>
<mo>(</mo>
<mrow>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<mi>F</mi>
</mrow>
<mrow>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<msub>
<mi>a</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>
<mi>G</mi>
</mrow>
<mrow>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<msub>
<mi>b</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>
<mi>F</mi>
</mrow>
<mrow>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<msub>
<mi>b</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>
<mi>G</mi>
</mrow>
<mrow>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<msub>
<mi>a</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>k</mi>
</mrow>
</msub>
</mrow>
</mfrac>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \{F,G\}_{ab}:=\sum _{k=1}^{s}\left({\frac {\partial F}{\partial a_{k}}}{\frac {\partial G}{\partial b_{k}}}-{\frac {\partial F}{\partial b_{k}}}{\frac {\partial G}{\partial a_{k}}}\right)}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a6c737b52e22df2656a4bf24d1df3795c8d6e320.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:39.909ex; height:6.843ex;" alt="{\displaystyle \{F,G\}_{ab}:=\sum _{k=1}^{s}\left({\frac {\partial F}{\partial a_{k}}}{\frac {\partial G}{\partial b_{k}}}-{\frac {\partial F}{\partial b_{k}}}{\frac {\partial G}{\partial a_{k}}}\right)}" loading="lazy"></span>.</dd></dl>
<p>Man sagt <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="./_assets_/eb734a37dd21ce173a46342d1cc64c92/545fd099af8541605f7ee55f08225526be88ce57.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}" loading="lazy"></span> und <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="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy"></span> <i>Poisson-kommutieren</i>, wenn die Poisson-Klammer verschwindet (<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="./_assets_/eb734a37dd21ce173a46342d1cc64c92/088faf34498919d4282bd41a081fad563cccedc8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.187ex; height:2.843ex;" alt="{\displaystyle \{F,G\}=0}" 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 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="./_assets_/eb734a37dd21ce173a46342d1cc64c92/545fd099af8541605f7ee55f08225526be88ce57.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}" loading="lazy"></span> und <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="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy"></span> stehen dann auch <i>in Involution</i>, weil die Größen, die durch diese Funktionen beschrieben werden, unabhängig voneinander sind und sich in ihrer Entwicklung nicht gegenseitig beeinflussen. Eine Funktion <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="./_assets_/eb734a37dd21ce173a46342d1cc64c92/545fd099af8541605f7ee55f08225526be88ce57.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}" loading="lazy"></span>, die mit der <a href="Hamilton-Funktion" title="Hamilton-Funktion">Hamilton-Funktion</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 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="./_assets_/eb734a37dd21ce173a46342d1cc64c92/75a9edddcca2f782014371f75dca39d7e13a9c1b.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> Poisson-kommutiert, ist eine <a href="Erhaltungsgr%C3%B6%C3%9Fe" class="mw-redirect" title="Erhaltungsgröße">Erhaltungsgröße</a>.
</p>
<div class="mw-heading mw-heading2"><h2 id="Eigenschaften">Eigenschaften</h2></div>
<ul><li><a href="Bilineare_Abbildung" title="Bilineare Abbildung">Bilinearität</a></li></ul>
<dl><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 \,\{c_{1}f_{1}+c_{2}f_{2},g\}=c_{1}\{f_{1},g\}+c_{2}\{f_{2},g\}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mspace width="thinmathspace"></mspace>
<mo fence="false" stretchy="false">{</mo>
<msub>
<mi>c</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<msub>
<mi>f</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<mo>+</mo>
<msub>
<mi>c</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msub>
<msub>
<mi>f</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msub>
<mo>,</mo>
<mi>g</mi>
<mo fence="false" stretchy="false">}</mo>
<mo>=</mo>
<msub>
<mi>c</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<mo fence="false" stretchy="false">{</mo>
<msub>
<mi>f</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<mo>,</mo>
<mi>g</mi>
<mo fence="false" stretchy="false">}</mo>
<mo>+</mo>
<msub>
<mi>c</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msub>
<mo fence="false" stretchy="false">{</mo>
<msub>
<mi>f</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msub>
<mo>,</mo>
<mi>g</mi>
<mo fence="false" stretchy="false">}</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \,\{c_{1}f_{1}+c_{2}f_{2},g\}=c_{1}\{f_{1},g\}+c_{2}\{f_{2},g\}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7341448f8ec99f582d00537e963e6e48ecc05edd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:39.609ex; height:2.843ex;" alt="{\displaystyle \,\{c_{1}f_{1}+c_{2}f_{2},g\}=c_{1}\{f_{1},g\}+c_{2}\{f_{2},g\}}" loading="lazy"></span></dd></dl>
<ul><li><a href="Antisymmetrische_Bilinearform" class="mw-redirect" title="Antisymmetrische Bilinearform">Antisymmetrie</a></li></ul>
<dl><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="./_assets_/eb734a37dd21ce173a46342d1cc64c92/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>, insbesondere <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,f\}=0}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo fence="false" stretchy="false">{</mo>
<mi>f</mi>
<mo>,</mo>
<mi>f</mi>
<mo fence="false" stretchy="false">}</mo>
<mo>=</mo>
<mn>0</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \{f,f\}=0}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9091706a9a5bfcae472e4565cb18c7f272f6982d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.177ex; height:2.843ex;" alt="{\displaystyle \{f,f\}=0}" loading="lazy"></span></dd></dl>
<ul><li><a href="Produktregel" title="Produktregel">Produktregel</a></li></ul>
<dl><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,gh\}=\{f,g\}h+g\{f,h\}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mspace width="thinmathspace"></mspace>
<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>
<mo fence="false" stretchy="false">{</mo>
<mi>f</mi>
<mo>,</mo>
<mi>g</mi>
<mo fence="false" stretchy="false">}</mo>
<mi>h</mi>
<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>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \,\{f,gh\}=\{f,g\}h+g\{f,h\}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a132051cfd88a2a9beeac0e129545225d8086a80.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.603ex; height:2.843ex;" alt="{\displaystyle \,\{f,gh\}=\{f,g\}h+g\{f,h\}}" loading="lazy"></span></dd></dl>
<ul><li><a href="Jacobi-Identit%C3%A4t" title="Jacobi-Identität">Jacobi-Identität</a></li></ul>
<dl><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\}\}+\{h,\{f,g\}\}+\{g,\{h,f\}\}=0}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mspace width="thinmathspace"></mspace>
<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>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>
<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>
<mn>0</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \,\{f,\{g,h\}\}+\{h,\{f,g\}\}+\{g,\{h,f\}\}=0}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e8461326b4a546bb82c9456ab0c23f00404bd019.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:41.683ex; height:2.843ex;" alt="{\displaystyle \,\{f,\{g,h\}\}+\{h,\{f,g\}\}+\{g,\{h,f\}\}=0}" loading="lazy"></span></dd></dl>
<ul><li>Invarianz unter <a href="Kanonische_Transformation" title="Kanonische Transformation">kanonischen Transformationen</a></li></ul>
<dl><dd>Physikalisch liegt es nahe, anzunehmen, dass die <a href="Zeitentwicklung" title="Zeitentwicklung">Zeitentwicklung</a> einer Eigenschaft eines Systems nicht von den verwendeten Koordinaten abhängen sollte; damit sollten auch die Poisson-Klammern unabhängig von den verwendeten kanonischen Koordinaten sein. Seien <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 (\mathbf {q} ,\mathbf {p} )}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<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>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle (\mathbf {q} ,\mathbf {p} )}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/917eec2ea801fbe8ff991069c91b113cbb297c43.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.744ex; height:2.843ex;" alt="{\displaystyle (\mathbf {q} ,\mathbf {p} )}" loading="lazy"></span> und <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 (\mathbf {Q} ,\mathbf {P} )}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<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>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle (\mathbf {Q} ,\mathbf {P} )}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9b4936bb741c87de424d8253f36249622c2ffd98.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.678ex; height:2.843ex;" alt="{\displaystyle (\mathbf {Q} ,\mathbf {P} )}" loading="lazy"></span> zwei verschiedene Sätze von Koordinaten, die durch kanonische Transformationen ineinander übergehen, so gilt:</dd></dl>
<dl><dd><dl><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\}_{\mathbf {qp} }=\{f,g\}_{\mathbf {QP} }=\{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>
<msub>
<mo fence="false" stretchy="false">}</mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">q</mi>
<mi mathvariant="bold">p</mi>
</mrow>
</mrow>
</msub>
<mo>=</mo>
<mo fence="false" stretchy="false">{</mo>
<mi>f</mi>
<mo>,</mo>
<mi>g</mi>
<msub>
<mo fence="false" stretchy="false">}</mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">Q</mi>
<mi mathvariant="bold">P</mi>
</mrow>
</mrow>
</msub>
<mo>=</mo>
<mo fence="false" stretchy="false">{</mo>
<mi>f</mi>
<mo>,</mo>
<mi>g</mi>
<mo fence="false" stretchy="false">}</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \{f,g\}_{\mathbf {qp} }=\{f,g\}_{\mathbf {QP} }=\{f,g\}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a095a40af59cee4ecbb12fe69dcc9d0385a6e45d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:28.684ex; height:3.009ex;" alt="{\displaystyle \{f,g\}_{\mathbf {qp} }=\{f,g\}_{\mathbf {QP} }=\{f,g\}}" loading="lazy"></span>.</dd></dl></dd></dl>
<div class="mw-heading mw-heading3"><h3 id="Fundamentale_Poisson-Klammern">Fundamentale Poisson-Klammern</h3></div>
<p>Für die kanonische Mechanik wichtig sind die fundamentalen Poisson-Klammern
</p>
<dl><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 \left\{q_{k},q_{l}\right\}=0}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow>
<mo>{</mo>
<mrow>
<msub>
<mi>q</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>k</mi>
</mrow>
</msub>
<mo>,</mo>
<msub>
<mi>q</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>l</mi>
</mrow>
</msub>
</mrow>
<mo>}</mo>
</mrow>
<mo>=</mo>
<mn>0</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \left\{q_{k},q_{l}\right\}=0}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8be711a4dab0399adf234e8a19b506bff840cedd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.505ex; height:2.843ex;" alt="{\displaystyle \left\{q_{k},q_{l}\right\}=0}" loading="lazy"></span></dd>
<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 \left\{p_{k},p_{l}\right\}=0}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow>
<mo>{</mo>
<mrow>
<msub>
<mi>p</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>k</mi>
</mrow>
</msub>
<mo>,</mo>
<msub>
<mi>p</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>l</mi>
</mrow>
</msub>
</mrow>
<mo>}</mo>
</mrow>
<mo>=</mo>
<mn>0</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \left\{p_{k},p_{l}\right\}=0}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d7b06d270792dcd44283735685b16ca15841527b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.77ex; height:2.843ex;" alt="{\displaystyle \left\{p_{k},p_{l}\right\}=0}" loading="lazy"></span></dd>
<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 \left\{q_{k},p_{l}\right\}=\delta _{kl}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow>
<mo>{</mo>
<mrow>
<msub>
<mi>q</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>k</mi>
</mrow>
</msub>
<mo>,</mo>
<msub>
<mi>p</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>l</mi>
</mrow>
</msub>
</mrow>
<mo>}</mo>
</mrow>
<mo>=</mo>
<msub>
<mi>δ<!-- δ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>k</mi>
<mi>l</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \left\{q_{k},p_{l}\right\}=\delta _{kl}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fd2e6f0a509d21df43b88f5ea923a57a5a42c4e5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.086ex; height:2.843ex;" alt="{\displaystyle \left\{q_{k},p_{l}\right\}=\delta _{kl}}" loading="lazy"></span> (<a href="Kronecker-Delta" title="Kronecker-Delta">Kronecker-Delta</a>)</dd></dl>
<p>Sie folgen aus den trivialen Beziehungen
</p>
<dl><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 {\begin{alignedat}{2}&{\frac {\partial q_{k}}{\partial q_{l}}}=\delta _{kl}\quad &&{\frac {\partial p_{k}}{\partial q_{l}}}=0\\&{\frac {\partial q_{k}}{\partial p_{l}}}=0\quad &&{\frac {\partial p_{k}}{\partial p_{l}}}=\delta _{kl}\end{alignedat}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mtable columnalign="right left right left" rowspacing="3pt" columnspacing="0em 0em 0em 0em" displaystyle="true">
<mtr>
<mtd></mtd>
<mtd>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<msub>
<mi>q</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>k</mi>
</mrow>
</msub>
</mrow>
<mrow>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<msub>
<mi>q</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>l</mi>
</mrow>
</msub>
</mrow>
</mfrac>
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<mo>=</mo>
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<annotation encoding="application/x-tex">{\displaystyle {\begin{alignedat}{2}&{\frac {\partial q_{k}}{\partial q_{l}}}=\delta _{kl}\quad &&{\frac {\partial p_{k}}{\partial q_{l}}}=0\\&{\frac {\partial q_{k}}{\partial p_{l}}}=0\quad &&{\frac {\partial p_{k}}{\partial p_{l}}}=\delta _{kl}\end{alignedat}}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ebaeffe1788be9d5c152c4b8b8d851395aac68c8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.505ex; width:23.186ex; height:12.176ex;" alt="{\displaystyle {\begin{alignedat}{2}&{\frac {\partial q_{k}}{\partial q_{l}}}=\delta _{kl}\quad &&{\frac {\partial p_{k}}{\partial q_{l}}}=0\\&{\frac {\partial q_{k}}{\partial p_{l}}}=0\quad &&{\frac {\partial p_{k}}{\partial p_{l}}}=\delta _{kl}\end{alignedat}}}" loading="lazy"></span></dd></dl>
<div class="mw-heading mw-heading2"><h2 id="Anwendung">Anwendung</h2></div>
<ul><li>Mithilfe der Poisson-Klammer kann die <a href="Evolution_(Mathematik)" title="Evolution (Mathematik)">Zeitevolution</a> einer beliebigen <a href="Observable" title="Observable">Observablen</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(q_{k},p_{k},t)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>f</mi>
<mo stretchy="false">(</mo>
<msub>
<mi>q</mi>
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<mi>k</mi>
</mrow>
</msub>
<mo>,</mo>
<msub>
<mi>p</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>k</mi>
</mrow>
</msub>
<mo>,</mo>
<mi>t</mi>
<mo stretchy="false">)</mo>
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<annotation encoding="application/x-tex">{\displaystyle f(q_{k},p_{k},t)}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/76f1bb0879537a28bea13d7e2afcf89d99ccf875.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.379ex; height:2.843ex;" alt="{\displaystyle f(q_{k},p_{k},t)}" loading="lazy"></span> eines <a href="Hamiltonsches_System" class="mw-redirect" title="Hamiltonsches System">Hamiltonschen Systems</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 H(q_{k},p_{k})}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>H</mi>
<mo stretchy="false">(</mo>
<msub>
<mi>q</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>k</mi>
</mrow>
</msub>
<mo>,</mo>
<msub>
<mi>p</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>k</mi>
</mrow>
</msub>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle H(q_{k},p_{k})}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/32d028e88cab77ef30259a6b60907e92daf57490.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.291ex; height:2.843ex;" alt="{\displaystyle H(q_{k},p_{k})}" loading="lazy"></span> ausgedrückt werden (<a href="Hamiltonsche_Bewegungsgleichungen" class="mw-redirect" title="Hamiltonsche Bewegungsgleichungen">Hamiltonsche Bewegungsgleichungen</a>):</li></ul>
<dl><dd><dl><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 {\frac {\mathrm {d} f}{\mathrm {d} t}}=\{f,H\}+{\frac {\partial f}{\partial t}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
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<mi mathvariant="normal">d</mi>
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<mi>f</mi>
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<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="normal">d</mi>
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<mi>t</mi>
</mrow>
</mfrac>
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<mo>=</mo>
<mo fence="false" stretchy="false">{</mo>
<mi>f</mi>
<mo>,</mo>
<mi>H</mi>
<mo fence="false" stretchy="false">}</mo>
<mo>+</mo>
<mrow class="MJX-TeXAtom-ORD">
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<mi mathvariant="normal">∂<!-- ∂ --></mi>
<mi>f</mi>
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<mi>t</mi>
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</mfrac>
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</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\frac {\mathrm {d} f}{\mathrm {d} t}}=\{f,H\}+{\frac {\partial f}{\partial t}}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d6aae7373e9c833417472d3fc4ab9b31da107bcb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:19.48ex; height:5.676ex;" alt="{\displaystyle {\frac {\mathrm {d} f}{\mathrm {d} t}}=\{f,H\}+{\frac {\partial f}{\partial t}}}" loading="lazy"></span>.</dd></dl></dd></dl>
<ul><li>Dual zur <a href="Bewegungsgleichung" title="Bewegungsgleichung">Bewegungsgleichung</a> der <a href="Observable" title="Observable">Observablen</a> ist die <a href="Liouville-Gleichung" title="Liouville-Gleichung">Liouville-Gleichung</a>, die die Dynamik der <a href="Verteilungsdichte" class="mw-redirect" title="Verteilungsdichte">Verteilungsdichte</a> in der <a href="Statistische_Mechanik" title="Statistische Mechanik">statistischen Mechanik</a> beschreibt:</li></ul>
<dl><dd><dl><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 {\dot {\rho }}=\{H,\rho \}.}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>ρ<!-- ρ --></mi>
<mo>˙<!-- ˙ --></mo>
</mover>
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</mrow>
<mo>=</mo>
<mo fence="false" stretchy="false">{</mo>
<mi>H</mi>
<mo>,</mo>
<mi>ρ<!-- ρ --></mi>
<mo fence="false" stretchy="false">}</mo>
<mo>.</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\dot {\rho }}=\{H,\rho \}.}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3bf328cac5c0663db43ad2fe4f0203b0219675b6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.745ex; height:2.843ex;" alt="{\displaystyle {\dot {\rho }}=\{H,\rho \}.}" loading="lazy"></span></dd></dl></dd></dl>
<ul><li>In der <a href="Quantenmechanik" title="Quantenmechanik">Quantenmechanik</a> wird im Rahmen der kanonischen <a href="Quantisierung_(Physik)" title="Quantisierung (Physik)">Quantisierung</a> die Poisson-Klammer ersetzt durch <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 \textstyle \left(-{\frac {\rm {i}}{\hbar }}\right)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mstyle displaystyle="false" scriptlevel="0">
<mrow>
<mo>(</mo>
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<mo>−<!-- − --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="normal">i</mi>
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<mi class="MJX-variant">ℏ<!-- ℏ --></mi>
</mfrac>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
</mstyle>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \textstyle \left(-{\frac {\rm {i}}{\hbar }}\right)}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7884b5b53d4a9fa3774ce8f669660d242f01a44a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:6.344ex; height:4.843ex;" alt="{\displaystyle \textstyle \left(-{\frac {\rm {i}}{\hbar }}\right)}" loading="lazy"></span> multipliziert mit dem <a href="Kommutator_(Mathematik)" title="Kommutator (Mathematik)">Kommutator</a>:<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup></li></ul>
<dl><dd><dl><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 \{H,f\}\rightarrow -{\frac {i}{\hbar }}[{\hat {H}},{\hat {f}}]}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo fence="false" stretchy="false">{</mo>
<mi>H</mi>
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<mo fence="false" stretchy="false">}</mo>
<mo stretchy="false">→<!-- → --></mo>
<mo>−<!-- − --></mo>
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<annotation encoding="application/x-tex">{\displaystyle \{H,f\}\rightarrow -{\frac {i}{\hbar }}[{\hat {H}},{\hat {f}}]}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8f8e4fdbbf82a122028f2cbf2359b43fcd3f9c86.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:20.356ex; height:5.343ex;" alt="{\displaystyle \{H,f\}\rightarrow -{\frac {i}{\hbar }}[{\hat {H}},{\hat {f}}]}" loading="lazy"></span></dd></dl></dd></dl>
<dl><dd>Außerdem werden Observablen durch <a href="Operator_(Mathematik)" title="Operator (Mathematik)">Operatoren</a> dargestellt. Die oben angeführte Gleichung der Zeitevolution einer Observablen führt so auf die Zeitevolution von Operatoren eines quantenmechanischen Systems mit <a href="Hamiltonoperator" title="Hamiltonoperator">Hamiltonoperator</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 {\hat {H}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>H</mi>
<mo stretchy="false">^<!-- ^ --></mo>
</mover>
</mrow>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\hat {H}}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6bb06de5217295d7fbdbf68fb9c5309a513fc99e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.843ex;" alt="{\displaystyle {\hat {H}}}" loading="lazy"></span> im <a href="Heisenberg-Bild" title="Heisenberg-Bild">Heisenberg-Bild</a>. Diese Bewegungsgleichung heißt <a href="Heisenbergsche_Bewegungsgleichung" class="mw-redirect" title="Heisenbergsche Bewegungsgleichung">Heisenbergsche Bewegungsgleichung</a>. Die Liouville-Gleichung findet ihre Entsprechung dabei in der <a href="Dichtematrix#Zeitentwicklung" class="mw-redirect" title="Dichtematrix">Von-Neumann’schen Bewegungsgleichung</a>.</dd></dl>
<ul><li>Sowohl die Phasenraumfunktionen der kanonischen Mechanik als auch die <a href="Operator_(Mathematik)" title="Operator (Mathematik)">Operatoren</a> der Quantenmechanik bilden mit ihren Klammern jeweils eine <a href="Lie-Algebra" title="Lie-Algebra">Lie-Algebra</a>.</li></ul>
<ul><li>Allgemein definiert man auf einer <a href="Symplektische_Mannigfaltigkeit" title="Symplektische Mannigfaltigkeit">symplektischen Mannigfaltigkeit</a> mit symplektischer Form, die in lokalen Koordinaten gegeben ist durch <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 \textstyle \omega =\sum _{ij}\omega _{ij}\,\mathrm {d} x^{i}\wedge \mathrm {d} x^{j}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mstyle displaystyle="false" scriptlevel="0">
<mi>ω<!-- ω --></mi>
<mo>=</mo>
<munder>
<mo>∑<!-- ∑ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
<mi>j</mi>
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<msub>
<mi>ω<!-- ω --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
<mi>j</mi>
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</msub>
<mspace width="thinmathspace"></mspace>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="normal">d</mi>
</mrow>
<msup>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msup>
<mo>∧<!-- ∧ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="normal">d</mi>
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<msup>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>j</mi>
</mrow>
</msup>
</mstyle>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \textstyle \omega =\sum _{ij}\omega _{ij}\,\mathrm {d} x^{i}\wedge \mathrm {d} x^{j}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d05d20e4b77c45d2c025f93aeb049305f69ab77f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:21.709ex; height:3.509ex;" alt="{\displaystyle \textstyle \omega =\sum _{ij}\omega _{ij}\,\mathrm {d} x^{i}\wedge \mathrm {d} x^{j}}" loading="lazy"></span>, die Poisson-Klammer der Funktionen <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="./_assets_/eb734a37dd21ce173a46342d1cc64c92/132e57acb643253e7810ee9702d9581f159a1c61.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> und <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>
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<annotation encoding="application/x-tex">{\displaystyle g}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/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> durch:</li></ul>
<dl><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\}=\sum _{ij}\omega ^{ij}\,\partial _{i}f\,\partial _{j}g\,.}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo fence="false" stretchy="false">{</mo>
<mi>f</mi>
<mo>,</mo>
<mi>g</mi>
<mo fence="false" stretchy="false">}</mo>
<mo>=</mo>
<munder>
<mo>∑<!-- ∑ --></mo>
<mrow class="MJX-TeXAtom-ORD">
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<msup>
<mi>ω<!-- ω --></mi>
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<mspace width="thinmathspace"></mspace>
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<mi mathvariant="normal">∂<!-- ∂ --></mi>
<mrow class="MJX-TeXAtom-ORD">
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<mspace width="thinmathspace"></mspace>
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<mi mathvariant="normal">∂<!-- ∂ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>j</mi>
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<mi>g</mi>
<mspace width="thinmathspace"></mspace>
<mo>.</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \{f,g\}=\sum _{ij}\omega ^{ij}\,\partial _{i}f\,\partial _{j}g\,.}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/80d17b10fe0ade88e94db7bb681c2ca9c769bc19.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:23.898ex; height:5.843ex;" alt="{\displaystyle \{f,g\}=\sum _{ij}\omega ^{ij}\,\partial _{i}f\,\partial _{j}g\,.}" loading="lazy"></span></dd></dl>
<ul><li>Koordinatenunabhängig lässt sich die Poisson-Klammer wie folgt darstellen: es sei <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:T^{*}M\rightarrow TM}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>J</mi>
<mo>:</mo>
<msup>
<mi>T</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>∗<!-- ∗ --></mo>
</mrow>
</msup>
<mi>M</mi>
<mo stretchy="false">→<!-- → --></mo>
<mi>T</mi>
<mi>M</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle J:T^{*}M\rightarrow TM}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1bd241020d2d915130f87ac10f25fcc3c26cc06f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:16.318ex; height:2.343ex;" alt="{\displaystyle J:T^{*}M\rightarrow TM}" loading="lazy"></span> der durch <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^{-1}(v)(w)=\omega (v,w)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>J</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>−<!-- − --></mo>
<mn>1</mn>
</mrow>
</msup>
<mo stretchy="false">(</mo>
<mi>v</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">(</mo>
<mi>w</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi>ω<!-- ω --></mi>
<mo stretchy="false">(</mo>
<mi>v</mi>
<mo>,</mo>
<mi>w</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle J^{-1}(v)(w)=\omega (v,w)}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/cfe778383ca63f6d473100333620fe04b08cf9fe.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.448ex; height:3.176ex;" alt="{\displaystyle J^{-1}(v)(w)=\omega (v,w)}" loading="lazy"></span> beschriebene <a href="Isomorphismus" title="Isomorphismus">Isomorphismus</a>. Weiter sei für eine Funktion <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>
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<annotation encoding="application/x-tex">{\displaystyle f}</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/132e57acb643253e7810ee9702d9581f159a1c61.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> das Vektorfeld <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_{f}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>f</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle X_{f}}</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bf089f5145a366dc825adfcc157c0a20820a3a0b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.061ex; height:2.843ex;" alt="{\displaystyle X_{f}}" loading="lazy"></span> definiert als <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(\mathrm {d} f)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>J</mi>
<mo stretchy="false">(</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="normal">d</mi>
</mrow>
<mi>f</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle J(\mathrm {d} f)}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/036fc30302a955dfc0e384e2d6116c13e45ce5f5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.852ex; height:2.843ex;" alt="{\displaystyle J(\mathrm {d} f)}" loading="lazy"></span>. Damit gilt dann</li></ul>
<dl><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\}=\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>
<mi>g</mi>
<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>
</mrow>
</msub>
<mo>,</mo>
<msub>
<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>g</mi>
</mrow>
</msub>
<mo stretchy="false">)</mo>
<mo>.</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \{f,g\}=\omega (X_{f},X_{g}).}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c64141b31b0b9fcc0d4e50d8273086a5c54f5531.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:19.794ex; height:3.009ex;" alt="{\displaystyle \{f,g\}=\omega (X_{f},X_{g}).}" loading="lazy"></span></dd></dl>
<div class="mw-heading mw-heading2"><h2 id="Weblinks">Weblinks</h2></div>
<ul><li><a href="Eric_Weisstein" title="Eric Weisstein">Eric W. Weisstein</a>: <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/PoissonBracket.html"><i>Poisson Bracket</i>.</a> In: <i><a href="MathWorld" title="MathWorld">MathWorld</a></i> (englisch).</li></ul>
<div class="mw-heading mw-heading2"><h2 id="Einzelnachweise">Einzelnachweise</h2></div>
<ol class="references">
<li id="cite_note-1"><span class="mw-cite-backlink"><a href="#cite_ref-1">↑</a></span> <span class="reference-text">Hong-Tao Zhang: <i>A Simple Method of Calculating Commutators in Hamilton System with Mathematica Software</i>, <a href="ArXiv" title="ArXiv">arxiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/quant-ph/0204081">quant-ph/0204081</a></span>
</li>
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<div class="klappleiste-kopf"><a href="Symplektische_Topologie" class="mw-redirect" title="Symplektische Topologie">Symplektische Topologie</a><div class="erweiterte-navigationsleiste-quicklinks" style="float:left; font-weight:normal; font-size:75%; margin-left:1em; margin-right:2em; display:none;"><span title="Vorlage anzeigen">V</span> </div></div>
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<td class="erw-nav-gruppe" style="white-space: nowrap;text-align: right;border: 1px solid transparent;border-top: 1px solid #FFF;border-bottom: 2px solid #FFF;padding: 0 1em;"><b>Mannigfaltigkeiten</b>
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<p><a href="Symplektische_Mannigfaltigkeit" title="Symplektische Mannigfaltigkeit">Symplektische Mannigfaltigkeit</a> | <a href="Lagrangesche_Untermannigfaltigkeit" title="Lagrangesche Untermannigfaltigkeit">Lagrangesche Untermannigfaltigkeit</a> | <a class="mw-selflink selflink">Poisson-Klammer</a>
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<p><a href="Symplektische_Abbildung" title="Symplektische Abbildung">Symplektische Abbildung</a> | <a href="Symplektomorphismus" title="Symplektomorphismus">Symplektomorphismus</a> | <a href="Hamiltonscher_Symplektomorphismus" title="Hamiltonscher Symplektomorphismus">Hamiltonscher Symplektomorphismus</a> | <a href="Impulsabbildung" title="Impulsabbildung">Impulsabbildung</a>
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<p><a href="Symplektisches_Vektorfeld" title="Symplektisches Vektorfeld">Symplektisches Vektorfeld</a> | <a href="Hamiltonsches_Vektorfeld" title="Hamiltonsches Vektorfeld">Hamiltonsches Vektorfeld</a>
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