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Micron Document

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<title>Hamilton-Funktion</title>

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<h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Hamilton-Funktion</span></h1>

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<p>Die <b>Hamilton-Funktion</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 {\mathcal {H}}({\vec {q}}_{1},{\vec {q}}_{2},\ldots ,{\vec {p}}_{1},{\vec {p}}_{2},\ldots ,t)}">

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {H}}({\vec {q}}_{1},{\vec {q}}_{2},\ldots ,{\vec {p}}_{1},{\vec {p}}_{2},\ldots ,t)}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3784df546320fc36f6c9454830d957784d02b800.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.521ex; height:2.843ex;" alt="{\displaystyle {\mathcal {H}}({\vec {q}}_{1},{\vec {q}}_{2},\ldots ,{\vec {p}}_{1},{\vec {p}}_{2},\ldots ,t)}" loading="lazy"></span> eines Systems von <a href="Teilchen" title="Teilchen">Teilchen</a>, ist deren Gesamt<a href="Energie" title="Energie">energie</a>, als Funktion der <a href="Generalisierte_Koordinate" title="Generalisierte Koordinate">verallgemeinerten Orte</a> und <a href="Generalisierte_Koordinate" title="Generalisierte Koordinate">Impulse</a> dieser Teilchen und ggf. der Zeit, sofern „skleronome“, d. h. nicht zeitabhängige <a href="Zwangsbedingung" title="Zwangsbedingung">Zwangsbedingungen</a> vorliegen. Sie ist nach <a href="William_Rowan_Hamilton" title="William Rowan Hamilton">William Rowan Hamilton</a> benannt und wird (aus dem Englischen übernommen) auch als <b>Hamiltonian</b> bezeichnet. Sie ist eine <a href="Legendre-Transformation" title="Legendre-Transformation">Legendre-Transformierte</a> der <a href="Lagrange-Funktion" class="mw-redirect" title="Lagrange-Funktion">Lagrange-Funktion</a> des Systems.

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<div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2></div>

<p>Die Hamilton-Funktion ist definiert durch

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {H}}(\mathbf {q} ,\mathbf {p} ,t):=\left\{\sum _{i=1}^{n}{\dot {q}}_{i}p_{i}\right\}-{\mathcal {L}}(\mathbf {q} ,{\dot {\mathbf {q} }},t),{\text{ mit }}{\dot {\mathbf {q} }}={\dot {\mathbf {q} }}(\mathbf {q} ,\mathbf {p} ,t)}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b0fd6492c4eeff5388d66ac55c92b4cdd2ea611d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:56.183ex; height:7.509ex;" alt="{\displaystyle {\mathcal {H}}(\mathbf {q} ,\mathbf {p} ,t):=\left\{\sum _{i=1}^{n}{\dot {q}}_{i}p_{i}\right\}-{\mathcal {L}}(\mathbf {q} ,{\dot {\mathbf {q} }},t),{\text{ mit }}{\dot {\mathbf {q} }}={\dot {\mathbf {q} }}(\mathbf {q} ,\mathbf {p} ,t)}" loading="lazy"></span></dd></dl>

<p>und hängt ab von

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<annotation encoding="application/x-tex">{\displaystyle t}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/65658b7b223af9e1acc877d848888ecdb4466560.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>,</li>

<li>den <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 \mathbf {q} =(q_{1},q_{2},\dotsc ,q_{n})}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {q} =(q_{1},q_{2},\dotsc ,q_{n})}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b4deffe2450cc56b9a9b72b1f372d18e172b39ca.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.974ex; height:2.843ex;" alt="{\displaystyle \mathbf {q} =(q_{1},q_{2},\dotsc ,q_{n})}" loading="lazy"></span> und</li>

<li>den <a href="Generalisierter_Impuls" title="Generalisierter Impuls">generalisierten Impulsen</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 \mathbf {p} =(p_{1},p_{2},\dotsc ,p_{n})}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {p} =(p_{1},p_{2},\dotsc ,p_{n})}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/464db915e9b330558ad9eec0fae1ee0b3ecc3aba.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.44ex; height:2.843ex;" alt="{\displaystyle \mathbf {p} =(p_{1},p_{2},\dotsc ,p_{n})}" loading="lazy"></span>.</li></ul>

<p>Sie geht hervor aus einer <a href="Legendre-Transformation" title="Legendre-Transformation">Legendre-Transformation</a> der <a href="Lagrange-Funktion" class="mw-redirect" title="Lagrange-Funktion">Lagrange-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 {\mathcal {L}}(t,\mathbf {q} ,{\dot {\mathbf {q} }})}">

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {L}}(t,\mathbf {q} ,{\dot {\mathbf {q} }})}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8395694df348460ac0c09ae5752ea0e8a08e1ba4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.152ex; height:2.843ex;" alt="{\displaystyle {\mathcal {L}}(t,\mathbf {q} ,{\dot {\mathbf {q} }})}" loading="lazy"></span> bezüglich der generalisierten Geschwindigkeiten, die von den generalisierten Koordinaten und ihren Geschwindigkeiten <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 {\mathbf {q} }}=({\dot {q}}_{1},{\dot {q}}_{2},\dotsc ,{\dot {q}}_{n})}">

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<annotation encoding="application/x-tex">{\displaystyle {\dot {\mathbf {q} }}=({\dot {q}}_{1},{\dot {q}}_{2},\dotsc ,{\dot {q}}_{n})}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {H}}(t,\mathbf {q} ,\mathbf {p} )=\left\{\sum _{i=1}^{n}{\dot {q}}_{i}\,p_{i}\right\}-{\mathcal {L}}(t,\mathbf {q} ,{\dot {\mathbf {q} }})}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/293c14bec7dbd59bdaf3d38a17a7d2af3b37f5ee.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:36.694ex; height:7.509ex;" alt="{\displaystyle {\mathcal {H}}(t,\mathbf {q} ,\mathbf {p} )=\left\{\sum _{i=1}^{n}{\dot {q}}_{i}\,p_{i}\right\}-{\mathcal {L}}(t,\mathbf {q} ,{\dot {\mathbf {q} }})}" loading="lazy"></span></dd></dl>

<p>Dabei sind auf der rechten Seite mit den Geschwindigkeiten <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 {\mathbf {q} }}}">

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<annotation encoding="application/x-tex">{\displaystyle {\dot {\mathbf {q} }}}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle {\dot {\mathbf {q} }}(t,\mathbf {q} ,\mathbf {p} )}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5dba090355990b915dfaa267b1332118fffa14cd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.033ex; height:2.843ex;" alt="{\displaystyle {\dot {\mathbf {q} }}(t,\mathbf {q} ,\mathbf {p} )}" loading="lazy"></span></dd></dl>

<p>gemeint, die man erhält, wenn man die Definition der generalisierten Impulse

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<annotation encoding="application/x-tex">{\displaystyle p_{i}(\mathbf {q} ,\mathbf {p} ,t):={\frac {\partial {\mathcal {L}}}{\partial {\dot {q}}_{i}}}(\mathbf {q} ,{\dot {\mathbf {q} }},t),\quad i=1,\dots ,n}</annotation>

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<p>nach den Geschwindigkeiten auflöst.

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<div class="mw-heading mw-heading2"><h2 id="Eigenschaften">Eigenschaften</h2></div>

<div class="mw-heading mw-heading3"><h3 id="Ableitung">Ableitung</h3></div>

<p>Das <a href="Totales_Differential" title="Totales Differential">totale Differential</a> der Hamilton-Funktion lautet:

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<annotation encoding="application/x-tex">{\displaystyle \mathrm {d} {\mathcal {H}}=\sum _{i=1}^{n}{\frac {\partial {\mathcal {H}}}{\partial q_{i}}}\mathrm {d} q_{i}+\sum _{i=1}^{n}{\frac {\partial {\mathcal {H}}}{\partial p_{i}}}\mathrm {d} p_{i}+{\frac {\partial {\mathcal {H}}}{\partial t}}\mathrm {d} t}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6e7708a087a373a3e61adccd7bbe1508122b857b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:40.402ex; height:6.843ex;" alt="{\displaystyle \mathrm {d} {\mathcal {H}}=\sum _{i=1}^{n}{\frac {\partial {\mathcal {H}}}{\partial q_{i}}}\mathrm {d} q_{i}+\sum _{i=1}^{n}{\frac {\partial {\mathcal {H}}}{\partial p_{i}}}\mathrm {d} p_{i}+{\frac {\partial {\mathcal {H}}}{\partial t}}\mathrm {d} t}" loading="lazy"></span></dd></dl>

<p>Aufgrund der <a href="Produktregel" title="Produktregel">Produktregel</a> erhält man

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<annotation encoding="application/x-tex">{\displaystyle \mathrm {d} {\mathcal {H}}=\sum _{i=1}^{n}\left(p_{i}\mathrm {d} {\dot {q}}_{i}+{\dot {q}}_{i}\mathrm {d} p_{i}-{\frac {\partial {\mathcal {L}}}{\partial q_{i}}}\mathrm {d} q_{i}-{\frac {\partial {\mathcal {L}}}{\partial {\dot {q}}_{i}}}\mathrm {d} {\dot {q}}_{i}\right)-{\frac {\partial {\mathcal {L}}}{\partial t}}\mathrm {d} t,}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4670472684153ab71d1aa12a058a669777a8115e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:57.214ex; height:6.843ex;" alt="{\displaystyle \mathrm {d} {\mathcal {H}}=\sum _{i=1}^{n}\left(p_{i}\mathrm {d} {\dot {q}}_{i}+{\dot {q}}_{i}\mathrm {d} p_{i}-{\frac {\partial {\mathcal {L}}}{\partial q_{i}}}\mathrm {d} q_{i}-{\frac {\partial {\mathcal {L}}}{\partial {\dot {q}}_{i}}}\mathrm {d} {\dot {q}}_{i}\right)-{\frac {\partial {\mathcal {L}}}{\partial t}}\mathrm {d} t,}" loading="lazy"></span></dd></dl>

<p>wobei wegen der Definition des verallgemeinerten Impulses <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 {\partial {\mathcal {L}}}{\partial {\dot {q}}_{i}}}=p_{i}}">

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<annotation encoding="application/x-tex">{\displaystyle {\frac {\partial {\mathcal {L}}}{\partial {\dot {q}}_{i}}}=p_{i}}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8bdd58f147f8024cd29800832f5774c7505129e1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:9.399ex; height:6.009ex;" alt="{\displaystyle {\frac {\partial {\mathcal {L}}}{\partial {\dot {q}}_{i}}}=p_{i}}" loading="lazy"></span> die ersten und letzten Terme in den Klammern die Summe 0 haben, sodass gilt:

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<annotation encoding="application/x-tex">{\displaystyle \mathrm {d} {\mathcal {H}}=\sum _{i=1}^{n}\left({\dot {q}}_{i}\mathrm {d} p_{i}-{\frac {\partial {\mathcal {L}}}{\partial q_{i}}}\mathrm {d} q_{i}\right)-{\frac {\partial {\mathcal {L}}}{\partial t}}\mathrm {d} t}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d609e6c58be540bce3591d42e2cdb6202c27ac7f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:37.647ex; height:6.843ex;" alt="{\displaystyle \mathrm {d} {\mathcal {H}}=\sum _{i=1}^{n}\left({\dot {q}}_{i}\mathrm {d} p_{i}-{\frac {\partial {\mathcal {L}}}{\partial q_{i}}}\mathrm {d} q_{i}\right)-{\frac {\partial {\mathcal {L}}}{\partial t}}\mathrm {d} t}" loading="lazy"></span></dd></dl>

<p>Mit der obigen Schreibweise des totalen Differentials folgen hieraus die <a href="Partielle_Ableitung" title="Partielle Ableitung">partiellen Ableitungen</a> der Hamilton-Funktion:

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<annotation encoding="application/x-tex">{\displaystyle {\frac {\partial {\mathcal {H}}}{\partial p_{i}}}={\dot {q}}_{i}}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/11912cc12f32c0b83bce14ee52e9d5c186cfbac5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:9.399ex; height:5.843ex;" alt="{\displaystyle {\frac {\partial {\mathcal {H}}}{\partial p_{i}}}={\dot {q}}_{i}}" loading="lazy"></span></dd>

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<annotation encoding="application/x-tex">{\displaystyle {\frac {\partial {\mathcal {H}}}{\partial q_{i}}}=-{\frac {\partial {\mathcal {L}}}{\partial q_{i}}}=-{\dot {p}}_{i}}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/45e2e4514a8faeb15cbcc5cfae72d3f3244400ad.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:20.081ex; height:5.843ex;" alt="{\displaystyle {\frac {\partial {\mathcal {H}}}{\partial q_{i}}}=-{\frac {\partial {\mathcal {L}}}{\partial q_{i}}}=-{\dot {p}}_{i}}" loading="lazy"></span></dd>

<mfrac>

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<annotation encoding="application/x-tex">{\displaystyle {\frac {\partial {\mathcal {H}}}{\partial t}}=-{\frac {\partial {\mathcal {L}}}{\partial t}}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bfe03eb46210b459be26b476580cab5d111898a0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:12.782ex; height:5.509ex;" alt="{\displaystyle {\frac {\partial {\mathcal {H}}}{\partial t}}=-{\frac {\partial {\mathcal {L}}}{\partial t}}}" loading="lazy"></span></dd></dl>

<div class="mw-heading mw-heading3"><h3 id="Erhaltungsgröße"><span id="Erhaltungsgr.C3.B6.C3.9Fe"></span>Erhaltungsgröße</h3></div>

<p>Die totale Ableitung der Hamilton-Funktion nach der Zeit ist identisch mit der partiellen:

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\frac {\mathrm {d} {\mathcal {H}}}{\mathrm {d} t}}&=\sum _{i=1}^{f}\left({\frac {\partial {\mathcal {H}}}{\partial p_{i}}}{\dot {p}}_{i}+{\frac {\partial {\mathcal {H}}}{\partial q_{i}}}{\dot {q}}_{i}\right)+{\frac {\partial {\mathcal {H}}}{\partial t}}\\&=\sum _{i=1}^{f}\left({\dot {q}}_{i}{\dot {p}}_{i}-{\dot {p}}_{i}{\dot {q}}_{i}\right)+{\frac {\partial {\mathcal {H}}}{\partial t}}\\&={\frac {\partial {\mathcal {H}}}{\partial t}}\end{aligned}}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e578f46e5c8a06b98ea00b508d524fe4dc317d24.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -9.838ex; width:37.481ex; height:20.843ex;" alt="{\displaystyle {\begin{aligned}{\frac {\mathrm {d} {\mathcal {H}}}{\mathrm {d} t}}&=\sum _{i=1}^{f}\left({\frac {\partial {\mathcal {H}}}{\partial p_{i}}}{\dot {p}}_{i}+{\frac {\partial {\mathcal {H}}}{\partial q_{i}}}{\dot {q}}_{i}\right)+{\frac {\partial {\mathcal {H}}}{\partial t}}\\&=\sum _{i=1}^{f}\left({\dot {q}}_{i}{\dot {p}}_{i}-{\dot {p}}_{i}{\dot {q}}_{i}\right)+{\frac {\partial {\mathcal {H}}}{\partial t}}\\&={\frac {\partial {\mathcal {H}}}{\partial t}}\end{aligned}}}" loading="lazy"></span></dd></dl>

<p>Wenn die Hamilton-Funktion also nicht explizit von der Zeit <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}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle t}</annotation>

</semantics>

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<mfrac>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {H}}\neq {\mathcal {H}}(t)\Rightarrow {\frac {\mathrm {d} {\mathcal {H}}}{\mathrm {d} t}}={\frac {\partial {\mathcal {H}}}{\partial t}}=0\Rightarrow {\mathcal {H}}=konst.}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9fa386b8bf264136b823b2f6dcb41de12cd06e5e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:43.845ex; height:5.509ex;" alt="{\displaystyle {\mathcal {H}}\neq {\mathcal {H}}(t)\Rightarrow {\frac {\mathrm {d} {\mathcal {H}}}{\mathrm {d} t}}={\frac {\partial {\mathcal {H}}}{\partial t}}=0\Rightarrow {\mathcal {H}}=konst.}" loading="lazy"></span></dd></dl>

<div class="mw-heading mw-heading3"><h3 id="Implikationen">Implikationen</h3></div>

<p>Die Hamilton-Funktion bestimmt die <a href="Zeitentwicklung" title="Zeitentwicklung">zeitliche Entwicklung</a> der Teilchenorte und -impulse durch die <a href="Hamiltonsche_Bewegungsgleichung" class="mw-redirect" title="Hamiltonsche Bewegungsgleichung">Hamiltonschen Bewegungsgleichungen</a>:

</p>

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<annotation encoding="application/x-tex">{\displaystyle {\dot {q}}_{k}={\frac {\partial {\mathcal {H}}}{\partial p_{k}}}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fb23c229c31acc91963ee58b81b914179abe1bb2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:9.977ex; height:5.843ex;" alt="{\displaystyle {\dot {q}}_{k}={\frac {\partial {\mathcal {H}}}{\partial p_{k}}}}" loading="lazy"></span></dd>

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<annotation encoding="application/x-tex">{\displaystyle {\dot {p}}_{k}=-{\frac {\partial {\mathcal {H}}}{\partial q_{k}}}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/48620c2e15615cc7dcbb5cbddd1ace3b4f98648f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; margin-left: -0.089ex; width:11.724ex; height:5.843ex;" alt="{\displaystyle {\dot {p}}_{k}=-{\frac {\partial {\mathcal {H}}}{\partial q_{k}}}}" loading="lazy"></span></dd></dl>

<p>Ebenso bestimmt der <a href="Hamiltonoperator" title="Hamiltonoperator">Hamiltonoperator</a> die Zeitentwicklung in der <a href="Quantenmechanik" title="Quantenmechanik">Quantenmechanik</a>. Man erhält ihn in vielen Fällen aus der Hamiltonfunktion durch <a href="Erste_Quantisierung" class="mw-redirect" title="Erste Quantisierung">kanonische Quantisierung</a>, indem man den algebraischen Ausdruck für <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}}(t,\mathbf {q} ,\mathbf {p} )}">

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {H}}(t,\mathbf {q} ,\mathbf {p} )}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/13d6189220822aa9f2d9e0fa483f2edc54f42f2e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.581ex; height:2.843ex;" alt="{\displaystyle {\mathcal {H}}(t,\mathbf {q} ,\mathbf {p} )}" loading="lazy"></span> als Funktion von <a href="Operator_(Mathematik)" title="Operator (Mathematik)">Operatoren</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 \mathbf {q} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {q} }</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7be005a326b7ac3fe4c24bca391369f44c4c2876.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.416ex; height:2.009ex;" alt="{\displaystyle \mathbf {q} }" 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 {p} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {p} }</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/dd73e3862cb92b016721b8c492eadb4e8a577527.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.485ex; height:2.009ex;" alt="{\displaystyle \mathbf {p} }" loading="lazy"></span> liest, die den <a href="Kanonische_Vertauschungsrelationen" class="mw-redirect" title="Kanonische Vertauschungsrelationen">kanonischen Vertauschungsrelationen</a> genügen.

</p>

<div class="mw-heading mw-heading2"><h2 id="Beispiele">Beispiele</h2></div>

<div class="mw-heading mw-heading3"><h3 id="Massenpunkt">Massenpunkt</h3></div>

<p>Bei einem Teilchen der Masse <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}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle m}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0a07d98bb302f3856cbabc47b2b9016692e3f7bc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}" loading="lazy"></span>, das sich nichtrelativistisch in einem Potential <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}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle V}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/af0f6064540e84211d0ffe4dac72098adfa52845.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}" loading="lazy"></span> bewegt, setzt sich die Hamilton-Funktion aus kinetischer und potentieller Energie zusammen:

</p>

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {H}}(t,{\vec {q}},{\vec {p}})={\frac {{\vec {p}}^{2}}{2\,m}}+V({\vec {q}})}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0eb1e9c1e6ce6875c56ebd7fd62e2bb0def92748.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:24.584ex; height:6.009ex;" alt="{\displaystyle {\mathcal {H}}(t,{\vec {q}},{\vec {p}})={\frac {{\vec {p}}^{2}}{2\,m}}+V({\vec {q}})}" loading="lazy"></span></dd></dl>

<p>Für ein relativistisches, <a href="Freies_Teilchen" title="Freies Teilchen">freies Teilchen</a> mit der <a href="Energie-Impuls-Relation" class="mw-redirect" title="Energie-Impuls-Relation">Energie-Impuls-Beziehung</a>

</p>

<msup>

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<annotation encoding="application/x-tex">{\displaystyle E^{2}-{\vec {p}}^{2}\,c^{2}=m^{2}\,c^{4}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1b52761554808eb9ae2bf678f8943bf9ff58b1ba.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:19.157ex; height:3.176ex;" alt="{\displaystyle E^{2}-{\vec {p}}^{2}\,c^{2}=m^{2}\,c^{4}}" loading="lazy"></span></dd></dl>

<p>gilt für die Hamilton-Funktion<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>

</p>

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {H}}(t,{\vec {q}},{\vec {p}})={\sqrt {m^{2}\,c^{4}+{\vec {p}}^{2}\,c^{2}}}.}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/81641eec319e43d03cf3965471179be83465f6d5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.505ex; width:28.594ex; height:4.843ex;" alt="{\displaystyle {\mathcal {H}}(t,{\vec {q}},{\vec {p}})={\sqrt {m^{2}\,c^{4}+{\vec {p}}^{2}\,c^{2}}}.}" loading="lazy"></span></dd></dl>

<p>Beim freien relativistischen Teilchen mit der Lagrangefunktion<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup>

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {L}}=-m\,c^{2}{\sqrt {1-{\dot {\vec {q}}}^{2}/c^{2}}}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6005d1d3a99d4fe09b833d97727069f741acab38.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:22.936ex; height:4.843ex;" alt="{\displaystyle {\mathcal {L}}=-m\,c^{2}{\sqrt {1-{\dot {\vec {q}}}^{2}/c^{2}}}}" loading="lazy"></span></dd></dl>

<p>hängt der generalisierte Impuls <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 {\vec {p}}={\frac {\partial {\mathcal {L}}}{\partial {\dot {\vec {q}}}}}}">

<mover>

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<annotation encoding="application/x-tex">{\displaystyle {\vec {p}}={\frac {\partial {\mathcal {L}}}{\partial {\dot {\vec {q}}}}}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fd2f23a716e4e14a73552d380d26b647ba84b838.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; margin-left: -0.089ex; width:8.271ex; height:6.509ex;" alt="{\displaystyle {\vec {p}}={\frac {\partial {\mathcal {L}}}{\partial {\dot {\vec {q}}}}}}" loading="lazy"></span> gemäß

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<annotation encoding="application/x-tex">{\displaystyle {\vec {p}}={\frac {m{\dot {\vec {q}}}}{\sqrt {1-{\dot {\vec {q}}}^{2}/c^{2}}}}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/43e0779badf398e8187a9dab4298cea952772d04.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.671ex; margin-left: -0.089ex; width:17.286ex; height:8.843ex;" alt="{\displaystyle {\vec {p}}={\frac {m{\dot {\vec {q}}}}{\sqrt {1-{\dot {\vec {q}}}^{2}/c^{2}}}}}" loading="lazy"></span></dd></dl>

<p>von der Geschwindigkeit <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 {\vec {q}}}}">

<mover>

<mover>

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</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\dot {\vec {q}}}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ae0db77c8f7c6467b328955f7b2607d9529d7ba5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.332ex; height:3.176ex;" alt="{\displaystyle {\dot {\vec {q}}}}" loading="lazy"></span> ab. Umgekehrt ist die Geschwindigkeit daher die Funktion

</p>

<mover>

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<annotation encoding="application/x-tex">{\displaystyle {\dot {\vec {q}}}={\frac {{\vec {p}}\,c^{2}}{\sqrt {m^{2}\,c^{4}+{\vec {p}}^{2}\,c^{2}}}}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9b520a6a9bdefada90687ccb31cf721db4d4cf2d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.671ex; width:20.801ex; height:8.509ex;" alt="{\displaystyle {\dot {\vec {q}}}={\frac {{\vec {p}}\,c^{2}}{\sqrt {m^{2}\,c^{4}+{\vec {p}}^{2}\,c^{2}}}}}" loading="lazy"></span></dd></dl>

<p>des Impulses.

</p>

<div class="mw-heading mw-heading3"><h3 id="Harmonischer_Oszillator">Harmonischer Oszillator</h3></div>

<p>Die Hamilton-Funktion eines eindimensionalen <a href="Harmonischer_Oszillator" title="Harmonischer Oszillator">harmonischen Oszillators</a> ist gegeben durch<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup>:

</p>

</mrow>

<mover>

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<mover>

</mover>

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<msup>

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</msup>

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<msup>

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</msup>

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {H}}(x,p)={\dot {x}}p-{\mathcal {L}}(x,{\dot {x}})={\frac {p^{2}}{2m}}+{\frac {m}{2}}\omega _{0}^{2}x^{2}=T+V=E}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a152f4c8929fb66bae91be14c4e95d6a33df944f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:54.825ex; height:5.676ex;" alt="{\displaystyle {\mathcal {H}}(x,p)={\dot {x}}p-{\mathcal {L}}(x,{\dot {x}})={\frac {p^{2}}{2m}}+{\frac {m}{2}}\omega _{0}^{2}x^{2}=T+V=E}" loading="lazy"></span></dd></dl>

<div class="mw-heading mw-heading3"><h3 id="Geladenes_Teilchen_im_elektromagnetischen_Feld">Geladenes Teilchen im elektromagnetischen Feld</h3></div>

<p>In kartesischen Koordinaten (<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 {\vec {q}}={\vec {x}}}">

<mover>

</mover>

</mrow>

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\vec {q}}={\vec {x}}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3f8538c9168c845e7a95af6aa7d368f7ccff0df2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.737ex; height:2.676ex;" alt="{\displaystyle {\vec {q}}={\vec {x}}}" loading="lazy"></span>) lautet die <a href="Lagrange-Funktion" class="mw-redirect" title="Lagrange-Funktion">Lagrange-Funktion</a> eines Teilchens der Ladung <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}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle q}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/06809d64fa7c817ffc7e323f85997f783dbdf71d.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>, das sich durch ein elektromagnetisches Feld bewegt<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup>,

</p>

</mrow>

<mfrac>

</mfrac>

</mrow>

<msup>

<mover>

<mover>

</mover>

</mrow>

</mover>

</mrow>

</mrow>

</msup>

<mrow>

<mrow>

<mover>

<mover>

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</mover>

</mrow>

<mover>

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</mrow>

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</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {L}}={\frac {1}{2}}m{\dot {\vec {x}}}^{2}+q\left({\dot {\vec {x}}}\cdot {\vec {A}}\right)-q\phi }</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/34098e4340f22a77ff41cc74c85661c37b71973d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:28.245ex; height:5.176ex;" alt="{\displaystyle {\mathcal {L}}={\frac {1}{2}}m{\dot {\vec {x}}}^{2}+q\left({\dot {\vec {x}}}\cdot {\vec {A}}\right)-q\phi }" loading="lazy"></span></dd></dl>

<p>Dabei ist <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 }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \phi }</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/72b1f30316670aee6270a28334bdf4f5072cdde4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.385ex; height:2.509ex;" alt="{\displaystyle \phi }" loading="lazy"></span> das <a href="Elektrisches_Potential" title="Elektrisches Potential">elektrische Potential</a> 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 {\vec {A}}}">

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\vec {A}}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/391292ffadc65b0cde3e96f23afcdb811619dd95.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:3.009ex;" alt="{\displaystyle {\vec {A}}}" loading="lazy"></span> das <a href="Magnetisches_Vektorpotential" title="Magnetisches Vektorpotential">Vektorpotential</a> des magnetischen Feldes. Der <a href="Generalisierter_Impuls" title="Generalisierter Impuls">kanonische Impuls</a> ist

</p>

<mover>

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<mfrac>

<mrow>

</mrow>

<mrow>

<mover>

<mover>

</mover>

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</mover>

</mrow>

</mfrac>

</mrow>

<mover>

<mover>

</mover>

</mrow>

</mover>

</mrow>

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\vec {p}}={\frac {\partial {\mathcal {L}}}{\partial {\dot {\vec {x}}}}}=m{\dot {\vec {x}}}+q{\vec {A}}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c6e80cdd9d58dc01474362f7424b7c2d24086216.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; margin-left: -0.089ex; width:20.393ex; height:6.176ex;" alt="{\displaystyle {\vec {p}}={\frac {\partial {\mathcal {L}}}{\partial {\dot {\vec {x}}}}}=m{\dot {\vec {x}}}+q{\vec {A}}}" loading="lazy"></span></dd></dl>

<p>Diese Gleichung kann so umgestellt werden, dass die Geschwindigkeit durch den Impuls ausgedrückt wird:

</p>

<mover>

<mover>

</mover>

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</mover>

</mrow>

<mfrac>

</mfrac>

</mrow>

<mrow>

<mrow>

<mover>

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<mover>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\dot {\vec {x}}}={\frac {1}{m}}\left({\vec {p}}-q{\vec {A}}\right)}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b1b11fbdef33c61fcd579f52b19c6375419d8949.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:17.445ex; height:5.176ex;" alt="{\displaystyle {\dot {\vec {x}}}={\frac {1}{m}}\left({\vec {p}}-q{\vec {A}}\right)}" loading="lazy"></span></dd></dl>

<p>Wird der Ausdruck für <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 {\vec {x}}}}">

<mover>

<mover>

</mover>

</mrow>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\dot {\vec {x}}}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2ed483ca94407c179bfb9a9e7b2818a14fc830dc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:2.843ex;" alt="{\displaystyle {\dot {\vec {x}}}}" 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 {\vec {p}}}">

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\vec {p}}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/84fee53c81592db54e0fe6c6f9eba002bb1dc74b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.415ex; height:2.676ex;" alt="{\displaystyle {\vec {p}}}" loading="lazy"></span> in die Definition der Hamilton-Funktion eingesetzt, ergibt sich diese zu:

</p>

<mtr>

<mtd>

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</mtd>

<mtd>

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<mover>

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<mtr>

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<mfrac>

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</mrow>

<msup>

<mrow>

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</msup>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\mathcal {H}}&={\dot {\vec {x}}}\cdot {\vec {p}}-{\mathcal {L}}={\frac {\vec {p}}{m}}\cdot \left({\vec {p}}-q{\vec {A}}\right)-{\frac {m}{2}}{\frac {1}{m^{2}}}\left({\vec {p}}-q{\vec {A}}\right)^{2}-{\frac {q}{m}}\left({\vec {p}}-q{\vec {A}}\right)\cdot {\vec {A}}+q\phi \\&={\frac {1}{m}}\left({\vec {p}}-q{\vec {A}}\right)^{2}-{\frac {1}{2m}}\left({\vec {p}}-q{\vec {A}}\right)^{2}+q\phi ={\frac {1}{2m}}\left({\vec {p}}-q{\vec {A}}\right)^{2}+q\phi \end{aligned}}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/171a4663bd2fafb2c16585aaa04fc291db1fa297.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.005ex; width:77.03ex; height:11.176ex;" alt="{\displaystyle {\begin{aligned}{\mathcal {H}}&={\dot {\vec {x}}}\cdot {\vec {p}}-{\mathcal {L}}={\frac {\vec {p}}{m}}\cdot \left({\vec {p}}-q{\vec {A}}\right)-{\frac {m}{2}}{\frac {1}{m^{2}}}\left({\vec {p}}-q{\vec {A}}\right)^{2}-{\frac {q}{m}}\left({\vec {p}}-q{\vec {A}}\right)\cdot {\vec {A}}+q\phi \\&={\frac {1}{m}}\left({\vec {p}}-q{\vec {A}}\right)^{2}-{\frac {1}{2m}}\left({\vec {p}}-q{\vec {A}}\right)^{2}+q\phi ={\frac {1}{2m}}\left({\vec {p}}-q{\vec {A}}\right)^{2}+q\phi \end{aligned}}}" loading="lazy"></span></dd></dl>

<div class="mw-heading mw-heading2"><h2 id="Literatur">Literatur</h2></div>

<ul><li>Herbert Goldstein, Charles P. Poole, Jr., John L. Safko: <cite style="font-style:italic">Klassische Mechanik</cite>. 3. Auflage. Wiley-VCH, Weinheim 2006, ISBN 3-527-40589-5.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Hamilton-Funktion&rft.au=Herbert+Goldstein%2C+Charles+P.+Poole%2C+Jr.%2C+...&rft.btitle=Klassische+Mechanik&rft.date=2006&rft.edition=3&rft.genre=book&rft.isbn=3527405895&rft.place=Weinheim&rft.pub=Wiley-VCH" style="display:none"> </span></li>

<li>Wolfgang Nolting: <cite style="font-style:italic">Grundkurs Theoretische Physik 2. Analytische Mechanik</cite>. 7. Auflage. Springer, Heidelberg 2006, ISBN 3-540-30660-9.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Hamilton-Funktion&rft.au=Wolfgang+Nolting&rft.btitle=Grundkurs+Theoretische+Physik+2.+Analytische+Mechanik&rft.date=2006&rft.edition=7&rft.genre=book&rft.isbn=3540306609&rft.place=Heidelberg&rft.pub=Springer" style="display:none"> </span></li></ul>

<div class="mw-heading mw-heading2"><h2 id="Einzelnachweise">Einzelnachweise</h2></div>

<li id="cite_note-1"><span class="mw-cite-backlink"><a href="#cite_ref-1">↑</a></span> <span class="reference-text">L. D. Landau, E. M. Lifschitz: <cite style="font-style:italic">Lehrbuch der theoretischen Physik, Band 2, Klassische Feldtheorie -</cite>. 8. Auflage. Akademie Verlag, Berlin 1981, <span style="white-space:nowrap">S.<span style="display:inline-block;width:.2em"> </span>32</span>.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Hamilton-Funktion&rft.au=L.+D.+Landau%2C+E.+M.+Lifschitz&rft.btitle=Lehrbuch+der+theoretischen+Physik%2C+Band+2%2C+Klassische+Feldtheorie+-&rft.date=1981&rft.edition=8.&rft.genre=book&rft.pages=32&rft.place=Berlin&rft.pub=Akademie+Verlag" style="display:none"> </span></span>

</li>

<li id="cite_note-2"><span class="mw-cite-backlink"><a href="#cite_ref-2">↑</a></span> <span class="reference-text">L. D. Landau, E. M. Lifschitz: <cite style="font-style:italic">Lehrbuch der theoretischen Physik, Band 2, Klassische Feldtheorie -</cite>. 8. Auflage. Akademie Verlag, Berlin 1981, <span style="white-space:nowrap">S.<span style="display:inline-block;width:.2em"> </span>30</span>.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Hamilton-Funktion&rft.au=L.+D.+Landau%2C+E.+M.+Lifschitz&rft.btitle=Lehrbuch+der+theoretischen+Physik%2C+Band+2%2C+Klassische+Feldtheorie+-&rft.date=1981&rft.edition=8.&rft.genre=book&rft.pages=30&rft.place=Berlin&rft.pub=Akademie+Verlag" style="display:none"> </span></span>

</li>

<li id="cite_note-3"><span class="mw-cite-backlink"><a href="#cite_ref-3">↑</a></span> <span class="reference-text">Torsten Fließbach: <cite style="font-style:italic">Mechanik - Lehrbuch zur Theoretischen Physik I</cite>. 6. Auflage. Spektrum Akademischer Verlag, Berlin 2009, ISBN 978-3-8274-2148-7, <span style="white-space:nowrap">S.<span style="display:inline-block;width:.2em"> </span>247</span>.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Hamilton-Funktion&rft.au=Torsten+Flie%C3%9Fbach&rft.btitle=Mechanik+-+Lehrbuch+zur+Theoretischen+Physik+I&rft.date=2009&rft.edition=6.&rft.genre=book&rft.isbn=9783827421487&rft.pages=247&rft.place=Berlin&rft.pub=Spektrum+Akademischer+Verlag" style="display:none"> </span></span>

</li>

<li id="cite_note-4"><span class="mw-cite-backlink"><a href="#cite_ref-4">↑</a></span> <span class="reference-text">Torsten Fließbach: <cite style="font-style:italic">Mechanik - Lehrbuch zur Theoretischen Physik I</cite>. 6. Auflage. Spektrum Akademischer Verlag, Berlin 2009, ISBN 978-3-8274-2148-7, <span style="white-space:nowrap">S.<span style="display:inline-block;width:.2em"> </span>73</span>.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Hamilton-Funktion&rft.au=Torsten+Flie%C3%9Fbach&rft.btitle=Mechanik+-+Lehrbuch+zur+Theoretischen+Physik+I&rft.date=2009&rft.edition=6.&rft.genre=book&rft.isbn=9783827421487&rft.pages=73&rft.place=Berlin&rft.pub=Spektrum+Akademischer+Verlag" style="display:none"> </span></span>

</li>

</ol></div><div><div class="zim-footer">

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