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<title>Einteilchenproblem</title>

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<h1 id="firstHeading" class="firstHeading mw-first-heading">Einteilchenproblem</h1>

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Das Einteilchenproblem behandelt im einfachsten Fall die <a href="Physik" title="Physik">physikalische</a> <a href="Fundamentale_Wechselwirkung" title="Fundamentale Wechselwirkung">Wechselwirkung</a> eines <a href="Teilchen" title="Teilchen">Teilchens</a> mit einem <a href="Kraftfeld_(Physik)" class="mw-redirect" title="Kraftfeld (Physik)">Kraftfeld</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {F}}}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ef40edff397a115ecdce7d3518001dfcc7f37d9e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.771ex; height:2.843ex;" alt="{\displaystyle {\vec {F}}}" loading="lazy">. Die konservativen Kräfte hängen nur vom <a href="Geometrischer_Ort" title="Geometrischer Ort">Ort</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {r}}}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6aec3c9ce13b53e9e24c98e7cce4212627884c91.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.223ex; height:2.343ex;" alt="{\displaystyle {\vec {r}}}" loading="lazy"> ab und haben ein skalares <a href="Potential_(Physik)" title="Potential (Physik)">Potential</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V({\vec {r}})}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/896062530de98a0b0831a83edabfddd2fa49d356.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.82ex; height:2.843ex;" alt="{\displaystyle V({\vec {r}})}" loading="lazy">, so dass <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle {\vec {F}}=-{\frac {\partial V({\vec {r}})}{\partial {\vec {r}}}}}">

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<annotation encoding="application/x-tex">{\displaystyle \textstyle {\vec {F}}=-{\frac {\partial V({\vec {r}})}{\partial {\vec {r}}}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7bdc21a7ffa132d9d502149e73050d04ef3d3df0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:11.891ex; height:5.009ex;" alt="{\displaystyle \textstyle {\vec {F}}=-{\frac {\partial V({\vec {r}})}{\partial {\vec {r}}}}}" loading="lazy"> gilt<a href="#cite_note-1">[1]</a>. Dabei wird angenommen, dass das Feld unabhängig vom Teilchen existiert und nicht durch die Bewegung des Teilchens beeinflusst wird. In einer <a href="Dimension_(Mathematik)" title="Dimension (Mathematik)">Dimension</a> kann das Einteilchenproblem mit dem <a href="Energiesatz" class="mw-redirect" title="Energiesatz">Energiesatz</a> durch eine einfache <a href="Integralrechnung" title="Integralrechnung">Integration</a> durch Trennung der Veränderlichen und anschließende <a href="Umkehrfunktion" title="Umkehrfunktion">Inversion</a> gelöst werden<a href="#cite_note-2">[2]</a>:

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\frac {1}{2}}m{\dot {x}}^{2}+V(x)&=E\\\Leftrightarrow {\frac {dx}{dt}}&={\sqrt {{\frac {2}{m}}\left[E-V(x)\right]}}\\\Rightarrow \int _{x_{0}}^{x}{\frac {dx'}{\sqrt {{\frac {2}{m}}\left[E-V(x')\right]}}}&=t-t_{0}\end{aligned}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9de0b371ecab5189d11b763d1edb3a703f55acd8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -9.186ex; margin-bottom: -0.318ex; width:44.612ex; height:20.176ex;" alt="{\displaystyle {\begin{aligned}{\frac {1}{2}}m{\dot {x}}^{2}+V(x)&=E\\\Leftrightarrow {\frac {dx}{dt}}&={\sqrt {{\frac {2}{m}}\left[E-V(x)\right]}}\\\Rightarrow \int _{x_{0}}^{x}{\frac {dx'}{\sqrt {{\frac {2}{m}}\left[E-V(x')\right]}}}&=t-t_{0}\end{aligned}}}" loading="lazy"></dd></dl>

Der Punkt über dem Buchstaben <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy"> ist die Newtonsche Schreibweise für die Zeitableitung, hier der Geschwindigkeit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\dot {x}}}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a82c85f33714da82ab42d6b69eae07ab7e5e234b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:2.176ex;" alt="{\displaystyle {\dot {x}}}" loading="lazy">. Die Gesamtenergie <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E=V(x_{0})}">

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<annotation encoding="application/x-tex">{\displaystyle E=V(x_{0})}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e4c7f43806f1c155c031e6e52177e2d15b986ca2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.855ex; height:2.843ex;" alt="{\displaystyle E=V(x_{0})}" loading="lazy"> und die Startzeit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t_{0}}">

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/02d3006c4190b1939b04d9b9bb21006fb4e6fa4a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.894ex; height:2.343ex;" alt="{\displaystyle t_{0}}" loading="lazy"> sind die beiden freien Konstanten in der Lösung der Bewegungsgleichung. Da die kinetische Energie <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\textstyle {\frac {1}{2}}}m{\dot {x}}^{2}}">

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<annotation encoding="application/x-tex">{\displaystyle {\textstyle {\frac {1}{2}}}m{\dot {x}}^{2}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3fe7a4e037cb2bb088d5e938db74a6e03af18664.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:6.082ex; height:3.509ex;" alt="{\displaystyle {\textstyle {\frac {1}{2}}}m{\dot {x}}^{2}}" loading="lazy"> positiv ist, existiert die Bewegung des Teilchens nur in Bereichen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E>V(x)}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/84c685d488512e9cb6f971a21422e9084ed99a19.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.8ex; height:2.843ex;" alt="{\displaystyle E>V(x)}" loading="lazy">. In der Skizze wären dies für die Energie <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{1}}">

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<annotation encoding="application/x-tex">{\displaystyle E_{1}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7ac42446bcd2cbb76ec8fe2895635d328da22e26.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.769ex; height:2.509ex;" alt="{\displaystyle E_{1}}" loading="lazy"> die Strecke <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {ab}}}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bc7a721561c238ec4b91e1e85628ef8ffd98b389.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.342ex; height:3.009ex;" alt="{\displaystyle {\overline {ab}}}" loading="lazy"> und der Abschnitt rechts von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/86a67b81c2de995bd608d5b2df50cd8cd7d92455.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.007ex; height:1.676ex;" alt="{\displaystyle c}" loading="lazy">. Die Geschwindigkeit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\dot {x}}\sim {\sqrt {E-V(x)}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\dot {x}}\sim {\sqrt {E-V(x)}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/14e7b89f70365d2a7ff8b175f3ed9ecbe1d8964d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:16.294ex; height:4.843ex;" alt="{\displaystyle {\dot {x}}\sim {\sqrt {E-V(x)}}}" loading="lazy"> ist umso größer, je kleiner das Potential <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V(x)}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4ab3e825c2bf9c80d11d12e070a4626d48e03c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.926ex; height:2.843ex;" alt="{\displaystyle V(x)}" loading="lazy"> ist<a href="#cite_note-3">[3]</a>. Das lokale Maximum <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S_{1}}">

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<annotation encoding="application/x-tex">{\displaystyle S_{1}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5bf84e7fd4fb8259a9b37f956afdf83ee2a020f9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.479ex; height:2.509ex;" alt="{\displaystyle S_{1}}" loading="lazy"> des Potentials in der Skizze ist instabil, während das Minimum <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S_{0}}">

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<annotation encoding="application/x-tex">{\displaystyle S_{0}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ebe0ac45a38c4437bd2689a14ec434cd499e7e49.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.479ex; height:2.509ex;" alt="{\displaystyle S_{0}}" loading="lazy"> eine stabile Gleichgewichtslage darstellt<a href="#cite_note-4">[4]</a>.

Aus der Zeitunabhängigkeit der Energie folgt die <a href="Newtonsche_Gesetze" title="Newtonsche Gesetze">Newtonsche Bewegungsgleichung</a><a href="#cite_note-5">[5]</a>:

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\frac {{\text{d}}E}{{\text{d}}t}}=0&={\frac {{\text{d}}\,\,\,}{{\text{d}}t}}\left[{\frac {m}{2}}\left({\frac {{\text{d}}x}{{\text{d}}t}}\right)^{2}+V(x)\right]={\frac {{\text{d}}x}{{\text{d}}t}}\left(m{\frac {{\text{d}}^{2}x}{{\text{d}}t^{2}}}+{\frac {{\text{d}}V(x)}{{\text{d}}x}}\right)\\&\Rightarrow \,m{\frac {{\text{d}}^{2}x}{{\text{d}}t^{2}}}=-{\frac {{\text{d}}V(x)}{{\text{d}}x}}\end{aligned}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b2ab9d3b5f42e0b92c94d27b62e8572d6ba360a7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.997ex; margin-bottom: -0.174ex; width:63.793ex; height:13.509ex;" alt="{\displaystyle {\begin{aligned}{\frac {{\text{d}}E}{{\text{d}}t}}=0&={\frac {{\text{d}}\,\,\,}{{\text{d}}t}}\left[{\frac {m}{2}}\left({\frac {{\text{d}}x}{{\text{d}}t}}\right)^{2}+V(x)\right]={\frac {{\text{d}}x}{{\text{d}}t}}\left(m{\frac {{\text{d}}^{2}x}{{\text{d}}t^{2}}}+{\frac {{\text{d}}V(x)}{{\text{d}}x}}\right)\\&\Rightarrow \,m{\frac {{\text{d}}^{2}x}{{\text{d}}t^{2}}}=-{\frac {{\text{d}}V(x)}{{\text{d}}x}}\end{aligned}}}" loading="lazy"></dd></dl>

Diese Bewegungsgleichung ergibt sich auch aus dem <a href="Prinzip_der_kleinsten_Wirkung" class="mw-redirect" title="Prinzip der kleinsten Wirkung">Prinzip der kleinsten Wirkung</a>. Nach <a href="Joseph-Louis_Lagrange" title="Joseph-Louis Lagrange">Lagrange</a> existiert ein Skalar <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\cal {L}}}">

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/cddbb21ad79aa4e70f27927e433fd985873a3b6a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.604ex; height:2.176ex;" alt="{\displaystyle {\cal {L}}}" loading="lazy"> als Differenz von kinetischer Energie <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\textstyle {\frac {1}{2}}}m{\dot {x}}^{2}}">

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<annotation encoding="application/x-tex">{\displaystyle {\textstyle {\frac {1}{2}}}m{\dot {x}}^{2}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3fe7a4e037cb2bb088d5e938db74a6e03af18664.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:6.082ex; height:3.509ex;" alt="{\displaystyle {\textstyle {\frac {1}{2}}}m{\dot {x}}^{2}}" loading="lazy"> und potentieller Energie <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V(x)}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4ab3e825c2bf9c80d11d12e070a4626d48e03c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.926ex; height:2.843ex;" alt="{\displaystyle V(x)}" loading="lazy">:

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<annotation encoding="application/x-tex">{\displaystyle {\cal {L}}(x,{\dot {x}})={\textstyle {\frac {1}{2}}}m{\dot {x}}^{2}-V(x)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/85293c90f706de2a9efc46a29b35ea9a23c1db6c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:24.054ex; height:3.509ex;" alt="{\displaystyle {\cal {L}}(x,{\dot {x}})={\textstyle {\frac {1}{2}}}m{\dot {x}}^{2}-V(x)}" loading="lazy"> </div>

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Zu den Zeitpunkten <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t_{1}}">

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/cb0768c0bd659f2f84fb5ef9f4b74f336123d915.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.894ex; height:2.343ex;" alt="{\displaystyle t_{1}}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t_{2}}">

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<annotation encoding="application/x-tex">{\displaystyle t_{2}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/749fee708b41e7079eabd50d61c8bf3e965db16f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.894ex; height:2.343ex;" alt="{\displaystyle t_{2}}" loading="lazy"> liegen feste Zustände <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{(1)}}">

<msup>

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<annotation encoding="application/x-tex">{\displaystyle x^{(1)}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/75f307045280ef28c6edaa460cf3c6f566c2c087.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.663ex; height:2.843ex;" alt="{\displaystyle x^{(1)}}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{(2)}}">

<msup>

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<annotation encoding="application/x-tex">{\displaystyle x^{(2)}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d4239b9b121db1ade2429afeddae5eddfefd3df7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.663ex; height:2.843ex;" alt="{\displaystyle x^{(2)}}" loading="lazy"> des Systems vor. Das System entwickelt sich so, dass die <a href="Wirkung_(Physik)" title="Wirkung (Physik)">Wirkung</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\cal {S}}}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/27962ddb2d082ea88e141807e88ee5afdfbfaa67.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.492ex; height:2.176ex;" alt="{\displaystyle {\cal {S}}}" loading="lazy"> als weiterer Skalar das Zeitintegral von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\cal {L}}}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/cddbb21ad79aa4e70f27927e433fd985873a3b6a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.604ex; height:2.176ex;" alt="{\displaystyle {\cal {L}}}" loading="lazy">

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<annotation encoding="application/x-tex">{\displaystyle {\cal {S=\int _{t_{1}}^{t_{2}}{\cal {L(x,{\dot {x}},t){\text{d}}t}}}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/238114d791a69bc4889556cf59db55c61ce55fd8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:20.135ex; height:6.343ex;" alt="{\displaystyle {\cal {S=\int _{t_{1}}^{t_{2}}{\cal {L(x,{\dot {x}},t){\text{d}}t}}}}}" loading="lazy"></dd></dl>

minimiert<a href="#cite_note-6">[6]</a>.

Das Verschwinden der <a href="Variationsrechnung" title="Variationsrechnung">Variation</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta {\cal {S}}=0}">

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<annotation encoding="application/x-tex">{\displaystyle \delta {\cal {S}}=0}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1b012c440dcfa0540c529b4959d533b3b0b741e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.802ex; height:2.343ex;" alt="{\displaystyle \delta {\cal {S}}=0}" loading="lazy"> von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\cal {S}}}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/27962ddb2d082ea88e141807e88ee5afdfbfaa67.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.492ex; height:2.176ex;" alt="{\displaystyle {\cal {S}}}" loading="lazy"> führt auf die Lagrangesche Differentialgleichung<a href="#cite_note-Landau-4-7">[7]</a>

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<annotation encoding="application/x-tex">{\displaystyle \delta {\cal {S}}=0\quad \Rightarrow \quad {\frac {{\text{d}}\,\,\,}{{\text{d}}t}}{\frac {\partial {\cal {L}}}{\partial {\dot {x}}}}-{\frac {\partial {\cal {L}}}{\partial x}}=0}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e689f0b0fcf8991a2b77c879a7ef9bf83c88f8c0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:32.968ex; height:5.509ex;" alt="{\displaystyle \delta {\cal {S}}=0\quad \Rightarrow \quad {\frac {{\text{d}}\,\,\,}{{\text{d}}t}}{\frac {\partial {\cal {L}}}{\partial {\dot {x}}}}-{\frac {\partial {\cal {L}}}{\partial x}}=0}" loading="lazy"> </div>

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</td></tr></tbody></table>

<td>Kurzer Beweis

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<td>

Gesucht ist die Funktion<a href="#cite_note-8">[8]</a> der Ortskoordinate <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x(t)}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d54c275db3a1e620737b58e143b0818107fa5f5c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.979ex; height:2.843ex;" alt="{\displaystyle x(t)}" loading="lazy">, die die Wirkung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\cal {S}}}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/27962ddb2d082ea88e141807e88ee5afdfbfaa67.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.492ex; height:2.176ex;" alt="{\displaystyle {\cal {S}}}" loading="lazy"> minimiert. Für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x(t)+\delta x(t)}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a40f5972baa49bc8d4fe734e567ffdce5e474ff1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.846ex; height:2.843ex;" alt="{\displaystyle x(t)+\delta x(t)}" loading="lazy"> wächst <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\cal {S}}}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/27962ddb2d082ea88e141807e88ee5afdfbfaa67.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.492ex; height:2.176ex;" alt="{\displaystyle {\cal {S}}}" loading="lazy">. Die Variation <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta x(t)}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3fedba5adb6ef89e9eb381a63bd2424e16dd8f20.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.027ex; height:2.843ex;" alt="{\displaystyle \delta x(t)}" loading="lazy"> der Funktion <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x(t)}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d54c275db3a1e620737b58e143b0818107fa5f5c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.979ex; height:2.843ex;" alt="{\displaystyle x(t)}" loading="lazy"> soll klein sein und an den Integrationsgrenzen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t_{1}}">

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<annotation encoding="application/x-tex">{\displaystyle t_{1}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/cb0768c0bd659f2f84fb5ef9f4b74f336123d915.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.894ex; height:2.343ex;" alt="{\displaystyle t_{1}}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t_{2}}">

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<annotation encoding="application/x-tex">{\displaystyle t_{2}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/749fee708b41e7079eabd50d61c8bf3e965db16f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.894ex; height:2.343ex;" alt="{\displaystyle t_{2}}" loading="lazy"> verschwinden: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta x(t_{1})=\delta x(t_{2})=0}">

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<annotation encoding="application/x-tex">{\displaystyle \delta x(t_{1})=\delta x(t_{2})=0}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3c4aeed6b27426f187894f33042c784de8fa1245.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.522ex; height:2.843ex;" alt="{\displaystyle \delta x(t_{1})=\delta x(t_{2})=0}" loading="lazy">. Alle Vergleichsfunktionen müssen an den Endpunkten <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{(1)}}">

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<annotation encoding="application/x-tex">{\displaystyle x^{(1)}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/75f307045280ef28c6edaa460cf3c6f566c2c087.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.663ex; height:2.843ex;" alt="{\displaystyle x^{(1)}}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{(2)}}">

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<annotation encoding="application/x-tex">{\displaystyle x^{(2)}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d4239b9b121db1ade2429afeddae5eddfefd3df7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.663ex; height:2.843ex;" alt="{\displaystyle x^{(2)}}" loading="lazy"> die gleichen Werte annehmen. Dieser Zuwachs der Wirkung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\cal {S}}}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/27962ddb2d082ea88e141807e88ee5afdfbfaa67.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.492ex; height:2.176ex;" alt="{\displaystyle {\cal {S}}}" loading="lazy"> durch <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x(t)+\delta x(t)}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a40f5972baa49bc8d4fe734e567ffdce5e474ff1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.846ex; height:2.843ex;" alt="{\displaystyle x(t)+\delta x(t)}" loading="lazy"> lautet:

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<annotation encoding="application/x-tex">{\displaystyle \delta {\cal {S}}=\int _{t_{1}}^{t_{2}}{\cal {L(x+\delta x,{\dot {x}}+\delta {\dot {x}},t){\text{d}}t-\int _{t_{1}}^{t_{2}}{\cal {L(x,{\dot {x}},t){\text{d}}t}}}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/57d4f2bd903d50af6b6ba76680d21854b4106bff.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:50.006ex; height:6.509ex;" alt="{\displaystyle \delta {\cal {S}}=\int _{t_{1}}^{t_{2}}{\cal {L(x+\delta x,{\dot {x}}+\delta {\dot {x}},t){\text{d}}t-\int _{t_{1}}^{t_{2}}{\cal {L(x,{\dot {x}},t){\text{d}}t}}}}}" loading="lazy"></dd></dl>

Die Entwicklung der Differenz nach Potenzen von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta x(t)}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3fedba5adb6ef89e9eb381a63bd2424e16dd8f20.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.027ex; height:2.843ex;" alt="{\displaystyle \delta x(t)}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta {\dot {x}}(t)}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7f54a46ff06c3ee66ce0f587ea3721f0f035f19c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.027ex; height:2.843ex;" alt="{\displaystyle \delta {\dot {x}}(t)}" loading="lazy"> im Integranden beginnt mit den Termen erster Ordnung. Eine notwendige Bedingung dafür, dass <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\cal {S}}}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/27962ddb2d082ea88e141807e88ee5afdfbfaa67.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.492ex; height:2.176ex;" alt="{\displaystyle {\cal {S}}}" loading="lazy"> ein Minimum (oder allgemeiner ein Extremum) annimmt, ist, dass die Summe dieser Terme verschwindet. Diese Summe nennt man die erste Variation des Integrals. Auf diese Weise kann das Prinzip der kleinsten Wirkung wie folgt ausgedrückt werden:

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<annotation encoding="application/x-tex">{\displaystyle \delta {\cal {S}}=\delta \int _{t_{1}}^{t_{2}}{\cal {L}}(x,{\dot {x}},t){\text{d}}t=\int _{t_{1}}^{t_{2}}\left({\frac {\partial {\cal {L}}}{\partial x}}\delta x(t)+{\frac {\partial {\cal {L}}}{\partial {\dot {x}}}}\delta {\dot {x}}(t)\right){\text{d}}t=0}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3d1d7cb9f0f7e47e0e8c5f80b9412e8c1ee0181e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:60.915ex; height:6.509ex;" alt="{\displaystyle \delta {\cal {S}}=\delta \int _{t_{1}}^{t_{2}}{\cal {L}}(x,{\dot {x}},t){\text{d}}t=\int _{t_{1}}^{t_{2}}\left({\frac {\partial {\cal {L}}}{\partial x}}\delta x(t)+{\frac {\partial {\cal {L}}}{\partial {\dot {x}}}}\delta {\dot {x}}(t)\right){\text{d}}t=0}" loading="lazy"></dd></dl>

Mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta {\dot {x}}(t)={\textstyle {\frac {{\text{d}}\,\,\,}{{\text{d}}t}}}\delta x(t)}">

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<annotation encoding="application/x-tex">{\displaystyle \delta {\dot {x}}(t)={\textstyle {\frac {{\text{d}}\,\,\,}{{\text{d}}t}}}\delta x(t)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8ee2217b13e5e0815eefbbf9a1cd9a4170bf94f3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:16.064ex; height:3.843ex;" alt="{\displaystyle \delta {\dot {x}}(t)={\textstyle {\frac {{\text{d}}\,\,\,}{{\text{d}}t}}}\delta x(t)}" loading="lazy"> und partieller Integration des zweiten Terms erhält man

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<annotation encoding="application/x-tex">{\displaystyle \delta {\cal {S}}={\frac {\partial {\cal {L}}}{\partial x}}\delta x(t){\bigg |}_{t_{1}}^{t_{2}}+\int _{t_{1}}^{t_{2}}\left({\frac {\partial {\cal {L}}}{\partial x}}-{\frac {{\text{d}}\,\,\,}{{\text{d}}t}}{\frac {\partial {\cal {L}}}{\partial {\dot {x}}}}\right)\delta x(t){\text{d}}t=0}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/18c7da15df5b26fed941143480ad44804a944645.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:52.953ex; height:6.509ex;" alt="{\displaystyle \delta {\cal {S}}={\frac {\partial {\cal {L}}}{\partial x}}\delta x(t){\bigg |}_{t_{1}}^{t_{2}}+\int _{t_{1}}^{t_{2}}\left({\frac {\partial {\cal {L}}}{\partial x}}-{\frac {{\text{d}}\,\,\,}{{\text{d}}t}}{\frac {\partial {\cal {L}}}{\partial {\dot {x}}}}\right)\delta x(t){\text{d}}t=0}" loading="lazy"></dd></dl>

Da <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x(t)}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d54c275db3a1e620737b58e143b0818107fa5f5c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.979ex; height:2.843ex;" alt="{\displaystyle x(t)}" loading="lazy"> in den Endpunkten <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t_{1},t_{2}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

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<annotation encoding="application/x-tex">{\displaystyle t_{1},t_{2}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7e76daf1a59dca26c96dbca2863a1c236b15b5a1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.822ex; height:2.343ex;" alt="{\displaystyle t_{1},t_{2}}" loading="lazy"> fixiert ist, verschwindet der erste Summand und das Integral kann für jedes <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta x(t)}">

</mstyle>

</mrow>

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3fedba5adb6ef89e9eb381a63bd2424e16dd8f20.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.027ex; height:2.843ex;" alt="{\displaystyle \delta x(t)}" loading="lazy"> nur dann Null werden, wenn der Integrand verschwindet. Dies ist die Lagrange-Gleichung der Mechanik<a href="#cite_note-Landau-4-7">[7]</a>.

<mfrac>

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</mrow>

<mfrac>

<mrow>

</mrow>

<mrow>

<mover>

</mover>

</mrow>

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</mrow>

<mfrac>

<mrow>

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<mrow>

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<annotation encoding="application/x-tex">{\displaystyle {\frac {{\text{d}}\,\,\,}{{\text{d}}t}}{\frac {\partial {\cal {L}}}{\partial {\dot {x}}}}-{\frac {\partial {\cal {L}}}{\partial x}}=0}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ed8b6ecc7d423c11b9de2dd7691e35383fc5da86.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:17.907ex; height:5.509ex;" alt="{\displaystyle {\frac {{\text{d}}\,\,\,}{{\text{d}}t}}{\frac {\partial {\cal {L}}}{\partial {\dot {x}}}}-{\frac {\partial {\cal {L}}}{\partial x}}=0}" loading="lazy"></dd></dl>

</td></tr></tbody></table>

Im <a href="Lagrange-Formalismus" title="Lagrange-Formalismus">Lagrange-Formalismus</a> der Mechanik wird die Bahn <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x(t)}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d54c275db3a1e620737b58e143b0818107fa5f5c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.979ex; height:2.843ex;" alt="{\displaystyle x(t)}" loading="lazy"> eines Systems durch den <a href="Konfigurationsraum" title="Konfigurationsraum">Konfigurationsraum</a> beschrieben. Die Bewegungsgleichungen sind Differentialgleichungen zweiter Ordnung. Das bedeutet, dass ein Punkt im Konfigurationsraum den Zustand eines mechanischen Systems nicht vollständig beschreibt. Zur Lösung müssen die Anfangskoordinaten und die Anfangsgeschwindigkeiten bekannt sein<a href="#cite_note-Susskind-116-9">[9]</a>.

In der <a href="Hamiltonsche_Mechanik" title="Hamiltonsche Mechanik">Hamilton-Formulierung</a> wird der <a href="Phasenraum" title="Phasenraum">Phasenraum</a> betrachtet, der gemeinsame Raum der Koordinaten <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy"> und des konjugierten Impulses <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/81eac1e205430d1f40810df36a0edffdc367af36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" loading="lazy">. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

</mstyle>

</mrow>

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}">

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</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/81eac1e205430d1f40810df36a0edffdc367af36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" loading="lazy"> werden gleich behandelt und die Bewegung des Systems wird durch eine Bahndarstellung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x(t)}">

</mstyle>

</mrow>

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d54c275db3a1e620737b58e143b0818107fa5f5c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.979ex; height:2.843ex;" alt="{\displaystyle x(t)}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p(t)}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9b827c545ca1487214f0c498131228ef87718ece.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:3.908ex; height:2.843ex;" alt="{\displaystyle p(t)}" loading="lazy"> beschrieben. Der Phasenraum ist zweidimensional, die Bewegungsgleichungen sind dann Differentialgleichungen erster Ordnung, die Zukunft wird durch den Anfangspunkt im Phasenraum bestimmt<a href="#cite_note-Susskind-116-9">[9]</a>.

Der <a href="Generalisierter_Impuls" title="Generalisierter Impuls">kanonisch konjugierte Impuls</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}">

</mstyle>

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/81eac1e205430d1f40810df36a0edffdc367af36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" loading="lazy"> ist die Ableitung der Lagrange-Funktion (1) nach der Geschwindigkeit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\dot {x}}}">

<mover>

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</mstyle>

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a82c85f33714da82ab42d6b69eae07ab7e5e234b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:2.176ex;" alt="{\displaystyle {\dot {x}}}" loading="lazy">:

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<annotation encoding="application/x-tex">{\displaystyle {\frac {{\text{d}}{\cal {L}}}{{\text{d}}{\dot {x}}}}={\frac {{\text{d}}\,\,\,}{{\text{d}}{\dot {x}}}}\left[{\textstyle {\frac {1}{2}}}m{\dot {x}}^{2}-V(x)\right]=m{\dot {x}}=:p}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7de6e4806ed642e01a7baad6124d6226ed6d8194.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:37.848ex; height:5.509ex;" alt="{\displaystyle {\frac {{\text{d}}{\cal {L}}}{{\text{d}}{\dot {x}}}}={\frac {{\text{d}}\,\,\,}{{\text{d}}{\dot {x}}}}\left[{\textstyle {\frac {1}{2}}}m{\dot {x}}^{2}-V(x)\right]=m{\dot {x}}=:p}" loading="lazy"></dd></dl>

Aus der <a href="Lagrange-Funktion" class="mw-redirect" title="Lagrange-Funktion">Lagrange-Funktion</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\cal {L}}}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/cddbb21ad79aa4e70f27927e433fd985873a3b6a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.604ex; height:2.176ex;" alt="{\displaystyle {\cal {L}}}" loading="lazy"> ergibt sich mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\dot {x}}=p/m}">

<mover>

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/033044889c2c2ada6c909e70fe11ca38cb7b3dab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.8ex; height:2.843ex;" alt="{\displaystyle {\dot {x}}=p/m}" loading="lazy"> die <a href="Hamilton-Funktion" title="Hamilton-Funktion">Hamilton-Funktion</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\cal {H}}}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4aae16db8596127825ed2547f9e49290df120084.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.964ex; height:2.176ex;" alt="{\displaystyle {\cal {H}}}" loading="lazy"> als weiterer Skalar<a href="#cite_note-10">[10]</a>

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<annotation encoding="application/x-tex">{\displaystyle {\cal {H}}:=p\cdot {\dot {x}}-{\cal {L}}={\frac {p^{2}}{m}}-\left({\frac {p^{2}}{2m}}-V(x)\right)={\frac {p^{2}}{2m}}+V(x)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b04f3ead221aaf28082f070b0269c9046067e377.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:53.461ex; height:6.343ex;" alt="{\displaystyle {\cal {H}}:=p\cdot {\dot {x}}-{\cal {L}}={\frac {p^{2}}{m}}-\left({\frac {p^{2}}{2m}}-V(x)\right)={\frac {p^{2}}{2m}}+V(x)}" loading="lazy"></dd></dl>

Die Hamilton-Funktion <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\cal {H}}}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4aae16db8596127825ed2547f9e49290df120084.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.964ex; height:2.176ex;" alt="{\displaystyle {\cal {H}}}" loading="lazy"> ist die Summe aus der kinetischen Energie <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\textstyle {\frac {1}{2m}}}p^{2}}">

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<annotation encoding="application/x-tex">{\displaystyle {\textstyle {\frac {1}{2m}}}p^{2}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a8c99d92dc8e4ae4c99783c7ab37e08ec20883ca.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:5.325ex; height:3.509ex;" alt="{\displaystyle {\textstyle {\frac {1}{2m}}}p^{2}}" loading="lazy"> und der potentiellen Energie <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V(x)}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4ab3e825c2bf9c80d11d12e070a4626d48e03c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.926ex; height:2.843ex;" alt="{\displaystyle V(x)}" loading="lazy"> und damit die Gesamtenergie<a href="#cite_note-11">[11]</a>.

Die Lagrangesche Differentialgleichung (2) ist dann äquivalent zu<a href="#cite_note-12">[12]</a>:

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}p&=m{\frac {{\text{d}}x}{{\text{d}}t}}={\frac {{\text{d}}{\cal {L}}}{{\text{d}}{\dot {x}}}}\\{\frac {{\text{d}}p}{{\text{d}}t}}&={\frac {{\text{d}}\,\,\,}{{\text{d}}t}}{\frac {{\text{d}}{\cal {L}}}{{\text{d}}{\dot {x}}}}{\underset {\text{(2)}}{=}}{\frac {{\text{d}}{\cal {L}}}{{\text{d}}x}}=-{\frac {{\text{d}}V(x)}{{\text{d}}x}}\end{aligned}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/349bd99acff6bfc7e27224bc6769a8e741e9b99b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.171ex; width:31.966ex; height:11.509ex;" alt="{\displaystyle {\begin{aligned}p&=m{\frac {{\text{d}}x}{{\text{d}}t}}={\frac {{\text{d}}{\cal {L}}}{{\text{d}}{\dot {x}}}}\\{\frac {{\text{d}}p}{{\text{d}}t}}&={\frac {{\text{d}}\,\,\,}{{\text{d}}t}}{\frac {{\text{d}}{\cal {L}}}{{\text{d}}{\dot {x}}}}{\underset {\text{(2)}}{=}}{\frac {{\text{d}}{\cal {L}}}{{\text{d}}x}}=-{\frac {{\text{d}}V(x)}{{\text{d}}x}}\end{aligned}}}" loading="lazy"></dd></dl>

Mit dem kanonisch konjugierten Impuls <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle p={\frac {{\text{d}}{\cal {L}}}{{\text{d}}{\dot {x}}}}}">

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<annotation encoding="application/x-tex">{\displaystyle \textstyle p={\frac {{\text{d}}{\cal {L}}}{{\text{d}}{\dot {x}}}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e372a7b9ce38d7512bddbc46441a0fd7ab600fdf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; margin-left: -0.089ex; width:7.241ex; height:4.176ex;" alt="{\displaystyle \textstyle p={\frac {{\text{d}}{\cal {L}}}{{\text{d}}{\dot {x}}}}}" loading="lazy">, der Hamilton-Funktion <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\cal {H}}}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4aae16db8596127825ed2547f9e49290df120084.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.964ex; height:2.176ex;" alt="{\displaystyle {\cal {H}}}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle {\dot {x}}={\frac {p}{m}}={\frac {{\text{d}}{\cal {H}}}{{\text{d}}p}}}">

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<annotation encoding="application/x-tex">{\displaystyle \textstyle {\dot {x}}={\frac {p}{m}}={\frac {{\text{d}}{\cal {H}}}{{\text{d}}p}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ee307237f845568bc41fdfdf20070327fd53da41.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:12.944ex; height:4.176ex;" alt="{\displaystyle \textstyle {\dot {x}}={\frac {p}{m}}={\frac {{\text{d}}{\cal {H}}}{{\text{d}}p}}}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle {\frac {{\text{d}}{V}}{{\text{d}}x}}={\frac {{\text{d}}{\cal {H}}}{{\text{d}}x}}}">

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<annotation encoding="application/x-tex">{\displaystyle \textstyle {\frac {{\text{d}}{V}}{{\text{d}}x}}={\frac {{\text{d}}{\cal {H}}}{{\text{d}}x}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/492e6c5ab4f3eca6f25f64c6de478c1553c8f19f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:9.251ex; height:3.843ex;" alt="{\displaystyle \textstyle {\frac {{\text{d}}{V}}{{\text{d}}x}}={\frac {{\text{d}}{\cal {H}}}{{\text{d}}x}}}" loading="lazy"> wird das obige Differentialgleichungssystem zu

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\frac {{\text{d}}x}{{\text{d}}t}}&={\frac {{\text{d}}{\cal {H}}}{{\text{d}}p}}\\{\frac {{\text{d}}p}{{\text{d}}t}}&=-{\frac {{\text{d}}{\cal {H}}}{{\text{d}}x}}\end{aligned}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e3a3eb8b39b4cc36906460e71c25c404fcdc800e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.901ex; margin-bottom: -0.27ex; width:13.209ex; height:11.509ex;" alt="{\displaystyle {\begin{aligned}{\frac {{\text{d}}x}{{\text{d}}t}}&={\frac {{\text{d}}{\cal {H}}}{{\text{d}}p}}\\{\frac {{\text{d}}p}{{\text{d}}t}}&=-{\frac {{\text{d}}{\cal {H}}}{{\text{d}}x}}\end{aligned}}}" loading="lazy"></dd></dl>

Es handelt sich um eine symmetrische Gruppe von Differentialgleichungen erster Ordnung. Sie werden Hamilton-Gleichungen genannt. In jeder Richtung des Phasenraums <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x,p)}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/28bd549c036cae5ca70a8579886756839f82e10c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.342ex; height:2.843ex;" alt="{\displaystyle (x,p)}" loading="lazy"> muss eine Differentialgleichung erster Ordnung erfüllt sein. Die Bahnkurve der Teilchen erhält man durch schrittweise Integration der Hamilton-Gleichungen.

Die <a href="Poisson-Klammer" title="Poisson-Klammer">Poisson-Klammer</a>

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<annotation encoding="application/x-tex">{\displaystyle [u,v]_{x,p}={\frac {{\text{d}}u}{{\text{d}}x}}{\frac {{\text{d}}v}{{\text{d}}p}}-{\frac {{\text{d}}u}{{\text{d}}p}}{\frac {{\text{d}}v}{{\text{d}}x}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/148d8b700c5eaaefcde49b5097e165abc776a758.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:26.854ex; height:5.843ex;" alt="{\displaystyle [u,v]_{x,p}={\frac {{\text{d}}u}{{\text{d}}x}}{\frac {{\text{d}}v}{{\text{d}}p}}-{\frac {{\text{d}}u}{{\text{d}}p}}{\frac {{\text{d}}v}{{\text{d}}x}}}" loading="lazy"></dd></dl>

stellt die Bewegungsgleichungen sehr symmetrisch dar<a href="#cite_note-13">[13]</a>

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<annotation encoding="application/x-tex">{\displaystyle {\dot {x}}=[x,{\cal {H}}]_{x,p}\quad {\text{und}}\quad {\dot {p}}=[p,{\cal {H}}]_{x,p}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2ce091670e140998bffba1e39f62c236b85a02f3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:33.404ex; height:3.009ex;" alt="{\displaystyle {\dot {x}}=[x,{\cal {H}}]_{x,p}\quad {\text{und}}\quad {\dot {p}}=[p,{\cal {H}}]_{x,p}}" loading="lazy"></dd></dl>

In höheren Dimensionen lässt sich dieser Trick anwenden, wenn weitere <a href="Symmetrie_(Physik)" title="Symmetrie (Physik)">Symmetrien</a> und daraus folgende <a href="Erhaltungsgr%C3%B6%C3%9Fe" class="mw-redirect" title="Erhaltungsgröße">Erhaltungsgrößen</a> existieren.

Für die Bewegung eines materiellen Punktes der Masse <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m}">

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

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</math><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"> unter dem Einfluss der Gravitation als Zentralkraft <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle -{\frac {\partial U}{\partial {\vec {r}}}}}">

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<annotation encoding="application/x-tex">{\displaystyle \textstyle -{\frac {\partial U}{\partial {\vec {r}}}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e62976d2eb0909d852419f07dec46e25e8aa74dc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:4.837ex; height:4.176ex;" alt="{\displaystyle \textstyle -{\frac {\partial U}{\partial {\vec {r}}}}}" loading="lazy"> bleibt der <a href="Drehimpuls" title="Drehimpuls">Drehimpuls</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {L}}}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a0c139fc28d6ca3873993892f44e7331e5ff18fd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.843ex;" alt="{\displaystyle {\vec {L}}}" loading="lazy"> erhalten<a href="#cite_note-14">[14]</a>. Damit ändert sich im <a href="Keplerproblem" class="mw-redirect" title="Keplerproblem">Keplerproblem</a> der Abstand vom Massenmittelpunkt in gleicher Weise wie <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0d1ecb613aa2984f0576f70f86650b7c2a132538.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}" loading="lazy"> im eindimensionalen Problem mit dem Potential<a href="#cite_note-15">[15]</a>

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<annotation encoding="application/x-tex">{\displaystyle V(r)=U(r)+{\frac {|{\vec {L}}|^{2}}{2mr^{2}}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/37c9937d3990d160c0a62a19f6c760f5d5a6345e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:21.366ex; height:6.509ex;" alt="{\displaystyle V(r)=U(r)+{\frac {|{\vec {L}}|^{2}}{2mr^{2}}}}" loading="lazy"></dd></dl>

Die Lösung lautet<a href="#cite_note-16">[16]</a>

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<annotation encoding="application/x-tex">{\displaystyle t_{\text{max}}-t_{\text{min}}=\int _{r_{\text{min}}}^{r_{\text{max}}}{\frac {{\text{d}}r'}{\sqrt {{\frac {2}{m}}\left[E-V(r')\right]}}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3a7f43eec8c49b58a70c20e86224f208376848c3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.671ex; width:37.42ex; height:8.343ex;" alt="{\displaystyle t_{\text{max}}-t_{\text{min}}=\int _{r_{\text{min}}}^{r_{\text{max}}}{\frac {{\text{d}}r'}{\sqrt {{\frac {2}{m}}\left[E-V(r')\right]}}}}" loading="lazy"></dd></dl>

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

<li id="cite_note-1"><a href="#cite_ref-1">↑</a> Friedhelm Kuypers: <cite style="font-style:italic">Klassische Mechanik - mit über 300 Beispielen und Aufgaben mit Lösungen</cite>. 9. Auflage. WILEY-VCH, Weinheim 2010, ISBN 978-3-527-40989-1, S. 7.<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:Einteilchenproblem&rft.au=Friedhelm+Kuypers&rft.btitle=Klassische+Mechanik+-+mit+%C3%BCber+300+Beispielen+und+Aufgaben+mit+L%C3%B6sungen&rft.date=2010&rft.edition=9.&rft.genre=book&rft.isbn=9783527409891&rft.pages=7&rft.place=Weinheim&rft.pub=WILEY-VCH" style="display:none">

</li>

<li id="cite_note-2"><a href="#cite_ref-2">↑</a> L. D. Landau, E. M. Lifschitz: <cite style="font-style:italic">Lehrbuch der theoretischen Physik, Band 1, Mechanik -</cite>. 9. Auflage. Akademie Verlag, Berlin 1979, S. 30.<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:Einteilchenproblem&rft.au=L.+D.+Landau%2C+E.+M.+Lifschitz&rft.btitle=Lehrbuch+der+theoretischen+Physik%2C+Band+1%2C+Mechanik+-&rft.date=1979&rft.edition=9.&rft.genre=book&rft.pages=30&rft.place=Berlin&rft.pub=Akademie+Verlag" style="display:none">

</li>

<li id="cite_note-3"><a href="#cite_ref-3">↑</a> V. I. Arnol’d: <cite style="font-style:italic">Mathematical Methods of Classical Mechanics -</cite>. 1. Auflage. Springer-Verlag, Berlin Heidelberg New York 1978, ISBN 3-540-90314-3, S. 17.<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:Einteilchenproblem&rft.au=V.+I.+Arnol%E2%80%99d&rft.btitle=Mathematical+Methods+of+Classical+Mechanics+-&rft.date=1978&rft.edition=1.&rft.genre=book&rft.isbn=3540903143&rft.pages=17&rft.place=Berlin+Heidelberg+New+York&rft.pub=Springer-Verlag" style="display:none">

</li>

<li id="cite_note-4"><a href="#cite_ref-4">↑</a> V. I. Arnol’d: <cite style="font-style:italic">Mathematical Methods of Classical Mechanics -</cite>. 1. Auflage. Springer-Verlag, Berlin Heidelberg New York 1978, ISBN 3-540-90314-3, S. 18.<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:Einteilchenproblem&rft.au=V.+I.+Arnol%E2%80%99d&rft.btitle=Mathematical+Methods+of+Classical+Mechanics+-&rft.date=1978&rft.edition=1.&rft.genre=book&rft.isbn=3540903143&rft.pages=18&rft.place=Berlin+Heidelberg+New+York&rft.pub=Springer-Verlag" style="display:none">

</li>

<li id="cite_note-5"><a href="#cite_ref-5">↑</a> V. I. Arnol’d: <cite style="font-style:italic">Mathematical Methods of Classical Mechanics -</cite>. 1. Auflage. Springer-Verlag, Berlin Heidelberg New York 1978, ISBN 3-540-90314-3, S. 16.<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:Einteilchenproblem&rft.au=V.+I.+Arnol%E2%80%99d&rft.btitle=Mathematical+Methods+of+Classical+Mechanics+-&rft.date=1978&rft.edition=1.&rft.genre=book&rft.isbn=3540903143&rft.pages=16&rft.place=Berlin+Heidelberg+New+York&rft.pub=Springer-Verlag" style="display:none">

</li>

<li id="cite_note-6"><a href="#cite_ref-6">↑</a> L. D. Landau, E. M. Lifschitz: <cite style="font-style:italic">Lehrbuch der theoretischen Physik, Band 1, Mechanik -</cite>. 9. Auflage. Akademie Verlag, Berlin 1979, S. 2.<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:Einteilchenproblem&rft.au=L.+D.+Landau%2C+E.+M.+Lifschitz&rft.btitle=Lehrbuch+der+theoretischen+Physik%2C+Band+1%2C+Mechanik+-&rft.date=1979&rft.edition=9.&rft.genre=book&rft.pages=2&rft.place=Berlin&rft.pub=Akademie+Verlag" style="display:none">

</li>

<li id="cite_note-Landau-4-7">↑ <a href="#cite_ref-Landau-4_7-0">a</a> <a href="#cite_ref-Landau-4_7-1">b</a> L. D. Landau, E. M. Lifschitz: <cite style="font-style:italic">Lehrbuch der theoretischen Physik, Band 1, Mechanik -</cite>. 9. Auflage. Akademie Verlag, Berlin 1979, S. 4.<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:Einteilchenproblem&rft.au=L.+D.+Landau%2C+E.+M.+Lifschitz&rft.btitle=Lehrbuch+der+theoretischen+Physik%2C+Band+1%2C+Mechanik+-&rft.date=1979&rft.edition=9.&rft.genre=book&rft.pages=4&rft.place=Berlin&rft.pub=Akademie+Verlag" style="display:none">

</li>

<li id="cite_note-8"><a href="#cite_ref-8">↑</a> L. D. Landau, E. M. Lifschitz: <cite style="font-style:italic">Lehrbuch der theoretischen Physik, Band 1, Mechanik -</cite>. 9. Auflage. Akademie Verlag, Berlin 1979, S. 3.<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:Einteilchenproblem&rft.au=L.+D.+Landau%2C+E.+M.+Lifschitz&rft.btitle=Lehrbuch+der+theoretischen+Physik%2C+Band+1%2C+Mechanik+-&rft.date=1979&rft.edition=9.&rft.genre=book&rft.pages=3&rft.place=Berlin&rft.pub=Akademie+Verlag" style="display:none">

</li>

<li id="cite_note-Susskind-116-9">↑ <a href="#cite_ref-Susskind-116_9-0">a</a> <a href="#cite_ref-Susskind-116_9-1">b</a> Leonard Susskind, George E. Hrabovsky: <cite style="font-style:italic">Klassische Mechanik - Das Theoretische Minimum : Alles, was Sie brauchen, um Physik zu treiben.</cite> 1. Auflage. Springer, Berlin 2019, ISBN 978-3-662-60333-8, S. 116.<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:Einteilchenproblem&rft.au=Leonard+Susskind%2C+George+E.+Hrabovsky&rft.btitle=Klassische+Mechanik+-+Das+Theoretische+Minimum+%3A+Alles%2C+was+Sie+brauchen%2C+um+Physik+zu+treiben.&rft.date=2019&rft.edition=1.&rft.genre=book&rft.isbn=9783662603338&rft.pages=116&rft.place=Berlin&rft.pub=Springer" style="display:none">

</li>

<li id="cite_note-10"><a href="#cite_ref-10">↑</a> Leonard Susskind, George E. Hrabovsky: <cite style="font-style:italic">Klassische Mechanik - Das Theoretische Minimum : Alles, was Sie brauchen, um Physik zu treiben.</cite> 1. Auflage. Springer, Berlin 2019, ISBN 978-3-662-60333-8, S. 114.<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:Einteilchenproblem&rft.au=Leonard+Susskind%2C+George+E.+Hrabovsky&rft.btitle=Klassische+Mechanik+-+Das+Theoretische+Minimum+%3A+Alles%2C+was+Sie+brauchen%2C+um+Physik+zu+treiben.&rft.date=2019&rft.edition=1.&rft.genre=book&rft.isbn=9783662603338&rft.pages=114&rft.place=Berlin&rft.pub=Springer" style="display:none">

</li>

<li id="cite_note-11"><a href="#cite_ref-11">↑</a> Ágoston Budó: <cite style="font-style:italic">Theoretische Mechanik</cite>. 4. Auflage. VEB Deutscher Verlag der Wissenschaften, Berlin 1967, § 35, S. 176.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abookitem&rfr_id=info:sid/de.wikipedia.org:Einteilchenproblem&rft.atitle=%C2%A7+35&rft.au=%C3%81goston+Bud%C3%B3&rft.btitle=Theoretische+Mechanik&rft.date=1967&rft.edition=4&rft.genre=bookitem&rft.pages=176&rft.place=Berlin&rft.pub=VEB+Deutscher+Verlag+der+Wissenschaften" style="display:none">

</li>

<li id="cite_note-12"><a href="#cite_ref-12">↑</a> V. I. Arnol’d: <cite style="font-style:italic">Mathematical Methods of Classical Mechanics -</cite>. 1. Auflage. Springer-Verlag, Berlin Heidelberg New York 1978, ISBN 3-540-90314-3, S. 60.<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:Einteilchenproblem&rft.au=V.+I.+Arnol%E2%80%99d&rft.btitle=Mathematical+Methods+of+Classical+Mechanics+-&rft.date=1978&rft.edition=1.&rft.genre=book&rft.isbn=3540903143&rft.pages=60&rft.place=Berlin+Heidelberg+New+York&rft.pub=Springer-Verlag" style="display:none">

</li>

<li id="cite_note-13"><a href="#cite_ref-13">↑</a> Klaus Lichtenegger: <cite style="font-style:italic">Schlüsselkonzepte zur Physik : Von den Newton-Axiomen bis zur Hawking-Strahlung</cite>. 1. Auflage. Springer Spektrum, Berlin 2015, ISBN 978-3-8274-2384-9, S. 35.<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:Einteilchenproblem&rft.au=Klaus+Lichtenegger&rft.btitle=Schl%C3%BCsselkonzepte+zur+Physik+%3A+Von+den+Newton-Axiomen+bis+zur+Hawking-Strahlung&rft.date=2015&rft.edition=1.&rft.genre=book&rft.isbn=9783827423849&rft.pages=35&rft.place=Berlin&rft.pub=Springer+Spektrum" style="display:none">

</li>

<li id="cite_note-14"><a href="#cite_ref-14">↑</a> V. I. Arnol’d: <cite style="font-style:italic">Mathematical Methods of Classical Mechanics -</cite>. 1. Auflage. Springer-Verlag, Berlin Heidelberg New York 1978, ISBN 3-540-90314-3, S. 31.<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:Einteilchenproblem&rft.au=V.+I.+Arnol%E2%80%99d&rft.btitle=Mathematical+Methods+of+Classical+Mechanics+-&rft.date=1978&rft.edition=1.&rft.genre=book&rft.isbn=3540903143&rft.pages=31&rft.place=Berlin+Heidelberg+New+York&rft.pub=Springer-Verlag" style="display:none">

</li>

<li id="cite_note-15"><a href="#cite_ref-15">↑</a> V. I. Arnol’d: <cite style="font-style:italic">Mathematical Methods of Classical Mechanics -</cite>. 1. Auflage. Springer-Verlag, Berlin Heidelberg New York 1978, ISBN 3-540-90314-3, S. 33.<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:Einteilchenproblem&rft.au=V.+I.+Arnol%E2%80%99d&rft.btitle=Mathematical+Methods+of+Classical+Mechanics+-&rft.date=1978&rft.edition=1.&rft.genre=book&rft.isbn=3540903143&rft.pages=33&rft.place=Berlin+Heidelberg+New+York&rft.pub=Springer-Verlag" style="display:none">

</li>

<li id="cite_note-16"><a href="#cite_ref-16">↑</a> V. I. Arnol’d: <cite style="font-style:italic">Mathematical Methods of Classical Mechanics -</cite>. 1. Auflage. Springer-Verlag, Berlin Heidelberg New York 1978, ISBN 3-540-90314-3, S. 34.<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:Einteilchenproblem&rft.au=V.+I.+Arnol%E2%80%99d&rft.btitle=Mathematical+Methods+of+Classical+Mechanics+-&rft.date=1978&rft.edition=1.&rft.genre=book&rft.isbn=3540903143&rft.pages=34&rft.place=Berlin+Heidelberg+New+York&rft.pub=Springer-Verlag" style="display:none">

</li>

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