Editing Forminvarianz der Lagrangegleichungen
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<noinclude>{{Scripthinweis|Mechanik|2|4}}</noinclude> | <noinclude>{{Scripthinweis|Mechanik|2|4}}</noinclude> | ||
Eine schwächere Form der Invarianz (als die Eichinvarianz) ist die Forminvarianz. | Eine schwächere Form der Invarianz ( als die Eichinvarianz) ist die Forminvarianz. | ||
Dabei gilt als Forminvarianz: | Dabei gilt als Forminvarianz: | ||
<math>\frac{\partial L}{\partial {{q}_{k}}}-\frac{d}{dt}\frac{\partial L}{\partial {{{\dot{q}}}_{k}}}=0\Rightarrow \frac{\partial L}{\partial {{Q}_{k}}}-\frac{d}{dt}\frac{\partial L}{\partial {{{\dot{Q}}}_{k}}}=0</math> | |||
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<math>F:\left\{ {{q}_{k}} \right\}\to \left\{ {{Q}_{k}} \right\}</math> | |||
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Sei | Sei | ||
<math>F:\left\{ {{q}_{k}} \right\}\to \left\{ {{Q}_{k}} \right\}</math> | |||
ein C²- Diffeomorphismus, | ein C²- Diffeomorphismus, | ||
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<math>F,{{F}^{-1}}</math> | |||
beide zweimal stetig differenzierbar, dann ist | beide zweimal stetig differenzierbar, dann ist | ||
<math>\left\{ {{Q}_{k}}(t) \right\}</math> | |||
Lösung der Lagrangegleichung zur transformierten Lagrangefunktion: | Lösung der Lagrangegleichung zur transformierten Lagrangefunktion: | ||
<math>\tilde{L}({{Q}_{k}},{{\dot{Q}}_{k}},t):=L({{f}_{k}}({{Q}_{i}},t),\sum\limits_{i}{{}}\frac{\partial {{f}_{k}}}{\partial {{Q}_{i}}}{{\dot{Q}}_{i}}+\frac{\partial {{f}_{k}}}{\partial t},t)</math> | |||
mit | |||
<math>\begin{align} | |||
& {{f}_{k}}({{Q}_{i}},t)={{q}_{k}} \\ | & {{f}_{k}}({{Q}_{i}},t)={{q}_{k}} \\ | ||
& \sum\limits_{i}{{}}\frac{\partial {{f}_{k}}}{\partial {{Q}_{i}}}{{{\dot{Q}}}_{i}}+\frac{\partial {{f}_{k}}}{\partial t}={{{\dot{q}}}_{k}} \\ | & \sum\limits_{i}{{}}\frac{\partial {{f}_{k}}}{\partial {{Q}_{i}}}{{{\dot{Q}}}_{i}}+\frac{\partial {{f}_{k}}}{\partial t}={{{\dot{q}}}_{k}} \\ | ||
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<math>\left\{ {{q}_{k}}(t) \right\}</math> | |||
sind Lösung der Lagrangegleichungen zu | sind Lösung der Lagrangegleichungen zu | ||
<math>L({{q}_{k}},{{\dot{q}}_{k}},t)</math> | |||
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<math>\frac{d}{dt}\frac{\partial \tilde{L}}{\partial {{{\dot{Q}}}_{k}}}=\sum\limits_{l=1}^{f}{\frac{d}{dt}\frac{\partial L}{\partial {{{\dot{q}}}_{l}}}\frac{\partial {{{\dot{q}}}_{l}}}{\partial {{{\dot{Q}}}_{k}}}=}\sum\limits_{l=1}^{f}{\frac{d}{dt}\left( \frac{\partial L}{\partial {{{\dot{q}}}_{l}}}\frac{\partial {{q}_{l}}}{\partial {{Q}_{k}}} \right)}</math> | |||
wegen | |||
<math>\begin{align} | |||
& {{f}_{k}}({{Q}_{i}},t)={{q}_{k}} \\ | & {{f}_{k}}({{Q}_{i}},t)={{q}_{k}} \\ | ||
& \sum\limits_{i}{{}}\frac{\partial {{f}_{k}}}{\partial {{Q}_{i}}}{{{\dot{Q}}}_{i}}+\frac{\partial {{f}_{k}}}{\partial t}={{{\dot{q}}}_{k}} \\ | & \sum\limits_{i}{{}}\frac{\partial {{f}_{k}}}{\partial {{Q}_{i}}}{{{\dot{Q}}}_{i}}+\frac{\partial {{f}_{k}}}{\partial t}={{{\dot{q}}}_{k}} \\ | ||
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<math>\begin{align} | |||
& \frac{d}{dt}\frac{\partial \tilde{L}}{\partial {{{\dot{Q}}}_{k}}}=\sum\limits_{l=1}^{f}{\left\{ \left[ \frac{d}{dt}\left( \frac{\partial L}{\partial {{{\dot{q}}}_{l}}} \right) \right]\frac{\partial {{q}_{l}}}{\partial {{Q}_{k}}}+\frac{\partial L}{\partial {{{\dot{q}}}_{l}}}\frac{d}{dt}\left( \frac{\partial {{q}_{l}}}{\partial {{Q}_{k}}} \right) \right\}} \\ | & \frac{d}{dt}\frac{\partial \tilde{L}}{\partial {{{\dot{Q}}}_{k}}}=\sum\limits_{l=1}^{f}{\left\{ \left[ \frac{d}{dt}\left( \frac{\partial L}{\partial {{{\dot{q}}}_{l}}} \right) \right]\frac{\partial {{q}_{l}}}{\partial {{Q}_{k}}}+\frac{\partial L}{\partial {{{\dot{q}}}_{l}}}\frac{d}{dt}\left( \frac{\partial {{q}_{l}}}{\partial {{Q}_{k}}} \right) \right\}} \\ | ||
& =\sum\limits_{l=1}^{f}{\left\{ \left[ \frac{d}{dt}\left( \frac{\partial L}{\partial {{{\dot{q}}}_{l}}} \right) \right]\frac{\partial {{q}_{l}}}{\partial {{Q}_{k}}}+\frac{\partial L}{\partial {{{\dot{q}}}_{l}}}\left( \frac{\partial {{{\dot{q}}}_{l}}}{\partial {{Q}_{k}}} \right) \right\}} \\ | & =\sum\limits_{l=1}^{f}{\left\{ \left[ \frac{d}{dt}\left( \frac{\partial L}{\partial {{{\dot{q}}}_{l}}} \right) \right]\frac{\partial {{q}_{l}}}{\partial {{Q}_{k}}}+\frac{\partial L}{\partial {{{\dot{q}}}_{l}}}\left( \frac{\partial {{{\dot{q}}}_{l}}}{\partial {{Q}_{k}}} \right) \right\}} \\ | ||
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<math>\frac{\partial \tilde{L}}{\partial {{Q}_{k}}}=\sum\limits_{l=1}^{f}{\left( \frac{\partial L}{\partial {{q}_{l}}}\frac{\partial {{q}_{l}}}{\partial {{Q}_{k}}}+\frac{\partial L}{\partial {{{\dot{q}}}_{l}}}\left( \frac{\partial {{{\dot{q}}}_{l}}}{\partial {{Q}_{k}}} \right) \right)}</math> | |||
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<math>\begin{align} | |||
& \frac{d}{dt}\frac{\partial \tilde{L}}{\partial {{{\dot{Q}}}_{k}}}-\frac{\partial \tilde{L}}{\partial {{Q}_{k}}}=\sum\limits_{l=1}^{f}{\left\{ \left[ \frac{d}{dt}\left( \frac{\partial L}{\partial {{{\dot{q}}}_{l}}} \right) \right]\frac{\partial {{q}_{l}}}{\partial {{Q}_{k}}}+\frac{\partial L}{\partial {{{\dot{q}}}_{l}}}\left( \frac{\partial {{{\dot{q}}}_{l}}}{\partial {{Q}_{k}}} \right)-\left( \frac{\partial L}{\partial {{q}_{l}}}\frac{\partial {{q}_{l}}}{\partial {{Q}_{k}}}+\frac{\partial L}{\partial {{{\dot{q}}}_{l}}}\left( \frac{\partial {{{\dot{q}}}_{l}}}{\partial {{Q}_{k}}} \right) \right) \right\}} \\ | & \frac{d}{dt}\frac{\partial \tilde{L}}{\partial {{{\dot{Q}}}_{k}}}-\frac{\partial \tilde{L}}{\partial {{Q}_{k}}}=\sum\limits_{l=1}^{f}{\left\{ \left[ \frac{d}{dt}\left( \frac{\partial L}{\partial {{{\dot{q}}}_{l}}} \right) \right]\frac{\partial {{q}_{l}}}{\partial {{Q}_{k}}}+\frac{\partial L}{\partial {{{\dot{q}}}_{l}}}\left( \frac{\partial {{{\dot{q}}}_{l}}}{\partial {{Q}_{k}}} \right)-\left( \frac{\partial L}{\partial {{q}_{l}}}\frac{\partial {{q}_{l}}}{\partial {{Q}_{k}}}+\frac{\partial L}{\partial {{{\dot{q}}}_{l}}}\left( \frac{\partial {{{\dot{q}}}_{l}}}{\partial {{Q}_{k}}} \right) \right) \right\}} \\ | ||
& =\sum\limits_{l=1}^{f}{\left\{ \left[ \frac{d}{dt}\left( \frac{\partial L}{\partial {{{\dot{q}}}_{l}}} \right) \right]\frac{\partial {{q}_{l}}}{\partial {{Q}_{k}}}-\left( \frac{\partial L}{\partial {{q}_{l}}}\frac{\partial {{q}_{l}}}{\partial {{Q}_{k}}} \right) \right\}}=\sum\limits_{l=1}^{f}{\frac{\partial {{q}_{l}}}{\partial {{Q}_{k}}}\left\{ \left[ \frac{d}{dt}\left( \frac{\partial L}{\partial {{{\dot{q}}}_{l}}} \right) \right]-\left( \frac{\partial L}{\partial {{q}_{l}}} \right) \right\}} \\ | & =\sum\limits_{l=1}^{f}{\left\{ \left[ \frac{d}{dt}\left( \frac{\partial L}{\partial {{{\dot{q}}}_{l}}} \right) \right]\frac{\partial {{q}_{l}}}{\partial {{Q}_{k}}}-\left( \frac{\partial L}{\partial {{q}_{l}}}\frac{\partial {{q}_{l}}}{\partial {{Q}_{k}}} \right) \right\}}=\sum\limits_{l=1}^{f}{\frac{\partial {{q}_{l}}}{\partial {{Q}_{k}}}\left\{ \left[ \frac{d}{dt}\left( \frac{\partial L}{\partial {{{\dot{q}}}_{l}}} \right) \right]-\left( \frac{\partial L}{\partial {{q}_{l}}} \right) \right\}} \\ | ||
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<math>\frac{\partial {{q}_{l}}}{\partial {{Q}_{k}}}</math> | |||
die Transformationsmatrix, die nichtsingulär sein muss, also | die Transformationsmatrix, die nichtsingulär sein muss, also | ||
<math>\det \frac{\partial {{q}_{l}}}{\partial {{Q}_{k}}}\ne 0</math> | |||
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Sei | Sei | ||
<math>F:\left\{ {{q}_{k}} \right\}\to \left\{ {{Q}_{k}} \right\}</math> | |||
ein C²- Diffeomorphismus, | ein C²- Diffeomorphismus, | ||
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<math>F,{{F}^{-1}}</math> | |||
beide zweimal stetig differenzierbar. | beide zweimal stetig differenzierbar. | ||
Nur dann ist | Nur dann ist | ||
<math>\left\{ {{Q}_{k}}(t) \right\}</math> | |||
Lösung der Lagrangegleichung zur transformierten Lagrangefunktion. | Lösung der Lagrangegleichung zur transformierten Lagrangefunktion. | ||
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<math>\begin{align} | |||
& {{Q}_{i}}={{F}_{i}}({{q}_{1}},...{{q}_{f}},t) \\ | & {{Q}_{i}}={{F}_{i}}({{q}_{1}},...{{q}_{f}},t) \\ | ||
& {{q}_{k}}={{f}_{k}}({{Q}_{1}},...,{{Q}_{f}},t)\quad mit\quad \det \frac{\partial {{f}_{k}}}{\partial {{Q}_{i}}}\ne 0 \\ | & {{q}_{k}}={{f}_{k}}({{Q}_{1}},...,{{Q}_{f}},t)\quad mit\quad \det \frac{\partial {{f}_{k}}}{\partial {{Q}_{i}}}\ne 0 \\ | ||
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<math>\frac{d}{dt}\frac{\partial \tilde{L}}{\partial {{{\dot{Q}}}_{k}}}-\frac{\partial \tilde{L}}{\partial {{Q}_{k}}}</math> | |||
ist kovariant unter diffeomorphen Transformationen der generalisierten Koordinaten | ist kovariant unter diffeomorphen Transformationen der generalisierten Koordinaten | ||
Also gibt es auch unendlich viele äquivalente Sätze generalisierter Koordinaten. | Also gibt es auch unendlich viele äquivalente Sätze generalisierter Koordinaten. |