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Display information for equation id:math.1909.13 on revision:1909
* Page found: Forminvarianz der Lagrangegleichungen (eq math.1909.13)
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TeX (original user input):
\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\}} \\
& =\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\}} \\
\end{align}
TeX (checked):
{\begin{aligned}&{\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\}}\\\end{aligned}}
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data-mjx-texclass="ORD"><mi>k</mi></mrow></msub></mrow></mrow></mfrac></mrow><mo>−</mo><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>∂</mi><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mi>L</mi><mo>~</mo></mover></mrow></mrow></mrow></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>∂</mi><msub><mi>Q</mi><mrow data-mjx-texclass="ORD"><mi>k</mi></mrow></msub></mrow></mrow></mfrac></mrow><mo>=</mo><munderover><mo form="prefix" texclass="OP">∑</mo><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>l</mi><mo>=</mo><mn>1</mn></mrow></mrow><mrow data-mjx-texclass="ORD"><mi>f</mi></mrow></munderover><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">{</mo><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">[</mo><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mi>d</mi></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>d</mi><mi>t</mi></mrow></mrow></mfrac></mrow><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>∂</mi><mi>L</mi></mrow></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>∂</mi><msub><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mi>q</mi><mo>˙</mo></mover></mrow></mrow><mrow data-mjx-texclass="ORD"><mi>l</mi></mrow></msub></mrow></mrow></mfrac></mrow><mo data-mjx-texclass="CLOSE">)</mo></mrow><mo data-mjx-texclass="CLOSE">]</mo></mrow><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>∂</mi><msub><mi>q</mi><mrow data-mjx-texclass="ORD"><mi>l</mi></mrow></msub></mrow></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>∂</mi><msub><mi>Q</mi><mrow data-mjx-texclass="ORD"><mi>k</mi></mrow></msub></mrow></mrow></mfrac></mrow><mo>+</mo><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>∂</mi><mi>L</mi></mrow></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>∂</mi><msub><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mi>q</mi><mo>˙</mo></mover></mrow></mrow><mrow data-mjx-texclass="ORD"><mi>l</mi></mrow></msub></mrow></mrow></mfrac></mrow><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>∂</mi><msub><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mi>q</mi><mo>˙</mo></mover></mrow></mrow><mrow data-mjx-texclass="ORD"><mi>l</mi></mrow></msub></mrow></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>∂</mi><msub><mi>Q</mi><mrow data-mjx-texclass="ORD"><mi>k</mi></mrow></msub></mrow></mrow></mfrac></mrow><mo data-mjx-texclass="CLOSE">)</mo></mrow><mo>−</mo><mrow 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data-mjx-texclass="ORD"><mover><mi>q</mi><mo>˙</mo></mover></mrow></mrow><mrow data-mjx-texclass="ORD"><mi>l</mi></mrow></msub></mrow></mrow></mfrac></mrow><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>∂</mi><msub><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mi>q</mi><mo>˙</mo></mover></mrow></mrow><mrow data-mjx-texclass="ORD"><mi>l</mi></mrow></msub></mrow></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>∂</mi><msub><mi>Q</mi><mrow data-mjx-texclass="ORD"><mi>k</mi></mrow></msub></mrow></mrow></mfrac></mrow><mo data-mjx-texclass="CLOSE">)</mo></mrow><mo data-mjx-texclass="CLOSE">)</mo></mrow><mo data-mjx-texclass="CLOSE">}</mo></mrow></mtd></mtr><mtr><mtd></mtd><mtd><mo>=</mo><munderover><mo form="prefix" texclass="OP">∑</mo><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>l</mi><mo>=</mo><mn>1</mn></mrow></mrow><mrow data-mjx-texclass="ORD"><mi>f</mi></mrow></munderover><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">{</mo><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">[</mo><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mi>d</mi></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>d</mi><mi>t</mi></mrow></mrow></mfrac></mrow><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>∂</mi><mi>L</mi></mrow></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>∂</mi><msub><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mi>q</mi><mo>˙</mo></mover></mrow></mrow><mrow data-mjx-texclass="ORD"><mi>l</mi></mrow></msub></mrow></mrow></mfrac></mrow><mo data-mjx-texclass="CLOSE">)</mo></mrow><mo data-mjx-texclass="CLOSE">]</mo></mrow><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>∂</mi><msub><mi>q</mi><mrow data-mjx-texclass="ORD"><mi>l</mi></mrow></msub></mrow></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>∂</mi><msub><mi>Q</mi><mrow data-mjx-texclass="ORD"><mi>k</mi></mrow></msub></mrow></mrow></mfrac></mrow><mo>−</mo><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>∂</mi><mi>L</mi></mrow></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>∂</mi><msub><mi>q</mi><mrow data-mjx-texclass="ORD"><mi>l</mi></mrow></msub></mrow></mrow></mfrac></mrow><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>∂</mi><msub><mi>q</mi><mrow data-mjx-texclass="ORD"><mi>l</mi></mrow></msub></mrow></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>∂</mi><msub><mi>Q</mi><mrow data-mjx-texclass="ORD"><mi>k</mi></mrow></msub></mrow></mrow></mfrac></mrow><mo data-mjx-texclass="CLOSE">)</mo></mrow><mo data-mjx-texclass="CLOSE">}</mo></mrow><mo>=</mo><munderover><mo form="prefix" texclass="OP">∑</mo><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>l</mi><mo>=</mo><mn>1</mn></mrow></mrow><mrow data-mjx-texclass="ORD"><mi>f</mi></mrow></munderover><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>∂</mi><msub><mi>q</mi><mrow data-mjx-texclass="ORD"><mi>l</mi></mrow></msub></mrow></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>∂</mi><msub><mi>Q</mi><mrow data-mjx-texclass="ORD"><mi>k</mi></mrow></msub></mrow></mrow></mfrac></mrow><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">{</mo><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">[</mo><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mi>d</mi></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>d</mi><mi>t</mi></mrow></mrow></mfrac></mrow><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>∂</mi><mi>L</mi></mrow></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>∂</mi><msub><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mi>q</mi><mo>˙</mo></mover></mrow></mrow><mrow data-mjx-texclass="ORD"><mi>l</mi></mrow></msub></mrow></mrow></mfrac></mrow><mo data-mjx-texclass="CLOSE">)</mo></mrow><mo data-mjx-texclass="CLOSE">]</mo></mrow><mo>−</mo><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>∂</mi><mi>L</mi></mrow></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>∂</mi><msub><mi>q</mi><mrow data-mjx-texclass="ORD"><mi>l</mi></mrow></msub></mrow></mrow></mfrac></mrow><mo data-mjx-texclass="CLOSE">)</mo></mrow><mo data-mjx-texclass="CLOSE">}</mo></mrow></mrow></mtd></mtr><mtr><mtd></mtd></mtr></mtable></mrow></mstyle></mrow></math>
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