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Display information for equation id:math.1905.10 on revision:1905
* Page found: Eichtransformation der Lagrangefunktion (eq math.1905.10)
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TeX (original user input):
\begin{align}
& 0=\frac{\partial L}{\partial {{q}_{k}}}-\frac{d}{dt}\frac{\partial L}{\partial {{{\dot{q}}}_{k}}}=m{{{\ddot{q}}}_{k}}+e\left( \frac{\partial }{\partial t}{{A}_{k}}+\left( \dot{\bar{q}}\cdot \nabla \right){{A}_{k}} \right)-e\left[ \frac{\partial }{\partial {{q}_{k}}}\left( \dot{\bar{q}}\cdot \bar{A} \right)-\frac{\partial }{\partial {{q}_{k}}}\Phi \right] \\
& =m{{{\ddot{q}}}_{k}}+e\left( \frac{\partial }{\partial t}{{A}_{k}}+\frac{\partial }{\partial {{q}_{k}}}\Phi \right)+e\left[ -\frac{\partial }{\partial {{q}_{k}}}\left( \dot{\bar{q}}\cdot \bar{A} \right)+\left( \dot{\bar{q}}\cdot \nabla \right){{A}_{k}} \right] \\
& =m{{{\ddot{q}}}_{k}}-e{{E}_{k}}-{{\left[ e\dot{\bar{q}}\times \left( \nabla \times \bar{A} \right) \right]}_{k}} \\
& =m{{{\ddot{q}}}_{k}}-e{{E}_{k}}-{{\left[ e\dot{\bar{q}}\times \bar{B} \right]}_{k}} \\
\end{align}
TeX (checked):
{\begin{aligned}&0={\frac {\partial L}{\partial {{q}_{k}}}}-{\frac {d}{dt}}{\frac {\partial L}{\partial {{\dot {q}}_{k}}}}=m{{\ddot {q}}_{k}}+e\left({\frac {\partial }{\partial t}}{{A}_{k}}+\left({\dot {\bar {q}}}\cdot \nabla \right){{A}_{k}}\right)-e\left[{\frac {\partial }{\partial {{q}_{k}}}}\left({\dot {\bar {q}}}\cdot {\bar {A}}\right)-{\frac {\partial }{\partial {{q}_{k}}}}\Phi \right]\\&=m{{\ddot {q}}_{k}}+e\left({\frac {\partial }{\partial t}}{{A}_{k}}+{\frac {\partial }{\partial {{q}_{k}}}}\Phi \right)+e\left[-{\frac {\partial }{\partial {{q}_{k}}}}\left({\dot {\bar {q}}}\cdot {\bar {A}}\right)+\left({\dot {\bar {q}}}\cdot \nabla \right){{A}_{k}}\right]\\&=m{{\ddot {q}}_{k}}-e{{E}_{k}}-{{\left[e{\dot {\bar {q}}}\times \left(\nabla \times {\bar {A}}\right)\right]}_{k}}\\&=m{{\ddot {q}}_{k}}-e{{E}_{k}}-{{\left[e{\dot {\bar {q}}}\times {\bar {B}}\right]}_{k}}\\\end{aligned}}
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<math class="mwe-math-element" xmlns="http://www.w3.org/1998/Math/MathML"><mrow data-mjx-texclass="ORD"><mstyle displaystyle="true" scriptlevel="0"><mrow data-mjx-texclass="ORD"><mtable columnalign="right left right left right left right left right left right left" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true" rowspacing="3pt"><mtr><mtd></mtd><mtd><mn>0</mn><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><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"><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="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>k</mi></mrow></msub></mrow></mrow></mfrac></mrow><mo>=</mo><mi>m</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>k</mi></mrow></msub><mo>+</mo><mi>e</mi><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mi>∂</mi></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>∂</mi><mi>t</mi></mrow></mrow></mfrac></mrow><msub><mi>A</mi><mrow data-mjx-texclass="ORD"><mi>k</mi></mrow></msub><mo>+</mo><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mrow data-mjx-texclass="ORD"><mrow 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data-mjx-texclass="ORD"><mover><mi>q</mi><mo>¯</mo></mover></mrow></mrow><mo>˙</mo></mover></mrow></mrow><mo>⋅</mo><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mi>A</mi><mo>¯</mo></mover></mrow></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"><mrow data-mjx-texclass="ORD"><mover><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mi>q</mi><mo>¯</mo></mover></mrow></mrow><mo>˙</mo></mover></mrow></mrow><mo>⋅</mo><mi mathvariant="normal">∇</mi><mo data-mjx-texclass="CLOSE">)</mo></mrow><msub><mi>A</mi><mrow data-mjx-texclass="ORD"><mi>k</mi></mrow></msub><mo data-mjx-texclass="CLOSE">]</mo></mrow></mtd></mtr><mtr><mtd></mtd><mtd><mo>=</mo><mi>m</mi><msub><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mi>q</mi><mo>¨</mo></mover></mrow></mrow><mrow 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