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Display information for equation id:math.1325.64 on revision:1325

* Page found: Der Hamiltonsche kanonische Formalismus (eq math.1325.64)

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TeX (original user input):

\begin{align}
  & {{p}_{k}}=\frac{\partial L(\bar{q},\dot{\bar{q}},t)}{\partial {{{\dot{q}}}_{k}}}=m{{{\dot{q}}}_{\acute{\ }k}}+e{{A}_{k}}(\bar{q},t) \\ 
 & \Rightarrow {{{\dot{q}}}_{k}}=\frac{1}{m}\left( {{p}_{k}}-e{{A}_{k}} \right) \\ 
 & H=\sum\limits_{k=1}^{3}{{{p}_{k}}}{{{\dot{q}}}_{k}}-L=\sum\limits_{k=1}^{3}{{{p}_{k}}}\frac{1}{m}\left( {{p}_{k}}-e{{A}_{k}} \right)-\frac{1}{2m}\sum\limits_{k=1}^{3}{{}}{{\left( {{p}_{k}}-e{{A}_{k}} \right)}^{2}}-\sum\limits_{k=1}^{3}{{}}\frac{e}{m}\left( {{p}_{k}}-e{{A}_{k}} \right){{A}_{k}}+e\Phi  \\ 
 & H\left( \bar{q},\bar{p},t \right)=\frac{1}{2m}{{\left( {{{\bar{p}}}_{{}}}-e\bar{A}{{(\bar{q},t)}_{{}}} \right)}^{2}}+e\Phi (\bar{q},t) \\ 
\end{align}

TeX (checked):

{\begin{aligned}&{{p}_{k}}={\frac {\partial L({\bar {q}},{\dot {\bar {q}}},t)}{\partial {{\dot {q}}_{k}}}}=m{{\dot {q}}_{{\acute {\ }}k}}+e{{A}_{k}}({\bar {q}},t)\\&\Rightarrow {{\dot {q}}_{k}}={\frac {1}{m}}\left({{p}_{k}}-e{{A}_{k}}\right)\\&H=\sum \limits _{k=1}^{3}{{p}_{k}}{{\dot {q}}_{k}}-L=\sum \limits _{k=1}^{3}{{p}_{k}}{\frac {1}{m}}\left({{p}_{k}}-e{{A}_{k}}\right)-{\frac {1}{2m}}\sum \limits _{k=1}^{3}{}{{\left({{p}_{k}}-e{{A}_{k}}\right)}^{2}}-\sum \limits _{k=1}^{3}{}{\frac {e}{m}}\left({{p}_{k}}-e{{A}_{k}}\right){{A}_{k}}+e\Phi \\&H\left({\bar {q}},{\bar {p}},t\right)={\frac {1}{2m}}{{\left({{\bar {p}}_{}}-e{\bar {A}}{{({\bar {q}},t)}_{}}\right)}^{2}}+e\Phi ({\bar {q}},t)\\\end{aligned}}

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\begin{aligned} p_{k} = \frac{\partial L (\bar{q}, \dot{\bar{q}}, t)}{\partial {\dot{q}}_{k}} = m {\dot{q}}_{\overset{´}{} k} + e A_{k} (\bar{q}, t) \\ \Rightarrow {\dot{q}}_{k} = \frac{1}{m} (p_{k} - e A_{k}) \\ H = \sum_{k = 1}^{3} p_{k} {\dot{q}}_{k} - L = \sum_{k = 1}^{3} p_{k} \frac{1}{m} (p_{k} - e A_{k}) - \frac{1}{2 m} \sum_{k = 1}^{3} {(p_{k} - e A_{k})}^{2} - \sum_{k = 1}^{3} \frac{e}{m} (p_{k} - e A_{k}) A_{k} + e Φ \\ H (\bar{q}, \bar{p}, t) = \frac{1}{2 m} {({\bar{p}}_{} - e \bar{A} {(\bar{q}, t)}_{})}^{2} + e Φ (\bar{q}, t) \end{aligned}

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Identifiers

$p_{k}$

$L$

$\bar{q}$

$\dot{\bar{q}}$

$t$

${\dot{q}}_{k}$

$m$

$\dot{q}$

$\overset{´}{}$

$k$

$e$

$A_{k}$

$\bar{q}$

$t$

${\dot{q}}_{k}$

$m$

$p_{k}$

$e$

$A_{k}$

$H$

$k$

$p_{k}$

${\dot{q}}_{k}$

$L$

$k$

$p_{k}$

$m$

$p_{k}$

$e$

$A_{k}$

$m$

$k$

$p_{k}$

$e$

$A_{k}$

$k$

$e$

$m$

$p_{k}$

$e$

$A_{k}$

$A_{k}$

$e$

$Φ$

$H$

$\bar{q}$

$\bar{p}$

$t$

$m$

$\bar{p}$

$e$

$\bar{A}$

$\bar{q}$

$t$

$e$

$Φ$

$\bar{q}$

$t$

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