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Display information for equation id:math.1326.53 on revision:1326

* Page found: Der Hamiltonsche kanonische Formalismus (eq math.1326.53)

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Die Hamiltonschen Gleichungen Eq: math.1948.27

Hash: 46c890c0ec6399e5452fb9f954276d87

TeX (original user input):

\begin{align}
  & {{{\dot{p}}}_{k}}=-\frac{\partial H}{\partial {{q}_{k}}} \\ 
 & \frac{\partial H}{\partial {{p}_{k}}}={{{\dot{q}}}_{k}}\quad k=1,...,f \\ 
 & \dot{r}=\frac{\partial H}{\partial {{p}_{r}}}=\frac{{{p}_{r}}}{m},\dot{\phi }=\frac{\partial H}{\partial {{p}_{\phi }}}=\frac{{{p}_{\phi }}}{m{{r}^{2}}},\dot{z}=\frac{\partial H}{\partial {{p}_{z}}}=\frac{{{p}_{z}}}{m} \\ 
 & {{{\dot{p}}}_{r}}=-\frac{\partial H}{\partial r}=\frac{{{p}_{\phi }}^{2}}{m{{r}^{3}}}-\frac{\partial V}{\partial r},{{{\dot{p}}}_{\phi }}=-\frac{\partial H}{\partial \phi }=-\frac{\partial V}{\partial \phi },{{{\dot{p}}}_{z}}=-\frac{\partial H}{\partial z}=-\frac{\partial V}{\partial z} \\ 
\end{align}

TeX (checked):

{\begin{aligned}&{{\dot {p}}_{k}}=-{\frac {\partial H}{\partial {{q}_{k}}}}\\&{\frac {\partial H}{\partial {{p}_{k}}}}={{\dot {q}}_{k}}\quad k=1,...,f\\&{\dot {r}}={\frac {\partial H}{\partial {{p}_{r}}}}={\frac {{p}_{r}}{m}},{\dot {\phi }}={\frac {\partial H}{\partial {{p}_{\phi }}}}={\frac {{p}_{\phi }}{m{{r}^{2}}}},{\dot {z}}={\frac {\partial H}{\partial {{p}_{z}}}}={\frac {{p}_{z}}{m}}\\&{{\dot {p}}_{r}}=-{\frac {\partial H}{\partial r}}={\frac {{{p}_{\phi }}^{2}}{m{{r}^{3}}}}-{\frac {\partial V}{\partial r}},{{\dot {p}}_{\phi }}=-{\frac {\partial H}{\partial \phi }}=-{\frac {\partial V}{\partial \phi }},{{\dot {p}}_{z}}=-{\frac {\partial H}{\partial z}}=-{\frac {\partial V}{\partial z}}\\\end{aligned}}

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MathML (6.245 KB / 566 B) :

\begin{aligned} {\dot{p}}_{k} = - \frac{\partial H}{\partial q_{k}} \\ \frac{\partial H}{\partial p_{k}} = {\dot{q}}_{k} k = 1, . . ., f \\ \dot{r} = \frac{\partial H}{\partial p_{r}} = \frac{p_{r}}{m}, \dot{ϕ} = \frac{\partial H}{\partial p_{ϕ}} = \frac{p_{ϕ}}{m r^{2}}, \dot{z} = \frac{\partial H}{\partial p_{z}} = \frac{p_{z}}{m} \\ {\dot{p}}_{r} = - \frac{\partial H}{\partial r} = \frac{{p_{ϕ}}^{2}}{m r^{3}} - \frac{\partial V}{\partial r}, {\dot{p}}_{ϕ} = - \frac{\partial H}{\partial ϕ} = - \frac{\partial V}{\partial ϕ}, {\dot{p}}_{z} = - \frac{\partial H}{\partial z} = - \frac{\partial V}{\partial z} \end{aligned}

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Translations to Computer Algebra Systems

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Identifiers

${\dot{p}}_{k}$

$H$

$q_{k}$

$H$

$p_{k}$

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

$k$

$f$

$\dot{r}$

$H$

$p_{r}$

$p_{r}$

$m$

$\dot{ϕ}$

$H$

$p_{ϕ}$

$p_{ϕ}$

$m$

$r$

$\dot{z}$

$H$

$p_{z}$

$p_{z}$

$m$

${\dot{p}}_{r}$

$H$

$r$

$p_{ϕ}$

$m$

$r$

$V$

$r$

${\dot{p}}_{ϕ}$

$H$

$ϕ$

$V$

$ϕ$

${\dot{p}}_{z}$

$H$

$z$

$V$

$z$

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