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Display information for equation id:math.1966.13 on revision:1966

* Page found: Der Satz von Liouville (eq math.1966.13)

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Der Satz von Liouville Eq: math.1970.13

Hash: 0749d75494648eb3e9dbae7b90aa134a

TeX (original user input):

\begin{align}
  & \det \left( D{{\Phi }_{t,{{t}_{0}}}} \right)=\left| \frac{\partial {{\Phi }^{i}}_{t,{{t}_{0}}}({{{\bar{x}}}_{0}})}{\partial {{x}_{0}}^{k}} \right|=1+(t-{{t}_{0}})\sum\limits_{i=1}^{2f}{{}}\frac{\partial {{{\bar{F}}}^{i}}({{{\bar{x}}}_{0}},t)}{\partial {{x}_{0}}^{i}}(t-{{t}_{0}})+O({{(t-{{t}_{0}})}^{2}}) \\ 
 & \sum\limits_{i=1}^{2f}{{}}\frac{\partial {{{\bar{F}}}^{i}}({{{\bar{x}}}_{0}},t)}{\partial {{x}_{0}}^{i}}=div\bar{F}=\frac{\partial }{\partial q}\frac{\partial H}{\partial p}-\frac{\partial }{\partial p}\frac{\partial H}{\partial q}=0 \\ 
\end{align}

TeX (checked):

{\begin{aligned}&\det \left(D{{\Phi }_{t,{{t}_{0}}}}\right)=\left|{\frac {\partial {{\Phi }^{i}}_{t,{{t}_{0}}}({{\bar {x}}_{0}})}{\partial {{x}_{0}}^{k}}}\right|=1+(t-{{t}_{0}})\sum \limits _{i=1}^{2f}{}{\frac {\partial {{\bar {F}}^{i}}({{\bar {x}}_{0}},t)}{\partial {{x}_{0}}^{i}}}(t-{{t}_{0}})+O({{(t-{{t}_{0}})}^{2}})\\&\sum \limits _{i=1}^{2f}{}{\frac {\partial {{\bar {F}}^{i}}({{\bar {x}}_{0}},t)}{\partial {{x}_{0}}^{i}}}=div{\bar {F}}={\frac {\partial }{\partial q}}{\frac {\partial H}{\partial p}}-{\frac {\partial }{\partial p}}{\frac {\partial H}{\partial q}}=0\\\end{aligned}}

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\begin{matrix} det (D Φ_{t, t_{0}}) = | \frac{\partial {Φ^{i}}_{t, t_{0}} ({\bar{x}}_{0})}{\partial {x_{0}}^{k}} | = 1 + (t - t_{0}) \sum_{i = 1}^{2 f} \frac{\partial {\bar{F}}^{i} ({\bar{x}}_{0}, t)}{\partial {x_{0}}^{i}} (t - t_{0}) + O ({(t - t_{0})}^{2}) \\ \sum_{i = 1}^{2 f} \frac{\partial {\bar{F}}^{i} ({\bar{x}}_{0}, t)}{\partial {x_{0}}^{i}} = d i v \bar{F} = \frac{\partial}{\partial q} \frac{\partial H}{\partial p} - \frac{\partial}{\partial p} \frac{\partial H}{\partial q} = 0 \end{matrix}

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Identifiers

$D$

$Φ_{t, t_{0}}$

$Φ$

$i$

$t$

$t_{0}$

${\bar{x}}_{0}$

$x_{0}$

$k$

$t$

$t_{0}$

$i$

$f$

$\bar{F}$

$i$

${\bar{x}}_{0}$

$t$

$x_{0}$

$i$

$t$

$t_{0}$

$O$

$t$

$t_{0}$

$i$

$f$

$\bar{F}$

$i$

${\bar{x}}_{0}$

$t$

$x_{0}$

$i$

$d$

$i$

$v$

$\bar{F}$

$q$

$H$

$p$

$p$

$H$

$q$

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