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Display information for equation id:math.1968.22 on revision:1968

* Page found: Der Satz von Liouville (eq math.1968.22)

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Hash: 2355113bdf7c146ff91c32cf543baab6

TeX (original user input):

\begin{align}
  & {{x}^{i}}(t)=\sum\limits_{k=1}^{2}{{{P}_{ik}}{{x}_{0}}^{k}} \\ 
 & P=\left( \begin{matrix}
   \cos {{\omega }_{0}}(t-{{t}_{0}}) & \frac{1}{{{\omega }_{0}}}\sin {{\omega }_{0}}(t-{{t}_{0}})  \\
   -{{\omega }_{0}}\sin {{\omega }_{0}}(t-{{t}_{0}}) & \cos {{\omega }_{0}}(t-{{t}_{0}})  \\
\end{matrix} \right) \\ 
 & \det P={{\cos }^{2}}{{\omega }_{0}}(t-{{t}_{0}})+{{\sin }^{2}}{{\omega }_{0}}(t-{{t}_{0}})=1 \\ 
 & d{{x}^{1}}d{{x}^{2}}=(\det P)d{{x}_{0}}^{1}d{{x}_{0}}^{2}=d{{x}_{0}}^{1}d{{x}_{0}}^{2} \\ 
\end{align}

TeX (checked):

{\begin{aligned}&{{x}^{i}}(t)=\sum \limits _{k=1}^{2}{{{P}_{ik}}{{x}_{0}}^{k}}\\&P=\left({\begin{matrix}\cos {{\omega }_{0}}(t-{{t}_{0}})&{\frac {1}{{\omega }_{0}}}\sin {{\omega }_{0}}(t-{{t}_{0}})\\-{{\omega }_{0}}\sin {{\omega }_{0}}(t-{{t}_{0}})&\cos {{\omega }_{0}}(t-{{t}_{0}})\\\end{matrix}}\right)\\&\det P={{\cos }^{2}}{{\omega }_{0}}(t-{{t}_{0}})+{{\sin }^{2}}{{\omega }_{0}}(t-{{t}_{0}})=1\\&d{{x}^{1}}d{{x}^{2}}=(\det P)d{{x}_{0}}^{1}d{{x}_{0}}^{2}=d{{x}_{0}}^{1}d{{x}_{0}}^{2}\\\end{aligned}}

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\begin{matrix} x^{i} (t) = \sum_{k = 1}^{2} P_{i k} {x_{0}}^{k} \\ P = (\begin{matrix} \cos ω_{0} (t - t_{0}) & \frac{1}{ω_{0}} \sin ω_{0} (t - t_{0}) \\ - ω_{0} \sin ω_{0} (t - t_{0}) & \cos ω_{0} (t - t_{0}) \end{matrix}) \\ det P = \cos^{2} ω_{0} (t - t_{0}) + \sin^{2} ω_{0} (t - t_{0}) = 1 \\ d x^{1} d x^{2} = (det P) d {x_{0}}^{1} d {x_{0}}^{2} = d {x_{0}}^{1} d {x_{0}}^{2} \end{matrix}

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Identifiers

$x$

$i$

$t$

$k$

$P_{i k}$

$x_{0}$

$k$

$P$

$ω_{0}$

$t$

$t_{0}$

$ω_{0}$

$ω_{0}$

$t$

$t_{0}$

$ω_{0}$

$ω_{0}$

$t$

$t_{0}$

$ω_{0}$

$t$

$t_{0}$

$P$

$ω_{0}$

$t$

$t_{0}$

$ω_{0}$

$t$

$t_{0}$

$d$

$x$

$d$

$x$

$P$

$d$

$x_{0}$

$d$

$x_{0}$

$d$

$x_{0}$

$d$

$x_{0}$

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