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Display information for equation id:math.1989.32 on revision:1989
* Page found: Hamilton-Jacobische Differenzialgleichung (eq math.1989.32)
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Hash: 32cbae6f774239257449a2d8767c7af4
TeX (original user input):
\begin{align}
& Q=\left( \frac{\partial S(q,P,t)}{\partial \alpha } \right)=-t+\frac{1}{\omega }\int{dq}{{\left( \frac{2\alpha }{m{{\omega }^{2}}}-{{q}^{2}} \right)}^{-\frac{1}{2}}}=\beta \\
& Q=\beta =-t+\frac{1}{\omega }\arcsin \left( q\sqrt{\frac{m{{\omega }^{2}}}{2\left| \alpha \right|}} \right) \\
& \Rightarrow q=\frac{1}{\omega }\sqrt{\frac{2\alpha }{m}}\sin \left( \omega (t+\beta ) \right) \\
\end{align}
TeX (checked):
{\begin{aligned}&Q=\left({\frac {\partial S(q,P,t)}{\partial \alpha }}\right)=-t+{\frac {1}{\omega }}\int {dq}{{\left({\frac {2\alpha }{m{{\omega }^{2}}}}-{{q}^{2}}\right)}^{-{\frac {1}{2}}}}=\beta \\&Q=\beta =-t+{\frac {1}{\omega }}\arcsin \left(q{\sqrt {\frac {m{{\omega }^{2}}}{2\left|\alpha \right|}}}\right)\\&\Rightarrow q={\frac {1}{\omega }}{\sqrt {\frac {2\alpha }{m}}}\sin \left(\omega (t+\beta )\right)\\\end{aligned}}
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