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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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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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Q=(∂S(q,P,t)∂α)=−t+1ω∫dq(2αmω2−q2)−12=βQ=β=−t+1ωarcsin⁡(qmω22|α|)⇒q=1ω2αmsin⁡(ω(t+β))
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