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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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MathML (33.905 KB / 3.979 KB) :

Q = ( ∂ ⁡ S ⁢ ( q , P , t ) ∂ ⁡ α ) = - t + 1 ω ⁢ ∫ 𝑑 q ⁢ ( 2 ⁢ α m ⁢ ω 2 - q 2 ) - 1 2 = β Q = β = - t + 1 ω ⁢ arcsin ⁡ ( q ⁢ m ⁢ ω 2 2 ⁢ | α | ) ⇒ q = 1 ω ⁢ 2 ⁢ α m ⁢ sin ⁡ ( ω ⁢ ( t + β ) ) missing-subexpression 𝑄 𝑆 𝑞 𝑃 𝑡 𝛼 𝑡 1 𝜔 differential-d 𝑞 superscript 2 𝛼 𝑚 superscript 𝜔 2 superscript 𝑞 2 1 2 𝛽 missing-subexpression 𝑄 𝛽 𝑡 1 𝜔 𝑞 𝑚 superscript 𝜔 2 2 𝛼 missing-subexpression ⇒ absent 𝑞 1 𝜔 2 𝛼 𝑚 𝜔 𝑡 𝛽 {\displaystyle{\displaystyle\begin{aligned} &\displaystyle 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\\ &\displaystyle Q=\beta=-t+\frac{1}{\omega}\arcsin\left(q\sqrt{\frac{m{{\omega}% ^{2}}}{2\left|\alpha\right|}}\right)\\ &\displaystyle\Rightarrow q=\frac{1}{\omega}\sqrt{\frac{2\alpha}{m}}\sin\left(% \omega(t+\beta)\right)\\ \end{aligned}}}

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MathML (3.563 KB / 574 B) :

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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