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Display information for equation id:math.1340.42 on revision:1340

* Page found: Die Hamilton-Jacobi-Theorie (eq math.1340.42)

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\begin{align}
  & {{p}_{j}}=\frac{\partial W}{\partial {{q}_{j}}} \\ 
 & {{Q}_{j}}=\frac{\partial W}{\partial {{P}_{j}}} \\ 
 & \bar{H}=H=E \\ 
 & \Rightarrow {{{\dot{Q}}}_{j}}=\frac{\partial \bar{H}}{\partial {{P}_{j}}}=\frac{\partial E}{\partial {{\alpha }_{j}}}={{\omega }_{j}}\Rightarrow {{Q}_{j}}={{\omega }_{j}}t+{{\beta }_{j}}=\frac{\partial W}{\partial {{P}_{j}}} \\ 
\end{align}

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MathML (26.594 KB / 2.947 KB) :

p j = ∂ ⁡ W ∂ ⁡ q j Q j = ∂ ⁡ W ∂ ⁡ P j H ¯ = H = E ⇒ Q ˙ j = ∂ ⁡ H ¯ ∂ ⁡ P j = ∂ ⁡ E ∂ ⁡ α j = ω j ⇒ Q j = ω j ⁢ t + β j = ∂ ⁡ W ∂ ⁡ P j missing-subexpression subscript 𝑝 𝑗 𝑊 subscript 𝑞 𝑗 missing-subexpression subscript 𝑄 𝑗 𝑊 subscript 𝑃 𝑗 missing-subexpression ¯ 𝐻 𝐻 𝐸 missing-subexpression ⇒ absent subscript ˙ 𝑄 𝑗 ¯ 𝐻 subscript 𝑃 𝑗 𝐸 subscript 𝛼 𝑗 subscript 𝜔 𝑗 ⇒ subscript 𝑄 𝑗 subscript 𝜔 𝑗 𝑡 subscript 𝛽 𝑗 𝑊 subscript 𝑃 𝑗 {\displaystyle{\displaystyle\begin{aligned} &\displaystyle{{p}_{j}}=\frac{% \partial W}{\partial{{q}_{j}}}\\ &\displaystyle{{Q}_{j}}=\frac{\partial W}{\partial{{P}_{j}}}\\ &\displaystyle\bar{H}=H=E\\ &\displaystyle\Rightarrow{{{\dot{Q}}}_{j}}=\frac{\partial\bar{H}}{\partial{{P}% _{j}}}=\frac{\partial E}{\partial{{\alpha}_{j}}}={{\omega}_{j}}\Rightarrow{{Q}% _{j}}={{\omega}_{j}}t+{{\beta}_{j}}=\frac{\partial W}{\partial{{P}_{j}}}\\ \end{aligned}}}

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pj=∂W∂qjQj=∂W∂PjH¯=H=E⇒Q˙j=∂H¯∂Pj=∂E∂αj=ωj⇒Qj=ωjt+βj=∂W∂Pj

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