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Display information for equation id:math.1982.42 on revision:1982

* Page found: Hamilton-Jacobische Differenzialgleichung (eq math.1982.42)

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Hamilton-Jacobische Differenzialgleichung Eq: math.1989.42

Hash: 87506e00d04ad65e208dc193b6d41276

TeX (original user input):

\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}

TeX (checked):

{\begin{aligned}&{{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{aligned}}

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MathML (3.052 KB / 406 B) :

\begin{matrix} 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 α_{j}} = ω_{j} \Rightarrow Q_{j} = ω_{j} t + β_{j} = \frac{\partial W}{\partial P_{j}} \end{matrix}

<math xmlns="http://www.w3.org/1998/Math/MathML" class="mwe-math-element mwe-math-element-inline"><mrow data-mjx-texclass="ORD"><mstyle displaystyle="true" scriptlevel="0"><mtable displaystyle="true"><mtr><mtd class="mwe-math-columnalign-r"></mtd><mtd class="mwe-math-columnalign-l"><msub><mi>p</mi><mrow data-mjx-texclass="ORD"><mi>j</mi></mrow></msub><mo stretchy="false">=</mo><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>∂</mi><mi>W</mi></mrow></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>∂</mi><msub><mi>q</mi><mrow data-mjx-texclass="ORD"><mi>j</mi></mrow></msub></mrow></mrow></mfrac></mrow></mtd></mtr><mtr><mtd class="mwe-math-columnalign-r"></mtd><mtd class="mwe-math-columnalign-l"><msub><mi>Q</mi><mrow data-mjx-texclass="ORD"><mi>j</mi></mrow></msub><mo stretchy="false">=</mo><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>∂</mi><mi>W</mi></mrow></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>∂</mi><msub><mi>P</mi><mrow data-mjx-texclass="ORD"><mi>j</mi></mrow></msub></mrow></mrow></mfrac></mrow></mtd></mtr><mtr><mtd class="mwe-math-columnalign-r"></mtd><mtd class="mwe-math-columnalign-l"><mover><mi>H</mi><mo>¯</mo></mover><mo stretchy="false">=</mo><mi>H</mi><mo stretchy="false">=</mo><mi>E</mi></mtd></mtr><mtr><mtd class="mwe-math-columnalign-r"></mtd><mtd class="mwe-math-columnalign-l"><mo stretchy="false">⇒</mo><msub><mover><mi>Q</mi><mo>˙</mo></mover><mrow data-mjx-texclass="ORD"><mi>j</mi></mrow></msub><mo stretchy="false">=</mo><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>∂</mi><mover><mi>H</mi><mo>¯</mo></mover></mrow></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>∂</mi><msub><mi>P</mi><mrow data-mjx-texclass="ORD"><mi>j</mi></mrow></msub></mrow></mrow></mfrac></mrow><mo stretchy="false">=</mo><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>∂</mi><mi>E</mi></mrow></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>∂</mi><msub><mi>α</mi><mrow data-mjx-texclass="ORD"><mi>j</mi></mrow></msub></mrow></mrow></mfrac></mrow><mo stretchy="false">=</mo><msub><mi>ω</mi><mrow data-mjx-texclass="ORD"><mi>j</mi></mrow></msub><mo stretchy="false">⇒</mo><msub><mi>Q</mi><mrow data-mjx-texclass="ORD"><mi>j</mi></mrow></msub><mo stretchy="false">=</mo><msub><mi>ω</mi><mrow data-mjx-texclass="ORD"><mi>j</mi></mrow></msub><mi>t</mi><mo stretchy="false">+</mo><msub><mi>β</mi><mrow data-mjx-texclass="ORD"><mi>j</mi></mrow></msub><mo stretchy="false">=</mo><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>∂</mi><mi>W</mi></mrow></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>∂</mi><msub><mi>P</mi><mrow data-mjx-texclass="ORD"><mi>j</mi></mrow></msub></mrow></mrow></mfrac></mrow></mtd></mtr><mtr><mtd class="mwe-math-columnalign-r"></mtd></mtr></mtable></mstyle></mrow></math>

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Identifiers

$p_{j}$

$W$

$q_{j}$

$Q_{j}$

$W$

$P_{j}$

$\bar{H}$

$H$

$E$

${\dot{Q}}_{j}$

$\bar{H}$

$P_{j}$

$E$

$α_{j}$

$ω_{j}$

$Q_{j}$

$ω_{j}$

$t$

$β_{j}$

$W$

$P_{j}$

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