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Display information for equation id:math.1989.21 on revision:1989
* Page found: Hamilton-Jacobische Differenzialgleichung (eq math.1989.21)
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Hash: 2c1423b83a9bef842c74cff1dc6501bc
TeX (original user input):
\begin{align}
& \frac{dS}{dt}=\sum\limits_{j}{{}}\frac{\partial S}{\partial {{q}_{j}}}{{{\dot{q}}}_{j}}+\frac{\partial S}{\partial t}=\sum\limits_{j}{{}}{{p}_{j}}{{{\dot{q}}}_{j}}+\frac{\partial S}{\partial t} \\
& \frac{\partial S}{\partial t}=\bar{H}-H=-H \\
& \frac{dS}{dt}=\sum\limits_{j}{{}}{{p}_{j}}{{{\dot{q}}}_{j}}-H=L \\
& \Rightarrow S=\int{Ldt} \\
\end{align}
TeX (checked):
{\begin{aligned}&{\frac {dS}{dt}}=\sum \limits _{j}{}{\frac {\partial S}{\partial {{q}_{j}}}}{{\dot {q}}_{j}}+{\frac {\partial S}{\partial t}}=\sum \limits _{j}{}{{p}_{j}}{{\dot {q}}_{j}}+{\frac {\partial S}{\partial t}}\\&{\frac {\partial S}{\partial t}}={\bar {H}}-H=-H\\&{\frac {dS}{dt}}=\sum \limits _{j}{}{{p}_{j}}{{\dot {q}}_{j}}-H=L\\&\Rightarrow S=\int {Ldt}\\\end{aligned}}
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MathML (3.423 KB / 455 B) :
<math class="mwe-math-element" xmlns="http://www.w3.org/1998/Math/MathML"><mrow data-mjx-texclass="ORD"><mstyle displaystyle="true" scriptlevel="0"><mrow data-mjx-texclass="ORD"><mtable columnalign="right left right left right left right left right left right left" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true" rowspacing="3pt"><mtr><mtd></mtd><mtd><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>d</mi><mi>S</mi></mrow></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>d</mi><mi>t</mi></mrow></mrow></mfrac></mrow><mo>=</mo><munder><mo form="prefix" texclass="OP">∑</mo><mrow data-mjx-texclass="ORD"><mi>j</mi></mrow></munder><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>∂</mi><mi>S</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><msub><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mi>q</mi><mo>˙</mo></mover></mrow></mrow><mrow data-mjx-texclass="ORD"><mi>j</mi></mrow></msub><mo>+</mo><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>∂</mi><mi>S</mi></mrow></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>∂</mi><mi>t</mi></mrow></mrow></mfrac></mrow><mo>=</mo><munder><mo form="prefix" texclass="OP">∑</mo><mrow data-mjx-texclass="ORD"><mi>j</mi></mrow></munder><msub><mi>p</mi><mrow data-mjx-texclass="ORD"><mi>j</mi></mrow></msub><msub><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mi>q</mi><mo>˙</mo></mover></mrow></mrow><mrow data-mjx-texclass="ORD"><mi>j</mi></mrow></msub><mo>+</mo><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>∂</mi><mi>S</mi></mrow></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>∂</mi><mi>t</mi></mrow></mrow></mfrac></mrow></mtd></mtr><mtr><mtd></mtd><mtd><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>∂</mi><mi>S</mi></mrow></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>∂</mi><mi>t</mi></mrow></mrow></mfrac></mrow><mo>=</mo><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mi>H</mi><mo>¯</mo></mover></mrow></mrow><mo>−</mo><mi>H</mi><mo>=</mo><mo>−</mo><mi>H</mi></mtd></mtr><mtr><mtd></mtd><mtd><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>d</mi><mi>S</mi></mrow></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>d</mi><mi>t</mi></mrow></mrow></mfrac></mrow><mo>=</mo><munder><mo form="prefix" texclass="OP">∑</mo><mrow data-mjx-texclass="ORD"><mi>j</mi></mrow></munder><msub><mi>p</mi><mrow data-mjx-texclass="ORD"><mi>j</mi></mrow></msub><msub><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mi>q</mi><mo>˙</mo></mover></mrow></mrow><mrow data-mjx-texclass="ORD"><mi>j</mi></mrow></msub><mo>−</mo><mi>H</mi><mo>=</mo><mi>L</mi></mtd></mtr><mtr><mtd></mtd><mtd><mo>⇒</mo><mi>S</mi><mo>=</mo><mo texclass="OP">∫</mo><mrow data-mjx-texclass="ORD"><mi>L</mi><mi>d</mi><mi>t</mi></mrow></mtd></mtr><mtr><mtd></mtd></mtr></mtable></mrow></mstyle></mrow></math>
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