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Display information for equation id:math.1961.13 on revision:1961
* Page found: Symplektische Struktur des Phasenraums (eq math.1961.13)
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Hash: 4ff69f22e143d5b37020fa0fd7898257
TeX (original user input):
\begin{align}
& {{M}_{3}}(\bar{p},\bar{Q},t)={{M}_{1}}(\bar{q},\bar{Q},t)-\sum\limits_{j=1}^{f}{{}}\frac{\partial {{M}_{1}}}{\partial {{q}_{j}}}{{q}_{j}} \\
& \Rightarrow {{q}_{j}}=\frac{\partial {{M}_{3}}}{\partial {{p}_{j}}} \\
& {{P}_{j}}=-\frac{\partial {{M}_{3}}}{\partial {{Q}_{j}}} \\
& \Rightarrow \frac{\partial {{q}_{j}}}{\partial {{Q}_{k}}}=-\frac{{{\partial }^{2}}{{M}_{3}}}{\partial {{Q}_{k}}\partial {{p}_{j}}}=\frac{\partial {{P}_{k}}}{\partial {{p}_{j}}} \\
\end{align}
TeX (checked):
{\begin{aligned}&{{M}_{3}}({\bar {p}},{\bar {Q}},t)={{M}_{1}}({\bar {q}},{\bar {Q}},t)-\sum \limits _{j=1}^{f}{}{\frac {\partial {{M}_{1}}}{\partial {{q}_{j}}}}{{q}_{j}}\\&\Rightarrow {{q}_{j}}={\frac {\partial {{M}_{3}}}{\partial {{p}_{j}}}}\\&{{P}_{j}}=-{\frac {\partial {{M}_{3}}}{\partial {{Q}_{j}}}}\\&\Rightarrow {\frac {\partial {{q}_{j}}}{\partial {{Q}_{k}}}}=-{\frac {{{\partial }^{2}}{{M}_{3}}}{\partial {{Q}_{k}}\partial {{p}_{j}}}}={\frac {\partial {{P}_{k}}}{\partial {{p}_{j}}}}\\\end{aligned}}
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<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><msub><mi>M</mi><mrow data-mjx-texclass="ORD"><mn>3</mn></mrow></msub><mo stretchy="false">(</mo><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mi>p</mi><mo>¯</mo></mover></mrow></mrow><mo>,</mo><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mi>Q</mi><mo>¯</mo></mover></mrow></mrow><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mi>M</mi><mrow data-mjx-texclass="ORD"><mn>1</mn></mrow></msub><mo stretchy="false">(</mo><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mi>q</mi><mo>¯</mo></mover></mrow></mrow><mo>,</mo><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mi>Q</mi><mo>¯</mo></mover></mrow></mrow><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo><mo>−</mo><munderover><mo form="prefix" texclass="OP">∑</mo><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>j</mi><mo>=</mo><mn>1</mn></mrow></mrow><mrow data-mjx-texclass="ORD"><mi>f</mi></mrow></munderover><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>∂</mi><msub><mi>M</mi><mrow data-mjx-texclass="ORD"><mn>1</mn></mrow></msub></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><mi>q</mi><mrow data-mjx-texclass="ORD"><mi>j</mi></mrow></msub></mtd></mtr><mtr><mtd></mtd><mtd><mo>⇒</mo><msub><mi>q</mi><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><msub><mi>M</mi><mrow data-mjx-texclass="ORD"><mn>3</mn></mrow></msub></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></mtd><mtd><msub><mi>P</mi><mrow data-mjx-texclass="ORD"><mi>j</mi></mrow></msub><mo>=</mo><mo>−</mo><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>∂</mi><msub><mi>M</mi><mrow data-mjx-texclass="ORD"><mn>3</mn></mrow></msub></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></mtd><mtd><mo>⇒</mo><mrow data-mjx-texclass="ORD"><mfrac><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><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>∂</mi><msub><mi>Q</mi><mrow data-mjx-texclass="ORD"><mi>k</mi></mrow></msub></mrow></mrow></mfrac></mrow><mo>=</mo><mo>−</mo><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><msup><mi>∂</mi><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></msup><msub><mi>M</mi><mrow data-mjx-texclass="ORD"><mn>3</mn></mrow></msub></mrow></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>∂</mi><msub><mi>Q</mi><mrow data-mjx-texclass="ORD"><mi>k</mi></mrow></msub><mi>∂</mi><msub><mi>p</mi><mrow data-mjx-texclass="ORD"><mi>j</mi></mrow></msub></mrow></mrow></mfrac></mrow><mo>=</mo><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>∂</mi><msub><mi>P</mi><mrow data-mjx-texclass="ORD"><mi>k</mi></mrow></msub></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></mtd></mtr></mtable></mrow></mstyle></mrow></math>
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