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Display information for equation id:math.1325.135 on revision:1325
* Page found: Der Hamiltonsche kanonische Formalismus (eq math.1325.135)
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Hash: c75d29133b30b400161196accfcdcb83
TeX (original user input):
\begin{align}
& H=\frac{{{p}^{2}}}{2m}+\frac{m{{\omega }^{2}}}{2}{{q}^{2}} \\
& {{M}_{1}}(q,Q)=\frac{m\omega }{2}{{q}^{2}}\cot Q \\
& \Rightarrow p=\frac{\partial {{M}_{1}}}{\partial q}=m\omega q\cot Q \\
& P=-\frac{\partial {{M}_{1}}}{\partial Q}=\frac{m\omega }{2}\frac{{{q}^{2}}}{{{\sin }^{2}}Q} \\
& q={{\left( \frac{2}{m\omega }P \right)}^{\frac{1}{2}}}\sin Q \\
& p={{\left( 2m\omega P \right)}^{\frac{1}{2}}}\cos Q \\
& \\
\end{align}
TeX (checked):
{\begin{aligned}&H={\frac {{p}^{2}}{2m}}+{\frac {m{{\omega }^{2}}}{2}}{{q}^{2}}\\&{{M}_{1}}(q,Q)={\frac {m\omega }{2}}{{q}^{2}}\cot Q\\&\Rightarrow p={\frac {\partial {{M}_{1}}}{\partial q}}=m\omega q\cot Q\\&P=-{\frac {\partial {{M}_{1}}}{\partial Q}}={\frac {m\omega }{2}}{\frac {{q}^{2}}{{{\sin }^{2}}Q}}\\&q={{\left({\frac {2}{m\omega }}P\right)}^{\frac {1}{2}}}\sin Q\\&p={{\left(2m\omega P\right)}^{\frac {1}{2}}}\cos Q\\&\\\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><mi>H</mi><mo>=</mo><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><msup><mi>p</mi><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></msup></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mn>2</mn><mi>m</mi></mrow></mrow></mfrac></mrow><mo>+</mo><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>m</mi><msup><mi>ω</mi><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></msup></mrow></mrow><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></mfrac></mrow><msup><mi>q</mi><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></msup></mtd></mtr><mtr><mtd></mtd><mtd><msub><mi>M</mi><mrow data-mjx-texclass="ORD"><mn>1</mn></mrow></msub><mo stretchy="false">(</mo><mi>q</mi><mo>,</mo><mi>Q</mi><mo stretchy="false">)</mo><mo>=</mo><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>m</mi><mi>ω</mi></mrow></mrow><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></mfrac></mrow><msup><mi>q</mi><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></msup><mi>cot</mi><mo>⁡</mo><mi>Q</mi></mtd></mtr><mtr><mtd></mtd><mtd><mo>⇒</mo><mi>p</mi><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>1</mn></mrow></msub></mrow></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>∂</mi><mi>q</mi></mrow></mrow></mfrac></mrow><mo>=</mo><mi>m</mi><mi>ω</mi><mi>q</mi><mi>cot</mi><mo>⁡</mo><mi>Q</mi></mtd></mtr><mtr><mtd></mtd><mtd><mi>P</mi><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>1</mn></mrow></msub></mrow></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>∂</mi><mi>Q</mi></mrow></mrow></mfrac></mrow><mo>=</mo><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>m</mi><mi>ω</mi></mrow></mrow><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></mfrac></mrow><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><msup><mi>q</mi><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></msup></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><msup><mi>sin</mi><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></msup><mi>Q</mi></mrow></mrow></mfrac></mrow></mtd></mtr><mtr><mtd></mtd><mtd><mi>q</mi><mo>=</mo><msup><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>m</mi><mi>ω</mi></mrow></mrow></mfrac></mrow><mi>P</mi><mo data-mjx-texclass="CLOSE">)</mo></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mn>1</mn></mrow><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></mfrac></mrow></mrow></msup><mi>sin</mi><mo>⁡</mo><mi>Q</mi></mtd></mtr><mtr><mtd></mtd><mtd><mi>p</mi><mo>=</mo><msup><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mn>2</mn><mi>m</mi><mi>ω</mi><mi>P</mi><mo data-mjx-texclass="CLOSE">)</mo></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mn>1</mn></mrow><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></mfrac></mrow></mrow></msup><mi>cos</mi><mo>⁡</mo><mi>Q</mi></mtd></mtr><mtr><mtd></mtd><mtd></mtd></mtr><mtr><mtd></mtd></mtr></mtable></mrow></mstyle></mrow></math>
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