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Display information for equation id:math.1326.206 on revision:1326
* Page found: Der Hamiltonsche kanonische Formalismus (eq math.1326.206)
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TeX (original user input):
\begin{align}
& {{x}^{i}}(t)=\sum\limits_{k=1}^{2}{{{P}_{ik}}{{x}_{0}}^{k}} \\
& P=\left( \begin{matrix}
\cos {{\omega }_{0}}(t-{{t}_{0}}) & \frac{1}{{{\omega }_{0}}}\sin {{\omega }_{0}}(t-{{t}_{0}}) \\
-{{\omega }_{0}}\sin {{\omega }_{0}}(t-{{t}_{0}}) & \cos {{\omega }_{0}}(t-{{t}_{0}}) \\
\end{matrix} \right) \\
& \det P={{\cos }^{2}}{{\omega }_{0}}(t-{{t}_{0}})+{{\sin }^{2}}{{\omega }_{0}}(t-{{t}_{0}})=1 \\
& d{{x}^{1}}d{{x}^{2}}=(\det P)d{{x}_{0}}^{1}d{{x}_{0}}^{2}=d{{x}_{0}}^{1}d{{x}_{0}}^{2} \\
\end{align}
TeX (checked):
{\begin{aligned}&{{x}^{i}}(t)=\sum \limits _{k=1}^{2}{{{P}_{ik}}{{x}_{0}}^{k}}\\&P=\left({\begin{matrix}\cos {{\omega }_{0}}(t-{{t}_{0}})&{\frac {1}{{\omega }_{0}}}\sin {{\omega }_{0}}(t-{{t}_{0}})\\-{{\omega }_{0}}\sin {{\omega }_{0}}(t-{{t}_{0}})&\cos {{\omega }_{0}}(t-{{t}_{0}})\\\end{matrix}}\right)\\&\det P={{\cos }^{2}}{{\omega }_{0}}(t-{{t}_{0}})+{{\sin }^{2}}{{\omega }_{0}}(t-{{t}_{0}})=1\\&d{{x}^{1}}d{{x}^{2}}=(\det P)d{{x}_{0}}^{1}d{{x}_{0}}^{2}=d{{x}_{0}}^{1}d{{x}_{0}}^{2}\\\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><msup><mi>x</mi><mrow data-mjx-texclass="ORD"><mi>i</mi></mrow></msup><mo stretchy="false">(</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>k</mi><mo>=</mo><mn>1</mn></mrow></mrow><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></munderover><mrow data-mjx-texclass="ORD"><msub><mi>P</mi><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>i</mi><mi>k</mi></mrow></mrow></msub><msup><msub><mi>x</mi><mrow data-mjx-texclass="ORD"><mn>0</mn></mrow></msub><mrow data-mjx-texclass="ORD"><mi>k</mi></mrow></msup></mrow></mtd></mtr><mtr><mtd></mtd><mtd><mi>P</mi><mo>=</mo><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mrow data-mjx-texclass="ORD"><mtable columnspacing="1em" rowspacing="4pt"><mtr><mtd><mi>cos</mi><mo>⁡</mo><msub><mi>ω</mi><mrow data-mjx-texclass="ORD"><mn>0</mn></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo>−</mo><msub><mi>t</mi><mrow data-mjx-texclass="ORD"><mn>0</mn></mrow></msub><mo stretchy="false">)</mo></mtd><mtd><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mn>1</mn></mrow><mrow data-mjx-texclass="ORD"><msub><mi>ω</mi><mrow data-mjx-texclass="ORD"><mn>0</mn></mrow></msub></mrow></mfrac></mrow><mi>sin</mi><mo>⁡</mo><msub><mi>ω</mi><mrow data-mjx-texclass="ORD"><mn>0</mn></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo>−</mo><msub><mi>t</mi><mrow data-mjx-texclass="ORD"><mn>0</mn></mrow></msub><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mo>−</mo><msub><mi>ω</mi><mrow data-mjx-texclass="ORD"><mn>0</mn></mrow></msub><mi>sin</mi><mo>⁡</mo><msub><mi>ω</mi><mrow data-mjx-texclass="ORD"><mn>0</mn></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo>−</mo><msub><mi>t</mi><mrow data-mjx-texclass="ORD"><mn>0</mn></mrow></msub><mo stretchy="false">)</mo></mtd><mtd><mi>cos</mi><mo>⁡</mo><msub><mi>ω</mi><mrow data-mjx-texclass="ORD"><mn>0</mn></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo>−</mo><msub><mi>t</mi><mrow data-mjx-texclass="ORD"><mn>0</mn></mrow></msub><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd></mtd></mtr></mtable></mrow><mo data-mjx-texclass="CLOSE">)</mo></mrow></mtd></mtr><mtr><mtd></mtd><mtd><mi>det</mi><mo>⁡</mo><mi>P</mi><mo>=</mo><msup><mi>cos</mi><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></msup><msub><mi>ω</mi><mrow data-mjx-texclass="ORD"><mn>0</mn></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo>−</mo><msub><mi>t</mi><mrow data-mjx-texclass="ORD"><mn>0</mn></mrow></msub><mo stretchy="false">)</mo><mo>+</mo><msup><mi>sin</mi><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></msup><msub><mi>ω</mi><mrow data-mjx-texclass="ORD"><mn>0</mn></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo>−</mo><msub><mi>t</mi><mrow data-mjx-texclass="ORD"><mn>0</mn></mrow></msub><mo stretchy="false">)</mo><mo>=</mo><mn>1</mn></mtd></mtr><mtr><mtd></mtd><mtd><mi>d</mi><msup><mi>x</mi><mrow data-mjx-texclass="ORD"><mn>1</mn></mrow></msup><mi>d</mi><msup><mi>x</mi><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></msup><mo>=</mo><mo stretchy="false">(</mo><mi>det</mi><mo>⁡</mo><mi>P</mi><mo stretchy="false">)</mo><mi>d</mi><msup><msub><mi>x</mi><mrow data-mjx-texclass="ORD"><mn>0</mn></mrow></msub><mrow data-mjx-texclass="ORD"><mn>1</mn></mrow></msup><mi>d</mi><msup><msub><mi>x</mi><mrow data-mjx-texclass="ORD"><mn>0</mn></mrow></msub><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></msup><mo>=</mo><mi>d</mi><msup><msub><mi>x</mi><mrow data-mjx-texclass="ORD"><mn>0</mn></mrow></msub><mrow data-mjx-texclass="ORD"><mn>1</mn></mrow></msup><mi>d</mi><msup><msub><mi>x</mi><mrow data-mjx-texclass="ORD"><mn>0</mn></mrow></msub><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></msup></mtd></mtr><mtr><mtd></mtd></mtr></mtable></mrow></mstyle></mrow></math>
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