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Display information for equation id:math.1948.2 on revision:1948

* Page found: Die Hamiltonschen Gleichungen (eq math.1948.2)

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Die Hamiltonschen Gleichungen Eq: math.1948.2

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TeX (original user input):

\begin{align}
  & L=L(q,\dot{q},t):dL=\frac{\partial L}{\partial q}dq+\frac{\partial L}{\partial \dot{q}}d\dot{q}+\frac{\partial L}{\partial t}dt \\ 
 & H=H(q,p,t):dH=\frac{\partial H}{\partial q}dq+\frac{\partial H}{\partial p}dp+\frac{\partial H}{\partial t}dt \\ 
 & H=\dot{q}p-L\Rightarrow dH=\dot{q}dp+pd\dot{q}-dL=\dot{q}dp+pd\dot{q}-\frac{\partial L}{\partial q}dq-\frac{\partial L}{\partial \dot{q}}d\dot{q}-\frac{\partial L}{\partial t}dt \\ 
\end{align}

TeX (checked):

{\begin{aligned}&L=L(q,{\dot {q}},t):dL={\frac {\partial L}{\partial q}}dq+{\frac {\partial L}{\partial {\dot {q}}}}d{\dot {q}}+{\frac {\partial L}{\partial t}}dt\\&H=H(q,p,t):dH={\frac {\partial H}{\partial q}}dq+{\frac {\partial H}{\partial p}}dp+{\frac {\partial H}{\partial t}}dt\\&H={\dot {q}}p-L\Rightarrow dH={\dot {q}}dp+pd{\dot {q}}-dL={\dot {q}}dp+pd{\dot {q}}-{\frac {\partial L}{\partial q}}dq-{\frac {\partial L}{\partial {\dot {q}}}}d{\dot {q}}-{\frac {\partial L}{\partial t}}dt\\\end{aligned}}

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\begin{matrix} L = L (q, \dot{q}, t) : d L = \frac{\partial L}{\partial q} d q + \frac{\partial L}{\partial \dot{q}} d \dot{q} + \frac{\partial L}{\partial t} d t \\ H = H (q, p, t) : d H = \frac{\partial H}{\partial q} d q + \frac{\partial H}{\partial p} d p + \frac{\partial H}{\partial t} d t \\ H = \dot{q} p - L \Rightarrow d H = \dot{q} d p + p d \dot{q} - d L = \dot{q} d p + p d \dot{q} - \frac{\partial L}{\partial q} d q - \frac{\partial L}{\partial \dot{q}} d \dot{q} - \frac{\partial L}{\partial t} d t \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"><mi>L</mi><mo stretchy="false">=</mo><mi>L</mi><mo stretchy="false">(</mo><mi>q</mi><mo>,</mo><mover><mi>q</mi><mo>˙</mo></mover><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">:</mo><mi>d</mi><mi>L</mi><mo stretchy="false">=</mo><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>∂</mi><mi>L</mi></mrow></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>∂</mi><mi>q</mi></mrow></mrow></mfrac></mrow><mi>d</mi><mi>q</mi><mo stretchy="false">+</mo><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>∂</mi><mi>L</mi></mrow></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>∂</mi><mover><mi>q</mi><mo>˙</mo></mover></mrow></mrow></mfrac></mrow><mi>d</mi><mover><mi>q</mi><mo>˙</mo></mover><mo stretchy="false">+</mo><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>∂</mi><mi>L</mi></mrow></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>∂</mi><mi>t</mi></mrow></mrow></mfrac></mrow><mi>d</mi><mi>t</mi></mtd></mtr><mtr><mtd class="mwe-math-columnalign-r"></mtd><mtd class="mwe-math-columnalign-l"><mi>H</mi><mo stretchy="false">=</mo><mi>H</mi><mo stretchy="false">(</mo><mi>q</mi><mo>,</mo><mi>p</mi><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">:</mo><mi>d</mi><mi>H</mi><mo stretchy="false">=</mo><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>∂</mi><mi>H</mi></mrow></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>∂</mi><mi>q</mi></mrow></mrow></mfrac></mrow><mi>d</mi><mi>q</mi><mo stretchy="false">+</mo><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>∂</mi><mi>H</mi></mrow></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>∂</mi><mi>p</mi></mrow></mrow></mfrac></mrow><mi>d</mi><mi>p</mi><mo stretchy="false">+</mo><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>∂</mi><mi>H</mi></mrow></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>∂</mi><mi>t</mi></mrow></mrow></mfrac></mrow><mi>d</mi><mi>t</mi></mtd></mtr><mtr><mtd class="mwe-math-columnalign-r"></mtd><mtd class="mwe-math-columnalign-l"><mi>H</mi><mo stretchy="false">=</mo><mover><mi>q</mi><mo>˙</mo></mover><mi>p</mi><mo stretchy="false">−</mo><mi>L</mi><mo stretchy="false">⇒</mo><mi>d</mi><mi>H</mi><mo stretchy="false">=</mo><mover><mi>q</mi><mo>˙</mo></mover><mi>d</mi><mi>p</mi><mo stretchy="false">+</mo><mi>p</mi><mi>d</mi><mover><mi>q</mi><mo>˙</mo></mover><mo stretchy="false">−</mo><mi>d</mi><mi>L</mi><mo stretchy="false">=</mo><mover><mi>q</mi><mo>˙</mo></mover><mi>d</mi><mi>p</mi><mo stretchy="false">+</mo><mi>p</mi><mi>d</mi><mover><mi>q</mi><mo>˙</mo></mover><mo stretchy="false">−</mo><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>∂</mi><mi>L</mi></mrow></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>∂</mi><mi>q</mi></mrow></mrow></mfrac></mrow><mi>d</mi><mi>q</mi><mo stretchy="false">−</mo><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>∂</mi><mi>L</mi></mrow></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>∂</mi><mover><mi>q</mi><mo>˙</mo></mover></mrow></mrow></mfrac></mrow><mi>d</mi><mover><mi>q</mi><mo>˙</mo></mover><mo stretchy="false">−</mo><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>∂</mi><mi>L</mi></mrow></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>∂</mi><mi>t</mi></mrow></mrow></mfrac></mrow><mi>d</mi><mi>t</mi></mtd></mtr><mtr><mtd class="mwe-math-columnalign-r"></mtd></mtr></mtable></mstyle></mrow></math>

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