– PhysikWiki

General

Display information for equation id:math.1947.16 on revision:1947

* Page found: Die Hamiltonschen Gleichungen (eq math.1947.16)

Occurrences on the following pages:

Die Hamiltonschen Gleichungen Eq: math.1948.38

Hash: 0bb4a78ce0bf8d4ffd1e4d0e9e6bcdcb

TeX (original user input):

\begin{align}
  & {{p}_{k}}=\frac{\partial L(\bar{q},\dot{\bar{q}},t)}{\partial {{{\dot{q}}}_{k}}}=m{{{\dot{q}}}_{\acute{\ }k}}+e{{A}_{k}}(\bar{q},t) \\ 
 & \Rightarrow {{{\dot{q}}}_{k}}=\frac{1}{m}\left( {{p}_{k}}-e{{A}_{k}} \right) \\ 
 &  H=\sum\limits_{k=1}^{3}{{{p}_{k}}}{{{\dot{q}}}_{k}}-L=\sum\limits_{k=1}^{3}{{{p}_{k}}}\frac{1}{m}\left( {{p}_{k}}-e{{A}_{k}} \right)-\frac{1}{2m}\sum\limits_{k=1}^{3}{{}}{{\left( {{p}_{k}}-e{{A}_{k}} \right)}^{2}}-\sum\limits_{k=1}^{3}{{}}\frac{e}{m}\left( {{p}_{k}}-e{{A}_{k}} \right){{A}_{k}}+e\Phi  \\ 
 & H\left( \bar{q},\bar{p},t \right)=\frac{1}{2m}{{\left( {{{\bar{p}}}_{{}}}-e\bar{A}{{(\bar{q},t)}_{{}}} \right)}^{2}}+e\Phi (\bar{q},t) \\ 
\end{align}

TeX (checked):

{\begin{aligned}&{{p}_{k}}={\frac {\partial L({\bar {q}},{\dot {\bar {q}}},t)}{\partial {{\dot {q}}_{k}}}}=m{{\dot {q}}_{{\acute {\ }}k}}+e{{A}_{k}}({\bar {q}},t)\\&\Rightarrow {{\dot {q}}_{k}}={\frac {1}{m}}\left({{p}_{k}}-e{{A}_{k}}\right)\\&H=\sum \limits _{k=1}^{3}{{p}_{k}}{{\dot {q}}_{k}}-L=\sum \limits _{k=1}^{3}{{p}_{k}}{\frac {1}{m}}\left({{p}_{k}}-e{{A}_{k}}\right)-{\frac {1}{2m}}\sum \limits _{k=1}^{3}{}{{\left({{p}_{k}}-e{{A}_{k}}\right)}^{2}}-\sum \limits _{k=1}^{3}{}{\frac {e}{m}}\left({{p}_{k}}-e{{A}_{k}}\right){{A}_{k}}+e\Phi \\&H\left({\bar {q}},{\bar {p}},t\right)={\frac {1}{2m}}{{\left({{\bar {p}}_{}}-e{\bar {A}}{{({\bar {q}},t)}_{}}\right)}^{2}}+e\Phi ({\bar {q}},t)\\\end{aligned}}

LaTeXML (experimentell; verwendet MathML) rendering

MathML (0 B / 8 B) :

SVG image empty. Force Re-Rendering

SVG (0 B / 8 B) :

MathML (experimentell; keine Bilder) rendering

MathML (6.261 KB / 671 B) :

\begin{matrix} p_{k} = \frac{\partial L (\bar{q}, \dot{\bar{q}}, t)}{\partial {\dot{q}}_{k}} = m {\dot{q}}_{\overset{´}{} k} + e A_{k} (\bar{q}, t) \\ \Rightarrow {\dot{q}}_{k} = \frac{1}{m} (p_{k} - e A_{k}) \\ H = \sum_{k = 1}^{3} p_{k} {\dot{q}}_{k} - L = \sum_{k = 1}^{3} p_{k} \frac{1}{m} (p_{k} - e A_{k}) - \frac{1}{2 m} \sum_{k = 1}^{3} {(p_{k} - e A_{k})}^{2} - \sum_{k = 1}^{3} \frac{e}{m} (p_{k} - e A_{k}) A_{k} + e Φ \\ H (\bar{q}, \bar{p}, t) = \frac{1}{2 m} {({\bar{p}}_{} - e \bar{A} {(\bar{q}, t)}_{})}^{2} + e Φ (\bar{q}, 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"><msub><mi>p</mi><mrow data-mjx-texclass="ORD"><mi>k</mi></mrow></msub><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><mo stretchy="false">(</mo><mover><mi>q</mi><mo>¯</mo></mover><mo>,</mo><mover><mover><mi>q</mi><mo>¯</mo></mover><mo>˙</mo></mover><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>∂</mi><msub><mover><mi>q</mi><mo>˙</mo></mover><mrow data-mjx-texclass="ORD"><mi>k</mi></mrow></msub></mrow></mrow></mfrac></mrow><mo stretchy="false">=</mo><mi>m</mi><msub><mover><mi>q</mi><mo>˙</mo></mover><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mtext>&#160;</mtext><mo data-mjx-pseudoscript="true">´</mo></mover><mi>k</mi></mrow></mrow></msub><mo stretchy="false">+</mo><mi>e</mi><msub><mi>A</mi><mrow data-mjx-texclass="ORD"><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mover><mi>q</mi><mo>¯</mo></mover><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd class="mwe-math-columnalign-r"></mtd><mtd class="mwe-math-columnalign-l"><mo stretchy="false">⇒</mo><msub><mover><mi>q</mi><mo>˙</mo></mover><mrow data-mjx-texclass="ORD"><mi>k</mi></mrow></msub><mo stretchy="false">=</mo><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mn>1</mn></mrow><mrow data-mjx-texclass="ORD"><mi>m</mi></mrow></mfrac></mrow><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><msub><mi>p</mi><mrow data-mjx-texclass="ORD"><mi>k</mi></mrow></msub><mo stretchy="false">−</mo><mi>e</mi><msub><mi>A</mi><mrow data-mjx-texclass="ORD"><mi>k</mi></mrow></msub><mo data-mjx-texclass="CLOSE">)</mo></mrow></mtd></mtr><mtr><mtd class="mwe-math-columnalign-r"></mtd><mtd class="mwe-math-columnalign-l"><mi>H</mi><mo stretchy="false">=</mo><munderover><mo form="prefix" movablelimits="false" stretchy="false">∑</mo><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>k</mi><mo stretchy="false">=</mo><mn>1</mn></mrow></mrow><mrow data-mjx-texclass="ORD"><mn>3</mn></mrow></munderover><msub><mi>p</mi><mrow data-mjx-texclass="ORD"><mi>k</mi></mrow></msub><msub><mover><mi>q</mi><mo>˙</mo></mover><mrow data-mjx-texclass="ORD"><mi>k</mi></mrow></msub><mo stretchy="false">−</mo><mi>L</mi><mo stretchy="false">=</mo><munderover><mo form="prefix" movablelimits="false" stretchy="false">∑</mo><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>k</mi><mo stretchy="false">=</mo><mn>1</mn></mrow></mrow><mrow data-mjx-texclass="ORD"><mn>3</mn></mrow></munderover><msub><mi>p</mi><mrow data-mjx-texclass="ORD"><mi>k</mi></mrow></msub><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mn>1</mn></mrow><mrow data-mjx-texclass="ORD"><mi>m</mi></mrow></mfrac></mrow><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><msub><mi>p</mi><mrow data-mjx-texclass="ORD"><mi>k</mi></mrow></msub><mo stretchy="false">−</mo><mi>e</mi><msub><mi>A</mi><mrow data-mjx-texclass="ORD"><mi>k</mi></mrow></msub><mo data-mjx-texclass="CLOSE">)</mo></mrow><mo stretchy="false">−</mo><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mn>1</mn></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mn>2</mn><mi>m</mi></mrow></mrow></mfrac></mrow><munderover><mo form="prefix" movablelimits="false" stretchy="false">∑</mo><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>k</mi><mo stretchy="false">=</mo><mn>1</mn></mrow></mrow><mrow data-mjx-texclass="ORD"><mn>3</mn></mrow></munderover><msup><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><msub><mi>p</mi><mrow data-mjx-texclass="ORD"><mi>k</mi></mrow></msub><mo stretchy="false">−</mo><mi>e</mi><msub><mi>A</mi><mrow data-mjx-texclass="ORD"><mi>k</mi></mrow></msub><mo data-mjx-texclass="CLOSE">)</mo></mrow><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></msup><mo stretchy="false">−</mo><munderover><mo form="prefix" movablelimits="false" stretchy="false">∑</mo><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>k</mi><mo stretchy="false">=</mo><mn>1</mn></mrow></mrow><mrow data-mjx-texclass="ORD"><mn>3</mn></mrow></munderover><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mi>e</mi></mrow><mrow data-mjx-texclass="ORD"><mi>m</mi></mrow></mfrac></mrow><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><msub><mi>p</mi><mrow data-mjx-texclass="ORD"><mi>k</mi></mrow></msub><mo stretchy="false">−</mo><mi>e</mi><msub><mi>A</mi><mrow data-mjx-texclass="ORD"><mi>k</mi></mrow></msub><mo data-mjx-texclass="CLOSE">)</mo></mrow><msub><mi>A</mi><mrow data-mjx-texclass="ORD"><mi>k</mi></mrow></msub><mo stretchy="false">+</mo><mi>e</mi><mi>Φ</mi></mtd></mtr><mtr><mtd class="mwe-math-columnalign-r"></mtd><mtd class="mwe-math-columnalign-l"><mi>H</mi><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mover><mi>q</mi><mo>¯</mo></mover><mo>,</mo><mover><mi>p</mi><mo>¯</mo></mover><mo>,</mo><mi>t</mi><mo data-mjx-texclass="CLOSE">)</mo></mrow><mo stretchy="false">=</mo><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mn>1</mn></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mn>2</mn><mi>m</mi></mrow></mrow></mfrac></mrow><msup><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><msub><mover><mi>p</mi><mo>¯</mo></mover><mrow data-mjx-texclass="ORD"></mrow></msub><mo stretchy="false">−</mo><mi>e</mi><mover><mi>A</mi><mo>¯</mo></mover><msub><mrow data-mjx-texclass="ORD"><mo stretchy="false">(</mo><mover><mi>q</mi><mo>¯</mo></mover><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo></mrow><mrow data-mjx-texclass="ORD"></mrow></msub><mo data-mjx-texclass="CLOSE">)</mo></mrow><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></msup><mo stretchy="false">+</mo><mi>e</mi><mi>Φ</mi><mo stretchy="false">(</mo><mover><mi>q</mi><mo>¯</mo></mover><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd class="mwe-math-columnalign-r"></mtd></mtr></mtable></mstyle></mrow></math>

Translations to Computer Algebra Systems

Translation to Maple

In Maple:

Translation to Mathematica

In Mathematica:

Identifiers

$p_{k}$

$L$

$\bar{q}$

$\dot{\bar{q}}$

$t$

${\dot{q}}_{k}$

$m$

$\dot{q}$

$\overset{´}{}$

$k$

$e$

$A_{k}$

$\bar{q}$

$t$

${\dot{q}}_{k}$

$m$

$p_{k}$

$e$

$A_{k}$

$H$

$k$

$p_{k}$

${\dot{q}}_{k}$

$L$

$k$

$p_{k}$

$m$

$p_{k}$

$e$

$A_{k}$

$m$

$k$

$p_{k}$

$e$

$A_{k}$

$k$

$e$

$m$

$p_{k}$

$e$

$A_{k}$

$A_{k}$

$e$

$Φ$

$H$

$\bar{q}$

$\bar{p}$

$t$

$m$

$\bar{p}$

$e$

$\bar{A}$

$\bar{q}$

$t$

$e$

$Φ$

$\bar{q}$

$t$

MathML observations

0results

no statistics present please run the maintenance script ExtractFeatures.php

0 results

General

LaTeXML (experimentell; verwendet MathML) rendering

MathML (experimentell; keine Bilder) rendering

Translations to Computer Algebra Systems

Translation to Maple

Translation to Mathematica

Similar pages

Identifiers

MathML observations

Navigationsmenü

General

LaTeXML (experimentell; verwendet MathML) rendering

MathML (experimentell; keine Bilder) rendering

Translations to Computer Algebra Systems

Translation to Maple

Translation to Mathematica

Similar pages

Identifiers

MathML observations

Navigationsmenü

Suche