– PhysikWiki

General

Display information for equation id:math.1326.33 on revision:1326

* Page found: Der Hamiltonsche kanonische Formalismus (eq math.1326.33)

Occurrences on the following pages:

Hash: d29cbc7f039e6c450cec8add44042cd3

TeX (original user input):

\begin{align}
  & dH=\sum\limits_{k}{{}}\left( \frac{\partial H}{\partial {{q}_{k}}}d{{q}_{k}}+\frac{\partial H}{\partial {{p}_{k}}}d{{p}_{k}} \right)+\frac{\partial H}{\partial t}dt=\sum\limits_{k}{{{{\dot{q}}}_{k}}d{{p}_{k}}}+\sum\limits_{k}{{{p}_{k}}d{{{\dot{q}}}_{k}}}-\sum\limits_{k}{\frac{\partial L}{\partial {{{\dot{q}}}_{k}}}d{{{\dot{q}}}_{k}}}-\sum\limits_{k}{\frac{\partial L}{\partial {{q}_{k}}}d{{q}_{k}}-\frac{\partial L}{\partial t}dt} \\ 
 & =\sum\limits_{k}{{{{\dot{q}}}_{k}}d{{p}_{k}}}-\sum\limits_{k}{\frac{\partial L}{\partial {{q}_{k}}}d{{q}_{k}}-\frac{\partial L}{\partial t}dt} \\ 
 & \Rightarrow \frac{\partial H}{\partial {{q}_{k}}}=-\frac{\partial L}{\partial {{q}_{k}}} \\ 
 & \frac{\partial L}{\partial {{q}_{k}}}=\frac{d}{dt}\frac{\partial L}{\partial {{{\dot{q}}}_{k}}}=\frac{d}{dt}{{p}_{k}} \\ 
 & \frac{\partial H}{\partial {{p}_{k}}}={{{\dot{q}}}_{k}};\frac{\partial H}{\partial t}=-\frac{\partial L}{\partial t} \\ 
\end{align}

TeX (checked):

{\begin{aligned}&dH=\sum \limits _{k}{}\left({\frac {\partial H}{\partial {{q}_{k}}}}d{{q}_{k}}+{\frac {\partial H}{\partial {{p}_{k}}}}d{{p}_{k}}\right)+{\frac {\partial H}{\partial t}}dt=\sum \limits _{k}{{{\dot {q}}_{k}}d{{p}_{k}}}+\sum \limits _{k}{{{p}_{k}}d{{\dot {q}}_{k}}}-\sum \limits _{k}{{\frac {\partial L}{\partial {{\dot {q}}_{k}}}}d{{\dot {q}}_{k}}}-\sum \limits _{k}{{\frac {\partial L}{\partial {{q}_{k}}}}d{{q}_{k}}-{\frac {\partial L}{\partial t}}dt}\\&=\sum \limits _{k}{{{\dot {q}}_{k}}d{{p}_{k}}}-\sum \limits _{k}{{\frac {\partial L}{\partial {{q}_{k}}}}d{{q}_{k}}-{\frac {\partial L}{\partial t}}dt}\\&\Rightarrow {\frac {\partial H}{\partial {{q}_{k}}}}=-{\frac {\partial L}{\partial {{q}_{k}}}}\\&{\frac {\partial L}{\partial {{q}_{k}}}}={\frac {d}{dt}}{\frac {\partial L}{\partial {{\dot {q}}_{k}}}}={\frac {d}{dt}}{{p}_{k}}\\&{\frac {\partial H}{\partial {{p}_{k}}}}={{\dot {q}}_{k}};{\frac {\partial H}{\partial t}}=-{\frac {\partial L}{\partial 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 (8.307 KB / 588 B) :

\begin{aligned} d H = \sum_{k} (\frac{\partial H}{\partial q_{k}} d q_{k} + \frac{\partial H}{\partial p_{k}} d p_{k}) + \frac{\partial H}{\partial t} d t = \sum_{k} {\dot{q}}_{k} d p_{k} + \sum_{k} p_{k} d {\dot{q}}_{k} - \sum_{k} \frac{\partial L}{\partial {\dot{q}}_{k}} d {\dot{q}}_{k} - \sum_{k} \frac{\partial L}{\partial q_{k}} d q_{k} - \frac{\partial L}{\partial t} d t \\ = \sum_{k} {\dot{q}}_{k} d p_{k} - \sum_{k} \frac{\partial L}{\partial q_{k}} d q_{k} - \frac{\partial L}{\partial t} d t \\ \Rightarrow \frac{\partial H}{\partial q_{k}} = - \frac{\partial L}{\partial q_{k}} \\ \frac{\partial L}{\partial q_{k}} = \frac{d}{d t} \frac{\partial L}{\partial {\dot{q}}_{k}} = \frac{d}{d t} p_{k} \\ \frac{\partial H}{\partial p_{k}} = {\dot{q}}_{k}; \frac{\partial H}{\partial t} = - \frac{\partial L}{\partial t} \end{aligned}

<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>d</mi><mi>H</mi><mo>=</mo><munder><mo form="prefix" texclass="OP">&#x2211;</mo><mrow data-mjx-texclass="ORD"><mi>k</mi></mrow></munder><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>&#x2202;</mi><mi>H</mi></mrow></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>&#x2202;</mi><msub><mi>q</mi><mrow data-mjx-texclass="ORD"><mi>k</mi></mrow></msub></mrow></mrow></mfrac></mrow><mi>d</mi><msub><mi>q</mi><mrow data-mjx-texclass="ORD"><mi>k</mi></mrow></msub><mo>+</mo><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>&#x2202;</mi><mi>H</mi></mrow></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>&#x2202;</mi><msub><mi>p</mi><mrow data-mjx-texclass="ORD"><mi>k</mi></mrow></msub></mrow></mrow></mfrac></mrow><mi>d</mi><msub><mi>p</mi><mrow data-mjx-texclass="ORD"><mi>k</mi></mrow></msub><mo data-mjx-texclass="CLOSE">)</mo></mrow><mo>+</mo><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>&#x2202;</mi><mi>H</mi></mrow></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>&#x2202;</mi><mi>t</mi></mrow></mrow></mfrac></mrow><mi>d</mi><mi>t</mi><mo>=</mo><munder><mo form="prefix" texclass="OP">&#x2211;</mo><mrow data-mjx-texclass="ORD"><mi>k</mi></mrow></munder><mrow data-mjx-texclass="ORD"><msub><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mi>q</mi><mo>˙</mo></mover></mrow></mrow><mrow data-mjx-texclass="ORD"><mi>k</mi></mrow></msub><mi>d</mi><msub><mi>p</mi><mrow data-mjx-texclass="ORD"><mi>k</mi></mrow></msub></mrow><mo>+</mo><munder><mo form="prefix" texclass="OP">&#x2211;</mo><mrow data-mjx-texclass="ORD"><mi>k</mi></mrow></munder><mrow data-mjx-texclass="ORD"><msub><mi>p</mi><mrow data-mjx-texclass="ORD"><mi>k</mi></mrow></msub><mi>d</mi><msub><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mi>q</mi><mo>˙</mo></mover></mrow></mrow><mrow data-mjx-texclass="ORD"><mi>k</mi></mrow></msub></mrow><mo>&#x2212;</mo><munder><mo form="prefix" texclass="OP">&#x2211;</mo><mrow data-mjx-texclass="ORD"><mi>k</mi></mrow></munder><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>&#x2202;</mi><mi>L</mi></mrow></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>&#x2202;</mi><msub><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mi>q</mi><mo>˙</mo></mover></mrow></mrow><mrow data-mjx-texclass="ORD"><mi>k</mi></mrow></msub></mrow></mrow></mfrac></mrow><mi>d</mi><msub><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mi>q</mi><mo>˙</mo></mover></mrow></mrow><mrow data-mjx-texclass="ORD"><mi>k</mi></mrow></msub></mrow><mo>&#x2212;</mo><munder><mo form="prefix" texclass="OP">&#x2211;</mo><mrow data-mjx-texclass="ORD"><mi>k</mi></mrow></munder><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>&#x2202;</mi><mi>L</mi></mrow></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>&#x2202;</mi><msub><mi>q</mi><mrow data-mjx-texclass="ORD"><mi>k</mi></mrow></msub></mrow></mrow></mfrac></mrow><mi>d</mi><msub><mi>q</mi><mrow data-mjx-texclass="ORD"><mi>k</mi></mrow></msub><mo>&#x2212;</mo><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>&#x2202;</mi><mi>L</mi></mrow></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>&#x2202;</mi><mi>t</mi></mrow></mrow></mfrac></mrow><mi>d</mi><mi>t</mi></mrow></mtd></mtr><mtr><mtd></mtd><mtd><mo>=</mo><munder><mo form="prefix" texclass="OP">&#x2211;</mo><mrow data-mjx-texclass="ORD"><mi>k</mi></mrow></munder><mrow data-mjx-texclass="ORD"><msub><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mi>q</mi><mo>˙</mo></mover></mrow></mrow><mrow data-mjx-texclass="ORD"><mi>k</mi></mrow></msub><mi>d</mi><msub><mi>p</mi><mrow data-mjx-texclass="ORD"><mi>k</mi></mrow></msub></mrow><mo>&#x2212;</mo><munder><mo form="prefix" texclass="OP">&#x2211;</mo><mrow data-mjx-texclass="ORD"><mi>k</mi></mrow></munder><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>&#x2202;</mi><mi>L</mi></mrow></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>&#x2202;</mi><msub><mi>q</mi><mrow data-mjx-texclass="ORD"><mi>k</mi></mrow></msub></mrow></mrow></mfrac></mrow><mi>d</mi><msub><mi>q</mi><mrow data-mjx-texclass="ORD"><mi>k</mi></mrow></msub><mo>&#x2212;</mo><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>&#x2202;</mi><mi>L</mi></mrow></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>&#x2202;</mi><mi>t</mi></mrow></mrow></mfrac></mrow><mi>d</mi><mi>t</mi></mrow></mtd></mtr><mtr><mtd></mtd><mtd><mo>&#x21D2;</mo><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>&#x2202;</mi><mi>H</mi></mrow></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>&#x2202;</mi><msub><mi>q</mi><mrow data-mjx-texclass="ORD"><mi>k</mi></mrow></msub></mrow></mrow></mfrac></mrow><mo>=</mo><mo>&#x2212;</mo><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>&#x2202;</mi><mi>L</mi></mrow></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>&#x2202;</mi><msub><mi>q</mi><mrow data-mjx-texclass="ORD"><mi>k</mi></mrow></msub></mrow></mrow></mfrac></mrow></mtd></mtr><mtr><mtd></mtd><mtd><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>&#x2202;</mi><mi>L</mi></mrow></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>&#x2202;</mi><msub><mi>q</mi><mrow data-mjx-texclass="ORD"><mi>k</mi></mrow></msub></mrow></mrow></mfrac></mrow><mo>=</mo><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mi>d</mi></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>d</mi><mi>t</mi></mrow></mrow></mfrac></mrow><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>&#x2202;</mi><mi>L</mi></mrow></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>&#x2202;</mi><msub><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mi>q</mi><mo>˙</mo></mover></mrow></mrow><mrow data-mjx-texclass="ORD"><mi>k</mi></mrow></msub></mrow></mrow></mfrac></mrow><mo>=</mo><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mi>d</mi></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>d</mi><mi>t</mi></mrow></mrow></mfrac></mrow><msub><mi>p</mi><mrow data-mjx-texclass="ORD"><mi>k</mi></mrow></msub></mtd></mtr><mtr><mtd></mtd><mtd><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>&#x2202;</mi><mi>H</mi></mrow></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>&#x2202;</mi><msub><mi>p</mi><mrow data-mjx-texclass="ORD"><mi>k</mi></mrow></msub></mrow></mrow></mfrac></mrow><mo>=</mo><msub><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mi>q</mi><mo>˙</mo></mover></mrow></mrow><mrow data-mjx-texclass="ORD"><mi>k</mi></mrow></msub><mo>;</mo><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>&#x2202;</mi><mi>H</mi></mrow></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>&#x2202;</mi><mi>t</mi></mrow></mrow></mfrac></mrow><mo>=</mo><mo>&#x2212;</mo><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>&#x2202;</mi><mi>L</mi></mrow></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>&#x2202;</mi><mi>t</mi></mrow></mrow></mfrac></mrow></mtd></mtr><mtr><mtd></mtd></mtr></mtable></mrow></mstyle></mrow></math>

Translations to Computer Algebra Systems

Translation to Maple

In Maple:

Translation to Mathematica

In Mathematica:

Identifiers

$d$

$H$

$k$

$H$

$q_{k}$

$d$

$q_{k}$

$H$

$p_{k}$

$d$

$p_{k}$

$H$

$t$

$d$

$t$

$k$

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

$d$

$p_{k}$

$k$

$p_{k}$

$d$

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

$k$

$L$

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

$d$

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

$k$

$L$

$q_{k}$

$d$

$q_{k}$

$L$

$t$

$d$

$t$

$k$

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

$d$

$p_{k}$

$k$

$L$

$q_{k}$

$d$

$q_{k}$

$L$

$t$

$d$

$t$

$H$

$q_{k}$

$L$

$q_{k}$

$L$

$q_{k}$

$d$

$d$

$t$

$L$

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

$d$

$d$

$t$

$p_{k}$

$H$

$p_{k}$

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

$H$

$t$

$L$

$t$

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