– PhysikWiki

General

Display information for equation id:math.1256.151 on revision:1256

* Page found: Das d'Alembertsche Prinzip (eq math.1256.151)

Occurrences on the following pages:

Normalschwingungen Eq: math.1515.12

Hash: d1e74e9e01cf1d60127b346f3df1f0cd

TeX (original user input):

\begin{align}
  & T=\frac{1}{2}\sum\limits_{j,k}{{{T}_{jk}}}{{{\dot{q}}}_{j}}{{{\dot{q}}}_{k}} \\
 & {{T}_{jk}}={{T}_{kj}}\approx \sum\limits_{i}{{{m}_{i}}{{\left( \frac{\partial {{{\vec{r}}}_{i}}}{\partial {{q}_{j}}} \right)}_{0}}{{\left( \frac{\partial {{{\vec{r}}}_{i}}}{\partial {{q}_{j}}} \right)}_{0}}} \\
 & {{T}_{jk}}=m\left[ \left( \frac{\partial x}{\partial {{q}_{j}}} \right)\left( \frac{\partial x}{\partial {{q}_{k}}} \right)+\left( \frac{\partial y}{\partial {{q}_{j}}} \right)\left( \frac{\partial y}{\partial {{q}_{k}}} \right)+\left( \frac{\partial z}{\partial {{q}_{j}}} \right)\left( \frac{\partial z}{\partial {{q}_{k}}} \right) \right] \\
\end{align}

TeX (checked):

{\begin{aligned}&T={\frac {1}{2}}\sum \limits _{j,k}{{T}_{jk}}{{\dot {q}}_{j}}{{\dot {q}}_{k}}\\&{{T}_{jk}}={{T}_{kj}}\approx \sum \limits _{i}{{{m}_{i}}{{\left({\frac {\partial {{\vec {r}}_{i}}}{\partial {{q}_{j}}}}\right)}_{0}}{{\left({\frac {\partial {{\vec {r}}_{i}}}{\partial {{q}_{j}}}}\right)}_{0}}}\\&{{T}_{jk}}=m\left[\left({\frac {\partial x}{\partial {{q}_{j}}}}\right)\left({\frac {\partial x}{\partial {{q}_{k}}}}\right)+\left({\frac {\partial y}{\partial {{q}_{j}}}}\right)\left({\frac {\partial y}{\partial {{q}_{k}}}}\right)+\left({\frac {\partial z}{\partial {{q}_{j}}}}\right)\left({\frac {\partial z}{\partial {{q}_{k}}}}\right)\right]\\\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 (5.752 KB / 577 B) :

\begin{aligned} T = \frac{1}{2} \sum_{j, k} T_{j k} {\dot{q}}_{j} {\dot{q}}_{k} \\ T_{j k} = T_{k j} \approx \sum_{i} m_{i} {(\frac{\partial {\vec{r}}_{i}}{\partial q_{j}})}_{0} {(\frac{\partial {\vec{r}}_{i}}{\partial q_{j}})}_{0} \\ T_{j k} = m [(\frac{\partial x}{\partial q_{j}}) (\frac{\partial x}{\partial q_{k}}) + (\frac{\partial y}{\partial q_{j}}) (\frac{\partial y}{\partial q_{k}}) + (\frac{\partial z}{\partial q_{j}}) (\frac{\partial z}{\partial q_{k}})] \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>T</mi><mo>=</mo><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><munder><mo form="prefix" texclass="OP">&#x2211;</mo><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>j</mi><mo>,</mo><mi>k</mi></mrow></mrow></munder><msub><mi>T</mi><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>j</mi><mi>k</mi></mrow></mrow></msub><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>j</mi></mrow></msub><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></mtd></mtr><mtr><mtd></mtd><mtd><msub><mi>T</mi><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>j</mi><mi>k</mi></mrow></mrow></msub><mo>=</mo><msub><mi>T</mi><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>k</mi><mi>j</mi></mrow></mrow></msub><mo>&#x2248;</mo><munder><mo form="prefix" texclass="OP">&#x2211;</mo><mrow data-mjx-texclass="ORD"><mi>i</mi></mrow></munder><mrow data-mjx-texclass="ORD"><msub><mi>m</mi><mrow data-mjx-texclass="ORD"><mi>i</mi></mrow></msub><msub><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><msub><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mi>r</mi><mo>→</mo></mover></mrow></mrow><mrow data-mjx-texclass="ORD"><mi>i</mi></mrow></msub></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>j</mi></mrow></msub></mrow></mrow></mfrac></mrow><mo data-mjx-texclass="CLOSE">)</mo></mrow><mrow data-mjx-texclass="ORD"><mn>0</mn></mrow></msub><msub><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><msub><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mi>r</mi><mo>→</mo></mover></mrow></mrow><mrow data-mjx-texclass="ORD"><mi>i</mi></mrow></msub></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>j</mi></mrow></msub></mrow></mrow></mfrac></mrow><mo data-mjx-texclass="CLOSE">)</mo></mrow><mrow data-mjx-texclass="ORD"><mn>0</mn></mrow></msub></mrow></mtd></mtr><mtr><mtd></mtd><mtd><msub><mi>T</mi><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>j</mi><mi>k</mi></mrow></mrow></msub><mo>=</mo><mi>m</mi><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">[</mo><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>x</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>j</mi></mrow></msub></mrow></mrow></mfrac></mrow><mo data-mjx-texclass="CLOSE">)</mo></mrow><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>x</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 data-mjx-texclass="CLOSE">)</mo></mrow><mo>+</mo><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>y</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>j</mi></mrow></msub></mrow></mrow></mfrac></mrow><mo data-mjx-texclass="CLOSE">)</mo></mrow><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>y</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 data-mjx-texclass="CLOSE">)</mo></mrow><mo>+</mo><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>z</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>j</mi></mrow></msub></mrow></mrow></mfrac></mrow><mo data-mjx-texclass="CLOSE">)</mo></mrow><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>z</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 data-mjx-texclass="CLOSE">)</mo></mrow><mo data-mjx-texclass="CLOSE">]</mo></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

$T$

$j$

$k$

$T_{j k}$

${\dot{q}}_{j}$

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

$T_{j k}$

$T_{k j}$

$i$

$m_{i}$

${\vec{r}}_{i}$

$q_{j}$

${\vec{r}}_{i}$

$q_{j}$

$T_{j k}$

$m$

$x$

$q_{j}$

$x$

$q_{k}$

$y$

$q_{j}$

$y$

$q_{k}$

$z$

$q_{j}$

$z$

$q_{k}$

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