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Maxwell- Gleichungen im Vakuum Eq: math.2110.17

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\begin{align}
& \frac{d}{dt}\frac{\partial L(\bar{r},\bar{v},t)}{\partial {{v}_{k}}}=m{{{\ddot{x}}}_{k}}+q\left( \frac{\partial }{\partial t}{{A}_{k}}(\bar{r},t)+\frac{\partial {{A}_{k}}(\bar{r},t)}{\partial {{x}_{l}}}\frac{\partial {{x}_{l}}}{\partial t} \right)=m{{{\ddot{x}}}_{k}}+q\left( \frac{\partial }{\partial t}+\bar{v}\cdot \nabla  \right){{A}_{k}}(\bar{r},t) \\
& \frac{\partial L(\bar{r},\bar{v},t)}{\partial {{x}_{k}}}=q\left[ \frac{\partial }{\partial {{x}_{k}}}\left( \bar{v}\bar{A} \right)-\frac{\partial }{\partial {{x}_{k}}}\Phi  \right] \\
& \Rightarrow 0=\frac{d}{dt}\frac{\partial L(\bar{r},\bar{v},t)}{\partial {{v}_{k}}}-\frac{\partial L(\bar{r},\bar{v},t)}{\partial {{x}_{k}}}=m{{{\ddot{x}}}_{k}}+q\left( \frac{\partial }{\partial t}+\bar{v}\cdot \nabla  \right){{A}_{k}}(\bar{r},t)-q\left[ \frac{\partial }{\partial {{x}_{k}}}\left( \bar{v}\bar{A} \right)-\frac{\partial }{\partial {{x}_{k}}}\Phi  \right] \\
& =m{{{\ddot{x}}}_{k}}+q\frac{\partial }{\partial t}{{A}_{k}}(\bar{r},t)+q\left[ \left( \bar{v}\cdot \nabla  \right){{A}_{k}}(\bar{r},t)-\frac{\partial }{\partial {{x}_{k}}}\left( \bar{v}\bar{A} \right) \right]+q\frac{\partial }{\partial {{x}_{k}}}\Phi  \\
& \left[ \left( \bar{v}\cdot \nabla  \right){{A}_{k}}(\bar{r},t)-\frac{\partial }{\partial {{x}_{k}}}\left( \bar{v}\bar{A} \right) \right]=-{{\left[ \bar{v}\times \left( \nabla \times \bar{A} \right) \right]}_{k}} \\
& \Rightarrow 0=m\ddot{\bar{r}}+q\frac{\partial }{\partial t}A(\bar{r},t)-q\left[ \bar{v}\times \left( \nabla \times \bar{A} \right) \right]+q\nabla \Phi =m\ddot{\bar{r}}+q\left[ \frac{\partial }{\partial t}A(\bar{r},t)+\nabla \Phi -\left[ \bar{v}\times \left( \nabla \times \bar{A} \right) \right] \right] \\
\end{align}

TeX (checked):

{\begin{aligned}&{\frac {d}{dt}}{\frac {\partial L({\bar {r}},{\bar {v}},t)}{\partial {{v}_{k}}}}=m{{\ddot {x}}_{k}}+q\left({\frac {\partial }{\partial t}}{{A}_{k}}({\bar {r}},t)+{\frac {\partial {{A}_{k}}({\bar {r}},t)}{\partial {{x}_{l}}}}{\frac {\partial {{x}_{l}}}{\partial t}}\right)=m{{\ddot {x}}_{k}}+q\left({\frac {\partial }{\partial t}}+{\bar {v}}\cdot \nabla \right){{A}_{k}}({\bar {r}},t)\\&{\frac {\partial L({\bar {r}},{\bar {v}},t)}{\partial {{x}_{k}}}}=q\left[{\frac {\partial }{\partial {{x}_{k}}}}\left({\bar {v}}{\bar {A}}\right)-{\frac {\partial }{\partial {{x}_{k}}}}\Phi \right]\\&\Rightarrow 0={\frac {d}{dt}}{\frac {\partial L({\bar {r}},{\bar {v}},t)}{\partial {{v}_{k}}}}-{\frac {\partial L({\bar {r}},{\bar {v}},t)}{\partial {{x}_{k}}}}=m{{\ddot {x}}_{k}}+q\left({\frac {\partial }{\partial t}}+{\bar {v}}\cdot \nabla \right){{A}_{k}}({\bar {r}},t)-q\left[{\frac {\partial }{\partial {{x}_{k}}}}\left({\bar {v}}{\bar {A}}\right)-{\frac {\partial }{\partial {{x}_{k}}}}\Phi \right]\\&=m{{\ddot {x}}_{k}}+q{\frac {\partial }{\partial t}}{{A}_{k}}({\bar {r}},t)+q\left[\left({\bar {v}}\cdot \nabla \right){{A}_{k}}({\bar {r}},t)-{\frac {\partial }{\partial {{x}_{k}}}}\left({\bar {v}}{\bar {A}}\right)\right]+q{\frac {\partial }{\partial {{x}_{k}}}}\Phi \\&\left[\left({\bar {v}}\cdot \nabla \right){{A}_{k}}({\bar {r}},t)-{\frac {\partial }{\partial {{x}_{k}}}}\left({\bar {v}}{\bar {A}}\right)\right]=-{{\left[{\bar {v}}\times \left(\nabla \times {\bar {A}}\right)\right]}_{k}}\\&\Rightarrow 0=m{\ddot {\bar {r}}}+q{\frac {\partial }{\partial t}}A({\bar {r}},t)-q\left[{\bar {v}}\times \left(\nabla \times {\bar {A}}\right)\right]+q\nabla \Phi =m{\ddot {\bar {r}}}+q\left[{\frac {\partial }{\partial t}}A({\bar {r}},t)+\nabla \Phi -\left[{\bar {v}}\times \left(\nabla \times {\bar {A}}\right)\right]\right]\\\end{aligned}}

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\begin{aligned} \frac{d}{d t} \frac{\partial L (\bar{r}, \bar{v}, t)}{\partial v_{k}} = m {\ddot{x}}_{k} + q (\frac{\partial}{\partial t} A_{k} (\bar{r}, t) + \frac{\partial A_{k} (\bar{r}, t)}{\partial x_{l}} \frac{\partial x_{l}}{\partial t}) = m {\ddot{x}}_{k} + q (\frac{\partial}{\partial t} + \bar{v} \cdot \nabla) A_{k} (\bar{r}, t) \\ \frac{\partial L (\bar{r}, \bar{v}, t)}{\partial x_{k}} = q [\frac{\partial}{\partial x_{k}} (\bar{v} \bar{A}) - \frac{\partial}{\partial x_{k}} Φ] \\ \Rightarrow 0 = \frac{d}{d t} \frac{\partial L (\bar{r}, \bar{v}, t)}{\partial v_{k}} - \frac{\partial L (\bar{r}, \bar{v}, t)}{\partial x_{k}} = m {\ddot{x}}_{k} + q (\frac{\partial}{\partial t} + \bar{v} \cdot \nabla) A_{k} (\bar{r}, t) - q [\frac{\partial}{\partial x_{k}} (\bar{v} \bar{A}) - \frac{\partial}{\partial x_{k}} Φ] \\ = m {\ddot{x}}_{k} + q \frac{\partial}{\partial t} A_{k} (\bar{r}, t) + q [(\bar{v} \cdot \nabla) A_{k} (\bar{r}, t) - \frac{\partial}{\partial x_{k}} (\bar{v} \bar{A})] + q \frac{\partial}{\partial x_{k}} Φ \\ [(\bar{v} \cdot \nabla) A_{k} (\bar{r}, t) - \frac{\partial}{\partial x_{k}} (\bar{v} \bar{A})] = - {[\bar{v} \times (\nabla \times \bar{A})]}_{k} \\ \Rightarrow 0 = m \ddot{\bar{r}} + q \frac{\partial}{\partial t} A (\bar{r}, t) - q [\bar{v} \times (\nabla \times \bar{A})] + q \nabla Φ = m \ddot{\bar{r}} + q [\frac{\partial}{\partial t} A (\bar{r}, t) + \nabla Φ - [\bar{v} \times (\nabla \times \bar{A})]] \end{aligned}

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data-mjx-texclass="ORD"><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mi>r</mi><mo>¯</mo></mover></mrow></mrow><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo><mo>&#x2212;</mo><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mi>&#x2202;</mi></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>&#x2202;</mi><msub><mi>x</mi><mrow data-mjx-texclass="ORD"><mi>k</mi></mrow></msub></mrow></mrow></mfrac></mrow><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mi>v</mi><mo>¯</mo></mover></mrow></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mi>A</mi><mo>¯</mo></mover></mrow></mrow><mo data-mjx-texclass="CLOSE">)</mo></mrow><mo data-mjx-texclass="CLOSE">]</mo></mrow><mo>=</mo><mo>&#x2212;</mo><msub><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">[</mo><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mi>v</mi><mo>¯</mo></mover></mrow></mrow><mo>&#x00D7;</mo><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mi mathvariant="normal">&#x2207;</mi><mo>&#x00D7;</mo><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mi>A</mi><mo>¯</mo></mover></mrow></mrow><mo data-mjx-texclass="CLOSE">)</mo></mrow><mo data-mjx-texclass="CLOSE">]</mo></mrow><mrow data-mjx-texclass="ORD"><mi>k</mi></mrow></msub></mtd></mtr><mtr><mtd></mtd><mtd><mo>&#x21D2;</mo><mn>0</mn><mo>=</mo><mi>m</mi><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mi>r</mi><mo>¯</mo></mover></mrow></mrow><mo>¨</mo></mover></mrow></mrow><mo>+</mo><mi>q</mi><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mi>&#x2202;</mi></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>&#x2202;</mi><mi>t</mi></mrow></mrow></mfrac></mrow><mi>A</mi><mo stretchy="false">(</mo><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mi>r</mi><mo>¯</mo></mover></mrow></mrow><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo><mo>&#x2212;</mo><mi>q</mi><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">[</mo><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mi>v</mi><mo>¯</mo></mover></mrow></mrow><mo>&#x00D7;</mo><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mi mathvariant="normal">&#x2207;</mi><mo>&#x00D7;</mo><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mi>A</mi><mo>¯</mo></mover></mrow></mrow><mo data-mjx-texclass="CLOSE">)</mo></mrow><mo data-mjx-texclass="CLOSE">]</mo></mrow><mo>+</mo><mi>q</mi><mi mathvariant="normal">&#x2207;</mi><mi mathvariant="normal">&#x03A6;</mi><mo>=</mo><mi>m</mi><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mi>r</mi><mo>¯</mo></mover></mrow></mrow><mo>¨</mo></mover></mrow></mrow><mo>+</mo><mi>q</mi><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">[</mo><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mi>&#x2202;</mi></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>&#x2202;</mi><mi>t</mi></mrow></mrow></mfrac></mrow><mi>A</mi><mo stretchy="false">(</mo><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mi>r</mi><mo>¯</mo></mover></mrow></mrow><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo><mo>+</mo><mi mathvariant="normal">&#x2207;</mi><mi mathvariant="normal">&#x03A6;</mi><mo>&#x2212;</mo><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">[</mo><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mi>v</mi><mo>¯</mo></mover></mrow></mrow><mo>&#x00D7;</mo><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mi mathvariant="normal">&#x2207;</mi><mo>&#x00D7;</mo><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mi>A</mi><mo>¯</mo></mover></mrow></mrow><mo data-mjx-texclass="CLOSE">)</mo></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

$d$

$d$

$t$

$L$

$\bar{r}$

$\bar{v}$

$t$

$v_{k}$

$m$

${\ddot{x}}_{k}$

$q$

$t$

$A_{k}$

$\bar{r}$

$t$

$A_{k}$

$\bar{r}$

$t$

$x_{l}$

$x_{l}$

$t$

$m$

${\ddot{x}}_{k}$

$q$

$t$

$\bar{v}$

$A_{k}$

$\bar{r}$

$t$

$L$

$\bar{r}$

$\bar{v}$

$t$

$x_{k}$

$q$

$x_{k}$

$\bar{v}$

$\bar{A}$

$x_{k}$

$Φ$

$d$

$d$

$t$

$L$

$\bar{r}$

$\bar{v}$

$t$

$v_{k}$

$L$

$\bar{r}$

$\bar{v}$

$t$

$x_{k}$

$m$

${\ddot{x}}_{k}$

$q$

$t$

$\bar{v}$

$A_{k}$

$\bar{r}$

$t$

$q$

$x_{k}$

$\bar{v}$

$\bar{A}$

$x_{k}$

$Φ$

$m$

${\ddot{x}}_{k}$

$q$

$t$

$A_{k}$

$\bar{r}$

$t$

$q$

$\bar{v}$

$A_{k}$

$\bar{r}$

$t$

$x_{k}$

$\bar{v}$

$\bar{A}$

$q$

$x_{k}$

$Φ$

$\bar{v}$

$A_{k}$

$\bar{r}$

$t$

$x_{k}$

$\bar{v}$

$\bar{A}$

$\bar{v}$

$\bar{A}$

$k$

$m$

$\ddot{\bar{r}}$

$q$

$t$

$A$

$\bar{r}$

$t$

$q$

$\bar{v}$

$\bar{A}$

$q$

$Φ$

$m$

$\ddot{\bar{r}}$

$q$

$t$

$A$

$\bar{r}$

$t$

$Φ$

$\bar{v}$

$\bar{A}$

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LaTeXML (experimentell; verwendet MathML) rendering

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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

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