"Lagrangian formulation of electromagetism"의 두 판 사이의 차이
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imported>Pythagoras0 (새 문서: ==introduction== * Lagrangian for a charged particle in an electromagnetic field <math>L=T-V</math> :<math>L(q,\dot{q})=m||\dot{q}||-e\phi+eA_{i}\dot{q}^{i}</math><br> * action :<ma...) |
imported>Pythagoras0 |
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| 1번째 줄: | 1번째 줄: | ||
==introduction== | ==introduction== | ||
* Lagrangian for a charged particle in an electromagnetic field <math>L=T-V</math> | * Lagrangian for a charged particle in an electromagnetic field <math>L=T-V</math> | ||
| − | :<math>L(q,\dot{q})=m||\dot{q}||-e\phi+eA_{i}\dot{q}^{i}</math | + | :<math>L(q,\dot{q})=m||\dot{q}||-e\phi+eA_{i}\dot{q}^{i}</math> |
* action | * action | ||
| − | :<math>S=-\frac{1}{4}\int F^{\alpha\beta}F_{\alpha\beta}\,d^{4}x</math | + | :<math>S=-\frac{1}{4}\int F^{\alpha\beta}F_{\alpha\beta}\,d^{4}x</math> |
* Euler-Lagrange equations | * Euler-Lagrange equations | ||
:<math>p_{i}=\frac{\partial{L}}{\partial{\dot{q}^{i}}}=m\frac{\dot{q}_{i}}{||\dot{q}_{i}||}+eA_{i}=mv_{i}+eA_{i}</math> | :<math>p_{i}=\frac{\partial{L}}{\partial{\dot{q}^{i}}}=m\frac{\dot{q}_{i}}{||\dot{q}_{i}||}+eA_{i}=mv_{i}+eA_{i}</math> | ||
| 9번째 줄: | 9번째 줄: | ||
F_{i}=\frac{\partial{L}}{\partial{q^{i}}}=\frac{\partial}{\partial{{q}^{i}}}(eA_{j}\dot{q}^{j})=e\frac{\partial{A_{j}}}{\partial{q}^{i}}\dot{q}^{j} | F_{i}=\frac{\partial{L}}{\partial{q^{i}}}=\frac{\partial}{\partial{{q}^{i}}}(eA_{j}\dot{q}^{j})=e\frac{\partial{A_{j}}}{\partial{q}^{i}}\dot{q}^{j} | ||
$$ | $$ | ||
| − | * equation of motion | + | * equation of motion<math>\dot{p}=F</math> Therefore we get |
| − | :<math>m\frac{dv_{i}}{dt}=eF_{ij}\dot{q}^{j}</math>. This is what we call the Lorentz force law. | + | :<math>m\frac{dv_{i}}{dt}=eF_{ij}\dot{q}^{j}</math>. This is what we call the Lorentz force law. |
* force on a particle is same as <math>e\mathbf{E}+e\mathbf{v}\times \mathbf{B}</math> | * force on a particle is same as <math>e\mathbf{E}+e\mathbf{v}\times \mathbf{B}</math> | ||
| + | |||
| + | ==expositions== | ||
* THOMAS YU [http://math.uchicago.edu/~may/REU2012/REUPapers/Yu.pdf LAGRANGIAN FORMULATION OF THE ELECTROMAGNETIC FIELD] | * THOMAS YU [http://math.uchicago.edu/~may/REU2012/REUPapers/Yu.pdf LAGRANGIAN FORMULATION OF THE ELECTROMAGNETIC FIELD] | ||
| − | * http://dexterstory.tistory.com/888 | + | * http://www.lecture-notes.co.uk/susskind/classical-mechanics/lecture-8/the-electromagnetic-lagrangian/ |
| − | * [[path integral formulation of quantum mechanics|path integral]] | + | * http://dexterstory.tistory.com/888 |
| + | |||
| + | ==related items== | ||
| + | * [[path integral formulation of quantum mechanics|path integral]] | ||
2013년 3월 23일 (토) 10:17 판
introduction
- Lagrangian for a charged particle in an electromagnetic field \(L=T-V\)
\[L(q,\dot{q})=m||\dot{q}||-e\phi+eA_{i}\dot{q}^{i}\]
- action
\[S=-\frac{1}{4}\int F^{\alpha\beta}F_{\alpha\beta}\,d^{4}x\]
- Euler-Lagrange equations
\[p_{i}=\frac{\partial{L}}{\partial{\dot{q}^{i}}}=m\frac{\dot{q}_{i}}{||\dot{q}_{i}||}+eA_{i}=mv_{i}+eA_{i}\] $$ F_{i}=\frac{\partial{L}}{\partial{q^{i}}}=\frac{\partial}{\partial{{q}^{i}}}(eA_{j}\dot{q}^{j})=e\frac{\partial{A_{j}}}{\partial{q}^{i}}\dot{q}^{j} $$
- equation of motion\(\dot{p}=F\) Therefore we get
\[m\frac{dv_{i}}{dt}=eF_{ij}\dot{q}^{j}\]. This is what we call the Lorentz force law.
- force on a particle is same as \(e\mathbf{E}+e\mathbf{v}\times \mathbf{B}\)
expositions
- THOMAS YU LAGRANGIAN FORMULATION OF THE ELECTROMAGNETIC FIELD
- http://www.lecture-notes.co.uk/susskind/classical-mechanics/lecture-8/the-electromagnetic-lagrangian/
- http://dexterstory.tistory.com/888