"Lagrangian formulation of electromagetism"의 두 판 사이의 차이
둘러보기로 가기
검색하러 가기
imported>Pythagoras0 |
imported>Pythagoras0 |
||
12번째 줄: | 12번째 줄: | ||
:<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> | ||
+ | |||
+ | |||
+ | |||
+ | ==memo== | ||
+ | * http://dexterstory.tistory.com/888 | ||
+ | |||
+ | |||
+ | |||
+ | ==related items== | ||
+ | * [[path integral formulation of quantum mechanics|path integral]] | ||
+ | |||
18번째 줄: | 29번째 줄: | ||
* [http://www.physics.sfsu.edu/~lea/courses/grad/fldlagr.PDF The field Lagrangian] | * [http://www.physics.sfsu.edu/~lea/courses/grad/fldlagr.PDF The field Lagrangian] | ||
* http://www.lecture-notes.co.uk/susskind/classical-mechanics/lecture-8/the-electromagnetic-lagrangian/ | * http://www.lecture-notes.co.uk/susskind/classical-mechanics/lecture-8/the-electromagnetic-lagrangian/ | ||
− | |||
− | == | + | |
− | * | + | ==questions== |
+ | * http://physics.stackexchange.com/questions/3005/derivation-of-maxwells-equations-from-field-tensor-lagrangian?rq=1 |
2013년 3월 23일 (토) 23:24 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 \(e\mathbf{E}+e\mathbf{v}\times \mathbf{B}\)
memo
expositions
- THOMAS YU LAGRANGIAN FORMULATION OF THE ELECTROMAGNETIC FIELD
- The field Lagrangian
- http://www.lecture-notes.co.uk/susskind/classical-mechanics/lecture-8/the-electromagnetic-lagrangian/