Lagrangian formulation of electromagetism

Lagrangian for a particle

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 $e\mathbf{E}+e\mathbf{v}\times \mathbf{B}$

Lagrangian for electromagnetic tensor

상호작용이 없는 전자기장의 라그랑지안은 다음과 같다

$$\mathcal{L}_{\text{EM}}= - \frac{1}{4}F_{\mu\nu}F^{\mu\nu}$$ 이 때 $F_{\mu\nu} = \partial_\mu A_\nu - \partial_\nu A_\mu \,\!$는 전자기텐서, $A=(A_{\mu})$는 전자기 포텐셜