Lagrangian formulation of electromagetism

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
• The field Lagrangian
• http://www.lecture-notes.co.uk/susskind/classical-mechanics/lecture-8/the-electromagnetic-lagrangian/
• http://dexterstory.tistory.com/888

related items

• path integral