# Full Lagrangian for Electrodynamics

1. Jan 18, 2014

### dEdt

The full Lagrangian for electrodynamics $\mathcal{L}$ can be expressed as $\mathcal{L}=\mathcal{L}_\textrm{field}+ \mathcal{L}_\textrm{interaction}+\mathcal{L}_\textrm{matter}$. Practically every textbook on relativity shows that $\mathcal{L}_\textrm{field}=-\frac{1}{4}F_{\alpha\beta}F^{\alpha\beta}$ and that $\mathcal{L}_\textrm{interaction}=-A_\alpha J^\alpha$. However, the expression for $\mathcal{L}_\textrm{matter}$ is almost never included, except for the simple case of a point particle. This term is needed in order to solve for the motion of a charge distribution. Presumably applying the Euler-Lagrange equations to the full Lagrangian will yield $f^\alpha=F^{\alpha\beta}J_\beta$, where $f^\alpha$ is the 4-force density.

So what is $\mathcal{L}_\textrm{matter}$ for a general charge distribution?

2. Jan 21, 2014