# Full Lagrangian for Electrodynamics

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?