Full Lagrangian for Electrodynamics

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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?
 

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Hi,
The point is that to write down the Lagrangian of a matter one has to specify the type of matter. The Lagrangian of a pressure-less fluid of charged particles is simply the sum of the Lagrangian of each particle. But if you want to take into account the interaction between particles , something like shear or pressure, you may need to describe the system phenomenologically and principally the dynamics of such fluids could not be driven from a variational concept and a Lagrangian.Instead the dynamics of the Electromagnetic field and matter flow in the level of Classical relativistic physics could be explained in terms of second law of Newton without the need of a Lagrangian.
 

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