# Euler-Lagrange equations for electrodynamics: matter vs. gauge field change

I know how to derive Maxwell's equations from a Lagrangian density via the Euler-Lagrange equations, but I'm bothered by one little detail.

Variation of the Lagrangian seems only to be done with respect to $$A_\mu$$ and $$A_\mu,_\nu$$. I haven't seen $$j_\mu$$ varied in the papers I've read on the subject. As a result, Maxwell's equations seem to ignore the effect the field will have on the source current density. It seems to me that $$j_\mu$$ is as much part of the whole field/current density system as $$A_\mu$$ and $$A_\mu,_\nu$$ are, so why isn't it treated the same way?

I suppose that to include $$j_\mu$$ as a variable on par with $$A_\mu$$ and $$A_\mu,_\nu$$, it would be necessary to know just what $$j_\mu$$ is made of, such as electrons in a conductor, or a "liquid" or "dust" composed of particles having a specific charge-to-mass ratio.

So, my guess at an answer is just that my exposure to electrodynamics has been too narrowly focused at vacuum fields; and that the Lagrangian and the equations of motion look a lot different in real, practical systems. I hope someone can clarify this for me. A link to a paper explaining it would be a big help.