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.