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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.

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  1. OP is right: In the pure Maxwell action with source term, the gauge fields are the dynamical fields while the matter (e.g. electron field, point charges) are the background. The EL eqs. are the Maxwell eqs.

  2. We can also have the opposite situation: A matter action (e.g. Dirac action) with a source term, where the matter are the dynamical variables and the gauge fields are background fields. The EL eqs. are the matter eqs. (e.g. Dirac eqs.).

  3. And finally, we can consider the sum of the Maxwell & matter action (minimally coupled to preserve gauge symmetry), where both gauge fields & matter are dynamical variables.

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