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This is a followup to the old thread What are the FULL classical electrodynamic equations? which never really provided a satisfactory answer.

I have decided to phrase it perhaps in a more straightforward manner. Given that we have the EM field, or the equivalent potential field, and charged matter. What is the lagrangian for this system?

I will be happy if given in terms of point charges or charge density. But it has to include both the action of the field on the charges and the fact that the sources of the field are those same charges. Writing down the lagrangian for an EM field with fixed sources or point particles influenced by a fixed field is easy and in every textbook.

Has anyone ever even

For an example of what I am talking about, here is the lagrangian density for spin 0 charges + EM field from quantum field theory. (This is just conceptual. Signs and constants might be wrong. H-bar and c are set to 1.)

[tex] (i\partial_{\mu}\phi^{\dag}-eA_{\mu})(i\partial^{\mu}\phi-eA^{\mu}) + m\phi^{\dag}\phi -\frac{1}{16}F^{\alpha\beta}F_{\alpha\beta}[/tex]

I am looking for the classical analog to this lagrangian.

I have decided to phrase it perhaps in a more straightforward manner. Given that we have the EM field, or the equivalent potential field, and charged matter. What is the lagrangian for this system?

I will be happy if given in terms of point charges or charge density. But it has to include both the action of the field on the charges and the fact that the sources of the field are those same charges. Writing down the lagrangian for an EM field with fixed sources or point particles influenced by a fixed field is easy and in every textbook.

Has anyone ever even

*seen*what I am asking for here?For an example of what I am talking about, here is the lagrangian density for spin 0 charges + EM field from quantum field theory. (This is just conceptual. Signs and constants might be wrong. H-bar and c are set to 1.)

[tex] (i\partial_{\mu}\phi^{\dag}-eA_{\mu})(i\partial^{\mu}\phi-eA^{\mu}) + m\phi^{\dag}\phi -\frac{1}{16}F^{\alpha\beta}F_{\alpha\beta}[/tex]

I am looking for the classical analog to this lagrangian.

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