Deriving energy conservation from EM Lagrangian

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I'm trying to derive the conservaton of energy for electromagnetic fields with currents from the action principle, but I have some trouble understanding how the interaction term in the Lagrangian fits into this.

The approach I have seen so far has been to express the Lagrangian density as $$\mathcal{L}(x^\alpha, A_\alpha, \partial_\beta A_\alpha) = \mathcal{L}_{field} + \mathcal{L}_{int} = -\frac {1} {4\mu_0}F^{\alpha \beta}F_{\alpha \beta} - A_\alpha J^\alpha$$ and then derive the equations of motion from that in the usual way. This leads to Maxwell's equations.

The problem I have with this approach is that ##J^\alpha(x)## depends on the space-time coordinates. This means that the Lagrangian is not invariant with respect to time and I cannot derive energy conservation using time translational symmetry. Without the interaction term, this works fine.

The above Lagrangian only describes the motion of ##A_\alpha##. Is there a way to formulate a Lagrangian that describes how ##A_\alpha## and ##J^\alpha## evolves together?
 

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vanhees71
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Sure, there's exchange of energy, momentum, and angular momentum between the em. field and the charges. You can derive the expressions for the appropriate energy-momentum and angular-momentum densities (modulo total divergences which are fixed by the demand of gauge invariance, which leads from the canonical to the Belinfante energy-momentum tensor and the usual relation of it to the angular-momentum tensor) of the em. field. Then including the interactions with the charges leads to the additional terms in the energy-momentum-angular-momentum balance equations of the electromagnetic field, leading to the correct Lorentz-force form of the equation of motion.
 
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Thanks for your reply. Unfortunately, it is a little over my head. In particular, I had not heard of the Belinfante energy-momentum tensor until just now. I know how to derive the canonical energy-momentum tensor from the lagrangian density though. I am basically at the level where I can understand the "Theoretical Minimum" lectures or "The Variational Principles of Mechanics" as well as some differential geometry.

Do you mind elaborating a bit or provide pointers where I can find more information?
 

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