# Deriving energy conservation from EM Lagrangian

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## Main Question or Discussion Point

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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