# 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?

## Answers and Replies

vanhees71
Science Advisor
Gold Member
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.

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?