progato
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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?
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?