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The action is the integral of the Lagrangian over time, or the integral of the Lagrangian Density over both time and space.
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Help with Invariance of Euler-Lagrange equations under a gauge transformation?
Ok, so after many months I could finally come back to this and answer the problematic section properly.
Starting from the transformed Lagrangian:
$$L' = L + e(\partial_{t}\phi + \vec \nabla\phi \cdot …
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Help with Invariance of Euler-Lagrange equations under a gauge transformation?
Explain this invariance from the point of view of the action.
As far as I know, $V, A, \phi$ depend on $(r(t), t)$- it really isn't explicit. …