The discussion confirms that while actions are invariant under gauge transformations, the equations of motion are generally covariant rather than invariant. Specifically, in electrodynamics, the equations of motion maintain invariance under gauge transformations, whereas in Yang-Mills theories, they are only covariant. Additionally, the Einstein-Hilbert action is invariant under general coordinate transformations, but the Einstein equations of motion are also merely covariant.
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Thanks!Demystifier said:No, in general they are just covariant. For electrodynamics the equations of motion are invariant under gauge transformations, but for Yang-Mills theories they are just covariant. Similarly, the Einstein-Hilbert action is invariant under general coordinate transformations, but the Einstein equation of motion is just covariant.