A Are equations of motion invariant under gauge transformations?

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The discussion centers on the invariance of equations of motion under gauge transformations. It confirms that while the action is invariant, the equations of motion are generally covariant rather than invariant. Specifically, in electrodynamics, the equations of motion are invariant 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 just covariant. Thus, the distinction between invariance and covariance is crucial in these theories.
Baela
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We know that all actions are invariant under their gauge transformations. Are the equations of motion also invariant under the gauge transformations?

If yes, can you show a mathematical proof (instead of just saying in words)?
 
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Yes. Since the action is the same the path of least action is also the same.
 
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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.
 
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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.
Thanks!
 
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