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## Main Question or Discussion Point

I'm hoping this is a really simple question, but I can't seem to find a definitive answer anywhere!

The trouble I'm having relates to SU(N) yang-mills theories where I'm finding that even if the action is invariant under the SU(N) local gauge transformation, the equations of motion are not. I believe this is related to the fact that the lagrangian is

*edit* also, if someone could explain the physical significance of this that would be amazing, because I am completely confused. For example in the SU(N) theory, the right side of the EOM (which is just the source term) is "rotated" after the transformation. i.e. [itex][D_\nu,F^{\mu\nu}]=J^\mu \rightarrow [D_\nu,F^{\mu\nu}]=U^\dagger J^\mu U[/itex]

**If the action is invariant under some symmetry transformation, do the equations of motion need to be invariant as well?**The trouble I'm having relates to SU(N) yang-mills theories where I'm finding that even if the action is invariant under the SU(N) local gauge transformation, the equations of motion are not. I believe this is related to the fact that the lagrangian is

**not**invariant, but changes by a total derivative term. However I'm not sure if I'm doing something wrong or this is actually possible...*edit* also, if someone could explain the physical significance of this that would be amazing, because I am completely confused. For example in the SU(N) theory, the right side of the EOM (which is just the source term) is "rotated" after the transformation. i.e. [itex][D_\nu,F^{\mu\nu}]=J^\mu \rightarrow [D_\nu,F^{\mu\nu}]=U^\dagger J^\mu U[/itex]