Actions, symmetries, and gauge theories


Suppose that we have some theory which is invariant under the action of
a gauge group, G. Since this theory is a gauge theory, it can be
derived from a singular Lagrangian or Lagrangian density, L. Now
suppose that the action for this theory,

S = \int L(q,\dot{q})

is invariant under the action of some group H. What is the relationship
between the groups G and H?

I guess the reason I'm interested in asking this is to find out whether
or not the symmetries of an action functional must necessarily be the
same as the symmetries of the resultant equations of motion for the
theory. As an extension, I wonder if the symmetries of an action could
generate a different symmetry group in the equations of motion. It
strikes me that this is an obvious question, but I can't seem to find
any (rigorous) information on it.


--
coalquay404
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