Unless for some reason you absolutely must work within the Hamiltonian approach, you can just directly look for the complete set of (infinitesimal Lie point) symmetries of the Euler--Lagrange equations or of the action itself. The procedure is standard and described in many good books. For instance, you can look into those by Olver (more math-y) or Stephani (somewhat closer to physics). Using the theory from this book these books you can also verify whether the transformation at the end of your question is indeed a symmetry.
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Unless for some reason you absolutely must work within the Hamiltonian approach, you can just directly look for the complete set of (infinitesimal Lie point) symmetries of the Euler--Lagrange equations or of the action itself. The procedure is standard and described in many good books. For instance, you can look into those by Olver (more math-y) or Stephani (somewhat closer to physics). Using the theory from this book you can also verify whether the transformation at the end of your question is indeed a symmetry. |
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