Normal forms of symmetric Hamiltonian systems (Q1332164)

A method of reduction of a Hamiltonian to an invariant normal form is presented in the formal category. It is proved that under certain conditions on symmetry group \(\Gamma\) of Hamiltonian its invariant normal form is \(\Gamma\)-invariant too and reduction to it can be realized via a \(\Gamma\)-equivariant contactomorphism. Conditions under which there exist a \(\Gamma\)-equivariant contactomorphism reducing a \(\Gamma\)-invariant Hamiltonian to a \(\Gamma\)-invariant Birkhoff normal form are found. It is shown that in resonance cases the Birkhoff normal forms can be simplified.