Normal Forms of Symmetrical Hamiltonian Systems

We study the following question: to what simplest normal form can a Hamiltonian with a symmetry group Γ be reduced by a Γ-equivariant contactomorphism (a contactomorphism conjugated with each transformation from Γ). In particular, we point out conditions under which there exists a Γ-equivariant contactomorphism reducing a Γ-invariant Hamiltonian to a Γ-equivariant Birkhoff normal form. In resonance cases the Birkhoff normal form can be simplified. We present a method of reduction to an invariant normal form, independent of information on symmetries. At the same time under certain conditions the invariant normal form of a Γ-invariant Hamiltonian is also Γ-invariant and the reduction to it can be realized via a Γ-equivariant contactomorphism. We understand the word "invariant" in the following sense: two Hamiltonians (Γ-invariant) are equivalent (under the action of the group of Γ-equivariant contactomorphisms) if and only if their invariant normal forms coincide.