NORMAL FORMS OF SYMMETRICAL HAMILTONIAN-SYSTEMS

We study the following question: to what simplest normal form can a Ha miltonian with a symmetry group GAMMA be reduced by a GAMMA-equivarian t contactomorphism (a contactomorphism conjugated with each transforma tion from GAMMA). In particular, we point out conditions under which t here exists a GAMMA-equivariant contactomorphism reducing a GAMMA-inva riant Hamiltonian to a GAMMA-equivariant Birkhoff normal form. In reso nance cases the Birkhoff normal form can be simplified. We present a m ethod of reduction to an invariant normal form, independent of informa tion on symmetries. At the same time under certain conditions the inva riant normal form of a GAMMA-invariant Hamiltonian is also GAMMA-invar iant and the reduction to it can be realized via a GAMMA-equivariant c ontactomorphism. We understand the word ''invariant'' in the following sense: two Hamiltonians (GAMMA-invariant) are equivalent (under the a ction of the group of r-equivariant contactomorphisms) if and only if their invariant normal forms coincide. (C) 1994 Academic Press, Inc.