Normal Forms of Symmetrical Hamiltonian Systems

✍ Scribed by M. Zhitomirskii

• Publisher: Elsevier Science
• Year: 1994
• Tongue: English
• Weight: 826 KB
• Volume: 111
• Category: Article
• ISSN: 0022-0396
• DOI: 10.1006/jdeq.1994.1075

✦ Synopsis

We study the following question: to what simplest normal form can a Hamiltonian with a symmetry group (\Gamma) be reduced by a (\Gamma)-equivariant contactomorphism (a contactomorphism conjugated with each transformation from (\Gamma) ). In particular, we point out conditions under which there exists a (\Gamma)-equivariant contactomorphism reducing a (\Gamma)-invariant Hamiltonian to a (\Gamma)-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 (\Gamma)-invariant Hamiltonian is also (\Gamma)-invariant and the reduction to it can be realized via a (\Gamma)-equivariant contactomorphism. We understand the word "invariant" in the following sense: two Hamiltonians ( (\Gamma)-invariant) are equivalent (under the action of the group of (\Gamma)-equivariant contactomorphisms) if and only if their invariant normal forms coincide. 1994 Academic Press, Inc.