Computation of normal forms of Hamiltonian systems in the presence of Poisson commuting integrals

Abstract

The aim of this paper is to show the role of first integrals in further reducing the normal form unfolding of Hamiltonian systems. Based on a work by Cicogna and Gaeta, the joint normal form approach for Hamiltonian vector fields is considered. This normal form procedure, couched in a Lie-Poincare scheme, allows us to see that we can reduce simultaneously the Hamiltonian together with its Poisson commuting integrals to a simplified normal form - a joint normal form - which is given a simple characterization. In this algorithmic procedure, approximate first integrals can be constructed (and used to simplify the normal form) at the same time that we bring the Hamiltonian to normal form. Further, following Walcher, we show that we can derive the joint normal form via a structure preserving transformation (in a sense to be specified). The approach is discussed from an implementational standpoint and illustrated by a Liouville-integrable Henon-Heiles system.