On the Global Structure of Normal Forms for Slow-Fast Hamiltonian Systems

TL;DR

This paper investigates the global structure of normal forms in slow-fast Hamiltonian systems with periodic fast flows, using Lie transforms and averaging, revealing an intrinsic splitting related to the Hannay-Berry connection.

Contribution

It introduces a novel intrinsic splitting of the first-order normal form in systems with non-free S^1-actions, expanding the understanding of Hamiltonian normal forms.

Findings

Derived an intrinsic splitting of the second term in the normal form.

Connected the splitting to the Hannay-Berry connection.

Extended normal form theory to non-free S^1-actions.

Abstract

In the framework of Lie transform and the global method of averaging, the normal forms of a multidimensional slow-fast Hamiltonian system are studied in the case when the flow of the unperturbed (fast) system is periodic and the induced S^1-action is not necessarily free and trivial. An intrinsic splitting of the second term in a S^1-invariant normal form of first order is derived in terms of the Hannay-Berry connection associated with the periodic flow.