The Lie Transform Method for perturbations of contravariant antisymmetric tensor fields and its applications to Hamiltonian dynamics

TL;DR

This paper extends the Lie transform method to contravariant antisymmetric tensor fields, enabling analysis of perturbations in Hamiltonian systems and applications to Lie coalgebras and Dirac brackets.

Contribution

It introduces a novel application of the Lie transform method to tensor fields and Hamiltonian dynamics, utilizing Schouten calculus and Poisson cohomology for perturbation analysis.

Findings

Derived infinitesimal generators for Lie transformations.

Applied methods to perturbed Euler equations on Lie coalgebras.

Analyzed Hamiltonian systems with Dirac brackets.

Abstract

By means of the Schouten calculus for contravariant antisymmetric tensor fields, we apply the Lie transform method to investigate smooth deformations of tensor fields and, in particular, to perturbations of Hamiltonian systems generated by deformations of the Poisson bracket. Using results by Karasev and Vorobiev on the computation of Poisson cohomology we describe infinitesimal generators for the Lie transformations. We give applications to perturbed Euler equations on six dimensional Lie coalgebras and to Hamiltonian systems on Poisson manifolds equipped with Dirac brackets.