Normal forms for Lie symmetric cotangent bundle systems with free and proper actions

TL;DR

This paper develops a method for analyzing Hamiltonian systems with symmetries by constructing symplectic slice coordinates and an iterative scheme for normal form computation, facilitating the study of dynamics near relative equilibria.

Contribution

It introduces a new approach to normal form calculations for symmetric Hamiltonian systems using derivatives at equilibria, avoiding explicit coordinate transformations.

Findings

Constructed symplectic slice coordinates for SO(3) symmetry.

Outlined an iterative scheme for truncated normal form computation.

Provided a method to analyze dynamics near relative equilibria.

Abstract

We consider free and proper cotangent-lifted symmetries of Hamiltonian systems. For the special case of G = SO(3), we construct symplectic slice coordinates around an arbitrary point. We thus obtain a parametrisation of the phase space suitable for the study of dynamics near relative equilibria, in particular for the Birkhoff-Poincare normal form method. For a general symmetry group, we observe that for the calculation of the truncated normal forms, one does not need an explicit coordinate transformation but only its higher derivatives at the relative equilibrium. We outline an iterative scheme using these derivatives for the computation of truncated Birkhoff-Poincare normal forms.