Hamiltonian Perturbation Theory on a Lie Algebra. Application to a non-autonomous Symmetric Top

TL;DR

This paper introduces a perturbation algorithm for Hamiltonian systems on Lie algebras, enabling the preservation of subalgebras under perturbation, demonstrated through a non-autonomous symmetric top example.

Contribution

It develops a novel algebraic transformation method to maintain subalgebra invariance in perturbed Hamiltonian systems, extending applicability to non-canonical cases.

Findings

Successfully preserves subalgebra structure up to quadratic perturbation terms.

Applies the method to analyze the dynamics of a non-autonomous symmetric rigid body.

Demonstrates the algebraic transform's role similar to the Iterative Lemma in KAM theory.

Abstract

We propose a perturbation algorithm for Hamiltonian systems on a Lie algebra $\mathbb{V}$, so that it can be applied to non-canonical Hamiltonian systems. Given a Hamiltonian system that preserves a subalgebra $\mathbb{B}$ of $\mathbb{V}$, when we add a perturbation the subalgebra $\mathbb{B}$ will no longer be preserved. We show how to transform the perturbed dynamical system to preserve $\mathbb{B}$ up to terms quadratic in the perturbation. We apply this method to study the dynamics of a non-autonomous symmetric Rigid Body. In this example our algebraic transform plays the role of Iterative Lemma in the proof of a KAM-like statement.