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Discontinuity, Nonlinearity, and Complexity 10(3) (2021) 347--367 | DOI:10.5890/DNC.2021.09.001

Lorenzo Valvo$^1$ , Michel Vittot$^2$

$^1$ Dipartimeno di Matematica, Universita degli Studi di Roma Tor Vergata, Via della Ricerca Scientifica 1 00133 Roma, Italy

$^2$ Aix Marseille Univ, Universite de Toulon, CNRS, CPT, Marseille, France

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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.

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