Stability transitions for axisymmetric relative equilibria of Euclidean symmetric Hamiltonian systems

TL;DR

This paper investigates stability mechanisms of axisymmetric relative equilibria in Euclidean symmetric Hamiltonian systems, highlighting how stability types transition and applying findings to underwater vehicle models with numerical analysis.

Contribution

It introduces a detailed analysis of stability transitions for axisymmetric relative equilibria, including energy-momentum confinement, KAM, and Nekhoroshev stability, with application to underwater vehicle dynamics.

Findings

Identification of stability transition mechanisms

Application to Kirchhoff underwater vehicle model

Numerical study of dissipation-induced instability

Abstract

In the presence of noncompact symmetry, the stability of relative equilibria under momentum-preserving perturbations does not generally imply robust stability under momentum-changing perturbations. For axisymmetric relative equilibria of Hamiltonian systems with Euclidean symmetry, we investigate different mechanisms of stability: stability by energy-momentum confinement, KAM, and Nekhoroshev stability, and we explain the transitions between these. We apply our results to the Kirchhoff model for the motion of an axisymmetric underwater vehicle, and we numerically study dissipation induced instability of KAM stable relative equilibria for this system.