Nonlinear Stability of Relative Equilibria and Isomorphic Vector Fields

TL;DR

This paper explores the use of isomorphic vector fields to analyze the nonlinear stability of relative equilibria, providing new criteria and an alternative proof for Hamiltonian systems.

Contribution

It introduces the application of isomorphic vector fields to nonlinear stability analysis and offers a new proof for existing stability criteria.

Findings

Established a criterion for nonlinear stability using isomorphic vector fields.

Provided an alternative proof for the stability of Hamiltonian relative equilibria.

Demonstrated the usefulness of replacing equivariant vector fields with isomorphic ones.

Abstract

We present applications of the notion of isomorphic vector fields to the study of nonlinear stability of relative equilibria. Isomorphic vector fields were introduced by Hepworth [Theory Appl. Categ. 22 (2009), 542-587] in his study of vector fields on differentiable stacks. Here we argue in favor of the usefulness of replacing an equivariant vector field by an isomorphic one to study nonlinear stability of relative equilibria. In particular, we use this idea to obtain a criterion for nonlinear stability. As an application, we offer an alternative proof of Montaldi and Rodr\'iguez-Olmos's criterion [arXiv:1509.04961] for stability of Hamiltonian relative equilibria.