Nonlinear Stability of Relative Equilibria and Isomorphic Vector Fields

Stefan Klajbor-Goderich

Department of Mathematics, University of Illinois at Urbana-Champaign, 1409 W. Green Street, Urbana, IL 61801 USA

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 Rodríguez-Olmos's criterion [arXiv:1509.04961] for stability of Hamiltonian relative equilibria.

Keywords: equivariant dynamics; relative equilibria; orbital stability; Hamiltonian systems.

MSC: 37J25; 57R25; 37J15; 53D20

Received: October 31, 2017; in final form March 9, 2018; Published online March 14, 2018

Language: English

DOI: 10.3842/SIGMA.2018.021

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