Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 14 (2018), 021, 37 pages      arXiv:1707.02828      https://doi.org/10.3842/SIGMA.2018.021

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

Received October 31, 2017, in final form March 09, 2018; Published online March 14, 2018

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.

Key words: equivariant dynamics; relative equilibria; orbital stability; Hamiltonian systems.

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