Explicit Symmetry Breaking and Hamiltonian Systems

Abstract

The central topic of this thesis is the study of persistence of stationnary motion under explicit symmetry breaking perturbations in Hamiltonian systems. Explicit symmetry breaking occurs when a dynamical system having a certain symmetry group is perturbed in a way that the perturbation preserves only some symmetries of the original system. We give a geometric approach to study this phenomenon in the setting of equivariant Hamiltonian systems. A lower bound for the number of orbits of equilibria and orbits of relative equilibria which persist after a small perturbation is given. This bound is given in terms of the equivariant Lyusternik-Schnirelmann category of the group orbit. We also find a localization formula for this category in terms of the closed orbit-type strata. We show that this formula holds for topological spaces admitting a particular cover, made of tubular neighbourhoods of their minimal orbit-type strata. Finally we propose a construction of symplectic slices for subgroup actions.

Date of Award	1 Aug 2018
Original language	English
Awarding Institution	The University of Manchester
Supervisor	James Montaldi (Supervisor) & Hovhannes Khudaverdyan (Supervisor)