Explicit symmetry breaking and Hamiltonian systems

Fontaine, Marine (2017) Explicit symmetry breaking and Hamiltonian systems. Doctoral thesis, University of Manchester.

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The central topic of this thesis is the study of persistence of stationary 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.

Item Type: Thesis (Doctoral)
Uncontrolled Keywords: Symmetry breaking, momentum map, bifurcations
Subjects: MSC 2010, the AMS's Mathematics Subject Classification > 22 Topological groups, Lie groups
MSC 2010, the AMS's Mathematics Subject Classification > 37 Dynamical systems and ergodic theory
MSC 2010, the AMS's Mathematics Subject Classification > 70 Mechanics of particles and systems
Depositing User: Dr James Montaldi
Date Deposited: 06 Feb 2018 11:43
Last Modified: 06 Feb 2018 11:43
URI: http://eprints.maths.manchester.ac.uk/id/eprint/2619

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