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

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## Abstract

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) |
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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: | https://eprints.maths.manchester.ac.uk/id/eprint/2619 |

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