Semidirect Products and Applications to Geometric Mechanics

Arathoon, Philip (2019) Semidirect Products and Applications to Geometric Mechanics. Doctoral thesis, University of Manchester.

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In this thesis we provide an overview of themes in geometric mechanics and apply them to the study of adjoint and coadjoint orbits of a semidirect product, and to the two-body problem on a sphere. Firstly, we show the existence of a geometrically defined bijection between the sets of adjoint and coadjoint orbits for a particular class of semidirect product. We demonstrate the bijection for the examples of the affine linear group and the Poincaré group. Additionally, we prove that any two orbits paired between this bijection are homotopy equivalent. Secondly, a correspondence is found between the two-body problem on a three- dimensional sphere and the four-dimensional Lagrange top. This correspondence establishes an equivalence between the two problems after reduction, and allows us to treat both reduced problems simultaneously. We implement a semidirect product reduction by stages to exhibit the reduced spaces as coadjoint orbits of a special Euclidean group, and then reduce by a further symmetry to obtain a full reduced system. This allows us to fully classify the relative equilibria for both problems and describe their stability.

Item Type: Thesis (Doctoral)
Subjects: MSC 2010, the AMS's Mathematics Subject Classification > 37 Dynamical systems and ergodic theory
Divisions: Manchester Institute for the Mathematical Sciences
Depositing User: Mr Philip Arathoon
Date Deposited: 07 Jan 2020 11:36
Last Modified: 07 Jan 2020 11:36

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