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 PoincarÃ© 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.
|Date of Award||1 Aug 2020|
- The University of Manchester
|Supervisor||James Montaldi (Supervisor) & Yuri Bazlov (Supervisor)|
SEMIDIRECT PRODUCTS AND APPLICATIONS TO GEOMETRIC MECHANICS
Arathoon, P. (Author). 1 Aug 2020
Student thesis: Phd