# Algebraic and Geometric Methods in Mechanics

• Nataliya Balabanova

Student thesis: Phd

### Abstract

This work consists of five chapters; in each one we approach a different prob- lem. Chapter 1 is an introduction, providing the basic definitions and tools we use throughout the thesis; each of the chapters has a specific introduction with the re- quired background theory. The second chapter is dedicated to investigating the two-body problem on a sphere in the presence of a magnetic monopole, with the focus on proving existence of relative equilibria in the general case. When the two bodies are identical, we locate all relative equilibria and determine their linear stability. Additionally, we demonstrate how the gravitational two-body problem on a sphere appears as a limit of the magnetic one. Chapters 3 and 4 are concerned with point vortex dynamics on non-orientable manifolds; we introduce the theoretical framework in the beginning of Chapter 3 and proceed to look in detail at the case of the Mobius band. In Chapter 4 we investigate point vortex motion on the Klein bottle. For both surfaces, we pose the same questions: write the explicit form of the Hamiltonian, local equations of motion, momentum map and investigate certain relative equilibria as well as the motion of small numbers of vortices. The last part of Chapter 4 is dedicated to using twisted forms to write twisted analogues of Hamiltonian equations on non-orientable manifolds Analogously to real forms of a complex Lie algebra, in Chapter 5 we introduce the notion of real forms of a complex Hamiltonian system and complexification of a real Hamiltonian system; we find out that various real forms of the same complex one can be vastly different. However, it has been observed that certain properties 'survive' complexification and, therefore, are shared by all real forms: integrability, collectiveness of the Hamiltonian, etc. The momentum map also undergoes very predictable transformations. We complexify and construct alternative real forms of the two well-known real systems: the two point vortices on a sphere and the two- body problem on a sphere. Additionally, we look in detail at the case of various real forms of n-dimensional complex rigid body; in particular, we draw comparisons between so(2, 1) and so(3,R) cases and investigate relative equilibria and their stability for the so(2, 2) and so(3, 1)-rigid bodies.
Date of Award 1 Aug 2022 English The University of Manchester James Montaldi (Supervisor) & Alexander Premet (Supervisor)

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