Monodromy, Chern Classes, and their Physical Significance
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Date
2016-05
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The Ohio State University
Abstract
The purpose of this thesis is to explore the mathematics behind these action-angle coordinates, and examine what happens in the "global" case. We begin with the analytic formulation, within which we follow the historical and physical development of actions and angles. We then turn to the underlying topology, and examine actions and angle in this context. In particular, we will examine the two main topological restrictions that exist when attempting to construct global action-angle coordinates: Chern classes and monodromy. This is followed by a discussion of the spherical pendulum. This system is interesting because it is a nice enough mechanical system for which our topological obstructions do in fact keep us from constructing global action angle coordinates. Finally, we give a taste of some of the more modern accounts of this theory. In particular we briefly touch on Lie algebra actions on symplectic manifolds, and relate them to the other discussions in the thesis.
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differential geometry, mathematical physics, classical mechanics, algebraic topology