O'Callaghan, Matthew
(2021)
Symmetry and Symplectic Reduction.
[Preprint]
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Abstract
In the last hundred years, Classical Mechanics has been dwarfed by the development of Relativity and
Quantum Mechanics. However, significant mathematical developments have been made to Classical Mechanics in the same time frame and this essay aims to explore some of the recent advances. The purpose of this
essay is to understand Hamiltonian dynamics, Noether’s theorem and symmetry in a geometric framework
using Poisson geometry. In particular, we want to analyse the case when the natural configuration space of
a system is a Lie group G and transformations on the space correspond to the Lie group acting on itself.
In this case, the Lie-Poisson reduction theorem will say that the reduced phase space T
∗G/G is a Poisson
manifold diffeomorphic to g
∗
. We will begin Chapter 1 by outlining the core concepts required to understand
symmetry, in particular, we will discuss the notion of a Lie group, its Lie algebra and how each of these
objects acts on certain manifolds. In Chapter 2 we will define Poisson and Symplectic manifolds and use
them to derive equations of motion and dynamics. Chapter 3 will discuss the geometric understanding of
Noether’s theorem by looking at momentum maps. Finally, Chapter 4 will discuss the Lie-Poisson reduction
theorem. Throughout the essay, we will aim to illustrate the concepts using the physically relevant cases
which arise when the configuration space G is a three-dimensional Lie group over the reals.
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