My bachelor thesis on symplectic geometry. Thesis was uploaded on October 16th 2024 and slides were presented on October 30th. I might update the LaTeX sources to make them more readable some time in the future.

Abstract

Symplectic geometry is a branch of differential geometry which aims to generalize the geometric (coordinate-independent) properties of the motion of physical systems. This work aims to explore its physical roots and give an example of its applciations in this field. The newtonian, lagrangian and hamiltonian formulations of classical mechanics are reviewed and put on a formal basis. A geometrical formulation of the mechanics of unconstrained systems is given. Subsequently, the theory of differential manifolds is explored and vectors, differential forms and operations on these objects are defined. The canonical symplectic structure of cotangent bundles is then defined and some of its properties are reviewed. The theory of Lie groups and associated algebras is introduced. The moment map is defined, providing a way to associate symmetries of a system to quantities conserved in its evolution. The symplectic reduction theorem is formulated, allowing one to exploit these conserved quantities to reduce the dimensionality of the motion problem. Finally, hamiltonian mechanics is reformulated in the language of symplectic geometry. The symplectic reduction theorem is applied to the analysis of the motion of the free rigid body.