Infinite dimensional symplectic reduction and the dynamics of a rigid body moving in a perfect fluid

The purpose of this thesis is to give an introduction to the theory of infinite dimensional Hamiltonian systems and infinite dimensional symplectic reduction. These topics often arise in applications, when dealing with Hamiltonian PDEs. In the first part, we will see how most of the standard constructions of differential geometry and Hamiltonian systems extend naturally from the finite dimensional to the infinite dimensional Banach setting, while, in the last part, an example of infinite dimensional symplectic reduction is presented. We will recall the basic properties of smooth manifolds and Lie groups modeled on Banach spaces. Furthermore, an introduction to theory of smooth actions of (infinite dimensional) Lie groups on (infinite dimensional) manifolds is presented. Also, the notion of infinite dimensional Hamiltonian systems with symmetries is studied which will be complemented by an infinite dimensional adaptation of the celebrated symplectic reduction theorem of Marsden and Weinstein. In the final part of the thesis, we will study the equations of motion for a planar rigid body moving in a potential two dimensional fluid in absence of external forces (namely the so called Kirchhoff's equations of hydrodynamics) from the perspective of symplectic reduction. We will see how these equations can be interpreted as the result of a two-stage Hamiltonian reduction procedure, involving an infinite dimensional symmetry.