Symplectic Geometry, Contrained Systems and Shape Dynamics

Seminar Talk in Achim Kempf’s group, University of Waterloo, Department of Applied Mathematics

Part I (Symplectic Geometry): I will review Symplectic Geometry which serves as the mathematical framework for Hamiltonian formulation of classical mechanics. I will define a symplectic manifold (or phase space), how we can associate a vector field to every phase space function and how this turns the space of smooth phase space functions into a Lie algebra using the Poisson brackets. Finally, I hope to connect with present knowledge by applying the structure to the Harmonic oscillator and the phase space diagram of the pendulum.
Part II (Constrained Systems): In the second part of my talk, I want to explain how constraints can arise after performing the Legrendre transform to restrict the physical phase space to a submanifold. This enables me to explain certain aspects of the Dirac procedure and how we can classify constraints. Finally, I hope to give new insights for the understanding of gauge degrees of freedom in the Hamiltonian picture.
Part III (Shape Dynamics): In the third part of my talk, I will summarize how Shape Dynamics can be defined from General Relativity using a Linking Theory in the Hamiltonian formulation. It can be shown that Shape Dynamics is a theory of gravity which has the Dirac observables and encodes the same dynamics as General Relativity, but has conformal gauge orbits instead of local time reparametrization.