Analytical Mechanics and Symplectic Geometry

27/04/2012 | PSIminar Talk, Perimeter Institute for Theoretical Physics

This PSIminar will be organized as a 60 minutes workshop: We will split the audience into five groups to have more intense discussion while a member of the reading group is presenting the material.
Symplectic geometry provides a geometrical picture for the Hamiltonian formalism of classical mechanics. It is well-suited for quantization and for numerical methods. Furthermore, it is used in topological quantum field theories and in non-Abelian gauge theories to understand ghosts based on the BRST formalism. This makes symplectic geometry a powerful tool in both, classical mechanics and modern research.
We will give a basic introduction to symplectic geometry and its connection to the Hamiltonian formulation of classical mechanics. By doing so we will especially cover
– Symplectic manifold (manifold with symplectic form)
– Hamiltonian vector field
– Darboux theorem
– Liouville theorem
– Conserved quantities
– Poisson brackets induced by vector field commutator and symplectic form
– Phase space as cotangent bundle of configuration space
– Example: Harmonic Oscillator
Depending on personal taste of the groups we might also cover some more advanced topics.