Symplectic Geometry

Canonical mechanics is most naturally and intrinsically described in the language of symplectic geometry. In this chapter we review the basic facts about symplectic geometry including: Lagrangian submanifolds, symplectomorphisms and their generating functions, Darboux theorem, Poisson brackets, momentum maps, and the symplectic reduction with the Marsden–Weinstein–Meyer quotient. In the last section we introduce contact geometry and the related notions: the Reeb vector, the characteristic curves, and the tube of characteristics with the associated Poincaré theorem.