Symplectic Manifolds

This chapter discusses the study of symplectic manifolds and their connection with Hamiltonian systems. It details the study of some properties of one-parameter groups of diffeomorphisms or flow, Lie derivative, interior product and Cartan's formula. The chapter deals with the study of a central theorem of symplectic geometry, namely Darboux's theorem: the symplectic manifolds ( M , ω ) of dimension 2 m are locally isomorphic to (R 2m , ω ). It then examines how to determine a symplectic structure on the orbit of the coadjoint representation of a Lie group. The chapter also explains the explicit determination of symplectic structures on adjoint and coadjoint orbits of a Lie group SO(n). The Darboux theorem plays a central role in symplectic geometry; the symplectic manifolds ( M , ω ) of dimension 2 m are locally isomorphic to (R 2m , ω ).