Integrable Systems

1Symplectic Manifolds

1.1. Introduction

This chapter is devoted to the study of symplectic manifolds and their connection with Hamiltonian systems. It is well known that symplectic manifolds play a crucial role in classical mechanics, geometrical optics and thermodynamics, and currently have conquered a rich territory, asserting themselves as a central branch of differential geometry and topology. In addition to their activity as an independent subject, symplectic manifolds are strongly stimulated by important interactions with many mathematical and physical specialties, among others. The aim of this chapter is to study some properties of symplectic manifolds and Hamiltonian dynamical systems, and to review some operations on these manifolds.

This chapter is organized as follows. In the second section, we begin by briefly recalling some notions about symplectic vector spaces. The third section defines and develops explicit calculation of symplectic structures on a differentiable manifold and studies some important properties. The forth section is devoted to the study of some properties of one-parameter groups of diffeomorphisms or flow, Lie derivative, interior product and Cartan’s formula. We review some interesting properties and operations on differential forms. The fifth section deals with the study of a central theorem of symplectic geometry, namely Darboux’s theorem: the symplectic manifolds (M, ω) of dimension 2m are locally isomorphic to (ℝ^2m, ω). The sixth section ...