Symplectic Manifolds

Abstract

In this final chapter we introduce a new kind of geometric structure on manifolds, called a symplectic structure, which is superficially similar to a Riemannian metric but turns out to have profoundly different properties. It is simply a choice of a closed, nondegenerate 2-form. Symplectic structures have surprisingly varied applications in mathematics and physics, including partial differential equations, differential topology, and classical mechanics, among many other fields. After defining symplectic structures, we give a proof of the important Darboux theorem, which shows that every symplectic form can be put into canonical form locally by a choice of smooth coordinates. Then we give a brief introduction to Hamiltonian systems, which are central to the study of classical mechanics, and to an odd-dimensional analogue of symplectic structures, called contact structures. At the end of the chapter, we show how symplectic and contact geometry can be used to construct solutions to first-order partial differential equations.