Symplectic Manifolds
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 close
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Symplectic Manifolds
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. The motivation for the definition may not be evident at first, but it will emerge gradually as we see how these properties are used. For now, suffice it to say that nondegeneracy is important because it yields a tangentcotangent isomorphism like that provided by a Riemannian metric, and closedness is important because it leads to a deep relationship between smooth functions and flows (see the discussions of Hamiltonian vector fields and Poisson brackets later in this chapter). Symplectic structures have surprisingly varied applications in mathematics and physics, including partial differential equations, differential topology, and classical mechanics, among many other fields. In this chapter, we can give only a quick overview of the subject of symplectic geometry. We begin with a discussion of the algebra of nondegenerate alternating 2tensors on a finite-dimensional vector space, and then turn our attention to symplectic structures on manifolds. The most important example is a canonically defined symplectic structure on the cotangent bundle of each smooth manifold. 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, so, unlike the situation for Riemannian metrics, there is no local obstruction to “flatness” of symplectic structures. Then we explore one of the most important applications of symplectic structures. Any smooth real-valued function on a symplectic manifold gives rise to a canonical system of ordinary differential equations called a Hamiltonian system. These systems are central to the study of classical mechanics. After treating Hamiltonian systems, we give a brief introduction to an odddimensional analogue of symplectic structures, called contact structures. Then 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. J.M. Lee, Introduction to Smooth Manifolds, Graduate Texts in Mathematics 218, 564 DOI 10.1007/978-1-4419-9982-5_22, © Springer Science+Business Media New York 2013
Symplectic Tensors
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Symplectic Tensors We begin with linear algebra. A 2-covector ! on a finite-dimensional vector space V y Dv³! is said to be nondegenerate if the linear map ! y W V ! V defined by !.v/ is invertible. I Exercise 22.1. Show that the following are equivalent for 2-covector ! on a finitedimensional vector space V : (a) ! is nondegenerate. (b) For each nonzero v 2 V , there exists w 2 V such that !.v; w/ ¤ 0. (c) In terms of some (hence every) basis, the matrix .!ij / representing ! is nonsingular.
A nondegenerate 2-covector is called a symplectic tensor. A vector space V endowed with a specific symplect
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