# Symplectic Geometry

• Gerd Rudolph
• Matthias Schmidt
Part of the Theoretical and Mathematical Physics book series (TMP)

## Abstract

In this chapter, we study symplectic manifolds. We start with the Theorem of Darboux, which states that all symplectic structures of a given dimension are locally equivalent. Thus, in sharp contrast to the situation in Riemannian geometry, symplectic manifolds of the same dimension can at most differ globally. The second important observation is that a symplectic structure provides a duality between smooth functions and certain vector fields, called Hamiltonian vector fields. As a consequence, one obtains the notion of Poisson structure. Given the great importance of Poisson structures both in mathematics and in physics, we go beyond the symplectic case and give a brief introduction to general Poisson manifolds, including a proof of the Symplectic Foliation Theorem. Two classes of symplectic manifolds are discussed in detail: cotangent bundles, because they serve as a mathematical model of phase space, and orbits of the coadjoint representation of a Lie group, because they show up in the study of systems with symmetries. Moreover, we show that the coadjoint orbits coincide with the symplectic leaves of the Lie-Poisson structure. Next, we discuss coisotropic submanifolds, present a number of natural generalizations of the Darboux Theorem and give an introduction to general symplectic reduction. We introduce the concept of generating function and make some elementary remarks on the group of symplectomorphisms. The last section is devoted to an introduction to Morse theory, which can be naturally formulated in the language of symplectic geometry. Methods of Morse theory are of special importance in the study of Hamiltonian systems, in particular, for the discussion of qualitative dynamics.

## Keywords

Symplectic Manifold Poisson Structure Cotangent Bundle Morse Function Coadjoint Orbit
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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