symplectic manifold
Symplectic manifolds^{} constitute the mathematical structure for modern Hamiltonian mechanics. Symplectic manifolds can also be seen as even dimensional analogues to contact manifolds.
Definition 1.
A symplectic manifold is a pair $\mathrm{(}M\mathrm{,}\omega \mathrm{)}$ consisting of a smooth manifold^{} $M$ and a closed 2form (http://planetmath.org/DifferentialForms) $\omega \mathrm{\in}{\mathrm{\Omega}}^{\mathrm{2}}\mathit{}\mathrm{(}M\mathrm{)}$, that is nondegenerate at each point. Then $\omega $ is called a symplectic form for $M$.
Properties

1.
Every symplectic manifold is even dimensional. This is easy to understand in view of the physics. In Hamilton equations, location and momentum vectors always appear in pairs.

2.
A form $\omega \in {\mathrm{\Omega}}^{2}(M)$ on a $2n$dimensional manifold $M$ is nondegenerate if and only if the $n$fold product ${\omega}^{n}=\omega \wedge \mathrm{\cdots}\wedge \omega $ is nonzero.

3.
As a consequence of the last , every symplectic manifold is orientable.
Let $(M,\omega )$ and $(N,\eta )$ be symplectic manifolds. Then a diffeomorphism $f:M\to N$ is called a symplectomorphism if ${f}^{*}\eta =\omega $, that is, if the symplectic form on $N$ pulls back to the form on $M$.
Notes
A symplectomorphism is also known as a canonical transformation. This is mostly used in the mechanics literature.
Title  symplectic manifold 
Canonical name  SymplecticManifold 
Date of creation  20130322 13:12:18 
Last modified on  20130322 13:12:18 
Owner  matte (1858) 
Last modified by  matte (1858) 
Numerical id  11 
Author  matte (1858) 
Entry type  Definition 
Classification  msc 53D05 
Related topic  ContactManifold 
Related topic  KahlerManifold 
Related topic  HyperkahlerManifold 
Related topic  MathbbCIsAKahlerManifold 
Defines  symplectic form 
Defines  symplectomorphism 
Defines  canonical transformation 