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 2-form (http://planetmath.org/DifferentialForms) $\omega \mathrm{\in }{\mathrm{\Omega }}^{\mathrm{2}}\mathit{}\mathrm{(}M\mathrm{)}$, that is non-degenerate 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 non-degenerate if and only if the $n$-fold product ${\omega }^n=\omega \wedge \mathrm{\cdots }\wedge \omega$ is non-zero.

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	2013-03-22 13:12:18
Last modified on	2013-03-22 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