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 $(M,\omega)$ consisting of a smooth manifold $M$ and a closed 2-form (http://planetmath.org/DifferentialForms) $\omega\in\Omega^{2}(M)$, that is non-degenerate at each point. Then $\omega$ is called a symplectic form for $M$.

Properties

1. 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. 2.

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

3. 3.

As a consequence of the last , every symplectic manifold is orientable.

Let $(M,\omega)$ and $(N,\eta)$ be symplectic manifolds. Then a diffeomorphism $f\colon M\to N$ 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