symplectic manifold

Symplectic manifoldsMathworldPlanetmath 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,ω) consisting of a smooth manifoldMathworldPlanetmath M and a closed 2-form ( ωΩ2(M), that is non-degenerate at each point. Then ω is called a symplectic form for M.


  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 ωΩ2(M) on a 2n-dimensional manifold M is non-degenerate if and only if the n-fold product ωn=ωω is non-zero.

  3. 3.

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

Let (M,ω) and (N,η) be symplectic manifolds. Then a diffeomorphism f:MN is called a symplectomorphism if f*η=ω, that is, if the symplectic form on N pulls back to the form on M.


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