symplectic manifold
Symplectic manifolds constitutethe mathematical structure for modern Hamiltonian mechanics.Symplectic manifolds can also be seen as even dimensionalanalogues to contact manifolds.
Definition 1.
A symplectic manifold is a pair consistingof a smooth manifold and aclosed 2-form (http://planetmath.org/DifferentialForms), that is non-degenerateat each point.Then is called a symplecticform for .
Properties
- 1.
Every symplectic manifold is even dimensional. This iseasy to understand in view of the physics. In Hamiltonequations, location and momentum vectors always appear in pairs.
- 2.
A form on a -dimensionalmanifold is non-degenerate if and only if the-fold product is non-zero.
- 3.
As a consequence of the last , every symplectic manifoldis orientable.
Let and be symplectic manifolds. Then a diffeomorphism iscalled a symplectomorphism if , that is, if the symplectic form on pulls back to the form on .
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 |