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 consisting of a smooth manifold and a closed 2-form (http://planetmath.org/DifferentialForms) , that is non-degenerate at each point. Then is called a symplectic form for .
Properties
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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.
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2.
A form on a -dimensional manifold is non-degenerate if and only if the -fold product is non-zero.
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3.
As a consequence of the last , every symplectic manifold is orientable.
Let and be symplectic manifolds. Then a diffeomorphism is called 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 |