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,ω) consisting
of a smooth manifold M and a
closed 2-form (http://planetmath.org/DifferentialForms)
ω∈Ω2(M), that is non-degenerate
at each point.
Then ω 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 ω∈Ω2(M) on a 2n-dimensional manifold M is non-degenerate if and only if the n-fold product ωn=ω∧⋯∧ω is non-zero.
-
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:M→N is called a symplectomorphism if f*η=ω, 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 |