PlanetMath: symplectic manifold

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symplectic manifold

(Definition)

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 and a closed 2-form $\omega\in\Omega^2(M)$ , that is non-degenerate at each point. Then $\omega$ is called a symplectic form for .

Properties

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.
A form $\omega\in \Omega^2(M)$ on a -dimensional manifold is non-degenerate if and only if the -fold product $\omega^n= \omega\wedge \cdots \wedge \omega$ is non-zero.
As a consequence of the last property, 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 pulls back to the form on .

Notes

A symplectomorphism is also known as a canonical transformation. This term is mostly used in the mechanics literature.

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"symplectic manifold" is owned by matte. [ full author list (3) | owner history (1) ]

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See Also: contact manifold, Kähler manifold, hyperkähler manifold, $\mathbb{C}$ as a Kähler manifold

Also defines:

symplectic form, symplectomorphism, canonical transformation

Attachments:: examples of symplectic manifolds (Example) by mathcam
every symplectic manifold has even dimension (Theorem) by cvalente

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Cross-references: diffeomorphism, orientable, consequence, product, vectors, Hamilton equations, point, non-degenerate, closed, smooth manifold, contact manifolds, even, Hamiltonian, structure
There are 24 references to this entry.

This is version 8 of symplectic manifold, born on 2002-12-05, modified 2006-07-09.
Object id is 3667, canonical name is SymplecticManifold.
Accessed 8508 times total.

Classification:

AMS MSC:

53D05 (Differential geometry :: Symplectic geometry, contact geometry :: Symplectic manifolds, general)

Pending Errata and Addenda

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