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单词 SymplecticManifold
释义

symplectic manifold


Symplectic manifoldsMathworldPlanetmath 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 (M,ω) consistingof a smooth manifoldMathworldPlanetmath M and aclosed 2-form (http://planetmath.org/DifferentialForms)ωΩ2(M), that is non-degenerateat each point.Then ω is called a symplecticform for M.

Properties

  1. 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. 2.

    A form ωΩ2(M) on a 2n-dimensionalmanifold M is non-degenerate if and only if then-fold product ωn=ωωis non-zero.

  3. 3.

    As a consequence of the last , every symplectic manifoldis orientable.

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

Notes

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

Titlesymplectic manifold
Canonical nameSymplecticManifold
Date of creation2013-03-22 13:12:18
Last modified on2013-03-22 13:12:18
Ownermatte (1858)
Last modified bymatte (1858)
Numerical id11
Authormatte (1858)
Entry typeDefinition
Classificationmsc 53D05
Related topicContactManifold
Related topicKahlerManifold
Related topicHyperkahlerManifold
Related topicMathbbCIsAKahlerManifold
Definessymplectic form
Definessymplectomorphism
Definescanonical transformation
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