nLab
symplectic vector field
Context
Symplectic geometry
symplectic geometry

higher symplectic geometry

Background
geometry

differential geometry

Basic concepts
almost symplectic structure , metaplectic structure , metalinear structure

symplectic form , n-plectic form

symplectic Lie n-algebroid

symplectic infinity-groupoid

symplectomorphism , symplectomorphism group

Hamiltonian action , moment map

symplectic reduction , BRST-BV formalism

isotropic submanifold , Lagrangian submanifold , polarization

Classical mechanics and quantization
Hamiltonian mechanics

quantization

deformation quantization ,

geometric quantization , higher geometric quantization

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Contents
Definition
For symplectic manifolds
For $(X, \omega)$ a symplectic manifold , a vector field $v \in \Gamma(T X)$ on $X$ is called a symplectic vector field if its Lie derivative preserves the symplectic form :

$\mathcal{L}_v \omega = 0
\,.$

For $n$ -plectic manifolds
The analogous definition applies to n-plectic geometry .

Properties
Relation to symplectomorphisms
The flow generated by a symplectic vector field is an auto-symplectomorphism .

Relation to Hamiltonian vector fields
By Cartan's magic formula and using that $\omega$ is by definition a closed form, the equation $\mathcal{L}_v \omega = 0$ is equivalent to

$d_{dR} \iota_v \omega = 0
\,,$

hence equivalent to the condition that the contraction of $v$ in $\omega$ is a closed differential form . If this contraction even is an exact differential form? in that there is a function $h \in C^\infty(X)$ such that

$\iota_v \omega = d_{dR} h
\,,$

then the symplectic vector field $v$ is called a Hamiltonian vector field and $h$ is called its Hamiltonian function.

Created on February 26, 2012 at 16:29:56.
See the history of this page for a list of all contributions to it.