nLab
symplectic vector field

Contents

Definition

For symplectic manifolds

For (X,ω)(X, \omega) a symplectic manifold, a vector field vΓ(TX)v \in \Gamma(T X) on XX is called a symplectic vector field if its Lie derivative preserves the symplectic form:

vω=0. \mathcal{L}_v \omega = 0 \,.

For nn-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 ω\omega is by definition a closed form, the equation vω=0\mathcal{L}_v \omega = 0 is equivalent to

d dRι vω=0, d_{dR} \iota_v \omega = 0 \,,

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

ι vω=d dRh, \iota_v \omega = d_{dR} h \,,

then the symplectic vector field vv is called a Hamiltonian vector field and hh is called its Hamiltonian function.

Created on February 26, 2012 16:29:56 by Urs Schreiber (82.113.121.217)