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 16:29:56 by Urs Schreiber (82.113.121.217)