nLab
symplectic vector field

Definition

For symplectic manifolds

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

$${ℒ}_v\omega =0.$$\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 ${ℒ}_v\omega =0$ is equivalent to

$$d_{\mathrm{dR}}{\iota }_v\omega =0,$$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 $h\in C^{\mathrm{\infty }}(X)$ such that

$${\iota }_v\omega =d_{\mathrm{dR}}h,$$\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.