nLab symplectic vector field

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.