Smooth vector fields are in a one-to-one relationship with flows $\Phi: D \subseteq M\times \mathbb{R} \rightarrow M$, $$X_m = {\frac{d}{d t}}_{t=0} \Phi(m, t),$$ and by the symplectic form also with 1-forms $$X \longleftrightarrow \; - \iota_X \omega. $$

The following is well known (e. g. Abraham Marsden, Proposition 3.3.6):

$X$ is local Hamiltonian iff $\iota_X \omega$ is closed iff the flow $\Phi$ of $X$ is symplectic.

$X$ is Hamiltonian iff $\iota_X \omega$ is exact iff ??.

**Question**: What is the corresponding property for flows of (global) Hamiltonian vector fields?

I am a bit confused about this, as the existence of flows can be guaranteed only for small times and only in neighborhoods of each point (so it is a strongly local object), but the difference between closed and exact forms is determined by global/topological characteristics.

A short proof of the cited proposition about local Hamiltonian vector fields: The flow $\Phi$ leaves the symplectic form invariant iff $\Phi_t^* \omega = \omega$ iff $0 = L_X \omega = d(\iota_X \omega)$ iff $\iota_X \omega$ is closed.