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.




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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.



Flow of a Hamiltonian vector field
The diffeomorphisms which are generated by (time-dependent) Hamiltonian vector fields are said to be Hamiltonian diffeomorphisms. Hamiltonian diffeomorphisms form a subgroup of the group of symplectic diffeomorphisms (actually, they are a subgroup of the connected component of the identity). 


As you observe, locally they cannot be distinguished from symplectic diffeomorphisms. But they are a much smaller class. For instance, the Hamiltonian diffeomorphisms of $\mathbb{T}^2$ are exactly those symplectic (i.e. area-preserving) diffeomorphisms which have a lift $\varphi:\mathbb{R}^2 \rightarrow \mathbb{R}^2$ such that $\varphi(x) = x + \psi(x)$ where $\psi$ is $\mathbb{Z}^2$-periodic and satisfies
$$
\int\_{[0,1]^2} \psi(x) dx =0.
$$
In particular, nontrivial translations on $\mathbb{T}^2$ are symplectic but not Hamiltonian (the latter fact is true also for $\mathbb{T}^{2n}$).


As for your doubts related to local existence: if $M$ is a compact manifold you of course have global existence, but these definitions make sense also on non-compact manifolds. Indeed, the non-exactness of $\imath\_X \omega$ can be detected on a compact subset $K$ of $M$ (a circle is enough) and you can find $\tau>0$ such that the flow of a neighborhood of $K$ exists up to time $\tau$. 


