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.
Best Answer
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 $\tau$.