# Flow of a Hamiltonian vector field

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.

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 $\tau$.
• @Tobias: Your intuition that the difference between Hamiltonian and symplectic diffeomorphisms is a topological one and Alberto's statement that the Hamiltonian diffeos form a much smaller class can be made precise in terms of the flux homomorphism. For instance on a manifold with $b_1=0$ symplectic diffeos are Hamiltonian. – Jonny Evans Apr 29 '12 at 20:41
• (My comment only concerns symplectic diffeos isotopic to the identity - there are of course interesting symplectic mapping classes on many manifolds with $b_1=0$) – Jonny Evans Apr 29 '12 at 20:50
To answer the question of 'What does the flow of a Hamiltonian vector field correspond to', it's useful to think physically. Exactness means we have some function $H$ which corresponds to the vector field $X_H$. If you take a Darboux coordinate system, then the flow of the Hamiltonian vector field is exactly the solution to Hamilton's equations. Hamilton's principle basically states that the real trajectory of a physical system follows Hamilton's equations, where we interpret $H$ as something like an energy function (equivalent to minimising an energy functional). I hope this helps, it's probably more useful from a physical point of view than a pure mathematical one.