Flow of a Hamiltonian vector field

Question

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.

Answer 1

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 $\tau$.

Answer 2

Given a symplectic vetor field, you may try to build a hamoltonian function the following way. Fix some point P. For any point Q, choose some curve from P to Q, and define H(Q) to be the flux of the vector field across the chosen curve (infinitesimal symplectic area swept by the curve when you push it by the flow during an infinitesimal amount of time). If you choose a second curve which is homotopic to the first one, you get the same value for H(Q), because your vector filed is symplectic. But this may not be the case if you choose an other curve which is not homotopic to the first one (think of a translation of the 2-torus). Thus the field is hamiltonian if and only if the flux across every closed curve is zero. (In dimension two, the flux is called the mean rotation vector).

Answer 3

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.