hamiltonian formalism - Does Hamilton Mechanics give a general phase-space conserving flux?

Hamiltonian dynamics fulfil the Liouville's theorem, which means that one can imagine the flux of a phase space volume under a Hamiltonian theory like the flux of an ideal fluid, which doesn't change it's volume. But is it possible to reproduce every phase space conserving flux with an appropriate Hamiltonian?

So can I simply imagine the entity of all possible Hamiltonian dynamics as all possible phase space conserving fluxes? Or are Hamiltonian dynamics a special case for phase space conserving fluxes? If they are a special case, what would be an example for a phase space conserving flux for which there is no Hamiltonian that can produce it?

Answer

First, let's take a look at one-dimensional systems with phase space dimension $2$.

The volume form is just the symplectic one, ie any volume-preserving flow is symplectic and thus at least locally Hamiltonian (but not necessarily globally so).

Now, consider an arbitrary phase space of dimension $2n\geq4$ with canonical coordinates $q^i,p^i$.

Up to a constant factor, the volume form is

$$\Omega = dq^1\wedge\cdots\wedge dq^n\wedge dp^1\wedge\cdots\wedge dp^n$$ and the symplectic form $$\omega = \sum_i dp^i\wedge dq^i$$ Let's take a look at the vector field $X$ given by $$X=q^1\frac{\partial}{\partial q^2}$$ As $$\mathcal L_X\Omega = 0$$ phase space volume will be preserved, but as $$\mathcal L_X\omega \not= 0$$ the vector field is not symplectic and thus also not Hamiltonian.