  The paper reviews the notion of $n+\frac{1}{2}$D non-autonomous Hamiltonian
systems, portraying their dynamics as the flow of the Reeb field related to a
closed two-form of maximal rank on a cosymplectic manifold, and naturally
decomposing into time-like and Hamiltonian components. The paper then
investigates the conditions under which the field-line dynamics of a
(tangential) divergence-free vector field on a connected compact three-manifold
(possibly with boundary) diffeomorphic to a trivial fibre bundle over the
circle can be conversely identified as a non-autonomous $1\frac{1}{2}$D
Hamiltonian system. Under the assumption that the field is transverse to a
global compact Poincar\'e section, an adaptation of Moser's trick shows that
all such fields are locally-Hamiltonian. A full identification is established
upon further assuming that the Poincar\'e sections are planar, which crucially
implies (together with Dirichlet boundary conditions) that the cohomology class
of the generating closed one-forms on each section is constant. By reviewing
the classification of fibre bundles over the circle using the monodromy
representation, it is remarked that as soon as the Poincar\'e section of an
alleged field is diffeomorphic to a disk or an annulus, the domain is
necessarily diffeomorphic to a solid or hollow torus, and thus its field-line
dynamics can always be identified as a non-autonomous Hamiltonian system.
