  Let $(M,\omega)$ be an almost symplectic manifold ($\omega$ is a non
degenerate, not closed, 2-form). We say that a vector field $X$ of $M$ is
locally Hamiltonian if $L_X\omega=0,d(i(X)\omega)=0$, and it is Hamiltonian if,
furthermore, the 1-form $i(X)\omega$ is exact. Such vector fields were
considered in a 2007 paper by F. Fasso and N. Sansonetto, under the name of
strongly Hamiltonian, and a corresponding action-angle theorem was proven.
Almost symplectic manifolds may have few, non-zero, Hamiltonian vector fields
or even none. Therefore, it is important to have examples and it is our aim to
provide such examples here. We also obtain some new general results. In
particular, we show that the locally Hamiltonian vector fields generate a Dirac
structure on $M$ and we state a reduction theorem of the Marsden-Weinstein
type. A final section is dedicated to almost symplectic structures on tangent
bundles.
