Hamiltonian vector fields on almost symplectic manifolds

Let (M, ω) be an almost symplectic manifold (ω is a nondegenerate, not closed, 2-form). We say that a vector field X of M is locally Hamiltonian if LXω = 0, d(i(X)ω) = 0, and it is Hamiltonian if, furthermore, the 1-form i(X)ω is exact. Such vector fields were considered in Fassò and Sansonetto ["Integrable almost-symplectic Hamiltonian systems," J. Math. Phys.48, 092902 (2007)]10.1063/1.2783937, under the name of strongly Hamiltonian, and a corresponding action-angle theorem was proven. Almost symplectic manifolds may have few, nonzero, 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.