  We determine the Hamiltonian vector field on an odd dimensional manifold
endowed with almost cosymplectic structure. This is a generalization of the
corresponding Hamiltonian vector field on manifolds with almost transitive
contact structures, which extends the contact Hamiltonian systems. Applications
are presented to the equations of motion on a particular five-dimensional
manifold, the extended Siegel-Jacobi upper-half plane
$\tilde{\mathcal{X}}^J_1$. The $\tilde{\mathcal{X}}^J_1$ manifold is endowed
with a generalized transitive almost cosymplectic structure, an almost
cosymplectic structure, more general than transitive almost contact structure
and cosymplectic structure.The equations of motion on $\tilde{\mathcal{X}}^J_1$
extend the Riccati equations of motion on the four-dimensional Siegel-Jacobi
manifold $\mathcal{X}^J_1$ attached to a linear Hamiltonian in the generators
of the real Jacobi group $G^J_1(\mathbb{R})$.
