Hamiltonian systems on almost cosymplectic manifolds

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