Hamiltonian systems on almost cosymplectic manifolds

Abstract

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 X˜1J. The X˜1J 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 X˜1J extend the Riccati equations of motion on the four-dimensional Siegel-Jacobi manifold X1J attached to a linear Hamiltonian in the generators of the real Jacobi group G1J(R).