Second order Hamiltonian vector fields on tangent bundles

Let M be a differentiable manifold. A vector field Γ of TM which corresponds to a system of second order ordinary differential equations on M is called a second order Hamiltonian vector field if it is the Hamiltonian field of a function F ε{lunate} C^∞(TM) with respect to a Poisson structure P of TM. We formulate the direct problem as that of finding Γ if P is given, and the inverse problem as that of finding P if Γ is given. We show the solution of the direct problem if P is a symplectic structure such that the fibers of TM are Lagrangian submanifolds. For the inverse problem we generalize Henneaux' method of looking for a solution by studying a corresponding system of linear algebraic restrictions.