We study Hamiltonian systems which generate extremal flows of regularvariational problems on smooth manifolds and demonstrate thatnegativity of the generalized curvature of such a system impliesthe existence of a global smooth optimal synthesis for the infinitehorizon problem.We also show that in the Euclidean case negativity of the generalized curvature is a consequence ofthe convexity of the Lagrangian with respect to the pair of arguments. Finally, we give a generic classification for 1-dimensional problems.
