Well-posed infinite horizon variational problems on a compact manifold

We give an effective suffcient condition for a variational problem with infinite horizon on a Riemannian manifold M to admit a smooth optimal synthesis, i. e. a smooth dynamical system on M whose positive semi-trajectories are solutions to the problem. To realize the synthesis we construct a well-projected to M invariant Lagrange submanifold of the extremals' flow on the cotangent bundle T^¤M. The construction uses the curvature of the flow on the cotangent bundle and some ideas of hyperbolic dynamics.