2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/73176The {\it curvature} and the {\it reduced curvature} are basic differential invariants of the pair: (Hamiltonian system, Lagrange distribution) on the symplectic manifold. We show that negativity of the curvature implies that any bounded semi-trajectory of the Hamiltonian system tends to a hyperbolic equilibrium, while negativity of the reduced curvature implies the hyperbolicity of any compact invariant set of the Hamiltonian flow restricted to a prescribed energy level. Last statement generalizes a well-known property of the geodesic flows of Riemannian manifolds with negative sectional curvatures.5 pagesDynamical SystemsSymplectic Geometry37D05; 37D40text