Hamiltonian systems of negative curvature are hyperbolic

Details Hamiltonian systems of negative curvature are hyperbolic Hamiltonian systems of negative curvature are hyperbolic

2004-11-10

arxiv.org/abs/math/0411224v1

Mathematics

Dynamical Systems

5 pages

Scientific paper

The {\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.

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