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Hamiltonian systems of negative curvature are hyperbolic

Andrei A. Agrachev·Natalia N. Chtcherbakova
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Abstract

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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Andrei A. Agrachev, Natalia N. Chtcherbakova. 2004-11-10. Hamiltonian systems of negative curvature are hyperbolic. https://arxiv.org/abs/math/0411224

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