Hamiltonian systems of negative curvature are hyperbolic

TL;DR

This paper demonstrates that Hamiltonian systems with negative curvature exhibit hyperbolic behavior, with bounded semi-trajectories tending to hyperbolic equilibria and invariant sets being hyperbolic, generalizing properties of negatively curved geodesic flows.

Contribution

It establishes a link between negative curvature invariants and hyperbolicity in Hamiltonian systems, extending known results from Riemannian geometry.

Findings

Bounded semi-trajectories tend to hyperbolic equilibria.

Negative reduced curvature implies hyperbolicity of invariant sets.

Generalizes properties of negatively curved geodesic flows.

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.