 
Summary: Hamiltonian Systems of Negative Curvature are
Hyperbolic
A. A. Agrachev
N. N. Chtcherbakova
Abstract
The curvature and the reduced curvature are basic differential in
variants of the pair: Hamiltonian system, Lagrange distribution on
the symplectic manifold. We show that negativity of the curvature
implies that any bounded semitrajectory of the Hamiltonian system
tends to a hyperbolic equilibrium, while negativity of the reduced cur
vature implies the hyperbolicity of any compact invariant set of the
Hamiltonian flow restricted to a prescribed energy level. Last state
ment generalizes a wellknown property of the geodesic flows of Rie
mannian manifolds with negative sectional curvatures.
1 Regularity and Monotonicity
Smooth objects are supposed to be C in this note; the results remain valid
for the class Ck with a finite and not large k but we prefer not to specify
the minimal possible k.
Let M be a 2ndimensional symplectic manifold endowed with a sym
plectic form . A Lagrange distribution TM is a smooth vector sub
