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Summary: Hamiltonian Systems of Negative Curvature are
Hyperbolic
A. A. Agrachev N. N. Chtcherbakova
39/2004/M
Abstract
The curvature and the reduced curvature are basic di erential invariants
of the pair: hHamiltonian system, Lagrange distributioni on the symplec-
tic 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 hyper-
bolicity of any compact invariant set of the Hamiltonian
ow restricted to
a prescribed energy level. Last statement generalizes a well-known prop-
erty of the geodesic
ows of Riemannian manifolds with negative sectional
curvatures.
1 Regularity and Monotonicity
Smooth objects are supposed to be C 1 in this note; the results remain valid
for the class C k with a nite and not large k but we prefer not to specify
the minimal possible k.
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