Hamiltonian Systems of Negative Curvature are Hyperbolic

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 semi-trajectory 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 well-known 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 2n-dimensional symplectic manifold endowed with a sym-plectic form σ. A Lagrange distribution ∆ ⊂ TM is a smooth vector sub-bundle of TM such that each fiber ∆x = ∆ ∩ TxM, x ∈ M, is a Lagrange subspace of the symplectic space TxM; in other words, dim∆x = n and σx(ξ, η) = 0 ∀ξ, η ∈ ∆x. Basic examples are cotangent bundles endowed with the standard sym-plectic structure and the “vertical ” distribution: M = T ∗N, ∆x = Tx(T ∗qN), ∀x ∈ T ∗qN, q ∈ N. (1) Let h ∈ C∞(M); then ~h ∈ VecM is the associated to h Hamiltonian vector field: dh = σ(·,~h). We assume that ~h is a complete vector field