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Hamiltonian Systems of Negative Curvature are A. A. Agrachev

Summary: Hamiltonian Systems of Negative Curvature are
A. A. Agrachev
N. N. Chtcherbakova
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 . A Lagrange distribution TM is a smooth vector sub-


Source: Agrachev, Andrei - Functional Analysis Sector, Scuola Internazionale Superiore di Studi Avanzati (SISSA)


Collections: Engineering; Mathematics