Geodesic
The Curvature and Hyperbolicity of Hamiltonian Systems
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Dynamical systems and optimization, Tome 256 (2007), pp. 31-53.

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Curvature-type invariants of Hamiltonian systems generalize sectional curvatures of Riemannian manifolds: the negativity of the curvature is an indicator of the hyperbolic behavior of the Hamiltonian flow. In this paper, we give a self-contained description of the related constructions and facts; they lead to a natural extension of the classical results about Riemannian geodesic flows and indicate some new phenomena. 
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A. A. Agrachev. The Curvature and Hyperbolicity of Hamiltonian Systems. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Dynamical systems and optimization, Tome 256 (2007), pp. 31-53. http://geodesic.mathdoc.fr/item/TM_2007_256_a1/

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