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The Curvature and Hyperbolicity of Hamiltonian Systems

Summary: The Curvature and Hyperbolicity of
Hamiltonian Systems
A. A. Agrachev #
Curvature­type invariants of Hamiltonian systems generalize sec­
tional curvatures of the Riemannian manifolds: negativity of the cur­
vatures is an indicator of the hyperbolic behavior of the Hamiltonian
flow. In this paper, we give a self­contained description of the related
constructions and facts; they lead to a natural extension of classi­
cal results about Riemannian geodesic flows and indicate some new
This paper is especially written to the 70th anniversary of Dmitrij Anosov.
One of the goals of the paper is to explain that classical Anosov's results
about geodesic flows of the negative curvature Riemannian manifolds can be
actually applied to the essentially larger class of flows than it is normally
Needless to say, I am not at all expert in the hyperbolic dynamics, but I
was obliged, as a member of the MIAN's department of di#erential equations,
to attend the seminar guided by professor Anosov. I learned first definitions


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


Collections: Engineering; Mathematics