Riemannian geometry of Hamiltonian chaos: Hints for a general theory

Year: 2008

Authors: Cerruti-Sola M., Ciraolo G., Franzosi R., Pettini M.

Autors Affiliation: Istituto Nazionale di Astrofisica (INAF), Osservatorio Astrofisico di Arcetri, Largo E. Fermi 5, I-50125 Firenze, Italy; Ecole Centrale de Marseille, MSNM-GP (L3 M) UMR 6181, IMT Technopôle de Chateau-Gombert, rue Frédéric Joliot Curie, 13451 Marseille, Cedex 20, France; Dipartimento di Fisica Università di Firenze, C.N.R.-I.N.F.M., UdR di Firenze, via G. Sansone 1, I-50019 Sesto Fiorentino, Italy; I.N.F.N., Sezione di Firenze, Italy

Abstract: We aim at assessing the validity limits of some simplifying hypotheses that, within a Riemmannian geometric framework, have provided an explanation of the origin of Hamiltonian chaos and have made it possible to develop a method of analytically computing the largest Lyapunov exponent of Hamiltonian systems with many degrees of freedom. Therefore, a numerical hypotheses testing has been performed for the Fermi-Pasta-Ulam beta model and for a chain of coupled rotators. These models, for which analytic computations of the largest Lyapunov exponents have been carried out in the mentioned Riemannian geometric framework, appear as paradigmatic examples to unveil the reason why the main hypothesis of quasi-isotropy of the mechanical manifolds sometimes breaks down. The breakdown is expected whenever the topology of the mechanical manifolds is nontrivial. This is an important step forward in view of developing a geometric theory of Hamiltonian chaos of general validity.

Journal/Review: PHYSICAL REVIEW E

Volume: 78 (4)      Pages from: 046205-1  to: 046205-17

KeyWords: Chaos theory; Computation theory; Degrees of freedom (mechanics); Differential equations; Geometry; Lyapunov functions; Lyapunov methods, Analytic computations; Geometric framework; Geometric theory; Hamiltonian chaos; Hamiltonian systems; Hypotheses testing; Largest Lyapunov exponent; Riemannian geometry, Hamiltonians
DOI: 10.1103/PhysRevE.78.046205

Citations: 9
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