Hamiltonian chaos and differential geometry of configuration space-time

PHYSICA D-NONLINEAR PHENOMENA

卷:422

Article

Di Cairano, Loris^1,2,3  Gori, Matteo⁴  Pettini, Giulio^5,6  Pettini, Marco^7,8 

[1] Forschungszentrum Julich, Inst Neurosci & Med INM 9, D-52428 Julich, Germany

[2] Forschungszentrum Julich, Inst Adv Simulat IAS 5, D-52428 Julich, Germany

[3] Rhein Westfal TH Aachen, Fac Math Comp Sci & Nat Sci, Dept Phys, Aachen, Germany

[4] Univ Luxembourg, Phys & Mat Sci Res Unit, L-1511 Luxembourg, Luxembourg

[5] Univ Firenze, Dipartimento Fis, Via G Sansone 1, I-50019 Sesto Fiorentino, Italy

[6] Ist Nazl Fis Nucl, Sez Firenze, Via G Sansone 1, I-50019 Sesto Fiorentino, Italy

[7] Aix Marseille Univ, Univ Toulon, CNRS, Marseille, France

[8] CNRS Ctr Phys Theor UMR7332, F-13288 Marseille, France

关键词: Hamiltonian chaos;    Differential geometry;    Eisenhart metric;   

DOI  :  10.1016/j.physd.2021.132909

来源: Elsevier

【 摘 要 】

This paper tackles Hamiltonian chaos by means of elementary tools of Riemannian geometry. More precisely, a Hamiltonian flow is identified with a geodesic flow on configuration space-time endowed with a suitable metric due to Eisenhart. Until now, this framework has never been given attention to describe chaotic dynamics. A gap that is filled in the present work. In a Riemannian-geometric context, the stability/instability of the dynamics depends on the curvature properties of the ambient manifold and is investigated by means of the Jacobi-Levi-Civita (JLC) equation for geodesic spread. It is confirmed that the dominant mechanism at the ground of chaotic dynamics is parametric instability due to curvature variations along the geodesics. A comparison is reported of the outcomes of the JLC equation written also for the Jacobi metric on configuration space and for another metric due to Eisenhart on an extended configuration space-time. This has been applied to the Henon-Heiles model, a two-degrees of freedom system. Then the study has been extended to the 1D classical Heisenberg XY model at a large number of degrees of freedom. Both the advantages and drawbacks of this geometrization of Hamiltonian dynamics are discussed. Finally, a quick hint is put forward concerning the possible extension of the differential-geometric investigation of chaos in generic dynamical systems, including dissipative ones, by resorting to Finsler manifolds. (C) 2021 Elsevier B.V. All rights reserved.

【 授权许可】

Free