Coherent Riemannian-geometric description of Hamiltonian order and chaos with Jacobi metric

Description cohérente en géométrie Riemannienne de l'ordre hamiltonien et du chaos avec métrique de Jacobi

Loris Di Cairano , Matteo Gori (1, 2) , Marco Pettini (1, 2)

1 CPT - Centre de Physique Théorique - UMR 7332
2 CPT - E7 Systèmes dynamiques : théories et applications

By identifying Hamiltonian flows with geodesic flows of suitably chosen Riemannian manifolds, it is possible to explain the origin of chaos in classical Newtonian dynamics and to quantify its strength. There are several possibilities to geometrize Newtonian dynamics under the action of conservative potentials and the hitherto investigated ones provide consistent results. However, it has been recently argued that endowing configuration space with the Jacobi metric is inappropriate to consistently describe the stability/instability properties of Newtonian dynamics because of the non-affine parametrization of the arc length with physical time. To the contrary, in the present paper, it is shown that there is no such inconsistency and that the observed instabilities in the case of integrable systems using the Jacobi metric are artefacts.

Mots clés: Hamiltonian Chaos; Differential Geometry

Domaines: Dynamique Chaotique [nlin.CD]

Loris Di Cairano, Matteo Gori, Marco Pettini. Coherent Riemannian-geometric description of Hamiltonian order and chaos with Jacobi metric. Chaos: An Interdisciplinary Journal of Nonlinear Science, 2019, 29 (12), pp.123134. ⟨10.1063/1.5119797⟩. ⟨hal-02161168⟩