On Integrability of Hirota--Kimura-Type Discretizations: Experimental Study of the Discrete Clebsch System

Petrera, Matteo ; Pfadler, Andreas ; Suris, Yuri B.

Experiment. Math., Tome 18 (2009) no. 1, p. 223-247 / Harvested from Project Euclid

Résumé

R. Hirota and K. Kimura discovered integrable discretizations of the Euler and the Lagrange tops, given by birational maps. Their method is a specialization to the integrable context of a general discretization scheme introduced by W. Kahan and applicable to any vector field with a quadratic dependence on phase variables. According to a proposal by T. Ratiu, discretizations of Hirota--Kimura type can be considered for numerous integrable systems of classical mechanics. Due to a remarkable and not well understood mechanism, such discretizations seem to inherit the integrability for all algebraically completely integrable systems. We introduce an experimental method for a rigorous study of integrability of such discretizations. ¶ Application of this method to the Hirota--Kimura-type discretization of the Clebsch system leads to the discovery of four functionally independent integrals of motion of this discrete-time system, which turn out to be much more complicated than the integrals of the continuous-time system. Further, we prove that every orbit of the discrete-time Clebsch system lies in an intersection of four quadrics in the six-dimensional phase space. Analogous results hold for the Hirota--Kimura-type discretizations for all commuting flows of the Clebsch system, as well as for the so(4) Euler top.

Détail

Publié le : 2009-05-15
Classification:  Integrable discretization,  computer-assisted proof,  birational dynamics,  Clebsch system,  integrable tops,  14E05,  14H70,  37J35,  37M15,  39A12,  70E40