On integrability of Hirota-Kimura type discretizations. Experimental study of the discrete Clebsch system

TL;DR

This paper introduces an experimental approach to study the integrability of Hirota-Kimura type discretizations, demonstrating their integrability for the discrete Clebsch system and related models by identifying conserved quantities and geometric structures.

Contribution

It develops a novel experimental method for analyzing integrability of Hirota-Kimura discretizations and applies it to establish integrability of the discrete Clebsch system with explicit integrals.

Findings

Discovered four independent integrals of motion for the discrete Clebsch system.

Proved that orbits lie in intersections of four quadrics in phase space.

Extended results to related systems like the $so(4)$ Euler top.

Abstract

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 the 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.