On the Hamiltonian structure of Hirota’s discretization of the Euler top

This talk deals with a remarkable integrable discretization of the SO(3) Euler top introduced in [1] by R. Hirota and K. Kimura. A class of implicit discretizations of the Euler top sharing the integrals of motion with the continuous system has been presented and studied in [2]. The Hirota-Kimura discretization of the Euler top leads to an explicit map. Its integrability is proven by finding two independent integrals of motion and a solution in terms of elliptic functions [1]. Nevertheless the Hamiltonian formulation has not been mentioned by them. We shall give a simplified and streamlined presentation of their results and we provide a bi-Hamiltonian structure for this discretization, thus proving its integrability in the standard Liouville-Arnold sense. The Hirota-Kimura discretization of the Lagrange top [3], as well as some preliminary results on other bilinear systems of classical mechanics, indicate the existence of a huge collection of discrete-time integrable models.