New results on integrability of the Kahan-Hirota-Kimura discretizations
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. We report several novel observations regarding integrability of the Kahan-Hirota-Kimura discretization. For several of the most complicated cases for which integrability is known (Clebsch system, Kirchhoff system, and Lagrange top), - we give nice compact formulas for some of the more complicated integrals of motion and for the density of the invariant measure, and - we establish the existence of higher order Wronskian Hirota-Kimura bases, generating the full set of integrals of motion. While the first set of results admits nice algebraic proofs, the second one relies on computer algebra.
Keywords
Cite
@article{arxiv.1805.12490,
title = {New results on integrability of the Kahan-Hirota-Kimura discretizations},
author = {Matteo Petrera and Yuri B. Suris},
journal= {arXiv preprint arXiv:1805.12490},
year = {2019}
}
Comments
28 pp