Open Access
Translator Disclaimer
2009 On Integrability of Hirota--Kimura-Type Discretizations: Experimental Study of the Discrete Clebsch System
Matteo Petrera, Andreas Pfadler, Yuri B. Suris
Experiment. Math. 18(2): 223-247 (2009).


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.


Download Citation

Matteo Petrera. Andreas Pfadler. Yuri B. Suris. "On Integrability of Hirota--Kimura-Type Discretizations: Experimental Study of the Discrete Clebsch System." Experiment. Math. 18 (2) 223 - 247, 2009.


Published: 2009
First available in Project Euclid: 25 November 2009

zbMATH: 1178.14007
MathSciNet: MR2549692

Primary: 14E05 , 14H70
Secondary: 37J35 , 37M15 , 39A12 , 70E40

Keywords: birational dynamics , Clebsch system , computer-assisted proof , Integrable discretization , integrable tops

Rights: Copyright © 2009 A K Peters, Ltd.


Vol.18 • No. 2 • 2009
Back to Top