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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).

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 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.

Citation

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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.

Information

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

zbMATH: 1178.14007
MathSciNet: MR2549692

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

Rights: Copyright © 2009 A K Peters, Ltd.

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Vol.18 • No. 2 • 2009
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