On integrability of hirota–kimura-type discretizations: Experimental study of the discrete clebsch system

Matteo Petrera, Andreas Pfadler, Yuri B. Suris

Research output: Contribution to journalArticlepeer-review

27 Scopus citations

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 ofmotion 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.

Original languageEnglish
Pages (from-to)223-247
Number of pages25
JournalExperimental Mathematics
Volume18
Issue number2
DOIs
StatePublished - 2009
Externally publishedYes

Keywords

  • Birational dynamics
  • Clebsch system
  • Computer-assisted proof
  • Integrable discretization
  • Integrable tops

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