Euler equations on the general linear group, cubic curves, and inscribed hexagons

Aleshkin, Konstantin; Izosimov, Anton

doi:10.4171/lem/62-1/2-9

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The Kirchhoff Case1

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2018

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“…, we see that the fourth integral H 1 can be replaced just by m 2 3 . Wronskian relation satisfied on solutions of (84):…”

Section: The Kirchhoff Casementioning

confidence: 95%

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New results on integrability of the Kahan-Hirota-Kimura discretizations

Petrera,

Suris

2018

Preprint

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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.

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“…, we see that the fourth integral H 1 can be replaced just by m 2 3 . Wronskian relation satisfied on solutions of (84):…”

Section: The Kirchhoff Casementioning

confidence: 95%

“…The only counterexample we are aware of, is given by the system L = [L 2 , A] with a general 3 × 3 matrix L and a constant diagonal matrix A [2]. For this system, polarization applied to quadratic-fractional integrals of the Kahan-Hirota-Kimura discretization (there are two independent such integrals) does not lead to integrals of motion.…”

mentioning

confidence: 99%