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A rigid body dynamics derived from a class of extended Gaudin models : an integrable discretization

F. Musso·M. Petrera·O. Ragnisco·G. Satta
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Abstract

We consider a hierarchy of classical Liouville completely integrable models sharing the same (linear) $r$--matrix structure obtained through an $N$--th jet--extension of $\mathfrak{su}(2)$ rational Gaudin models. The main goal of the present paper is the study of the integrable model corresponding to N=3, since the case N=2 has been considered by the authors in separate papers, both in the one--body case (Lagrange top) and in the $n$--body one (Lagrange chain). We now obtain a rigid body associated with a Lie--Poisson algebra which is an extension of the Lie--Poisson structure for the two--field top, thus breaking its semidirect product structure. In the second part of the paper we construct an integrable discretization of a suitable continuous Hamiltonian flow for the system. The map is constructed following the theory of Bäcklund transformations for finite--dimensional integrable systems developed by V.B. Kuznetsov and E.K. Sklyanin.

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F. Musso, M. Petrera, O. Ragnisco, G. Satta. 2005-03-01. A rigid body dynamics derived from a class of extended Gaudin models : an integrable discretization. https://arxiv.org/abs/math-ph/0503002

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