Orlando Ragnisco


Musso F., Petrera M., Ragnisco O., Satta G.
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.
Keywords: models, Bäcklund transformations, spinning tops
Citation: Musso F., Petrera M., Ragnisco O., Satta G.,  A rigid body dynamics derived from a class of extended Gaudin models: an integrable discretization , Regular and Chaotic Dynamics, 2005, vol. 10, no. 4, pp. 363-380
Ballesteros A., Musso F., Ragnisco O.
We review here a method, recently introduced by the authors, that can be used to construct completely integrable Classical and Quantum Hamiltonian Systems from representations of coalgebras with Casimir element(s). As a prototype example, we choose the spin $1/2$ Calogero–Gaudin system and its $q$-deformation. Possible drawbacks and generalizations of the method are outlined.
Citation: Ballesteros A., Musso F., Ragnisco O.,  Classical and Quantum Integrable Systems: the Coalgebra Approach, Regular and Chaotic Dynamics, 2002, vol. 7, no. 4, pp. 393-398

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