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The problem of integrable discretization: Hamiltonian approach
 Progress in Mathematics, Volume 219. Birkhäuser
"... this paper. Hence, the nature of this discretization as a member of the Toda hierarchy was not understood properly. A complete account was given in Suris (1995). The work Rutishauser (1954) contains not only the denition of the qd algorithm from the viewpoint of the numerical analist, but also the e ..."
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this paper. Hence, the nature of this discretization as a member of the Toda hierarchy was not understood properly. A complete account was given in Suris (1995). The work Rutishauser (1954) contains not only the denition of the qd algorithm from the viewpoint of the numerical analist, but also the equations of motion of the Toda lattice (under the name of a \continuous analogue of the qd algorithm")! The relation of the qd algorithm to integrable systems might have important implications for the numerical analysis, cf. Deift et al. (1991), Nagai and Satsuma (1995).
Discretetime RuijsenaarsSchneider system and Lagrangian 1form structure. arXiv:1112.4576 [nlin.SI
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Symmetry, Integrability and Geometry: Methods and Applications
"... Abstract. The original continuoustime “goldfish ” dynamical system is characterized by two neat formulas, the first of which provides the N Newtonian equations of motion of this dynamical system, while the second provides the solution of the corresponding initialvalue problem. Several other, more ..."
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Abstract. The original continuoustime “goldfish ” dynamical system is characterized by two neat formulas, the first of which provides the N Newtonian equations of motion of this dynamical system, while the second provides the solution of the corresponding initialvalue problem. Several other, more general, solvable dynamical systems “of goldfish type ” have been identified over time, featuring, in the righthand (“forces”) side of their Newtonian equations of motion, in addition to other contributions, a velocitydependent term such as that appearing in the righthand side of the first formula mentioned above. The solvable character of these models allows detailed analyses of their behavior, which in some cases is quite remarkable (for instance isochronous or asymptotically isochronous). In this paper we introduce and discuss various discretetime dynamical systems, which are as well solvable, which also display interesting behaviors (including isochrony and asymptotic isochrony) and which reduce to dynamical systems of goldfish type in the limit when the discretetime independent variable ℓ = 0, 1, 2,... becomes the standard continuoustime independent variable t, 0 ≤ t < ∞. Key words: nonlinear discretetime dynamical systems; integrable and solvable maps; isochronous discretetime dynamical systems; discretetime dynamical systems of goldfish type 2010 Mathematics Subject Classification: 37J35; 37C27; 70F10; 70H06 1
email: ragnisco @ roma1.infn.it and
, 1996
"... Abstract. Integrable discretizations are introduced for the rational and hyperbolic spin Ruijsenaars–Schneider models. These discrete dynamical systems are demonstrated to belong to the same integrable hierarchies as their continuous–time counterparts. Explicit solutions are obtained for arbitrary f ..."
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Abstract. Integrable discretizations are introduced for the rational and hyperbolic spin Ruijsenaars–Schneider models. These discrete dynamical systems are demonstrated to belong to the same integrable hierarchies as their continuous–time counterparts. Explicit solutions are obtained for arbitrary flows of the hierarchies, The famous Calogero–Moser many–particle model possesses the most ramified tree of all possible kinds of generalizations. Even at the classical (non– quantum) level, one has three types of the usual Calogero–Moser (CM) systems: rational, hyperbolic/trigonometric, and elliptic [1], [2]. They may be