**Author(s): ** Francesco Calogero**Journal: ** Symmetry, Integrability and Geometry : Methods and Applications ISSN 1815-0659

**Volume: ** 8;

**Start page: ** 046;

**Date: ** 2012;

Original page**Keywords: ** nonlinear discrete-time dynamical systems |

integrable and solvable maps |

isochronous discrete-time dynamical systems |

discrete-time dynamical systems of goldfish type**ABSTRACT**

A new solvable many-body problem is identified. It is characterized by nonlinear Newtonian equations of motion (''acceleration equal force'') featuring one-body and two-body velocity-dependent forces ''of goldfish type'' which determine the motion ofan arbitrary number $N$ of unit-mass point-particles in a plane. The $N$ (generally complex) values $z_{n}(t)$ at time $t$ ofthe $N$ coordinates of these moving particles are given by the $N$eigenvalues of a time-dependent $Nimes N$ matrix $U(t)$explicitly known in terms of the $2N$ initial data $z_{n}(0)$and $dot{z}_{n}(0) $. This model comes in two dif/ferentvariants, one featuring 3 arbitrary coupling constants, the other only 2; for special values of these parameters all solutions are completely periodic with the same period independent of the initial data (''isochrony''); for other special values of these parameters this property holds up to corrections vanishing exponentially as $tightarrow infty$ (''asymptotic isochrony''). Other isochronous variants of these models are also reported. Alternative formulations, obtained by changing the dependent variables from the $N$ zeros of a monic polynomial of degree $N$ to its $N$ coefficients, are also exhibited. Some mathematical findings implied by some of these results - such as Diophantine properties of the zeros of certain polynomials - are outlined, but their analysis is postponed to a separate paper.

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