### Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 8 (2012), 046, 17 pages      arXiv:1207.4850      https://doi.org/10.3842/SIGMA.2012.046
Contribution to the Special Issue “Geometrical Methods in Mathematical Physics”

### Another New Solvable Many-Body Model of Goldfish Type

Francesco Calogero
Physics Department, University of Rome ''La Sapienza'', Istituto Nazionale di Fisica Nucleare, Sezione di Roma, Italy

Received May 03, 2012, in final form July 17, 2012; Published online July 20, 2012

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 of an arbitrary number $N$ of unit-mass point-particles in a plane. The $N$ (generally complex) values $z_{n}( t)$ at time $t$ of the $N$ coordinates of these moving particles are given by the $N$ eigenvalues of a time-dependent $N\times N$ 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 different variants, 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 $t\rightarrow \infty$ (''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.

Key words: nonlinear discrete-time dynamical systems; integrable and solvable maps; isochronous discrete-time dynamical systems; discrete-time dynamical systems of goldfish type.

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