Another new solvable many-body model of goldfish type

Francesco Calogero

Symmetry, integrability and geometry: methods and applications, Tome 8 (2012)

Résumé

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\to\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.

Détail

Keywords: 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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