Another New Solvable Many-Body Model of Goldfish Type

• Published in: Symmetry, Integrability and Geometry: Methods and Applications
• Date: 2012
• Main Author: Calogero, F.
• Format: Article
• Language: English
• Published: Інститут математики НАН України 2012
• Online Access: https://nasplib.isofts.kiev.ua/handle/123456789/148460
• Journal Title: Digital Library of Periodicals of National Academy of Sciences of Ukraine
• Cite this: Another New Solvable Many-Body Model of Goldfish Type / C. Calogero // Symmetry, Integrability and Geometry: Methods and Applications. — 2012. — Т. 8. — Бібліогр.: 16 назв. — англ.

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 zn(t) at time t of the N coordinates of these moving particles are given by the N eigenvalues of a time-dependent N×N matrix U(t) explicitly known in terms of the 2N initial data zn(0) and 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→∞ (''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.