Another New Solvable Many-Body Model of Goldfish Type

A new solvable many-body problem is identified. It is characterized by nonlinear Newtonian equations of motion (“acceleration equal force”) featuring one-body and twobody 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 żn(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.