Novel solvable extensions of the goldfish many-body model

Abstract

A novel solvable extension of the goldfish N-body problem is presented. Its Newtonian equations of motion read ζ̈n=2aζ̇nζn+2∑m=1,m≠nN(ζ̇n−aζn2)(ζ̇m−aζm2)∕(ζn−ζm), n=1,…,N, where a is an arbitrary (nonvanishing) constant and the rest of the notation is self-evident. The isochronous version of this model is characterized by the Newtonian equations of motion z̈n−3iωżn−2ω2zn=2a(żn−iωzn)zn+2∑m=1,m≠nN(żn−iωzn−azn2)(żm−iωzm−azm2)∕(zn−zm), n=1,…,N, where ω is an arbitrary positive constant and the points zn(t) move now necessarily in the complex z-plane. The generic solution of this second model is completely periodic with a period Tk=kT which is an integer multiple k (not larger than N!, indeed generally much smaller) of the basic period T=2π∕ω and which is independent of the initial data (for sufficiently small, but otherwise arbitrary, changes of such data). These many-body models have an intriguing variety of equilibrium configurations (genuine: with no two particles sitting at the same place), but only for small values of N (N=2,3,4 for the first model, N=2,3,4,5 for the second). Other versions of these models are also discussed. The study of the behavior of the second, isochronous model around its equilibrium configurations yields some amusing diophantine results.