Solution of the Goldfish N-Body Problem in the Plane with (Only) Nearest-Neighbor Coupling Constants All Equal to Minus One Half

Abstract

The (Hamiltonian, rotation- and translation-invariant) “goldfish” N-body problem in the plane is characterized by the Newtonian equations of motion written here in their complex version, entailing the identification of the real “physical” plane with the complex plane. In this paper we exhibit in completely explicit form the solution of the initial-value problem for this N-body model in the special case in which the two-body interaction only acts among “nearest neighbors” (namely, only among particles whose labels differ by one unit: a n,m = 0 unless |n − m| = 1) and the corresponding coupling constants all equal minus one half, a n,n+1 = a n+1,n = −1/2, n = 1, 2, ..., N − 1. This result implies that, if ω is a real nonvanishing constant, say, without loss of generality, ω > 0, then all the solutions of this N-body model are completely periodic indeed isochronous with period T = 2π / ω. An analogous conclusion holds as well for the model in which also the first and last particle interact with the same coupling constant, namely a 1,N = a N,1 = −1/2 (rather than vanishing).