Equilibria of a solvable N-body problem and related properties of the N numbers xn at which the Jacobi polynomial of order N has the same value

Abstract

The class of solvable N-body problems of “goldfish” type has been recently extended by including (the additional presence of) three-body forces. In this paper we show that the equilibria of some of these systems are simply related to the N roots xn of the polynomial equation (x)= w, where (x) is the Jacobi polynomial of order N, the parameters α and β are related to parameters of the N-body problem (which can be arbitrarily assigned) and w is an arbitrary number. By investigating the behavior of these solvable N-body systems in the infinitesimal neighborhood of these equilibria, the eigenvalues associated to certain N × N matrices explicitly given in terms of the N numbers xn (and of additional free parameters of the N-body problem) are explicitly identified. In some cases—corresponding to isochronous N-body problems—these findings have a Diophantine connotation, inasmuch as these eigenvalues are then rational numbers.