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

DOI

10.1080/14029251.2013.868267How to use a DOI?

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 x_n of the polynomial equation PN(α,β) (x)= w, where PN(α,β) (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 x_n (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.

Copyright

© 2013 The Authors. Published by Atlantis Press and Taylor & Francis

Open Access

This is an open access article distributed under the CC BY-NC 4.0 license (http://creativecommons.org/licenses/by-nc/4.0/).

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TY  - JOUR
AU  - Oksana Bihun
AU  - Francesco Calogero
PY  - 2021
DA  - 2021/01/06
TI  - 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
JO  - Journal of Nonlinear Mathematical Physics
SP  - 539
EP  - 551
VL  - 20
IS  - 4
SN  - 1776-0852
UR  - https://doi.org/10.1080/14029251.2013.868267
DO  - 10.1080/14029251.2013.868267
ID  - Bihun2021
ER  -