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
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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.
- © 2013 The Authors. Published by Atlantis Press and Taylor & Francis
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Cite this article
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 -