New Solvable Many-Body Model of Goldfish Type

A new solvable N-body model of goldfish type is identified. Its Newtonian equations of motion read as follows: \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\eqalign{\ddot{z}_n = - 6\dot{z}_n}{z_n} - 4z_n^3 + {3 \over 2}(\dot{z}_n + 2z_n^2)\sum\limits_{k = 1}^N {\left( {{{{{\dot{z}}_k}} \over {{z_k}}} + 2{z_k}} \right)} + 2\sum\limits_{\ell = 1,\ell \ne n}^N {\left[ {{{({{\dot{z}}_n} + 2z_n^2)({{\dot{z}}_\ell } + 2z_\ell ^2)} \over {{z_n} - {z_\ell }}}} \right]} , n = 1, \ldots ,N, $$\end{document} where zn ≡ zn(t) are the N dependent variables (with N an arbitrary positive integer), t is the independent variable (“time”) and the dots indicate time-differentiations. Its isochronous variant is also obtained and discussed. Other new solvable models of goldfish type characterize the behavior of these systems in the immediate neighborhood of their equilibria.