Linear Stability of the n-gon Relative Equilibria of the (1 + n)-Body Problem

We consider the linear stability of the regular n-gon relative equilibria of the (1 + n)-body problem. It is shown that there exist at most two kinds of infinitesimal bodies arranged alternatively at the vertices of a regular n-gon when n is even, and only one set of identical infinitesimal bodies when n is odd. In the case of n even, the regular n-gon relative equilibrium is shown to be linearly stable when \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${n \geqslant 14}$$\end{document}. In each case of n = 8, 10 and 12, linear stability can also be preserved if the ratio of two kinds of masses belongs to an open interval. When n is odd, the related conclusion on the linear stability is recalled.