, Volume 12, Issue 1, pp 255-271
Date: 14 Aug 2012

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

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Abstract

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 \({n \geqslant 14}\) . 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.