We study the linear stability of regular n-gon rotating equilibria in the n-body problem with logarithm interaction. We find that linear stability is insured if a central mass M if M is bounded below and above by constants depending on the number and mass of the (equal) outer n bodies. Moreover, we provide explicit formulae for these bounds. In the absence of a central mass, we find that the regular n-gon is linearly stable for \(n =2,3,\ldots 6\) only.

中文翻译：

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具有对数引力相互作用的正方旋转平衡

我们研究了具有对数相互作用的n体问题中规则n边形旋转平衡的线性稳定性。我们发现，如果中心质量M的上下边界取决于（相等的）外部n个物体的数量和质量，则线性稳定性得到保证。此外，我们为这些界限提供了明确的公式。在没有中心质量的情况下，我们发现规则的n边形仅对于\(n =2,3,\ldots 6\)是线性稳定的。