Regular n-gon rotating equilibria with logarithm gravitational interaction

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 \(n =2,3,\ldots 6\) $n=2,3,\dots 6$ only.