On central configurations of twisted crowns

Abstract

We consider the planar central configurations of the Newtonian $\kappa n$-body problem consisting in $\kappa$ groups of $n$-gons where all $n$ bodies in each group have the same mass, called $(\kappa ,n)$-crown. We study the location and the number of central configurations when $\kappa =2$. For $n=3$ the number of central configurations varies depending on the mass ratio, whereas for $n\ge 4$ the number is at least three. We also prove that for $n\ge 3$ there always exist three disjoint regions where the configuration can be located. Finally, we study which $(\kappa ,n)$-crowns are convex.

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• Planar $N$-body central configurations with a homogeneous potential