Spiderweb central configurations

Olivier Hénot; Christiane Rousseau

doi:10.1007/s12346-019-00330-y

paper

Spiderweb central configurations

arXiv:1810.09915 · doi:10.1007/s12346-019-00330-y

Abstract

In this paper we study spiderweb central configurations for the $N$ -body problem, i.e configurations given by $N = n \times ℓ + 1$ masses located at the intersection points of $ℓ$ concurrent equidistributed half-lines with $n$ circles and a central mass $m_{0}$ , under the hypothesis that the $ℓ$ masses on the $i$ -th circle are equal to a positive constant $m_{i}$ ; we allow the particular case $m_{0} = 0$ . We focus on constructive proofs of the existence of spiderweb central configurations, which allow numerical implementation. Additionally, we prove the uniqueness of such central configurations when $ℓ \in {2, \dots, 9}$ and arbitrary $n$ and $m_{i}$ ; under the constraint $m_{1} \geq m_{2} \geq \dots \geq m_{n}$ we also prove uniqueness for $ℓ \in {10, \dots, 18}$ and $n$ not too large. We also give an algorithm providing a rigorous proof of the existence and local unicity of such central configurations when given as input a choice of $n$ , $ℓ$ and $m_{0}, ..., m_{n}$ . Finally, our numerical simulations highlight some interesting properties of the mass distribution.