Spiderweb central configurations

TL;DR

This paper investigates the existence and uniqueness of spiderweb central configurations in the N-body problem, providing constructive proofs, algorithms, and numerical insights into mass distributions.

Contribution

It offers constructive proofs and algorithms for the existence and uniqueness of spiderweb central configurations, with numerical simulations revealing properties of mass distribution.

Findings

Existence of spiderweb configurations proven constructively.

Uniqueness established for certain parameter ranges.

Numerical simulations show properties of mass distribution.

Abstract

In this paper we study spiderweb central configurations for the $N$-body problem, i.e configurations given by $N=n \times \ell+1$ masses located at the intersection points of $\ell$ concurrent equidistributed half-lines with $n$ circles and a central mass $m_0$, under the hypothesis that the $\ell$ 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 $\ell \in \{2,\dots,9\}$ and arbitrary $n$ and $m_i$; under the constraint $m_1\geq m_2\geq \ldots \geq m_n$ we also prove uniqueness for $\ell \in \{10,\dots,18\}$ 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 $n$, $\ell$ and $m_0, . . . ,m_n$. Finally, our numerical simulations highlight some interesting properties of the mass distribution.