Spiderweb Central Configurations

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 implementations. Additionally, we prove by a rigorous numerical method the uniqueness of such central configurations when $$\ell \in \{2,\ldots ,9\}$$ and arbitrary n and $$m_i$$ ; under the constraint $$m_1\ge m_2\ge \cdots \ge m_n$$ we also prove uniqueness for $$\ell \in \{10,\ldots ,18\}$$ 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 $$m_0, \ldots ,m_n$$ . Finally, our numerical simulations highlight some interesting properties of the mass distribution.

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