On the Number of Equilibria Balancing Newtonian Point Masses with a Central Force

TL;DR

This paper investigates the number of equilibrium points in a planar system of Newtonian point masses with a quadratic term, establishing bounds and analyzing specific configurations to understand how many equilibria can exist.

Contribution

It provides new bounds on the number of equilibria, including a sharp lower bound and an exponential upper bound, and introduces the analysis of ring configurations for maximum equilibria.

Findings

Number of equilibria is finite for generic parameters.

Lower bound of n+1 equilibria, sharp in some cases.

Upper bound grows exponentially with n.

Abstract

We consider the critical points (equilibria) of a planar potential generated by $n$ Newtonian point masses augmented with a quadratic term (such as arises from a centrifugal effect). Particular cases of this problem have been considered previously in studies of the circular restricted $n$-body problem. We show that the number of equilibria is finite for a generic set of parameters, and we establish estimates for the number of equilibria. We prove that the number of equilibria is bounded below by $n+1$, and we provide examples to show that this lower bound is sharp. We prove an upper bound on the number of equilibria that grows exponentially in $n$. In order to establish a lower bound on the maximum number of equilibria, we analyze a class of examples, referred to as ``ring configurations'', consisting of $n-1$ equal masses positioned at vertices of a regular polygon with an additional mass located at the center. Previous numerical observations indicate that these configurations can produce as many as $5n-5$ equilibria. We verify analytically that the ring configuration has at least $5n-5$ equilibria when the central mass is sufficiently small. We conjecture that the maximum number of equilibria grows linearly with the number of point masses. We also discuss some mathematical similarities to other equilibrium problems in mathematical physics, namely, Maxwell's problem from electrostatics and the image counting problem from gravitational lensing.