Open Problem 1: Are Palmore’s “Ignored Estimates” on the Number of Planar Central Configurations Correct?

Abstract

Abstract. We wish to draw attention on estimates on the number of relative equilibria in the Newtonian n-body problem that Julian Palmore published in 1975.Julian Palmore published in [12] a first estimate on the number of planar central configurations with given number of bodies n and given positive masses m 1 , . . . , m n . If these central configurations are non-degenerate, they are at least (3n − 4)(n − 1)!/2.The word "configuration" is ambiguous. Here we consider two "configurations" which are deduced one from the other by homothety and rotation as the same configuration. But we consider that applying a reflection to a non-collinear configuration produces a distinct configuration. We also consider that the bodies are "distinguishable": configurations which differ only in the numbering of the bodies are nevertheless considered as distinct.Palmore published in [13] a second estimate for the same number, under the same non-degeneracy hypothesis. There are at least (n − 2)!(2 n−1 (n − 2) + 1) planar central configurations of n bodies. We call this estimate the "ignored" Palmore estimate, because the subsequent authors on lower bounds did not even mention it (maybe because the proof is missing). Palmore also gave detailed estimates, i.e. a lower bound on the number of central configuration with given index. We do not know if these estimates are true. They seem however compatible with all the known results.When we speak of degeneracy or index, we think of central configurations as critical points of a function. This function is the Newtonian potential U = i<j m i m j /r ij , where r ij is the distance from body i to body j. The potential U is defined on the configuration space, and should be restricted to the quadric I = 1, where I = i<j m i m j r 2 ij /(m 1 + · · · + m n ). One should quotient out the rotations and the translations to get the good concept of configuration. The non-degeneracy can only occur at the level of the quotient space. Our convention is that a local minimum has zero index.

Central Configurations and Mutual Differences

“…Another easy consequence of Lemma 4.1 is the following proposition (see [1,11,17] for its importance in estimating the number non-degenerate planar central configurations of four bodies). Proof .…”

Pfaffians and the inverse problem for collinear central configurations

“…, n. For α = 1, it is the Newtonian gravitational interaction. There have been an active line of research on central configurations since decades: see for example [17], [4] [22]), [23], [15], [12], [2], [16], [1]. The purpose of this article is to survey some recent results (mostly of the author) in this field, with applications of fixed point theory.…”