Pfaffians and the inverse problem for collinear central configurations

TL;DR

This paper investigates the inverse problem for collinear central configurations, providing new estimates on pfaffian positivity and identifying regions where solutions do not exist for positive masses.

Contribution

It offers new bounds on Albouy-Moeckel pfaffians and characterizes configuration regions lacking solutions for the inverse problem with positive masses.

Findings

Pfaffians are positive for n ≤ 6 or n ≤ 10 with α=1 (computer-assisted)

Explicit regions of the configuration space have no solutions for the inverse problem when n ≥ 4

New estimates improve understanding of the inverse problem for collinear configurations

Abstract

We consider, after Albouy-Moeckel, the inverse problem for collinear central configurations: given a collinear configuration of $n$ bodies, find positive masses which make it central. We give some new estimates concerning the positivity of Albouy-Moeckel pfaffians: we show that for any homogeneity $\alpha$ and $n\leq 6$ or $n\leq 10$ and $\alpha=1$ (computer-assisted) the pfaffians are positive. Moreover, for the inverse problem with positive masses, we show that for any homogeneity and $n\geq 4$ there are explicit regions of the configuration space without solutions of the inverse problem.