Fixed points and the inverse problem for central configurations

TL;DR

This paper explores how central configurations in the n-body problem can be identified as fixed points in shape space and discusses the inverse problem of determining masses for given collinear configurations, using fixed point theory.

Contribution

It introduces a novel approach to find central configurations as fixed points and addresses the inverse problem in one dimension, connecting these to fixed point theory.

Findings

Central configurations can be characterized as projective fixed points.

The inverse problem in 1D involves determining masses for given collinear configurations.

Application of fixed point theory provides new insights into the inverse problem.

Abstract

Central configurations play an important role in the dynamics of the $n$-body problem: they occur as relative equilibria and as asymptotic configurations in colliding trajectories. We illustrate how they can be found as projective fixed points of self-maps defined on the shape space, and some results on the inverse problem in dimension $1$, i.e. finding (positive or real) masses which make a given collinear configuration central. This survey article introduces readers to the recent results of the author, also unpublished, showing an application of the fixed point theory.