Saari's Homographic Conjecture of the Three-Body Problem

TL;DR

This paper proves Saari's homographic conjecture for large initial condition sets in three-body problems with homogeneous potentials, including Newtonian gravity, showing that constant configurational measure implies homographic solutions.

Contribution

The paper provides a proof of Saari's homographic conjecture for broad classes of three-body problems with homogeneous potentials, extending previous partial results.

Findings

Confirmed the conjecture for large initial condition sets in three-body problems.

Extended the validity of the conjecture to some n-body problems with n ≥ 3.

Applicable to systems with Newtonian and other homogeneous potentials.

Abstract

Saari's homographic conjecture, which extends a classical statement proposed by Donald Saari in 1970, claims that solutions of the Newtonian $n$-body problem with constant configurational measure are homographic. In other words, if the mutual distances satisfy a certain relationship, the configuration of the particle system may change size and position but not shape. We prove this conjecture for large sets of initial conditions in three-body problems given by homogeneous potentials, including the Newtonian one. Some of our results are true for $n\ge 3$.