Saari's homographic conjecture for general masses in planar three-body problem under Newton potential and a strong force potential

TL;DR

This paper proves Saari's homographic conjecture for the planar three-body problem with general masses under Newton and strong force potentials, confirming that constant configurational measure implies homographic motion.

Contribution

It extends the proof of Saari's conjecture to the case of general masses for both Newton and strong force potentials in the three-body problem.

Findings

Confirmed Saari's conjecture for Newton potential.

Confirmed Saari's conjecture for strong force potential.

Applicable to general positive masses in the three-body problem.

Abstract

Saari's homographic conjecture claims that, in the N-body problem under the homogeneous potential, $U=\alpha^{-1}\sum m_i m_j/r_{ij}^\alpha$ for $\alpha\ne 0$, a motion having constant configurational measure $\mu=I^{\alpha/2}U$ is homographic, where $I$ represents the moment of inertia defined by $I=\sum m_i m_j r_{ij}^2/\sum m_k$, $m_i$ the mass, and $r_{ij}$ the distance between particles. We prove this conjecture for general masses $m_k>0$ in the planar three-body problem under Newton potential ($\alpha=1$) and a strong force potential ($\alpha=2$).