Saari's homographic conjecture for planar equal-mass three-body problem under a strong force potential

TL;DR

This paper proves Saari's homographic conjecture for the planar equal-mass three-body problem under a strong force potential, showing that constant configurational measure implies fixed shape of the system.

Contribution

The paper confirms Saari's conjecture specifically for the three-body problem with equal masses under a strong force potential, a case not previously established.

Findings

Saari's conjecture holds for the specified case.

Constant configurational measure implies fixed shape.

The proof is specific to the strong force potential case.

Abstract

Donald Saari conjectured that the $N$-body motion with constant configurational measure is a motion with fixed shape. Here, the configurational measure $\mu$ is a scale invariant product of the moment of inertia $I=\sum_k m_k |q_k|^2$ and the potential function $U=\sum_{i<j} m_i m_j/|q_i-q_j|^\alpha$, $\alpha >0$. Namely, $\mu = I^{\alpha/2}U$. We will show that this conjecture is true for planar equal-mass three-body problem under the strong force potential $\sum_{i<j} 1/|q_i-q_j|^2$.