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

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 ${\sum }_{i<j}1/|q_i-q_j{|}^2$.

References in corpus (1)

• Saari's Homographic Conjecture of the Three-Body Problem