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

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 ($\alpha =2$).

Submitted to Journal of Physics A

References in corpus (4)

• Saari's Conjecture for the Collinear $n$-Body Problem
• Saari's Homographic Conjecture of the Three-Body Problem
• Evolution of the Moment of Inertia of Three-Body Figure-Eight Choreography
• Kinematic geometry of triangles and the study of the three-body problem

Cited by in corpus (1)

• On the sectional curvature along central configurations