Saari’s conjecture for the collinear $n$-body problem

Abstract

Abstract. In 1970 Don Saari conjectured that the only solutions of the Newtonian n-body problem that have constant moment of inertia are the relative equilibria. We prove this conjecture in the collinear case for any potential that involves only the mutual distances. Furthermore, in the case of homogeneous potentials, we show that the only collinear and non-zero angular momentum solutions are homographic motions with central configurations.

N-BODY SOLUTIONS AND COMPUTING GALACTIC MASSES

“…3 Both conjectures have been verified for any collinear N-body problem (Diacu et al 2005;Saari 2005 (Section 2.4.2)), and for the general three-body problem (Moeckel 2005(Moeckel , 2008. Moreover, the conjectures can be expected to hold in general; e.g., the first one holds for generic settings of N-body systems (Schmah & Stoica 2007).…”

On the singularities of the curved $n$-body problem

“…The reason for having to exclude collision-antipodal configurations from the hypothesis of Lemma 2 is connected to a property proved in Theorem 1 (iii) of [4]. We showed there that there are choices of masses and initial conditions for which a 3-body problem taking place in S 2 1 can have finite forces and velocities at a collision-antipodal configuration; in other words the solution remains analytic at t * . For instance, this is the case when two bodies of mass 4m and a third body of mass m move on a great circle of S 2 1 , forming at each moment an isosceles triangle, and such that the larger bodies collide in finite time while the smaller body reaches the diametrically opposed side of the circle.…”

“…We showed there that there are choices of masses and initial conditions for which a 3-body problem taking place in S 2 1 can have finite forces and velocities at a collision-antipodal configuration; in other words the solution remains analytic at t * . For instance, this is the case when two bodies of mass 4m and a third body of mass m move on a great circle of S 2 1 , forming at each moment an isosceles triangle, and such that the larger bodies collide in finite time while the smaller body reaches the diametrically opposed side of the circle. So there are orbits that do not experience a singularity at t * but for which lim inf t→t * min 1<i≤j<n |κq i ⊙ q j − 1| = 0.…”

Saari's conjecture is true for generic vector fields

“…The conjecture has been proven for the planar 3-body problem with equal masses by C. McCord [McC04] and J. Llibre and E. Piña [LlP02]. F. Diacu, E. Pérez-Chavela, and M. Santoprete [DPS05] have proven the conjecture for the collinear N -body problem. Recently R. Moeckel [Moe05a,Moe05b] has given a computeraided proof for the general 3-body conjecture in R d .…”