Saari's Conjecture for the Collinear $n$-Body Problem

Saari猜想的证明

A proof of Saari's conjecture

本文证明了Saari猜想，指出具有恒定极惯性矩的经典$n$体问题解必表现为旋转刚体，并指出该猜想可推广至经典$n$体问题范畴之外。

Saari's homographic conjecture for planar equal-mass three-body problem in Newton gravity

Saari's homographic conjecture in N-body problem under the Newton gravity is the following; configurational measure μ=\sqrt{I}U, which is the product of square root of the moment of inertia I=(\sum m_k)^{-1}\sum m_i m_j r_{ij}^2 and the potential function U=\sum m_i m_j/r_{ij}, is constant if and only if the motion is homographic. Where m_k represents mass of body k and r_{ij} represents distance between bodies i and j. We prove this conjecture for planar equal-mass three-body problem. In this work, we use three sets of shape variables. In the first step, we use ζ=3q_3/(2(q_2-q_1)) where q_k \…

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

Saari's homographic conjecture claims that, in the N-body problem under the homogeneous potential, $U=α^{-1}\sum m_i m_j/r_{ij}^α$ for $α\ne 0$, a motion having constant configurational measure $μ=I^{α/2}U$ 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 ($α=1$) and a strong force potential ($α=2$).

On the n-body problem on surfaces of revolution

We explore the $n$-body problem, $n\geq 3,$ on a surface of revolution with a general interaction depending on the pairwise geodesic distance. Using the geometric methods of classical mechanics we determine a large set of properties. In particular, we show that Saari's conjecture fails on surfaces of revolution admitting a geodesic circle. We define homographic motions and, using the discrete symmetries, prove that when the masses are equal, they form an invariant manifold. On this manifold the dynamics are reducible to a one-degree of freedom system. We also find that for attractive interacti…

The n-body problem in spaces of constant curvature

We generalize the Newtonian n-body problem to spaces of curvature k=constant, and study the motion in the 2-dimensional case. For k>0, the equations of motion encounter non-collision singularities, which occur when two bodies are antipodal. This phenomenon leads, on one hand, to hybrid solution singularities for as few as 3 bodies, whose corresponding orbits end up in a collision-antipodal configuration in finite time; on the other hand, it produces non-singularity collisions, characterized by finite velocities and forces at the collision instant. We also point out the existence of several cla…

沿直线的点电荷势的临界点

Critical points of point charge potentials along lines

该研究证明了Gabrielov-Novikov-Shapiro猜想1.9，即n个点电荷的势在直线上的非常数限制至多有2n-1个临界点，证明采用对偶性与莫尔斯理论，且该界对正电荷也紧。