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

摘要

Saari's homographic conjecture claims that in the N-body problem under the homogeneous potential, U = alpha(-1)Sigma m(i)m(j)/r(ij)(alpha) for alpha not equal 0, a motion with the constant configurational measure mu = (IU)-U-alpha/2 is homographic, where I represents the moment of inertia defined by I = Sigma m(i)m(j)r(ij)(2)/Sigma m(k), m(i) is the mass, and r(ij) is the distance between particles. We prove this conjecture for general masses, m(k) {L-End} > 0, in the planar three-body problem under the Newton potential (alpha = 1) and a strong force potential (alpha = 2).