Geometric Characterizations for Variational Minimization Solutions of the 3-Body Problem

In this paper, we prove that for any given positive masses the variational minimization solutions of the 3-body problem in R 3 or R 2 are precisely the planar equilateral triangle circular solutions found by J. Lagrange in 1772, and that the variational minimization solutions of the circular restricted 3-body problem in R 3 or R 2 are also planar equilateral triangle circular solutions.