Geometric Characterization for Variational Minimization Solutions of the 3-Body Problem with Fixed Energy

In this paper, for any given positive masses we prove that the constrained variational minimization solutions of the fixed energy or fixed mean potential energy problem for the 3-body type problems with homogeneous potentials in R3 or R2 are precisely the planar equilateral triangle circular solutions found by J. Lagrange in 1772.