On a Periodic Solution of the 4-Body Problems

We study the necessary and sufficient conditions on the masses for the periodic solution of planar 4-body problems, where three particles locate at the vertices of an equilateral triangle and rotate with constant angular O-ring velocity about a resting particle.We prove that the above periodic motion is a solution of Newtonian 4-body problems if and only if the resting particle is at the origin and the masses of the other three particles are equal and their ART angular velocity satisfies a special condition.