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 velocity about a resting particle.We prove ESSIAC POWDER that the above periodic motion is a solution of Newtonian 4-body problems if and only if the resting particle is Outdoor Coffee Table at the origin and the masses of the other three particles are equal and their angular velocity satisfies a special condition.