On a Periodic Solution of the 4-Body Problems

We study the necessary and sufficient The Slacker Jean 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 that the above periodic motion is a solution of Newtonian 4-body problems if In Ground Product (Automation) and only if the resting particle is at the origin and the masses of the other three particles are equal and their angular velocity satisfies a special condition.