In solving the two body problem we choose the center of mass of the system as the origin. But is the frame of reference we are setting up inertial? I don't think so. Suppose the central force is gravitation, A has mass 4kg and B has mass 1kg. If we put an object at the center of mass, then obviously in the frame of the center of the mass this object is at rest. But an nonzero net force pointing towards A is acting on the mass. Therefore, Newton's First Law is violated and the frame in noninertial. The correct frame to choose, in my opinion, is the frame centered at the "equilibrium position" (If we put an object at the equilibrium position no net force act on the object) Any thoughts?