Symmetries in the n -body problem yield global integrals of motion that simplify the problem. Translational symmetry of the problem results in the center of mass C = ∑ ∑ i = 1 n m i q i ∑ ∑ i = 1 n m i moving with constant velocity, so that C = L 0 t + C 0, where L 0 is the linear velocity and C 0 is the initial position. The constants of motion L 0 and C 0 represent six integrals of the motion. Rotational symmetry results in the total angular momentum being constant A = ∑ ∑ i = 1 n q i × × p i , where × is the cross product. The three components of the total angular momentum A yield three more constants of the motion. The last general constant of the motion is given by the conservation of energy H. Hence, every n -body problem has ten integrals of motion.