The n -body problem considers n point masses m i, i = 1, 2, …, n in an inertial reference frame in three dimensional space ℝ moving under the influence of mutual gravitational attraction. Each mass m i has a position vector q i. Newton's second law says that mass times acceleration m i ⁠ d q i / dt ⁠ is equal to the sum of the forces on the mass. Newton's law of gravity says that the gravitational force felt on mass m i by a single mass m j is given by F i j = G m i m j ‖ q j − − q i ‖ 2 ⋅ ⋅ ( q j − − q i ) ‖ q j − − q i ‖ = G m i m j ( q j − − q i ) ‖ q j − − q i ‖ 3 , where G is the gravitational constant and ‖ q j − q i ‖ is the magnitude of the distance between q i and q j (metric induced by the l 2 norm).