For a collection of N point particles, the scalar moment of inertia I about the origin is defined by the equation I = ∑ ∑ k = 1 N m k | r k | 2 = ∑ ∑ k = 1 N m k r k 2 where m k and r k represent the mass and position of the k th particle. r k = | r k | is the position vector magnitude. The scalar G is defined by the equation G = ∑ ∑ k = 1 N p k ⋅ ⋅ r k where p k is the momentum vector of the k th particle. Assuming that the masses are constant, G is one-half the time derivative of this moment of inertia 1 2 d I d t = 1 2 d d t ∑ ∑ k = 1 N m k r k ⋅ ⋅ r k = ∑ ∑ k = 1 N m k d r k d t ⋅ ⋅ r k = ∑ ∑ k = 1 N p k ⋅ ⋅ r k = G . In turn, the time derivative of G can be written d G d t = ∑ ∑ k = 1 N p k ⋅ ⋅ d r k d t + ∑ ∑ k = 1 N d p k d t ⋅ ⋅ r k = ∑ ∑ k = 1 N m k d r k d t ⋅ ⋅ d r k d t + ∑ ∑ k = 1 N F k ⋅ ⋅ r k = 2 T + ∑ ∑ k = 1 N F k ⋅ ⋅ r k where m k is the mass of the k th particle, F k = ⁠ d p k / dt ⁠ is the net force on that particle, and T is the total kinetic energy of the system according to the v k = ⁠ d r k / dt ⁠ velocity of each particle T = 1 2 ∑ ∑ k = 1 N m k v k 2 = 1 2 ∑ ∑ k = 1 N m k d r k d t ⋅ ⋅ d r k d t .