<P> the rate of change of the angular momentum L equals the net torque N </P> <Dl> <Dd> N = d L d t = r _̇ × μ r _̇ + r × μ r _̈, (\ displaystyle \ mathbf (N) = (\ frac (d \ mathbf (L)) (dt)) = (\ dot (\ mathbf (r))) \ times \ mu (\ dot (\ mathbf (r))) + \ mathbf (r) \ times \ mu (\ ddot (\ mathbf (r))) \,) </Dd> </Dl> <Dd> N = d L d t = r _̇ × μ r _̇ + r × μ r _̈, (\ displaystyle \ mathbf (N) = (\ frac (d \ mathbf (L)) (dt)) = (\ dot (\ mathbf (r))) \ times \ mu (\ dot (\ mathbf (r))) + \ mathbf (r) \ times \ mu (\ ddot (\ mathbf (r))) \,) </Dd> <P> and using the property of the vector cross product that v × w = 0 for any vectors v and w pointing in the same direction, </P>

Vector two body orbital mechanics equation of motion and it's closed form solution