Taking the second derivative of r with respect to time and using Hamilton's equation of motion for p r yields the radial-force equation d 2 r d t 2 = 1 m d p r d t = − − 1 m ( ∂ ∂ H ∂ ∂ r ) = p φ φ 2 m 2 r 3 − − 1 m d U d r = L 2 m 2 r 3 + 1 m F ( r )