Derivatives of r with respect to time may be rewritten as derivatives of u = 1 / r with respect to angle: d u d θ θ = d d t ( 1 r ) d t d θ θ = − − r ˙ ˙ r 2 θ θ ˙ ˙ = − − r ˙ ˙ h d 2 u d θ θ 2 = − − 1 h d r ˙ ˙ d t d t d θ θ = − − r ¨ ¨ h θ θ ˙ ˙ = − − r ¨ ¨ h 2 u 2