\frac{-\omega q\ddot{q}}{\dot{q}^2+\omega^2q^2} + \frac{\omega \dot{q}^2}{\dot{q}^2+\omega^2q^2} -\omega = & \omega\frac{\omega^2q^2+\dot{q}^2}{\dot{q}^2+\omega^2q^2} - \omega = 0\\