ta = u_0 - mgl \sin\theta -
      b\dot\theta,$$ this time with $b = 0$.  This time the system dynamics are
      truly second-order.  We can always think of any second-order system, $$\ddot{\theta} = f(\theta,\dot\theta),$$ as
      (coupled) first-order system with twice as many variables. This system is equivalent to the
      two-dimensional first-order system \begin{align*} \dot x_1 =& x_2 \\ \dot
      x_2 =& f(x_1,x_2), \end{align*} where $x_1 = \theta$ and $x_2 = \dot \theta$.
      Therefore, the g