Simple Pendulum

Lagrangian

$$\begin{equation*} \begin{split} x &= l \sin \theta \\ y &= -l \cos \theta \\ \dot{x} &= l\frac{\partial}{\partial t} \sin \theta = l \cos \theta \dot{\theta} \\ \dot{y} &= l\frac{\partial}{\partial t} \cos \theta = -l \sin \theta \dot{\theta} \\ \end{split} \end{equation*}$$

$$\begin{equation*} \begin{split} L &= T - U \\ U &= mgh = mg(l- l \cos \theta) = mgl - mgl \cos \theta \\ T &= \frac{1}{2}m(\dot{x}^2 + \dot{y}^2) \\ &= \frac{1}{2}m(l^2{\cos}^2 \theta {\dot{\theta}^2} + l^2{\sin}^2 \theta {\dot{\theta}^2} ) \\ &= \frac{1}{2}ml^2{\dot{\theta}^2} \\ L &= \frac{1}{2}ml^2{\dot{\theta}^2} -mgl + mgl \cos \theta \\ \end{split} \end{equation*}$$

$$\begin{equation*} \begin{split} \frac{\partial L}{\partial \theta} &= -mgl \sin \theta \\ \frac{\partial L}{\partial \dot{\theta}} &= ml^2 \dot{\theta} \\ \frac{\partial }{\partial t} \frac{\partial L}{\partial \dot{\theta}} &= ml^2 \ddot{\theta} \\ \frac{\partial }{\partial t} \frac{\partial L}{\partial \dot{\theta}} - \frac{\partial L}{\partial \theta} &= Q \\ ml^2 \ddot{\theta} + mgl \sin \theta &= Q \\ \text{where } Q = -b \dot{\theta} + u \\ \\ \text{when } Q = 0, \\ \ddot{\theta} &= -\frac{g}{l} \sin \theta \\ \text{when } b \dot{\theta} = 0, \\ \ddot{\theta} &= -\frac{g}{l} \sin \theta + \frac{1}{ml^2}u\\ \text{else}, \\ \ddot{\theta} &= -\frac{g}{l} \sin \theta - \frac{b}{ml^2} \dot{\theta} + \frac{1}{ml^2}u\\ \end{split} \end{equation*}$$