\[V=mgL\sin{\theta}\]
\[E=\frac{1}{2}mv^2+\frac{1}{2}I\omega^2\]
\[L=V-E=mgL\sin{\theta}-\left(\frac{1}{2}mv^2+\frac{1}{2}I\omega^2\right)=mgL\sin{\theta}-\frac{3}{4}mv^2=mgL\sin{\theta}-\frac{3}{4}m\left(\frac{dx}{dt}\right)^2\]
\[\frac{d}{dt}\left(\frac{\partial L}{\partial \dot{x}}\right)-\frac{\partial L}{\partial x}=0\]
\[E=\frac{1}{2}mv^2+\frac{1}{2}I\omega^2\]
\[L=V-E=mgL\sin{\theta}-\left(\frac{1}{2}mv^2+\frac{1}{2}I\omega^2\right)=mgL\sin{\theta}-\frac{3}{4}mv^2=mgL\sin{\theta}-\frac{3}{4}m\left(\frac{dx}{dt}\right)^2\]
\[\frac{d}{dt}\left(\frac{\partial L}{\partial \dot{x}}\right)-\frac{\partial L}{\partial x}=0\]
