A pendulum is constructed from a bob of mass \(m\) on one end of a light string of length \(D\). The other end of string is attached to the top of a circular cylinder of radius \(R\) \((R<2 D / \pi)\). The string makes an angle \(\theta\) with the vertical, as shown in the figure, as the pendulum swings in the plane. There is a uniform gravity \(g\) directed downward.

(a) Find the Lagrangian of the bob, using \(\theta\) as the generalized coordinate.

(b) Identify any first integrals of motion.

(c) Find the frequency of small oscillations about the stable equilibrium point.