A pendulum consists of a plumb bob of mass \(m\) on the end of a string that swings back and forth in a plane. The upper end of the string passes through a small hole in the ceiling, and the angle \(\theta\) of the bob relative to the vertical changes with time as it swings back and forth. The string is pulled upward at constant rate through the hole, so the length \(R\) of the pendulum decreases at a constant rate, with \(d R / d t=-\alpha\).

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

(b) Find the Hamiltonian \(H\). Is \(H\) equal to the energy \(E\) ? Why or why not?

(c) Is either \(H\) or \(E\) conserved? Why or why not?