A plane pendulum consists of a rod of length \(R\) and negligible mass supporting a plumb bob of mass \(m\) that swings back and forth in a uniform gravitational field \(g\). The point of support at the top end of the rod is forced to oscillate vertically up and down with \(y=A \cos \omega t\). Using the angle \(\theta\) of the rod from the vertical as the coordinate,

(a) find the Lagrangian of the bob.

(b) Find the Hamiltonian \(H\). Is \(H=E\), the energy? Is either one or both conserved?

(c) Write out Hamilton's equations of motion.