A plane pendulum consists of a string supporting a plumb bob of mass \(m\) free to swing in a vertical plane and free to swing subject to uniform gravity \(g\). The upper end of the string is threaded through a hole in the ceiling and steadily pulled upward, so the length of the string beneath the point in the ceiling is \(\ell(t)=\ell_{0}-\alpha t\), where \(\alpha\) is a positive constant.

(a) Find the Lagrangian of the plumb bob.

(b) Find its Hamiltonian \(H\). Is \(H=E\), the energy of the bob?

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

(d) Solve them assuming \(l(t)\) is changing slowly and the angle of the pendulum remains small.