A thin, stiff metal ring of radius \(R\) is placed in a vertical plane, and made to spin with constant angular velocity \(\omega\) about a vertical axis that passes through the center of the ring. A bead of mass \(m\) is free to slide around the ring, with its position defined by its angle \(\theta\) up from the bottom of the ring.

(a) Find the Hamiltonian of the bead and write out the corresponding Hamilton-Jacobi equation.

(b) Show that Hamilton's principal function can be separated into \(S=S_{\theta}(\theta)+S_{t}(t)\), and find \(S_{t}(t)\) explicitly and \(S_{\theta}(\theta)\) as an integral over a function of \(\theta\).