A massless unstretchable string is slung over a massless pulley. A weight of mass \(2 m\) is attached to one end of the string and a weight of mass \(m\) is attached to the other end. One end of a spring of force constant \(k\) is attached beneath \(m\), and a second weight of mass \(m\) is hung on the spring. Using the distance \(x\) of the weight \(2 m\) beneath the pulley and the stretch \(y\) of the spring as generalized coordinates, find the Hamiltonian of the system.

(a) Show that one of the two coordinates is ignorable (i.e., cyclic.) To what symmetry does this correspond?

(b) If the system is released from rest with \(y(0)=0\), find \(x(t)\) and \(y(t)\).