A rod is bent in the middle by angle \(\alpha\). The bottom portion is kept vertical and the top portion is therefore oriented at angle \(\alpha\) to the vertical. A bead of mass \(m\) is slipped onto the top portion and the bottom portion is forced by a motor to rotate at constant angular speed \(\omega\) about the vertical axis.

(a) Define a generalized coordinate for the bead and write down the Lagrangian.

(b) Identify any conserved quantity or quantities and explain why it (or they) are conserved.

(c) Find the generalized momentum of the bead and the Hamiltonian.

(d) Are there any equilibrium points of the bead? If so, are they stable or unstable?