A spring pendulum features a pendulum bob of mass \(m\) attached to one end of a spring of force-constant \(k\) and unstretched length \(R\). The other end of the spring is attached to a fixed point on the ceiling. The pendulum is allowed to swing in a plane. Use \(r\), the distance of the bob from the fixed point, and \(\theta\), the angle of the spring relative to the vertical, as generalized coordinates.

(a) Find the kinetic and potential energies of the bob in terms of the generalized coordinates and velocities.

(b) Find the Lagrangian. Are there any ignorable coordinates?

(c) Are there any conserved quantities?

(d) Find a complete set of equations of motion, including as many first integrals as possible.