A spherical pendulum consists of a particle of mass \(m\) on the end of a string of length \(R\). The position of the particle can be described by a polar angle \(\theta\) and an azimuthal angle \(\varphi\). The length of the string decreases at the rate \(d R / d t=-f(t)\), where \(f(t)\) is a positive function of time.

(a) Find the Lagrangian of the particle, using \(\theta\) and \(\varphi\) as generalized coordinates.

(b) Find the Hamiltonian \(H\). Is \(H\) equal to the energy? Why of why not?

(c) Is either \(E\) or \(H\) conserved? Why or why not?