(i) An axisymmetric bowling ball of mass $M$ has the shape of a sphere of radius $a$. However, it is biased so that the centre of mass is located a distance $a / 2$ away from the centre, along the symmetry axis.

The three principal moments of inertia about the centre of mass are $(A, A, C)$. The ball starts out in a stable equilibrium at rest on a perfectly frictionless flat surface with the symmetry axis vertical. The symmetry axis is then tilted through $\theta_{0}$, the ball is spun about this axis with an angular velocity $n$, and the ball is released.

Explain why the centre of mass of the ball moves only in the vertical direction during the subsequent motion. Write down the Lagrangian for the ball in terms of the usual Euler angles $\theta, \phi$ and $\psi$.

(ii) Show that there are three independent constants of the motion. Eliminate two of the angles from the Lagrangian and find the effective Lagrangian for the coordinate $\theta$.

Find the maximum and minimum values of $\theta$ in the motion of the ball when the quantity $\frac{C^{2} n^{2}}{A M g a}$ is (a) very small and (b) very large.