Past exam of the mathematics course of the University of Cambridge / 2011 / ii / Paper 3 / 9C / Solution

Noether theorem associates a conserved quantity ${\sum }_i{p}_i\delta q_i$ to a continuous symmetry of the Lagrangian (with the usual boundary-term correction if the Lagrangian changes by a total derivative). The angles $\psi$ and $\varphi$ are cyclic coordinates, so for the heavy symmetric topTime-translation symmetry gives conserved energy, whose Hamiltonian expression isIt is quadratic in the momenta; $p_{\theta }=I_1\dot{\theta }$. Eliminating the cyclic velocities and putting $\mu =\cos \theta$, so ${\dot{\mu }}^2={\sin }^2\theta {\dot{\theta }}^2$, yieldsThe first term has degree three and the second degree two, displaying the cubic governing nutation.