Past exam of the mathematics course of the University of Cambridge / 2015 / ii / Paper 2 / 7D / a / Solution

The two cyclic Euler angles give conserved canonical momentaHere $L_3$ is the body-axis angular momentum and $L_z$ the vertical angular momentum. Thus ${\omega }_3=L_3/I_3$ is constant. Since the Lagrangian has no explicit time dependence, the energy is also conserved:These are the three independent conserved quantities furnished by the continuous symmetries of the heavy symmetric top.