Past exam of the theoretical physics I course of the University of Cambridge / 2014 / Paper 1 / 1 / d

The new kinetic matrix is diagonal, givingFor the first mode, $\varphi =0$: the line between the centres stays vertical, both centres translate together, and $\dot{\psi }=2\dot{\theta }$. For the second, $\theta =0$: the outer hoop stays still while the inner hoop rolls within it, with $\dot{\psi }=-\dot{\varphi }$. These coordinate motions are independent only at the retained quadratic order.