A particle of mass $m_{1}$ is constrained to move in the horizontal $(x, y)$ plane, around a circle of fixed radius $r_{1}$ whose centre is at the origin of a Cartesian coordinate system $(x, y, z)$. A second particle of mass $m_{2}$ is constrained to move around a circle of fixed radius $r_{2}$ that also lies in a horizontal plane, but whose centre is at $(0,0, a)$. It is given that the Lagrangian $L$ of the system can be written as

$$L=\frac{m_{1}}{2} r_{1}^{2} \dot{\phi}_{1}^{2}+\frac{m_{2}}{2} r_{2}^{2} \dot{\phi}_{2}^{2}+\omega^{2} r_{1} r_{2} \cos \left(\phi_{2}-\phi_{1}\right)$$

using the particles' cylindrical polar angles $\phi_{1}$ and $\phi_{2}$ as generalized coordinates. Deduce the equations of motion and use them to show that $m_{1} r_{1}^{2} \dot{\phi}_{1}+m_{2} r_{2}^{2} \dot{\phi}_{2}$ is constant, and that $\psi=\phi_{2}-\phi_{1}$ obeys an equation of the form

$$\ddot{\psi}=-k^{2} \sin \psi$$

where $k$ is a constant to be determined.

Find two values of $\psi$ corresponding to equilibria, and show that one of the two equilibria is stable. Find the period of small oscillations about the stable equilibrium.