(a) The action for a system with a generalized coordinate $q$ is given by

$$S=\int_{t_{1}}^{t_{2}} L(q, \dot{q}, t) d t$$

State the Principle of Least Action and state the Euler-Lagrange equation.

(b) Consider a light rigid circular wire of radius $a$ and centre $O$. The wire lies in a vertical plane, which rotates about the vertical axis through $O$. At time $t$ the plane containing the wire makes an angle $\phi(t)$ with a fixed vertical plane. A bead of mass $m$ is threaded onto the wire. The bead slides without friction along the wire, and its location is denoted by $A$. The angle between the line $O A$ and the downward vertical is $\theta(t)$.

Show that the Lagrangian of this system is

$$\frac{m a^{2}}{2} \dot{\theta}^{2}+\frac{m a^{2}}{2} \dot{\phi}^{2} \sin ^{2} \theta+m g a \cos \theta .$$

Calculate two independent constants of the motion, and explain their physical significance.