The action for a system with generalized coordinates $q_{i}(t)$ for a time interval $\left[t_{1}, t_{2}\right]$ is given by

$$S=\int_{t_{1}}^{t_{2}} L\left(q_{i}, \dot{q}_{i}, t\right) d t$$

where $L$ is the Lagrangian. The end point values $q_{i}\left(t_{1}\right)$ and $q_{i}\left(t_{2}\right)$ are fixed.

Derive Lagrange's equations from the principle of least action by considering the variation of $S$ for all possible paths.

Define the momentum $p_{i}$ conjugate to $q_{i}$. Derive a condition for $p_{i}$ to be a constant of the motion.

A symmetric top moves under the action of a potential $V(\theta)$. The Lagrangian is given by

$$L=\frac{1}{2} I_{1}\left(\dot{\theta}^{2}+\dot{\phi}^{2} \sin ^{2} \theta\right)+\frac{1}{2} I_{3}(\dot{\psi}+\dot{\phi} \cos \theta)^{2}-V$$

where the generalized coordinates are the Euler angles $(\theta, \phi, \psi)$ and the principal moments of inertia are $I_{1}$ and $I_{3}$.

Show that $\omega_{3}=\dot{\psi}+\dot{\phi} \cos \theta$ is a constant of the motion and give expressions for two others. Show further that it is possible for the top to move with both $\theta$ and $\dot{\phi}$ constant provided these satisfy the condition

$$I_{1} \dot{\phi}^{2} \sin \theta \cos \theta-I_{3} \omega_{3} \dot{\phi} \sin \theta=\frac{d V}{d \theta}$$