(i) The action for a system with generalized coordinates $\left(q_{a}\right)$ is given by

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

Derive Lagrange's equations from the principle of least action by considering all paths with fixed endpoints, $\delta q_{a}\left(t_{1}\right)=\delta q_{a}\left(t_{2}\right)=0$.

(ii) A pendulum consists of a point mass $m$ at the end of a light rod of length $l$. The pivot of the pendulum is attached to a mass $M$ which is free to slide without friction along a horizontal rail. Choose as generalized coordinates the position $x$ of the pivot and the angle $\theta$ that the pendulum makes with the vertical.

Write down the Lagrangian and derive the equations of motion.

Find the frequency of small oscillations around the stable equilibrium.

Now suppose that a force acts on the pivot causing it to travel with constant acceleration in the $x$-direction. Find the equilibrium angle $\theta$ of the pendulum.