A pendulum of length $\ell$ oscillates in the $x y$ plane, making an angle $\theta(t)$ with the vertical $y$ axis. The pivot is attached to a moving lift that descends with constant acceleration $a$, so that the position of the bob is

$$x=\ell \sin \theta, \quad y=\frac{1}{2} a t^{2}+\ell \cos \theta .$$

Given that the Lagrangian for an unconstrained particle is

$$L=\frac{1}{2} m\left(\dot{x}^{2}+\dot{y}^{2}\right)+m g y,$$

determine the Lagrangian for the pendulum in terms of the generalized coordinate $\theta$. Derive the equation of motion in terms of $\theta$. What is the motion when $a=g$ ?

Find the equilibrium configurations for arbitrary $a$. Determine which configuration is stable when

$$\text { (i) } a<g$$

and when

$$\text { (ii) } a>g \text {. }$$