Lagrange’s equations — UPSC IFoS 2019 Physics PYQ

A particle of mass m rests on a smooth plane. The plane is then raised to an inclination angle \theta at a constant rate \alpha (\theta = 0 at t = 0), causing the particle to move down the plane, as shown in the figure below.

Write down the Lagrangian and determine the equations of motion. Solve the resulting equation for "r" to obtain the expression for r(t).