A pendulum consisting of a ball at the end of a rope swings back and forth in a two dimensional vertical plane, with the angle \(\theta\) between the rope and the vertical evolving in time. The rope is pulled upward at a constant rate so that the length \(l\) of the pendulum's arm is decreasing according to \(d l / d t=-\alpha \equiv\) constant.

(a) Find the Lagrangian for the system with respect to the angle \(\theta\).

(b) Write the corresponding equations of motion.

(c) Repeat parts

(a) and

(b) using Lagrange multipliers .