A pendulum system consists of a mass-less rod of length ?, and a bob of mass m at the bottom end which is free to move in a vertical plane. The top end of the rod is forced to move vertically, so that its displacement at any time t is given by y = - h(t), where h is some known function. If ? is the angle the rod makes with the downward vertical direction, the position of the bob in Cartesian coordinates is x = ? sin ?, and y = - h - ? cos ?.

(a) Find the Lagrangian of this system in terms of ?.

0 (0,-h(t)) 1 m X