This problem will familiarize you with the procedure of "quantization" of a classical system by identifying "phase space" variables (coordinates and momenta), finding the hamiltonian, turning phase space variables into operators and poisson brackets into commutators, and representing these operators as coordinates and their derivatives a plane pendulum has kinetic energy t = 102 and potential energy v =-u cos θ where θ is its angle of deviation from the vertical. 1 is its moment of inertia and ụ-mg1 is a constant that depends on its weight mg and distance of its center of mass from the pivot point l. (overdot denotes time derivative, as usual.) a) write down the lagrangian l of the pendulum and derive its equation of motion b) find the canonical momentum j conjugate to the coordinate θ and write the hamil- tonian of the pendulum. derive the equations of motion in hamiltonian form (in terms of θ and j c) the pendulum is quite small (actually, a molecule) so it must be treated quantum mechanically. define operators θ and j corresponding to the phase space variables θ and j write their commutation relation and write the hamiltonian of the pendulum in terms of θ and j.