4. A spherical pendulum where a bob of mass m attached to a light inextensible rod length ( free to swing in any direction about a fixed pivot is shown in Figure 1(b). In spherical co-ordinates, x = l sin d cos 0, y = l sin o sin 0, z = a + lcoso.Appropriate generalized coordinates are Q = (ab, 6). Kinetic energy: The kinetic energy T = m 3'32 + Q2 + 2'2 in s herical coordinates is 2 P ml?2 T = 7% + 92 sin2 (1;). The potential energy is U = mgz + constant so U = mg(a+ cos). (a) Find the associated differential equations of motion. Deduce that 7711?2(sin2 (b)? = const. (which corresponds to conservation of angular momentum about the zaxis. (b) Determine the conjugate momenta P E (194,, 199). (c) Express Q in terms of t, Q, P, and nd the Hamiltonian H(t, (,'b, 6,194), p9). (d) What are the canonical equations of motion? Compare with the equations of motion found in part (a). (e) Describe a strategy for solving the canonical equations, but do not solve them. \f