Consider the spherical pendulum subject to gravity g.

(a) Using spherical coordinates (r, , ) write down the Lagrangian.

Derive the Euler-Lagrange equations for (r, , ), subject to the constraint r = l where l is the length of the pendulum. After substitution of r=l, integrate once the and equations to express d/dt and d/dt in terms of integration constants j (azimuthal angular momentum) and total energy E.

(b) Sketch qualitatively the effective potential Veff() for large and small j.

Find the equilibrium position o such that dVeff()/d = 0 in terms of E and j.

Find the frequency of small oscillations around o. How about the frequency as o go to 0 and /2?

(c) Use the Lagrange multiplier for the r equation and Find the tension in the rod at theequilibrium position o.