a) Write down the Lagrangian for a particle moving in

QUESTION:

a) Write down the Lagrangian for a particle moving in three dimensions under the influence of a conservative central force with potential energy U (r), using spherical polar coordinates (r, 0, 4'). (b) Write down the three Lagrange equations and explain their significance in terms of radial accelera- tion, angular momentum, and so forth. (The 0 equation is the tricky one, since you will find it implies that the 0 component of varies with time, which seems to contradict conservation of angular mo- mentum. Remember, however, that to is the component of in a variable direction.) (c) Suppose that initially the motion is in the equatorial plane (that is, 00 = 7r/2 and 60 = 0). Describe the subsequent motion. (d) Suppose instead that the initial motion is along a line of longitude (that is, 4o = 0). Describe the subsequent motion.