2. A particle of mass m is confined to a parabolic surface of rotation z = a?2, where ? = ?x2 + y2. The graviational potential is U = mgz.
a) Show the Lagrangian is
L = 1
2
m ?
z?
2 + ??
2 + ?2
??2
?
? mgz
subject to the constraint that z = a?2, i.e. the particle remains on the surface of parabolic
rotation.