Action angles result from a type-2 canonical transformation where the generating function is Hamilton's characteristic function W ( q ) (not Hamilton's principal function S ). Since the original Hamiltonian does not depend on time explicitly, the new Hamiltonian K ( w , J ) is merely the old Hamiltonian H ( q , p ) expressed in terms of the new canonical coordinates, which we denote as w (the action angles, which are the generalized coordinates) and their new generalized momenta J . We will not need to solve here for the generating function W itself; instead, we will use it merely as a vehicle for relating the new and old canonical coordinates.