A natural perturbation formulation involves amplitudes and angles. Averaging over the angles produces valid approximations, but special problems and new phenomena in certain parts of phase space, the so-called resonance manifolds, will arise. A full description of the dynamics in resonance manifolds usually requires the construction of a second order approximation. Several examples illustrate the dynamics, and in Chap. 12 the theory is applied to the dynamics of a flywheel on an elastic foundation.