On the averaging principle for one-frequency systems. Seminorm estimates for the error

We extend some previous results of ours (J. Phys. A: Math. Gen. 39 (2006) 3673–3702) on the error of the averaging method, in the one-frequency case. The new error estimates apply to any separating family of seminorms on the space of the actions; they generalize our previous estimates given in terms of the Euclidean norm. For example, one can use the new approach to get separate error estimates for each action coordinate. An application to rigid body under damping is presented. In a companion paper (Nonlinear Dyn. 58 (2009) 273-294), the same method is applied to the motion of a satellite around an oblate planet.