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

We extend some previous results of our work (Morosi and Pizzocchero in J. Phys. A: Math. Gen. 39:3673–3702, 2006) 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 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, the same method will be applied to the motion of a satellite around an oblate planet.