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We extend some previous results of our work [1] on the error of the averagingmethod, in the one-frequency case. The new error estimates apply to anyseparating family of seminorms on the space of the actions; they generalize ourprevious estimates in terms of the Euclidean norm. For example, one can use thenew approach to get separate error estimates for each action coordinate. Anapplication to rigid body under damping is presented. In a companion paper [2],the same method will be applied to the motion of a satellite around an oblateplanet.Comment: LaTeX, 23 pages, 4 figures. The final version published in Nonlinear  Dynamic

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