Averaging methods in celestial mechanics

Abstract

The purpose of this paper is to study the relationships among the classical Delaunay theory, the Diliberto periodic surface theory, and the Krylov-Bogoliubov method of averaging. It is shown that the von Zeipel simplification of the Delaunay method and the canonical method of averaging produce the same second order approximations. Furthermore, it follows from the existence of a formal expansion of the von Zeipel determining function that if a monotonicity condition is satisfied, then there exists a formal expansion of a family of periodic two surfaces.