Exact and approximate integrals of some canonical systems

Abstract

In the first part of this paper we discuss two cases in which the celebrated Poincaré’s theorem is not applicable, but it still holds. In the second part, the formal integral of the restricted problem of three bodies proposed by Contopoulos is discussed. We show that without violating Contopoulos’ rule his integral 00 [fórmula] may be reduced to [fórmula], if use is made of the integrals of the osculating problem. Moreover, we give a very simple example showing that the approximate integral to the second order in μ, obtained by applying Contopoulos’ rule, may be in error of order μ2.