On the elimination of the critical terms of a first order theory of Jupiter perturbed by Saturn carried out through Hori's method and Poincar� canonical variables

Abstract

The elimination of the critical terms inside the Hamiltonian of a first order theory of Jupiter perturbed by Saturn is carried out through the Poincare canonical variables and the Hori's procedure. Powers of the eccentricities and the sines of inclinations which are>3 are neglected. The Poincare variablesL1,H1,P1,λ1,K1,Q1 of Jupiter which result from a previous elimination of the short period terms are expressed in terms of the Poincare canonical variablesL u ′ ,H u ′ ,P u ′ ,λ u ′ ,Q u ′ ;u=1, 2; index 1 Jupiter, index 2 Saturn resulting from the elimination of the short period and critical terms. The differential equations inH u ′ ,P u ′ ,K u ′ ,Q u ′ are solved through the method of Lagrange and the analytical expressions ofL1,H1,P1,λ1,K1,Q1 as functions of timet are finally obtained.