1 Introduction

Lagrange (1867-1892) showed that the three-body problem has five relative equilibrium configurations. In the restricted version (i.e. when one of the bodies is massless) it is useful to view the motion of the test particle in a frame co-rotating with the massive bodies; in this case the relative equilibrium configurations correspond to fixed points two of which, named L₄ and L₅, are linearly stable when the mass ratio is small enough (i.e. when $m_1/m_{\rm c}<0.0385$). These are also known as the triangular equilibrium points and correspond to configurations where the test particle is in the exact 1:1 mean motion resonance, i.e. it is co-orbital with m₁ but leading or trailing it by $60^{\circ}$. In this paper, Trojan-type motion refers to the tadpole-shaped librations around L₄ or L₅(in analogy with the Trojan asteroids which occupy tadpole orbits in the Sun-Jupiter system) and also to the horseshoe-shaped librations that enclose both triangular points as well as the unstable co-linear point L₃.

In Morais (1999), hereafter Paper I, we constructed a secular theory for Trojan-type motion in the framework of the restricted three-body problem. A secular solution was derived based on the heuristic assumption that the terms in the averaged disturbing potential depending on the relative mean longitude do not have a net effect on the evolution of the eccentricity or inclination. Here we show how we can re-derive this secular solution applying first-order canonical adiabatic theory (Lichtenberg & Lieberman 1983). Moreover, we show that this methodology allows us to construct a complete secular theory which includes the effect of an oblate central mass and the secular perturbations from additional massive bodies, and we are thus able to explain rigorously some preliminary results obtained in Paper I.