The Trojan Problem from a Hamiltonian Perturbative Perspective

Abstract

The study of the Trojan problem (i.e. the motion in the vicinity of the equilateral Lagrangian points L4 or L5) has a long history in the literature. Starting from a representation of the Elliptic Restricted 3-Body Problem in terms of modified Delaunay variables, we propose a sequence of canonical transformations leading to a Hamiltonian decomposition in the three degrees of freedom (fast, synodic and secular). From such a decomposition, we introduce a model called the ‘basic Hamiltonian’ H b , corresponding to the part of the Hamiltonian independent of the secular angle. Averaging over the fast angle, the 〈H b 〉 turns to be an integrable Hamiltonian, yet depending on the value of the primary’s eccentricity e′. This allows to formally define action-angle variables for the synodic degree of freedom, even when e′ ≠ 0. In addition, we introduce a method for locating the position of secondary resonances between the synodic libration frequency and the fast frequency, based on the use of the normalized 〈H b 〉. We show that the combination of a suitable normalization scheme and the representation by the H b is efficient enough so as to allow to accurately locate secondary resonances as well as higher order resonances involving also the very slow secular frequencies.