The trojan problem from a hamiltonian perturbative perspective

The study of the Trojan problem (i.e. the motion in the vicinity of the equilateral Lagrangian points L4or L5) has a long history in the literature. Starting froma representation of the Elliptic Restricted 3-Body Problem in terms ofmodified 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’ Hb, corresponding to the part of the Hamiltonian independent of the secular angle. Averaging over the fast angle, the (Hb) 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 (Hb). We show that the combination of a suitable normalization scheme and the representation by the Hbis efficient enough so as to allow to accurately locate secondary resonances as well as higher order resonances involving also the very slow secular frequencies.