A Perturbative Method for Problems with Two Critical Arguments

Abstract

Resonance in the restricted three body problem usually lead by averaging to one-degree of freedom Hamiltonian systems described by the Hamiltonian H_o (p,P).

Consideration of a further degree of freedom, such as a fourth body or the ellipticity of the orbit of the perturbing body, may introduce a second critical argument (q) in the averaged system which will then be described by the Hamiltonian H = H_o (p,P) + ε H₁ (p,q,P,Q).

In a recent paper (Henrard-Lemaître, 1986) we have developed a semi-numerical perturbation method to deal with such systems even when the “unperturbed” Hamiltonian H_O possesses critical curves in the region of interest.

This method is based upon the fact that the solution of the Hamilton-Jacobi equation by which one usually introduce the action-angle variables do have a geometrical interpretation. The action variable is an area and the angular variable is a normalized time along the orbit. These quantities can be computed numerically even for fairly complex “unperturbed” systems.

This method is used by A. Lemaître to unravel the complexity of the motion of asteroids in the Jovian resonance 2/1 taking into account the eccentricity of Jupiter.