Normal Form of the Equations of Perturbed Motion near Triangular Libration Points at Third-Order Resonances

Abstract

A treatment is given of the spatial restricted elliptic problem of three bodies interacting under Newtonian gravity. The problem depends on two parameters: the ratio between the masses of the main attracting bodies and the eccentricity of their elliptic orbits. The eccentricity is assumed to be small. Nonlinear equations of motion of the test mass near a triangular libration point are analyzed. It is assumed that the parameters of the problem lie on the curves of third-order resonances corresponding to the planar restricted problem. In addition to these resonances (their number is equal to five), the spatial problem has a resonance that takes place at any parameter values since the the frequency of small linear oscillations of the test mass along the axis perpendicular to the plane of the orbit of the main bodies is equal to the frequency of Keplerian motion of these bodies. In this paper, the normal form of the Hamiltonian function of perturbed motion through fourth-degree terms relative to deviations from the libration point is obtained. Explicit expressions for the coefficients of normal form up to and including the second degree of eccentricity are found.