On the Stability of Lagrange Solutions in the Spatial Near-Circular Restricted Three-Body Problem

The restricted problem of three bodies (material points) is considered. The orbits of the main attracting bodies are assumed to be ellipses of small eccentricity, and the passively gravitating body during its motion can leave the plane of the orbits of the main bodies (spatial problem). The stability of body motion corresponding to triangular Lagrangian libration points is investigated. A characteristic feature of the spatial problem under study is the presence of resonance due to the equality of the Keplerian motion period of the main bodies and the linear oscillation period of the passively gravitating body in the direction perpendicular to the plane of their orbits. Using the methods of classical perturbation theory, Kolmogorov-Arnold-Moser (KAM) theory and computer algebra algorithms, the nonlinear problem of stability for most (in the Lebesgue-measure sense) initial conditions and formal stability (stability in any arbitrarily high finite approximation with respect to the coordinates and impulses of perturbed motion) are investigated.