On some aspects of the restricted three dimensional three-body problem
In this paper we consider the three body problem in the variant, suggested by Kolmogorov,
when two points of equal masses in the plane follow the elliptic orbits, symmetric with respect to the axis orthogonal
to the plane; the third point of zero mass belongs to this axis during the motion. Considering the eccentricity as a small parameter
of our problem, we prove the existence of infinitely many families (with respect to eccentricity) of regular long-periodic
solutions. It means, the studied system is in a compound chaotic behavior. The elliptic solutions are stable not only in linear approximation,
but they are orbitally stable in Lyapunov sense.