Stationary and Periodic Solutions for the Restricted Problem of Three Bodies in Three-Dimensional Space

Abstract

In this paper the restricted problem of three bodies in the three-dimensional space, that is, the motion of an asteroid in the gravitational field of the sun and Jupiter moving on circular orbits is treated. The equations of motion for the asteroid can be expressed by the following canonical variables; $$ \begin{array}{*{20}{c}} {L = k\sqrt {{a,}} \quad \;} \\ {G = L\sqrt {{1 - {{e}^{2}},}} } \\ {H = G\cos i\;\;\;\;} \\ \end{array} \begin{array}{*{20}{c}} {l:mean\;anomaly } \\ {g:argument\;of\;perihelion,} \\ {h = \Omega - \lambda '\quad \quad \quad \quad \quad \quad } \\ \end{array} $$ , where a, e, i, Ω, and λ’ are, respectively, the semi-major axis, the eccentricity, the inclination to Jupiter’s orbital plane, the longitude of the ascending node for the asteroid, and Jupiter’s longitude, and k is the gravitational constant of Gauss.