On integrability of evolutionary equations in the restricted three-body problem with variable masses

Abstract

The satellite version of the restricted three-body problem formulated on the basis of classical Gylden-Meshcherskii problem is considered. Motion of the point P2 of infinitesimal mass about the point P0 is described in the first approximation in terms of the osculating elements of the aperiodic quasi-conical motion, and an influence of the point P1 gravity on this motion is analyzed. Long-term evolution of the orbital elements is determined by the differential equations written in the Hill approximation and averaged over the mean anomalies of points P1 and P2. Integrability of the evolutionary equations is analyzed, and the laws of mass variation have been found for which the evolutionary equations are integrable. All relevant symbolic calculations and visualizations are done with the computer algebra system Mathematica. © 2014 Springer International Publishing.