Dynamical evolution of weakly disturbed two-planetary system on cosmogonic time scales: The Sun-Jupiter-Saturn system

In the present paper, we used the Hori-Deprit method to construct the averaged Hamiltonian of the two-planetary problem accurate to the second order of a small parameter, the generating function of the transform, the change of variables formulas, and the right-hand sides of the equations in average elements. The evolution of the two-planet Sun-Jupiter-Saturn system was investigated by numerical integration over 10 billion years. The motion of the planets has an almost periodic character. The eccentricities and inclinations of Jupiter’s and Saturn’s orbits remain small but different from zero. The short-term disturbances remain small over the entire period considered in the study.