The orbital evolution of the Sun-Jupiter-Saturn-Uranus-Neptune system on long time scales

The averaged semi-analytical motion theory of the four-planetary problem is constructed up to the third order in planetary masses and the sixth degree in the orbital eccentricities and inclinations. The second system of Poincare elements and the Jacobi coordinate system are used for the construction of the Hamiltonian expansion. The averaged Hamiltonian is obtained in the third approximation by the Hori-Deprit method. All analytical transformations are performed by using CAS Piranha. The constructed equations of motion in averaged elements are numerically integrated by the different methods for the giant planets of the Solar System over a time interval of up to 10 Gyr. The planetary motion is quasi-periodic, and the short-term perturbations of the orbital elements conserve small values in the modeling process. The comparison of obtained amplitudes and periods of the change of the orbital elements with numerical motion theories shows an excellent agreement with them. The properties of the planetary motion are given. The short-periodic perturbations and the precision of the integration are estimated.