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.

中文翻译：

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太阳-木星-土星-天王星-海王星系统在长时间尺度上的轨道演化

四行星问题的平均半解析​​运动理论被构建到行星质量的三阶和轨道偏心率和倾角的六阶。第二个庞加莱元素系统和雅可比坐标系统用于构造哈密顿展开式。平均哈密顿量是通过 Hori-Deprit 方法在第三次近似中获得的。所有分析转换均使用 CAS Piranha 进行。平均元素中构建的运动方程通过不同方法在高达 10 Gyr 的时间间隔内对太阳系巨行星进行数值积分。行星运动是准周期性的，轨道元素的短期扰动在建模过程中保留了很小的值。获得的轨道元素变化幅度和周期与数值运动理论的比较表明，它们具有极好的一致性。给出了行星运动的特性。估计短期扰动和积分精度。