ISSN 0038-0946, Solar System Research, 2007, Vol. 41, No. 4, pp. 265–300. © Pleiades Publishing, Inc., 2007. Original Russian Text © K.V. Kholshevnikov, E.D. Kuznetsov, 2007, published in Astronomicheskii Vestnik, 2007, Vol. 41, No. 4, pp. 291–329. 265 INTRODUCTION The advancement of observing and computing tech- niques has led to noticeable progress in studying the motion of main bodies in the Solar system (the Sun and the major planets) in two interrelated directions. The first direction is the representation of motion with the highest possible accuracy on a short time scale (10–10 3 years). The second direction is a qualitative description of the main properties of motion on cosmogonic time scales (~10 4 –10 10 years). In this review, without having any pretensions to completeness, we will describe the most important achievements in this area of science. We will focus our attention on the results of the last decades, but will also touch on the last century and even ancient times. From Chaldeans and Greeks to Kepler inclusive, theoretical astronomy had no chance to rely on physics (because the latter was underdeveloped) and was based exclusively on mathematics. From the viewpoint of a modern mathematician, an expert on the theory of approximation, astronomers then constructed mathe- matical models that, in a sense, represented best the observations. Interestingly, these models are very simi- lar (from the viewpoint of the mentioned mathemati- cian), although almost the entire literature on the his- tory of astronomy suggests the opposite. First, each model represented the motion of planets in a bilaterally infinite time interval. Second, the motion was described by P. Bohl’s quasi-periodic (Levitan, 1953) function of time, i.e., H. Bohr’s quasi-periodic function with a finite set of basic frequencies. In the theories of Ptolemy, Copernicus, and Tycho Brahe, these were the frequencies of revolution in deferents and epicycles. The frequency basis increased as their number grew. In the theory of Kepler, this basis decreased to the number of planets N, each revolving around the Sun with its own frequency. The question of whether the planetary orbits are stable was not on the agenda: in a quasi-peri- odic motion, everything falls back into place. It is interesting to note the following paradox. Kepler’s mathematical theory, which served as the foundation of Newton’s physics, describes the motion of planets with a limited accuracy, with the discrepan- cies between the theory and the observations increasing with time to unacceptable values (say, to 180° for the longitude difference). The main reason is that the basis Review of the Works on the Orbital Evolution of Solar System Major Planets K. V. Kholshevnikov 1 and E. D. Kuznetsov 2 1 Astronomical Institute, St. Petersburg State University, Universitetskii pr. 28, St. Petersburg, 198504 Russia 2 Ural State University, Yekaterinburg, Russia Received July 31, 2006 Abstract—The cognition history of the basic laws of motion of Solar system major planets is presented. Before Newton, the description of motion was purely kinematic, without relying on physics in view of its underdevelop- ment. From the standpoint of the modern mathematical theory of approximation, all of the models from Ptole- my’s predecessors to Kepler inclusive differ only in details. The mathematical theory worked on an infinite time scale; the motion was represented by P. Bohl’s quasi-periodic functions (a special case of H. Bohr’s quasi-pe- riodic functions). After Newton, the mathematical description of motion came to be based on physical princi- ples and took the form of ordinary differential equations. The advent of General Relativity (GR) and other rel- ativistic theories of gravitation in the 20th century changed little the mathematical situation in the field under consideration. Indeed, the GR effects in the Solar system are so small that the post-post-Newtonian approxima- tion is sufficient. Therefore, the mathematical description using ordinary differential equations is retained. Moreover, the Lagrangian and Hamiltonian forms of the equations are retained. From the 18th century until the mid-20th century, all the theories of planetary motion needed for practice were constructed analytically by the small parameter method. In the early 20th century, Lyapunov and Poincaré established the convergence of the corresponding series for a sufficiently small time interval. Subsequently, K. Kholshevnikov estimated this in- terval to be on the order of several tens of thousands of years, which is in agreement with numerical experi- ments. The first works describing analytically (in the first approximation) the evolution on cosmogonic time scales appeared in the first half of the 19th century (Laplace, Lagrange, Gauss, Poisson). The averaging method was developed in the early 20th century based on these works. Powerful analytical and numerical methods that have allowed us to make significant progress in describing the orbital evolution of Solar system major planets appeared in the second half of the 20th century. This paper is devoted to their description. DOI: 10.1134/S0038094607040016