IAU Symposium No. 81, “Dynamics of the Solar System”, was held at the Hydrographic Office, Tokyo, Japan, during 23–26 May 1978. The Symposium was cosponsored by COSPAR and IUTAM, and generous financial support was also provided by the Japan Society for the Promotion of Science. IAU sponsorship was through Commissions 4, 7 and 20, and the Scientific Organizing Committee consisted of the current Presidents, Vice Presidents and immediate Past Presidents of these Commissions: V. K. Abalakln, R. L. Duncombe, Y. Kozai, L. Kresfik, B. G. Marsden (Chairman), P. J. Message, A. M. Sinzi, G. Sitarski and V. G. Szebehely.

The aim of celestial mechanics is to derive all solutions of the differential equations of the three- and n-body problems and to reveal the interrelation among all such solutions with various values of the masses and initial conditions, not only to predict the positions of celestial bodies at any instant but to think over the cosmogony of stellar systems and to ascertain the law of universal gravitation in view of the time-space picture of physical theory.

This paper reviews the present status of research on the problem of stability of satellite and planetary systems in general. In addition new results concerning the stability of the solar system are described. Hill's method is generalized and related to bifurcation (or catastrophe) theory. The general and the restricted problems of three bodies are used as dynamical models. A quantitative measure of stability is introduced by establishing the differences between the actual behavior of the dynamical system as given today and its critical state. The marginal stability of the lunar orbit is discussed as well as the behavior of the Sun-Jupiter-Saturn system. Numerical values representing the measure of stability of several components of the solar system are given, indicating in the majority of cases bounded behavior.

In this paper, recent numerical and analytical results of the author and others, concerning the stability of the solar system are presented. The results indicate that the actual planetary system could possibly have stable and bounded motion.

It is known that families of periodic orbits in the general N-body problem (N≥3) exist, in a rotating frame of reference (Hadjidemetriou 1975, 1977). A special case of the above families of periodic orbits are the periodic orbits of the planetary type. In this latter case only one body, which we shall call sun, is the more massive one and the rest N-1 bodies, which we shall call planets, have small but not negligible masses. The aim of this paper is to study the properties of the families of periodic planetary-type orbits, with particular attention to stability. To make the presentation clearer, we shall start first with the case N=3 and we shall extend the results to N>3. We shall discuss planar orbits only.

Usual periodic orbits have periods of the order of magnitude of a few revolutions. However if we consider much longer periods it is possible to find, for three given masses, periodic orbits in any neighbourhood of arbitrary given initial eccentricities and inclinations provided that the distance of the outer body is sufficiently large with respect to the mutual distance of the two inner bodies.

Let us consider motion of an infinitesimal body in the gravitational field of two homogeneous spheroids with coinoident equatorial planes. Let mass of the infinitesimal body M - a material point - be negligible if compared with masses of the spheroids S1, S2 respectively. We will study the motion of M in the common equatorial plane provided that the spheroids move along circular orbits (in an inertial frame of reference) - see Kondurar', 1952.

The article studies collisions with the smaller primary, m2, in the planar restricted problem of three bodies. The formulation is in polar coordinates and the corresponding momenta. The results are based on the fact that the collisions correspond to a solution of a partial differential equation.

In the circular plane restricted three-body problem, we study the stable large retrograde non-periodic satellite orbits. We use rotating axes with the origin in the body around which turns the satellite, called its primary. We choose the initial conditions such as Yo=0 and Uo=0, so that an orbit can be represented by a point in the (Xo,Vo) plane. In this plane, the set of stable orbits is represented by a limited region, which we call the stability zone. This zone is composed in general by a large continental region, approximately limited by Lagrange points, and a peninsula more or less elongated.

The motion of a particle of mass m according to the central force of Newton, is denoted by where K is a constant. x=0 corresponds to singular points of this equations. The domain of (1), denoted by = (R3 −{0})×R3, is called the phase space of the Kepler motion. In the sequel we set m=K=1 for simplicity and also transform the independent variable from t to s by dt =|x|ds (x≠0), then the Kepler motion in the phase space is written as Further, we shall confine the following discussion to the case of the negative energy value, except the preliminary discussion.

Some non-stationary problems of celestial mechanics can be described in an inertial system of right-angled coordinates with gravitational potential of the form:

where is a sufficiently arbitrary function of time and is the meaning of in the initial epoch For example, in a two-body problem of variable mass we have:

Classical mechanics failed to solve two problems in own defensive area, namely the motion of Mercury's perihelion, and the high-velcity motion of a charged particle. Today it is generally believed that the concepts of classical mechanics are completely invalid in a treatment of these problems. In this paper, however, we discuss these problems throughly with the concepts of classical mechanics — Euclidean space-time, point of mass and central force. Thereat we introduce a new concept “absolute mass variation”, with which we extend the Newton's second law of motion. In following chapters we show that this extended equation explains the motion of Mercury's perihelion, and that it throws new light on the atomic physics. We also make a study of reconstruction of internal structure of mechanics. We discuss the possibility of the revival of the principal frame of Newton's mechanics.

The location and the stability of the libration points in the restricted problem have been studied when there are perturbations in the potentials between the bodies. It is seen that if the perturbing functions involving the parameters α,α1,α2 satisfy certain conditions, there are five libration points, two triangular and three collinear. It is further observed that the collinear points are unstable and for the triangular points, the range of stability increases or decreases depending upon whether the perturbation point (α,α1,α2) lies on one or the other side of the plane Aα + Bα1 + Cα2 = 0, and it remains the same if the point lies on the plane, where A,B,C depend on the perturbations. The theory is verified in the following four cases: (1) there are no perturbations in the potentials (classical problem), (2) only the bigger primary is an oblate spheroid, (3) both the primaries are oblate spheroids, and (4) the primaries are spherical in shape and the bigger is a source of radiation.

At the Bureau of Longitudes the construction of planetary theories have been developed in three directions: A general theory of the motion of the four largest planets in the solar system is in the course of development at the Faculty of Sciences at Lille by L. Duriez (1977) following the methods of V. A. Brumberg and J. Chapront (1973). Theories of the classical type with secular variations of the motions of all of the planets from Mercury to Neptune are being completed at the Bureau of Longitudes. They are constructed by P. Bretagnon and J. L. Simon (1975, 1978). The numerical complement to all of these studies, numerical integration, a representation of the solution by Tchebychev series, are being carried out by P. Rocher as concerns the motions of minor planets, and by J. Piranx for the action of Pluto on Uranus and Neptune in the framework of theories with secular variations.

A new approach to the planetary theory is examined under the following procedure: 1) we use a canonical perturbation method based on the averaging principle; 2) we adopt Charlier's canonical relative coordinates fixed to the Sun, and the equations of motion of planets can be written in the canonical form; 3) we adopt some devices concerning the development of the disturbing function. Our development can be applied formally in the case of nearly intersecting orbits as the Neptune-Pluto system. Procedure 1) has been adopted by Message (1976).

The theory of the motion of the Moon can be approached by two types of methods. In the analytical methods, the solution is usually expanded in formal power series of the small quantities m, e and γ. It is known that the expansions in m are slowly convergent and that, in order to obtain an accuracy of the order of the meter, the expansions have to be carried to very high order (beyond 20) involving several hundred thousands terms.

In the lunar or satellite theory, the disturbing function is developed in the form in which each term consists of two factors. The first factor depends on the position of the sun, and the second one that of the satellite.

The article studies the application of the Lagrange series inversion formula to perturbation theories. One of the main problems is to generalize the Lagrange formula to several variables. The article also shows the need for introducing some operators to simplify the notation. To terminate, an application to the Von Zeipel-Brouwer satellite theory is given.

A new form to determine the contribution of some special small divisors in perturbation theory is presented in this paper. We can avoid to calculate all the kλ+k′λ as it has to be done normally (λ and λ′ designate the mean longitudes of the two regarded planets). For a chosen k and k′ we calculate with a very high precision the contribution to the perturbation of the elements with the aid of the Hansen's coefficients.