This expository lecture surveys recent progress of the stability theory in Celestial Mechanics with emphasis on the analytical problems. In particular, the old question of convergence of perturbation series are discussed and positive results obtained, in the light of the work by Kolmogorov Arnold and Moser. For the three body problem, classes of quasi-periodic solutions and doubly asymptotic (or homoclinic) orbits are discussed.

In order to analyse generic or typical properties of dynamical systems we consider the space of all C1-vector fields on a fixed differentiable manifold M. In the C1-metric, assuming M is compact, is a complete metric space and a generic subset is an open dense subset or an intersection of a countable collection of such open dense subsets of . Some generic properties (i.e. specifying generic subsets) in are described. For instance, generic dynamic systems have isolated critical points and periodic orbits each of which is hyperbolic. If M is a symplectic manifold we can introduce the space of all Hamiltonian systems and study corresponding generic properties.

In this paper the point of view is taken that the distribution of orbital elements in the solar system should be discussed first on a purely gravitational basis, i.e. on the basis of a set of particles entirely under gravitational interaction, before hydromagnetic and other effects are taken in consideration too. One might indeed assume that there has been a time in the history of the solar system from when on hydromagnetic and gas laws ceased to play an important role in comparison to gravity. In the epoch since that time the solar system might have developed from a set of a large number of smaller particles into the present solar system by way of transitions which these particles made to preferential orbital elements, and by accretion. Means had been found to handle the development of this set of particles under gravitational interaction, by defining the set appropriately in terms of a statistical distribution. In considering the problem of the evolution of the solar system, such a gravitational approach, which was encouraged by Einstein, seems the reasonable first step.

It is suggested that the general form of the constant of quantization, K in Schrödinger's equation, is not h/2π, but K=2sα-k, with s being the spin of the orbiting object, α the fine structure constant (1/137.0361), and k a small positive integer, or zero. For atoms k = 0; for planets and satellites k = 2, 3 or 4; for the solar system as a whole, revolving around the center of the Galaxy, k = 6. The probability that 16 objects of the solar system would follow this quantum rule by chance alone is 1 in 1016, suggestive that quantum mechanics, as we know it today, can be seen as a special case of a more general quantum mechanics of the future; it also supports the view expressed by Dirac, that h is probably not a fundamental constant.

Section 1 contains the basic idea which induced me to undertake an investigation of a relationship between rotational and orbital angular momenta of planets; Sections 2–7 contain the experimental data, the application of the new quantum rule and the statistical evaluation whether the relationship proposed in Section 1, could have occurred by chance alone. The results obtained in Sections 2–7 are noteworthy in themselves, independently whether the basic idea is accepted or not.

The depletion of an initially uniform distribution of asteroids extending from Mars to Saturn, caused by the gravitational perturbations of Jupiter and Saturn, is calculated by numerical integration of the asteroid orbits. Almost all (about 85%) the asteroids between Jupiter and Saturn are ejected in the first 6000 years. Most of the asteroids between the 2/3 Jupiter resonance (4.0 AU) and Jupiter are ejected in the first 2400 years with the exception of the stable librators (e.g., the Hilda group). Interior to the 2/3 resonance the depletion was small, and interior to the 1/2 resonance (3.3 AU), no asteroids were ejected in the first 2400 years.

The distribution of orbits in the satellite systems of the major planets is definitely nonrandom as there is a marked preference for commensurability among pairs of mean motions in these systems. If this preference is not the result of a formation process then it follows that since the time of satellite formation dynamical evolution has occurred. In this paper (a full version of which is to be submitted to another journal) I show that there is a bulk of evidence in favour of the hypothesis that orbital evolution has occurred as the result of tidal dissipation in the planets. This evidence can be listed as follows:

(i) In a tidally evolved satellite system there should be a linear correlation between log (orbital radius) and log (satellite mass). The expected correlation is observed among the inner satellites of Saturn and Uranus, the slope of the log-log plots being consistent with an amplitude and frequency independent Q (the tidal dissipation function). The correlation in Saturn's system must be largely due to either a formation process or possibly an early amplitude and frequency dependent tidal process, but certainly the present satellite distribution is completely consistent with the tidal hypothesis.

(ii) If Q is amplitude- and frequency-independent then the Mimas-Tethys and Enceladus-Dione resonances are stable under the action of tidal forces. This is an important result as it indicates that the necessary tidal dissipation occurred in the solid parts of the planets. Whether or not these planets have any solid parts at present is a matter of dispute.

(iii) The tidal hypothesis can account for the formation (capture into libration) of the Mimas-Tethys and Enceladus-Dione resonances.

(iv) If it is allowed that the age of the Mimas-Tethys resonance is not small (> 108 yr) in comparison with that of the solar system and that orbital evolution since the time of satellite formation has been appreciable then it can be shown that tidal forces (with Q amplitude- and frequency-independent) can account for the present large amplitude of libration of this resonance.

(v) If tidal evolution in the satellite systems of Jupiter and Saturn has been appreciable then from the present orbits of Io and Mimas it can be deduced that Q (Jupiter) ~1.1 × 105 and Q (Saturn) ~1.2 × 105. As the masses of the two satellites differ by a factor ~2000 and the amplitudes of the tides they raise on their respective planets differ by a factor ~ 5000 this agreement in the Q-values is somewhat remarkable. But as the stability of the Mimas-Tethys and Enceladus-Dione resonances demands that Q is amplitude- and frequency-independent and as the chemical compositions and structures of Jupiter and Saturn are probably similar then one should expect the Q's of these planets to be similar and thus the result supports the tidal hypothesis.

(vi) Finally, as the values of Q are so very high it would be remarkable if tidal evolution had not taken place.

From extended statistical investigations of the orbital elements of asteroids and of their dispersions it is found that the relations for some of the dispersions are stable and are regarded to be invariable. By assuming that the orbital elements derived by putting the dispersions equal to zero correspond to those of the mother planet of the asteroid ring, it is found that the mother planet is Mars.

In this paper the orbital evolution of Trojan asteroids are studied by integrating numerically the equations of motion over the interval 1660–2060, perturbations from Venus to Pluto being taken into account. The comparison of the actual motion of Trojans in the solar system with the theory based on the restricted three-body problem are given.

Saturn's satellites Mimas and Tethys appear to be involved in a unique resonance. All orbit-orbit resonances, by definition, have the satellites' conjunction librating about some specific longitude. Equivalently, their orbital periods, measured relative to that longitude, are commensurable. Most orbit-orbit resonances are of the eccentricity-type; the conjunctions librate about the longitude of an apse of one orbit. The Mimas-Tethys resonance is of the inclination type.

There exist gaps in the frequency distribution of the osculating mean motions of the asteroids. In these gaps, the mean motions of the asteroids are commensurable to Jupiter's mean motion.

The first-order theory of secular perturbations for an asteroid indicates that the secular motions of the longitudes of the perihelion and of the ascending node have an equal absolute value with opposite signs, that is, , where b is a function of the semimajor axis and has the disturbing mass as a factor, if the eccentricity and the inclination are sufficiently small so that their squares are of the order of the disturbing mass and if there is no commensurable relation between the mean motions of the asteroid and the disturbing planet. Thus Π + Θ is a stable quantity according to the first-order theory, therefore, the origin of families of asteroids has been discussed by using the data for distribution of Π + Θ as well as those for dispersions of other orbital elements of asteroids belonging to a family.

We integrate numerically the evolution of a three-dimensional system of particles with finite dimensions, which bounce inelastically upon each other. The particles are subjected to the attraction of a central mass; their mutual attraction is neglected. This model is used to study the evolution of Saturn's ring. The first results are presented: such a collision mechanism can flatten very quickly the Saturn's ring and the system tends towards a final equilibrium state.

A general theory of stationary solutions of the averaged N-body problem is briefly described. Numerical results in some particular cases (general three-body problem in the nonresonant case and restricted circular problem both in the nonresonant and resonant ones) are presented. Some applications into the problem of planetary stability are developed.

The evolutions of satellite's orbit under the influence of the third disturbing body moving on an elliptic orbit and the oblate planet are studied by averaging the disturbing function, and integrable cases are classified in seven cases.

The problem of the critical inclination in the artificial satellite theory can be regarded as a problem of resonance arising from the near-commensurability 1:1 between the anomalistic and the draconitic frequencies of the motion, n1 and n2. Let us define the main problem by the potential function (1) with (2)

Characteristics and stability of simple-periodic retrograde satellites of the lighter body are presented for Hill's case and for all values of the mass ratio m2/(m1+m2) between 0 and 0.5.

The equations of variation of the three-dimensional elliptic restricted three-body problem corresponding to the equilibrium solutions (the libration points) have been separated into three Hill's equations. As regards the equation ‘corresponding’ to the motion of the infinitesemal body in the z-axis (perpendicular to the plane of motion of the primaries), the matter is trivial one since the initial equation - as known - reads d2z/dv2 + (Ai + e cosv)/(l + e cosv) = 0 (e, 0<e< 1, and v are the eccentricity and the true anomaly of the relative motion of the primaries) with Ai > 1 for the straight-line libration points Li (i= 1, 2, 3) and Ai=l for the triangular libration points Li, i=4, 5. As concerns the remaining two components, x and y, of the motion of the infinitesimal body (x, y and z are the Nechvíle's variables), in the case of the straight-line libration points, L1, L2 and L3, the corresponding equations of variation have been transformed and separated into two further - mutually independent - Hill's equations without any limitation. In the case of the equilateral triangle libration points, L4 and L5, the separation has been found only when the eccentricity e and the dimensionless mass μ, 0<μ≦1/2 of the ‘minor’ primary satisfy the additional conditions: Let us write the latter two Hill's equations obtained in the form where Ik, k= 1, 2, are 2π-periodic even functions of the true anomaly v. The functions Ik, k = 1, 2, are real functions in the case of the straight-line libration points, L1, L2 and L3, without a limitation but in the case of the triangular libration points, L4 and L5, they are real only if Provided the functions Ik, k= 1, 2, are complex-valued functions of the real variable v.