Miscellanea from the history of celestial mechanics

The KS-transformation introduced by P. Kustaanheimo and E. Stiefel into celestial mechanics is derived straight from the Kepler Formulas. There follow the treatment of the inverse Newton problem comprising the derivation of the differential equations of mechanics by J. Hermann and L. Euler and also remarks concerning the fundamental papers by Euler about the planet problem and then-body problem. The conclusion is a simple example given by A. Voss and H. Liebmann, for the differential equations of mechanics with non-holonomic condition, which is of pseudoplanetary quality.This paper originated from several stays at the Eidgenössische Technische Hochschule, Zürich; Seminar of Professor E. Stiefel in 1973/74.     

Libration celestial mechanics experiment

The exploration of planet moons and minor bodies (Avduevskii et al., 1996) is a basic task for comprehending the nature of the processes occurring in our Solar System. Knowing the current state of the moons, we can better describe their past and look into the future. This knowledge is important, first of all, for understanding the origin of the Solar System. Interest in the Martian moon Phobos has been displayed during recent decades. The interest is caused by some questions to which there have been no answers up until now (Sagdeev et al., 1988; 1989). For example, there is a question regarding the origin of the moon: whether it is an asteroid captured by Mars’ gravitational field or it is an accumulated body in the Martian orbit. In connection with this, it is interesting to conduct studies aimed at answering this question. If Phobos appears to be an asteroid, then investigations regarding the chemical and isotopic compositions of the moon as the primary matter of the Solar System as well as its evolution are of great interest.     

Perturbation equations of celestial mechanics

Perturbation equations of celestial mechanics in terms of orbital elements are completely derived in application to the motion of interplanetary dust particle in the gravational field of the Sun and under the action of disturbing forces. Consideration of change of mass of interplanetary dust particle is the most important feature of this derivation. The results obtained are completely general in the case of constant masses.     

TheN-body problem of celestial mechanics

A selective survey of then-body problem of celestial mechanics is given where the emphasis is on the asymptotic behavior of all solutions ast, the possible configurations the particles can assume in phase space and in physical space, and collision and non-collision singularities.Supported in part by NSF Grant MPS 71-03407 A03.     

On resonance in celestial mechanics

This brief survey of the author's contribution to the theory of resonance in celestial mechanics begins with the genesis of the Small Divisor. The fundamental distinction between theshallow anddeep resonance is illustrated by the 52 Jupiter-Saturn and the 3-2 Neptune-Pluto resonances in the planetary system.The search for aglobal solution through a removal of the small divisor is put into a historical perspective through the work of Laplace, Bohlin, and Poincaré. The author's own contribution to the methodology is the formulation and the solution of the Ideal Resonance Problem. If the resonance issimple, all the singularities in the solution are removed by means of aregularizing function. On the other hand, if the resonance isdouble, the second critical divisor seems irremovable, and a global solution may be precluded.Invited paper, IAU 1979, Commission 7, Montreal, Canada.     

Generalized perturbation equations of celestial mechanics

Generalized perturbation equations of celestial mechanics in terms of orbital elements are derived. The most general case is considered: Keplerian motion of two bodies caused by gravitational forces between them is disturbed by disturbing acceleration acting on each of the bodies separately and by changes of masses of these bodies. It is also pointed out why derivation presented in Klaka (1992a) is completely physically correct only for constant masses.     

Applications of computers to celestial mechanics

The development of digital computers induced major new developments in Celestial Mechanics. At present, one can hardly mention a project in Celestial Mechanics that does not use computers as the principal tool. One can distinguish many different manners of using computers in Celestial Mechanics. Among them, the following are presented and typical examples are given: algebraic manipulations for literal and semi-numerical theories, numerical integration of the equations of motion, determination of physical parameters, numerical checks of analytical results, studies of families of solutions, search for new conjectures, scanning the phase space. In all these approaches, and in others omitted here, major scientific achievements were obtained in the last 20 years and new problems can now be envisaged that were unaccessible even a few years ago.     

On the use of STF-tensors in celestial mechanics

The purpose of this article is to emphasize the usefulness of STF-tensors in celestial mechanics. Using STF-mass multipole moments and Cartesian coordinates the derivations of equations of motion, the interaction- and tidal-potentials for an isolated system ofN arbitrarily shaped and composed, purely gravitationally interacting bodies are particularly simple. Using simple relations between STF-tensors and spherical harmonics it is shown how all Cartesian formulas can be converted easily into the usual spherical representations. Some computational aspects of STF-tensors and spherical harmonics are discussed. A list of useful formulas for STF-tensors is provided.     

Kepler discretization in regular celestial mechanics

In this paper, a special extrapolation method for the numerical integration of perturbed Kepler problems (given in KS-formulation) is worked out and analyzed in detail. The underlying so-called Kepler discretization isexact for the pure (elliptic) Kepler motion. A numerically stable realization is presented together with a backward error analysis: this analysis shows that the effect of the arising rounding errors can be regarded as a small perturbation inferior to the physical perturbation. For test purposes, a well-known example describing the motion of an artificial Earth satellite in an equator plane subject to the oblateness perturbation is used to demonstrate the efficiency of the new extrapolation method.Proceedings of the Sixth Conference on Mathematical Methods in Celestial Mechanics held at Oberwolfach (West Germany) from 14 to 19 August, 1978.     

Variable constant of gravitation and celestial mechanics

In the present paper the celestial mechanics consequences (light deflection, radar ranging of the planet, geodetic precession and secular effects in the orbital elements in the two-body problem) for the class of the theories based on the vacuum Jordan's Lagrangian has been considered. In these theories the gravitational constantG is proportional to ^–, being a scalar field and , some dimensionless parameter and the local law of conservation of the energy-momentum tensor holds. Of all these theories with different the most interesting one is that corresponding to =0. In the postnewtonian approximation this gravitational theory is completely equivalent to the general theory of relativity.     

Specialized celestial mechanics systems for symbolic manipulation

Construction and application of the current high accuracy analytical theories of motion of celestial bodies necessitates the development of specialized software for the implementation of analytical algorithms of celestial mechanics. This paper describes a typical software package of this kind. This package includes a universal Poisson processor for the rational functions of many variables, a tensorial processor for purposes of relativistic celestial mechanics, a Keplerian processor valid for the solutions of the two body problem in the form of a Poisson series, Taylor expansions in powers of time and closed expressions, and an analytical generator of celestial mechanics functions, facilitating the immediate implementation of the present analytical methods of celestial mechanics. The package is completed with a numerical-analytical interface designed, in particular, for the fast evaluation of the long Poisson series.     

The Lyapunov characteristic exponents-applications to celestial mechanics

After a presentation of Lyapunov characteristic exponents (LCE) we recall their basic properties and numerical methods of computation. We review some numerical computations which are concerned with LCEs and mainly related to celestial mechanics problems.     

Numerical integration of the equations of motion of celestial mechanics

Recently a number of new techniques have been developed for the numerical solution of the differential equations governing the motion of bodies in the Solar System, moving under their mutual gravitational forces. Some of these new methods are tested against each other and against more traditional methods and conclusions are made as to under what circumstances any of these methods should be used to produce optimum results.     

Boundary manifolds for energy surfaces in celestial mechanics

We complete Mc Gehee's picture of introducing a boundary (total collision) manifold to each energy surface. This is done by constructing the missing components of its boundary as other submanifolds. representing now the asymptotic behavior at infinity.It is necessary to treat each caseh=0,h>0 orh<0 separately. In the first case, we repeat the known result that the behavior at total escape is the same as in total collision. In particular, we explain why the situation is radically different in theh>0 case compared with the zero energy case. In the caseh<0 we have many infinity manifold components. and the general situation is not quite well understood.Finally, our results forh0 are shown to be valid for general homogeneous potentials.Paper presented at the 1981 Oberwolfach Conference on Mathematical Methods in Celestial Mechanics.The research conducted in this paper has been partially supported by CONACYT (México), under grant PCCBNAL 790178.Partially supported by an Ajut a l'Investigacio of the University of Barcelona.     

Symplectic integrators for long-term integrations in celestial mechanics

We discuss the use of symplectic integration algorithms in long-term integrations in the field of celestial mechanics. The methods' advantages and disadvantages (with respect to more common integration methods) are discussed. The numerical performance of the algorithms is evaluated using the 2-body and circular restricted 3-body problems. Symplectic integration methods have the advantages of linear phase error growth in the 2-body problem (unlike most other methods), good conservation of the integrals of the motion, good performance for moderately eccentric orbits, and ease of use. Its disadvantages include a relatively large number of force evaluations and an inability to continuously vary the step size.     

Poincaré's hydrodynamic analogy in celestial mechanics

The analogy between the differential equations describing dynamical systems of two degrees of freedom on one hand and two and three dimensional flow on the other hand is investigated in some detail.Presented at the Conference on Celestial Mechanics, Oberwolfach, Germany, August 17–23, 1969.