Problems Of Celestial Mechanics Research Articles

The real singularities of theN-body problem in celestial mechanics

N-body problem real singularities in celestial mechanics, considering present status and holomorphic motion

On the Ideal Resonance Problem--II

A number of problems in celestial mechanics, as well as other branches of applied mathematics, possess an ideal resonance structure. That is, the Hamiltonian F , of the dynamical system, may be written in the form |$-F\,=\,B(x)\,+\,2\mu A(x)\,\text{sin}^{2}\,y$| , where x and y are the canonically conjugate momentum and coordinate variables respectively and μ is a small constant parameter. Resonance is associated with the vanishing of the first derivative of B for some value of x . It is well known that the phase-plane of such a system is separated into libration and circulation regions by the limiting orbit. The author has previously published a paper which presented a new method of solution for the region of libration. In the present publication the general nature of the higher order solution is discussed. The motivation for developing the new solution, characterized by a transformation due to Poincaré, is explained. It is found that, to any power of μ½ , an asymptotic solution can be constructed, free from singularities and mixed secular terms. General expressions are given which demonstrate that the solution can always be expressed in terms of a small number of elliptic functions and fundamental integrals, defined in the text.

FORMAc Language and its Application to Celestial Mechanics

The IBM FORMAC language, a computer system for the symbolic manipulation of formulas and ex- pressions, is described and several applications to problems of celestial mechanics are outlined. These applications include the derivation of power-series solutions to the equations of motion, the generation and application of high-order time derivatives of arbitrary powers of the radius vector, and the extension of Cayley's tables.

The binary collision in the N-body problem

The binary collision in the N-body problem of celestial mechanics and its regularization are studied from the point of view of analytic differential equations, i.e., for complex values of the time and of the coordinates as independent and dependent variables. Based on a suitable definition of a binary collision, a proof is given for the classical result that a binary collision corresponds to an algebraic branch point of the first or second order. The convergence of Sundman's integral and the regularization of the binary collision are briefly discussed also for more general problems not necessarily possessing an energy integral; this situation is present, e.g., in the reduced three-body problem.

Contributions to the elliptic restricted three-body problem

Two contributions are given for the elliptic case of the restricted three-body problem of Celestial Mechanics. The first contribution consists in ascertaining analytically the eigenvalues about a Lagrangian triangular point as a power series of the eccentricity. This has entailed the determination of a coordinate transformation, whose coefficients are functions of the true anomaly, for the purpose of reducing the linear portion of the differential system to diagonal form. For each power of the eccentricity, a linear system of differential equations was found that yielded the coefficients of the coordinate transformation. The eigenvalues were calculated by imposing the absence of pure secular terms. The second contribution is the establishing of series solutions for the equations of motion about a triangular point by appropriately modifying a procedure devised by this author for the circular problem. Solutions are given in the form of double power series, whose coefficients (functions of the true anomaly) can be found recursively by solving linear differential systems. The two variables of the power series can be obtained in a similar way as solutions of an auxiliary nonlinear differential system of appropriate form. It was further required that the distance between the Lagrangian point and the initial point on the orbit be considered a small parameter.

Ascent or Descent from Satellite Orbit by Low Thrust

The problem of ascent or descent from an initially Keplerian orbit by a constant low thrust is investigated by the two-variable expansion procedure. The slowly varying elements of the Keplerian orbit are explicitly computed, and a simple integral relating the eccentricity and semimajor axis is derived and used to justify the omission of higher order terms in the eccentricity for ascent from almost circular initial orbits. The corresponding assumption for descent from an almost circular orbit is shown to be inconsistent. The present higher order theory that is valid for arbitrary initial eccentricities is shown to contain earlier results for circular starting orbits and gives simple explicit formulas for the solution at small eccentricities. The present results do, however, suffer from the deficiency of earlier solutions in that the validity is restricted to periods before escape velocity is reached for the case of ascending orbits. HE problem of ascent from circular orbit by a small thrust in either the radial or circumferenti al directions was first investigated by Tsien. 1 Tsien gave an exact solution for the case of radial thrust and developed an approximate first-order solution for the circumferential thrust case. Recently, Ting and Brofman2 extended Tsien's solution to one higher order and allowed the thrust to be arbitrarily oriented while remaining constant. Their approach consists of splitting the dependence of the radius into oscillatory and non-oscillatory parts and using an expansion procedure combining the features of singular perturbations and the method of averaging. More recently, Nayfeh3 also developed a solution for the case treated by Ting and Brofman, using a more systematic asymptotic method. The quoted results were all concerned with the case of circular initial orbits and do not shed any light on the more interesting case of an arbitrary elliptic starting orbit. In the closely related problem of ascent by tangential thrust, Zee4 and Cohen5 attempted to solve for the oscillatory terms in the radius contributed by the ellipticity of the initial orbit. Their solutions suffer from the presence of unnecessary assumptions that restrict the validity of the results to small eccentricity in spite of the complicated approaches proposed. In another recent paper, King6 considered the question of descent from a parking orbit by low thrust, but did not discuss the analytical implications of this reversal of the usual role of low thrust. In this paper, the following generalizations are made. The initial elliptic orbit is not restricted in any way, and the implications of negative thrust (i.e., thrust opposing the motion) are fully investigated. The solution is developed to second order by the two-variable expansion procedure developed by Cole and Kevorkian7 and Kevorkian.8 This method, which has been applied to two other problems in celestial mechanics by Eckstein, Shi and Kevorkian,9'10 is a generalized asymptotic expansion procedure that provides uniformly valid developments for the coordinates appropriate to satellite problems. At this point, a brief review of the diverse methods applied to the solution of this problem is appropriate. In the dis

On the problem op n-stationary centers

Among the various problems of celestial mechanics related to the n-body problem, a certain amount of interest attaches to the specific situation wherein a passive gravitational point mass M moves under the assumption that the relative disposition of the other active gravitational masses experiences no large changes. If the attracting masses are regarded as points and if changes in the relative disposition of the attracting bodies are neglected, one arrives at the problem of the motion of the point M in a field produced by n-stationary attracting centers (the point M here represents the ( n+ l)-th body). In addition to the problem of central motion ( n = 1), soluble dynamics problems of this category include Euler's case [1] of two ( n= 2) stationary Newtonian attracting centers. This problem, which for a long time was of solely theoretical Interest as an example of an integrable Liouville system [2], has recently been attracting attention in connection with the mechanics of artificial satellites, particularly after it was shown that the potential of an oblate spheroid can be approximated by the potential of two specifically chosen stationary Newtonian attracting centers [3 and 4]. The solution of the problem for n-attracting centers for n ≥ 3 is unknown, except for a single special case of three centers pointed out by Lagrange and considered In greater detail by J.A. Serre [5]. We shall concern ourselves here with problems on the existence of periodic trajectories in the case of n-attracting centers. An arbitrary and not necessarily Newtonian gravitational law will be assumed. Our analysis is based on the theory of quasiindices of singular force field points as set forth in [60].

A new method of perturbation theory and its application to the satellite problem of celestial mechanics.

Article A new method of perturbation theory and its application to the satellite problem of celestial mechanics. was published on January 1, 1966 in the journal Journal für die reine und angewandte Mathematik (volume 1966, issue 221).

Using Rumyantsev's theorem on stability in some variables in celestial mechanics problems

Using Rumyantsev's theorem on stability in some variables in celestial mechanics problems

Neuere numerische methoden in der himmelsmechanik

The main problem of celestial mechanics is to compute the movement of a point mass in space. The force entering the relevant differential equations can often be split into two parts, one of which can be considered as a small perturbation. This paper reviews some methods useful for the solution of such equations. Grobner's non-linear method is briefly mentioned. Further, a regularization technique, appropriate for the perturbed Kepler problem, is discussed and compared with other methods. Finally, the treatment in the case of forced perturbations which are of interest for satellites, is briefly outlined.

The pseudo-integral of a system of differential equations

First we shall briefly review what is meant by saying that a function is an integral for a system of differential equations. Next we shall introduce a generalization of this idea by modifying two conditions on the function to obtain what in this paper shall be called the pseudo-integral. Finally we shall apply this new concept to some problems in celestial mechanics to demonstrate its usefulness in obtaining information about the solution of a system of differential equations.

A class of periodic solutions for the restricted three-body problem

Periodic orbits have been established in the planar case of the restricted three-body problem of Celestial Mechanics, which can be generated from circular Keplerian orbits and are valid for any mass ratio of the finite bodies. Extending a procedure by Siegel, already applied by this author to Hill's lunar theory, a periodic series solution has been obtained for the equation of motion, whose numerical coefficients can be calculated from a consistent set of recurrent relations. Fourier series representations have been given for the coordinates of the third body with respect to the synodic reference, where the coefficients of the trigonometric functions appear as power series of a parameter related to the period of the orbit. Approximate expansions for the radius vector, angular velocity, magnitude of velocity vector, and the Jacobi constant of motion have been furnished which exhibit how the motion occuring along a periodic orbit of this class deviates from a uniform, circular one. Using the majorant method of series and adapting a theorem on implicit functions, the series solution has been proved to be convergent for small values of the period but for an arbitrary mass ratio of the two primary bodies. The results include as a limiting case Hill's periodic solutions. The procedure is fit for extensive numerical work.

Complete Averaging of Canonical Problem of Celestial Mechanics With Several Intermediate Elements

Complete Averaging of Canonical Problem of Celestial Mechanics With Several Intermediate Elements

Interpolation Averaged Variants of the Canonical Problem of Celestial Mechanics

Interpolation Averaged Variants of the Canonical Problem of Celestial Mechanics

Proceedings of the Celestial Mechanics Conference: The effects of imperfect elasticity in problems of celestial mechanics

Proceedings of the Celestial Mechanics Conference: The effects of imperfect elasticity in problems of celestial mechanics

On the Role of First Integrals in the Perturbation of Periodic Solutions

In two previous papers methods were given for appraising the range of the parameter in the theory of perturbation of periodic solutions, originated by Poincare. In the first of these [3 of the bibliography], only the simplest case was considered, in which the variational equations are assumed to have no periodic solutions not identically zero. In the second paper [4] the methods of the first paper were modified so as to be applicable to what may be considered the general case in which the variational equations have one or more linearly independent periodic solutions. The results of neither of these papers are, however, applicable to systems admitting first integrals, unless the first integrals are first used to lower the order of the original system. This can lead to formal complications, which it might be desirable to by-pass. The purpose of this paper is to show how this by-pass can be accomplished in those cases where the number of first integrals is equal to the number of linearly independent periodic solutions of the variational equations. In its reliance upon the methods of integral equations, successive approximations, generalized Green's matrices, and the discussion of the so called bifurcation equations (Verzweigungsgleichungen), this paper stands in intimate relationship with a magnificent paper by E. H6lder [2]. This relationship is considerably closer than in the preceding papers of the present writer; and hence it would be well to point out the two principal contrasts between Holder's work and our own. First, the main purpose of the present paper is to propose a method for appraising the interval for the parameter. Holder makes no mention of the feasibility of doing this, nor does his paper contain those inequalities which might enable the reader easily to form his own estimates. Secondly, whereas Holder focuses attention on the equations of certain problems in celestial mechanics, we wish in this paper to present the theory in a more general setting. Of particular significance is the fact that Holder bases his discussion of the bifurcation equations on the invariance of Hamilton's action integral with respect to a certain group of transformations (rotations in space and translations in time). In this paper a like discussion is based on first integrals. As is generally well known [6] and as H6lder himself explicitly recognized [p. 231, loc. cit.], the invariance of the action integral under a continuous group leads (under suitable hypotheses of regularity) to certain first integrals, namely those * This research was supported by the United States Air Force through the Office of

An Examination of the Quantum Theories III

The Bohr formulation—the original one—is now largely a matter of history. The usual career of a theory was its fate. It explained much, but it failed at the first signs of complication, even for the simplest molecules as hydrogen and helium. Some results were either only approximate in a loose or incomplete fashion, e.g. in the application of the correspondence principle to intensities, the only reliable predictions being only for an absence of certain lines, or they quite disagreed with experiment, e.g. in the case of the hydrogen molecule and the hydrogen molecule ion, the normal and excited states of helium, the anomalous Zeeman effect, the Ramsauer effect (the phenomenon of the deflection of electrons at low speeds being less than at higher speeds in atomic collisions), and in the treatment of problems where more than one electron was involved. In the last mentioned cases the perturbation method, so successful in solving “many-body” problems in celestial mechanics, failed completely because systems of several electrons, possessing frequencies of the order of light waves, cannot interact classically. A bit of much forced pragmatism was practiced in treating the many electron atom as a variation of the hydrogen atom. Besides, there were technical oddities, like ½ quantum numbers, and impenetrated anologies between visible and X-ray spectra.

Upon a Theory of Infinite Systems of Non-Linear Implicit and Differential Equations

Introduction. There has arisen, in recent years, a large literature upon infinite systems of non-linear implicit and differential equations.' The methods wrhich are employed have become classical in the treatment of finite systems. However these methods (employed in the treatment of finite systems) may be extended only to infinite systems of very special type, restricted by inequalities, which for many reasons are not fulfilled in possible applications of the method of infinitely many variables to various classical problems in analysis. Thus the theorems which have been proved by me a few years ago2 are the only ones to my knowledge which have found an application to concrete problems, in particular to the problems of Celestial Mechanics or the Calculus of Variations. I shall give here a comprehensive review of these questions without giving any essential extensions of my results as set forth in my previous papers and without assuming any previous knowledge of the theory of infinitely many variables. The applications will be excluded and the reader referred in this connection to some earlier papers appearing in the Mathemcbtische Annalen and in the Mathematische Zeitschrift. A characteristic application can be found in the paper of Martin appearing in this number of the Journal. The infinite system which I shall treat is composed exclusively of power series (and not of more general functions) of the infinite sequence of variables (as is indeed always the case in concrete applications). My problem was to introduce a method of treatment in which the power series are not subjected to inequalities which would not be fulfilled in concrete applications but which would be necessary in the classical modes of treatment. On page 252 there will be given a number of examples, of most important character, which will serve to illustratewhy the usual methods must fail. Inasmuch as the nature of the problems appears most clearly in its application to the complex space of Hilbert, the present paper will be restricted in its treatment to this space. It is of course possible by using the principles of General Analysis to further extend the methods here set forth. There is no difficulty in using other spaces ' than that of Hilbert. In the articles cited above also spaces other than that of Hilbert are treated; cf. the above mentioned paper of Martin. 241

On certain determinants connected with a problem in celestial mechanics

On certain determinants connected with a problem in celestial mechanics