On some applications of the problem of many fixed centres to geophysics

In the present paper, using as examples the problems of four, five and six fixed centres, some applications of the problem of many fixed centres to geophysics are given.The paper was presented at the workshop on the theme The Use of Observations of the Artificial Earth Satellites in Geodesy, held in Slavsk (U.S.S.R.) from 14 to 19 April, 1980.     

The intermediate orbit of resonance asteroids of the Hecuba family constructed on the basis of solution of the internal variant of the generalized problem of the three fixed centres

On the basis of the solution of the internal variant of the generalized problem of three fixed centres (taking as an example a minor planet of 108 Hecuba) an intermediate orbit of the resonance asteroids of the Hecuba family has been constructed. Comparison of the results obtained from the formulae with observations and also with the results of Isaeva (1976) showed that in investigating the motion of the celestial bodies it would be reasonable to use the orbits of the internal variant of the generalized problem of three fixed centres.     

On the global solution of the N-body problem

In connection with the publication (Wang Qiu-Dong, 1991) the Poincaré type methods of obtaining the maximal solution of differential equations are discussed. In particular, it is shown that the Wang Qiu-Dong'sglobal solution of the N-body problem has been obtained in Babadzanjanz (1979). First the more general results on differential equations have been published in Babadzanjanz (1978).     

On the generalized restricted problem of three bodies

The particular case of the complete generalized three-body problem (Duboshin, 1969, 1970) where one of the body-points does not exert influence on the other two is analysed. These active material points act on the passive point and also on each other with forces (attraction or repulsion), proportional to the product of masses of both points and a certain function of the time, their mutual distances and their first and second derivatives. Furthermore it is not supposed that generally the third axiom of mechanics (action=reaction) takes place.Here under these more general assumptions the equations of motion of the active masses and the passive point, as well as the diverse transformations of these equations are analogous of the same transformations which are made in the classical case of the restricted three-body problem.Then we determine conditions for some particular solutions which exist, when the three points form the equilateral triangle (Lagrangian solutions) or remain always on a straight line (Eulerian solutions).Finally, assuming that some particular solutions of the above kind exist, the character of solutions near this particular one is envisaged. For this purpose the general variational equations are composed and some conclusions on the Liapunov stability in the simplest cases are made.     

On oscillatory motion in the problem of three bodies

In the case of oscillatory motion in the problem of three bodies it is shown that ast the mutual distances between particles cannot separate faster thanCt ^2/3 whereC is some positive constant. As bounding functions of time exist for the other classifications of motion in the three body problem, it follows in general that the mutual distances between particles is 0(t) ast.     

On the accuracy in the numerical solution of theN-body problem

There exist several widely used methods that give a qualitative estimation of the accuracy of the results in the numerical solution of theN-body problem. The reverse and closure tests are examined here critically. The author has developed a method for the estimation of global errors propagated in the numerical solution of ordinary and partial differential equations that has proven to be rather efficient in numerous cases (see P. E. Zadunaisky [17]). Applications of the method to several cases of theN-body problem are presently made and the advantages and limitations of the method are shown in a set of examples.     

Anisotropic Kepler and anisotropic two fixed centres problems

In this paper we show that the anisotropic Kepler problem is dynamically equivalent to a system of two point masses which move in perpendicular lines (or planes) and interact according to Newton’s law of universal gravitation. Moreover, we prove that generalised version of anisotropic Kepler problem as well as anisotropic two centres problem are non-integrable. This was achieved thanks to investigation of differential Galois groups of variational equations along certain particular solutions. Properties of these groups yield very strong necessary integrability conditions.     

On the restricted three body problem with oblate primaries

We present a study of the Lagrangian triangular equilibria in the planar restricted three body problem where the primaries are oblate homogeneous spheroids steadily rotating around their axis of symmetry and whose equatorial planes coincide throughout their motion.     

Periodic solutions about the collinear lagrangian solution in the general problem of three bodies

The article describes the solutions near Lagrange's circular collinear configuration in the planar problem of three bodies with three finite masses. The article begins with a detailed review of the properties of Lagrange's collinear solution. Lagrange's quintic equation is derived and several expressions are given for the angular velocity of the rotating frame.The equations of motion are then linearized near the circular collinear solution, and the characteristic equation is also derived in detail. The different types of roots and their corresponding solutions are discussed. The special case of two equal outer masses receives special attention, as well as the special case of two small outer masses.Finally, the fundamental family of periodic solutions is extended by numerical integration all the wap up to and past a binary collision orbit. The stability and the bifurcations of this family are briefly enumerated.     

On lagrange solutions in the problem of three rigid bodies

The present paper is a direct continuation of the paper (Duboshin, 1973) in which was proved the existence of one kind of Lagrange (triangle) and Euler (rectilinear) solutions of the general problem of the motion of three finite rigid bodies assuming different laws of interaction between the elementary particles of the rigid bodies. In particular, Duboshin found that the general problem of three rigid bodies permits such solutions in which the centres of mass of the bodies always form an equilateral triangle or always remain on one straight line, and each body possesses an axial symmetry and a symmetry with respect to the plane of the centres of mass and rotates uniformly around its axis orthogonal to this plane. The conditions for the existence of such solutions have also been found. The results in Duboshin's paper have greatly interested the author of the present paper. In another paper (Kondurar and Shinkarik, 1972) considering a more special problem, when two of the three bodies are spheres, either homogeneous or possessing a spherically symmetric distribution of the densities or of the material points, and the third is an axially symmetrical body possessing equatorial symmetry, the present author obtained analogous solutions of the ‘float’ type describing the motion of the indicated dynamico-symmetrical body in assuming its passive gravitation. In the present paper new Lagrange solutions of the considered general problems of three rigid bodies of ‘level’ type are found when the axes of geometrical and mechanical symmetry of all three bodies always lie in the triangle plane, and the bodies themselves rotate inertially around the symmetry axis, independently of the parameters of the orbital motion of the centres of mass as in the ‘float’ case. The study of particular solutions of the general problem of the translatory-rotary motion of three rigid bodies, which are a generalization of Lagrange solutions, is in the author's opinion, a novelty of some interest for both theoretical and practical divisions of celestial mechanics. For example, in recent times the problem of the libration points of the Earth-Moon system has acquired new interest and value. A possible application which should be mentioned is that to the orbits of artificial satellites near the triangular libration points to serve as observation stations with the aim of specifying the physical parameters in the Earth-Moon system (e.g., the relation of the Earth's mass to the Moon's mass for investigating the orientation of the satellite, solar radiation, etc.).     

On the global solution in the resonance problem of Poincaré

Poincaré formulated a general problem of resonance in the case of a dynamical system which is reducible to one degree of freedom. He introduced the concept of a global solution; in essence, this means that the domain of the solution(s) covers the entire phase plane, comprising regions of libration and circulation. It is the author's opinion that the technique proposed by Poincaré for the construction of a global solution is impractical. Indeed, in §§201 and 211 ofLes méthodes nouvelles de la méchanique céleste, where he describes the passage from shallow resonance to deep resonance, Poincaré asserts an erroneous conclusion. An alternative procedure, which admits secular terms into the determining function and introduces a regularizing function, is outlined. The latter method has been successfully applied to the Ideal Resonance Problem, which is a special case of the more general problem considered by Poincaré, (Garfinkelet al. (1971); Garfinkel (1972).     

On the stability of Laplace's solutions of the unrestricted three body problem

The instability criterion of a nonlinear mechanical system neutral to the first approximation is formulated for the internal resonance case which is characterized by the existence of commensurabilities between the frequencies of the system.The criterion derived is used for determining the regions of instability of Laplace's constant triangular solutions of the unrestricted three-body problem. It is shown that in the region where necessary Routh-Joukovsky's stability conditions are satisfied there may exist eight resonanceunstable sets of the masses of the three bodies. These sets may be mechanically interpreted as follows: in the case of resonance instability the barycentre of the equilateral triangle formed by the three bodies is located on one of the eight circles constructed in the geometrical centre of this triangle.     

On the Hill stability in the general problem of three finite bodies

We construct zero-kinetic-energy surfaces and determine the regions where motion is possible. We show that for bodies with finite sizes, there are bounded regions of space within which a three-body system never breaks up. The Hill stability criterion is established.     

On the stability of the solutions of the general problem of three bodies

The following question is investigated: By how much may the initial conditions of a given three-body system be varied before the subsequent evolution of the new system completely differs from that of the original? Stated somewhat differently, how big is the ‘island’ in the phase space of initial conditions throughout which the parameters describing the evolution of the systems are continuous functions of the initial conditions? The extent of one such island is determined numerically and found to be surprisingly large. It is conjectured, however, that this result is due to the fact that the corresponding systems have very short disintegration times, so that the total motion is not very complex.     

An asymptotic solution for the Trojan case of the plane elliptic restricted problem of three bodies

The planar motion of a Trojan asteroid is considered within the framework of the elliptic restricted three-body problem. The solution is derived asymptotically to second order taking the square root of the Jupiter-Sun mass ratio and the orbital eccentricity of Jupiter as first order quantities. The results are given in explicit form for the coordinates as functions of the true anomaly of Jupiter including both short and long periodic terms resulting from the orbital accentricity of Jupiter.     

On the period of the periodic orbits of the restricted three body problem

We will show that the period T of a closed orbit of the planar circular restricted three body problem (viewed on rotating coordinates) depends on the region it encloses. Roughly speaking, we show that, \(2 T=k\pi +\int _\Omega g\) where k is an integer, \(\Omega \) is the region enclosed by the periodic orbit and \(g:{\mathbb {R}}^2\rightarrow {\mathbb {R}}\) is a function that only depends on the constant C known as the Jacobian constant; it does not depend on \(\Omega \). This theorem has a Keplerian flavor in the sense that it relates the period with the space “swept” by the orbit. As an application we prove that there is a neighborhood around \(L_4\) such that every periodic solution contained in this neighborhood must move clockwise. The same result holds true for \(L_5\).     

A nonlinear stability problem in the three dimensional restricted three body problem

The question of whether or not there is a transfer of energy between the in-plane motion and out-of-plane motion in the neighborhood ofL ₄ in the restricted problem of three bodies is investigated in this paper. The in-plane motion is assumed to be finite and the out-of-plane motion to be infinitesimal. The equation governing the out-of-plane motion becomes one with time varying coefficients. The stability of this equation is then investigated using Lie Series.Presented as a paper AAS No. 70-313, at the AAS/AIAA Astrodynamics Specialists Conference 1971 at Fort Lauderdale Fla., U.S.A.     

Towards solution of the pulsar problem

The 24-year-old pulsar problem is reconsidered. New results are obtained by replacing the assumption of steady-state discharges near the polar caps by oscillatory discharges, and by creating the neutral-excess pair plasma via inverse-Compton collisions rather than via curvature radiation. As a result, the electrons and positrons which compose the pulsar wind have different bulk velocities and an oscillating space density, and (strong) coherent curvature radiation is implied (without invoking the excitation of instabilities, and contrary to existing proofs of its impossibility). The magnetospheres of young pulsars are likely to have considerable higher-order multipole components, in particular octupole. Radiation transfer through the pulsar magnetosphere results in fan beams whose polarization is dictated by the bottom of the radiation zone, hence, looks like curvature radiation from dipole-like polar caps.Wind generation depends mainly on the quantityB² which takes similar values for the ms pulsars; the latter compensate for (somewhat) weaker fields by wider polar caps and smaller curvature radii.