The Lagrange-Jacobi equation in the finite-size many-body problem

We determine the values of the barycentric energy constant that necessarily result in collisions between bodies. The standard Hill stability regions in the problem of four or more bodies are shown to be located inside the regions where collisions are inevitable. Only in the problem of three finite bodies is part of the Hill stability region preserved where the bodies can move without colliding with one another. We point out possible astronomical applications of our results.     

Further analysis of the region of permissible motion in the problem of three bodies

In this paper we investigate the range of variation of the orbital inclination I between two bodies in the problem of three bodies. We give the conditions to determine whether the two bodies move in the same or opposite senses and give the maximum and minimum values of I when the three bodies are in collinear positions. We prove that the range of variation of I must attain maximum and minimum values when the bodies are collinear.     

Generalized hill's problem Lagrangian Hill's case

In this paper, we investigate a generalization of the Hill's problem to the case where no restriction is made about the nature of the field of force perturbing two small bodies in gravitational interaction. We apply the general equations obtained to the dynamics of two bodies located in the vicinity of the triangular lagrangian points of the restricted three-body problem.     

Symmetric planar central configurations of five bodies: Euler plus two

We study planar central configurations of the five-body problem where three of the bodies are collinear, forming an Euler central configuration of the three-body problem, and the two other bodies together with the collinear configuration are in the same plane. The problem considered here assumes certain symmetries. From the three bodies in the collinear configuration, the two bodies at the extremities have equal masses and the third one is at the middle point between the two. The fourth and fifth bodies are placed in a symmetric way: either with respect to the line containing the three bodies, or with respect to the middle body in the collinear configuration, or with respect to the perpendicular bisector of the segment containing the three bodies. The possible stacked five-body central configurations satisfying these types of symmetries are: a rhombus with four masses at the vertices and a fifth mass in the center, and a trapezoid with four masses at the vertices and a fifth mass at the midpoint of one of the parallel sides.     

Conservative two-body problem with variable masses

We consider the conservative two-body problem with a constant total mass, but with variable individual masses. The problem is shown to be completely integrable for any mass variation law. The Keplerian motion known for the classical two-body problem with constant masses remains valid for the relative motion of the bodies. The absolute motions of the bodies depend on the center-of-mass motion. Hitherto unknown quadratures that depend on the mass variation law were derived for the integrals of motion of the center of mass. We consider some of the laws that are of interest in studying the motion of close binary stars with mass transfer.     

The libration points of a spherical satellite in the photogravitational restricted problem of three bodies

In this paper the photogravitational circular restricted problem of three bodies is considered. We have assumed that one of the finite bodies be a spherical luminous and the other be a triaxial nonluminous body. The possibility of existence of the libration points be studied.     

Restricted four-body problem. The case of a central configuration: Libration points and their stability

We consider the problem of the motion of a zero-mass body in the vicinity of a system of three gravitating bodies forming a central configuration.We study the case where two gravitating bodies of equal mass lie on the same straight line and rotate around the central body with the same angular velocity. Equations for calculating the equilibrium positions in this system have been derived. The stability of the equilibrium points for a system of three gravitating bodies is investigated. We show that, as in the case of libration points for two bodies, the collinear points are unstable; for the triangular points, there exists a ratio of the mass of the central body to the masses of the extreme bodies, 11.720349, at which stability is observed.     

Derivation of a condition on the motion of two deformable bodies

We attempt to derive the conditions for which the motion of a system of two deformable (fluid or not) bodies can be reduced to the well-known two-body problem. The new condition is discussed for some pairs of such bodies existing in the natural world.     

Attitude equilibria of dual spin satellites subject to gravitational torques ofn bodies

We consider the motion of a dual spin satellite placed in the gravitational field of n material points, assuming that the satellite has no influence on the motion of these points. The main bodies are located at the libration points of the classical n bodies problem. We investigate the set of relative equilibria of the satellite. As in the elementary case of a gyrostat attracted by a single point, we show that this problem is equivalent to find the extremum of a quadratic function. We obtain all possible equilibria of the satellite by solving two algebraic equations. Sufficient conditions of stability of these relative equilibria are given.     

Equations of motion of elliptically-restricted problem of a body with variable mass and two tri-axial bodies

The differential equations of motion of the elliptic restricted problem of three bodies, an infinitesimal spherical body with decreasing mass and two tri-axial bodies are derived. We have applied Jeans's law and the space-time of Meshcherskii in the special case whenn=1,k=0,q=1/2. Also Nechvíle's transformation for the elliptic problem be applied for this case.     

Formation of dust ion-acoustic solitary waves in a dusty plasma with two-temperature trapped electrons

We look for particular solutions to the restricted three-body problem where the bodies are allowed to either lose or gain mass to or from a static atmosphere. In the case that all the masses are proportional to the same function of time, we find analogous solution to the five stationary solutions of the usual restricted problem of constant masses: the three collinear and the two triangular solutions, but now the relative distance of the bodies changes with time at the same rate. Under some restrictions, there are also coplanar, infinitely remote and ring solutions.     

Reduction,relative equilibria and potential in the two rigid bodies problem

In this paper the problem of two, and thus, after a generalization, of an arbitrary finite number, of rigid bodies is considered. We show that the Newton-Euler equations of motion are Hamiltonian with respect to a certain non-canonical structure. The system possesses natural symmetries. Using them we shown how to perform reduction of the number of degrees of freedom. We prove that on every stage of this process equations of motion are Hamiltonian and we give explicite form corresponding of non-canonical Poisson bracket. We also discuss practical consequences of the reduction. We prove the existence of 36 non-Lagrangean relative equilibria for two generic rigid bodies. Finally, we demonstrate that our approach allows to simplify the general form of the mutual potential of two rigid bodies.     

Hill stability in the many-body problem

We established a criterion for the Hill stability of motions in the problem of many spherical bodies with a spherical density distribution. The region of Hill stability was determined. The sizes of this region are comparable to the total volume of all of the bodies in the system, which sharply increases the probability of mutual collisions. This result may be considered as a confirmation that a supermassive core can be formed at the center of a globular star cluster. The motions in the n-body problem are shown to be unstable according to Hill.     

On the dynamics in the asteroids belt. Part I : General theory

The classical problem of the dynamics in the asteroids belt is revisited in the light of recently developed perturbation methods. We consider the spatial problem of three bodies both in the circular and in the elliptic case, looking for families of periodic or quasi periodic orbits. Some criteria for deciding the stability of these families are also indicated.     

Generalizations of the Jacobian integral

The restricted problem of three bodies is generalized to the restricted problem of 2+n bodies. Instead of one body of small mass and two primaries, the system is modified so that there are several gravitationally interacting bodies with small masses. Their motions are influenced by the primaries but they do not influence the motions of the primaries. Several variations of the classical problem are discussed. The separate Jacobian integrals of the minor bodies are lost but a conservative (time-independent) Hamiltonian of the system is obtained. For the case of two minor bodies, the five Lagrangian points of the classical problem are generalized and fourteen equilibrium solutions are established. The four linearly stable equilibrium solutions which are the generalizations of the triangular Lagrangian points are once again stable but only for considerably smaller values of the mass parameter of the primaries than in the classical problem.     

Equations of motion of elliptic restricted problem of three bodies with variable mass

The differential equations of motion of the elliptic restricted problem of three bodies with decreasing mass are derived. The mass of the infinitesimal body varies with time. We have applied Jeans' law and the space-time transformation of Meshcherskii. In this problem the space-time transformation is applicable only in the special case whenn=1,k=0,q=1/2. We have applied Nechvile's transformation for the elliptic problem. We find that the equations of motion of our problem differ from that of constant mass only by a small perturbing force.     

Analysis of some degenerate quadruple collisions

We consider the trapezoidal problem of four bodies. This is a special problem where only three degrees of freedom are involved. The blow up method of McGehee can be used to deal with the quadruple collision. Two degenerate cases are studied in this paper: the rectangular and the collinear problems. They have only two degrees of freedom and the analysis of total collapse can be done in a way similar to the one used for the collinear and isosceles problems of three bodies. We fully analyze the flow on the total collision manifold, reducing the problem of finding heteroclinic connections to the study of a single ordinary differential equation. For the collinear case, from which arises a one parameter family of equations, the analysis for extreme values of the parameter is done and numerical computations fill up the gap for the intermediate values. Dynamical consequences for possible motions near total collision as well as for regularization are obtained.Paper presented at the 1981 Oberwolfach Conference on Mathematical Methods in Celestial Mechanics.Dedicated to Prof. Szebehely on the occasion of his sixtieth birthday.     

Global bifurcation of planar and spatial periodic solutions in the restricted <Emphasis Type="Italic">n</Emphasis>-body problem

The paper deals with the study of a satellite attracted by n primary bodies, which form a relative equilibrium. We use orthogonal degree to prove global bifurcation of planar and spatial periodic solutions from the equilibria of the satellite. In particular, we analyze the restricted three body problem and the problem of a satellite attracted by the Maxwell’s ring relative equilibrium.     

Existence of libration points in the generalised photogravitational restricted problem of three bodies

In this paper we have proved the existence of libration points for the generalised photogravitational restricted problem of three bodies. We have assumed the infinitesimal mass of the shape of an oblate spheroid and both of the finite masses to be radiating bodies and the effect of their radiation pressure on the motion of the infinitesimal mass has also been taken into account. It is seen that there is a possibility of nine libration points for small values of oblateness, three collinear, four coplanar and two triangular.     

Sundman surfaces and Hill stability in the three-body problem

Using the famous Sundman inequality, we have constructed for the first time the surfaces for the general three-body problem that we suggest calling Sundman surfaces. These surfaces are a generalization of the widely known Hill surfaces in the restricted circular three-body problem. The Sundman surfaces are constructed in a rectangular coordinate system that uses the mutual distances between the bodies as the Cartesian rectangular coordinates. The singular points of the family of these surfaces have been determined. The possible and impossible regions of motion of the bodies have been constructed in the space of mutual distances. We have shown the existence of Hill stable motions and established sufficient criteria for Hill stability of motions. Some of the astronomical applications are considered.