Equilibrium Points and Central Configurations for the Lennard-Jones 2- and 3-Body Problems

In this paper we study the relative equilibria and their stability for a system of three point particles moving under the action of a Lennard-Jones potential. A central configuration is a special position of the particles where the position and acceleration vectors of each particle are proportional, and the constant of proportionality is the same for all particles. Since the Lennard-Jones potential depends only on the mutual distances among the particles, it is invariant under rotations. In a rotating frame the orbits coming from central configurations become equilibrium points, the relative equilibria. Due to the form of the potential, the relative equilibria depend on the size of the system, that is, depend strongly of the momentum of inertia I. In this work we characterize the relative equilibria, we find the bifurcation values of I for which the number of relative equilibria is changing, we also analyze the stability of the relative equilibria. This revised version was published online in July 2006 with corrections to the Cover Date.     

Linear Stability of the Lagrangian Triangle Solutions for Quasihomogeneous Potentials

In this paper, we study the linear stability of the relative equilibria for homogeneous and quasihomogeneous potentials. First, in the case the potential is a homogeneous function of degree −a, we find that any relative equilibrium of the n-body problem with a>2 is spectrally unstable. We also find a similar condition in the quasihomogeneous case. Then we consider the case of three bodies and we study the stability of the equilateral triangle relative equilibria. In the case of homogeneous potentials we recover the classical result obtained by Routh in a simpler way. In the case of quasihomogeneous potentials we find a generalization of Routh inequality and we show that, for certain values of the masses, the stability of the relative equilibria depends on the size of the configuration.     

Hamiltonian Dynamics of a Gyrostat in the N-Body Problem: Relative Equilibria

We consider the non-canonical Hamiltonian dynamics of a gyrostat in Newtonian interaction with n spherical rigid bodies. Using the symmetries of the system we carry out two reductions. Then, working in the reduced problem, we obtain the equations of motion, a Casimir function of the system and the equations that determine the relative equilibria. Global conditions for existence of relative equilibria are given. Besides, we give the variational characterization of these equilibria and three invariant manifolds of the problem; being calculated the equations of motion in these manifolds, which are described by means of a canonical Hamiltonian system. We give some Eulerian and Lagrangian equilibria for the four body problem with a gyrostat. Finally, certain classical problems of Celestial Mechanics are generalized.     

The two-body problem of a pseudo-rigid body and a rigid sphere

In this paper we consider the two-body problem of a spherical pseudo-rigid body and a rigid sphere. Due to the rotational and “re-labelling” symmetries, the system is shown to possess conservation of angular momentum and circulation. We follow a reduction procedure similar to that undertaken in the study of the two-body problem of a rigid body and a sphere so that the computed reduced non-canonical Hamiltonian takes a similar form. We then consider relative equilibria and show that the notions of locally central and planar equilibria coincide. Finally, we show that Riemann’s theorem on pseudo-rigid bodies has an extension to this system for planar relative equilibria.     

From the circular to the spatial elliptic restricted three-body problem

We deal with the study of the spatial restricted three-body problem in the case where the small particle is far from the primaries, that is, the so-called comet case. We consider the circular problem, apply double averaging and compute the relative equilibria of the reduced system. It appears that, in the circular problem, we find not only part of the equilibria existing in the elliptic case, but also new ones. These critical points are in correspondence with periodic and quasiperiodic orbits and invariant tori of the non-averaged Hamiltonian. We explain carefully the transition between the circular and the elliptic problems. Moreover, from the relative equilibria of elliptic type, we obtain invariant 3-tori of the original system.     

Stability of the planar full 2-body problem

The stability of the Full Two-Body Problem is studied in the case where both bodies are non-spherical, but are restricted to planar motion. The mutual potential is expanded up to second order in the mass moments, yielding a highly symmetric yet non-trivial dynamical system. For this system we identify all relative equilibria and determine their stability properties, with an emphasis on finding the energetically stable relative equilibria and conditions for Hill stability of the system. The energetically stable relative equilibria always correspond to the classical “gravity gradient” configuration with the long ends of the two bodies pointed at each other, however there always exists a second equilibrium in this configuration at a closer separation that is unstable. For our model system we precisely map out the relations between these different configurations at a given value of angular momentum. This analysis identifies the fundamental physical constraints and limitations that exist on such systems, and has immediate applications to the stability of asteroid systems that are fissioned due to a rapid spin rate. Specifically, we find that all contact binary asteroids which are spun to fission will initially lie in an unstable dynamical state and can always re-impact. If the total system energy is positive, the fissioned system can disrupt directly from this relative equilibrium, while if it is negative the system is bound together.     

Energy and stability in the Full Two Body Problem

The conditions for relative equilibria and their stability in the Full Two Body Problem are derived for an ellipsoid–sphere system. Under constant angular momentum it is found that at most two solutions exist for the long-axis solutions with the closer solution being unstable while the other one is stable. As the non-equilibrium problem is more common in nature, we look at periodic orbits in the F2BP close to the relative equilibrium conditions. Families of periodic orbits can be computed where the minimum energy state of one family is the relative equilibrium state. We give results on the relative equilibria, periodic orbits and dynamics that may allow transition from the unstable configuration to a stable one via energy dissipation.      

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.     

Minimum energy configurations in the N-body problem and the celestial mechanics of granular systems

Minimum energy configurations in Celestial Mechanics are investigated. It is shown that this is not a well defined problem for point-mass celestial mechanics but well-posed for finite density distributions. This naturally leads to a granular mechanics extension of usual Celestial Mechanics questions such as relative equilibria and stability. This paper specifically studies and finds all relative equilibria and minimum energy configurations for N?=?1, 2, 3 and develops hypotheses on the relative equilibria and minimum energy configurations for N ? 1 bodies.     

Dynamics of two spring-connected masses in orbit

This paper discusses relative equilibria (or steady motions) and their stability for the dynamics of the system of two spring-connected masses in a central gravitational field. The system can be regarded as a simplified model for the Tethered Satellite System (TSS), where the tether is modeled by a (linear or nonlinear) spring. In the previous studies of the TSS problem, it was typically assumed that the center of mass is located at the massive one of the two end-masses, and moves on a great-circle orbit. However, for the simple system treated in this paper, it is proved that nongreat-circle relative equilibria do exist. Some fundamental concepts of the dynamics of an arbitrary assembly moving in a central gravitational field are discussed. The notion of steady motions used in engineering literature is linked with the notion of relative equilibria in geometric mechanics. Numerical computations show some interesting nongreat-circle relative equilibria for the spring-connected system. Radial relative equilibria, which correspond to the station-keeping mode for TSS, are then introduced. Within the framework of symmetry and reduction, their stability properties are investigated by adopting the reduced energy-momentum method, which takes the advantage of the intrinsic symmetry structure. It is shown that for practical configurations, the system at radial relative equilibria is stable if some conditions are satisfied.This work was partially supported by the National Science Council, Republic of China, under grant NSC-83-0208-M-002-082. The authors would like to thank W.-T. Chou for some computational assistance.     

On the existence of the relative equilibria of a rigid body in the J 2 problem

The motion of a point mass in the J ₂ problem has been generalized to that of a rigid body in a J ₂ gravity field for new high-precision applications in the celestial mechanics and astrodynamics. Unlike the original J ₂ problem, the gravitational orbit-rotation coupling of the rigid body is considered in the generalized problem. The existence and properties of both the classical and non-classical relative equilibria of the rigid body are investigated in more details in the present paper based on our previous results. We nondimensionalize the system by the characteristic time and length to make the study more general. Through the study, it is found that the classical relative equilibria can always exist in the real physical situation. Numerical results suggest that the non-classical relative equilibria only can exist in the case of a negative J ₂, i.e., the central body is elongated; they cannot exist in the case of a positive J ₂ when the central body is oblate. In the case of a negative J ₂, the effect of the orbit-rotation coupling of the rigid body on the existence of the non-classical relative equilibria can be positive or negative, which depends on the values of J ₂ and the angular velocity Ω _e . The bifurcation from the classical relative equilibria, at which the non-classical relative equilibria appear, has been shown with different parameters of the system. Our results here have given more details of the relative equilibria than our previous paper, in which the existence conditions of the relative equilibria are derived and primarily studied. Our results have also extended the previous results on the relative equilibria of a rigid body in a central gravity field by taking into account the oblateness of the central body.     

Collinear relative equilibria of the planarN-body problem

The following theorem is proved. THEOREM.For any n2, the set of collinear relative equilibria classes of the n-body problem generates by analytical continuation a total of n!(n+3)/2 relative equilibria classes of the n+1 body problem.Together with Arenstorf's results we state a general theorem for the 4 body problem with 3 arbitrary masses and 1 inferior mass.Research supported in part by NSF grant MCS-78-00395 A01.Paper presented at the 1981 Oberwolfach Conference on Mathematical Methods in Celestial Mechanics.     

Influence of Constant Torque on Equilibria of Satellite in Circular Orbit

We consider the equilibria of a satellite in a circular orbit under the action of gravitational and constant torques. The number of equilibria depending on the parameters of the problem is found by the analysis of an algebraic equation of order 6. The domains with different numbers of equilibria are specified, and the equations of boundary curves are determined in function of values of the components of constant torque. Classification of different distributions of number of equilibria is made for arbitrary values of the parameters.     

Bifurcations of relative equilibria for one spheroidal and two spherical bodies

We discuss existence and bifurcations of non-collinear (Lagrangian) relative equilibria in a generalized three body problem. Specifically, one of the bodies is a spheroid (oblate or prolate) with its equatorial plane coincident with the plane of motion where only the “J ₂” term from its potential expansion is retained. We describe the bifurcations of relative equilibria as function of two parameters: J ₂ and the angular velocity of the system formed by the mass centers. We offer the values of the parameters where bifurcations in shape occur and discuss their physical meaning. We conclude with a general theorem on the number and the shape of relative equilibria.     

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.     

Reduction and relative equilibria for the two-body problem on spaces of constant curvature

We consider the two-body problem on surfaces of constant nonzero curvature and classify the relative equilibria and their stability. On the hyperbolic plane, for each \(q>0\) we show there are two relative equilibria where the masses are separated by a distance q. One of these is geometrically of elliptic type and the other of hyperbolic type. The hyperbolic ones are always unstable, while the elliptic ones are stable when sufficiently close, but unstable when far apart. On the sphere of positive curvature, if the masses are different, there is a unique relative equilibrium (RE) for every angular separation except \(\pi /2\). When the angle is acute, the RE is elliptic, and when it is obtuse the RE can be either elliptic or linearly unstable. We show using a KAM argument that the acute ones are almost always nonlinearly stable. If the masses are equal, there are two families of relative equilibria: one where the masses are at equal angles with the axis of rotation (‘isosceles RE’) and the other when the two masses subtend a right angle at the centre of the sphere. The isosceles RE are elliptic if the angle subtended by the particles is acute and is unstable if it is obtuse. At \(\pi /2\), the two families meet and a pitchfork bifurcation takes place. Right-angled RE are elliptic away from the bifurcation point. In each of the two geometric settings, we use a global reduction to eliminate the group of symmetries and analyse the resulting reduced equations which live on a five-dimensional phase space and possess one Casimir function.     

Saari's Conjecture for the Planar Three-Body Problem with Equal Masses

In the N-body problem, it is a simple observation that relative equilibria (planar solutions for which the mutual distances between the particles remain constant) have constant moment of inertia. In 1970, Don Saari conjectured that the converse was true: if a solution to the N-body problem has constant moment of inertia, then it must be a relative equilibrium. In this note, we confirm the conjecture for the planar three-body problem with equal masses. This revised version was published online in July 2006 with corrections to the Cover Date.     

Stability of the classical type of relative equilibria of a rigid body in the J 2 problem

The motion of a point mass in the J ₂ problem is generalized to that of a rigid body in a J ₂ gravity field. The linear and nonlinear stability of the classical type of relative equilibria of the rigid body, which have been obtained in our previous paper, are studied in the framework of geometric mechanics with the second-order gravitational potential. Non-canonical Hamiltonian structure of the problem, i.e., Poisson tensor, Casimir functions and equations of motion, are obtained through a Poisson reduction process by means of the symmetry of the problem. The linear system matrix at the relative equilibria is given through the multiplication of the Poisson tensor and Hessian matrix of the variational Lagrangian. Based on the characteristic equation of the linear system matrix, the conditions of linear stability of the relative equilibria are obtained. The conditions of nonlinear stability of the relative equilibria are derived with the energy-Casimir method through the projected Hessian matrix of the variational Lagrangian. With the stability conditions obtained, both the linear and nonlinear stability of the relative equilibria are investigated in details in a wide range of the parameters of the gravity field and the rigid body. We find that both the zonal harmonic J ₂ and the characteristic dimension of the rigid body have significant effects on the linear and nonlinear stability. Similar to the classical attitude stability in a central gravity field, the linear stability region is also consisted of two regions that are analogues of the Lagrange region and the DeBra-Delp region respectively. The nonlinear stability region is the subset of the linear stability region in the first quadrant that is the analogue of the Lagrange region. Our results are very useful for the studies on the motion of natural satellites in our solar system.     

Spectral stability of relative equilibria

The spectral stability of synchronous circular orbits in a rotating conservative force field is treated using a recently developed Hamiltonian method. A complete set of necessary and sufficient conditions for spectral stability is derived in spherical geometry. The resulting theory provides a general unified framework that encompasses a wide class of relative equilibria, including the circular restricted three-body problem and synchronous satellite motion about an aspherical planet. In the latter case we find an interesting class of stable nonequatorial circular orbits. A new and simplified treatment of the stability of the Lagrange points is given for the restricted three-body problem.