Generalized Problem of Two and Four Newtonian Centers

We consider integrable spherical analog of the Darboux potential, which appear in the problem (and its generalizations) of the planar motion of a particle in the field of two and four fixed Newtonian centers. The obtained results can be useful when constructing a theory of motion of satellites in the field of an oblate spheroid in constant curvature spaces.     

Perturbations of the Kepler Problem in Global Coordinates

The usual action-angle variables for the Kepler Problem (the Delaunay variables) are not globally defined, leaving out some orbits (circular orbits or those lying on the xy-plane). Moreover they are trascendental functions of the physical variables, making it quite difficult to write the perturbed Hamiltonian. The way-out proposed here is to pass to a 8-dimensional rank-6 Poisson manifold, that is, to parametrize the state of the Kepler Problem with two 4-dimensional vectors mutually orthogonal and of equal norm. This revised version was published online in July 2006 with corrections to the Cover Date.     

Perturbations of the Kepler Problem in Global Coordinates: A Program

The present paper is a brief report of how, along the lines of a previous paper, the implementation of the program KEPLER¹, for the numerical integration of the perturbations of the Kepler problem, has been carried out.     

Topology of the Two Fixed Centers Problem

In this paper we give a complete topological characterization of the Two Fixed Centers (TFC) problem flow. This characterization is based on the link formed by some basic periodic orbits. The Restricted Circular Three-Body (RCTB) problem is considered as a perturbation of the TFC in the case of two primaries with equal masses. The basic periodic orbits of the integrable problem can be continued in the non-integrable one.This revised version was published online in October 2005 with corrections to the Cover Date.     

Global Regularization of the Restricted Problem of Three Bodies

The global regularizing transformations of the planar, circular restricted problem of three bodies are studied. It is shown that all these transformations can be written in the same general form which is the solution of a first order ordinary differential equation. This revised version was published online in July 2006 with corrections to the Cover Date.     

On the Dynamics and Topology of the Elliptic Rectilinear Restricted 3-Body Problem

We study a symmetric collinear restricted 3-body problem, where the equal mass primaries perform elliptic collisions, while a third massless body moves in the line between the primaries, during the time between two consecutive elliptic collisions. After desingularizing binary and triple collisions, we prove the existence of a transversal heteroclinic orbit beginning and ending in triple collision. This orbit is the unique homothetic orbit that the problem possess. Finally, we describe the topology of the compact extended phase space. This revised version was published online in July 2006 with corrections to the Cover Date.     

Non-Integrability of the Generalized Two Fixed Centres Problem

We consider two fixed centres attracting a third body. Each centre is the source of a force field with potential V=−ar ⁻²ⁿ , where n is a real number. We prove that this generalization of the classical two fixed centres problem is non-integrable except when n= 0, 1/2, −1 and −2. This revised version was published online in July 2006 with corrections to the Cover Date.     

Stability of the Triangular Libration Points in the Unrestricted Planar Problem of a Symmetric Rigid Body and a Point Mass

In papers (Godziewski and Maciejewski, 1998a, b, 1999), we investigate unrestricted, planar problem of a dynamically symmetric rigid body and a sphere. Following the original statement of the problem by Kokoriev and Kirpichnikov (1988), we assume that the potential of the rigid body is approximated by the gravitational field of a dumb-bell. The model is described in terms of a 2D Hamiltonian depending on three parameters.In this paper, we investigate the stability of triangular equilibria permissible by the dynamics of the model, under the assumption of low-order resonances. We analyze all resonances of order smaller than four, and we examine the stability with application of theorems by Markeev and Sokolsky. These are the possible following cases: the non-diagonal resonance of the first order with two null characteristic frequencies (unstable); resonances of the first order with one nonzero frequency (diagonal and non-diagonal, stable and unstable); the second-order resonance, which is non-diagonal and stable, and the third-order resonance which is generically unstable, except for three points in the parameters' space, corresponding to stable equilibria.We discuss a perturbed version of Kokoriev and Kirpichnikov model, and we find that if the perturbation is small and depends on the coordinates only, the triangular equilibria persist, except if for the unperturbed equilibria the first-order resonance occurs. We show that the resonances of the order higher than two are also preserved if the perturbation acts.     

Using neural networks for the derivation of Runge-Kutta-Nyström pairs for integration of orbits

In this paper we present Runge-Kutta-Nyström (RKN) pairs of orders 4(3) and 6(4). We choose a test orbit from the Kepler problem to integrate for a specific tolerance. Then we train the free parameters of the above RKN4(3) and RKN6(4) families to perform optimally. For that we form a neural network approach and minimize its objective function using a differential evolution optimization technique. Finally we observe that the produced pairs outperform standard pairs from the literature for Pleiades orbits and Kepler problem over a wide range of eccentricities and tolerances.     

Non-Existence of the Modified First Integral by Symplectic Integration Methods II: Kepler Problem

Symplectic integration methods conserve the Hamiltonian quite well because of the existence of the modified Hamiltonian as a formal conserved quantity. For a first integral of a given Hamiltonian system, the modified first integral is defined to be a formal first integral for the modified Hamiltonian. It is shown that the Runge-Lenz vector of the Kepler problem is not well conserved by symplectic methods, and that the corresponding modified first integral does not exist. This conclusion is given for a one-parameter family of symplectic methods including the symplectic Euler method and the Störmer/Verlet method.     

Bifurcation of Relative Equilibria in the Main Problem of Artificial Satellite Theory for a Prolate Body

In this paper, we study circular orbits of the J ₂ problem that are confined to constant-z planes. They correspond to fixed points of the dynamics in a meridian plane. It turns out that, in the case of a prolate body, such orbits can exist that are not equatorial and branch from the equatorial one through a saddle-center bifurcation. A closed-form parametrization of these branching solutions is given and the bifurcation is studied in detail. We show both theoretically and numerically that, close to the bifurcation point, quasi-periodic orbits are created, along with two families of reversible orbits that are homoclinic to each one of them.     

Stable Orbits of Planets of a Binary Star System in the Three-Dimensional Restricted Problem

The present research was motivated by the recent discovery of planets around binary stars. Our initial intention was thus to investigate the 3-dimensional nearly circular periodic orbits of the circular restricted problem of three bodies; more precisely Stromgren's class L, (direct) and class m, (retrograde). We started by extending several of Hénon's vertical critical orbits of these 2 classes to three dimensions, looking especially for orbits which are near circular and have stable characteristic exponents.We discovered early on that the periodic orbits with the above two qualifications are fairly rare and we decided thus to undertake a systematic exploration, limiting ourselves to symmetric periodic orbits. However, we examined all 16 possible symmetry cases, trying 10000 sets of initial values for periodicity in each case, thus 160000 integrations, all with z _o or _o equal to 0.1 This gave us a preliminary collection of 171 periodic orbits, all fairly near the xy-plane, thus with rather low inclinations. Next, we integrated a second similar set of 160000 cases with z _o or _o equal to 0.5, in order to get a better representation of the large inclinations. This time, we found 167 periodic orbits, but it was later discovered that at least 152 of them belong to the same families as the first set with 0.1Our paper quickly describes the definition of the problem, with special emphasis on the symmetry properties, especially for the case of masses with equal primaries. We also allow a section to describe our approach to stability and characteristic exponents, following our paper on this subject, (Broucke, 1969). Then we describe our numerical results, as much as space permits in the present paper.We found basically only about a dozen families with sizeable segments of simple stable periodic orbits. Some of them are around one of the two stars only but we do not describe them here because of a lack of space. We extended about 170 periodic orbits to families of up to 500 members, (by steps of 0.005 in the parameter), although, in many cases, we do not know the real end of the families. We also give an overview of the different types of periodic orbits that are most often encountered. We describe some of the rather strange orbits, (some of which are actually stable).     

受摄限制性三体问题平动点线性稳定性的判断条件及在Robe问题的应用

给出了受摄限制性三体问题平动点线性稳定性的一个判断条件．条件只与平动点切映像的特征方程系数有关，使用方便．用判断条件，讨论了Robe问题平动点在阻力摄动下的线性稳定性，得到了Hallan等给出的Robe问题平动点在阻力摄动下的线性稳定范围．并改进了Giordanoc等的结果．     

Symmetries, Reduction and Relative Equilibria for a Gyrostat in the Three-body Problem

The problem of three bodies when one of them is a gyrostat is considered. Using the symmetries of the system we carry out two reductions. Global considerations about the conditions for relative equilibria are made. Finally, we restrict to an approximated model of the dynamics and a complete study of the relative equilibria is made.     

Properties of New Coordinates for the General Three-Body Problem

This is an analysis of the features of the new coordinate system given by the principal axes of inertia, as determined by Euler angles, and twodistances related to the inertia principal moments and an auxiliar angleas coordinates, for studying the general three-body problem, interactingthrough gravitational forces.The reduction of order is performed in these new coordinates by using the angular velocity vector or the Euler angles.The Eulerian case of collinear motion is revisited from our own perspective.The value of the auxiliar angle is computed for the Sun–Earth–Moon system.     

Periodic and Chaotic Trajectories of the Second Species for the n-Centre Problem

For the n-centre problem of one particle moving in the potential of attracting centres of small mass fixed in an arbitrary smooth potential and magnetic field, we prove the existence of periodic and chaotic trajectories shadowing sequences of collision orbits. In particular, we obtain large subshifts of solutions of this type for the circular restricted 3-body problem of celestial mechanics. Poincaré had conjectured existence of the periodic ones and given them the name ‘second species solutions’. This revised version was published online in July 2006 with corrections to the Cover Date.     

Simulation of the full two rigid body problem using polyhedral mutual potential and potential derivatives approach

Herein we investigate the coupled orbital and rotational dynamics of two rigid bodies modelled as polyhedra, under the influence of their mutual gravitational potential. The bodies may possess any arbitrary shape and mass distribution. A method of calculating the mutual potential’s derivatives with respect to relative position and attitude is derived. Relative equations of motion for the two body system are presented and an implementation of the equations of motion with the potential gradients approach is described. Results obtained with this dynamic simulation software package are presented for multiple cases to validate the approach and illustrate its utility. This simulation capability is useful both for addressing questions in dynamical astronomy and for enabling spacecraft missions to binary asteroid systems.     

Bounds on Rotation Periods of Disrupted Binaries in the Full 2-Body Problem

Constraints on the final rotation rates of a Hill unstable binary in the full 2-body problem are derived and analyzed. Application of these constraints are made to the problem of a constant density body spun to disruption. This analysis has relevance to the evolution of asteroid spin rates. This revised version was published online in July 2006 with corrections to the Cover Date.     

Some Equilibrium of the Restricted Three-Body Problem with Axial Symmetric Primaries and a Gyrostat as Infinitesimal Mass

The restricted three-body problem is reconsidered by replacing the point-like primaries of the classical problem by a pair of axisymmetric rigid bodies which have a plane of symmetry perpendicular to their axes, and the infinitesimal mass by a gyrostat. The conditions for the circular motion of the primaries around their center of mass are stated and they yield the classification of all possible orientations of these bodies into four groups according to the value of their angular velocity. Then the equations of motion of the gyrostat are derived and solved for the equilibrium configurations of the system.