KAM tori for N-body problems: a brief history

We review analytical (rigorous) results about the existence of invariant tori for planetary many-body problems.     

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.     

On the vertical families of two-dimensional tori near the triangular points of the Bicircular problem

This paper focuses on some aspects of the motion of a small particle moving near the Lagrangian points of the Earth–Moon system. The model for the motion of the particle is the so-called bicircular problem (BCP), that includes the effect of Earth and Moon as in the spatial restricted three body problem (RTBP), plus the effect of the Sun as a periodic time-dependent perturbation of the RTBP. Due to this periodic forcing coming from the Sun, the Lagrangian points are no longer equilibrium solutions for the BCP. On the other hand, the BCP has three periodic orbits (with the same period as the forcing) that can be seen as the dynamical equivalent of the Lagrangian points. In this work, we first discuss some numerical methods for the accurate computation of quasi-periodic solutions, and then we apply them to the BCP to obtain families of 2-D tori in an extended neighbourhood of the Lagrangian points. These families start on the three periodic orbits mentioned above and they are continued in the vertical (z and ż) direction up to a high distance. These (Cantor) families can be seen as the continuation, into the BCP, of the Lyapunov family of periodic orbits of the Lagrangian points that goes in the (z, ż) direction. These results are used in a forthcoming work [9] to find regions where trajectories remain confined for a very long time. It is remarkable that these regions seem to persist in the real system. This revised version was published online in July 2006 with corrections to the Cover Date.     

Dynamics of a satellite orbiting a planet with an inhomogeneous gravitational field

We study the dynamics of a satellite (artificial or natural) orbiting an Earth-like planet at low altitude from an analytical point of view. The perturbation considered takes into account the gravity attraction of the planet and in particular it is caused by its inhomogeneous potential. We begin by truncating the equations of motion at second order, that is, incorporating the zonal and the tesseral harmonics up to order two. The system is formulated as an autonomous Hamiltonian and has three degrees of freedom. After three successive Lie transformations, the system is normalised with respect to two angular co-ordinates up to order five in a suitable small parameter given by the quotient between the angular velocity of the planet and the mean motion of the satellite. Our treatment is free of power expansions of the eccentricity and of truncated Fourier series in the anomalies. Once these transformations are performed, the truncated Hamiltonian defines a system of one degree of freedom which is rewritten as a function of two variables which generate a phase space which takes into account all of the symmetries of the problem. Next an analysis of the system is achieved obtaining up to six relative equilibria and three types of bifurcations. The connection with the original system is established concluding the existence of various families of invariant 3-tori of it, as well as quasiperiodic and periodic trajectories. This is achieved by using KAM theory techniques.     

On the Centre Manifold of Collinear Points in the Planar Three-Body Problem

We study the existence of invariant tori in a neighbourhood of the collinear equilibrium points of the planar three-body problem. To this end some properties of the normal form of the Hamiltonian reduced to the 4D central manifold are proved. Using this normal form, we show that the nondegeneracy conditions of KAM theorem are satisfied for all positive masses, including the 2:1 resonance case. The evaluation of the conditions is done numerically.     

Surfaces of section in the Miranda-Umbriel 3:1 inclination problem

The recent numerical simulations of Tittemore and Wisdom (1988, 1989, 1990) and Dermottet al. (1988), Malhotra and Dermott (1990) concerning the tidal evolution through resonances of some pairs of Uranian satellites have revealed interesting dynamical phenomena related to the interactions between close-by resonances. These interactions produce chaotic layers and strong secondary resonances. The slow evolution of the satellite orbits in this dynamical landscape is responsible for temporary capture into resonance, enhancement of eccentricity or inclination and subsequent escape from resonance. The present contribution aims at developing analytical tools for predicting the location and size of chaotic layers and secondary resonances. The problem of the 3:1 inclination resonance between Miranda and Umbriel is analysed.     

Lagrangian coherent structures in the planar elliptic restricted three-body problem

This study investigates Lagrangian coherent structures (LCS) in the planar elliptic restricted three-body problem (ER3BP), a generalization of the circular restricted three-body problem (CR3BP) that asks for the motion of a test particle in the presence of two elliptically orbiting point masses. Previous studies demonstrate that an understanding of transport phenomena in the CR3BP, an autonomous dynamical system (when viewed in a rotating frame), can be obtained through analysis of the stable and unstable manifolds of certain periodic solutions to the CR3BP equations of motion. These invariant manifolds form cylindrical tubes within surfaces of constant energy that act as separatrices between orbits with qualitatively different behaviors. The computation of LCS, a technique typically applied to fluid flows to identify transport barriers in the domains of time-dependent velocity fields, provides a convenient means of determining the time-dependent analogues of these invariant manifolds for the ER3BP, whose equations of motion contain an explicit dependency on the independent variable. As a direct application, this study uncovers the contribution of the planet Mercury to the Interplanetary Transport Network, a network of tubes through the solar system that can be exploited for the construction of low-fuel spacecraft mission trajectories. Electronic supplementary material  The online version of this article (doi:) contains supplementary material, which is available to authorized users.     

Invariant Tori in the Secular Motions of the Three-body Planetary Systems

We consider the problem of the applicability of KAM theorem to a realistic problem of three bodies. In the framework of the averaged dynamics over the fast angles for the Sun–Jupiter–Saturn system we can prove the perpetual stability of the orbit. The proof is based on semi-numerical algorithms requiring both explicit algebraic manipulations of series and analytical estimates. The proof is made rigorous by using interval arithmetics in order to control the numerical errors.     

Invariant shape solutions of the spinning three craft Coulomb tether problem

In this paper we study shape-preserving formations of three spacecraft, where the formation keeping forces arise from the electric charges deposed on each craft. Inspired by Lagrange’s 3-body problem, the general conditions that guarantee preservation of the geometric shape of the electrically charged formation are derived. While the classical collinear configuration is a solution to the problem, the equilateral triangle configuration is found to only occur with unbounded relative motion. The three collinear spacecraft problem is analyzed and the possible solutions are categorized based on the spacecraft mass–charge ratio. Precise statements on the number of solutions associated with each category are provided. Finally, a methodology is proposed to study boundedness of the collinear solution that is inspired by past understanding and results for the 3-body problem. Given the initial position and the velocity vectors of each craft along with the charges, analytical solutions are provided describing the resulting relative motion.     

Boundary-value problem formulations for computing invariant manifolds and connecting orbits in the circular restricted three body problem

We demonstrate the remarkable effectiveness of boundary value formulations coupled to numerical continuation for the computation of stable and unstable manifolds in systems of ordinary differential equations. Specifically, we consider the circular restricted three-body problem (CR3BP), which models the motion of a satellite in an Earth–Moon-like system. The CR3BP has many well-known families of periodic orbits, such as the planar Lyapunov orbits and the non-planar vertical and halo orbits. We compute the unstable manifolds of selected vertical and halo orbits, which in several cases leads to the detection of heteroclinic connections from such a periodic orbit to invariant tori. Subsequent continuation of these connecting orbits with a suitable end point condition and allowing the energy level to vary leads to the further detection of apparent homoclinic connections from the base periodic orbit to itself, or the detection of heteroclinic connections from the base periodic orbit to other periodic orbits. Some of these connecting orbits are of potential interest in space mission design.     

Effect of oblateness on the non-linear stability of the triangular liberation points of the restricted three-body problem in the presence of resonances

The non-linear stability of the triangular libration points of the restricted three-body problem is studied under the presence of third and fourth order resonance's, when the more massive primary is an oblate spheroid. In this study Markeev's theorem are utilised with the help of KAM theorem. It is found that the stability of the triangular libration points are unstable in the third order resonance case and stable in the fourth order resonance case, for all the values of oblateness factor A₁. This revised version was published online in August 2006 with corrections to the Cover Date.     

A fourth-order solution of the ideal resonance problem

The second-order solution of the Ideal Resonance Problem, obtained by Henrard and Wauthier (1988), is developed further to fourth order applying the same method. The solutions for the critical argument and the momentum are expressed in terms of elementary functions depending on the time variable of the pendulum as independent variable. This variable is related to the original time variable through a Kepler-equation. An explicit solution is given for this equation in terms of elliptic integrals and functions. The fourth-order formal solution is compared with numerical solutions obtained from direct numerical integrations of the equations of motion for two specific Hamiltonians.     

Resonant motion in the restricted three body problem

The resonant structure of the restricted three body problem for the Sun- Jupiter asteroid system in the plane is studied, both for a circular and an elliptic orbit of Jupiter. Three typical resonances are studied, the 2 : 1, 3 : 1 and 4 : 1 mean motion resonance of the asteroid with Jupiter. The structure of the phase space is topologically different in these cases. These are typical for all other resonances in the asteroid problem. In each case we start with the unperturbed two-body system Sun-asteroid and we study the continuation of the periodic orbits when the perturbation due to a circular orbit of Jupiter is introduced. Families of periodic orbits of the first and of the second kind are presented. The structure of the phase space on a surface of section is also given. Next, we study the families of periodic orbits of the asteroid in the elliptic restricted problem with the eccentricity of Jupiter as a parameter. These orbits bifurcate from the families of the circular problem. Finally, we compare the above families of periodic orbits with the corresponding families of fixed points of the averaged problem. Different averaged Hamiltonians are considered in each resonance and the range of validity of each model is discussed.     

Stability of the triangular libration points of the circular restricted problem in the presence of resonances

The stability of triangular libration points, when the bigger primary is a source of radiation and the smaller primary is an oblate spheroid. has been investigated in the resonance cases ₁ = 2₂ and ₁ = 3₂. The motion is unstable for all the values of parameters q and A when ₁ = 2₂ and the motion is unstable and stable depending upon the values of the parameters q and A when ₁ = 3₂. Here q is the radiation parameter and A is the oblateness parameter.     

Construction of confidence regions in problem on probabilistic study into motion of minor bodies of the solar system

Different approaches to constructing regions of possible motions of minor bodies are examined. An economical approach to the minimization of the number of points of the initial region by means of defining this region with its boundary surface is offered, and the estimations of its efficiency are given. The methods for estimating the admissibility of the linear approach are considered. For this purpose, simple methods for calculating the nonlinear factors are offered, which makes it possible to classify a problem to be solved as either strongly or weakly nonlinear. Recommendations are given on the possibility of reducing the concrete estimation problem to a weakly nonlinear one, where the more economical linear approach can be used. The combined method of mapping the initial region in time is also offered which unites the linear and nonlinear approaches. By example of two asteroids, the area of applicability of linear mappings is estimated.     

Motion near the 3/1 resonance of the planar elliptic restricted three body problem

The global semi-numerical perturbation method proposed by Henrard and Lemaître (1986) for the 2/1 resonance of the planar elliptic restricted three body problem is applied to the 3/1 resonance and is compared with Wisdom's perturbative treatment (1985) of the same problem. It appears that the two methods are comparable in their ability to reproduce the results of numerical integration especially in what concerns the shape and area of chaotic domains. As the global semi-numerical perturbation method is easily adapted to more general types of perturbations, it is hoped that it can serve as the basis for the analysis of more refined models of asteroidal motion. We point out in our analysis that Wisdom's uncertainty zone mechanism for generating chaotic domains (also analysed by Escande 1985 under the name of slow Hamiltonian chaotic layer) is not the only one at work in this problem. The secondary resonance _p = 0 plays also its role which is qualitatively (if not quantitatively) important as it is closely associated with the random jumps between a high eccentricity mode and a low eccentricity mode.     

Periodic orbits of the general three-body problem for the Sun-Jupiter-Saturn system

Two families of symmetric periodic orbits of the planar, general, three-body problem are presented. The masses of the three bodies include ratios equal to the Sun-Jupiter-Saturn system and the periods of the orbits of Jupiter and Saturn are in a 25 resonance. The (linear) stability of the orbits are studied in relation to eccentricity and mass variations. The generation of the two families of periodic orbits follows a systematic approach and employs (numerical) continuation from periodic orbits of the first and second kind in the circular restricted problem to the elliptic restricted problem and from the circular and elliptic problems to the general problem through bifurcation phenomena relating the three dynamical systems. The approach also provides insight into the evolutionary process of periodic orbits continued from the restricted problems to the general problem.     

Periodic orbits of the elliptic restricted problem for the Sun-Jupiter-Saturn system

A systematic approach to generate periodic orbits in the elliptic restricted problem of three bodies in introduced. The approach is based on (numerical) continuation from periodic orbits of the first and second kind in the circular restricted problem to periodic orbits in the elliptic restricted problem. Two families of periodic orbits of the elliptic restricted problem are found by this approach. The mass ratio of the primaries of these orbits is equal to that of the Sun-Jupiter system. The sidereal mean motions between the infinitesimal body and the smaller primary are in a 2:5 resonance, so as to approximate the Sun-Jupiter-Saturn system. The linear stability of these periodic orbits are studied as functions of the eccentricities of the primaries and of the infinitesimal body. The results show that both stable and unstable periodic orbits exist in the elliptic restricted problem that are close to the actual Sun-Jupiter-Saturn system. However, the periodic orbit closest to the actual Sun-Jupiter-Saturn system is (linearly) stable.