Chaos in the Three Body Problem

We consider the general spatial three body problem and study the dynamics of planetary systems consisting of a star and two planets which evolve into 2/1 mean motion resonance and into inclined orbits. Our study is focused on the periodic orbits of the system given in a suitable rotating frame. The stability of periodic orbits characterize the evolution of any planetary system with initial conditions in their vicinity. Stable periodic orbits are associated with long term regular evolution, while unstable periodic orbits are surrounded by regions of chaotic motion. We compute many families of symmetric periodic orbits by applying two schemes of analytical continuation. In the first scheme, we start from the 2/1 (or 1/2) resonant periodic orbits of the restricted problem and in the second scheme, we start from vertical critical periodic orbits of the general planar problem. Most of the periodic orbits are unstable, but many stable periodic orbits have been, also, found with mutual inclination up to $50^\circ$ - $60^\circ$, which may be related with the existence of real planetary systems.

High-order resonant orbit manifold expansions for mission design in the planar circular restricted 3-body problem

This paper investigates the global structures of periodic orbits that appear in Rayleigh–Bénard convection, which is modelled by a two-dimensional perturbed Hamiltonian model, by focusing upon resonance, symmetry and bifurcation of the periodic orbits. First, we show the global structures of periodic orbits in the extended phase space by numerically detecting the associated periodic points on the Poincaré section. Then, we illustrate how resonant periodic orbits appear and specifically clarify that there exist some symmetric properties of such resonant periodic orbits which are projected on the phase space; namely, the period m and the winding number n become odd when an m-periodic orbit is symmetric with respect to the horizontal and vertical centre lines of a cell. Furthermore, the global structures of bifurcations of periodic orbits are depicted when the amplitude ɛ of the perturbation is varied, since in experiments the amplitude of the oscillation of the convection gradually increases when the Rayleigh number is raised.

We study the dynamics of 3:1 resonant motion for planetary systems with two planets, based on the model of the general planar three body problem. The exact mean motion resonance corresponds to periodic motion (in a rotating frame) and the basic families of symmetric and asymmetric periodic orbits are computed. Four symmetric families bifurcate from the family of circular orbits of the two planets. Asymmetric families bifurcate from the symmetric families, at the critical points, where the stability character changes. There exist also asymmetric families that are independent of the above mentioned families. Bounded librations exist close to the stable periodic orbits. Therefore, such periodic orbits (symmetric or asymmetric) determine the possible stable configurations of a 3:1 resonant planetary system, even if the orbits of the two planets intersect. For the masses of the system 55Cnc most of the periodic orbits are unstable and they are associated with chaotic motion. There exist however stable symmetric and asymmetric orbits, corresponding to regular trajectories along which the critical angles librate. The 55Cnc extra-solar system is located in a stable domain of the phase space, centered at an asymmetric periodic orbit.

9R7. Classical Many-Body Problems Amenable to Exact Treatments. (Solvable and/or Integrable and/or Linearizable…) in One-, Two-, and Three-Dimensional Space. Lecture Notes in Physics, Vol m66. - F Calogero (Dept of Phys, Univ of Rome “La Sapienza,” p Aldo Moro, Rome, 00185, Italy). Springer-Verlag, Berlin. 2001. 749 pp. ISBN 3-540-41764-8. $79.95.Reviewed by M Pascal (Lab de Modelisation en Mec, Univ Pierre et Marie Curie, Tour 66, 4 Place Jussieu, Paris, 75252 Cedex 05, France).This book is concerned with integrable problems in classical mechanics (excluding quantam or relativistic mechanics). These problems are related to the motion of a system of particles acted upon by several kinds of forces in one-, two-, or three-dimensions. More general Hamiltonian systems (not associated with a Lagrangian function) are also considered. In most cases, integrable Hamiltonian systems mean Liouville integrable systems. In almost all examples dealing with the motion of a set of particles, the masses are the same for all particles. Several methods to find integrable problems are described, but the main part of the book is devoted to the Lax pair technique. This book provides a rather exhaustive survey of results obtained in the past by scientists such as Moser, Toda, Flashka, and also by the author himself.The book involves five chapters followed by several appendices and a rather wide list of references. Each chapter includes a great amount of examples and exercises. Due to the rather abstract topics studied in the book, no figures are included in it. The first chapter is a short introduction giving some basic knowledge about Newtonian equations of motion, Hamiltonian formalism, and Liouville integrable systems. The main part of the book is contained in Chapter 2, which deals with one-dimensional motions of particles along a line or a circle. Liouville integrable problems are found by using the Lax pair technique. Several examples are obtained by solving three kinds of functional equations. From these integrable cases, other integrable systems are deduced by means of several transformations like change of variables, duplication and so on. Another method introduces a time dependant polynomial of degree N with respect to the variable X; the N zeros of this polynomial are interpreted as the positions of N particles (with the same masses) on a line. Assuming that the polynomial is solution of a linear partial derivative equation, several solvable many body problems are found. An extension of this approach is also presented, using the exact Lagrangian interpolation method. In Chapter 3, a generalized formulation of this last method is formulated for spaces of arbitrary dimensions. Chapter 4 deals with integrable systems in the plane. The method is based on the idea that integrable problems in two-dimensional space can be deduced from integrable problems in one-dimension assuming that the equations of motion hold also for complex values of the variables: it is the so called technique of complexification. A survey of integrable systems in the plane obtained by this method is given at the end of the chapter. The last chapter presents integrable many body systems in three-dimensional space. The idea is to identify some solvable evolution equations governing a time dependant matrix with the motion equations of a set of particles in three-dimensions. Again, several examples are presented. As a conclusion, Classical Many-Body Problems Amenable to Exact Treatments…, is not intended for a very large audience, but it provides a rather exhaustive survey about integrable problems in some special cases of particle dynamics. It can be useful for researchers involved in celestial mechanics or can be used as a background for teaching an undergraduate course.

We prove the existence of half-entire parabolic solutions, asymptotic to a prescribed central configuration, for the equation \begin{equation*} \ddot{x} = \nabla U(x) + \nabla W(t,x), \qquad x \in \mathbb{R}^{d}, \end{equation*} where $d \geq 2$, $U$ is a positive and positively homogeneous potential with homogeneity degree $-\alpha$ with $\alpha\in\mathopen{]}0,2\mathclose{[}$, and $W$ is a (possibly time-dependent) lower order term, for $\vert x \vert \to +\infty$, with respect to $U$. The proof relies on a perturbative argument, after an appropriate formulation of the problem in a suitable functional space. Applications to several problems of Celestial Mechanics (including the $N$-centre problem, the $N$-body problem and the restricted $(N+H)$-body problem) are given.

In this work, the perturbed equations of motion of the infinitesimal body are constructed in the framework of the circular restricted three-body problem when the main two bodies are oblate and radiating. Under the perturbations effects of the oblateness and the radiation pressure the positions of collinear Lagrange points are evaluated, the interior and exterior first-order resonant periodic orbits are also studied. In addition, the initial positions of the periodic orbits and the size of loops have been estimated under these effects. Thus, the characteristics of periodic orbits have been studied under the combine effects of two, three and four perturbations for all the possible combinations of the perturbed parameters. The different order of resonant periodic orbits have been also analysed under the effects of Jacobi constant, mass factor, order of resonance and number of loops.

In this work, our target is to analyze the dynamics around the $1:-1$ resonance which appears when a family of periodic orbits of a real analytic three-degree of freedom Hamiltonian system changes its stability from elliptic to a complex hyperbolic saddle passing through degenerate elliptic. Our analytical approach consists of computing, up to some given arbitrary order, the normal form around that resonant (or <em> critical</em>) periodic orbit. <br> Hence, dealing with the normal form itself and the differential equations related to it, we derive the generic existence of a two-parameter family of invariant 2D tori which bifurcate from the critical periodic orbit. Moreover, the coefficient of the normal form that determines the stability of the bifurcated tori is identified. This allows us to show the Hopf-like character of the unfolding: elliptic tori unfold "around'' hyperbolic periodic orbits (case of <em> direct</em> bifurcation) while normal hyperbolic tori appear "around'' elliptic periodic orbits (case of <em>inverse</em> bifurcation). Further, the parametrization of the main invariant objects as well as a global description of the dynamics of the normal form are also given.

We consider a planetary system consisting of two primaries, namely a star and a giant planet, and a massless secondary, say a terrestrial planet or an asteroid, which moves under their gravitational attraction. We study the dynamics of this system in the framework of the circular and elliptic restricted TBP, when the motion of the giant planet describes circular and elliptic orbits, respectively. Originating from the circular family, families of symmetric periodic orbits in the 3/2, 5/2, 3/1, 4/1 and 5/1 mean-motion resonances are continued in the circular and the elliptic problems. New bifurcation points from the circular to the elliptic problem are found for each of the above resonances and thus, new families, continued from these points are herein presented. Stable segments of periodic orbits were found at high eccentricity values of the already known families considered as whole unstable previously. Moreover, new isolated (not continued from bifurcation points) families are computed in the elliptic restricted problem. The majority of the new families mainly consist of stable periodic orbits at high eccentricities. The families of the 5/1 resonance are investigated for the first time in the restricted three-body problems. We highlight the effect of stable periodic orbits on the formation of stable regions in their vicinity and unveil the boundaries of such domains in phase space by computing maps of dynamical stability. The long-term stable evolution of the terrestrial planets or asteroids is dependent on the existence of regular domains in their dynamical neighbourhood in phase space, which could host them for long time spans. This study, besides other celestial architectures that can be efficiently modelled by the circular and elliptic restricted problems, is particularly appropriate for the discovery of terrestrial companions among the single-giant planet systems discovered so far.

Views Icon Views Article contents Figures & tables Video Audio Supplementary Data Peer Review Share Icon Share Twitter Facebook Reddit LinkedIn Tools Icon Tools Reprints and Permissions Cite Icon Cite Search Site Citation Ernesto A. Lacomba, Jaume Llibre; Periodic orbits and non integrability of autonomous Glydén‐like systems. AIP Conference Proceedings 7 October 2010; 1283 (1): 25–31. https://doi.org/10.1063/1.3506069 Download citation file: Ris (Zotero) Reference Manager EasyBib Bookends Mendeley Papers EndNote RefWorks BibTex toolbar search Search Dropdown Menu toolbar search search input Search input auto suggest filter your search All ContentAIP Publishing PortfolioAIP Conference Proceedings Search Advanced Search |Citation Search

In the following paper we tried to apply the Lie-formalism to the regularized restricted three body problem. It will be shown that this algorithm leads to a very simple structure program which is also fast.

The discovery of binary and triple asteroids in addition to the execution of space missions to minor celestial bodies in the past several years have focused increasing attention on periodic orbits around irregular-shaped celestial bodies. In the present work, we adopt a polyhedron shape model for providing an accurate representation of irregular-shaped bodies and employ the model to calculate their corresponding gravitational and effective potentials. We also investigate the characteristics of periodic orbit families and the continuation of periodic orbits. We prove a fact, which provides a conserved quantity that permits restricting the number of periodic orbits in a fixed energy curved surface about an irregular-shaped body. The collisions of Floquet multipliers are maintained during the continuation of periodic orbits around the comet 1P/Halley. Multiple bifurcations in the periodic orbit families about irregular-shaped bodies are also discussed. Three bifurcations in the periodic orbit family have been found around the asteroid 216 Kleopatra, which include two real saddle bifurcations and one period-doubling bifurcation.

Chapter 9 - The Restricted Three-Body Problem

We study the existence of quasi-periodic, invariant tori in a nearly integrable Hamiltonian system of high order proper degeneracy, i.e., the integrable part of the Hamiltonian involves several time scales and at each time scale the corresponding Hamiltonian depends on only part of the action variables. Such a Hamiltonian system arises frequently in problems of celestial mechanics, for instance, in perturbed Kepler problems like the restricted and non-restricted 3-body problems and spatial lunar problems in which several bodies with very small masses are coupled with two massive bodies and the nearly integrable Hamiltonian systems naturally involve different time scales. Using KAM method, we will show under certain higher order non-degenerate conditions of Bruno–Russmann type that the majority of quasi-periodic, invariant tori associated with the integrable part will persist after the non-integrable perturbation. This actually concludes the KAM metric stability for such a properly degenerate Hamiltonian system.

The purpose of this paper is to show that the explosion and fragmentation of the nucleus of a galaxy may produce circumstances theoretically resembling a generalization of a Restricted Problem in Celestial Mechanics.