Partial Reduction in the N-Body Planetary Problem using the Angular Momentum Integral

We present a new set of variables for the reduction of the planetary n-body problem, associated to the angular momentum integral, which can be of any use for perturbation theory. The construction of these variables is performed in two steps. A first reduction, called partial is based only on the fixed direction of the angular momentum. The reduction can then be completed using the norm of the angular momentum. In fact, the partial reduction presents many advantages. In particular, we keep some symmetries in the equations of motion (d'Alembert relations). Moreover, in the reduced secular system, we can construct a Birkhoff normal form at any order. Finally, the topology of this problem remains the same as for the non-reduced system, contrarily to Jacobi's reduction where a singularity is present for zero inclinations. For three bodies, these reductions can be done in a very simple way in Poincaré's rectangular variables. In the general n-body case, the reduction can be performed up to a fixed degree in eccentricities and inclinations, using computer algebra expansions. As an example, we provide the truncated expressions for the change of variable in the 4-body case, obtained using the computer algebra system TRIP.     

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.     

Wintner's collinear and flat solutions are nowhere dense

It is shown that in then-body problem with generalized attraction law, the sets of initial conditions which lead to Wintner's collinear, respectively flat, motion are nowhere dense relative to the set of initial conditions that define solutions inR ³.     

Global geometry of non-planar 3-body motions

The aim of this paper is to study the global geometry of non-planar 3-body motions in the realms of equivariant Differential Geometry and Geometric Mechanics. This work was intended as an attempt at bringing together these two areas, in which geometric methods play the major role, in the study of the 3-body problem. It is shown that the Euler equations of a three-body system with non-planar motion introduce non-holonomic constraints into the Lagrangian formulation of mechanics. Applying the method of undetermined Lagrange multipliers to study the dynamics of three-body motions reduced to the level of moduli space [`(M)]{\bar{M}} subject to the non-holonomic constraints yields the generalized Euler-Lagrange equations of non-planar three-body motions in [`(M)]{\bar{M}} . As an application of the derived dynamical equations in the level of [`(M)]{\bar{M}} , we completely settle the question posed by A. Wintner in his book [The analytical foundations of Celestial Mechanics, Sections 394–396, 435 and 436. Princeton University Press (1941)] on classifying the constant inclination solutions of the three-body problem.     

New formulation of the two body problem using a continued fractional potential

In the present work, the two body problem using a potential of a continued fractions procedure is reformulated. The equations of motion for two bodies moving under their mutual gravity is constructed. The integrals of motion, angular momentum integral, center of mass integral, total mechanical energy integral are obtained. New orbit equation is obtained. Some special cases are followed directly. Some graphical illustrations are shown. The only included constant of the continued fraction procedure is adjusted so as to represent the so called J ₂ perturbation term of the Earth’s potential.     

Conjugate Points in the Gravitational n-Body Problem

In an effort to understand the nature of almost periodic orbits in the n-body problem (for all time t) we look first to the more basic question of the oscillatory nature of solutions of this problem (on a half-line, usually taken as R ⁺). Intimately related to this is the notion of a conjugate point(due to A. Wintner) of a solution. Specifically, by rewriting the mass unrestricted general problem of n-bodies in a symmetric form we prove that in the gravitational Newtonian n-body problem with collisionless motions there exists arbitrarily large conjugate points in the case of arbitrary (positive) masses whenever the cube of the reciprocal of at least one of the mutual distances is not integrable at infinity. The implication of this result is that there are possibly many Wintner oscillatorysolutions in these cases (some of which may or may not be almost periodic). As a consequence, we obtain sufficient conditions for all continuable solutions (to infinity) to be either unbounded or to allow for near misses (at infinity). The results also apply to potentials other than Newtonian ones. Our techniques are drawn from results in systems oscillation theory and are applicable to more general situations. Dedicated to the memory of Robert M. (Bob) Kauffman, formerly Professor of the University of Alabama in Birmingham     

On the number of central configurations in the N-body problem

Central configurations are critical points of the potential function of the n-body problem restricted to the topological sphere where the moment of inertia is equal to constant. For a given set of positive masses m ₁,..., m _n we denote by N(m ₁, ..., m _n, k) the number of central configurations' of the n-body problem in ^k modulus dilatations and rotations. If m _{n 1},..., m _n, k) is finite, then we give a bound of N(m ₁,..., m _n, k) which only depends of n and k.     

Low Energy Transfer to the Moon

In 1991, the Japanese Hiten mission used a low energy transfer with a ballistic capture at the Moon which required less Vthan a standard Hohmann transfer. In this paper, we apply the dynamical systems techniques developed in our earlier work to reproduce systematically a Hiten-like mission. We approximate the Sun–Earth–Moon-spacecraft 4-body system as two 3-body systems. Using the invariant manifold structures of the Lagrange points of the 3-body systems, we are able to construct low energy transfer trajectories from the Earth which execute ballistic capture at the Moon. The techniques used in the design and construction of this trajectory may be applied in many situations.     

Earth–Mars transfers with ballistic escape and low-thrust capture

In this paper novel Earth–Mars transfers are presented. These transfers exploit the natural dynamics of n-body models as well as the high specific impulse typical of low-thrust systems. The Moon-perturbed version of the Sun–Earth problem is introduced to design ballistic escape orbits performing lunar gravity assists. The ballistic capture is designed in the Sun–Mars system where special attainable sets are defined and used to handle the low-thrust control. The complete trajectory is optimized in the full n-body problem which takes into account planets’ orbital inclinations and eccentricities. Accurate, efficient solutions with reasonable flight times are presented and compared with known results.     

Jacobi Dynamics And The N-Body Problem With Variable Masses

An appropriate generalization of the Jacobi equation of motion for the polar moment of inertia I is considered in order to study the N-body problem with variable masses. Two coupled ordinary differential equations governing the evolution of I and the total energy E are obtained. A regularization scheme for this system of differential equations is provided. We compute some illustrative numerical examples, and discuss an average method for obtaining approximate analytical solutions to this pair of equations. For a particular law of mass loss we also obtain exact analytical solutions. The application of these ideas to other kind of perturbed gravitational N-body systems involving drag forces or a different type of mass variation is also considered. This revised version was published online in July 2006 with corrections to the Cover Date.     

Accelerating universe with time variation of G and Λ

We study a gravitational model in which scale transformations play the key role in obtaining dynamical G and Λ. We take a non-scale invariant gravitational action with a cosmological constant and a gravitational coupling constant. Then, by a scale transformation, through a dilaton field, we obtain a new action containing cosmological and gravitational coupling terms which are dynamically dependent on the dilaton field with Higgs type potential. The vacuum expectation value of this dilaton field, through spontaneous symmetry breaking on the basis of anthropic principle, determines the time variations of G and Λ. The relevance of these time variations to the current acceleration of the universe, coincidence problem, Mach’s cosmological coincidence and those problems of standard cosmology addressed by inflationary models, are discussed. The current acceleration of the universe is shown to be a result of phase transition from radiation toward matter dominated eras. No real coincidence problem between matter and vacuum energy densities exists in this model and this apparent coincidence together with Mach’s cosmological coincidence are shown to be simple consequences of a new kind of scale factor dependence of the energy momentum density as ρ∼a ⁻⁴. This model also provides the possibility for a super fast expansion of the scale factor at very early universe by introducing exotic type matter like cosmic strings.     

Global stability and the restricted 3-body problem

Global stability regions are found for classi orbits of the circular restricted 3-body problem for primary masses equal and Jacobi constantK>15.5. As this constant decreases, the stability, region shrinks extremely rapidly.     

N-body simulation of two-dimensional stationary stellar distributions: Dissipation of energy and development of the field of velocity

The procedure for the simulation of two-dimensionalN-body configurations described in this paper has been devised to investigate the dissipation of energy and the development of velocity in given sub-regions of stellar systems, especially in the central region of a galaxy. In these simulations massive particles are injected into the given spatial region at a pre-determined constant time rate and at a given distribution of velocity vectors. The mass points are allowed to interact gravitationally and special attention is given to the influence of near-by encounters. After a certain period of time a stationary (equilibrium) situation settles down with a certain distribution of mass points in position and velocity space. Preliminary results obtained so far demonstrate the reliability of this procedure forN-body simulation and for the investigation of the dissipation of energy and of the development of velocity fields. Forthcoming results are hoped to deliver information on the role of three-body encounters in the processes of randomization at work in self-consistent stellar configurations.J. Polcher and W. Wanner also of the Mathematical Physics Group of the IFK took part in many of the discussions concerning the development of this programme.     

Relative equilibrium solutions in the four body problem

Beyond the casen=3 little was known about relative equilibrium solutions of then-body problem up to recent years. Palmore's work provides in the general case much useful information. In the casen=4 he gives the totality of solutions when the four masses are equal and studies some degeneracies. We present here a survey of solutions for arbitrary masses, discussing the manifolds of degeneracy. The ordering of restricted potentials allows a counting of the number of bifurcation sets and different invariant manifolds. An analysis of linear stability is done in the restricted and general cases. As a result, values of the masses ensuring linear stability are given.     

Motions close to escapes in the rhomboidal four body problem

In this work we study escape and capture orbits in the planar rhomboidal 4-body problem in a level of constant negative energy. There are only two different values of the masses here. By using numerical analysis, we show certain transversal intersections of the invariant manifolds of parabolic orbits. We then introduce Symbolic Dynamics when the mass ratio is small, and when it is close to one. In the first case the escapes or captures predominate in the direction of one of the diagonals of the rhombus, while in the second case we find solutions escaping or being captured in the direction of both possible diagonals.     

The global solution of the N-body problem

The problem of finding a global solution for systems in celestial mechanics was proposed by Weierstrass during the last century. More precisely, the goal is to find a solution of the n-body problem in series expansion which is valid for all time. Sundman solved this problem for the case of n = 3 with non-zero angular momentum a long time ago. Unfortunately, it is impossible to directly generalize this beautiful theory to the case of n > 3 or to n = 3 with zero-angular momentum.A new blowing up transformation, which is a modification of McGehee's transformation, is introduced in this paper. By means of this transformation, a complete answer is given for the global solution problem in the case of n > 3 and n = 3 with zero angular momentum.The main result in this paper has appeared in Chinese in Acta Astro. Sinica. 26 (4), 313–322. In this version some mistakes have been rectified and the problems we solved are now expressed in a much clearer fashion.     

Angular momentum flux from an isolated system ofN-bodies in higher multipole moments

The present paper contains a systematic study of the generalization of the angular momentum formula in higher multipole moments from aN-body isolated system in the linear approximation of general relativity.     

Applications to stellar and galactic dynamics

Computation and a wealth of new observational techniques have reinvigorated dynamical studies of galaxies and star clusters. These objects are examples of the gravitationaln-body problem withn in the range from a few hundred to 10¹¹. Relaxation effects dominate at the low end and are completely negligible at the high end. The gravitationaln-body problem is chaotic, and the principal challenge in doing physics where that problem is involved (whether computationally or with analytic theory) is to ensure that chaos has not vitiated the results. Enforcing a Liouville theorem accomplishes this with collision-free large-n problems, but equivalent recipes are not in common use for smallern. We describe some important insights and discoveries that have come from computation in stellar dynamics, discuss chaos briefly, and indicate the way the physics that comes up in different astronomical contexts is addressed in numerical methods currently in use. Graphics is a vital part of any computational approach. The long range prospects are very promising for continued high scientific productivity in stellar dynamics.     

Relative equilibria and the virial theorem

A solution of the Newtoniann-body problem for which the moment of inertia (with respect to the centre of mass) is constant is a solution of relative equilibrium.Research supported in part by NSF grant MCS-77-01716.