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Discrete & Continuous Dynamical Systems - S

2008 , Volume 1 , Issue 4

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Preface
Ernesto Perez Chavela , Clark Robinson and  Marian Gidea
2008, 1(4): i-i doi: 10.3934/dcdss.2008.1.4i +[Abstract](24) +[PDF](48.4KB)
Abstract:
This special issue of DCDS-S is in honor of the contribution of Donald Saari to the mathematical aspects of Celestial Mechanics over a period of 40 years. He had important contributions to the classification of the $n$-body problem as time goes to infinity, the measure of the set of initial conditions leading to collision, singularities that are not collisions for the $n$-body problem, and many other topics. Years ago, he made a conjecture about whether a system with constant moment of inertia must be a planar relative equilibrium. (See Saari's article in this issue for a further explanation.) This problem has been solved just for the three body problem and is still open for $n \geq 4$r. Several of the articles in this issue deal with this issue and related issues about central configurations (a generalization of relative equilibria). The recent monograph (D. Saari, "Collisions, rings, and other Newtonian $N$-body problems", CBMS Region Conference in Mathematics 104(2005), Amer. Math. Soc., Providence RI) gives an introduction into the full extent of the Saari's contribution to celestial mechanics. Besides the creative ideas Don brought to his work, his clear and inspiring talks and exposition has contributed to his impact on the field. We are sure that all the contributors to this issue join us in thanking Don for his contributions and inspiration to us all.
    The idea for this collection of papers related to the work of Saari originated at the Saarifest Conference held in 2005 in Guanajuato México. The papers in this collection vary between many that relate rather closely to the work of Saari to others that relate to his work in a broad sense. We hope that you the reader benefit from the work presented here and you are inspired to study some of his original papers and monographs.
On the spatial central configurations of the 5--body problem and their bifurcations
Martha Alvarez , Joaquin Delgado and  Jaume Llibre
2008, 1(4): 505-518 doi: 10.3934/dcdss.2008.1.505 +[Abstract](33) +[PDF](257.1KB)
Abstract:
Central configurations provide special solutions of the general $n$--body problem. Using the mutual distances between the $n$ bodies as coordinates we study the bifurcations of the spatial central configurations of the $5$--body problem going from the problem with equals masses to the $1+4$-- body problem which has one large mass and four infinitesimal equal masses. This study is made by giving a computer--aided proof.
Random walk in the three-body problem and applications
Edward Belbruno
2008, 1(4): 519-540 doi: 10.3934/dcdss.2008.1.519 +[Abstract](62) +[PDF](462.9KB)
Abstract:
The process of random walk is described, in general, and how it can be applied in the three-body problem in a systematic manner. Several applications are considered. The main one which is a focus of this paper is on the evolution of horseshoe orbits and their transition to breakout motion in the restricted three-body problem. This connection is related to their use for an Earth-impactor in a theory on the formation of the Moon. We briefly discuss another application on the instability of asteroid orbits.
Symbolic dynamics of the elliptic rectilinear restricted 3--body problem
Samuel R. Kaplan , Ernesto A. Lacomba and  Jaume Llibre
2008, 1(4): 541-555 doi: 10.3934/dcdss.2008.1.541 +[Abstract](39) +[PDF](361.8KB)
Abstract:
We apply symbolic dynamics to continue our previous study of 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. Based on properties of the homothetic orbit, which is a transversal heteroclinic orbit beginning and ending in triple collision hyperbolic equilibria and using a global Poincaré section, we describe the possible itineraries of binary collisions an orbit can have.
Oscillatory motions in the rectangular four body problem
Ernesto A. Lacomba and  Mario Medina
2008, 1(4): 557-587 doi: 10.3934/dcdss.2008.1.557 +[Abstract](32) +[PDF](1582.0KB)
Abstract:
In this paper we describe a symbolic dynamics for the rectangular four body problem by applying blow ups at total collisions and at infinity, studying the homoclinic or heteroclinic orbits obtained as intersection of corresponding two dimensional invariant submanifolds in a 3 dimensional energy level plus a convenient Poincaré map. With this tool we show the existence of a very rich dynamics and obtain the Main Theorem of this article. It gives the transition matrix for the symbolic dynamics of the images of conveniently chosen rectangles in the Poincaré section of the flow.
On the Dziobek configurations of the restricted $(N+1)$-body problem with equal masses
Eduardo S. G. Leandro
2008, 1(4): 589-595 doi: 10.3934/dcdss.2008.1.589 +[Abstract](30) +[PDF](134.0KB)
Abstract:
In this article we show that the $(N-1)$-dimensional central configurations of the restricted $(N+1)$-body problem with equal masses are symmetrical. As a consequence, we are able to prove finiteness and find upper and lower bounds for the number of central configurations.
The mass dependence of the period of the periodic solutions of the Sitnikov problem
Marcelo Marchesin
2008, 1(4): 597-609 doi: 10.3934/dcdss.2008.1.597 +[Abstract](25) +[PDF](176.8KB)
Abstract:
We study the problem in which $N$ bodies, called primaries, of equal masses $m$ are describing circular keplerian solutions in the $xy$ plane and a body $\mu$, of zero mass, moves on a line perpendicular to the plane of motion of the primaries and passing through their center of mass. We show that such a problem is equivalent to the Classical Circular Sitnikov Problem, in which $N=2$ and $m=\frac{1}{2}$. We also show that the main parameter in searching for periodic solutions is $M=mN$, the total mass of all the primaries. We add an analytic study of the period, $T(h)$, as a function of the negative energy $h$. We generalize some results of [2] and we show the dependence of $T(h)$ on the mass parameter $M$. Finally, we confirm, the expected result that the case of the Newtonian potential for a homogeneous circular ring of mass $M$ is just the limit case of the problem we have studied, in which we let $N$ go to infinity, while keeping the product $mN$ finite.
The 2-body problem under Fock's potential
Vasile Mioc and  Ernesto Pérez-Chavela
2008, 1(4): 611-629 doi: 10.3934/dcdss.2008.1.611 +[Abstract](43) +[PDF](302.9KB)
Abstract:
We study the two-body problem moving under the Fock's potential, where the global flow is fully described. The analysis is separately performed for negative, zero, and positive energy levels. Many kinds of orbits are found, some of them being of positive Lebesgue measure. We also show some unusual features as the coexistence of fundamentally different orbits for the same energy level and for the same angular momentum.
A proof of Saari's conjecture for the three-body problem in $\R^d$
Richard Moeckel
2008, 1(4): 631-646 doi: 10.3934/dcdss.2008.1.631 +[Abstract](37) +[PDF](201.2KB)
Abstract:
The well-known central configurations of the three-body problem give rise to periodic solutions where the bodies rotate rigidly around their center of mass. For these solutions, the moment of inertia of the bodies with respect to the center of mass is clearly constant. Saari conjectured that such rigid motions, called relative equilibrium solutions, are the only solutions with constant moment of inertia. This result will be proved here for the Newtonian three-body problem in $\R^d$ with three positive masses. The proof makes use of some computational algebra and geometry. When $d\le 3$, the rigid motions are the planar, periodic solutions arising from the five central configurations, but for $d\ge 4$ there are other possibilities.
Uniform subharmonic orbits for Sitnikov problem
Clark Robinson
2008, 1(4): 647-652 doi: 10.3934/dcdss.2008.1.647 +[Abstract](37) +[PDF](123.0KB)
Abstract:
We highlight the argument in Moser's monograph that the subharmonic periodic orbits for the Sitnikov problem exist uniformly for the eccentricity sufficiently small. We indicate how this relates to the uniformity of subharmonic periodic orbits for a forced Hamiltonian system of one degree of freedom with a symmetry.
Reflections on the "Saari Conjecture" and several new ones
Donald G. Saari
2008, 1(4): 653-665 doi: 10.3934/dcdss.2008.1.653 +[Abstract](24) +[PDF](202.1KB)
Abstract:
After explaining what motivated an earlier, yet unanswered conjecture whether a constant moment of inertia requires a relative equilibrium motion, several related conjectures follow and are described. One of them would generalize the $N$-body Virial Theorem. The mathematical obstacle hindering solution of all of these issues is identified and discussed.

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