New stacked central configurations for the planar 5-body problem

A stacked central configuration in the n-body problem is one that has a proper subset of the n-bodies forming a central configuration. In this paper we study the case where three bodies with masses m ₁, m ₂, m ₃ (bodies 1, 2, 3) form an equilateral central configuration, and the other two with masses m ₄, m ₅ are symmetric with respect to the mediatrix of the segment joining 1 and 2, and they are above the triangle generated by {1, 2, 3}. We show the existence and non-existence of this kind of stacked central configurations for the planar 5-body problem.     

Seven-body central configurations: a family of central configurations in the spatial seven-body problem

The main result of this paper is the existence of a new family of central configurations in the Newtonian spatial seven-body problem. This family is unusual in that it is a simplex stacked central configuration, i.e the bodies are arranged as concentric three and two dimensional simplexes.      

Central configurations of the five-body problem with equal masses

In this paper we present a complete classification of the isolated central configurations of the five-body problem with equal masses. This is accomplished by using the polyhedral homotopy method to approximate all the isolated solutions of the Albouy-Chenciner equations. The existence of exact solutions, in a neighborhood of the approximated ones, is then verified using the Krawczyk method. Although the Albouy-Chenciner equations for the five-body problem are huge, it is possible to solve them in a reasonable amount of time.     

Central configurations of the planar 1+N body problem

In this paper, we give a new derivation of the equations for the central configurations of the 1+n body problem. In the case of equal masses, we show that forn large enough there exists only one solution. Our lower bound forn improves by several orders of magnitude the one previously found by Hall.     

The restricted planar isosceles three-body problem with non-negative energy

We consider a restricted three-body problem consisting of two positive equal masses m ₁ = m ₂ moving, under the mutual gravitational attraction, in a collision orbit and a third infinitesimal mass m ₃ moving in the plane P perpendicular to the line joining m ₁ and m ₂. The plane P is assumed to pass through the center of mass of m ₁ and m ₂. Since the motion of m ₁ and m ₂ is not affected by m ₃, from the symmetry of the configuration it is clear that m ₃ remains in the plane P and the three masses are at the vertices of an isosceles triangle for all time. The restricted planar isosceles three-body problem describes the motion of m ₃ when its angular momentum is different from zero and the motion of m ₁ and m ₂ is not periodic. Our main result is the characterization of the global flow of this problem.     

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.     

Periodic orbits of the planar collision restricted 3-body problem

Using the continuation method we prove that the circular and the elliptic symmetric periodic orbits of the planar rotating Kepler problem can be continued into periodic orbits of the planar collision restricted 3-body problem. Additionally, we also continue to this restricted problem the so called “comet orbits”. An erratum to this article can be found at     

Inverse problem of central configurations and singular curve in the collinear 4-body problem

In this paper, we consider the inverse problem of central configurations of n-body problem. For a given \({q=(q_1, q_2, \ldots, q_n)\in ({\bf R}^d)^n}\), let S(q) be the admissible set of masses denoted \({ S(q)=\{ m=(m_1,m_2, \ldots, m_n)| m_i \in {\bf R}^+, q}\) is a central configuration for m}. For a given \({m\in S(q)}\), let S _m (q) be the permutational admissible set about m = (m ₁, m ₂, . . . , m _n ) denoted

$S_m(q)=\{m^\prime | m^\prime\in S(q),m^\prime \not=m \, {\rm and} \, m^\prime\,{\rm is\, a\, permutation\, of }\, m \}.$

The main discovery in this paper is the existence of a singular curve \({\bar{\Gamma}_{31}}\) on which S _m (q) is a nonempty set for some m in the collinear four-body problem. \({\bar{\Gamma}_{31}}\) is explicitly constructed by a polynomial in two variables. We proved:

1. (1)
   If \({m\in S(q)}\), then either ^# S _m (q) = 0 or ^# S _m (q) = 1.
    
2. (2)
   ^#S _m (q) = 1 only in the following cases:

   1. (i)
      If s = t, then S _m (q) = {(m ₄, m ₃, m ₂, m ₁)}.
       
   2. (ii)
      If \({(s,t)\in \bar{\Gamma}_{31}\setminus \{(\bar{s},\bar{s})\}}\), then either S _m (q) = {(m ₂, m ₄, m ₁, m ₃)} or S _m (q) = {(m ₃, m ₁, m ₄, m ₂)}.
       
    

     

Tame and chaotic behavior in the planar isosceles 3-body problem

We show that every planar isosceles solution of the three-body problem encounters a collision of the symmetric particles, either forwards or backwards in time. Regularizing analytically this collision, the solution has at least a syzygy configuration and/or leads to a total collapse. Some further simple results support the intuitive image on the tame local behavior of the motion as long as it does not lead to a triple collision. As a main result we prove that total collapse singularities, can be regularized in aC ¹-fashion with respect to time, for all values of the masses. Using symbolic dynamics, the chaotic character of theC ¹-regularized solutions is pointed out.     

A family of stacked central configurations in the planar five-body problem

We study planar central configurations of the five-body problem where three bodies, \(m_1, m_2\) and \(m_3\), are collinear and ordered from left to right, while the other two, \(m_4\) and \(m_5\), are placed symmetrically with respect to the line containing the three collinear bodies. We prove that when the collinear bodies form an Euler central configuration of the three-body problem with \(m_1=m_3\), there exists a new family, missed by Gidea and Llibre (Celest Mech Dyn Astron 106:89–107, 2010), of stacked five-body central configuration where the segments \(m_4m_5\) and \(m_1m_3\) do not intersect.     

A compact model for the planar rhomboidal 4-body problem

We obtain a compact model for the global study of the planar rhomboidal 4-body problem in a level of constant negative energy. This model is a variation of the non compact model obtained through a McGehee blow up transformation. but compactness permits to obtain results which are not clear in the other case.     

Stability of the planar full 2-body problem

The stability of the Full Two-Body Problem is studied in the case where both bodies are non-spherical, but are restricted to planar motion. The mutual potential is expanded up to second order in the mass moments, yielding a highly symmetric yet non-trivial dynamical system. For this system we identify all relative equilibria and determine their stability properties, with an emphasis on finding the energetically stable relative equilibria and conditions for Hill stability of the system. The energetically stable relative equilibria always correspond to the classical “gravity gradient” configuration with the long ends of the two bodies pointed at each other, however there always exists a second equilibrium in this configuration at a closer separation that is unstable. For our model system we precisely map out the relations between these different configurations at a given value of angular momentum. This analysis identifies the fundamental physical constraints and limitations that exist on such systems, and has immediate applications to the stability of asteroid systems that are fissioned due to a rapid spin rate. Specifically, we find that all contact binary asteroids which are spun to fission will initially lie in an unstable dynamical state and can always re-impact. If the total system energy is positive, the fissioned system can disrupt directly from this relative equilibrium, while if it is negative the system is bound together.     

On the central configurations of the planar 1 + 3 body problem

We consider the Newtonian four-body problem in the plane with a dominat mass M. We study the planar central configurations of this problem when the remaining masses are infinitesimal. We obtain two different classes of central configurations depending on the mutual distances between the infinitesimal masses. Both classes exhibit symmetric and non-symmetric configurations. And when two infinitesimal masses are equal, with the help of extended precision arithmetics, we provide evidence that the number of central configurations varies from five to seven.     

Central configurations for the planar Newtonian four-body problem

Planar central configurations of four different masses are analyzed theoretically and computed numerically. We follow Dziobek’s approach to four-body central configurations with a straightforward implicit (in the masses and distances) method of our own in which the fundamental quantities are each the quotient of a directed area divided by the corresponding mass. We apply a new simple numerical algorithm to construct general four-body central configurations. We use this tool to obtain new properties of the symmetric and non-symmetric central configurations. The explicit continuous connection between three-body and four-body central configurations where one of the four masses approaches zero is clarified. Some cases of coorbital 1+3 problems are also considered.     

On the planar syzygy solutions of the 3-body problem

Relations between the rectilinear, collinear and syzygy solutions of the N-body problem are first pointed out. It is shown that, along a solution, the set of the non-collinear syzygy configuration instants is formed by isolated points. Then we restrict the study to the planar 3-body problem and prove that for Dirichlet-stable solutions, a non-syzygy solution cannot be as close as possible to a syzygy one. It is also true that, in the case of a syzygy solution, the orbit of one particle crosses the line of the other two and can not be tangent to this line in the transition point. Finally we prove that the set of initial conditions leading to non-collinear syzygy solutions is non-empty and open.     

Dziobek’s configurations in restricted problems and bifurcation

We consider some questions on central configurations of five bodies in space. In the first one, we get a general result of symmetry for the restricted problem of n+1 bodies in dimension n-1. After that, we made the calculation of all c.c. for n=4. In our second result, we extend a theorem of symmetry due to [Albouy, A. and Libre, I.: 2002, Contemporary Math. 292, 1-16] on non-convex central configurations with 4 unit masses and an infinite central mass. We obtain similar results in the case of a big, but finite central mass. Finally, we continue the study by [Schmidt, D.S.: 1988, Contemporary Math. 81 ] of the bifurcations of the configuration with four unit masses located at the vertices of a equilateral tetrahedron and a variable mass at the barycenter. Using Liapunov-Schmidt reduction and a result on bifurcation equations, which appear in [Golubitsley, M., Stewart, L. and Schaeffer, D.: 1988, Singularties and Groups in Bifurcation Theory, Vol. II, Springer-Verlag, New York], we show that there exist indeed seven families of central configurations close to a regular tetrahedron parameterized by the value of central mass.     

Stability of motion in the Sitnikov 3-body problem

We study the stability of motion in the 3-body Sitnikov problem, with the two equal mass primaries (m ₁ = m ₂ = 0.5) rotating in the x, y plane and vary the mass of the third particle, 0 ≤ m ₃ < 10⁻³, placed initially on the z-axis. We begin by finding for the restricted problem (with m ₃ = 0) an apparently infinite sequence of stability intervals on the z-axis, whose width grows and tends to a fixed non-zero value, as we move away from z = 0. We then estimate the extent of “islands” of bounded motion in x, y, z space about these intervals and show that it also increases as |z| grows. Turning to the so-called extended Sitnikov problem, where the third particle moves only along the z-axis, we find that, as m ₃ increases, the domain of allowed motion grows significantly and chaotic regions in phase space appear through a series of saddle-node bifurcations. Finally, we concentrate on the general 3-body problem and demonstrate that, for very small masses, m ₃ ≈ 10⁻⁶, the “islands” of bounded motion about the z-axis stability intervals are larger than the ones for m ₃ = 0. Furthermore, as m ₃ increases, it is the regions of bounded motion closest to z = 0 that disappear first, while the ones further away “disperse” at larger m ₃ values, thus providing further evidence of an increasing stability of the motion away from the plane of the two primaries, as observed in the m ₃ = 0 case.     

Superhyperbolic expansion,noncollision singularities and symmetry configurations

A large class of symmetry solutions of the Newtonian n-body problem cannot end in a noncollision singularity nor expand faster than any constant multiple of time.Following a suggestion from Christian Marchal, we extended the original theme of this essay to include superhyperbolic motion. The work of D. Saari was supported by NSF Grant ISI 9103180; the work of F. Diacu was supported by NSERC Grant 3-48376     

The collinear central configuration of n equal masses

Moulton's Theorem says that given an ordering of masses m ₁, ..., m _nthere exists a unique collinear central configuration. The theorem allows us to ask the questions: What is the distribution of n equal masses in the collinear central configuration? What is the behavior of the distribution as n → ∞? These questions are due to R. Moekel (personal conversation). Central configurations are found to be attracting fixed points of a flow — a flow we might call an auxiliary flow (in the text it is denoted F(X)), since it has little to do with the equations of motion. This flow is studied in an effort to characterize the mass distribution. Specifically, for a collinear central configuration of n equal masses, a bound is found for the position of the masses furthest from the center of mass. Also some facts concerning the distribution of the inner masses are discovered.     

Symmetry, bifurcation and stacking of the central configurations of the planar 1+4 body problem

In this work we are interested in the central configurations of the planar $1+4$ body problem where the satellites have different infinitesimal masses and two of them are diametrically opposite in a circle. We can think of this problem as a stacked central configuration too. We show that the configurations are necessarily symmetric and the other satellites have the same mass. Moreover we prove that the number of central configurations in this case is in general one, two or three and, in the special case where the satellites diametrically opposite have the same mass, we prove that the number of central configurations is one or two and give the exact value of the ratio of the masses that provides this bifurcation.