An analytical proof on certain determinants connected with the collinear central configurations in the n-body problem

In this paper we give a short analytical proof of the inequalities proved by Albouy–Moeckel through computer algebra, in the cases $n=5$ and $n=6$ . These inequalities guarantee that, in the $n$ -body problem, the family of mass vectors making a given collinear configuration a central configuration is 2-dimensional. The induction techniques here may be used to prove the inequalities for general $n$ with more subtle estimation but currently the inequalities still remains unproved for $n\ge 7$ .     

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.      

Non-integrability of the equal mass n-body problem with non-zero angular momentum

We prove an integrability criterion and a partial integrability criterion for homogeneous potentials of degree ?1 which are invariant by rotation. We then apply it to the proof of the meromorphic non-integrability of the n-body problem with Newtonian interaction in the plane on a surface of equation (H, C) = (H ₀, C ₀) with (H ₀, C ₀) ?? (0, 0) where C is the total angular momentum and H the Hamiltonian, in the case where the n masses are equal. Several other cases in the 3-body problem are also proved to be non integrable in the same way, and some examples displaying partial integrability are provided.     

Finiteness of spatial central configurations in the five-body problem

We strengthen a generic finiteness result due to Moeckel by showing that the number of spatial central configurations of the Newtonian five-body problem with positive masses is finite, apart from some explicitly given special cases of mass values.     

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.     

New central configurations for the planar 5-body problem

In this paper we show the existence of three new families of planar central configurations for the 5-body problem with the following properties: three bodies are on the vertices of an equilateral triangle and the other two bodies are on a perpendicular bisector.     

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.     

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)}.
       
    

     

Some problems on the classical n-body problem

Our idea is to imitate Smale’s list of problems, in a restricted domain of mathematical aspects of Celestial Mechanics. All the problems are on the n-body problem, some with different homogeneity of the potential, addressing many aspects such as central configurations, stability of relative equilibrium, singularities, integral manifolds, etc. Following Steve Smale in his list, the criteria for our selection are: (1) Simple statement. Also preferably mathematically precise, and best even with a yes or no answer. (2) Personal acquaintance with the problem, having found it not easy. (3) A belief that the question, its solution, partial results or even attempts at its solution are likely to have great importance for the development of the mathematical aspects of Celestial Mechanics.     

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.     

Improbability of collinear solutions in the n-body problem with generalized attraction law

It is shown that in the n-body problem with generalized attraction law (inverse (α + l )-power of the distance, α > 0) the set of initial conditions which lead to collinear motion is of Lebesgue measure zero and nowhere dense with respect to the set of initial conditions that define solutions in R³ or R².     

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.     

On the variational approach to the periodic n-body problem

This expository paper gathers some of the results obtained by the author in recent works in collaboration with Davide Ferrario and Vivina Barutello, focusing on the periodic n-body problem from the perspective of the calculus of variations and minimax theory. These researches were aimed at developing a systematic variational approach to the equivariant periodic n-body problem in the two and three-dimensional space. The purpose of this paper is to expose the main problems and achievements of this approach. The material here was exposed in the talk that given at the Meeting CELMEC IV promoted by SIMCA (Società italiana di Meccanica Celeste).     

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.     

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.     

The equilibrium configurations of the three-dipole problem

Studying the general kinetic behaviour of a charged particle in the space of three celestial bodies electromagnetic field we give here for the first time the equilibrium configurations of the dynamical system with all the variational equations and integrals. Also we develop the procedure we follow for locating the planar and the three-dimensional equilibrium points demonstrating an extensive numerical investigation of them by giving their number under the influence of the variation of the magnetic field and in all the cases we list their coordinates and energy.