Topological bifurcations of spatial central configurations in the N-body problem

We study topological bifurcations of classes of spatial central configurations (s.c.c.) from the following highly symmetrical families: two nested regular tetrahedra, octahedra and cubes, two nested rotated regular tetrahedra and two dual regular polyhedra for 14 bodies. We prove the existence of local and global topological bifurcations of s.c.c. from these families. We seek new classes of s.c.c. by using equivariant bifurcation theory. It is worth pointing out that the shapes of the bifurcating families are less symmetrical than the shapes of the considered families of s.c.c.


Introduction
One of the most important problems of Celestial Mechanics is the problem of classification of all central configurations (c.c.) in the N -body problem. It has a long history and has been studied by various mathematicians. Collinear configurations for 3 bodies (see Euler 1767) discovered by Euler and an equilateral triangle with arbitrary positive masses at the vertices (see Lagrange 1772) found by Lagrange are the first known c.c. Examples of c.c. provide us with highly symmetrical configurations with restrictions on the masses. By using the symmetries of configurations and imposing restrictions on the masses the system of equations for c.c. can be reduced to the simpler one. We cannot find in the literature many of c.c. with an irregular shape, there are only a few non-symmetric ones known in the history of this problem. For example, Meyer and Schmidt (1988a, b)  In our paper we consider the following highly symmetrical families: • a spatial central configuration of two nested regular tetrahedra, octahedra and cubes studied by Corbera and Llibre (2008), • a spatial central configuration of two nested rotated regular tetrahedra studied by Corbera and Llibre (2009), • a spatial central configuration of two dual regular polyhedra for 14 bodies studied by Corbera et al. (2014).
Note that the studied families consist of non-collinear s.c.c. It is known that the isotropy group of any non-collinear spatial central configuration is trivial. Moreover, the collinear c.c. are non-degenerate as critical orbits of an appropriate potential (see Pacella 1987) and thus there are no topological bifurcations from these c.c. [see Theorem 3.1]. The existence of symmetric central configurations in general was considered in the recent paper of Montaldi (2015) where the uniform approach was given by using some variational methods in symmetric case. The principal conclusion of it is the existence of a central configuration of any symmetry type and for any symmetric distribution of masses. In particular, Montaldi proved the existence of families of s.c.c. mentioned above for any symmetric choice of masses. We study topological bifurcations from those families and use the precise formulas for positions and masses provided in the above listed papers of Corbera, Llibre and Pérez-Chavela. There are numerous reasons for studying c.c., the most important ones are the following: c.c. generate the unique explicit solutions in the N -body problem; planar central configurations (p.c.c.) also give rise to families of periodic solutions; c.c. are important in analysing total collision orbits (see Saari 1980;Wintner 1941); the hypersurfaces of constant energy in a level set of the angular momentum change their topology exactly at the energy level sets which contain c.c. (see Smale 1970).
C.c. are invariant under homotheties and rotations because of the homogeneity of the potential and the fact that the potential depends only on the mutual distances between the particles and not on their positions. Besides, Newton's equations are of gradient nature which allows us to apply methods which benefit from this.
In Kowalczyk (2015) we used equivariant versions of classical topological invariants such as the degree for equivariant gradient maps (see Balanov et al. 2006;Balanov et al. 2010;Gȩba 1997;Rybicki 2005) and the equivariant Conley index (see Bartsch 1993;Floer 1987;Gȩba 1997) to formulate abstract theorems giving necessary and sufficient conditions for the existence of topological bifurcations (local and global ones) in the vicinity of a given family (a so-called trivial family) of solutions of equivariant gradient equations [see Theorems 3.1, 3.2 and 3.3 of Kowalczyk (2015) in the case of global bifurcation and Theorems 3.1, 3.5 and 3.6 of Kowalczyk (2015) for local one]. So we proved some abstract results in equivariant bifurcation theory and applied them to the study of bifurcations of new families of p.c.c. [see Theorems 4.1 and 4.2 of Kowalczyk (2015)]. This approach using equivariant theory has been used in Maciejewski and Rybicki (2004) and Pérez-Chavela and Rybicki (2013). Moreover, we showed that the new families are less symmetrical than those trivial ones. Now, applying the above-mentioned theorems, we shall study classes of s.c.c. which are treated as SO(3)-orbits of critical points of SO(3)-invariant potentials. Similarly as in Kowalczyk (2015), we shall also address the shapes of the found families of s.c.c. and thus we prove that the bifurcating families have poorer symmetries or less regular shapes. Namely, we exclude that the bifurcating families of s.c.c. have the same type of symmetry as the trivial ones. We emphasise that the information about the shapes of new families of s.c.c. is only local.
It is worth pointing out that in the planar case the action of the symmetry group SO(2) is free, so a natural thing to consider is the quotient space /SO(2) of the action of the group SO(2) on the configuration space (see Lei and Santoprete 2006;Meyer 1987;Meyer and Schmidt 1988a, b). In the spatial case, the action of the symmetry group SO(3) is not free, i.e. there is a non-trivial isotropy group appearing in the configuration space. On the other hand, the only s.c.c. which provide non-trivial isotropy groups are collinear and, as we mentioned above, there are no topological bifurcations from the families of collinear s.c.c. Thus we get that the quotient space /SO(3) of the action of the group SO(3) on is a manifold if we exclude collinear s.c.c. from the configuration space. But Theorem 3.2 of Kowalczyk (2015) which we apply to seek bifurcations (global ones) was proved for orthogonal representations. We do not have the corresponding global bifurcation theorem on manifolds and so we study instead of the quotient space /SO(3) to study the existence of global bifurcations. This paper is organised as follows: in Sect. 2 we introduce the spatial N -body problem and investigate s.c.c. as the SO(3)-orbits of solutions of the equation (1.1) Next, we apply our abstract results of Kowalczyk (2015) to the spatial N -body problem and we prove the existence of topological bifurcations from the following families of s.c.c.: (2.4), (2.5), (2.6), (2.7) and (2.8). Tedious computations of this section are aided by the algebraic processor MAPLE™. Section 3 contains some facts of equivariant topology and theorems giving necessary and sufficient conditions for the existence of local and global bifurcations from a given family of s.c.c. It is worth pointing out that in this paper we study topological bifurcations [local and global ones, see Definition 3.1]. Namely, in the case of local ones, we consider known family of solutions of the Eq. (1.1) (the so-called trivial family of solutions) and seek non-trivial solutions nearby. In the case of global ones, we seek connected sets of non-trivial solutions nearby the trivial ones which satisfy some additional condition [see Definition 3.1(2)]. The notion of a topological bifurcation is different from the notion of a bifurcation in the sense of a change in the number of solutions of the Eq. (1.1). These two definitions, the topological bifurcation and the bifurcation in the sense of a change in the number of solutions, are independent. So, in other words, the first one can occur while the second one does not occur and inversely. To shorten the notation, throughout this paper, we will write bifurcations instead of topological bifurcations.

Bifurcations of spatial central configurations
In this section we present the existence of new families of s.c.c. which bifurcate from certain known families of s.c.c. We consider N bodies of positive masses m 1 , . . . , m N in the 3dimensional Euclidean vector space, whose positions are denoted by q 1 , . . . , q N ∈ R 3 . We define the configuration space ⊂ R 3N as follows By SO(3) we will understand the group of special orthogonal matrices of order 3. From now on we will treat the space R 3N as an SO(3)-representation V, i.e. a representation of the Lie group SO(3) which is a direct sum of N copies of the natural orthogonal SO(3)representation (i.e. the usual multiplication of a vector by a matrix). Moreover, R denotes the one-dimensional trivial representation of SO(3), i.e. R = (R, 1), where 1(g) = 1 for any g ∈ SO(3). The action of SO(3) on V × R is given by By q ∈ V we mean q ∈ R 3N and, for simplicity of notation, we write gq instead of g · q.
If q ∈ V then SO(3) q = {g ∈ SO(3) : gq = q} is the isotropy group of q and the set The equations of motion of the spatial N -body problem are given by where the Newtonian potential U : and without loss of generality the gravitational constant G can be taken equal to one. We will use the symbol ∇ q U to denote the gradient of U with respect to the first coordinate.
is a central configuration of the system (2.1) if there exists a positive constant λ such thatq = −λq.
Equivalently, the following condition is fulfilled: where I : × (0, +∞) N → R given by the formula I (q, m) = 1/2 N j=1 m j |q j | 2 is the moment of inertia. Note that the centre of mass of the bodies is at the origin of the coordinate chart for any solution of the Eq. (2.2). Moreover, one can show that λ = U (q, m)/(2I (q, m)).
First, observe that the set (3) and (q, m) ∈ × (0, +∞) N . In this approach our problem of studying s.c.c. becomes a problem of studying critical SO(3)-orbits of a smooth SO(3)-invariant potentialφ : ×(0, +∞) N → R given byφ(q, m) = U (q, m)+λI (q, m). Additionally, assume there exist continuous maps w : R → and m : R → (0, +∞) N such that ∇ qφ (w(ρ), m(ρ)) = 0. Now, define a potential ϕ : × R → R by ϕ(q, ρ) = ϕ(q, m(ρ)). Thus we will consider the following equation: The family F is called the trivial family of solutions of the Eq. (2.3) and for any subset Now, we will apply Theorems 3.1 and 3.2 to prove the existence of bifurcations from the following families of s.c.c. w : (2.4), (2.5), (2.6), (2.7) and (2.8). First, observe that, according to Lemma 3.1, for any parameter ρ we have dim ker ∇ 2 q ϕ(w(ρ), ρ) ≥ dim SO(3)(w(ρ)). Additionally, parameters which satisfied the necessary condition for the existence of local bifurcation, given by Theorem 3.1, are the ones for which the strict inequality holds. Next, Fig. 1 Central configurations of two nested regular polyhedra. a Eight bodies in nested regular tetrahedra configuration. b Twelve bodies in nested regular octahedra configuration. c Sixteen bodies in nested regular cubes configuration we will use Theorem 3.2 giving the sufficient conditions for the existence of local and global bifurcations. Note that for families w considered in this paper, for any parameter ρ, we have SO(3) w(ρ) = {I}, where by the symbol I we denote the identity matrix. Thus the matrix C(w(ρ)) [see a special form of ∇ 2 q ϕ(w(ρ), ρ) given by the formula (3.1)] is zerodimensional and in particular, its Morse index is equal to zero, i.e. the number of negative eigenvalues (counting multiplicities) of the matrix C(w(ρ)). Therefore for any ρ First, we consider the case of 8 bodies and address bifurcations from the families of s.c.c. which were studied by Corbera and Llibre (2009). This type of families consists of 8 bodies with the masses located at the vertices of two nested regular tetrahedra, where the masses on each tetrahedron are equal but the masses on different tetrahedra could be different. They have proved that there are only two different classes of s.c.c. of this type, either when one of the tetrahedra is homothetic to the other one (Type I) or when one of the tetrahedra is rotated with a rotation matrix of Euler angles α = γ = 0 and β = π (Type II).
Next, we will show bifurcations from the family of Type II (see Fig. 2), in which bodies are considered with the following positions: where a is a scale factor. This configuration was also studied by Corbera et al. (2014).
To prove the above lemma we apply an analysis similar to that used in the proof of Lemma 2.1.
It remains to prove that the bifurcating families are less symmetrical. To do this, notice that we can consider a subset of the full configuration space which is invariant for the gradient flow, i.e. the set of s.c.c. of two nested rotated tetrahedra type q 1 , . . . ,q 8 , M, M, M, M, m, m, m, m ((a, M, m) for short), then studying c.c. in this set becomes a problem of studying zeros of a function F : ((0, β 1 ) ∪ (α 1 , β 2 ) ∪ (α 2 , +∞)) × (0, +∞) 2 → R given by the formula F(a, M, m) = Mg(a) + m f (a) (see Corbera et al. 2014). For the trivial family of solutions (2.5), for any a ∈ (0, β 1 ) ∪ (α 1 , β 2 ) ∪ (α 2 , +∞), we have F (a, 1, −g(a)/ f (a)) = 0. Additionally, for any a ∈ (a 1 , a 2 ) ∪ (a 5 , a 6 ) ∪ (a 7 , a 8 ) ∪  (a 11 , a 12 ), we obtain numerically that F a (a, 1, −g(a)/ f (a)) = g (a)−(g(a)/ f (a)) f (a) < 0, so by the implicit function theorem there is no bifurcation of s.c.c. of two nested rotated tetrahedra type from these segments. From what has already been proved, we conclude that the families which bifurcate from the family (2.5) from those segments are not of two nested rotated tetrahedra type and the proof is complete.
(2) There exists a connected set of s.c.c. bifurcating from the family (2.7) from the segment (ρ 2 , ρ 3 ), i.e. a global bifurcation from the segment (ρ 2 , ρ 3 ) occurs. Moreover, the bifurcating families are locally less symmetrical. Proof We first show that (ρ 1 , ρ 2 ) ∩ BIF = ∅ and (ρ 2 , ρ 3 ) ∩ GLOB = ∅. As in the proof of Theorem 2.1, applying Lemma 2.4 and Theorem 3.2, we conclude that there exists a local bifurcation parameter in (ρ 1 , ρ 2 ) and a global one in (ρ 2 , ρ 3 ). Now, notice that we can consider a subset of the full configuration space which is invariant for the gradient flow, i.e. the set of s.c.c. of two nested cubes type q 1 , . . . ,q 16 , M, M, M, M, M, M, M, M, m, m, m, m, m, m, m, m) ((ρ, M, m) for short), then studying c.c. in this set becomes a problem of studying zeros of a function F : (α, +∞) × (0, +∞) 2 → R given by the formula F(ρ, M, m) = m f (ρ) − Mb(ρ) (see Corbera and Llibre 2008). For the trivial family of solutions (2.7), for any ρ ∈ (α, +∞), so by the implicit function theorem there is no bifurcation of s.c.c. of two nested cubes type from the trivial family. It follows that the families which bifurcate from the family (2.7) are not of two nested cubes type, which proves our assertion.

Let ⊂ V be an open and SO(3)-invariant subset of an SO
Lemma 3.1 Assume that φ ∈ C 2 SO(3) ( , R), q 0 ∈ (∇φ) −1 (0) and denote by H the isotropy group of q 0 . Then the Hessian According to Lemma 3.1, we always have that dim ker We now introduce the notions of local and global bifurcations of SO(3)-orbits of solutions of the Eq. (2.3), which are preceded by some additional notation. We will denote by C(ρ 0 ) the connected component of the set cl({(q, ρ) ∈ ( × R)\F : ∇ q ϕ(q, ρ) = 0}) containing Definition 3.1 Fix parameters ρ ± ∈ R such that ρ − < ρ + .
In the spatial N -body problem we have a presence of symmetry of the Lie group SO(3). Thus we formulate the necessary and sufficient conditions for the existence of local and global bifurcations from the segment of SO(3)-orbits of solutions of the Eq. (2.3). Abstract results for an arbitrary compact Lie group G can be found in Kowalczyk (2015) [for details see Theorems 3.1, 3.2, 3.3 and 3.5 of Kowalczyk (2015)]. The necessary condition for the Theorem 3.1 Fix a parameter ρ 0 ∈ R. If ρ 0 ∈ BIF, then dim ker ∇ 2 q ϕ(w(ρ 0 ), ρ 0 ) > dim SO(3)(w(ρ 0 )).
Let us introduce the notion of a conjugacy class. Two closed subgroups H and H of SO(3) are called conjugate if there exists g ∈ SO(3) such that H = g −1 H g. Notice that conjugacy is an equivalence relation and let (H ) SO(3) denote the conjugacy class of a closed subgroup H of SO(3). In the following theorem we give the sufficient conditions for the existence of local and global bifurcations. Theorem 3.2 states that a change of the Morse indices implies the existence of a local bifurcation and a change of parity of the Morse indices guarantees the existence of a global bifurcation.