Central configurations in planar nbody problem with equal masses for \(n=5,6,7\)
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Abstract
We give a computerassisted proof of the full listing of central configuration for nbody problem for Newtonian potential on the plane for \(n=5,6,7\) with equal masses. We show all these central configurations have a reflective symmetry with respect to some line. For \(n=8,9,10\), we establish the existence of central configurations without any reflectional symmetry.
Keywords
Central configurations Symmetries Interval arithmetic Krawczyk operator1 Introduction
A central configuration, denoted as CC, is an initial configuration \((q_1, \ldots , q_n)\) in the Newtonian nbody problem, such that if the particles were all released with zero velocity, they would collapse toward the center of mass c at the same time. In the planar case, CCs are initial positions for periodic solutions which preserve the shape of the configuration. CCs play also an important role in the study of the topology of integral manifolds in the nbody problem (see Moeckel and the references given there).

finding all CCs in the nbody problem on the plane (\(d = 2\)) with equal masses and

showing that each CC has a line of reflection symmetry.
1.1 The state of the art
The listing (apparently full for \(n \leqslant 9\) ) of central configurations with equal masses was given by Ferrario (2002) in unpublished notes for \(n \in \{3,\ldots ,10\}\) and by Moeckel for \(n \leqslant 8\). For \(n=4\), it was shown by Albouy (1995) that all CCs have some reflectional symmetry and later in Albouy (1996) with computer assistance the full list of central configurations was given. From numerical simulations (see for example Moeckel; Ferrario 2002), it is apparent that all CCs with equal masses have some reflectional symmetry for \(n=5,6,7\). Moeckel has found numerically some CCs without any symmetry for \(n=8\). Also for \(n=9\) Simo (2018) has found 2 families, nonequivalent, and without any symmetry. Some CCs without symmetry for \(n=10\) can be found also in Ferrario (2002).
The investigations of central configurations for equal masses are a subcase of more general problem of central configurations with arbitrary positive masses. The general conjecture of finiteness of central configurations (relative equilibria) in the nbody problem is stated in Wintner (1941) and appears as the sixth problem of Smale’s eighteen problems for the 21st century (Smale 1998). There are many works on the existence of some particular central configurations. Here we discuss only those papers which aim to more general statement about all CCs. The two most important works are Hampton and Moeckel (2005) and Albouy and Kaloshin (2012). In Hampton and Moeckel (2005), the finiteness of CC for \(n=4\) for any system of positive masses was proved with computer assistance. In Albouy and Kaloshin (2012), for \(n=4\) problem the finiteness of CC was proven without computer assistance. In the same paper, the finiteness for \(n=5\) was proven for arbitrary positive masses, except perhaps if the 5tuple of positive masses belongs to a given codimension 2 subvariety of the mass space. It is interesting to notice that the equal masses case treated in our paper belongs to this subvariety. For the spatial 5body problem, Moeckel (2001) established the generic finiteness of Dziobek’s CCs (CCs which are not planar). A computerassisted work by Hampton and Jensen (2011) strengthens this result by giving an explicit list of conditions for exceptional values of masses. A common feature of these works is that they give a quite poor estimate for the maximum number of central configurations. In this context, it is worth to mention the work of Simo, based on extensive numerical studies. In Simo (1978), he gives the number of CCs for all possible masses for \(n=4\).
In Lee and Santoprete (2009), the spatial 5body problem with equal masses was considered. A complete classification of the isolated central configurations of the 5body problem was given (this includes also planar isolated CCs). The approach has a numerical component; hence, it cannot be claimed to be fully rigorous. Also the proof does not exclude the possibility that a higherdimensional set of solutions exists. On the other hand, the existence of identified isolated CC has been proven using the Krawczyk’s operator, i.e., a tool from interval arithmetic we also use. Kotsireas (see Kotsireas 2000 and references given there) considers the 5body problem with equal masses. He gives computerassisted proof of a full list of all such configurations and shows that each of them posses some reflectional symmetry.
The abovementioned works study the polynomial equations derived from the equations for CC using the (real or complex) algebraic geometry tools. In contrast, we take a different approach: we use standard interval arithmetic tools; hence, in principle we can treat also other potentials which cannot be reduced to polynomial equations.
1.2 The main results
Theorem 1
There exist only a finite number of various types of CCs, for \(n=5,6,7\) the planar nbody Newtonian problem with equal masses. They are listed in Supplementary Material. Any CC can be obtained from one of them by suitable composition of translation, rescaling, rotation, reflection and permutation of bodies. Moreover, each of these central configurations has some reflectional symmetry.
Theorem 2
For \(n=8,9,10\) in the planar nbody Newtonian problem with equal masses, there exist CCs without any line of reflectional symmetry. They are listed in Supplementary Material.
In the case of equal masses, one can consider equivalences in two different ways: either one passes from a solution to another one by rotation (scaling is already taken into account) or one can also add permutations and reflections. For instance, for 4 bodies in the first criterion of equivalence there are 50 classes [see numerical work by Simo (1978)], while in the second only 4 classes. In this paper, we use this second criterion for equivalence.
Let us briefly describe our method. This is basically a brute force approach using standard interval arithmetic tools. Throughout the paper, we will use often box or \(cube \) to describe a set which is a product of intervals (some of them can be degenerated). The interval arithmetic allows to evaluate elementary functions on the box in a single call, i.e., the box is returned containing the true result for all points in the argument box (see for example Moore 1966; Neumeier 1990). When looking for CCs, we explore the whole configuration space (modulo some a priori bounds), and it is surprising that the most demanding part is to exclude the possibility of the existence of CC in a given box. Once we are ‘very close’ to an isolated CC, it is relatively easy to establish its existence and local uniqueness using the Krawczyk (1969) operator. The additional difficulty is that the potential contains singularity, which introduces some noncompactness in the domain to be covered. Our algorithm, which is more or less a binary search algorithm, scales poorly with n — this is the dimensionality curse (see Traub et al. 1988), which means that the complexity of our algorithm grows exponentially with n. For example, assume that we can decide if a box in the configuration space contains some CC, only when its diameter is less than \(10^{2}\) in each direction. Then adding a new body in \([1,1]\times [1,1]\) multiplies the number of boxes to be examined by \((2/10^{2})^2 = 4\cdot 10^4\). For this reason, we were not able to obtain a rigorous listing of CCs for \(n=8\). Note that for \(n=5\) the computations were done in 24 seconds, for \(n=6\) it took about one hour to get the result, while for \(n=7\) we needed almost a hundred hours (see Sect. 7.3 for more technical data).
For any CC from the listing in Moeckel or Ferrario (2002) for \(n\geqslant 8\) we have found no difficulty proving its existence and local uniqueness. In particular, we confirmed the existence of nonsymmetric planar CCs for \(n=8,9,10\) (see Theorem 2).
The paper is organized as follows. In Sect. 2 we recall the equations for the central configurations and their basic properties. In Sect. 3, we derive several a priori bounds for CC, so that we obtain a compact domain for our search algorithm. In Sect. 4, we discuss various tests which are used to show that a given box does not contain any CC. In Sect. 5, we derive a reduced set of equations for CC. This is necessary because to apply the Krawczyk’s method we need to ensure that the system of equations does not contain any degeneracies, which are due to symmetries of the original system of equations for CCs. In Sect. 6, we give assumptions and basic ideas concerning the computerassisted proofs of main Theorems 1 and 2 and we explain the Krawczyk’s method. Details of the algorithm are described in Sect. 7. In Sect. 8, we present an attempt to minimize the dependency problem in interval arithmetic in the evaluation of the gravitational force. In electronic supplementary material, we give an output of the program establishing Theorems 1 and 2 (also for \(n= 3,4\)) and pictures of all CCs found.
2 Equations for central configurations
In the paper by z, where \(z=(z_1,\ldots ,z_d) \in \mathbb {R}^d\), we will denote the euclidean norm of z, i.e., \(z=\left( \sum _{i=1}^d z_i^2\right) ^{1/2}\) and by (x, y), where \(x,y \in \mathbb {R}^d\), we will denote the standard scalar product, i.e., \((xy)=\sum _{i=1}^d x_iy_i\). We will often use \(z^2:=(zz)\).
In the planar case if we consider the bodies in a rotating system (with the center of mass at the origin) with constant angular velocity \(\omega =\sqrt{\lambda }\), the physical meaning of (2) is obvious: the gravitational attraction is compensated by the centrifugal force and the central configurations are fixed points in the rotating frame (see Moeckel and the references given there).
2.1 Some identities and conservation laws
But (6) and (7) can be seen also as the consequences of the symmetries of Newtonian nbody problem. According to Noether’s Theorem, by the translational symmetry we have a conservation of momentum, which is equivalent to (6), while the rotational symmetry implies the conservation of angular momentum, which is implied by (7).
2.2 Moment of inertia of central configurations
The important role of the moment of inertia in the investigation of central configurations is well known. In our context, it plays a crucial role in stating some a priori bounds for central configurations. We present, with proofs, some wellknown results on moment of inertia taken from the notes by Moeckel (2014) and the paper of Albouy and Kaloshin (2012).
Definition 1
Lemma 3
Proof
Lemma 4
Proof
3 A priori bounds for central configurations

two or more bodies might be arbitrary close to a collision,

some bodies might be arbitrary far from the origin.
3.1 Lower bound on the distances
It is well known that central configurations are away from the collision set (see Shub 1970 or Moeckel 2014, Prop. 15). However, in these works no quantitative statement directly applicable to system (3) has been given. Here we develop explicit a priori bounds.
The main idea is to use \(I(q)=U(q)\) (see Lemma 4) to show that some term(s) \(m_im_j/r_{ij}\) entering U(q) dominate and cannot be balanced, when bodies are very close. Observe that using \(I(q)=U(q)\) and positivity of all terms entering U(q) allows us to escape the discussion of large terms on the rhs in the system (3), which might cancel out or not etc. This is not the case in the framework of Albouy and Kaloshin (2012), where complex configurations and even complex masses have been considered, hence the positivity of I and U is lost.
Lemma 5
Proof
Since for any \(1 \leqslant i \leqslant n:q_i \leqslant R\), thus \( I(q) = \sum _{i} m_iq_i^2 \leqslant \sum _{i} m_iR^2 = MR^2. \) \(\square \)
Theorem 6
Proof
Below we establish a lower bound on the radius of ball centered at 0 and containing a central configuration in the case of equal masses.
Theorem 7
Proof
The size of a minimal ball containing all normalized central configurations with \(M=1\) for several n’s for equal masses case
n  \(\root 3 \of {\frac{n1}{4n}}\)  R 

3  0.550321  0.577350 
4  0.572357  0.620813 
5  0.584804  0.650513 
6  0.592816  0.672798 
In the next theorem, we do not assume that all masses are equal.
Theorem 8
Proof
Observe that the above estimate is optimal, because it is realized for the equilateral triangle for \(n=3\) and a tetrahedron (non planar CC) for \(n=4\). From the above theorem, we obtain the following lower bound for the size of a central configuration. Contrary to Theorem 7, we do not assume that all masses are equal.
Theorem 9
Assume that \(M=1\) and q is a normalized central configuration and \(q_i \leqslant R\) for \(i=1,\ldots ,n\). Then \(R \geqslant 1/2\).
Proof
For the proof by contradiction assume that \(R < 1/2\). Then for all pairs \(r_{ij} <1\). From Theorem 8, it follows that q is not central configuration. \(\square \)
3.2 The upper bound on the size of central configuration
The goal of this section is to give the upper bounds for the size of the central configuration. This time we exploit the fact that if the forces are bounded, then large \(q_i\)’s on the left hand side of the system (3) cannot be balanced. The obvious difficulty with the realization of this idea is: we can have a group of bodies with large norms which are close to each other in the central configuration, which produce large terms on rhs of the system (3). To overcome this, we consider clusters of points far from the origin and the resulting force on it. In such situation, mutual interactions between bodies in the cluster cancel out.
Lemma 10
Proof

\(i_0 \in \mathcal {C}\)

if \(j \in \mathcal {C}\) and \(q_k  q_j \leqslant \varepsilon \), then \(k \in \mathcal {C}\).
From Lemma 10, we obtain the following estimate on the size of any central configuration.
Theorem 11
Proof
For \(n\leqslant 10\) and the equal masses, the above estimate appears to be an overestimation. The largest possible size of CC found was slightly above 1 and is considerably smaller than the one established in the above theorem.
4 Exclusion tests for CCs
Assume that we have an interval set D (i.e., a box, a product of intervals) in which we would like to exclude the existence of CC. We do not assume that \(D \subset \,\mathrm{dom}\,F\) [see (5)] and this is an important point. The a priori estimates discussed in Sect. 3 allow to exclude D iff there is no point in D which satisfies the obtained bounds.
In the following, we discuss other exclusion tests.
4.1 Checking for zeros
 0.
given box D in the configuration space, such that \(D \subset \,\mathrm{dom}\,F_i\) (i.e., the ith does not have a collision with other bodies). Let \(D_i=\{q_i \,\, q \in D\}\).
 1.
compute the interval enclosure of \(f_i(q)\) for \(q \in D\), denoted by \(\langle f_i(D)\rangle \),
 2.
if \(D_i \cap \frac{1}{m_i}\langle f_i(D)\rangle =\emptyset \) [compare Eq. (3)], then D does not contain any normalized central configuration
 3.
if \(D_i \cap \frac{1}{m_i}\langle f_i(D)\rangle \ne \emptyset \), then we define \(D^{\mathrm {ref}}=\{q \in D\, \, q_i \in \frac{1}{m_i}\langle f_i(D)\rangle \}\).
4.2 The cluster of bodies—checking for zeros test
If it is not satisfied, then we conclude that D does not contain any CC. Again let us stress that the set D might contain some collisions, and this test is still applicable.
4.3 The cluster of bodies—test based on moment of inertia and potential
The important point is that we can compute the infimum in \(U_{\mathcal {C},Z}\) even if the set Z contains collisions. It makes sense to take as \(\mathcal {C}\) a cluster of close points (containing possible collisions and near collisions), so that there is no collision between bodies in \(\mathcal {C}\) and its complement. In such case, \(F_{\mathcal {C},Z}\) will be finite.
We have the following criterion for nonexistence of CC in Z:
Lemma 12
Proof
Observe that if \(\mathcal {C}=\{1,\ldots ,n\}\) the above lemma is reduced to checking whether \(\inf _{q \in Z} U(q) > \sup _{q \in Z} I(q) \).
5 The reduced system of equations for CC on the plane
5.1 Nondegenerate solutions of full and reduced systems of equations
Following Moeckel (2014), we state the definition.
Definition 2
We will say that a normalized central configuration \(q=(q_1,\ldots ,q_n)\) is nondegenerate if the rank of \(D\!F(q)\) is equal to \(dn\dim SO(d)\). Otherwise the configuration will be called degenerate.
The idea of the above notion of degeneracy is to allow only for the degeneracy related to the rotational symmetry of the problem, because by setting \(\lambda =1\) in (2) and keeping the masses fixed we removed the scaling symmetry.
5.2 The reduced system on the plane
We consider a planar case here, i.e., \(d=2\). The fact that the system of Eq. (3) is degenerate (each solution give rise to a circle of solutions) make this system not amenable for the use of standard interval arithmetic methods (see for example the Krawczyk operator discussed in Sect. 6.3) to rigorously count all possible solutions. We need to kill the SO(2)symmetry and then hope that all solutions will be nondegenerate. In this section, we show how to reduce the system (3) to an equivalent system amenable to the interval analysis tools.
The next theorem addresses the question: whether from the reduced system (32–34) we obtain the solution of (3)?
Theorem 13
Assume \(d = 2\). If \((q_1,\ldots ,q_{n1},q_n(q_1,\ldots ,q_{n1}))\) is a solution of the reduced system (32–34) and \(x_k \ne x_n\), then it is a normalized central configuration, i.e., it satisfies (3).
Proof
The system (32–33) contains \(2(n1)1\) equations in \(2(n1)\) variables and has O(2) symmetry (i.e., rotations around origin and reflection symmetries with respect to lines passing through the origin map solutions of this system into itself). In order to obtain a system with the same number of equations and variables, we can impose additional condition leading to the removal of \(y_k\) variable, so that the rotational symmetry will be broken. Obviously in the above setting we could drop the equation for \(x_k\) and we will obtain an analogous result.

we fix some hyperplane H, in the reduced (by the center of mass condition) configuration space \(\mathbb {R}^{2(n1)}\), so that H is transversal to the action SO(2) and k is such that \(v_k \in \{x_k,y_k\}\) can be computed in terms of other variables. This will induce an embedding, \(J_{k}:\mathbb {R}^{2(n1)\dim SO(2)} \rightarrow H\).
Theorem 14
Proof
The first part is obvious in view of Theorem 13 and condition \(x_k \ne x_n\) implies that it is a central configuration. The maximum rank in the reduced system gives the nondegeneracy of the configuration in the sense of Definition 2. \(\square \)
 we set \(k=2\) and we eliminate variable \(y_2\) by setting$$\begin{aligned} y_2=y_1, \end{aligned}$$(38)
 we set \(k=n1\) and we eliminate variable \(y_{n1}\) by setting$$\begin{aligned} y_{n1}=0. \end{aligned}$$(39)
Now, consider the condition (39). If we setup our computations so that \(q_{n1}\) body maximizes the distance from the origin for all bodies, then we have (40) satisfied, otherwise \(q_n\) will be further from zero. This observation does not prove that if q is a nondegenerate CC in the sense of Definition 2, then it is also a nondegenerate solution of the reduced system, but this appears to happen in our rigorous computation of central configurations so far.
In our proof, since we apply the Krawczyk method (see Sect. 6.3) to obtain the solutions of the reduced system, all CCs whose existence we establish are nondegenerate in the sense of Definition 2.
6 On the computerassisted proof
6.1 Equal masses case, the reduction in the configuration space for CCs

\(q_{n2}=(R,0)\) is the furthermost body from the origin

\(q_0\) is the leftmost with nonnegative ycoordinate

\(q_1\) has the smallest y coordinate

all other bodies have their x coordinates in the order of increasing/decreasing indices.
Times of asynchronous computations in minutes for different orderings (the computations were carried out on the computer Intel Core i75500U CPU @ 2.40 GHz \(\times \) 4 with 8 GB RAM; a single thread was used)
n  Increasing  Decreasing 

4  0.027170  0.026153 
5  3.369141  2.615399 
6  1103.138988  924.083085 
6.2 Outline of the approach
In the algorithm, we look for all zeros of the reduced system (41, 42), which under assumption \(x_{n1}\ne x_{n2}\) by Theorem 13 is equivalent to (3). For our algorithm, proving an existence of locally unique solution in some box is as important as proving that in a given box there is no solution.
For proving of the existence of the locally unique solution, we use the Krawczyk operator applied to the system (41, 42). To rule out the existence of solution, we use the exclusion tests discussed in Sect. 4 and also the Krawczyk operator.
The symmetry of CCs is established by proving the uniqueness in a suitable symmetric box (see Sect. 7.2).
6.3 The Krawczyk operator
The Krawczyk operator (Alefeld 1994; Krawczyk 1969; Neumeier 1990) is an interval analysis tool to establish the existence of unique zero for the system of n nonlinear equations in n variables. Below we briefly explain how the Krawczyk operator is derived, as it appears mysterious and little known outside the interval arithmetic community.
6.3.1 Motivation, heuristic derivation
6.3.2 The Krawczyk method

\([x] \subset \mathbb {R}^n\) be an interval set (i.e., product of intervals),

\(x_0 \in [x]\). Typically \(x_0\) is chosen to be midpoint of [x], we will denote this by \(x_0=mid([x])\).

\(C \in \mathbb {R}^{n \times n}\) be a linear isomorphism.
Theorem 15
 1.
If \(x^* \in [x]\) and \(F(x^*) =0\), then \(x^* \in K(x_0,[x],F) \).
 2.
If \(K(x_0,[x],F) \subset \mathrm{int}\,[x]\), then there exists in [x] exactly one solution of equation \(F(x)=0\). This solution is nondegenerate, i.e., dF(x) is an isomormophism.
 3.
If \(K(x_0,[x],F) \cap [x]=\emptyset \), then for all \(x \in [x]\) \(F(x) \ne 0\).
Observe that point 2. in the above theorem gives us the way to establish the existence of unique zero of F in [x], while point 3. rules out the existence of zero in [x], i.e., in the terminology of previous section this is the exclusion test. When [x] is close to a zero of F then \({<}F([x]){>}\) the evaluation of F([x]) in the interval arithmetic might produce such overestimates that \(0 \in {<}F([x]){>}\), while the Krawczyk operator will rule out the existence of 0 of F in [x]. This is in fact quite common phenomenon.
 0.
given \([x]_0 \subset \mathbb {R}^n\)
 1.
compute \([y] = K(mid([x]_k),[x]_k,F)\)
 2.
if \([y] \subset \mathrm{int}\,[x]_k\), then return success, a unique solution in \([x]_0\) was found
elseif \([x]_k \subset [y]\), then return failure
elseif set \([x]_{k+1}:=[y] \cap [x]_k\) and goto 1.
In our context, the only weakness of the Krawczyk operator is that it requires the sets of the diameter in each coordinate directions to be smaller than \(10^{2}\) to give us something. Above that threshold, we usually have \([x] \subset K(mid([x]),[x],F)\) and the Krawczyk method is useless.
7 The algorithm
The algorithm runs in the reduced configuration space which is a subset of \(\mathbb {R}^{2(n1)1}\), i.e., a configuration is represented by a point \((x_0, y_0,\ldots , x_{n3}, y_{n3}, x_{n2})\). Physically, we interpret such a configuration as \(n1\) bodies with \(q_i=(x_i, y_i)\) for \(i = 0, \ldots , n3\) and \(q_{n2}=(x_{n2},0)\). From (44) we obtain \(q_{n1}\) the position of the last body.
 1.
n—the number of bodies
 2.
some cube in the reduced configuration space.
The program is divided into two stages: searching finds solutions and testing tests them to distinguish different CC and to find the kind of symmetry (if any exists).
7.1 Searching stage
 (I)
if there is no solution in the cube return 0;
 (II)
if there is unique solution in the cube return 1;
 (III)
otherwise bisect the longest edge and recursively run the procedure for both parts;
 (IV)
return result.
7.1.1 Details and optimizations
 1.
checkAprioriBounds(bodies)—tests if bodies satisfy a priori bounds (see Theorem 11);
 2.
checkUEqI(bodies)—if there is no collision in bodies, tests if \(U(q) == I(q)\) (see Lemma 4);
 3.
 4.
 5.
checkZero(bodies, i)—computes functions of vector field [see Eq. (4)] and tests if it is possible to have \(F(q_0, \ldots , q_{N1}) = 0\) as discussed in Sect. 4.1.
Comparison of execution times for 5 bodies for different thresholds, where we start Krawczyk’s method (the computations were carried out on the computer Intel Core i75500U CPU @ 2.40 GHz \(\times \) 4 with 8 GB RAM; a single thread was used)
bias  Time m:s.d  Failed  No zero 

1e−4  4:27.65  0  32,095 
1e−3  3:09.13  0  25,846 
5e−3  2:15.54  232  32,580 
1e−2  1:59.44  12,886  48,446 
1e−1  3:29.29  585,151  98,946 
7.2 Testing stage
The main goal of this stage is to identify distinct solutions. Additionally, we check the symmetry of solutions. In this stage, we consider solutions in the full system.
 (1)
the only difference is the ordering of the bodies,
 (2)
the boxes defining them have nonempty intersection, having been obtained in different series of partitions.
It may happen that there exists a solution in the set unionBodies, but the set is too small to prove this using the Krawczyk operator, thus we ‘inflate’ it and retry the proof in the bigger set (the function blowUp(unionBodies)).
In line 2, we calculate the parameters of the possible reflection symmetry line, but the symmetry tested contains also a permutation of bodies, we construct a configuration symBodies considering all possible permutations of bodies. Note that lines 5–12 in the function checkSym are identical, up to the variable names, to lines 3–11 in theSameSolutions.
7.3 Technical data
Comparison of execution times for different number of bodies
No bodies  No CCs  Total no of CPUseconds  Elapsed time h:m:s.d  Average percentage of the CPU 

3  2  0.05  0:00.06  240 
4  4  3.07  0:00.69  666 
5  5  203.82  0:24.84  1023 
6  9  42,430.04  59:51.59  1203 
7  14  8,490,959.77  98:56:00.00  2531 
8 Minimizing dependency problem in gravitational force evaluation
8.1 Estimates for \(x^a / r^b\) and \(y^a / r^b\)
Note, that there is still a lot of room for further optimization, but for now only this version is implemented in the program.
Notes
Supplementary material
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