Spiderweb central configurations

In this paper we study spiderweb central configurations for the $N$-body problem, i.e configurations given by $N=n \times \ell+1$ masses located at the intersection points of $\ell$ concurrent equidistributed half-lines with $n$ circles and a central mass $m_0$, under the hypothesis that the $\ell$ masses on the $i$-th circle are equal to a positive constant $m_i$; we allow the particular case $m_0=0$. We focus on constructive proofs of the existence of spiderweb central configurations, which allow numerical implementation. Additionally, we prove the uniqueness of such central configurations when $\ell \in \{2,\dots,9\}$ and arbitrary $n$ and $m_i$; under the constraint $m_1\geq m_2\geq \ldots \geq m_n$ we also prove uniqueness for $\ell \in \{10,\dots,18\}$ and $n$ not too large. We also give an algorithm providing a rigorous proof of the existence and local unicity of such central configurations when given as input a choice of $n$, $\ell$ and $m_0, . . . ,m_n$. Finally, our numerical simulations highlight some interesting properties of the mass distribution.


Introduction
The N -body problem consists in describing the positions r 1 (t), . . . , r N (t) of N masses m 1 , . . . , m N interacting through Newton's gravitational law, which are solutions of the system of coupled non-linear equations for i = 1, . . . , N , with r = (r 1 , . . . , r N ) ∈ R 3N >0 , where G denotes the gravitational constant. Specific solutions, called central configurations, arise when the acceleration of each mass-particle is proportional to the position with the same constant of proportionality (depending on time) for all masses. In this paper we are interested in spiderweb configurations of N = n × + 1 masses, where the masses are located at the intersection points of concurrent equidistributed half-lines with n circles of radii r 1 < · · · < r n , and a central mass m 0 , under the hypothesis that the n masses on the i-th circle are equal to a positive constant m i , while the mass m 0 is allowed to vanish (see Figure 1).
The existence of such central configurations have been studied in the literature, often in the special case m 0 = 0, starting with Moulton in 1910 ([7]), which settled the case = 2 as a particular case of N aligned masses. The case n = 1 has been treated by Maxwell in the 19th century [4]. Later Moeckel and Simo [5] proved the existence and uniqueness of such a configuration in the case n = 2 and m 0 = 0. Corbera, Delgado and Llibre [1] considered the case of n and arbitrary with restrictions on the masses of the type m 1 . . . m n for m 0 = 0, while Saari treated the general case, releasing the restrictions on the masses with a different method in [9] and [10]. We became interested in the problem and decided to study numerically the distribution M (η) of mass depending of the distance η to the origin for large values of and n. At the same time we considered the proofs of the results appearing in [1], [9] and [10]: to our surprise, these proofs are incomplete and we started working on completing them. We could give some complete proofs, but not for all values of n, and of the masses. But our proofs are constructive and can be implemented numerically. Our numerical experimentations suggest the uniqueness of the central configurations (as claimed by Saari), and allow exploring the mass distribution in these configurations for large values of n and . By the time a first version of this paper was ready, we learnt of the general result of Montaldi [6], giving the existence of central configurations with a symmetrical mass distribution [6]. The proof of Montaldi, based on a variational formulation of the problem and using the principle of symmetric criticality of Palais, is very elegant. However, his proof is completely existential. Hence we believe that our proofs are a complement to the one provided by Montaldi in [6]. Additionally, for ∈ {2, . . . , 9} and a few other particular cases, we could prove the uniqueness of the central configurations.
The proof of Corbera, Delgado and Llibre is by induction on n. To go from n circles to n + 1 circles, the idea is to add an n+1-th circle with masses m n+1 = 0 and to allow the mass m n+1 to increase, via the implicit function theorem. The use of the implicit function theorem requires some invertible Jacobian. Since the authors could not prove that the Jacobian is invertible, they use an uniqueness argument to claim that the unique solution can be extended for nonzero values of m n+1 . The argument is not valid as shown by the following counter-example f (x, y, m) = (x 2 + y 2 + m 2 , y + m), which has a unique zero for m = 0 and no zero for m = 0. However the proof can easily be repaired and we include in the paper a proof of the existence of such configurations for m 1 . . . m n , which is much shorter than the one in [1].
By adapting the method of [1] in the spirit of Moulton [7], i.e. starting from a restricted N -body problem and following the solutions by the implicit function theorem for large values of the m i , we were able to prove the existence and uniqueness of spiderweb central configurations for ∈ {2, . . . , 9} and arbitrary n and m i . Under constraints on the mass distribution and the maximum number of circles, the uniqueness is also proven for ∈ {10, . . . , 18}.
In [10], Saari proposed a proof of the existence of spiderweb central configurations in the general case. There, again, the proof was by induction on n, and used continuity arguments, which had to rely on the implicit function theorem. But no checking of the hypothesis of the implicit function theorem could be found. Our checking of these hypotheses revealed much harder than expected, but we could adapt the method of Saari and prove the existence of spiderweb central configurations for arbitrary and m i and n ∈ {3, 4}.
To conclude, we give an algorithm providing a rigorous proof of the existence and local unicity of such central configurations when given as input a choice of n, and m 0 , ..., m n . The algorithm has been applied to all n ≤ 100 and all even values ≤ 200 when m 0 = 0 and all masses are equal. We have also applied it in the case of different masses. Our numerical explorations allowed us studying the profile of the function M (η) describing the distribution of mass at the distance η from the center of mass. This profile reveals universal features that are quite interesting.
The paper is organized as follows. Section 2 contains preliminaries. Section 3 shows the existence of spiderweb central configurations with N = n × + 1 or N = n × , and arbitrary n and . In Section 4 we prove the existence and uniqueness of spiderweb central configurations for ∈ {2, . . . , 9}, and arbitrary n and m i in the spirit of [7], while in Section 5 we give a constructive proof of the existence of spiderweb configurations for n ∈ {3, 4} and arbitrary . Finally Section 6 deals with the numerical algorithm providing rigorous proof of existence, while Section 7 studies the properties of the function M (η). (1.1) For simplicity, we translate the center of mass at the origin. Considering changes (r, m, t) → (Ar, Bm, Ct) in the units of length, mass and time satisfying A 3 B −1 C −2 = G scales G = 1. There remains two degrees of freedom: indeed additional changes preserve G = 1 provided A 3 = BC 2 .

Scalings and central configurations in
Definition 2.1. The configuration of N bodies is central at some time t * ifr(t * ) = λr(t * ) for some common λ, where r = (r 1 , . . . , r N ) ∈ R 3N >0 . Remark 2.2. The previous definition suggests that being a central configuration is a characteristic of the precise time t * . However, it is well-known that, for well-chosen initial velocities, the N bodies remain in a central configuration for all time t; during the motion of the N bodies, the common λ is a function of t.
It is easy to see that λ is a strictly negative value given by λ = U/I < 0 where I = m i r 2 i is the moment of inertia. A scaling in time allows to take λ = −1.
Moreover, by the definition of a central configuration and the equation of motion (1.1), any homothety and rotation of the positions, i.e. r = (r 1 , . . . , r N ) → A Ωr where A > 0 and Ω ∈ SO(3), yields a central configuration with λ/γ 3 . Hence, when discussing the uniqueness of central configurations we mean the uniqueness of the equivalence class for the equivalence relation (r, m, λ) ≡ A Ωr, Bm, λ (2.1)

Spiderweb configurations
We consider spiderweb central configurations formed by n × masses located at the intersection points of n circles centered at the origin of radii r 1 , . . . , r n , with half-lines starting at the origin, whose angle with the positive x-axis is θ k = 2πk/ for k = 0, . . . , − 1, together with a mass m 0 placed at the origin, under the hypothesis that the masses on the i-th circle are equal to a positive constant m i .
By symmetry, it is clear that the gravitational tug F 0 on the mass m 0 located at the origin is identically zero, and thusr 0 = λr 0 for any λ.
Rearranging the terms and using the symmetry in the equation (1.1) so that there are n bodies on the positive horizontal axis, it suffices to consider the following system for i = 1, . . . , n, with θ k = 2πk and r = (r 1 , . . . , r n ) ∈ R n .

Tools
Under the previous reduction to (2.2), the contribution of the gravitational force on the mass located at (r i , 0) is F i (r) = n j=1 F ij (r i , r j ), where F ij (r i , r j ) is the contribution of F i (r) coming from the attraction of the j-th circle, given by is analytic and all the coefficients of its power series expansion are positive. In particular, for 0 < x < 1, φ 1 (x) is analytic and all its derivative are positive.

We have
and the chain rule give For j > i, we have Now, because a 1 = 0, (xφ 1 (x) + 2φ 1 (x))/x is strictly positive and increasing. Since 5. Let 0 < k < j < i and y s = r k /r s for s = i, j. We have The function yφ 1 (y) + 2φ 1 (y) is strictly positive and increasing. Since y j < y i , we get ∂ r k λ ik < ∂ r k λ jk < 0.
We have the four following properties: There exists p ∈ R n , such that the n × + 1 spiderweb configuration respects Furthermore, in the case n = 4, we may choose (p 3 , p 4 ) such that Proof. We start with an initial circle located at p 1 ∈ R >0 . For n = 2, we have λ 1 (p 1 , +∞) < λ 2 (p 1 , +∞) = 0 and from the Point 4 of Lemma 2.4, this inequality is preserved when r 2 decreases. Repeating this exact argument for n = 3, 4 gives the expected result. Moreover, in the case n = 4, the radius p 4 can be taken sufficiently large so that λ 3 (p 1 , r 2 , p 3 , p 4 ) < λ 4 (p 1 , r 2 , p 3 , p 4 ) for all r 2 ∈ (p 1 , p 2 ].

Equations for spiderweb central configurations
We have seen in Section 2.1 that the common λ characterizing a central configuration may be fixed to any real strictly negative number, independently of n, and the mass distribution (m 0 , m).
Hence, depending of the method we will use, a spiderweb central configuration can be seen as a solution of Λ = 0 or as a zero of the map f : R n −→ R n given by

Existence of spiderweb central configurations with arbitrary n and
In this section we give a very short proof of the theorem announced in [1]. This requires introducing the tool of restricted spiderweb central configurations, which will be used also later in the paper.
Proof. By hypothesis, there exists r ∈ R n such that the n × + 1 spiderweb configuration is central.
On the one hand, adding particles of negligible mass bears no effect on the gravitational force felt by the particles on the n initial circle, that is λ 1 = . . . = λ n = λ < 0.
On the other hand, Lemma 2.4 gives the monotonous limits Consequently, there is a unique r n+1 in each case such that λ n+1 = λ.

Proof of the existence of central configurations
Theorem 3.2. Let n ∈ N, ∈ N ≥2 and (m 0 , m 1 ) ∈ R ≥0 × R >0 . There exists r ∈ R n and masses m n . . . m 2 m 1 giving a n × + 1 spiderweb central configuration.
Proof. The proof is by induction on n. Let m 0 ∈ R ≥0 and λ < 0. If n = 1, for any m 1 ∈ R >0 , according to equation (2.6), there exists a unique zero f (r 1 ) = f 1 (r 1 ) = 0 and the derivative never vanishes according to Lemma 2.3.
The implicit function theorem yields a neighborhood V of m n = 0 such that the function r = r(m n ) is a zero of f for all m n ∈ V . So, the condition m n m n−1 ensures the existence of the spiderweb central configuration.

Existence and uniqueness for circles of low density ( small)
Recall the map f given in (2.6), whose zeros give a spiderweb central configuration, and its Jacobian matrix D r f given in (3.1). In Theorem 3.2, the existence of a spiderweb central configuration is asserted under the condition m 1 . . . m n . However, for a system with circles of low density, namely small values of , the implicit function theorem may be used to extend the zeros of f for all positive value of the mass m n on the outermost circle. In such cases, iterating the argument allows us to construct a spiderweb central configuration for an arbitrary number n of circles and, moreover, to prove its uniqueness.    Proof. For n = 1, we have a regular -gon with a central mass. The equation (2.6) shows the existence of a unique r 1 ∈ R >0 such that the configuration is central.
We will prove the following: 1 . The jacobian matrix D r f ∈ M n (R), whose entries are given by (3.1), is invertible for all r ∈ R n and m n ∈ R ≥0 .
2 . All the radii remain bounded for all m n ∈ R ≥0 .
3 . All the radii remain distinct for all m n ∈ R ≥0 .
The first claim allows using the implicit function theorem to obtain a function r(m n ) such that f (r(m n ), m n ) = 0. Claims 2 and 3 allow concluding that r(m n ) can be uniquely extended for any value of m n ∈ R ≥0 , yielding its local uniqueness. The global uniqueness follows from the following argument: suppose there is an other function ψ(m n ) such that f (ψ(m n ), m n ) = 0, then it can be extended on R ≥0 . In particular, ψ(0) = r(0) because r(0) is unique. Hence, ψ(m n ) = r(m n ) for every m n ∈ R ≥0 .
Claim 1: Recall that a sufficient criterion for a matrix to be invertible is to be strictly diagonally dominant 1 .
We know that ζ is strictly positive and, by lemma 2.3, that ∂ i f i < 0 and ∂ j f i > 0 for j = i. Hence, we must show

Now, if we rewrite f i in terms of x j and φ defined respectively in (2.3) and (2.4)
− But, Noticing that the term for k = 0 is zero, we find Since λ < 0 and ζ > 0, the expression given in (4.1) is strictly positive if h (x j ) is positive, where the sign of the latter depends on the sign of the polynomial It is sufficient to show that the sign of h (x j ) is strictly positive in the case x j ∈ (0, 1), that is when j > i. Indeed, for x > 1, we have which are clearly strictly positive function for all x j ∈ (0, 1).
Therefore, we have established that −∂ i f i − n j=1 j =i ∂ j f i is a sum of positive terms, thus proving the first claim.
By the implicit function theorem, we conclude to the existence of a function r = r(m n ) defined on a neighborhood V of m n = 0, such that f (r(m n ), m n ) = 0 for all m n ∈ V .
Claim 2: Let {a k } k≥1 be a sequence in V such that lim k→+∞ a k = sup V .
Claim 3: Recall V and {a k } k≥1 previously defined.
Without loss of generality, suppose that for an index i we have lim k→+∞ r i (a k )/r j (a k ) = 1 with j > i. To preserve the equality f i (r(a k ), a k ) = 0, the equation (2.6) implies lim k→+∞ r j (a k )/r i (a k ) = 1 with j < i. Again, f j (r(a k ), a k ) = 0 requires that lim k→+∞ r j (a k )/r j (a k ) = 1 with j < j . Iterating this argument, we find lim k→+∞ r 1 (a k )/r 2 (a k ) = 1 yielding lim k→+∞ f 1 (r(a k ), a k ) = −∞. Contradiction.

Constructive proof of existence for n ∈ {3, 4} and arbitrary
The proof is a completion of Saari's proof given in [10] for the existence of spiderweb central configurations when n ∈ {3, 4}. It makes an essential use of all properties proved in Lemma 2.4 and Corollary 2.5.
Furthermore, in the case n = 2, it is the unique such configuration for a fix λ. (The result for n = 2 is already in [5]).
Proof. The proof consists in three parts, Part A (resp. Part B and Part C) sufficient for n = 2 (resp. for 3 and 4).

Part B
Let n = 3. Lemma 2.6 gives us an initial position such that the three circles satisfies λ 1 (p) < λ 2 (p) < λ 3 (p) < 0. We drop the dependency on p 1 as we keep the position of the first circle steady.
A quick look at the signs in shows that r 3 (r 2 ) > 0. Whence, for any r 2 ∈ V ∩ (p 1 ,r 2 ], we get r 3 (r 2 ) ∈ (r 2 , p 3 ]. Thus, (r 2 , ψ(r 2 )) is contained within the surface given by r 4 = z(r 2 , r 3 ) for (r 2 , r 3 ) ∈ (p 1 , p 2 ] × (r 2 , p 3 ], and this surface is included in R 3 by construction. Hence, provided inf V > p 1 , the radius of the third and fourth circles, given by ψ(r 2 ), cannot tend to the same limit as r 2 → inf V .
Remark 5.2. The proof requires multiple uses of the implicit function theorem and, in each case, it was easy to show that the corresponding Jacobian had a fixed sign for all values of the r i . Going to n > 4, there seems no easier way than lengthy calculations for each particular value of n to check the hypotheses of the implicit function theorem each time it is necessary to extend a solution by varying the r i .

Computer-assisted proof
Once again, we consider the map f defined in the Section 2.6, that is Let A : R n → R n be a linear operator and define T : Knowing an approximate zero of f , the radii polynomial approach gives bounds so that we may find a ball, centered at this approximation, on which T is a contraction to which we can apply the Banach fixed point theorem and A is non singular. Hence, it allows proving the existence and uniqueness of a true solutionr ∈ R n lying in this ball.
Due to the singularities in the equation (1.1) we must be careful in our numerical approach. We consider the local version in finite dimension of the radii polynomial approach established by Lessard, that is we introduce an upper bound ρ * for the radius of the ball in order to remain away from any singularities.
Theorem 6.1 (Radii polynomial [3]). Let U an open set of R n and f : U → R n a map of class C 1 . Let Define the radii polynomial by If there exists ρ 0 ∈ (0, ρ * ] such that p(ρ 0 ) < 0, then A is invertible and there exists a uniquẽ x ∈ B ρ 0 (x) satisfying f (x) = 0.
Remark 6.2. The choice of ρ * is quite arbitrary. We make an intial heuristic choice and check a posteriori that if there exists ρ 0 such that p(ρ 0 ) < 0 then ρ 0 < ρ * . Otherwise, we must increase the value of ρ * .
The quantityx corresponds to a numerical approximation of the zero of f obtained via Newton's method.

Computation of the bounds
Accordingly to Newton's method, we choose A to be the numerical inverse of D r f (r) wherer is the numerical zero of f .
To rigorously compute the bounds, we use techniques of interval arithmetic [8]. Let us work in (R n , · ∞ ). The bounds Y 0 and Z 0 can be found immediately from the theorem, knowing that the Jacobian matrix D r f is given by In the infinity norm, it is possible to obtain a general expression for the Z 2 bound, assuming f ∈ C 2 , by applying the mean value theorem. Lemma 6.3.
1≤k,m≤n 1≤j≤n We use this lemma to estimate the bound. The tensor D 2 r f is given by We unfold D 2 r f and represent it by the n × n 2 matrix Whence,

Numerical experimentations with circles of equal mass
We have tested successfully our algorithm with λ = −1 for all n ≤ 100, ≤ 200 even when m 0 = 0 and m 1 = . . . = m n . The bounds have been rigorously computed using interval arithmetic.
From our many numerical experimentations we are led to believe that the n × and n × + 1 spiderweb central configurations not only exist, but are unique in the sense of Section 2.1. Saari stated the same result in his papers [9] and [10].

Mass distribution
The numerical approach allows quantitative insights on spiderweb central configurations. All the profiles studied in this section are validated by applying Theorem 6.1 to each spiderweb central configuration.
For this purpose we introduce some invariants of the configurations. The first is the relative spacing between consecutive circles (see Figure 3) The second is the relative width of a spiderweb central configuration given by (see Figure 4) Depending on the context, we write explicitly the dependence on .
Conjecture 7.1. (See Figure 3) For circles of equal mass and any n ∈ N, ∈ N ≥2 , the sequence {a i } 1≤i≤n−1 is convex. When = 2 and only in this case, the sequence is strictly increasing.
Moreover, let a i * be the maximum of the a i . From the convexity we know that i * ∈ {1, n − 1} and, a priori, we cannot exclude that a 1 = a n−1 . Numerically however we have never seen a 1 = a n−1 . Hence we could have a unique i * . There exists an increasing function µ : N −→ N such that Numerically µ(n) seems small compared to n. Thus, i * = 1 whenever ≥ n.
Let χ(η) = #{j ∈ {1, . . . , n} : r j ≤ η}. The mass distribution of a spiderweb central configuration, with n circles, m 0 = 0 and equal masses per circle m 1 , . . . , m n , is given by By definition, M (η) is given for the values of λ and m 1 , . . . , m n chosen beforehand. What is remarkable is that while the sequence {a 1 , . . . , a n } can have a very wild behavior when the sequence {m 1 , . . . , m n } is irregular, the mass distribution M (η) looks very regular (see Figures 6 and 7). Indeed, to compensate for lighter masses on some circles, the neighbouring circles are closer.
This suggests that the general shape of the mass distribution is intrinsic to the spiderweb central configurations and deserves further study.