Topologically Distinct Collision-Free Periodic Solutions for the N -Center Problem

This work concerns the planar N -center problem with homogeneous potential of degree − α ( α ∈ [ 1 , 2 ) ). The existence of inﬁnitely many, topologically distinct, non-collision periodic solutions with a prescribed energy is proved. A notion of admissibility in the space of loops on the punctured plane is introduced so that in any admissible class and for any positive h the existence of a classical periodic solution with energy h for the N -center problem with α ∈ ( 1 , 2 ) is proven. In case α = 1 a slightly different result is shown: it is the case that there is either a non-collision periodic solution or a collision-reﬂection solution. The results hold for any position of the centres and it is possible to prescribe in advance the shape of the periodic solutions. The proof combines the topological properties of the space of loops in the punctured plane with variational and geometrical arguments.


Introduction
The planar N -center problem with homogeneous potential of degree −α, α ∈ [1,2) consists in the equation where the potential function is given by C def = {c 1 , . . . , c N } being the set of centres, c j ∈ R 2 and m j > 0 for all j = 1, . . . , N . Following the terminology usually adopted in celestial mechanics, a non collision periodic solution is a C 2 function q(t) so that q(t + T ) = q(t) for some period T > 0 and q(t) / ∈ C for all t.
This paper studies the existence of non-collision, periodic solutions of (1) satisfying the energy constraint for any h > 0. The gravitational case, i.e. α = 1, is surely the most interesting and widely studied case. When N = 1 it corresponds to the 2 body problem which is integrable and was solved by Newton. The 2-center problem, adopted as a model for one-electron molecules such as H + 2 and H He 2+ [41], was shown by Euler to be integrable in elliptic coordinates [42]. Except for the cases N = 1, 2, the gravitational N -center problem is not integrable [6,7]. In recent decades a quite broad literature has been produced dealing with the case N ≥ 3, both for planar and spatial motion, but a complete review of all the existing results is beyond the scope of this introduction. However, we do assemble some remarkable quantitative and qualitative results related to ours. In [6,8,9] it was shown that for N ≥ 3 the system is non-integrable on non-negative energy levels, and it has positive entropy. The authors introduced a topological (global) regularization by applying the local KS-regularization around each center. It results that the dynamics restricted on the energy level sets is a reparametrization of the geodesic flow on a non compact complete manifold. Homological properties of the space of loops of the regularized manifold allow us to estimate from below the geodesic and hence the topological entropy of the system. In [27,28] it was proved that the set of bounded trajectories has a Cantor structure which is analysed using symbolic dynamics. The paper [29] concerns the non existence of an analytic independent integral for the N ≥ 3 centre problem in the space, when the energy is larger than a threshold E th . In the case of negative energy it has been proved that there exists chaotic dynamics for the 3-center problem on small negative energy level sets when one centre is far away from the other two [10] or when the mass of one of the centres is much smaller then the others [19]. Both of these results rely on perturbative methods.
Most of the quoted works consider a regularisation of the singularities. The question about regularisation of the singular potential problem and the possible extension for the collision solutions has been proposed and discussed by different authors. We mention [20,30,31] for α = 1, and [4,12,32] for α = 1, [13][14][15]37] in case of logarithmic potential (α = 0).
A different approach to the N -center problem and, in general, to singular lagrangian systems, is that of variational arguments. In this context, the major difficulty is to prove that the minimiser is collision free. The Strong force case α ≥ 2, treated for instance in [1,5,22,24,33], is relatively easier, compared to the weak force case α ∈ (0, 2), where there is no control on the value of the functional whose critical points may correspond to trajectories passing through the singularities, see [3,18,34,35,38,39].
A variational technique has been exploited in [36] to prove the existence of infinitely many non collision periodic solutions with small negative energy for the N -center problem with α ∈ [1,2). Also, the symbolic dynamics of trajecto-ries having a prescribed topological characterisation with respect to the centres is discussed.
Motivated by this work, our contribution aims at answering the following question: which requirements, in terms of the shape of the orbit, are necessary to assure the existence of a collision-free solution? How many conditions are required when prescribing the behaviour of a non collision periodic solution? By prescribing the shape of the solution we mean to select the homotopy class [γ ] in π 1 (R 2 \ C) in which the solution belongs.
The answer to our question relies on Definition 2.15, where the concept of Admissible class is introduced, and on the notion of a collision-reflection solution. • q has a finite number of collisions, i.e. there exists a finite set of instants T c = (τ 1 , . . . , τ K ) ⊂ S such that q(τ i ) ∈ C for any i = 1, . . . , K ; • the restriction q| S\T c is a classical solution of q(t) = ∇V (q(t)); • the energy is preserved through collisions; • at any collision instant the trajectory is reflected: We prove the following: Before proceeding further we acknowledge that the result of Theorem 1.2 is not new and a different proof can be found in [27].
As will be made clear during the exposition, the conditions for a homotopy class [γ ] to be admissible are weak enough to ensure, for any h > 0, the existence of infinitely many homotopically distinct periodic solutions with energy h whenever the number N of the centres is larger than 2. In case N = 3 it is admissible any homotopy class [γ ] where γ is, for instance, any of the paths depicted in Fig. 1.
Furthermore, we introduce a set of symbols to encode the behaviour of a loop γ , so that to any finite but arbitrarily long sequence of symbols satisfying a certain rule there corresponds an admissible homotopy class. Following [36], the proofs of theorems 1.2, 1.3 are based on a constrained minimisation argument for the Maupertuis functional (the obstacle problem). Similar techniques have been exploited in the literature to solve minimisation problems in presence of singular potential, for instance in [11,16,17,40]. The core of the method consists in the analysis, close to the singularities, of a particular minimising sequence {u n } n . In order to prove the existence of collision-free orbits, it is required that any u n does not admit any loop enclosing one of the centres.
The occurrence of such a loop can be avoided by imposing symmetry constraints [40], or by performing the minimisation on spaces of intersection-free paths [16], or on spaces of paths whose vectors of winding numbers with respect to the centres have the same parity [36]. In any of the above mentioned cases, the periodic solution turns out to be intersection free. Instead of these assumptions, in this work the minimisation is performed on selected homotopy classes and we allow the minimisers to have self intersections and hence to design much more complicated dynamics around the centres.
The presence of self intersections on the minimising sequence represents a technical hurdle when applying the obstacle problem argument. Moreover, no information on the number and location of the self intersections of a loop γ , that is a local property, can be inferred by the knowledge of the homotopy class of γ ; that is rather a global property.
In spite of this, it is possible to control the change in number of the selfintersections of a curve, and intersections between different curves, along the homotopies. This result, although weak, reveals itself to be enough for our purposes.
The paper is organised as follows. In Section 2 we deal with the topological properties of the space of loops and we introduce the definition of Admissible class. Section 3 concerns the variational setting: we introduce the Maupertuis' functional M and the space of loops H [γ ] , which is where to look for minima. Also, useful properties of the Maupertuis' functional are discussed. In Section 4, by means of classical arguments we prove the existence of a minimiser for M on H [γ ] and, combining topological and variational arguments, we study geometrical features of the minimal path. Section 5 concerns the obstacle problem. We show the major steps of the procedure, while referring to [36] and [40] for the details. In Section 6 we provide the proofs of the theorems (1.2) (1.3). Finally, in Section 7 we discuss a method to generate, in terms of symbols, the admissible classes.
We hasten to remark that the dichotomy presented in Theorem 1.2 is easily solvable; indeed if the support of a collision-reflection solution does not belong to the boundary of the homotopy class [γ ], then the periodic solution is collision-free.
On the other side, the same variational method does not allow one to conclude the existence of non-collision solutions in the ultra-weak case α ∈ (0, 1), and nor does it for the logarithmic potential, α = 0.

Curves and Intersections
Let ⊂ R 2 and I ⊂ R any interval. A curve in is any continuos map γ (t) : I → . • γ is simple if γ is injective.
• γ has a self-intersection of order k at the point p * if there exist t 1 , . . . , t k ∈ I such that γ (t j ) = p * . If k = 2, for short, we simply refer to p * by the name of self-intersection.
. We say that the intersection is transversal if γ is C 1 -transversal (in the sense of differential topology) at the point p * as immersed manifold. We say that γ has a self-tangency at point p * if the tangent space of γ at p * is 1-dimensional.
Otherwise p * is a self-tangency point.
Let γ : I → be a curve and I 1 ⊆ I any subinterval. In the following we denote: • by γ | I 1 the restricted function, i.e. the curve γ 1 : all t ∈ I 1 • and by γ (I 1 ) the image of γ (t) for t ∈ I 1 .
For later purposes, we introduce the operation of the composition of curves: given two curves γ : [a, b] → and γ 0 : [c, d] → such that γ (b) = γ 0 (c), we denote by γ #γ 0 the curve obtained by gluing γ with γ 0 , that is Moreover we denote by inv(γ ) the curve When possible, in the following we adopt the notation γ −1 instead of inv(γ ).

Loops in the Punctured Plane
Denote by R 2 C = R 2 \ C the punctured plane, S := [0, 1]/{0, 1} the unit circle parameterized by t ∈ [0, 1] and let γ 1 , γ 2 ∈ C 0 (S, R 2 C ) be two loops in R 2 C . The two loops are said to be freely homotopic, and we write γ 1 ∼ γ 2 if there exists a continuous function F : for all t ∈ S and F(S, s) is a loop for any s ∈ [0, 1]. The relation ∼ is an equivalence relation. Denote by the set of equivalence classes and indicate with [γ ] an element of : The space R 2 C is path connected; given a loop γ and a point x 0 in R 2 C , there always exists a loopγ equivalent to γ and based in x 0 . Because of this, we can define the composition of two classes [γ ], [τ ] ∈ : if γ and τ represent two equivalence classes, without loss of generality we can assume that γ and τ are based at the same point, so we may define the operation [γ ] * [τ ] as naturally done in π 1 (R 2 C , x 0 ). Indeed the elements of correspond to conjugacy classes in the fundamental group π 1 (R 2 C , x 0 ), modulo changing of the base point x 0 and ( , * ) is a group. Since the space R 2 C is not simply connected, the set is not trivial. More precisely, since , for any N ≥ 1 ( N is the cardinality of the set of centers C), the set has cardinality Z N and it is isomorphic to the free group of N generators.
The unity of this group is given by the class of null-homotopic loops, that is, those loops homotopic to a point.
For any loop γ ∈ C 0 (S, R 2 C ), with the identification R 2 ∼ = C, it is well defined the index The j-th component of I nd(γ ) is the index with respect to the centre c j and it agrees with the winding number of γ with respect to c j . Remark 2.7. If a path γ is null-homotopic in R 2 C then the winding number of γ with respect to any center is zero, i.e. I nd(γ ) = 0.
The homotopy (t, s) is continuos in both the variable and (t, s) = c j for all t ∈ S, s ∈ [0, 1] and j = 1, . . . , N , thus the function (s) is continuos. Since the index function takes values in Z, it follows that (s) is constant. Definition 2.10. Let γ be a loop in R 2 C . We say that γ has a: i. singular 1-gon if there is a subinterval I of S such that γ identifies the end-points of I and γ | I defines a null-homotopic loop in R 2 C ( i.e. γ has a null-homotopic sub-loop); ii. singular 2-gon if there are disjoint intervals I 1 and I 2 of S such that γ identifies the end-points of I 1 and I 2 and γ | I 1 #γ | I 2 defines a null-homotopic loop on R 2 C .
With some abuse of terminology, we will say that the loop γ | I is the singular 1-gon ( in the first case) and γ | I 1 #γ | I 2 is the singular 2-gon.    A criteria for determining whether or not a given loop is taut is provided by the following theorem, [25].  Theorem 2.14. Let γ be an immersed curve which does not minimize the number of self intersections in its homotopy class. Then there exists an homotopy from γ to a immersed curve γ 1 which is taut such that the number of self-intersections of the curve s is not increasing with s. is a regular homotopy except for a finite number of times when a small loop in the curve shrinks to a point.
We are now in the position to introduce the admissible classes. Definition 2.15. Let [γ ] ∈ be an homotopy class and letγ be a taut representative of [γ ]. We say that the homotopy class [γ ] is admissible if the following property holds: [A] for any innermost loop j ofγ there exist at least two centers c j 1 , c j 2 so that j is not null-homotopic both in R 2 \ c j 1 and in R 2 \ c j 2 .
For Lemma 2.6, any innermost loop ofγ is simple, hence [A] simply means that at least two centers are in the region bounded by . The existence of a taut representative is implicit in the definition of minimal intersection number of a homotopy class. However, for any primitive element [γ ] in we can construct a taut representative as follows. We can think of small open balls B i in place of the centers c i and the configuration space Any free homotopy class of R 2 B contains exactly one geodesic and a primitive geodesic cannot have excess self intersections, see c.f. [21].
Moreover, the definition is well posed and does not depend on the particular taut representative. If γ and γ 1 are two taut curves in [γ ], for Theorem 2.13 the sup-loops of the two curves are in one-to-one correspondence with the property of enclosing the same centers.
Before proceeding, let us recall the definition of tied loop introduced by Gordon in [23].
Definition 2.17. Let B be a closed non empty subset of R 2 . A loop γ in R 2 \ B is said to be tied to B if γ cannot be continuously moved off to infinity without either crossing B or having its arc length become infinite.

Lemma 2.18. Let γ be a not null-homotopic loop in R 2
C . Then γ is tied to at last one of the center c j ∈ C.

Variational Setting
The natural setting, when addressing the existence of periodic solutions of the system (1) by means of the variational methods, is that of the Sobolev space endowed with the scalar product and norm given, respectively, by Introduce the space and the open subspace H ⊂ H of collision free periodic functions Since H is embedded into C 0 (S, R 2 ), the condition u(t) ∈ R 2 \ C makes sense. Following the notations introduced by authors in [36], we define as the set of all colliding paths in H 1 (S, R 2 ); clearly hence H is nothing but the weak H 1 -closure of the space H . From now on, suppose h > 0 is fixed. We introduce the Maupertuis functional Beside the space of periodic paths H and the functional M , for any given couple of points p 1 , p 2 , let us introduce the spaces and H p 1 , p 2 (a, b) as its weak H 1 closure. Let us define the functional (with [a, b] in place of S in the second case).
Let us now state some features of the Maupertuis functional. Then Because of the invariance of Maupertuis functional under time rescaling, the choice of the length of the time interval used in the definition of H and of M is irrelevant. In the above definition, the functional M is defined on paths u with Supp(u) = S. However, although v is a path with Supp(v) = S, with some abuse of notation we write Unlike other functionals, for instance the Lagrange action functional or the Jacoby length functional, often used in addressing similar problems, the Maupertuis functional is not additive. Indeed if u(t) = u 1 #u 2 , we may have ] is a minimiser for an additive functional then u minimises locally. This means that any restriction is the minimiser of the restricted functional among all paths that satisfy the end-points constraints. Nevertheless, although the Maupertuis functional is not additive, the minimisers still admit the same property.
Assume by contradiction that u 1 is not a minimiser for M p 1 , p 2 and let v 1 ∈ As before, let us write explicitly the kinetic and potential parts Hence To prove the second part of the statement, let us assume that As before denote by η a proper time rescaling of v 1 so that K η = K u 1 . It follows that V η = V u 1 and, denotingw = η#u 2 , we have M (w) = M (u). Again for the invariance of M under time rescaling, the path w ∈ H , obtained fromw by rescaling is such that The Maupertuis functional is differentiable over H and, up to reparametrization, its critical points are solutions to of the N -center problem.  (7). Then u ∈ C 2 (S) and solves the equation p 2 (a, b) be a critical point of M p 1 , p 2 at positive level and ω be as in (7) ((a, b)) and solves Eq. (9) for any t ∈ (a, b) and the function x(t) Integrating by parts and letting ε to go to zero, it follows that hence Eq. (9).
The same arguments apply for the fixed-end problem.
By regularity theory it turns out that a critical point u ∈ H of M is indeed a classical solution (at least C 2 ) of the differential equation in (1).
According to the previous lemma, we look for periodic solutions of the Ncenter problem as minimisers of the action functional M in H . The existence of minimisers will be proved following the direct method of calculus of variations. Rather than H , which is not weakly closed, a suitable closed subspaces H ⊂ H must be selected. Aiming at classical solutions, some topological constraints are required on the elements of H so as to rule out the occurrence of collisions on the minimising paths. Recalling As suggested in [36], as the set of paths that collide in the centre c j and that are homotopic to γ in R 2 minus the other centres. Similarly, for any Although reasonable, this construction leads to a set H [γ ] that is not the weak H 1 closure of H [γ ] , as the counterexample depicted in Fig. (5) shows.

Existence of Minimizers
Also, for Lemma 2.18 and according to Gordon [23], there exists at least one center c ∈ C so that the elements of [γ ] are tied to c. For the Cauchy-Schwarz inequality, be a sequence u n u weakly in H 1 . Thenu n u weakly in L 2 and, u n being a bounded sequence in H 1 for the Sobolev embedding, up to a subsequence, u n → u uniformly. The lower semicontinuity of the L 2 norm and Fatou's lemma imply Hence M is coercive and lower semicontinuous. By application of the direct method of calculus of variations, M attains its minimum in H [γ ] .
Denote byū a minimiser of M on H [γ ] :

Properties of the minimisers
In this section some analytical and geometrical properties of the minimiser are described. Concerning the occurrence of collisions, introduce the set of collision times: Since V is infinite on collisions and the value M (ū) is finite, it follows that T c (ū) = 0 is closed and of null measure. Hence the complement S \ T c (ū) is the union of a finite or countable number of open intervals.
Given a path u(t), define the inertia of u with respect to the j-th centre as (9), possibly with collisions. Then for

Lemma 4.2. Suppose u(t) is a solution of
Proof. For Theorem 3.3, In the neighbourhood of c j the first and the last term of the right hand side are bounded, while the second term tends to +∞ as |u(t) − c j | → 0. Hence there exists R j such thatÏ j (u)(t) > 0 for any t so that |u(t) − c j | ≤ R j .
The convexity of the inertia functionals in the neighborhood of the centers allows us to prove the following (see for instance [40]).

Lemma 4.3. The cardinality of T c (ū) is finite.
Let K = |T c (ū)| be the number of collisions ofū and denote by 0 < τ 1 < τ 2 < · · · < τ K < 1 the collision times. Denote by J (ū) = ( j 1 , . . . , j K ) ∈ {1, . . . , N } K the ordered sequence of the labels of the centers hit byū, that isū(τ i ) = c j i . Out of the collision instants, the minimiserū provides a classical solution for the N -center problem. More precisely, we have the following: For any (a, b) the restrictionū| (a,b) is a C 2 path and solves the equation Proof. As in the proof of Theorem 3.3, it is a consequence of the minimality of u with respect to variations with compact support in (a, b).
In what follows the geometric properties of the minimiser are analysed, with particular emphasis on the possible occurrence of 1-gons and 2-gons. For this, let us first recall the notion of homotopy relative to the boundary.  ∈ (a, b). Denote by p 1 , p 2 = H p 1 , p 2 ∼ rb the space of equivalence classes of H p 1 , p 2 with respect to the relative homotopy.
As for the space , the space p 1 , p 2 has infinitely many elements. Using the same notation adopted in case of loops, for an equivalence class [γ ] ∈ p 1 , p 2 we denote H [γ ] = H p 1 , p 2 ∩ [γ ] and H [γ ] as its closure in the weak H 1 topology. Also, since H [γ ] is weakly closed and M p 1 , p 2 is coercive and weakly lower semicontinuous, the direct method of calculus of variations assures the existence of a minimiser for M p 1 , p 2 in H [γ ] .
We remark that, up to time reparameterization, trajectories of the N -body problem with energy h are geodesics of the Jacobi-Maupertuis metric g h (u,u) = 2(V (u) + h)|u| 2 on R 2 \ C. For h > 0 the gaussian curvature K of g h is negative, hence, by the Gauss-Bonnet theorem, the minimiserū is unique, eventually with collisions, in any non-trivial homotopy class [γ ] ∈ . Since the minimiserū is unique in the closure of its homotopy class H [γ ] , it follows that any arcū| [a,b] , joining p 1 =ū(a) to p 2 =ū(b), p 1 , p 2 possibly singular points, is the unique minimiser for M p 1 , p 2 in the homotopy class [ū| [a,b] ] ∈ p 1 , p 2 .
The minimality condition implies that the solution curve can only have transversal self intersections and the number of intersections is minimised, as stated in the next proposition. 1. Let t a ∈ (τ i , τ i+1 ) and t b ∈ (τ j , τ j+1 ) be two non-collision instants. Ifū has a self tangency at point p = u(t a ) = u(t b ), thenū has a tangency at any t ∈ (τ i , τ i+1 ) and t ∈ (τ j , τ j+1 ). Suppose by contradiction thatū has a singular 2-gon. Denote by γ 1 and γ 2 the two geodesic arcs and by α, β > 0 the two angles formed at the intersection points. The region A bounded by the singular 2-gon is homeomorphic to a disc, hence for the Gauss-Bonnet theorem

The loopū does not admit any singular 1-gon neither singular 2-gon in R
Beside the singular 1-gon and singular 2-gon, we now discuss the possibility for the minimiserū to admits 1-gons and 2-gons whose edges intersect some of the centres. . Thenū does not admit any sub loop that intersects c j and that is a singular 1-gon in R 2 C ∪{c j } for one c j ∈ C. Proof. Arguing by contradiction, assume that =ū| [a,b] is a singular 1-gon in R 2 C ∪ c j and one of the following occurs: i.ū(a) =ū(b) = c j ; ii.ū(a) =ū(b) = p, p = c j and it exists τ ∈ (a, b) so thatū(τ ) = c j .
In the case i considerũ =ū| S\ [a,b] . Since is homotopic to the point c j , it follows thatũ ∈ H [γ ] and M (ũ) < M (ū), hence the contradiction.
In the case ii denote by γ 1 =ū| [a,τ ] and γ 2 =ū| [τ,b] . Let H [γ 1 ] ∈ p,c j be the homotopy class containing γ 1 and, up to time reversion, γ 2 . Let v 1 be the unique minimiser of M p,c j on H [γ 1 ] . Thus, for Lemma 3.2, the pathū is given byū = v 1 #(v 1 ) −1 #u| S\ [a,b] . Howeverū is not of class C 1 at p, hence the contradiction. The following corollaries concern a generalisation of the situations described in point i and ii of the previous lemma, see also Figure 6.
Proof. By assumption bothū andū| S\(a,b) #˜ belong to H [γ ] . Since˜ is homotopic to c j 0 , the pathũ =ū| S\(a,b) belongs to H [γ ] and M (ũ) < M (ū). Lemma 4.7 and related corollaries concern the presence of 1-gon on a minimal path. In practice, under certain assumptions, no 1-gon can be present on the minimal path.
The remainder of the section focuses on the occurrence of 2-gons. Under similar assumptions to those above, we prove that the 2-gons as well cannot be present in the minimal paths. Indeed the 2-gon is replaced by a geodesic arc covered twice. Suppose there exist c j 1 and c j 2 ( possibly equal) such that:  Thenū cannot admit any 2-gon γ = γ 1 #γ 2 with the following properties: d] . In particular the two arcs γ 1 , γ 2 cover the same path back and forth and intersect the same set of centres.
The above lemma concerns the case when the two arcs γ 1 and γ 2 join the points c j 1 and c j 2 in the opposite direction. By straighforward modification the same argument holds for the case when the two arcs have the same direction.

The Obstacle Problem
Recalling the definition of R j given in Lemma 4.2, let us define a radius R such that and the balls of radius R around the centers where the collisions occur, with the property that whenever there Note that R is defined small enough so that anytimeū enters the ball B j thenū collides with the centre c j before leaving the ball. For any k = 1, . . . , K , being K the number of collisions, let us define as the instants when the pathū enters and leaves the ball B j k just before and after the collision time τ k , and as the points where the minimiser enters and leaves B j k . (a k , b k ), it follows that τ k is the only collision time in the interval [a k , b k ] and [a k , b k ] ∩ [a j , b j ] = ∅ for any k = j. [γ ] . Assume thatū has collisions. Then an the following holds: eitherū is a collision reflection solution, or there is at least one k so that p 1 k = p 2 k .

Lemma 5.1. Letū be the minimizer for M in H
Proof. Supposeū does not reflect after the collision instant τ k ∈ T c . Arguing by contradiction, assume p 1 k = p 2 k . Then, for Lemma 4.7,ū is not a minimiser. Otherwise, if a reflection occurs at any collision instants, thenū is a collision reflection solution according to Definition 1.1.
Let us sketch the plan of the proof of the Main theorem as it will be given in Section 6: by contradiction assume thatū has collisions but it is not a collision reflection solution. For Lemma 5.1 there is at least one k so that p 1 k = p 2 k andū is not a collision reflection solution within the ball B k . The absurdity will arise by proving that p 1 k must be equal to p 2 k , that is, anytime a collision occurs, the pathū has to reflect. Therefore, either the minimizerū is collision free or it is a collision reflection solution.
The core of the proof is based on the so called Obstacle problem, which allows one to analyse the behaviour of the minimisers of a given functional in the neighbourhood of the collisions. It turns out that, if certain hypothesis are satisfied, whenever the minimiserū has a collision, it must be a collision reflection solution, at least locally.
In the following we briefly review the technique, mostly for the sake of stating the result and fixing the notation while referring to previous works for the details.
Henceforth we assume that u has K > 0 collisions, i.e. |T c | = K . HYP Without loss of generality, we apply the obstacle problem in the neighbourhood of τ 1 . Denote (12) Moreover let us define Since K [γ ] is weakly closed and M 1 is coercive and weakly lower semicontinuous, the direct method of the calculus of variations assures that M 1 admits a minimum. Lemma 3.2 implies that the arcū| [a 1 ,b 1 ] is a minimizer of M 1 . For ε > 0 let us define Since the weak convergence in H 1 implies the uniform convergence the set Proof. Look at [40,Lemma 17].
As a direct consequence of the previous lemma, there exist two sequences 0 < ε n <ε n , both vanishing, and a sequence {v n } n such that For any n, denote The next proposition summarizes the properties of the sequence {v n }. We refer to [36,40] for the proof. a C 2 function in any interval I ⊂ (a 1 , b 1

) \ T n and solves
iii. the set T n consists in a interval, and the sequence {w n } n is bounded above and uniformly bounded below by a positive constant.
Moreover, writing v n (t) = c j 1 + ρ n (t)e ıθ n (t) , it holds that: vi. the function θ n (t) is C 2 and strictly monotone in T n .
The convergence of the sequence {v n } n is now discussed.
Proof. Since M 1 (v n ) = d(ε n ) and ε n → 0, from Lemma 5.2, it follows that M 1 (ε n ) → d(0). The coercivity of M 1 implies that {v n } is bounded, hence, up to a subsequence still denoted by {v n }, it is weakly convergent to a arcũ 1 ∈ K [γ ] . The weak convergence in H 1 implies uniform convergence, thusũ 1 has to collide with c j 1 .
The last step of the obstacle problem technique consists in the analysis of the behaviour of the minimising sequence v n and of the limitũ 1 . Again, we do not provide the detailed proof, since it is a small modification of the proof given in [36]. The argument is the following. Denote by n the total variation for t ∈ [a 1 , b 1 ] of the angular coordinate θ n associated to v n . One shows that n ≥ n for any n and n → 2π 2−α as n → ∞, see also [38]. This means that for any α > 1 the path v n has to self intersect for n large enough. Therefore, it is enough to prove that all the v n 's are simple to obtain the contradiction. The case α = 1 is more delicate. Indeed, even under the hypothesis that v n are simple, the limitũ 1 could have the collision. However, if that happens,ũ 1 is proven to be a collision reflection solution. The idea is to combine the minimisation procedure with the Levi Civita regularisation [31]. In the neighbourhood of one of the centres, the potential V (x) is a perturbation of the one-centre keplerian potential and, by means of the classical Levi-Civita change of coordinates, the singular dynamical system is related to a regular one defined on a Riemann surface. As a result, the minimising sequence {v n } n is associated to a minimising sequence {q n } n of an equivalent obstacle problem in the regularised system. The weak limitq of the sequence q n is related to the arcũ 1 as follows.
Assuming that the centre c j 1 has been mapped to the origin in the Levi-Civita plane and denoting with s 0 the time whenq(s 0 ) = 0, if the limitq is a transmission solution, i.e. ifq(s 0 + s) = −q(s 0 − s), then the limitũ 1 (t) is a collision reflection solution. The pathq is a transmission solution if the angle swept by q n on the obstacle is not larger than π or, equivalently, if the angle n covered by v n (t) is not larger than 2π . In the gravitational case, α = 1, it is proved that n ≤ 2π for any n, as wanted.
Summarising, the following holds.
Proposition 5.6. Case α = 1. If for any n the path v n is simple and the limit u 1 (t) has a collision, thenũ 1 (t) is a collision reflection solution within the ball B 1 . In particular, p 1 1 = p 1 2 . Case 1 < α < 2. If for any n the path v n is simple, then the limitũ 1 (t) is collision-free.

Proof of the Theorems
We are now in the position to proof the Theorem 1.2. For clarity, let us recall the statement: Otherwise, suppose thatū has collisions. We aim to prove thatū parametrizes a collision reflection solution. Arguing by contradiction, suppose thatū is not a collision reflection solution. Then, recalling the definitions (10) and (11) of R and B j and of the couple ( p 1 k , p 2 k ), for Lemma 5.1 there exists at least one k * so that p 1 k * = p 2 k * . Without loss of generality, set k * = 1. We now apply the obstacle problem in the neighbourhood of τ 1 as described in Section 5. Assume for the moment that for any n the arc v n is simple. This fact will be proved afterwards in Theorem 6.3. According to Proposition 5.6, we conclude that p 1 1 = p 2 1 , hence the contradiction. Therefore, ifū has collisions,ū reflects at any collision instants and the path x(s) =ū(ws) is a collision reflection solution with energy h of the N -centre problem.
The proof of Theorem 1.3 is more immediate. Proof. As done in the proof of Theorem 1.2, letū be a minimiser of M [γ ] on H [γ ] . Ifū is collision free, then x(s) =ū(ws) provides the periodic solution of (1)-(3). Otherwise, assume by contradiction thatū has a collision and apply the obstacle problem around one of the collisions. Again, suppose that the paths v n are simple, then Proposition 5.6 provides the absurd.
It remains to prove that for any n the arc v n is simple. From Proposition 5.4 we remind that, for any n, the path v n ∈ C 1 ([a 1 , b 1 Lemma 6.1. we have that: i. The arcs v 1 n , v 2 n are simple. ii. The path v n does not admit any singular 1-gon neither singular 2-gon.
iii. v 1 n is nowhere tangent to v 2 n .
Proof. For point iv in Proposition 5.4, the two arcs are simple. This excludes the possibility for v n to perform singular 1-gon. Also, following the same arguments as in the proof of Proposition 4.6, the two arcs v 1 n and v 2 n don't have any tangencies, neither they can intersect each other so to create singular 2-gon.
Hence, the possible self intersections of v n are due to transversal intersections of v 1 n with v 2 n or to multiple covering of part of the obstacle (tangential self intersection). The remainder of this section aims at proving that both the transversal and tangential intersections cannot occur. The argument is based on the fact that the path γ n given in (13) is in the closure of H [γ ] and [γ ] is an admissible class.
First, let us construct a path γ * n , free of collisions, without self tangencies, so that γ * n ∈ [γ ] and γ * n visits the balls B j in the same order as γ n . If τ 1 is the only collision time forū then γ n (t) ∈ R 2 C and γ n ∈ [γ ]. In that case, define γ * n = γ n . Otherwise, let {τ 2 , . . . , τ K } and {c j 2 , . . . , c j K } be, respectively, the collision instants for γ n (t) and the centres where the collisions occur. Since H [γ ] is the weak H 1 closure of H [γ ] , for any 0 < ρ < R small enough, there exists a collection of K − 1 arcs Fig. 8. The self-tangency is removed by homotopy. It results in two arcs that intersect, at most, onceγ Clearly the choice of the arcû i is not unique. Following [25], the arcsû i can be selected so that: iv. for any i the arcû i does not have self tangencies andû i minimises the number of self intersections in its homotopy class ( with respect to the relative homotopy); v. for any i = k so that c j i = c j k , if any, the arcsû i andû k do not intersect tangentially and they minimise the number of intersections; vi. if c j i = c j 1 for some i ∈ {2, . . . , K }, thenû i is nowhere tangent to v n and it minimises the number of intersections with v n .
Sinceγ n is equal toū outside j B j ,γ n inherits the self tangencies fromū. As described in Proposition 4.6, Lemma 4.10 and Corollary 4.11, there may exist a set of indexes j 1 ), . . . , (i m , j m )}, for > 0 small enough define m homotopies relative to the boundaries, to be applied toγ in sequence, so that the tangent arcs are moved to a different configuration without tangencies and with the minimal number of transversal intersections, see Fig. 8. Indeed, the tangent arcs can be homotoped to two arcs with at most one intersection [2]. Also, the homotopies k can be defined so that the number of intersections between the two arcs involved in the homotopy with the remaining part of the curveγ n does not change.
Denote by γ * n (t) the path resulting after applying all the homotopies k . By construction, γ * n satisfies the following properties: Proof. Suppose by contradiction that for some n the path v n (t) is not simple, i.e., there exists at least one couple t * < t * such that v n (t * ) = v n (t * ).
For the moment assume that v n does not have any self-intersection in the interval [c, d] (later on we will come back to this eventuality). Hence, the self-intersections of v n are due to intersections of v 1 n with v 2 n . Denote by the instants of transversal intersections and by the points where they occur. For Proposition 5.4 the angular velocity of v 1 n , v 2 n is constant and v 1 n , v 2 n have finite length. Thus, the number of intersections must be finite.
Consider the previously introduced path γ * n . For Lemma 6.2, γ * n belongs to an admissible class. Note that γ * n has an innermost loop γ * n | [t m 1 ,t m 2 ] enclosing only one centre. Therefore, by definition of the admissible class, γ * n is not taut. For Theorem 2.12, γ * n has a singular 1-gon or singular 2-gon, denoted by . Since for Lemmas 6.1 and 6.2 the arcs v 1 n and v 2 n do not produce singular 1-gon nor 2gon and γ * n has no self-tangencies, it follows that the point p = v n (t 1 1 ) = v n (t 2 1 ) is necessarily one of the end-points of the singular object whose boundary lies partially inside B j 1 and partially outside B j 1 . We show that whatever singular object is (1-gon or 2-gon), a contradiction arises. Let us first consider the case that is a singular 1-gon and denote by = γ * n | [t 2 1 ,t 1 1 ] the null homotopic loop that bounds , as shown in Fig. 9. Shrinking to a point, we have that v n | [t 1 and v n | [t 1 1 ,t 2 1 ] is taut, since no singular 1-gon or 2-gon lies in B j 1 . This contradicts the definition of admissible class, since v n | [t 1 1 ,t 2 1 ] has an innermost loop containing only one centre. Suppose now that is a singular 2-gon. Denote by q the second end-point (the first one is p) and by α and β the edges of . For Lemma 6.1, at least one of the edges is not completely within the ball B j 1 . Without loss of generality, the case when one of the edges completely lies in the ball is depicted in Fig. 10a.
Looking at the definition ofγ n , we see α as the composition of three arcs: α = α 1 #α 2 #α 3 . The first of these is a portion of the arc v n from the point p up to the point p 2 1 on the boundary of B j 1 ; the second is made by part ofū and eventually someû i up to the point p j * 1 where α is back on B j 1 ; finally, α 3 is part ofû j * . Consider the arc =ū| [τ 1 ,b 1 ] #α 2 #ū| [a j * ,τ j * ] . The arc ∈ H c j 1 ,c j 1 is homotopic to the point c j 1 , hence the loopū| S\[τ 1 ,τ j * ] # belongs to H [γ ] . This contradicts Lemma 4.7 or the associated Corollary 4.8.
Consider now the case when neither of the edges -α or -β lies completely in the ball B j 1 . Depending on whether q belongs to B j 1 (case 1), to B j k , k = 1 (case 2), or outside any of the balls B j (case 3), the situation is depicted in the Fig. 11. Arguing as before, the same procedure leads to a configuration that contradicts Lemma 4.10 and related Corollary 4.11 in case 1 and case 2, and Lemma 4.9 in case 3.
It remains to consider the case when v n has tangential self-intersections on the obstacle. The idea is to replace v n with a homotopically equivalent path ν n without self-tangencies and equal to v n (t) for those t where v n (t) is far away from the obstacle. The path ν n has at least one self-intersection and it satisfies the properties stated in Lemma 6.1. Then, the same procedure as before can be applied Fig. 12. Relaxation of the self-tangency. By means of a homotpy , the self-tangent arc is moved to an arc with transversal self-intersections only Fig. 13. Construction of the path w n . The figure depicts the situation when v n turns around the obstacle only once. The same construction provides the contradiction even in the case where v n turns around the obstacle many times to ν n , proving that ν n cannot have self-intersections, hence the original v n must be simple.
Let us show how to construct ν n . Remember from Proposition 5.4 that the angular coordinate θ(t) is monotone for t ∈ [c, d]. As depicted in Fig. 12, definẽ R = |I tr m − c j 1 | as the distance between the centre c j 1 and the transversal selfintersection point of v n closest to the obstacle. ( In case v n does not have transversal self intersections, consider t m 1 = a 1 , t m 2 = b 1 andR = R.) For a choice ofε n ∈ (ε n ,R), ε n being the radius of the obstacle, definẽ  Fig. 12. Sinceṽ minimises the number of self-intersections,ṽ does not admit singular 1-gon nor singular 2-gon, [25]. However, there could be a singular 2-gon whose end-points are given by I tr m and I tg 1 . If that happens, the original path v n would behave as depicted in Fig. 13 and the curve w n drawn in Fig. 13 would be a minimiser like v n , without being C 1 , providing a contradiction.

Admissible Classes as a Sequence of Symbols
In the previous section we proved the existence of a periodic solution for the N centre problem in any admissible class [γ ]. According to the Definition 2.15, to test whether a given class is admissible one has to count the number of centres in any innermost loop of a taut representative of the homotopy class. Such a test could be time consuming and tedious, however it is straightforward thanks Theorem 2.12.
Nevertheless, a natural question arises as to whether it is possible to prescribe sufficient conditions to a loop γ so that its homotopy class [γ ] is admissible. That is exactly the content of this section: given the centres in any position on the plane we propose a method to draw infinitely many closed curves, each of them belonging to a different admissible class. The idea is to consider a simple polygonal having the centres as vertices and to construct different closed curves whose sequence of consecutive crossings with the edges of the polygonal are prescribed in advance. Certain conditions on the sequence of the consecutive crossings will imply the co-existence of at least two centres in any innermost loop of a taut representative, hence the admissibility of the homotopy class.
Let us introduce the technique in the case where N centres are placed at vertices of a regular N -gon; then we will show how to extend to the general case.
Let P be a regular N -gon of unit radius centreed in the origin of R 2 . Let c 1 , c 2 , . . . , c N be N -centres placed at the vertices of P and labelled in a counterclockwise orientation, i.e. c i = (cos( 2π N (i − 1)), sin( 2π N (i − 1))). Denote by L = {l 1 , l 2 , . . . , l N } the set of the edges of P, where l i = {λc i + (1 − λ)c i+1 , λ ∈ [0, 1]} is the edge joining c i and c i+1 , for i = 1, . . . , N . To simplify the exposition we will always assume the cyclic action of the labels on a set of N elements, so that, for instance, c N +1 = c 1 . Denote ∂P = l i . On L we consider the Lee distance Thinking at l i as letters, we introduce the words with alphabet L as follows: Definition 7.1. We define a word W with alphabet L as any possible string where w k ∈ L for any 1 ≤ k ≤ M and M < ∞. M is said to be the length of the word and we denote M = |W |.
In the next Definition we fix further notations and we introduce some useful operations. From now on we consider only irreducible words. Given a word W , we associate with W a path γ (t) in the punctured plane so that the sequence of consecutive crossings of γ (t) with the edges of P is exactly the one prescribed by W . Definition 7.3. For any given word W ∈ W e , define γ W any closed path with the following properties: Since we allow the path γ W (t) to only have transversal intersections with ∂P it turns out that γ W (t) has an even number of crossings with ∂P. That is the reason why we consider only words of even length.
The above definition is well posed in the sense that for any given W of even length it is always possible to define a path with properties i), ii) and iii). Clearly the path γ W is not unique, since, for instance, small perturbations produce paths with the same sequence of consecutive crossings with the edges of the polygon P. Furthermore, the same word W could produce paths γ W belonging to different homotopy classes; indeed any time the path γ W leaves P crossing the edge w i , there are two different directions it can follow to reach the next edge w i+1 , namely the clockwise or counterclockwise direction; it can even wind around the polygon P many times before entering. In general, a different choice of directions produces non homotopic curves, see Fig.14. Nevertheless, the following result holds: Theorem 7.4. Let N ≥ 4 centres be placed at the vertices of a regular N -gon and let W be any irreducible word of even length such that Let γ W (t) be any path associated to W according to Definition 7.3. Then, the homotopy class [γ W ] is admissible. Proof. First note that the irreducibility of W implies that γ W minimizes the number of possible intersections with ∂P in its homotopy class. Indeed suppose that there exists a homotopy φ s that moves γ W to φ 1 (γ W ) with less crossings of ∂P. Since the number of intersections between two closed curves only changes in correspondence to tangential crossings, the change of the number of crossings between φ s (γ W ) and ∂P occurs only at instants s i when φ s i (γ W ) is tangential to one of the edges l j . At the first of these instants, say s 1 , suppose that φ s 1 (γ W ) is tangential to l j . Then the path γ W = φ 0 (γ W ) has two consecutive crossings with l j . This contradicts the irreducibility of the word W . The definition of admissibility of class [γ W ] relies on some conditions being satisfied by a taut pathγ ∈ [γ W ]. The constructed curve γ W does not need to be taut, however there exists a taut pathγ with the same sequence of crossings of ∂P as γ W . As before, let φ s be the homotopy that moves γ W toγ : at any s the number of crossings of φ s (γ W ) with ∂P cannot be less then the number of crossings of γ W and it can increase only at instants of tangencies with one of the l j . Such tangencies could be avoided, since they produce null-homotopy 2-gons with the perimeter of P.
Suppose first thatγ is not simple and letγ (α) be an innermost loop. For Lemma (2.6)γ (α) is a simple closed curve and an alternative occurs:γ (α) does not intersect ∂P or it has an even number of crossings corresponding to a part of the word W . In the former case all the centres lie in the region bounded byγ (α) (it cannot be the opposite becauseγ (α) is not null-homotopic) so condition A1 in the Definition 2.15 holds. In the latter case the set C is split in two subsets C int and C ext of those centres that respectively lie in the bounded and unbounded regions of the plane separated byγ (α). Condition (15) assures that both C int and C ext contain at least two centres, thus condition A1 is still satisfied.
In case thatγ is simple, the innermost loop is given by the curve itself and the same argument as before applies.
The above construction of the symbolic dynamic is based on the fact that the edges {l i } i do not intersect with each other so it is possible to label them according to an orientation and to define the suitable distance. However the same theory applies anytime it is possible to define a simple polygonal joining the centres (i.e. a union of non-intersecting line segments that are joined pair-wise to form a closed path), no matter whethere or not it is a regular N -gon. Furthermore the edges {l i } i need not be segments; they could be smooth arcs joining two of the centers. This remark allows us to extend the construction of the symbolic dynamic to the general case, regardless of the location of the N centres. It is enough to draw a simple polygonal (even curvilinear) where each edge joins two of the centres.
Suppose first that all of the N centres are placed along a straight line r , as in Fig. 15a: then we can label them according to the direction of the line and we can define the curvilinear polygonal P with edges l i = c i+1 − c i for i = 1, . . . , N − 1 and the edge l N as any arc joining c 1 and c N without crossing r .
Suppose now that the N centres are not placed on the same straight line so that the interior of the convex hull Fig. 15b. Choose a point p in the interior of H C so that no two centres are aligned with p. (Such a point p always exists: indeed let S be the union of all possible straight lines passing through two of the centres and denote O =H C \ S,H C being the interior of H C . Any point of O satisfies the requirements for p and the set O is not empty; indeed it is of full measure inH C being S a finite union of one dimensional differentiable objects.) Then, consider R large enough so that all the centres {c j } j lie in the ball B R ( p) centreed at p and with radius R, and define the map The set ϕ(C) consists of N points on the circle ∂ B R ( p) so they can be labelled accordingly with an orientation. Denote by {c j } j such points and construct the polygonalP with {c j } j as vertices andl j =c j+1 −c j , j = 1, . . . , N as edges. It turns out that p is within the convex hull of ϕ(C) (indeed if p = i α i c i , α i = 1, it follows that p = i β i ϕ(c i ) where β i = α i |c i − p|/ j α j |c j − p|) and in particular it is inside the polygonalP. Thus, denoting by (ρ, θ ) a system of polar coordinates centreed at p, it follows that any edgel j stays in a strip (0, R) × j so that˚ j ∩˚ k = ∅. Hence, labelling the centres {c j } j so that ϕ(c j ) =c j , the polygonal P with edges l j = c j+1 − c j is simple because each l j lives in the strip (0, R) × j . Fig. 16. The three curves represent γ W in the cases where W is W 1 , W 2 or W 3 respectively For instance, let C = {c 1 , . . . , c 7 } be the set of seven centres, as in Fig. 15b. Then all of the following words: W 1 = l 1 l 3 l 7 l 4 l 2 l 6 , W 2 = l 2 l 5 l 3 l 7 , W 3 = l 2 l 6 l 2 l 7 l 5 l 1 l 6 l 3 l 5 l 7 W 4 = l 4 l 2 l 7 l 5 l 3 l 6 l 1 l 4 l 1 W 5 = l 3 l 6 l 3 l 7 l 5 l 1 l 5 l 1 , generate a curve γ W so that [γ W ] is admissible. Figure 16 depicts the loops γ W 1 , γ W 2 and γ W 3 .