On the regularization of the collision solutions of the one-center problem with weak forces

We study the possible regularization of collision solutions for one centre problems with a weak singularity. In the case of logarithmic singularities, we consider the method of regularization via smoothing of the potential. With this technique, we prove that the extended flow, where collision solutions are replaced with a transmission trajectory, is continuous, though not differentiable, with respect to the initial data.


Introduction
In this paper we deal with dynamical systems associated with conservative central forces which are singular at the origin. A classical solution does not interact with the singularity of the force, i.e., it is a path u ∈ C 2 (T, R 2 \ {0}) which fullfils the initial value problem where V (x) ∈ C 2 (R + , R) is the potential of the force and T denotes the maximal interval of existence. As well-known, the two-body problem with an interaction potential V can be reduced to a system of this form where u(t) denotes the position of one of the particle with respect to the center of mass. Accordingly, we shall term collision the configuration u(t) = 0. As the force field diverges at u = 0, collisions are among the main sources of non-completeness of the associated flow. This work studies the possible extensions of the flow through the collision that make it continuous with respect to the initial conditions. We are concerned with weak singularities of the potential, namely logarithms.
The regularization of total and partial collisions in the N -body problem is a very classical subject and, in the years, different strategies have been developed in order to extend motions beyond the singularity [7,6,9,8,3,11,10]. Very roughly, these classical methods rely upon suitable changes of space-time variables aimed at obtaining a smooth flow, possibly on an extended phase space; to this aim, the first step is to determine the asymptotic behaviour of the collision solutions and then the phase space is extended either by means of a double covering, or with the attachment of a collision manifold.
In this paper we consider a further, non classical way of extending the flow, related to the technique of regularization via smoothing of the potential introduced by Bellettini, Fusco, Gronchi [2]. This method consists in smoothing the singular potential and passing to the limit as the smoothing parameter ε and the angular momentum tend to zero simultaneously but in an independent manner (indeed we know that the only collision motions have vanishing angular momentum). This involves an in-depth analysis about the ways the smoothing of the potential coupled with the perturbation of initial conditions lead to define a global solution of the singular problem. This technique, when successful, has the advantage of being extremely robust with respect to the application of existence techniques such as the direct method of the calculus of variation. Let us mention that variational methods have been widely exploited in the recent literature in order to obtain selected symmetric trajectories for N -body problems with Kepler potentials ( [4]).
To begin with, we remove the singularity at x = 0 and we denote with V ε (x) the smoothed function defined as Then we look at the regularized problem P (ε) : Unlike (1), the differential equation (2) is no longer singular, so that the initial value problem admits a global solution in C ∞ ((−∞, +∞); R 2 ) for every choice of the datum (q 0 , p 0 ), provided ∇V (x) is sublinear at infinity 1 . Since we focus on the singularities due to collisions, we fix a ball B 0 (R) of radiusR centered at the origin, where the collision is the only singularity that system (1) can develop and we denote with S(V ) ⊂ R 2 × R 2 the set of initial conditions (q 0 , p 0 ) leading to collision for the system P with |q 0 | ≤R. For everyν ∈ S(V ) let uν(t) ∈ C 2 (T, R 2 ) be the collision solution where T denote the maximal interval of existence such that |uν(t)| ≤R. Denoting with u ε,ν (t) the solution of (2) with initial data ν, we investigate the existence of the asymptotic limit of the paths u ε,ν (t) as (ε, ν) → (0,ν), its relationship with the collision solution uν(t) of the singular system P and the continuity of the limit trajectory with respect to initial data. The definition of regularization considered in [2] is the following.
Definition 1.1. Let V (x) be a singular potential. We say that the problem (1) is weakly regularizable via smoothing of the potential in B 0 (R) if for everyν ∈ S(V ) there exist two sequences (ε k ) k , (ν k ) k tending to 0 andν respectively, such that there exists lim k→∞ u ε k ,ν k = u 0 and the flowũ is continuous with respect to ν.
In addition we say that Definition 1.2. The singular one centre problem (1) is strongly regularizable via smoothing of the potential if there existsR such that for everyν ∈ S(V ) there exists and the flowũ is continuous with respect to ν.
In both the definitions we mean that the limit of the regularizing paths u ε,ν (t) and the continuity of the extended flow are held in the ball B 0 (R).
In [2] the authors prove that in the case of homogeneous potential of degree α, V (x) = 1 |x| α , α > 0, the one-centre problem is weakly regularizable via smoothing of the potential if and only if α is in the form where n is a positive integer or α > 2. On the other hand they show that the homogeneous problem is never strongly regularizable via smoothing of the potential. Indeed, a necessary condition in order to achieve the uniform limit (3) is that the apsidal angle ∆θ l (u) of a solution of the system (1) has to converge to π 2 as the angular momentum l tends to zero (see the definition of apsidal angle in the next section). This condition is never satisfied by α-homogeneous potentials α > 0, since ∆θ l (u) → π 2−α > π 2 as l → 0 [2,12]. This explains why the one centre problem with homogeneous potential can not be regularized according to definition 1.2. Conversely, when the logarithmic potential is considered, it can be proved ( [12]) that the limiting apsidal angle do indeed converge to π/2. Then there is no obstruction and we could expect that the limit (3) is attained. This fact suggests to extend the motion after a collision by reflecting it about the origin. We will show that, in this way, not only for the logarithmic potential, but for a larger class V * of potentials, the problem is regularizable according to definition 1.2.
The sets of potential functions we will consider in this paper are the following.
is decreasing with respect to x and iv. d dx The properties iii,iv guarantee the existence of a T > 0 such that LetR := min{T, S}. The set V includes potentials having homogeneous singularities and weaker. For instance the logarithmic potential, V (x) = − log(x), as well as the homogeneous potentials, V (x) = |x| −α , provided α ∈ (0, 1), belong to V. On the other hand condition v. can be considered as a logarithmic type property or a zero-homogeneity property: indeed it is never satisfied by homogeneous potentials, while the logarithmic potential is a prototype of all the functions satisfying condition v..
Our main goal is the following: Theorem 1. For every V (x) ∈ V * the one centre problem is regularizable according to definition 1.2 whereR is given in (6).
In the particular case of logarithmic potential, V (x) = − log(x), one hasR = +∞, therefore The paper is organized as follows. In section 2 we follow the classical method for dealing with central problem based on first integrals and we derive the set S(V ) of initial conditions leading to the collisions for the unperturbed system. Next, in section 3, given any collision solution u(t), we set the initial dataν ∈ S(V ) and we define the family of paths u ε,ν (t). Section 4 contains the proof of the main theorem and the analysis of the regularity of the extended flow. The main part of the proof consists in proving the existence of the limit of the path u ε,ν (t) as (ε, ν) → (0,ν), especially for what that concerns the angular part, (Theorem 2). This is the most delicate step, for the it involves the uniformity of the limit as (ε, ν) → (0,ν), and it allows to conclude the strong regularizability of the problem.
It results that the natural extension of the collision solution is the transmission solution, see definition 4.1, obtained by reflecting the motion through the collision. The regularity of the extended flow is carried on in section 4.2: in theorems 3 and 4 the continuity of the Poincaré map and Poincaré section with respect initial data is achieved.
Furthermore, in order to have a complete picture of the problem, in section 5 we join a variational approach and we analyse the variational properties of the collision paths.

Preliminaries
For any choice of the potential function V (x) ∈ C 2 (R + , R) the one centre problem (1) is a hamiltonian system and admits two integrals of motion: the energy E and the angular momentum l: Since the conservation of the angular momentum implies the motion is planar, in the following l is used to denote the modulo of the angular momentum, rather than the vector. The radial symmetry of the equation of motion (1) suggests to introduce the polar coordinates in the plane (r, θ). In this setting the quantities E and l are expressed in the form We define f (r) = 2r 2 (E + V (r)) (8) then the relation (7) reads as l 2 = f (r) − (rṙ) 2 and shows that a solution of system (1) of energy E and angular momentum l exists only for those values of the radial coordinate r ≥ 0 such that f (r) ≥ l 2 .
For fixed values of E and l, we denote with R + , if it exists, the minimum positive value of r such that f (r) = l 2 and f ′ (r) < 0, R + = +∞ otherwise, and we denote with R − , if it exists,  the minimum positive value of r < R + such that f (r) = l 2 and f ′ (r) > 0, R − = 0 otherwise, see figure 1(a). By definition, it follows that R − = 0 for collision solutions and R + = +∞ for unbounded orbits. In any case we term R + and R − the maximal and minimal value of the angular coordinate and we refer to them as the apsidal values of the orbit. Moreover, following the terminology adopted in celestial mechanics, we sometimes refer to R + and R − respectively as the apocentre and the pericentre of the orbit. As it is well known, in case of non collision and bounded trajectories, the radial coordinate r(t) oscillates periodically between its extremal values R + and R − while the angular coordinate θ(t) covers an angle equal to between each singular oscillation of r(t). We term the angle ∆θ l (u) the apsidal angle, see figure  1(b). The knowledge of the apsidal values R + and R − and the value of the apsidal angle is sufficient to determine the behaviour of the solution since the whole trajectory is obtained repeating periodically the part of path between a point where r(t) is maximum and the following point where r(t) is minimal.
The definition of the apsidal angle extends in a natural way for unbounded and collision solutions: in the first case the orbits is composed by a single oscillation of the radial coordinate from infinity to R − and back to infinity and the apsidal angle represents the angle covered by the particle coming from infinity to the pericentre and it is obtained replacing R + = +∞ in the previous relation. On the other hand, if a collision occurs, the apsidal angle denotes the increment of the angular coordinate between the apocentre and the collision point.
In order to characterise the set S(V ) of initial data leading to a collision we give the following definition.
A similar classification of singular potentials can be found in a work of Gordon [5] where a potential V (x) is said to satisfy a strong force condition at a point x 0 if V (x) tends to infinity as x tends to x 0 and also there exists a function U (x) with infinitely deep wells at x 0 , such that in a neighbourhood of x 0 . We notice that, among the homogeneous potentials, the set of potentials with the property to be of strong type and the ones satisfying the Gordon's strong force condition coincide. Clearly, an α-homogeneous potential is of weak type if and only if α ∈ (0, 2) and in these cases a collision occurs only in zero angular momentum orbits [8], while if α ≥ 2 a collision solution exists also for non-zero values of the angular momentum [2,8]. The next proposition extends this result.

) is a potential of weak type, a solution u(t) of the dynamical systemü = ∇V (|u|) ends into a collision if and only if the angular momentum is zero. Proof
Denote with E and l the energy and the angular momentum of the solution u(t) and let f (r) as in (8). As mentioned before, a solution exists only for the values of radial coordinate r satisfying l 2 ≤ f (r). Suppose l = 0: since V (x) tends to infinity as x goes to zero, for every value of E there exists a neighbourhood of the origin where E + V (r) > 0 then the solution presents a collision.
Conversely, since V (x) is a weak type potential, it follows that f (r) → 0 as x → 0 + thus for every value of l = 0 there exist a neighbourhood of the origin where l 2 > f (r). Hence the collision can not be attained on solutions with non zero angular momentum.
for every ξ ∈ (0, T ). Therefore, again by integration, we infer and we conclude lim Propositions 2.1, 2.2 show that, for any choice of V (x) ∈ V, a solution of the system (1) ends into a collision if and only if the angular momentum is zero. Therefore the set S(V ) of initial conditionsν = (q 0 ,p 0 ) that make the solutions to be singular consists in those (q 0 ,p 0 ) satisfying l = |q 0 ∧p 0 | = 0.

Setting
For every fixed V (x) ∈ V letR be the quantity defined in (6) and let B 0 (R) be used to denote the ball of radiusR around the origin. The properties of the potential class V assure that the collision is the only source of singularity for the dynamical system inside B 0 (R). As discussed in the introduction, given a collision solutionū(t) for the one central problem (1), our intent is to define an extension in B 0 (R) for the solutionū(t) beyond the collision.
To this we first have to set the initial conditionsν = (q 0 , p 0 ) for the singular pathū(t): we denote with P the first positive solution of equation f (r) = 0, P := +∞ if such value does not exists. We remind that the angular momentum l is zero for collision solution, hence the orbits drawn byū(t) is a straight line joining some point in the plane with the origin.
An alternative occurs: The collision solutionū(t) is bounded in a ball centered in the origin of radius P <R and P represents the maximal value of the radial coordinate ( Figure 2). Without loss of generality, we can set the initial conditionν of the collision solutionū as Case2 P ≥R In this case the collision solution is not bounded in B 0 (R) and it could also be unbounded. We focus our analysis only on the portion of path bounded byR hence we select as initial condition forū(t) the coupleν = (q,p), |q| =R, where the initial velocityp is directed toward the center of attraction ( Figure 3). In both the cases, for ε > 0 let u ε,ν (t) be a solution of system (2) leading from an initial data ν = (q 0 , p 0 ). We refer to u ε,ν (t) as regularizing paths in order to underline the purpose to define the extension for the singular solutionū(t) as a limit of u ε,ν (t) as (ε, ν) → (0,ν). The smoothing of the potential does not affect the hamiltonian structure of the system, therefore the angular momentum l and the energy E ε,ν = 1 2 |u ε,ν | 2 − V ε (|u ε,ν |) are conserved along the solution u ε,ν (t). Decomposing the initial velocity p 0 in terms of the parallel and orthogonal component with respect to q 0 , It turns out that also the apsidal values of the regularizing paths have to converge to the corresponding ones ofū(t), indeed as (ε, ν) → (0,ν), the pericentre R − of the solution u ε,ν (t) tends to zero while the apocentre R + is bounded byR and tends to P in case 1, while R + >R and possibly R + = +∞ in case 2.
In the following sections we will deal with the existence and the property of the limit for the paths u ε,ν (t) as (ε, ν) → (0,ν). As discussed in [2], the behaviour of the angular coordinate of the regularizing paths plays a fundamental role for the existence of the limit of u ε,ν (t) uniformly as (ε, ν) → (0,ν) rather then for subsequences (ε k , ν k ) → (0,ν). For this reason and since we focus our analysis only inside the ball B 0 (R), we extend the definition of apsidal angle for the solution u ε,ν (t) as follows: we denote with ∆θ(u ε,ν ) the apsidal angle of the path u ε,ν (t) in case the trajectory is bounded byR, case1 otherwise, in case 2, we denote with ∆θ(u ε,ν ) the angle covered by the path u ε,ν (t) between the point where u ε,ν (t) enter in the ball B 0 (R) and the point of minimal distance from the origin 4 Proof of Main Theorem and property of the extended flow The proof of theorem 1 is composed by two parts: first, in section 4.1 we prove the existence of the limit of the trajectories u ε,ν (t) as (ε, ν) → (0,ν) and we define the extension of the singular solution, then in section 4.2 we study the regularity of the extended flow. A necessary condition for the existence of the limit (3) is the existence of the limit of the angular part of the regularized solutions. Theorem 2 concerns the asymptotic of ∆θ(u ε,ν ) as (ε, ν) → (0,ν): to this aim we first prove in lemma 4.4 the L 1 boundness of the integrand in (13) and (14) then we apply the dominated convergence theorem and pass to the limit under the integral sign. The boundness of the integrand is a consequence of a technical estimate stated in the proposition 4.1 and it is attained for every potential V (x) ∈ V, while the existence of the limit is a consequence of proposition 4.3 based on assumption v.. The result we obtain suggest to define the extension u 0 (t) of the collision solutionū(t) beyond the singularity as a transmission solution.
In order to gain the regularity of the extension, we analyze, in theorems 3 and 4, the continuity of the Poincaré map and Poincaré section of the extended flow in the phase space.

4.1
The existence of the limit of u ε,l (t) Proposition 4.1. Let V (x) ∈ V andR as in (6). Then for everyr <R there exists anε such that ∀ε <ε and for every 0 < y <r it holds for every y ≤ x ≤r.
The proof of the proposition 4.1 is split in two parts: we first show that there existsε > 0 such that for every ε <ε the function F (y,r) (ε, x) > F (y,r) (0, x) for every 0 < y ≤ x ≤r < S , then we show that F (y,r) (0, x) ≥r for every 0 < y ≤ x ≤r <R.
Proof By means of straightforward calculations and reminding the definition of smoothed potential V ε (·) = V ( (·) 2 + ε 2 ), the relation (16) is equivalent to For every s > 0 we define the function U (s) = V ( √ s). Obviously the function U (s) inherit property i. and property ii. for every s ∈ (0, S 2 ), while the relation is increasing for every s ∈ (0, S 2 ). The proof of the lemma follows proving that for every choice of 0 < y <r < S 2 there existsε > 0 such that ∀ε <ε and ∀x ∈ (y,r) it holds Let 0 < y <r < S 2 be fixed and let the function g(ε, x) be defined as g(ε, x) : The sign of the derivative is given by the sign of the function We observe that G (y,r) (y) = G (y,r) (r) = 0 and Since G (y,r) (x) is continuous, there exists at least one pointx ∈ (y,r) where dG (y,r) dx (x) = 0; the proof of the lemma follows once we prove the inequalities Indeed, if (20) hold, the function G (y,r) (x) and, consequently the derivative in (18) are negative for every x ∈ (y,r). To obtain the relations (20) we multiply both the sides in (19) for √ x and divide them for U ′ (x)(U (r) − U (y)). We obtain By definition ofx it follows N (x) = 0. Moreover, since √ s U ′′ (s) U ′ (s) is increasing for every s ∈ (0, S 2 ) and the factor (U ′ (y) − U ′ (r)) (U (r) − U (y)) is positive, we infer that N (x) is increasing in x. Therefore N (x) < 0 for x <x and N (x) > 0 otherwise and, since U ′ (x)(U (r) − U (y)) > 0, inequalities (20) hold.

Proof
Reminding the definition ofR and R + , we define β = min{R, R + } therefore, for every u ε,l (t), regardless they are bounded or not byR, we write In order to deal with the convergence of ∆θ(u ε,ν ) we first rewrite the integrand in a different way. From the conservation of energy it followṡ thus, replacing the radial velocity and extracting the roots R − and β from the denominator, we infer By means of the change of variables ρ = r R− we rewrite the integral in the form We proceed with the proof of the theorem as follows: first we exhibit, for every V (x) ∈ V, an uniform bound for the function K provided ε small enough is taken, then we apply the Lebesgue's theorem in order to obtain the limit of ∆θ(u ε,ν ) as (ε, ν) → (0,ν) and we'll prove that such limit exists if V (x) ∈ V * . Proof We observe that v is the zero in case β = R + . Subtracting the energy formula We note that for every 0 < a < b, the function f (z) = a+z b+z is increasing for positive z. The condition ii. in definition (1.3) implies that, for every small enough ε, the function V ε (x) is decreasing with respect to x for every x < β. This yields the relations − ρ 2 − 1 and, by means of the substitutions x = ρR − and y = R − Finally, applying proposition (4.1), we obtain the uniform bound Continue the proof of theorem 2. The boundness of the functions K 1 and K 2 and the formula implies that all the integral (24) is bounded by πβ.
This suggests to extend the collision solutionū(t) beyond the singularity replacing symmetrically the solution itself forward the collision point in the same direction.
In order to complete the proof of theorem 1 it remains to show that, for every t ∈ [0, 2T 0 ], the sequence {u ε,ν (t)} pointwise converges to u 0 (t) as (ε, ν) → (0,ν) and that the flow obtained replacing the collision solutionū with the transmission solution u 0 is continuous with respect initial data.
For every initial dataȳ = (q,p), |q| ≤R, leading to collision for the system P (0), let y(t) = (x(t),ẋ(t)) : [0, T 0 ) → F be the corresponding singular solution where T 0 denote the collision time and |x(t)| ≤R for every t ∈ [0, T 0 ). We extendȳ(t) according to the definition 4.1 defining y 0 (t) = (x 0 (t),ẋ 0 (t)) as The extension for the collision solutions allows to define the flow beyond the singularity, therefore for ε ≥ 0 we denote with Φ ε (y, t) : F × R + → F the flow associated to the system P (ε). Our aim is to study the continuity of Φ ε (y, t) with respect to initial data y ∈ F and ε as (y, ε) → (ȳ, 0); to this end we consider the Poincaré map Φ T (y, ε) defined as the solution at time T of the system P (ε) with initial value y and we show that We note that the continuity of the Poincaré map implies the proof of theorem 1.
Remark 4.1. We can not expect the continuity of the Poincaré map in T 0 because, even if the configurations x T0 (y, ε) would converge to x T0 (ȳ, 0), the limit can not be attained by the sequencė x T0 (y, ε), sinceẋ T0 (ȳ, 0) is unbounded.
If T < T 0 no problem arises, indeed the above limit comes from the classical theorem of continuity with respect initial data of ordinary differential equations.
On the other hand, for T > T 0 the continuity of the Poincaré map is stated in the following theorem.
Theorem 3. Let V (x) ∈ V * and supposeȳ = (q,p), |q| ≤R be an initial condition leading to collision for the system P (0) at time T 0 . Then for every T > T 0 such thatẋ(T ) = 0 .
In the proof of the theorem we will need the following classical lemma Proof of theorem (3) As usual we set (r, θ) the polar coordinates of the plane then a point y = (x,ẋ) in the phase space is replaced by x = (r cos θ, r sin θ) x = (ṙ cos θ − rθ sin θ,ṙ sin θ + rθ cos θ) From the definition of Φ T (y, ε) remain well defined the functions and r T (y, ε), θ T (y, ε),ṙ T (y, ε),θ T (y, ε) denoting, respectively, the values of the radial and angular coordinate and their velocity at time T = T 0 of a solution of system P (ε) with initial data y. The continuity of Φ T (y, ε) is equivalent to the continuity of each of the previous functions. Letȳ = (r,θ,ṙ,θ) be an initial data in the phase space leading to collision with nonzero initial velocity,ṙ < 0, and denote withĒ = 1 2ṙ 2 − V (r) the energy of the collision solution.
A point y ∈ F, y = (r 0 , θ 0 ,ṙ 0 ,θ 0 ), tends toȳ if it holds We start showing the continuity of the function r T (y, ε) as y →ȳ and ε → 0. The value of r T (y, ε) is governed by the differential equationṙ = 2(E + V ε (r)) − l 2 r 2 , thus r T (y, ε) is a function of the initial position r 0 , the couple E, l and the parameter ε. Define T 0 (E, l, r 0 , ε) as the time necessary to the solution r(y, t) to reach the minimal value R − = R − (E, l), then Claim The function T 0 and T are continuous respect to r 0 , E, l, ε. Suppose for the moment that the claim is true, since the function T is strictly increasing with respect to r T (y, ε), for lemma (4.5), the function r T (y, ε) is continuous with respect to the set of parameters E, l, r 0 , ε. Therefore lim y→ȳ ε→0 r T (y, ε) = r T (ȳ) .
The continuity of θ T (y, ε) is equivalent to the continuity of ∆θ T (y, ε) := θ T (y, ε) − θ 0 . For the definition of transmission solution ∆θ T (ȳ) = π, while, in theorem (2), we showed that the apsidal angle associated to a solution of the perturbed system tends to π 2 whenever the initial data tends to a colliding one and the parameter ε tends to zero. Thus we gain lim y→ȳ ε→0 The continuity ofṙ T (y, ε) andθ T (y, ε) follows immediately by the continuity of r T (y, ε) and relationsṙ Proof of the claim Denoting with p andp the sets of parameters p = (E, l, r 0 , ε) andp = (Ē, 0,r, 0), we have to show that lim We want to apply the dominated convergence theorem and pass the limit under the integral sign in To this aim we first exhibit an L 1 bound for the integrand function. We observe that the integrand become unbounded only for ρ approaching to R − since we have chosen the initial valueṙ < 0 and, as a consequence, we can supposeṙ 0 < 0. Let β ∈ (R − , r 0 ) and write Since for every ρ ∈ [β, r 0 ] the integrand is bounded, it follows II ≤ C 1 (r 0 − β). In order to show the L 1 boundness of the first integrand, using the definition of the energy integral, we rewrite I in the form Again from the definition of energy, evaluated in r 0 and in the pericentre R − , Figure 4: Poincaré section and, replacing the last relation in the integrand, we obtain For proposition 4.1 there exists a positive constant K such that, for every ε small enough, In order to obtain an uniform bound for every choice of R − , we perform the variable change z = ρ − R − and conclude We now apply the Lebesgue theorem and we obtain thus the function T 0 and, for the same reason, the function T are continuous with respect to the parameter p.
Let againȳ be an initial condition leading to collision for the system P (0), T 0 the collision instant and, for a choice of T > T 0 , we denote y 1 = Φ(ȳ, T ) according to the definition 4.1 of extension of collision solution. Given Σ an hyperplane in the phase space passing through y 1 and transversally to the flow, we show that for every initial data y nearȳ, there exists a time t = τ (y) when the trajectory Φ(y, t) intersects the hyperplane Σ. As the data y changes, the trace S(y) = Φ(y, τ (y)) drawn on Σ is called the Poincaré section of the flow on Σ, see figure 4. In the next theorem we prove the continuity of the map τ (y) and the continuity of the Poincaré section in a neighbourhood ofȳ. Theorem 4. Let V (x) ∈ V * . Then there exists a δ > 0 and a continuous function τ (y) defined in a δ-neighbourhood ofȳ, N δ (ȳ), such that τ (ȳ) = T and Φ(y, τ (y)) ∈ Σ Moreover the Poincaré section is continuous in N δ (ȳ).

Proof
Let F a vector in the phase space such that F · f (y 1 ) > 0 and consider the hyperplane By definition of y 1 it holds (Φ(ȳ, T ) − y1) · F = 0 and, since there exists an ξ > 0 such that For theorem (3) and for the sign permanence theorem it follows that there exists a δ > 0 such that for every |ȳ − y| < δ For every fixed y the function (Φ(y, t) − y1) · F is increasing in t: indeed thus the continuity of the vector space f (y) out of collision points and the continuity of the orbit Φ(y, t) with respect to both the variables imply that f (Φ(y, t)) · F > 0. Therefore for every y ∈ N δ (ȳ) there exists a time τ (y) such that Φ(y, τ (y)) ∈ Σ and τ (ȳ) = T . In order to prove the continuity of τ (y) we define H(y, τ ) = (Φ(y, τ ) − y 1 ) · F then τ (y) is the implicit solution of H(y, τ (y)) = 0. Since H is continuous in y and it is continuous and increasing with respect to τ , for lemma (4.5) the function τ (y) is continuous. Moreover, for composition of continuous functions we infer the continuity of the Poincaré section.

Variational property of the collision solutions
In this section we join a variational approach that consists in seeking solutions of the system (1) as critical points of the action functional where L(u,u) = 1 2 |u(t)| 2 + V (|u(t)| is the lagrangian function associated to the equation of motion. This method is well known in the literature and it has been extensively exploited in order to find periodic solutions for the Nbody problem, see for instance [1,4,5] and the references therein. Besides the discussion about the existence of minimal paths, an interesting question is whether the collisions are avoided by the minimal paths even in presence of weak potentials whose contribution one could expect to be negligible by a variational point of view.
In the following theorem we prove that, despite of the weakness of the singularity of every potential V (x) ∈ V * , a minimal path for the action functional can not have a collision in the interior of its domain.

Proof
In order to prove that u 0 (t) is not the minima for the action A among all the paths in Λ, we perform a variation on the trajectory u 0 that removes the collision and makes the action decrease. Let Let us compute the difference ∆A = A(u 0 ) − A(u 1 ) and show that for every δ sufficiently small ∆A is positive: this means that u 0 (t) is not a local minimum of the action functional. We study separately the kinetic and the potential contribute.
Since the variation v δ is directed orthogonally to u 0 , we gain |u 1 (t)| 2 = |u 0 (t)| 2 |t| ≤ T 1 |u 0 (t)| 2 + δ 2 (T −T1) 2 We show that, for every δ small enough, the contribution of the potential part is bigger than the penalising contribution, due to the kinetic part.
For every t fixed For δ sufficiently small the functions f δ (t) are positive and for Fatou's Lemma The last relation holds since the radial differential equationr(t) = −V ′ (r(t)) is satisfied, being the collision solution u 0 a radial solution. Combining (28), (29) and (30) we conclude for every δ small enough.