Periodic linear motions with multiple collisions in a forced Kepler type problem

In [ 7 ] the author proved the existence of multiple periodic linear motions with collisions for a periodically forced Kepler problem. We extend this result obtaining periodic solutions with multiple collisions for a forced Kepler type problem. In order to do that we apply the Poincare-Birkhoff theorem.


1.
Introduction. Isaac Newton related the acceleration of a body with the forces exerted over it. Making use of Kepler's laws of planetary motion he understood that the gravitational force exerted by a mass over another was proportional to the inverse square of their distances. Being r the position of one of the masses relative to the other, this givesr ∝ − r |r| 3 . (1) However, if we modify a little bit this equation, i.e., if we make a perturbation, the properties of its solutions are not fully understood, not even in the one dimensional case. In [7], R. Ortega considered the equation where p : R −→ R is a 2π-periodic continuously differentiable function and proved the existence of generalized periodic solutions. The generalized solutions are solutions with collisions, i.e., solutions which may attain the singularity in a discrete set of instants. The author proved the existence of two types of periodic solutions: solutions with exactly one collision in the interval [0, 2π] and, for each N > 1, solutions with exactly one collision in the interval [0, 2π[ and no collision on [2π, 2N π[. In a recent paper [11], L. Zhao proved the existence of generalized quasi-periodic solutions for (2) when p is sufficiently regular.
2. Existence of solutions with one collision. In this section we give some preliminaries essential in the following. Some of these preliminaries have proofs analogous to others obtained in [7] in the case α = 2 and hence we omit them in this paper but they can be found in [8]. The proofs we opted to give can be found in the Appendix. At the end of this section we give a result on the existence of periodic solutions of (3) with exactly one collision in the interval [0, 2M π], for some M ∈ N.
We are dealing with a non-autonomous differential equation. Let us define the energy function as the Hamiltonian of the equation without perturbations and denote it by h. Therefore, define Given a solution u of (3), we will denote h(t) = h(u(t),u(t)).
Following the work in [7], we will consider solutions with collisions, hence we deal with a wide set of solutions. We recall the concept of bouncing solution. Given suitable conditions, it is possible to state an existence and uniqueness theorem concerning these solutions. Below we will introduce the concepts of collision conditions and of generalized Cauchy problem. Bouncing solutions are also called generalized solutions. Proposition 1. Let u(·) be a classical solution of (3) defined in a maximal interval ]t 0 , t 1 [ and t 0 ∈ R. Then we have the following asymptotic expansions: Also we have If t 1 ∈ R a similar result holds when the solution approaches t 1 .
The limit (8) is a consequence of the asymptotic expansions (5) -(6) of the solution (see the deduction of these expansions in the Appendix).
Using the Sundman integral for the α − 1-potential we can remove the singularity in (3). Notice that from (5) we conclude that this is a convergent integral. Let us denote by T the inverse of S. If we consider a solution u of (3), the functions U (s) = u(T (s)) and H(s) = h(T (s)) then U, T and H satisfy Additionally, when U ≥ 0, this system admits the first integral By construction, a solution of (3) satisfying (7) and (8) gives rise to a solution of system (10), contained in I −1 (0) and satisfying the initial conditions Conversely, a nonnegative solution of (10) contained in I −1 (0) and satisfying (12) leads to a nonnegative solution of (3) for t ≥ t 0 satisfying (7) and (8). The details are easy to check and are analogous to those in [7]. In particular the fact that V (0) = 0 follows from Lemma 4.1 or from Proposition 1. Notice that system (10) may not be defined in R 4 for some values of α, which could possibly be an obstacle to the application of existence and uniqueness theorem to conditions (12). However, we can easily avoid the situation by considering the absolute value of U in the regularized system (10). Also we would define I with absolute values on U .
After that the problem would be overcome and we could proceed analogously, since solutions of equation (3) correspond to functions U that are positive and therefore satisfy system (10). Conversely a solution of (10) in an interval where U is positive satisfies (10) and corresponds to a solution of (3). Note that, as α = 2 or α ≥ 4, if p is a C 2 function then the vector field associated with system (10) is C 2 -differentiable in all its domain. We remark that if we had prescribed a value of α lying in (2,4) then this vector field would not be C 2 -differentiable everywhere anymore.
As a consequence of the existence and uniqueness for the Cauchy problem associated to system (10), we can prove that given t 0 and h 0 there exists a unique classical solution of (3) satisfying (7) and (8).
By gluing together consecutive classical solutions and imposing compatibility conditions on collision instants, it is possible to construct a bouncing solution. Of course it is necessary to exclude the possibility that the zeros accumulate at finite time. The solution obtained is by construction unique. This is the content of the following theorem: From now on, we will denote by u(·; t 0 , h 0 ) the unique solution described by the theorem. Now we introduce the successor map which we will denote by P. This map is analogous to the Poincaré map associated to a dynamical system, in the sense that for each (t 0 , h 0 ) such that P n (t 0 , h 0 ) = (t 0 + 2πM, h 0 ) we can associate a 2πM -periodic bouncing solution of (3) with n zeros in [0, 2πM [.
If this value is finite, then by Proposition 1 the energy function of the solution has a finite value at time t 1 (t 0 , h 0 ), which we will denote by h 1 (t 0 , h 0 ). The successor map is defined precisely using these two functions: It is fairly easy to show that this map satisfies the compatibility conditions due to periodicity of equation (3). Additionally, from existence and uniqueness of bouncing solutions we deduce that P is one-to-one. Moreover, we have the following result Proposition 2. There exists a function ψ : R → R ∪ {+∞} such that the domain of P is characterized by The function ψ is 2π-periodic, lower semi-continuous and Finally for each t 0 ∈ R, the function In the proof of this proposition Lemma 4.6 in the Appendix is fundamental. Also comparison with some autonomous equations is used and hence also Lemma 4.3 is applied. The theorem implies that P is a twist map.
If p is a C k function and α ≥ 2k, then P is C k -differentiable. This fact has a very delicate proof. Essentially, the idea is to work with the regularized system and to find the first return instant to the surface by means of the implicit function theorem. Then C k -differentiability of P is a consequence of C k -differentiability of this first return map, which in turn is a consequence of C k -regularity of (14). An analogous proof for the case k = 1 and α = 2 is given in [7]. In [8] it is proved, with an analogous proof to the one in [7] for the particular case, that P is exact symplectic, i.e., the differential form r 1 dθ 1 − rdθ is exact on the cylinder. This means there exists some function V ∈ C 2 (D) such that dV = r 1 dθ 1 − rdθ and V(θ + 2π, r) = V(θ, r), ∀ (θ, r) ∈ D.
We are now in position to apply a version of the Poincaré-Birkhoff theorem given in [7] and used there to prove the main theorem, which generalizes for α 2: 3. Main results. The previous discussion leads immediately to a naïve question. Is it possible, by applying similar methods, fixing M, n ∈ N to deduce the existence of periodic solutions with n collisions in the interval [0, 2M π[? The answer to this question is the ultimate goal of this discussion.
In order to prove there are periodic solutions with period 2M π and containing n collisions in [0, 2M π[, where n, M ∈ N, it suffices to prove that the equation: has a solution in R 2 . Now, the idea behind the proof is to apply a version of the Poincaré-Birkhoff theorem to the iterates P n of the successor map. However, two different questions come up. The first one is if we can apply the version of the Poincaré-Birkhoff theorem used to obtain the previous result. The second one is more delicate. What is the domain where these iterates are well-defined (see Figure 1)?
As it turns out, the answer to the first question is no, we cannot apply the same version of the Poincaré-Birkhoff theorem. The reason for this is that we cannot guarantee anymore that P n is a twist map in the points where it is well-defined. In fact, for example for n = 2, denoting in order to apply the monotonicity property.
Instead, we will use the the Poincaré-Birkhoff theorem in the cylinder (see for example [5]).
which is the lift of an area preserving homeomorphism S : Assume that S is isotopic by homeomorphisms to the identity in the cylinder and that Then S has at least two fixed points.
We recall that we say that two C k -embeddings are C k -isotopic if there exists an homotopy between these two maps, H : B × I → B such that, for each λ ∈ I, H λ is a C k -embedding. We will always denote the unit interval [0, 1] by I throughout the text.
We will follow the steps used by Marò in [6] to construct a map S to which we apply Theorem 3.1 in order to deduce the existence of periodic bouncing solutions with multiple collisions. More precisely we will construct a function S defined in a strip B and which is an extension of P |A where A is a strip contained in B.
To do that we will need to consider a family of equations where p λ depend continuously on λ and state some lemmas on continuous dependence of their solutions on λ. As the continuous dependence on λ is just a consequence of continuous dependence in the regularized system for (16), the proofs of these lemmas are analogous to the ones in [7] for the case without λ and hence we omit them.
is a continuous function of the four variables in the set As a simple corollary of the last lemma, we have the continuity of solutions of (20) on λ.
is continuous.
Also we need the following Then, the solution satisfies: To meet the theorem's conditions we will first prove the following lemma Lemma 3.5. The function P is isotopic to the inclusion in its domain.
Proof. We could prove this fact in different ways, we will proceed by construction. First of all, we note that the successor map associated to an autonomous equation is always isotopic to the inclusion. In fact, given the autonomous equation its successor map, which we will denote by A P (t 0 , h 0 ), is given by the expression This map is easily shown to be isotopic to the inclusion in R 2 , it suffices to consider the isotopy that "stops" time: Going back to equation (3), the idea is that its successor map P is isotopic in its domain D to A P where we choose P = − p ∞ . Then the conclusion that P is isotopic to the inclusion immediately follows.
Note that by Lemma 4.5, the domain of the successor map H λ of the equation where p λ (t) = λp(t) + (1 − λ)(− p ∞ ) and λ ∈ I, contains D. In what follows H λ denotes its restriction to D. Observe that H 1 = P and H 0 = A P and that 3962 CARLOTA REBELO AND ALEXANDRE SIMÕES p λ ∞ 2 p ∞ , for all λ ∈ I. We will prove that H is an isotopy between P and A P . First, let us discuss the continuity of H in all the three variables. We begin by proving that t 1 (t 0 , h 0 , λ) is a continuous function.
Notice that it is possible to choose k ∈ N in order that u n (τ ) andu n (τ ) satisfy (21), ∀ n k. Thus, by Lemma 3.4, we conclude that Now, it is left to check the continuity of the function h 1 (t 0 , h 0 , λ). Consider a point (t 0 , h 0 , λ) and a sequence {(t 0n , h 0n , λ n )} in the same conditions as before.
By simplicity, set h 1n := h 1 (t 0n , h 0n , λ n ) and and By continuous dependence, we can choose a number k 1 ∈ N such that Combining (23) and (24) we get To establish the continuity of h 1 (t 0 , h 0 , λ) we must prove that there is a number k ∈ N such that |h 1n − h 1 | < δ, ∀n k.
Since, by continuous dependence, there is a k 2 ∈ N such that the proof of the continuity is completed. Now, let us notice that for each λ ∈ I, the map H λ is one-to-one, continuous and differentiable since it is a successor map. Moreover, it is a differentiable embedding since its inverse is also a successor map. Therefore, H is an isotopy between P and A P .
In order to apply the Poincaré-Birkhoff theorem we will consider, as previously said, the successor map restricted to a suitable set. To define this set we will need the following lemma.
Proof. By Proposition 2 we know that for each t 0 there exists an unique c 1 (t 0 ) such that t 1 (t 0 , c 1 (t 0 )) = t 0 + 2M π + 1. Also it is clear that c 1 is T -periodic and it is easy to prove that it is continuous. Either t 0 → c 1 (t 0 ) is C ∞ or it is not but by the Fejer-Cesaro theorem we can approximate it by a C ∞ 2π-periodic function c satisfying the desired property. The existence of a 0 follows from the same results and also from the fact that for each t 0 the We can now consider the successor map P and the region contained in the domain of P.
The set A can be transformed into a strip via the area-preserving diffeomorphism 2π 0 c(s)ds. Therefore we will assume that A is the strip R×[−a 0 −1, µ] and keep the letter P to denote the composition CPC −1 . Note that points which verify (15) for this new P lead to points which verify (15) for the previous one. Set where we have chosen b > max{µ, a 0 + 1} and such that P(A) ⊆ B.
Since P is of class C 2 we can apply [6, Lemma 2.4. and the remarks above] and extend P| A : A → B to an area-preserving diffeomorphism g 1 : B → B which is isotopic to the identity, restricted to ∂B is the identity and it is the lift of a diffeomorphism in the cylinder B = S 1 × [−b, b]. Then we can alter it on the regions where its values do not agree with P. Let us write g 1 as where ϕ : B → R is a real function. As g 1 is a continuous extension of P, there exist ε 1 , ε 2 > 0 such that Consider two smooth positive functions on R such that and also the following translations defined in B Proof. The function f is the composition of area-preserving diffeomorphisms. Hence the only thing we need to check is that the points satisfying property (30) are such that their orbit along n-iterates of f , Orb n (x) = {y ∈ B : y = f k (x), k = 0, ..., n} ⊂ A.
Let us examine what is the orbit Orb n (x) of points outside A.
First we look at the image of these points by the map f . In the strip R×[µ+ε 1 , b], which we will refer to as region D 1 (see Figure 2), Analogously, on the region R × [−b, −a 0 − 1 − ε 2 ], which we will denote by D 4 , and thus T (t 0 , h 0 ) < t 0 .

On the strip D
n . Moreover, also by (27), points on this region are subject to the condition

Figure 2. Cylinder B
Now, let us denote f n (t 0 , h 0 ) = (T n (t 0 , h 0 ), R n (t 0 , h 0 )). Points in region D 4 such that their orbit remains in D 4 , cannot satisfy property (30) as T n (t 0 , h 0 ) < t 0 in this set of points. Therefore points in D 4 can satisfy property (30) only if their orbit leaves D 4 . We will return to this case in a moment but first consider what happens to points in the remaining regions. A point such that its orbit remains in D 3 cannot satisfy property (30) because T n (t 0 , h 0 ) < t 0 + 1 < t 0 + 2M π. The same happens trivially in regions D 1 and D 2 because T (t 0 , h 0 ) > t 0 + 2M π in these regions. Therefore we must examine orbits that leave its original region.
Observe that T n (t 0 , h 0 ) > 0 in region A. Therefore we may conclude that points in regions D 1 and D 2 could only satisfy property (30) if its orbit intersected regions D 3 or D 4 . However, we will show that points in regions D 3 and D 4 neither satisfy property (30) nor can their orbits intersect sets D 1 and D 2 in less than n-iterates.
By (27)  Finally points in A which satisfy (30) have n-orbits which cannot intersect D i for each i. In fact it is immediate to see that they cannot intersect D 1 and D 2 . Also they cannot intersect D 3 and D 4 , in fact in order that the orbits intersect these sets the points should lie in R × [−a 0 − 1, −a 0 ] and in this case by Lemma 3.6 they cannot satisfy (30).
We are finally in position to state and prove our main theorem. Proof. We apply Theorem 3.1 to the map S : B → B defined by This map is isotopic to the identity, it is area-preserving and satisfies the boundary twist conditions. Now by Lemma 3.7 the result follows.

4.
Appendix. In this appendix we give the proofs of auxiliary results.

Asymptotic expansions near a collision.
In order to obtain the asymptotic expansions near collisions for the solutions of equation (3), we adapted the analogous proof in [10]. We give here the details. Let us begin with two auxiliary lemmas. Proof. First we prove thatu maintains the sign in a neighbourhood of t 0 . Note that, since u(t) is positive in the interval ]t 0 , t 1 [ and it has a collision at t 0 , there We have just proved that there is a neighbourhood of t 0 in whichu is positive. Multiplying equation (3) byu(t) and integrating we find that To the last integral we apply the mean value theorem and conclude that and so, noting that ξ depends on t, we finḋ So in fact we havė where A is a constant depending just on t * . Therefore, Proof. We have thatḣ (t) =u(t)p(t) is the derivative of the energy function. As before, let us choose a point t * in a neighbourhood of t 0 , say ]t 0 , t 0 +ε[, in whichu is positive. Integrating the derivative of the energy between some t ∈]t 0 , t * [ and t * and then applying the mean value theorem we conclude As p is a bounded function and u stays bounded as t approaches t 0 we conclude that h(t) remains bounded as t approaches t 0 . Now we are ready to obtain the asymptotic expansions.
Proof of (5), (6). First note that We drop the argument on u and its derivatives for simplicity. We know thaṫ and thatü Inserting (33) and (34) in (32) we get Let us define R = u α and b(t) = 2α(α − 1)u α−2 h(t) + αu α−1 p(t). Note that b is a bounded function in a neighbourhood of t 0 .
Then we find thatR = α

Multiplying this equation byṘ and integrating we get
To the second integral, we apply the mean value theorem to conclude that there exists Again let us operate another change of variables. This time let Therefore, substituting in equation (35) we geṫ z = α + 1 2α Integrating between t 0 and t and using the fact that lim t→t0 z(t) = 0 it follows Notice that Furthermore, |ż(ξ)| is bounded in a neighbourhood of t 0 as the function b is also bounded and as z converges to 0 as t approaches t 0 . Therefore we conclude that Going back to the formula (37) we find

ds.
Then, as b is a bounded function, Expanding the integrand we deduce And again expanding the integral, it follows Finally, as u = z 2 α+1 we get after expanding the power We have found the Taylor expansion of u around t 0 . Now, we know thatu = 2 α+1 z 1−α α+1ż . Following the same procedure we used to obtain equation (38), we can write equation (36) in the forṁ Using equation (39) and expanding the power we geṫ

4.2.
Autonomous equation and comparison lemmas. Consider now P ∈ R and the autonomous equationü This reduces to a first order Hamiltonian system, with Hamiltonian function which is a first integral of the system. It is the sum of the kinetic energy K(v) and the potential energy V (u). If P is negative, then each classical solution is defined in a bounded maximal interval. But this is not the case when P is non-negative. If P is positive, classical solutions are defined in a bounded maximal interval if In every case, if the classical solution is defined in a bounded maximal interval then it has a unique maximum u max attained at the midpoint of its interval domain. The length of the domain is given by the positive function .
Also we have the following Proof. Given t 0 , h 0 ∈ R, let u be a classical solution of equation (40), satisfying the collision conditions (7) and (8) and having a bounded maximal interval of definition. First suppose that h 0 and P are both negative. Let us define and recall that u max is the maximum value of u in its interval of definition. As, in this case, V is an increasing function we have and then, by definition of V , Let us split τ (h 0 , P ) into two integrals

PERIODIC MOTIONS WITH MULTIPLE COLLISIONS 3971
Let us prove that the first integral, which we will refer to simply as A, vanishes when h 0 is sufficiently small. For simplicity, we will omit the constant 2 √ 2 for it will not take part in what follows.
Then, using (42) and (44) we conclude that It is not difficult to prove that when P is negative the following hold Therefore, lim Now let us prove that the second integral, to which we will simply refer to as B, also vanishes when h 0 is sufficiently small. Substituting Applying Lagrange's Theorem to the function f (u) = u α−1 we conclude Hence, as P (u − u max ) 0 in [u 0 , u max ] and by (47) we have Also, as u − u max 0 in [u 0 , u max ], we obtain a much simpler inequality Recall that as u has bounded maximal interval, then u P − 1 α , and note that V is strictly increasing on the interval ]0, P − 1 α [. Define the function As P is positive, V (u) < V (u), ∀ u ∈]0, +∞[. Moreover this function is strictly increasing on ]0, +∞[ as its derivative is strictly positive on this interval.
Therefore there exists a unique u 0 ∈]0, +∞[ such that However as h 0 = V (u max ) and V (u 0 ) < V (u 0 ), it follows that From the definition of u 0 and the fact that the function V attains the value h 0 at u max , we can extract formulas for u α−1 0 and u α−1 max , respectively. Dividing these formulas, we get By letting h 0 → −∞ and noting that the second limit in (45) still holds, we conclude that We prove now that the following integral vanishes when h 0 is sufficiently small. Let