The Circular Hill Problem Regarding Arbitrary Disturbing Forces: The Periodic Solutions that are Emerging from the Equilibria

: In this work, sufﬁcient conditions for computing periodic solutions have been obtained in the circular Hill Problem with regard to arbitrary disturbing forces. This problem will be solved by means of using the averaging theory for dynamical systems as the main mathematical tool that has been applied in this work.


Introduction
The three-body problem, as it is known, is a special case of the n−body problem where the motion of three point masses under their mutual gravitational interactions is described. This classical problem represents a large range of astronomical situations. Because a general solution of the problem exists but it is unusable in practice, some simplifications to it have been made. The Hill problem is an example of this. There are many studies on the Hill problem [1][2][3][4][5][6][7][8][9][10][11], and many modifications of this classic problem have appeared in the mathematical, astronomical, and mechanical literature. For example, in [6], Markellos and Roy introduced a photogravitational Hill problem as a simplified, non-trivial form of the photogravitational restricted three-body problem. In [8], Papadakis made a mainly numerical study of this problem on the planar case. On the other hand, in [2], the authors focused their attention on the case of a perturbed spatial Hill lunar problem using the averaging theory, and in [3], Chavineau and Mignard studied the trajectories of the Hill problem to describe the effect of solar perturbations on the relative motion of a binary asteroid. In a central force field, the binary objects case can be approached from potential [4].
Our main goal in this study was to find, using the averaging theory, sufficient conditions for the existence of periodic orbits that come from the Lagrangian points of the circular Hill problem with arbitrary disturbing forces.
Firstly, we assumed that the first two masses were small, compared to the third one (i.e., we assumed m 1 and m 2 were small as compared to M), and that the two small masses were close together, as compared to their separation from the third mass.
In the case of the circular Hill problem, the equations of the relative motion are [3,4]: where G is the gravitational constant, m = m 1 + m 2 , and ω is the angular velocity of the motion of the barycenter of the m 1 -m 2 system around M.
The equilibrium solutions of the differential system (1) are given by In this work, we provide sufficient conditions for the existence of periodic solutions coming from the equilibrium point, L 1 . The same conditions are valid for the equilibrium, L 2 .
Making the change of variables x = q 1 − Gm 3ω 2 1/3 , y = q 2 and z = q 3 , and linearizing, we can rewrite the unperturbed system of differential Equations (1) in the form: Using the averaging theory, we will provide a system of nonlinear equations whose simple zeros provide periodic solutions of the perturbed system: x − 9ω 2 x − 2ωẏ = εF 1 (t, x,ẋ, y,ẏ, z,ż), y + 2ωẋ + 3ω 2 y = εF 2 (t, x,ẋ, y,ẏ, z,ż), where the parameter ε is small, and F 1 , F 2 and F 3 are the smooth and periodic functions in the variable t in resonance p : q, (p, q positive integers relatively prime) with some of the periodic solutions for ε = 0.
As usual, we are going to call a simple zero of the system G(X 0 ) = 0 to any solution X 0 * = (X 0 * 1 , X 0 * 2 ), such that det ∂G ∂X 0 | X 0 =X 0 * = 0. If X 0 * is a simple zero, then the T 1 -periodic solution of the unperturbed system with X 0 * as the initial value gives a T 1 -periodic solution of the perturbed system (see Appendix A).
Given an initial condition, we can use this result in order to find sufficient conditions for the existence of families of periodic orbits. Theorem 1. Suppose that the functions of the equations of motion of (3), F 1 , F 2 , and F 3 are smooth and periodic functions of the pT 1 /q period in the variable t, with p and q being positive integers that are relatively prime, and where the ratio of the frequencies ω 2 /ω 1 are not resonant with π. If ε = 0 is sufficiently small, then for every simple zero that is not null, X 0 * , of the nonlinear system G(X 0 ) = 0, the system (3) has a periodic solution (x(t, ε), y(t, ε), z(t, ε)) such that (x(t, ε), y(t, ε), z(t, ε))→(x(t), y(t), z(t)) = (σ 1 1 (t), σ 1 3 (t), σ 1 5 (t))| X 0 =X 0 * , when → 0-that is, when the periodic solution for the unperturbed system (2) travelled p times.
Let G = (G 3 , G 4 ) be the Malkin bifurcation function (see [12]) for system (3) and , · · · , σ 2 6 (t)), i = 1, 2, and As has been stated before, if Y 0 * is a simple zero of G(Y 0 * ) = 0, then the T 2 -periodic solution of the unperturbed system with Y 0 * as the initial value gives a T 2 -periodic solution of the perturbed system (see Appendix A).

Theorem 2.
Suppose that the functions of the equations of motion of (3), F 1 , F 2 , and F 3 , are smooth and periodic functions of the pT 2 /q period in the variable t, with p and q positive integers relatively prime, and the ratio of the frequencies ω 2 /ω 1 not resonant with π. If ε = 0 is sufficiently small, then for every simple and not null zero, Y 0 * of the nonlinear system G(Y 0 ) = 0, the system (3) has a periodic solution ( when the periodic solution for the unperturbed system (2) travelled p times.
The proof of Theorem 2 is given in Section 2. In the next corollary, an application of Theorem 2 is shown, whose proof is implemented in Section 3.

Proof of Theorems 1 and 3
The differential system (3) must be written as a first-order differential system defined in R 6 . For that, the variables (x 1 , x 2 , y 1 , y 2 , z 1 , z 2 ) = (x,ẋ, y,ẏ, z,ż) have been introduced in the following form, this being: It must be remarked that if ε = 0, the differential system (5) is equivalent to the differential system (2), well-known as the unperturbed system. Otherwise, when ε = 0, this is known as the perturbed system The change of variables (x 1 ,x 2 ,ȳ 1 ,ȳ 2 ,z 1 ,z 2 ) → (x 1 , x 2 , y 1 , y 2 , z 1 , z 2 ) given by writes the linear part of the differential system (5) in its real Jordan normal form, and this system in the new variablesx = (x 1 ,x 2 ,ȳ 1 ,ȳ 2 ,z 1 ,z 2 ) becomeṡx where In order to prove Theorems 1 and 2, in the following lemma we are going to obtain the general expressions of the periodic orbits of the unperturbed system. Lemma 1. If = 0, the T 1 −periodic solutions of the system of the differential Equations (7) arē x 1 (t) = X 0 1 cos(ω 1 t) + X 0 2 sin(ω 1 t), x 2 (t) = X 0 2 cos(ω 1 t) − X 0 1 sin(ω 1 t), y 1 (t) = 0, and the T 2 −periodic solutions arex Proof of Lemma 1. If ε = 0, the system (7) is a system of linear differential equations, and the proof follows easily.
Proof of Theorem 1. Since the positive integers p and q are relatively prime, and the functions F i , i = 1, 2, 3, of the system (3) are pT 1 /q−periodic in the variable t, we can consider that the system (7) and the solutions (8) are pT 1 −periodic. Now, we shall study what periodic solutions of the unperturbed system (7) with ε = 0 of the type (8) persist as periodic solutions, with the perturbed one for ε = 0 being sufficiently small.
If we write the system (7) in the forṁ we can apply Theorem A1 of the Appendix A. Our starting point is the description of the different elements which appear in this theorem for the particular case of the differential system (7). In this case, Ω = R 6 , k = 2, and n = 6. Now, let r 1 > 0 be arbitrarily small, and let r 2 > 0 be arbitrarily large. If V is the open and bounded subset of the planex i = 0, i = 3, . . . , 6, of the form V = {(X 0 1 , X 0 2 , 0, 0, 0, 0) ∈ R 6 : where || · || is the Euclidean norm in R 2 . If we denote by Cl(V), the closure of V, the function β : Cl(V) → R 4 is β(α) = (0, 0, 0, 0). Using all these elements in our system, we have Now, for each z α ∈ Z, we consider the pT 1 −periodic solution x(t, z α ) = (X 1 (t), X 2 (t), 0, 0, 0, 0) given by (8).
Proof of Theorem 2. The proof is analogous to the proof of Theorem 1 changing the roles of T 1 for T 2 , and we obtain the periodic solution In consequence, a periodic solution (x, y, z)(t, ε) of system (3) has been obtained, so that when ε → 0.

Proof of Corollaries 1 and 2
Proof of Corollary 1. Under the assumptions of Corollary 1, we have the function and det ∂G ∂X 0 Therefore, the solution X 0 * = 0, 3 4 3ω of the nonlinear system G(X 0 ) = 0 is simple, and by Theorem 1 we only have one periodic solution for the system of this corollary.

Proof of Corollary 2.
Under the assumptions of Corollary 2, we have the function , 0 of the nonlinear system G(Y 0 ) = 0 is simple.
By means of Theorem 1, the system of this Corollary has only this periodic solution.
Author Contributions: All the authors have contributed in an equal way to the research in this paper.

Conflicts of Interest:
The authors declare no conflict of interest.

Appendix A. Basic Results on Averaging Theory
In this appendix we present the basic result from the averaging theory that we shall need for proving the main results of this paper.
We consider the problem of the bifurcation of T−periodic solutions from a differential system of the formẋ with = 0 sufficiently small. Here the functions G 0 , G 1 : R × Ω → R n and G 2 : R × Ω × (− 0 , 0 ) → R n are C 2 functions, T−periodic in the first variable, and Ω is an open subset of R n . The main assumption is that the unperturbed systemẋ has a submanifold of periodic solutions. A solution of this problem is given using the averaging theory. Malkin (see [13] and references therein) studied the bifurcation of T-periodic solutions in the T-periodic systemẋ = G 0 (t, x) + εG 1 (t, x, ε), whose unperturbed system has a family of T-periodic solutions with initial conditions given by a smooth function β : R k → R n , and proved that if the bifurcation function G(α) = T 0    < u 1 (t, α), g(t, x(t, β(α)), 0) > · · · < u k (t, α), g(t, x(t, β(α)), 0) >    dt, where u i , i = 1, . . . , k, are k linearly independent T-periodic solutions of the adjoint linearized differential system, has a simple zero α 0 such that det(G)| α=α 0 = 0, then for any ε > 0 sufficiently small, the systemẋ = G 0 (t, x) + εG 1 (t, x, ε) has an unique T-periodic solution x ε such that x ε (0) → β(α 0 ) as ε → 0.
This can be rephrased as follows: Let x(t, z, ) be the solution of the system (A2) such that x(0, z, ) = z. We write the linearization of the unperturbed system along a periodic solution x(t, z, 0) aṡ y = D x G 0 (t, x(t, z, 0))y.
In what follows we denote by M z (t) some fundamental matrix of the linear differential system (A3), and by ξ : R k × R n−k → R k the projection of R n onto its first k coordinates; i.e., ξ(x 1 , . . . , x n ) = (x 1 , . . . , x k ). We assume that there exists a k-dimensional submanifold Z of Ω filled with T−periodic solutions of (A2). Then an answer to the problem of bifurcation of T−periodic solutions from the periodic solutions contained in Z for system (A1) is given in the following result.
Theorem A1. Let V be an open and bounded subset of R k , and let β : Cl(V) → R n−k be a C 2 function. We assume that (i) Z = {z α = (α, β(α)), α ∈ Cl(V)} ⊂ Ω and that for each z α ∈ Z the solution x(t, z α ) of (A2) is T-periodic; (ii) for each z α ∈ Z there is a fundamental matrix M z α (t) of (A3) such that the matrix M −1 z α (0) − M −1 z α (T) has in the upper right corner the k × (n − k) zero matrix, and in the lower right corner a (n − k) × (n − k) matrix ∆ α with det(∆ α ) = 0.