SYMMETRIC PERIODIC ORBITS FOR THE COLLINEAR CHARGED 3-BODY PROBLEM

In this paper we study the existence of periodic symmetric orbits of the 3-body problem when each body possess mass and an electric charge. The main technique applied in this study is the continuation method of Poincaré.


Introduction
One of the most relevant objects to study in the theory of dynamical systems is the n-body problem and many work has been done for understanding its dynamics. Thus the study of its periodic orbits is one of the main objectives. In this paper we focus the attention on the periodic orbits of the 3-body problem when the three bodies are collinear and charged.
Recently many distinct techniques and methods have been used to prove the existence of periodic orbits for the n-body problem, for example, averaging theory, numerical analysis, Melnikov functions, normal forms, variational methods, among others. One of the first analytical studies of the existence of periodic orbits for the n-body problem was done by Poincaré in [13], and we apply his method to study the symmetric periodic orbits of the charged collinear 3-body problem.
There exists a large literature studying the existence of periodic solutions of the n-body problem, see [10] and [11] for example. More precisely, if we restrict our attention to the 3-body problem, Hénon in [8] has studied numerically the existence and stability of a class of symmetric rectilinear periodic orbits of the general problem of three bodies. In [2] the authors studied, numerically, families of symmetric periodic orbits for the collinear 3-body problem when the two non-central masses are equal. In [9] the singularity generated by the triple collision of bodies of the collinear 3-body problem is studied.
In [5] the authors studied the symmetric periodic orbits by the continuation method of Poincaré of the collinear 3-body problem when the bodies do not have electric charges. In the present paper we allow that the bodies posses electric charges.
The continuation method of Poincaré was originally presented in [13] and this method consists in given a periodic solution for the system with a parameter equal to zero and it provides conditions for extending this solution to small values of the parameter. For more details about this method see for example [6].
The organization of this paper is as follows. In Section 2 the equations that model the dynamic of the collinear charged 3-body problem are described, in Section 3 we present the main results, and in Section 4 we study the symmetries of the periodic solutions of this system. The study of symmetric periodic orbits for the parameter µ = 0 are done in Section 5, and in Section 6 we apply the continuation method of Poincaré to extend the periodic solution obtained in previous section for µ = 0 to small and positive values of the parameter µ. A brief conclusion and some comments comparing the periodic orbits of the charged with the ones of the uncharged system are presented in Section 7.

Equations of motion of the collinear charged 3-body problem
The charged n-body problem studies the dynamics of n particles with a positive mass and an electrostatic charge of any sign, moving under the influence of the respective Newtonian and Coulombian forces. There exist many studies for particular values of n, in [7] it is considered the charged rhomboidal four-body problem, in [1] the charged isosceles 3-body problem, in [4] the restricted charged four body problem and in [12] the central configurations of the charged 3-body problem, among others.
In this paper we study the charged collinear 3-body problem. Taking conveniently the units of mass and charges we can assume, without loss of generality, that the gravitational constant and the Coulomb's constant are equal to one. The particles posses masses m 1 , m 2 and m 3 with charges q 1 , q 2 and q 3 in the position x 1 , x 2 , x 3 ∈ R, respectively.
The differential equations that governs the motion are given by the symbol ∇ x i U denotes the gradient of U with respect to x i , λ ij = m i m j − q i q j for i, j = 1, 2, 3 and i = j, and d(x i , x j ) denotes the Euclidean distance between the points x i and x j . THE CHARGED 3-BODY PROBLEM   3 We denote the masses and electric charges of the bodies b 1 , b 2 and b 3 by m 1 = µ(1 − ν), m 2 = 1 − µ, m 3 = µν and λ 1 = µα, λ 2 = µ 2 β, λ 3 = µγ, respectively, where 0 ≤ µ < 1, 0 < ν < 1 and α, β, γ ∈ R. We assume that the three bodies are in position x i such that 0 < x 1 < x 2 < x 3 , see Figure  1. Consider the change of coordinates given by z 1 = x 2 −x 1 and z 2 = x 3 −x 2 , that denotes the distance between x 2 and x 1 and x 3 and x 2 . Then the kinetic T and the potential U energies of the 3-body problem are given by

SYMMETRIC PERIODIC ORBITS FOR
where C = m 1q1 + m 2q2 + m 3q3 is the linear momentum. With this notation we have that if λ ij > 0 then the resultant force between the particles i and j is attractive, and if λ ij < 0 then its repulsive. Without loss of generality we suppose that C = 0, i.e. the center mass is in rest. Writing L = T − U and denoting the new variables by we obtain the Hamiltonian H of the system given by Associated to the Hamiltonian (1) we have the system By the rescaling of the variables z 1 = µ 2 z 1 , z 2 = µ 2 z 2 , t = µ 3 t system (2) becomes where we omit the tilde in the variables. The Hamiltonian associated to system (3) is given by Similarly to the case of the uncharged particles system (3) possess one singularity at z 1 = 0, that corresponds to a binary collision between m 1 and m 2 , another at z 2 = 0 which corresponds to the collision between m 2 and m 3 , and finally one at z 1 + z 2 = 0 which represents a triple collision. Doing a Levi-Civita transformation, see [9], given by the trajectories of the collinear charged 3-body problem (3) in the new coordinates are the solution of the system (4) on the energy level H = h for some constant h. System (4) is a Hamiltonian system with a Hamiltonian G given by with G = 0 if and only if H = h. Note that system (4) is analytic except when ξ 2 1 + ξ 2 2 = 0, that corresponds to the triple collision. We study the periodic orbits of the collinear charged 3-body problem with binary collisions between m 1 , m 2 and m 2 , m 3 . Considering these periodic solutions, our objective is study the periodic solutions of system (4) for µ > 0 sufficiently small, satisfying the energy relation G = 0, more precisely we are interested in the symmetric periodic orbits of system (4). In this way will be necessary to study the symmetries involving this system.
The periodic solutions of the differential equation that governs the dynamic of the collinear charged 3-body can be simultaneously S 1 and S 2 symmetric, see Section 4. These kind of periodic solutions will be called S 12 -symmetric periodic solutions. In analogous way, we have S 13 -and S 23periodic solutions.
Our results on the S 12 -symmetric periodic solutions for small and positive values of µ are given in the next theorem. Theorem 1. Consider ν ∈ (0, 1), α > 0, γ > 0, h = h 1 + h 2 < 0 and p and q odd positive integers. Then the S 12 -symmetric periodic solutions of the charged 3-body problem (4) for µ = 0 and with the initial conditions , can be continued to a µ-parameter family of S 12 -symmetric periodic orbits of the charged 3-body problem (4) for µ > 0 and small.
For the periodic solutions having S 13 -symmetry we obtain the next result.
Finally for the S 23 -symmetric periodic solutions, we get: p is even and q is odd positive integers. Then the S 23 -symmetric periodic solutions of the charged 3-body problem (4) for µ = 0 with initial conditions , can be continued to a µ-parameter family of S 23 -symmetric periodic orbits of the charged 3-body problem (4) for µ > 0 and small.
Note that the set {Id, S 1 , . . . , S 7 } with the usual composition forms an abelian group isomorphic to Z 2 × Z 2 × Z 2 . This kind of symmetries usually appear in Hamiltonian systems, see for example [3] and [14]. Note that the symmetries S 1 , S 2 and S 3 generate the other ones. In fact, Here we only consider the symmetric periodic orbits with respect to the symmetries S 1 , S 2 and S 3 . The periodic solutions of system (4) can be simultaneously S 1 and S 2 symmetric. These kind of periodic solutions will be called S 12 -symmetric periodic solutions. In analogous way we have the S 13 -and the S 23 -periodic solutions. Using similar arguments to the ones presented in [3] we can prove the following propositions.
Furthermore the next result, proved in [5], shows that there are no symmetric periodic solutions having more than two symmetries.

Proposition 2.
There are no periodic solutions of system (4) which are simultaneously S i -symmetric for i = 1, 2, 3.

Symmetric periodic orbits of system
with Hamiltonian G given by In coordinates (z 1 , z 2 , p 1 , p 2 ) the Hamiltonian H, for µ = 0, is given by serve that the flow of system (5) on the energy level H = h is given by the flow of the Hamiltonian H 1 (z 1 , p 1 ) on the energy level H 1 = h 1 , and by the flow of the Hamiltonian H 2 (z 2 , p 2 ) on the energy In the Levi-Civita coordinates the Hamiltonians H 1 and H 2 are given by Consider the solution ϕ(s) = (ξ 1 (s), ξ 2 (s), η 1 (s), η 2 (s)) of system (5) satisfying the energy condition G = 0 (or equivalently H = h) and we define the new times σ and τ as follows Note that (ξ 1 , η 1 ) satisfies the system of differential equations and (ξ 2 , η 2 ) satisfies the system of differential equations Therefore considering the new times σ and τ the functions G 1 = G/ξ 2 2 and G 2 = G/ξ 2 1 are These two functions are the Hamiltonians of system (8) and (9) respectively. Our objective now is to study the periodic solutions of systems (8) and (9). Thus, fixing h 1 < 0 we can integrate system (8) directly with the initial conditions ξ 1 (0) = ξ 10 and η 1 (0) = η 10 , obtaining the solution (ξ 1 (σ), η 1 (σ)) given by (10) is a periodic solution of system (8) with period σ = 2π/ω 1 . Assuming that solution (10) satisfies the energy relation G 1 = 0 we obtain the following relationship between the initial conditions ξ 10 and η 10 : Moreover by the parametrization of the time given in (7) we obtain the period of this solution in terms of the time t, i.e.
for some p, q ∈ N coprime, then ϕ(s) is a periodic solution of system (5). (c) Let s(t) be the inverse function of For the h 1 given in (b) the period and the quarter of period in times σ, τ, t and s are given in Table 1.
Observe that by statement (c) of Proposition 3 we have that dt/ds > 0 when there are no collisions, and zero in the binary collisions. Therefore the inverse function s = s(t) exists always that the system has no triple collision, and it is differentiable if there is no binary collisions. As in the uncharged case studied in [5], page 128, the number p in Proposition 3 represents the number of binary collisions between m 1 and m 2 , and the number q the binary collisions between the particles m 2 and m 3 .
We stress that the main objective of this paper is to analyse the periodic orbits of system (4) satisfying the energy relation G = 0. So the next proposition provides the initial conditions in order to prove that the solutions of system (4) are symmetric. Consider the initial conditions given in Proposition 3.
is a S 23 -symmetric periodic solution.

Concluding remarks
In this paper we study the periodic solutions of the collinear charged 3-body problem which are S 12 -, S 13 -and S 23 -symmetric. Applying the continuation method of Poincaré we obtain that six families of symmetric periodic orbits can be extend from µ = 0 to small positive values of µ. In [5] it was studied a similar problem, but in that paper the bodies are uncharged and applying the continuation method of Poincaré only three families of periodic orbits can be extended from µ = 0 to µ small and positive.
If we consider that families the values of the charges q 1 , q 2 , q 3 tends to zero then, for each symmetry S ij considered, we observe that one of the families of periodic orbits converge continuously to one of the families of periodic orbits given in [5], and the other one the continuation method cannot be applied because the determinant of the partial derivatives of the system is zero. In fact, the expression of the charges q i in terms of the parameters