The symmetric periodic orbits for the two-electron atom

We analyse the existence of symmetric periodic orbits of the two-electron atom. The results obtained show that there exist six families of periodic orbits that can be prolonged from a continuum of periodic symmetric orbits. The main technique applied in this study is the continuation method of Poincaré.


Introduction
The estimate of the ground state of the energy of the helium atom was a cornerstone in the evolution of quantum mechanics. In 1980s and 1990s (see [5,6]), it was presented some shortcomings of the old quantum theory, and one of the most relevant subject was: to understand the behaviour of the periodic trajectories when the classical dynamics is non-integrable or even chaotic.
In this context, we study in this paper the existence of periodic trajectories of the system that governs the dynamics of the collinear two-electron atom and B Durval José Tonon djtonon@ufg.br Jaume Llibre jllibre@mat.uab.cat 1 Institute of Mathematics and Statistics, Federal University of Goiás, Avenida Esperança s/n, Campus Samambaia, Goiânia, Goiás CEP 74690-900, Brazil investigate when this periodic orbits can be prolonged, depending on a small parameter.
Many distinct techniques can be applied for studying the existence of periodic orbits, for example, numerical analysis, averaging theory, Melnikov functions, normal forms, variational methods, among others. However, one of the first analytical studies of the existence of periodic orbits was done by Poincaré in [9].
The continuation method of Poincaré, originally presented in [9], consists in providing conditions for extending a given periodic solution with a parameter equal to zero to a solution with small but positive values of the parameter. For more details about this method, see [3], for example.
In [4], [12] and [11] was presented the model of collinear helium atom, that consists of two electrons of mass m e and charge −e moving on a straight line with respect to a fixed positively charged nucleus of charge +2e. The Hamiltonian modelling the collinear helium system presented in [4], [12] and [11] is given by hereṙ 1 = p 1 ,ṙ 2 = p 2 and Z represents the charge of the nucleus of the helium atom.
Considering the change of coordinates given by r 1 = 2/Zr 1 , r 2 = 2/Zr 2 and the rescaling of time t = (Z /2) 2 t, we obtain that the Hamiltonian H 1

becomes the Hamiltonian
where μ = 2/Z , the derivative is computed considering the new time t and we omit the tilde over the variables.
In what follows we study the Hamiltonian system defined by the Hamiltonian H , assuming that μ = 0 that represents the critical case when the charge of the nucleus of the helium tends to infinity, and we obtain results about the existence of families of periodic orbits for small values of μ (that corresponds to large values of charge Z ).
In the previous paper [7], we applied the same approach to study the symmetric periodic orbits for the charged collinear 3-body problem, which consists of three different masses each one with different charge, and we do not have any of the three masses fixed, as the collinear helium system has. This is a strong difference between these two systems. This difference originates completely different Hamiltonian models. In fact, the Hamiltonian that models the collinear charged 3-body problem is of the form where the μ i 's are related to the masses of the three bodies and the e i j 's with their masses and charges. Moreover, we note that the terms in the Hamiltonian H 2 different from the term 1 2 μ 3ṙ1ṙ2 cannot be non-dimensionalized as in the Hamiltonian H because the masses m 1 and m 3 , as their charges are not equal as in H . In this way, the results about the existence and continuation of periodic orbits obtained in the paper [7] cannot be applied to the collinear two-electron atom. We organize this paper as follows: In Sect. 2 we provide the equations that model the dynamics of the two-electron atom; in Sect. 3 we study the symmetries of the periodic solutions of this system; and in Sect. 4 we present the main results. Considering the parameter μ = 0, the symmetric periodic orbits are studied in Sect. 5, and in Sect. 6 we apply the continuation method of Poincaré to extend the periodic solutions obtained in the previous section for μ = 0 to small positive values of the parameter μ. A brief conclusion and some comments comparing the periodic orbits of the uncharged, charged symmetric three-body problem with the ones of the two-electron atom are presented in Sect. 7.

Differential equations that govern the classical motion of the two-electron atom
In the literature, there are few rigorous results about the general three-body Coulomb problem and the main reason for this is that the equation that governs the motion are multidimensional, non-integrable and singular. In order to obtain a model of the equations that govern the motion, an essential ingredient for the classical analysis of the three-body Coulomb problem is the regularization of its equations of motion, see [12] and [11]. We assume that the particles are in position x 1 , x 2 , x 3 ∈ R, respectively, such that 0 < x 1 < x 2 < x 3 , and we 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 . Let L = T − U be the Lagrangian associated with this system, where T, U denotes the kinetic and potential energy, respectively. We introduce the variables Then the Hamiltonian that governs the two-electron atom is given by Associated with Hamiltonian (1), we have the system Applying the Levi-Civita transformation, see [8], given by the trajectories of the two-electron atom (2) in the new coordinates are the solution of the system on the energy level H = h for some constant h. System (3) is a Hamiltonian system with a Hamiltonian G given by with G = 0 if and only if H = h. System (3) is analytic except when ξ 2 1 + ξ 2 2 = 0, that corresponds to the triple collision.
We want to study the periodic orbits of the two-electron atom with binary collisions that correspond to the parameter μ = 0. Considering these periodic solutions, our objective is to study the periodic solutions of system (3) for μ > 0 sufficiently small, satisfying the energy relation G = 0. In this way in the following section we explore some symmetries involving this system.

Symmetries of the two-electron atom system
The results obtained in this section are similar to the ones given in [2,7] for the collinear uncharged and charged three-body systems, respectively. Consider the involutions We say that the solution ϕ(s) = (ξ 1 (s), ξ 2 (s), η 1 (s), η 2 (s)) of system (3) is invariant under the symmetry S i if S i (ϕ(s)) is also a solution of system (3) with i ∈ {1, . . . , 7}. We say that ϕ(s) is S i -symmetric if S i (ϕ(s)) = ϕ(s).
Observe that the set of involutions {I d, S 1 , . . . , S 7 } with the usual composition forms an abelian group isomorphic to Z 2 × Z 2 × Z 2 . These symmetries usually appear in Hamiltonian systems, see, for example, [1], [10] and [7]. Note that the symmetries S 1 , S 2 and S 3 generate the other ones. So we only consider the symmetric periodic orbits with respect to the symmetries S 1 , S 2 and S 3 . Using similar arguments to the ones presented in [1] and [7], the following proposition holds.
Moreover the following result, proved in [2], shows that there are no symmetric periodic solutions having more than two symmetries.

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

Main results about the symmetric periodic orbits
As explored in [7], the periodic solutions of the differential equation (3) of the twoelectron atom, which are simultaneously S 1 and S 2 symmetric, are denoted by S 12symmetric periodic solutions. Similarly, we have S 13 -and S 23 -periodic solutions.
The results on the S 12 -symmetric periodic solutions for small and positive values of μ are given in the next theorem.
Theorem 1 Consider h = h 1 + h 2 < 0, p and q odd positive integers. Then the S 12 -symmetric periodic solutions of the two-electron atom (3) for μ = 0 and with the initial conditions where h 1 = p q 2 3 h 2 , can be continued to a μ-parameter family of S 12 -symmetric periodic orbits of the two-electron atom (3) for μ > 0 small.
The S 13 -symmetry periodic solutions are given by the next result.
Theorem 2 Consider h = h 1 + h 2 < 0, p odd and q even positive integers. Then the S 13 -symmetric periodic solutions of the two-electron atom (3) for μ = 0 with initial conditions h 2 , can be continued to a μ-parameter family of S 13 -symmetric periodic orbits of the two-electron atom (3) for μ > 0 small.
Finally for the S 23 -symmetric periodic solutions, we obtain:

even and q odd positive integers. Then the S 23 -symmetric periodic solutions of the two-electron atom
where h 1 = p q 2 3 h 2 can be continued to a μ-parameter family of S 23 -symmetric periodic orbits of the two-electron atom (3) for μ > 0 small.

Symmetric periodic orbits for µ = 0
In the Levi-Civita coordinates (ξ 1 , ξ 2 , η 1 , η 2 ), system (3) with μ = 0 is given by with the Hamiltonian , that can be rewritten as if we rescale the time by 4ξ 2 1 ξ 2 2 . Note that the level of energy ) the solution of system (4) satisfying the energy condition H = h, we define the new times σ and τ : Considering these new times, we obtain the system of differential equations in coordinates (ξ 1 , η 1 ) and (ξ 2 , η 2 ) satisfies the system of differential equations Consider the functions G 1 = G/ξ 2 2 and G 2 = G/ξ 2 1 , i. e.

Proposition 3 Consider the periodic solutions
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. The number p in Proposition 3 represents the number of binary collisions between m 1 and m 2 , and q is the number of binary collisions between the particles m 2 and m 3 . We emphasize that our interest in this paper is to study the symmetric periodic orbits of system (3) satisfying the energy relation G = 0. So in the following we exhibit conditions under the initial points, given in Proposition 3, to get symmetric periodic orbits.

Proposition 4
The following statements hold.
is a S 13 -symmetric periodic solution. (c) If p is even and q is odd, then the solution ϕ(s) given in Proposition 3 with initial conditions is a S 23 -symmetric periodic solution.

Remark 1 Note that the Levi-Civita transformation duplicates the number of orbits.
Hence it is sufficient to consider the positive square roots of the initial conditions given in Proposition 4.

Applying the continuation method of Poincaré to obtain symmetric periodic solutions for µ > 0 small
In Proposition 4, we have the values for the initial conditions to get S 12 -, S 13 -, S 23symmetric periodic orbits for system (3) when μ = 0. We study the symmetric periodic orbits for small positive values of μ using the continuation method of Poincaré.

Case (b)
Similarly to the previous case the solution ϕ is a S 12  Again by statement (a) of Proposition 4 we obtain the initial conditions and the numbers p and q to have S 12 -symmetric periodic solutions (see Table 1 then there exist unique analytic functions ξ 10 = ξ 10 (μ) and S = S(μ) defined for μ ≥ 0 sufficiently small that satisfy (i) ξ 10 (0) = ξ * 10 and S(0) = S * , (ii) and ϕ(s; ξ 10 , 0, 0, η 20 , μ) is a S 12 -symmetric periodic solution of system (3) with period S = S(μ) satisfying the energy condition G = 0.

Concluding remarks
In this paper, we have studied the periodic solutions of the two-electron atom 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 [2] and [7], it was studied similar problems, but in that papers the authors studied the collinear 3-body problem in the cases where the bodies are uncharged and charged, respectively. In both cases, it was applied the continuation method of Poincaré. In the first one only three families of periodic orbits can be extended from μ = 0 to μ small and positive and in the case where the bodies are charged, similar as we obtain in this paper, six families of symmetric periodic orbits can be extended.