PERIODIC ORBITS OF THE PLANAR ANISOTROPIC KEPLER PROBLEM

In this paper we prove that at every energy level the anisotropic problem with small anisotropy has two periodic orbits which bifurcate from elliptic orbits of the Kepler problem with high eccentricity. Moreover we provide approximate analytic expressions for these periodic orbits. The tool for proving this result is the averaging theory.


Introduction
The anisotropic Kepler problem is a modified model of the Kepler problem.This model can be used to describe the motion of two-body in an anisotropic configuration plane under a mutual gravitational attraction described by Newtonian's universal law of gravitation.
The anisotropic Kepler problem comes originally from quantum mechanics, in this model the flat space is replaced with an anisotropic one, it was introduced by Gutzwiller in [8,9,10,11], and after studied by other authors see for instance Devaney [3,4,5], and Casasayas and Llibre [2].
Gutzwiller in his researches wanted to find an approximation of the quantum mechanical energy levels for a chaotic system.He chooses to study the anisotropic Kepler problem because it is a chaotic system and it is considered one of the most suitable models to study physical phenomena faced in semiconductors.
1.1.The planar anisotropic Kepler problem.The equations of motion of the planar anisotropic problem in Hamiltonian formulation are (1) q , where the potential is Here |ε| > 0 denotes a small parameter and the prime denotes derivative with respect to the time variable t.
We change these equations of motion to the McGehee coordinates (r, θ, v, u), more precisely we choose polar coordinates (r, θ) in the configuration space, together with scaled versions (u, v) of the radial and angular components of velocity as follows (q 1 , q 2 ) = r(cos θ, sin θ), for more details in these coordinates see [4,5,14].So in the McGehee coordinates the equations of motion (1) write ( 2) where and, the energy level The equations of motion (2) have a collision singularity at r = 0, which can be removed with the following change in the independent variable The equations of motion (2) in the new time τ goes over to here the dot denotes derivative with respect to τ .
In short, we have that if ε = 0 the equations of motion (4) corresponds to the planar Kepler problem.If |ε| > 0 and small, system (4) provides the equations of motion of the planar anisotropic Kepler problem.1.2.Our main result.In this paper we study analytically the periodic orbits of the planar anisotropic Kepler problem in its energy levels using the averaging theory.
Theorem 1.In every energy level (3) the anisotropic Kepler problem (4) has two periodic orbits bifurcating from two elliptic orbits of the Kepler problem with the same eccentricity 0.95502193.. and with the arguments of the pericenter given by θ 0 = 0.90077616.. and 2π − θ 0 radians.
The values of the eccentricity and of the argument of the pericenter which appear in the statement of Theorem 1 are zeros of explicit functions given in the proof of this theorem in section 2 We note that the existence of periodic orbits for the spatial anisotropic Kepler problem has been studied topologically by Devaney in [4] and Casasayas and Llibre in [2].As far as we know this is the unique article which studies the periodic orbits of the anisotropic Kepler problem analytically.Gutzwiller [8] has studied numerically some periodic solutions of the anisotropic Kepler problem.
Note that our definition of the anisotropic Kepler problem coincides with the one of Gutzwiller, Devaney, Casasayas and Llibre.There are other different problems also called anisotropic Kepler problem, as the ones studied by Diacu, Pérez-Chavela and Santoprete [6], Escalona-Buendía and Pérez-Chavela [7], and López, Martínez and Vera [12].These last authors also study analytically some periodic orbits of a different anisotropic Kepler problem using Delaunay variables.

Proof of Theorem 1
In what follows we shall study the periodic orbits of the anisotropic Kepler problem in the energy levels H = h ∈ R.So in every energy level H = h we compute r = r(θ, v, u, h), and we get Then the equations of motion (4) in the energy level H = h taking as independent variable the variable θ become which has the general solution This is the solution of the planar Kepler problem with eccentricity e and being θ 0 the angle of the direction of the pericenter, i.e. the argument of the pericenter.Of course, if e = 0 the solution is circular, if e ∈ (0, 1) the solution is elliptic, if e = 1 the solution is parabolic, and if e > 1 the solution is hyperbolic.
We are interested in knowing what are the periodic solutions of the Kepler problem which can be extended to periodic solutions of the anisotropic Kepler problem, i.e. what solutions (6) with eccentricity e ∈ [0, 1) can be extended.For doing this study we shall use the averaging theory of first order described in the appendix.
In order to apply the averaging theory to the differential system (5) we identify our variables with the ones of averaging theory given in the appendix, and we obtain where The first variational equation ( 11) of the unperturbed system (5) along the periodic solution ( 6) with e ∈ [0, 1) is where , and consequently The fundamental matrix M z (θ) of the differential system (7) such that M z (0) is the identity matrix is We compute the integral F (z) of ( 12) and we obtain Now we have to study the solutions (e, θ 0 ) of the system f 1 (e, θ 0 ) = 0, f 1 (e, θ 0 ) = 0 with e ∈ (0, 1).Note that since in the expressions of the functions f 1 (e, θ 0 ) and f 2 (e, θ 0 ) the e appears in the denominators we cannot consider e = 0, i.e. with the averaging theory of first order we cannot study if some circular periodic solutions of the Kepler problem can be extended to a periodic solution of the anisotropic Kepler problem.For solving the system f 1 (e, θ 0 ) = 0, f 2 (e, θ 0 ) = 0 with e ∈ (0, 1), we first solve the equation f 2 (e, θ 0 ) = 0 with respect to e, and we obtain Since e ∈ (0, 1), then the first two zeros e 1 and e 2 are not acceptable, and the third zero e 3 is only valid for the values of θ 0 satisfying the conditions sin(3θ 0 ) > sin θ 0 and sin(3θ 0 ) sin θ 0 > 0, which are equivalent to say that e 3 ∈ (0, 1).Now let g(e 3 , θ 0 ) be the numerator of the function f 1 (e, θ 0 ) evaluated at e = e 3 .This function has ten zeros in the interval θ 0 ∈ [0, 2π) as it is shown in Figure 1.These zeros are θ 1 0 , θ 2 0 , ...., θ 10 0 and the corresponding values of the eccentricity e 1 0 , e 2 0 , ...., e 10 0 related with these zeros are given in Table 1.It is noted that there are only two values of e 3 ∈ (0, 1), namely e 2 0 and e 7 0 , these two values coincide, i.e. e 2 0 = e 7 0 = 0.95502193.., while sin θ 7 0 = − sin θ 2 0 , and this means that θ 7 0 = 2π − θ 2 0 .

Appendix: Basic results on averaging theory
We want to provide sufficient conditions for the existence of T -periodic solutions for the differential systems being Ω an open subset of R n .We assume that the unperturbed system has a submanifold M of dimension n filled of periodic solutions of period T .
As usual we denote by M z (t) some fundamental matrix of the linear differential system (11).
Suppose that there exists an open set V with Cl(V ) ⊂ M such that for each z ∈ Cl(V ), x(t; z, 0) is T -periodic.Then the next result provides sufficient conditions in order that the some periodic solutions of V persist for values of ε = 0 sufficiently small.Theorem 2 (Perturbations of an isochronous set).We assume that there exists an open and bounded set V with Cl(V ) ⊂ M such that for each z ∈ Cl(V ), the solution x(t; z, 0) is T -periodic, then we define the function F : Cl(V ) → R n as (12) F (z) = 1 T T 0 M −1 z (t)F 1 (t, x(t; z, 0))dt.
Then the following statements hold.