THE 3 – DIMENSIONAL CORED AND LOGARITHM POTENTIALS : PERIODIC ORBITS

We study analytically families of periodic orbits for the cored and logarithmic Hamiltonians H(x, y, z, px, py, pz) = (px +p 2 y +p 2 z/q)/2+(1+x 2 + (y2 +z2)/q2)1/2, and H(x, y, z, px, py, pz) = (px +p 2 y +p 2 z/q)/2+(log(1+x 2 + (y2 + z2)/q2))/2, with 3 degrees of freedom, which are relevant in the analysis of the galactic dynamics. First, after introducing a scale transformation in the coordinates and momenta with a parameter ε, we show that both systems give essentially the same set of equations of motion up to first order in ε. Then the conditions for finding families of periodic orbits, using the averaging theory up to first order in ε, apply equally to both systems in every energy level H = h > 0. The averaging method used proves the existence of at most three periodic orbits, for ε small enough, and gives an analytic approximation for the initial conditions of these periodic orbits.


Introduction
In this paper we are interested in 3 degrees Hamiltonian systems of the form H(x, y, z, p x , p y , p z ) = 1 2 where V a smooth potential with an absolute minimum and a reflection symmetry with respect the three axes.The motivation for the choice of these symmetries becomes from the interest of these potentials in galactic dynamics.In particular, we considered the cored potential (1) and the logarithm potential (2) ) .
Our goal is to study the periodic orbits of the corresponding Hamiltonian differential system using the averaging theory.
Belmont et al. [8] applied the method of resonant detuned normal forms to investigate properties of the logarithmic galactic potential.These forms are obtained with a method based on the Lie transformation which is analogous to the Prendergast method applied by Contopoulos and Seimenis.In Pucacco et al. [7], they showed that it is possible to find periodic orbits with this method.
The averaging method has already been applied to others particular galactic potentials.Jiménez-Lara and Llibre [3] obtained two families of periodic orbits in the plane, when the parameter q is irrational.Lacomba and Llibre [4] studied the potential with a general perturbation of fourth order and obtained four families of periodic orbits in the plane, when the parameter q is equal a 1.The parameter q gives the ellipticity of the potential, which ranges in the interval [0.6, 1].
In this paper, we find new families of periodic orbits parameterized by the energy and depending on the parameter q.

Statement of the problem
We shall study the periodic orbits of the Hamiltonians systems associated to the (3) ) , called the cored and logarithm Hamiltnonians, respectively.
The cored Hamiltonian system is , and the logarithmic Hamiltonian system is ) , ) .
After introducing a non-canonical scale transformation with a small parameter ε > 0, Note that in what follows the six variables (x 1 , y 1 , z 1 , p x 1 , p y 1 , p z 1 ) are denoted again by (x, y, z, p x , p y , p z ) respectively.Thus both Hamiltonian systems ( 5) and ( 6) can be reduced to study the same differential system (7) This is the cored Hamiltonian system.The logarithm Hamiltonian system has the small modification that, instead of ε, it has 2ε.Then we proceed the study of the system (7) which includes both Hamiltonian systems.
We summarize our main results as follows.
We remark that the periodic solutions γ 1 (t, ε) of Theorems 1 and 2 bifurcate from the same unperturbed periodic solution in each energy level h, but the results are different because in Theorem 1, q is irrational and in Theorem 2, q is rational.Theorem 3. If q is irrational and q 3/2 is rational, then for ε > 0 sufficiently small, at every energy level H = h > 0 the perturbed differential system (7) has at least one periodic solution γ(t, ε) = (x(t, ε), y(t, ε), z(t, ε), p x (t, ε), p y (t, ε), p z (t, ε)) such that γ(0, ε) −→ (0, √ 2hq, 0, 0, 0, 0) when ε → 0.Moreover, the families of periodic solutions γ(t, ε) bifurcates from periodic solutions of system (7) with ε = 0 living in the plane (0, y, 0, 0, p y , 0).We cannot study using the averaging theory if from the families of periodic solutions living in the space (x, 0, z, p x , 0, p z ) of system (7) with ε = 0 under the assumptions of Theorem 3 bifurcate to ε > 0 same families of periodic solutions.The problem is that we cannot compute some integrals which appear when applying the averaging method.
On the other hand, the case that remains also not covered by Theorems 1, 2 and 3 is the case q and q 3/2 rationals.In this case, system (7) with ε = 0 has a family of periodic solutions which fills the space (x, y, z, p x , p y , p z ).But again in this case when we apply the averaging method appear integrals that we cannot compute.
We observe that the families of periodic solutions γ 1 and γ 2 of Theorem 1, and the one of Theorem 3 have already appeared in the work [3] where this problem was studied with only 2 degrees of freedom.

Proof of Theorem 1
We consider the case q, q 1/2 and q 3/2 are irrational and we obtain three family of periodic orbits.
Proof of Theorem 1 for case k = 1.We consider the unperturbed periodic solutions in the plane (x, 0, 0, p x , 0, 0).Note that these periodic solutions living in the plane (x, 0, 0, p x , 0, 0) are not in resonance with the other periodic solutions of system (7) for ε = 0 living in the planes (0, y, 0, 0, p y , 0, 0) and (0, 0, z, 0, 0, p z ) due to the fact that q and q 3/2 are irrational.The first integral (8) when ϵ = 0 takes on these periodic solutions the value We apply the First Order Averaging Theorem, see Theorem 4 of the appendix, to every fixed energy level, H = h > 0. This allows to eliminate one of the coordinates, in this case p x , and to reduce the study to dimension 5. Then p x at the energy level H = h with h given by ( 9) is where we choose the plus sign for the determination of the square root, but the results that we shall obtain will be the same choosing the minus sign.This will be the case in all the proofs of this paper and we do not mention this in the next proofs.
The equations of motion ( 7) on the energy level H = h > 0 are given by and Observe that the order of the variables is very important in the application of the Averaging Theorem (see Theorem 4 in the appendix).In what follows we use the notation introduced in the appendix.In this case we take the order {x, y, p y , z, p z } and have k = 1, n = 5, α = x 0 , β 0 : R −→ R 4 is β 0 (α) = (0, 0, 0, 0), z α = (x 0 , 0, 0, 0, 0) is the initial condition of the unperturbed periodic orbits, for each z x0 the solution x(t, z x 0 ) = (x 0 cos t + p x 0 sin t, 0, 0, 0, 0) is 2π-periodic, and ξ(x, y, p y , z, p z ) = x.
Let M z x 0 (t) be the fundamental matrix satisfying M z x 0 (0) = I solution of the variational equation ( 22) along the periodic solutions x(t, z x 0 ).Then M z x 0 (t) is given by . Now we verify the condition det∆ x 0 ̸ = 0, then we compute The function F 1 along the periodic orbit is given by 8(p x0 cos t − x 0 sin t) , 0, 0, 0, 0 ( p x 0 (x 0 cos t + p x 0 sin t) 4   8(p x 0 cos t − x 0 sin t) 2 , 0, 0, 0, 0 ) .
Thus, from (23) we have where p x0 = √ 2h − x 2 0 at the energy level H = h > 0. Thus The zeros of F(x 0 ) = 0 are x 0 = ± √ 2h, which implies p x0 = 0. Since F ′ (± √ 2h) ̸ = 0, both zeros of F(x 0 ) provide two initial conditions of the same periodic orbit for the perturbed differential system in the energy level H = h > 0. Hence the proof of Theorem 1 for case k = 1 follows.
Proof of Theorem 1 for k = 2.We have the unperturbed periodic solution in the plane (0, y, 0, 0, p y , 0).Note that these periodic solutions living in the plane (0, y, 0, 0, p y , 0, 0) are not in resonance with the other periodic solutions of system (7) for ε = 0 living in the planes (x, 0, 0, p x , 0, 0) and (0, 0, z, 0, 0, p z ) due to the fact that q and q 1/2 are irrational.The first integral (8) when ε = 0 takes on these periodic solutions the value ) .
Let M zy 0 (t) be the fundamental matrix satisfying M zy 0 (0) = I solution of the variational equation ( 22) along the periodic solutions x(t, z y 0 ).Then M z y 0 (t) is given by . Now we verify the condition det∆ y0 ̸ = 0, then we compute M −1 zy 0 (0) − M −1 zy 0 (2πq) and we get The function F 1 along the periodic orbit is given by ( p y 0 q cos( t q ) − y 0 sin( t q ) ) , ( p y0 q cos( t q ) − y 0 sin( t q ) ) 2 , 0, 0, 0, 0 From (23) we have where p y 0 = √ 2h − y 2 0 /q 2 at the energy level.Thus (15) The zeros of F(y 0 ) = 0 are y 0 = ±2 √ hq, which implies p y 0 = 0. Since F ′ (±2 √ hq) ̸ = 0, both zeros provide two initial conditions of the same periodic orbit for the perturbed differential system in the energy level H = h > 0. So the proof of Theorem 1 for case k = 2 is done.
Proof of Theorem 1 for k = 3.Now we consider, the unperturbed periodic solutions ) .

Proof of Theorem 2
We consider the case q is rational and q 3/2 is irrational and we shall obtain two families of periodic orbits.
in the subspace (x, y, 0, p x , p y , 0).Note that these periodic solutions living in the space (x, y, 0, p x , p y , 0) are not in resonance with the periodic solutions of system (7) for ε = 0 living in the plane (0, 0, z, 0, 0, p z ) because q is rational and q 3/2 are irrational.The first integral (8) when ε = 0 takes on these periodic solutions the value ) .
We apply the Averaging Theorem to every fixed energy level H = h > 0. This allows to eliminate one of the coordinates, in this case p x and to reduce the study The function F(α) = (F D , F E , F F ) where F D , F E , F F are the projections in the first, second and third component, respectively, of the integral of M −1 z α (t)F 1 (t, x(t, z α )) in one period, i. e., ) ̸ = 0, both zeros provide initial conditions of the same periodic orbit for the perturbed differential system in the energy level H = h > 0. This completes the proof of Theorem 2 for k = 1.