Periodic orbits of a perturbed 3-dimensional isotropic oscillator with axial symmetry

We study the periodic orbits of a generalized Yang–Mills Hamiltonian H\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {H}$$\end{document} depending on a parameter β\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta $$\end{document}. Playing with the parameter β\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta $$\end{document} we are considering extensions of the Contopoulos and of the Yang–Mills Hamiltonians in a 3-dimensional space. This Hamiltonian consists of a 3-dimensional isotropic harmonic oscillator plus a homogeneous potential of fourth degree having an axial symmetry, which implies that the third component N of the angular momentum is constant. We prove that in each invariant space H=h>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {H}=h>0$$\end{document} the Hamiltonian system has at least four periodic solutions if either β<0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta <0$$\end{document}, or β=5+13\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta = 5+\sqrt{13}$$\end{document}; and at least 12 periodic solutions if β>6\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta >6$$\end{document} and β≠5+13\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta \ne 5+\sqrt{13}$$\end{document}. We also study the linear stability or instability of these periodic solutions.


Introduction and statement of the main results
In this paper we study the periodic solutions of a generalized Yang-Mills Hamiltonian (see [16,19]), which consists of a 3-dimensional (or simply 3D) isotropic harmonic oscillator perturbed by a homogeneous potential of fourth degree Of course ε is a small parameter. This perturbation exhibits an axial symmetry with respect to the x 3 -axis which depends on one real parameters β, and thus its study can be reduced to a family of Hamiltonian systems with 2 degrees of freedom fixing the third component of the angular momentum. In [19] two Hamiltonians are studied whose motions take place in the plane (x 1 , x 2 ), one with a cubic potential and another with a quartic one, while we study a Hamiltonian in the space (x 1 , x 2 , x 3 ) whose motion takes place outside the plane (x 1 , x 2 ) when the third component of the angular momentum is not zero. In [16] there is also studied the Yang-Mills Hamiltonian in the plane (x 1 , x 2 ). When p 3 = x 3 = 0 and the perturbation is x 2 1 x 2 2 we obtain the so-called Contopoulos Hamiltonian, studied by him and coworkers during many years, see for instance [5][6][7]. The Contopoulos Hamiltonian studies the perturbed central part of an elliptical or barred galaxy without escapes. Several authors studied quartic homogeneous potentials (without quadratic terms), see for instance [1,2,11]. Some generalizations of the mechanical Yang-Mills Hamiltonian, with three, four or five quartic terms, have been considered in [3,9,10,17,19]. On the other hand in [12] is studied a similar problem, but the perturbation in the Hamiltonian (1) is given by a cubic potential instead of a quartic one, and the tools there used are essentially based on the normal form theory, while our tools are based on the averaging theory for studying periodic solutions, see for instance [14,15,18] where this technique is applied. Roughly speaking, if we want to study the periodic solutions of a differential system using the normal form theory, first this theory reduces the system to its easier expression conserving all its dynamics and after it allows to study its periodic solutions, while in general the averaging theory simplifies the system eliminating one variable (usually an angle) and reduces the study of the periodic solutions to study the zeros of a nonlinear function.
The Hamiltonian (1) can be written in the called nodal-Lissajous variables (l, g, ν, L , G, N ) (see Sect. 2 for details), where N = x 1 p 2 − x 2 p 1 is the first integral given by the third component of the angular momentum, and G = ||x × p|| is the first integral given by the total angular momemtum, where x = (x 1 , x 2 , x 3 ) and p = ( p 1 , p 2 , p 3 ). In these variables the Hamiltonian (1) writes Our main result on the periodic orbits of the Hamiltonian system associated with the Hamiltonian (1) is the following.
The two periodic orbits bifurcating from the two polar elliptic orbits of the 3D isotropic harmonic oscillator with initial conditions (h, G 0 , 0) and (h, G 0 , π) are linearly stable if β ≥ 6, and unstable if β < 0. The two periodic orbits bifurcating from the two polar elliptic orbits of the 3D isotropic harmonic oscillator with initial condi- If β > 6 and β = 5 + √ 13 there exist four 2π −periodic solutions (L(l, ε), G(l, ε), g(l, ε)) in the angular variable l such that The two periodic orbits bifurcating from the two inclined elliptic orbits of the 3D isotropic harmonic oscillator with initial conditions (h, G 0 , 0) and (h, G 0 , π) are linearly stable, and the two periodic orbits bifurcating from the inclined elliptic orbits of the 3D isotropic harmonic oscillator with initial conditions (h, G 0 , π/2) and (h, G 0 , 3π/2) are unstable.
Theorem 1 is proved in Sect. 3. From Theorem 1 given h > 0 we have four periodic solutions for each of the following values when β > 6 and β = 5 + √ 13. While we have only four periodic solutions when n = 0 and either β < 0, or β = 5 + √ 13. In short we have the following result. The blue, green and red graphics correspond to periodic orbits for ε = 0 (unperturbed periodic orbits), ε = 10 −8 and 10 −9 (per-turbed periodic orbits), respectively. The e which appears in the periodic orbits of the first row denotes the eccentricity of these unperturbed periodic orbits. The computations have been done with the values of n = 0 and h = 10. The periodic orbits have been computed with the algebraic manipulator mathematica with a precision of 10 −20

Corollary 1 The Hamiltonian system associated with the Hamiltonian (1) or (2) has at every energy level
In Celestial Mechanic the periodic orbits having the third component of the angular momentum N equal to zero are usually called polar periodic orbits. Let γ be a periodic orbit for which N and G are constant. Then the angle I defined as cos I = N /G with 0 ≤ I ≤ π is called the inclination angle of γ . So, when I is π/2 the periodic orbit γ is polar. When I ∈ (0, π) the periodic orbit γ is called inclined. Finally, when I is zero or π , the periodic orbit is called equatorial. See Figs. 1 and 2, respectively. In these figures the orbits of the first row (the blue ones) are periodic orbits of the unperturbed system (i.e., the Hamiltonian system with ε = 0), while the orbits of the second and third row (the green and the red ones) are obtained with the averaging theory from the ones of the first row according with the values of ε described in the caption of the figures.
For proving Theorem 1 we use in an essential way Corollary 2 given in Sect. 2. Thus with this corollary The blue and red periodic orbits correspond to periodic orbits for ε = 0 (unperturbed periodic orbits) and ε = 10 −9 (perturbed periodic orbits). The e which appears in the periodic orbits of the first row denotes the eccentricity of these unperturbed periodic orbits. The computations have been done with the value of n given by the formula (3) and h = 10. The periodic orbits have been computed with the algebraic manipulator mathematica with a precision of 10 −20 we can study the cases |N | < G of Theorem 1, i.e., Theorem 1 studies polar and inclined periodic orbits of the Hamiltonian system associated with the Hamiltonian (1). But unfortunately the case |N | = G, i.e., the equatorial periodic orbits, cannot be studied using the nodalLissajous variables used in this paper.
We remark that the inclined periodic orbits found in the statement (b) of Theorem 1 have inclination angle I satisfying cos I < 1/ √ 5. The angle I = 63 • .43 for which cos I = 1/ √ 5 coincides with the so-called critical inclination angle in the satellite theory, see for more details [4].
In Sect. 2 we recall some basic results that we shall need for proving Theorem 1.

Preliminary results
In this paper we study the periodic orbits of the Hamiltonian systems associated with Hamiltonians of the form using as main tool the next Theorem 2, based on the averaging theory. When we apply the results that we shall obtain for the Hamiltonian (4) to Hamiltonian (1), we must take ω = 1. The perturbations P 1 (x 2 1 +x 2 2 , x 3 ) of the 3D isotropic harmonic oscillator that we shall treat in the present work are invariant under the axial S 1 action Note that R θ stands for the rotation about the third coordinate axis. From Noether's theorem, or by direct verification, can be obtained that the third component N = x 1 p 2 − x 2 p 1 of the angular momentum is an integral of motion. This symmetry is a key point for simplifying the analysis of the family of Hamiltonian systems (4).
First we assume that the total angular momentum G = ||x × p|| does not vanish. This allows to choose the angular momentum vector x × p as the third coordinate axis. In the new coordinates the motion in the configuration space takes place in the (Q 1 , Q 2 )-plane, in particular we have G = |Q 1 P 2 − Q 2 P 1 |.
The angle ν determines the nodal line, i.e., ν measures the angle from the x 1 -axis to the straight line intersection between the (x 1 , x 2 )-plane and the (Q 1 , Q 2 )plane. Thus I is the angle between these two planes. The condition |N | < G ensures that x×p is not parallel to the x 3 -axis, so ν is well defined.
The Whittaker transformation is a canonical transformation, i.e., the symplectic structure remains the standard one. In the new coordinates the Hamiltonian (4) becomes By construction ν is a cyclic variable, thus the conjugate momentum N is a first integral of the corresponding Hamiltonian system. For a fixed value n of N we have reduced the original system to a system of two degrees of freedom. The Whittaker transformation is only defined on the open and dense set of the phase space where |N | < G. Now we shall write the Hamiltonian (5) in the Lissajous variables defined by g, ν, L , G, N ) where , , These variables were introduced by Deprit in [8]. The angle l describes the position on an ellipse at the configuration space, measured from its semi-minor axis. So l is named true anomaly, which measures the position on the ellipse from its semi-minor axis. The other variables define this ellipse, which is centered at the origin of coordinates. The angle g gives the position of the semi-minor axis reckoning from the Q 1 -axis, i.e., from the nodal line. The eccentricity of the ellipse is given by and determines its shape, while the size is encoded in the length of its semi-minor axis b given by The condition G < L ensures that the ellipse does not degenerate to a circle, and thus the angles g and l are well defined. Finally ν still represents the angle of the ascending node of the orbital plane, and the inclination of this plane with respect to the angular momentum vector is given by the angle I . The transformation given by L • W is known as the nodal-Lissajous transformation. The nodal-Lissajous variables are defined on {(x, p) ∈ T R 3 : |N | < G < L}. The Hamiltonian (5) expressed in the nodal-Lissajous variables has the form H = ωL + εP(L , G, N , l, g), (6) where P (L , G, N , l, g) is the perturbing function P 1 (x 2 1 + x 2 2 , x 3 ) expressed in the nodal-Lissajous variables. Since H and N are first integral we shall restrict our attention to the invariant sets H = h > 0 and N = n under the flow of the Hamiltonian systems associated with (4), (5) or (6).
As usual the Poisson bracket of the functions f (I 1 , I 2 , θ 1 , θ 2 ) and g (I 1 , I 2 , θ 1 , θ 2 ) is computed by The next result proved using the averaging theory for studying the periodic solutions of a differential system provides sufficient conditions for computing periodic orbits of the Hamiltonian system associated with the Hamiltonian (7). For more details on averaging theory see the books [20,21].

Theorem 2 (See Theorem 1 of [13]) We define
and we consider the differential system restricted to the energy level H = h with h ∈ R. The value h is such that the function H −1 0 is a neighborhood of h is a diffeomorphism. System (8) is a Hamiltonian system with Hamiltonian ε H 1 . If ε = 0 is sufficiently small, then for every equilibrium point p = (I 0 2 , θ 0 2 ) of system (8) satisfying that there exists a 2π -periodic solution γ ε (θ 1 ) = (I 1 (θ 1 , ε), ε)) of the Hamiltonian system associated with the Hamiltonian (7) taking as independent variable the angle θ 1 such that γ ε (0) → (H −1 0 (h), I 0 2 , θ 0 2 ) when ε → 0. The stability or instability of the periodic solution γ ε (θ 1 ) is given by the linear stability or instability of the equilibrium point p of system (8).
In fact, the equilibrium point p has the linear stability behavior of the Poincaré map associated with the periodic solution γ ε (θ 1 ). G, N , l, g)dl.

We define
Corollary 2 For ε = 0 sufficiently small the Hamiltonian system defined by the Hamiltonian (6) in the invariant set H = h > 0 and N = n ∈ R has a 2π −periodic solution γ ε (l) = (L(l, ε), G(l, ε), g(l, ε)) in the variable l such that (L(0, ε), G(0, ε), The stability or instability of the periodic solution γ ε (l) is given by the stability or instability of the equilibrium point p of the differential system In fact, the equilibrium point p has the stability behavior of the Poincaré map associated with the periodic solution γ ε (l).
Proof The corollary follows directly from Theorem 2.

Proof of Theorem 1
We shall apply Corollary 2 to the Hamiltonian (4) with the perturbation given by Applying the nodal-Lissajous transformation to the Hamiltonian (1) we get that the function P(L , G, N , l, g) is 1 4 cos(2I )(s sin(g + l) − d sin(g − l)) 2 where s = √ L + G, d = √ L − G and cos I = N /G. Therefore the function P defined in (9) The functions f 1 (G, g) and f 2 (G, g) of the differential system (11) are here L = h. The solutions (G 0 , g 0 ) of the system give rise to periodic orbits for each L = h > 0 and N = n ∈ R if the Jacobian (10) at (G 0 , g 0 ) is nonzero, see Corollary 2.
Proof (Proof of statement (a) of Theorem 1) Assume that N = 0 < G and H = h > 0. Then it is not difficult to check that system (12) has the four solutions (G 0 , g i ) given by For the solutions (G 0 , g i ) with i = 0, 1 we obtain 18+β(β−6)) 36 0 for β > 6, and and The Jacobian (10) at (G 0 , g i ) for i = 0, 2 is The graphics of the functions J i (β, h) for i = 1, 2 are given in Fig. 3. Therefore, by Corollary 2 the solutions (G 0 , g i ) for i = 0, 2 provide two periodic solutions linearly stable if β > 6, and unstable if β < 0.
On the other hand the Jacobian (10) at (G 0 , g i ) with i = 1, 3 is The graphics of the functions J i (β, h) for i = 3, 4 are given in Fig. 4. Hence the solutions (G 0 , g i ) for i = 1, 3 provide two periodic solutions linearly stable if β < 0, and unstable if β ≥ 6. This completes the proof of the statement (a) of Theorem 1. where For the solutions (G 0 , g i ) with i = 0, 1 we obtain Moreover for the solutions (G 0 , g i ) with i = 2, 3 we obtain The Jacobian (10) at (G 0 , g i ) for i = 0, 1 is h 1 (β, h) = δ 1 δ 2 with β > 6. The graphic of the function h 1 (β, h) is given in Fig. 5. Therefore, by Corollary 2 the solutions (G 0 , g i ) for i = 0, 1 provide two linearly stable periodic solutions if β > 6 and β = 5 + √ 13, this last value of β corresponds to the unique zero of the function h 1 (β, h) for β > 6.

Conclusion
In this paper we have studied the periodic orbits of a Hamiltonian system defined by the Hamiltonian (1) being ε a small parameter, this Hamiltonian is obtained perturbing the isotropic harmonic oscillator with a homogeneous potential of fourth degree. On the one hand such a Hamiltonian (1) is a generalization of the Yang-Mills Hamiltonian, and on the other hand that Hamiltonian also allows to study the motion of the central part of an elliptical or barred galaxy without escapes.
Our Hamiltonian system has two first integrals the Hamiltonian H and the third component of the angular momentum N . We prove that in each invariant space H = h > 0 the Hamiltonian system has at least 4 periodic solutions if either β < 0, or β = 5+ √ 13; and at least 12 periodic solutions if β > 6 and β = 5+ √ 13. We also have studied the linear stability or instability of these periodic solutions. More precisely, the two periodic orbits bifurcating from the two polar elliptic orbits of the 3D isotropic harmonic oscillator with initial conditions (h, G 0 , 0) and (h, G 0 , π) are linearly stable if β ≥ 6, and unstable if β < 0. The two periodic orbits bifurcating from the two polar elliptic orbits of the 3D isotropic harmonic oscillator with initial conditions (h, G 0 , π/2) and (h, G 0 , 3π/2) are unstable if β ≥ 6, and linearly stable if β < 0. The two periodic orbits bifurcating from the two inclined elliptic orbits of the 3D isotropic harmonic oscillator with initial conditions (h, G 0 , 0) and (h, G 0 , π) are linearly stable, and the two periodic orbits bifurcating from the inclined elliptic orbits of the 3D isotropic harmonic oscillator with initial conditions (h, G 0 , π/2) and (h, G 0 , 3π/2) are unstable.
It is well known that the averaging theory for finding periodic orbits of a differential system in general does not find all the periodic orbits. In the future it will be interesting to study, at least numerically, other families of periodic orbits of the Hamiltonian system here analyzed.