Periodic orbits for the generalized Yang–Mills Hamiltonian system in dimension 6

We apply the averaging theory to study a generalized Yang–Mills Hamiltonian system in dimension 6\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$6$$\end{document} with six parameters. We provide sufficient conditions on the six parameters of the system which guarantee the existence of continuous families of period orbits parameterized by the energy.


Introduction
We study the generalized classical Yang-Mills Hamiltonian system in dimension 6. It consists of a harmonic oscillator plus a homogenous potential of fourth degree with six real parameters a, b, c, d, e, and f .
When z = p z = 0 the previous Hamiltonian contains the planar classical Yang-Mills Hamiltonian system.
The periodic solutions of this system when z = p z = 0 were studied in [15]. Our aim is to study the periodic solutions in the different energy levels H = h of the Hamiltonian system associated to the Hamiltonian (1). The mentioned planar Hamiltonian system (z = p z = 0) for a = 0 was studied by Contopoulos and co-workers during many years, such a Hamiltonian is now known as the Contopoulos Hamiltonian which describes the perturbed central part of an elliptical or barred galaxy without escapes. For more details see Refs. [5][6][7]. See also the article of Deprit and Elipe [8] where several periodic orbits and bifurcations are studied for this planar Hamiltonian system. When the quadratic part (x 2 + y 2 )/2 = 0 and d = 0 we obtain the mechanical Yang-Mills Hamiltonian H = ( p 2 x + p 2 y )/2 + bx 2 y 2 /2; where the term x 2 y 2 characterizes the Yang-Mills potential, which arises in connection with the classical Yang-Mills field with gage group SU(2) for a homogeneous two-component field, see [11]. Several authors studied quartic homogeneous potentials (without quadratic terms), see for instance Refs [2,3,10]. Moreover, when b = 0 it is well known that the Hamiltonian of Yang-Mills is non-integrable and strongly chaotic. Others studies and investigations related with generalizations of the mechanical Yang-Mills Hamiltonian have treated quartic terms with three up to five terms in [4,9,13,16]. Maciejewski et al. [16] studied generalized Yang-Mills Hamiltonian systems, which have a quadratic potential plus a homogeneous of fourth degree potential with five parameters, and they proved the existence of connected branches of non-stationary periodic trajectories emanating from the origin. Caranicolas and Varvoglis [4] studied a Hamiltonian with a quartic potential of three parameters plus a quadratic harmonic potential with frequencies ω 1 and ω 2 being two extra parameters of the form and they calculated numerically families of periodic orbits and its characteristic curves. In [14] the generalization of the Yang-Mills potentials H = ( p 2 x + p 2 y + x 2 + y 2 )/2 + ax 4 /4 + bx 2 y 2 /2 was studied with two real parameters a and b, in order that the problem be tractable in a two-dimensional parameter space, although these calculations can be generalized to higher dimensional parameter spaces. Here we study a harmonic oscillator plus a homogenous potential of fourth degree with six real parameters. The Hamiltonian differential system associated to the Hamiltonian (1) iṡ The dot denotes derivative with respect to the independent variable t, the time.
We study the periodic orbits of the Yang-Mills Hamiltonian system by using the averaging theory, see Sect. 2 for more details about this tool. More precisely through the averaging method we will provide sufficient conditions on the six parameters a, b, c, d, e, and f for the existence of periodic orbits of our Yang-Mills system.
The periodic orbits studied in this paper are isolated in every energy level and they are of special interest because after the equilibrium points the periodic orbits are the most simple non-trivial solutions of the system, and their stability determines the kind of motion in their neighborhood.
The averaging method provides periodic orbits of a perturbed periodic non-autonomous differential system depending on a small parameter ε. Roughly speaking, the problem of finding periodic solutions of a differential system is reduced to find zeros of some convenient finite dimensional function. We check the conditions under which the averaging theory guarantees the existence of periodic orbits, and we find them as a function of the energy. In this way we can find analytically periodic orbits in any energy level as function of the six parameters of the Yang-Mills systems (3). We summarize our main result on the periodic orbits as follows.
(b) two periodic orbits if one of the following conditions hold: (c) eight periodic orbits if one of the following conditions hold: Theorem 1 is proved in Sect. 3.

The averaging theory of first order
Now we shall present the basic results from averaging theory that we need for proving the results of this paper. The next theorem provides a first order approximation for the periodic solutions of a periodic differential system, for the proof see Theorems 11.5 and 11.6 of Verhulst [17].
Consider the differential equatioṅ are T -periodic in t. We also consider in D the averaged differential equatioṅ where Under certain conditions, equilibrium solutions of the averaged equation turn out to correspond with Tperiodic solutions of Eq. (4).
Theorem 2 Consider the two initial value problems (4) and (5). Suppose: Then the following statements hold.
(b) The stability or instability of the limit cycle ϕ(t, ε) is given by the stability or instability of the equilibrium point p of the averaged system (5). In fact the singular point p has the stability behavior of the Poincaré map associated to the limit cycle ϕ(t, ε).
For a proof of Theorem 2, see Sections 6.3 and 11.8 in [17].
To apply the averaging theory we will do some transformations in the Hamiltonian differential system (3) in order to write it in the normal form of Eq. (4). So first we do a rescaling transformation with a factor √ ε in order to have a small parameter ε > 0 in the Hamiltonian system. Second, using a kind of generalized polar coordinates in R 6 and taking as new independent variable an angular coordinate instead of the time, we obtain a 2π -periodic differential system. Third, fixing the energy level and omitting a redundant variable in every energy level, we will get the differential system written in the normal form for applying the averaging theorem of first order (i.e., Theorem 2), and finally we shall prove the existence of some isolated periodic orbits in every energy level. This technique has been used for some of the authors in other Hamiltonian systems, see for instance [12].

Proof of Theorem 1
We do a rescaling using a small parameter ε > 0. In the Hamiltonian system (3) we change the variables In the new variables, This system again is Hamiltonian with Hamiltonian As the change of variables is only a scale transformation for all ε > 0, the original and the transformed systems (3) and (8) have the same topological phase portrait, and additionally system (8) for ε sufficiently small is close to an integrable one. The averaging theory needs the periodicity in the independent variable of the differential system, so we change the Hamiltonian (9) and the equations of motion (8) to a kind of generalized polar coordinates (r , θ , ρ, α, R, β) in R 6 . Thus the explicit change of variables is This change of variables is done when r ≥ 0, ρ ≥ 0, and R ≥ 0. The first integral H in the new coordinates becomes and the new equations of motion arė r = −εr cos θ sin θ a r 2 cos 2 θ We observe in this last system that if we take the variable θ as the new independent variable of the system instead of t, we will obtain the periodicity necessary for applying the averaging theory.
From now on the independent variable will be θ . This means that our system has now only five equations. We denote by a prime the derivative with respect to θ and we expand the previous system in Taylor series in powers of ε. Thus, the system (12) becomes r = ε r sin θ cos θ a r 2 cos 2 θ + bρ 2 cos 2 (θ + α) Now system (13) is 2π -periodic respect to the variable θ . In order to apply the averaging theory we must fix the value of the first integral H = h with h > 0. Otherwise when we apply the averaging theory to system (13) the Jacobian (7) will be zero because the periodic orbits are non-isolated living on a cylinder parameterized by the energy, see for more details [1]. So we shall restrict system (13) to every positive energy level H = h > 0.
By solving Eq. (11) for ρ we obtain two solutions, we choose the positive one, and expanding it in Taylor series in ε we have Substituting ρ in the system (13) we obtain the differential system r = ε r sin θ cos θ a r 2 cos 2 As we shall see in what follows the differential system (15) is in the normal form (4) for applying the averaging theory. Moreover, due to the changes of variables that we have done system (15) is exactly the Hamiltonian differential system (8) restricted to the energy level H = h > 0. So studying with the averaging theory the periodic solutions of system (15) we are studying the periodic solutions of the Hamiltonian system (8) in the energy level H = h > 0.
So, So, If we substitute α 2 in f 14 (0, α, R, β) we get So, The four zeros of (15) in the subcase R = 0 are Referencing to (10) we see clearly that the unique angle which plays a role in the initial conditions of the periodic solution is α because of the nullity of r and R so (15) Summarizing the results of this subcase, the system (15) admits two solutions with the following existence's conditions 3d−2b 1i (r, α, R, β), for i = 2, 4 and we obtain So, We substitute β 3 in f 12 and we have We substitute β 4 in f 12 and we have The four zeros of (15) are Referencing to (10) we see clearly that the unique angle which plays a role in the initial conditions of the periodic solution is β because of the nullity of r and ρ so (15) admits one of the above zeros if −2c Summarizing the conditions of the existence of the two solutions chosen we have We substitute α in f 1i , for i = 2, 4 and we obtain So solving f 12 (0, α, R, β) = 0 with respect to the variable R we get We substitute R in f 14 (0, α, R, β) and we have So solving f 14 (0, α, R, β) = 0 with respect to the variable β we obtain If we substitute β in the expression of R we find • The zeros of (15) for k = 0 • The zeros of (15) for k = 2 r * 2 = 0, Referencing to (10) Summarizing the conditions of the existence of this subcase • The zeros of (15) for k = 1 • The zeros of (15) for k = 3 Referencing to (10) we see clearly that the angles which play a role in the initial conditions of the periodic solution are β and α because the nullity of r so (15) admits two of the above where Ω = 32(3d − 2e + 3 f ) 3 .
The zeros of (15) in this subcase are Referencing to (10) we see clearly that (15) admits only one solution because the nullity of R and ρ.
Solving f 14 = 0 with respect to R we obtain Solving f 12 = 0 with respect to α we obtain Substituting α in r and R we obtain • the zeros of (15) for k = 1 Summarizing the existence's conditions of the solution,