The Perlick system type I: from the algebra of symmetries to the geometry of the trajectories

Physics Letters A 381 (2017) 3355–3363 Contents lists available at ScienceDirect Physics Letters A www.elsevier.com/locate/pla The Perlick system type I: From the algebra of symmetries to the geometry of the trajectories ¸S. Kuru a, J. Negro b,∗, O. Ragnisco c aDepartment of Physics, Faculty of Science, Ankara University, 06100 Ankara, Turkey bDepartamento de Física Teórica, Atómica y Óptica, Universidad de Valladolid, 47011 Valladolid, Spain cInstituto Nazionale di Fisica Nucleare, Sezione di Roma 3, Via della Vasca Navale 84, 00146 Roma, Italy a r t i c l e i n f o a b s t r a c t Article history: Received 13 June 2017 Accepted 17 August 2017 Available online 26 August 2017 Communicated by A.P. Fordy Keywords: Bertrand spaces Superintegrable systems Factorization method Constants of motion In this paper, we investigate the main algebraic properties of the maximally superintegrable system known as “Perlick system type I”. All possible values of the relevant parameters, Kand β, are considered. In particular, depending on the sign of the parameter Kentering in the metrics, the motion will take place on compact or non compact Riemannian manifolds. To perform our analysis we follow a classical variant of the so called factorization method. Accordingly, we derive the full set of constants of motion and construct their Poisson algebra. As it is expected for maximally superintegrable systems, the algebraic structure will actually shed light also on the geometric features of the trajectories, that will be depicted for different values of the initial data and of the parameters. Especially, the crucial role played by the rational parameter βwill be seen “in action”. ©2017 Elsevier B.V. All rights reserved. 1. Introduction The Perlick systems type I and II have been studied in a large number of papers, where several crucial features have been discovered (see for instance [1–10]). In particular, let us remind that in his original paper [1], V. Perlick constructed the most general class of three-dimensional spherically symmetric lagrangian systems complying with the requirements of the celebrated Bertrand’s Theorem [11], namely the fact that they possess stable circular orbits and all bounded trajectories are closed. Of course he had to abandon the Euclidean space framework in favor of more general conformally flat manifolds. In this way, he unveiled the deep connection existing between the metrics characterizing the manifold and the corresponding integrable potentials. He classified these generalized Bertrand systems in two multiparametric families, defined according to their Euclidean limit. Family I is the one yielding the Kepler–Coulomb system and Family II is the one leading to the radial harmonic oscillator [12–14]. In ref. [3] by means of the coalgebra approach the extension to hyperspherically symmetric systems in any dimension was performed, and an intrinsic characterization of the two families was proposed. Family I was denoted as “intrinsic Kepler–Coulomb” because, up to additive and multi- *Corresponding author. E-mail addresses: [email protected] (¸S. Kuru), [email protected] (J. Negro), [email protected] (O. Ragnisco). plicative constants (one could say, “up to affine transformations”), the pertaining class of potentials were discovered to be just the Green’s functions of the corresponding Laplace–Beltrami operators. Analogously, the potentials comprised in Family II, again up to affine transformations, were identified as the “inverse-squared” of those belonging to Family I and these have been called “intrinsic oscillator”. In a subsequent paper [4] the same authors gave a rigorous proof of the previously obtained results, including that of the periodicity of the bounded motions, and provided a closed expression for the corresponding trajectories. Thus, Perlick’s systems of type I and II are the most general classes of “maximally superintegrable” lagrangian (and consequently Hamiltonian) systems with (hyper-)spherical symmetry. To further clarify our previous statements, let us recall that a classical Hamiltonian system with ndegrees of freedom is said to be integrable if there are nfunctionally independent globally defined and single-valued integrals of motion in involution for a given Poisson bracket. If there exist k, with 0 <k ≤n −1, additional integrals (they must be functionally independent, but not all of them in involution), it is called superintegrable. When k =n −1, the system is maximally superintegrable. In this case, all bounded trajectories are closed and the motion is periodic [15–17]. The constants of motion of maximally superintegrable systems close an algebra and can be used to determine the trajectories in pure algebraic way. Kepler–Coulomb and (isotropic) harmonic oscillator systems are examples of spherically symmetric and maximally superintegrable systems in an ndimensional Euclidean space. In http://dx.doi.org/10.1016/j.physleta.2017.08.042 0375-9601/©2017 Elsevier B.V. All rights reserved. 3356 ¸S. Kuru et al. / Physics Letters A 381 (2017) 3355–3363 dealing here with a maximally superintegrable system such as Perlick system of type I, rather than relying on an analytic approach as in [4], we will follow an algebraic one, that can be generally called “factorization approach” or “factorization method”. Through the factorization method the algebra of the integrals of motions can be explicitly constructed, and its features will pave the way to identifying relevant geometric properties such as the shape of the closed orbits. We wish to remark that the factorization method applies both to classical and quantum systems. The method is quite simple in yielding the constants of motion (symmetries) and unveiling their underlying algebra [18–25]. We should mention that other methods have been successfully applied to find the constants of motion of superintegrable systems. One of them is based on the Hamilton–Jacobi equation, as it is shown in many references [14, 26,27]. Another interesting way to look for constants of motion is the method of coalgebras developed in [28,29]. A systematic algebraic method has also been used in other papers [30,31]. In this work, we will perform a systematic investigation of the constants of motion of the classical superintegrable three dimensional Perlick system type I by using the factorization method in order to derive the trajectories in a purely algebraic way. This paper is organized as follows. In section 2, we briefly recall the basic definition of the Perlick system of type I which depends essentially on two parameters βand K. We will introduce new variables more suitable to describe the compact or non compact manifolds where this system may be defined. In section 3, we construct the constants of motion by means of an extension of the factorization method to classical three dimensional systems (it can be applied to higher dimensions too). The trajectories of Perlick I system are derived, discussed and depicted, for different values of the initial conditions as well as of the relevant parameters of the system, i.e., Kand β=m/n. Both open and closed trajectories are presented. The crucial point of the whole treatment, namely the evaluation of the algebra of the constants of motion is also given here. Section 4is devoted to the special case of the motion in the plane. The key role played by the requirement that βbe a rational number becomes extremely clear in this somehow simplified setting. Finally, in section 5, we summarize the new results that we have obtained, emphasizing what are in our opinion the most relevant ones. As a closing comment, we will briefly outline the most promising developments. 2. The Perlick system type I The Hamiltonian for the Perlick system type I in spherical coordinates (r, θ, ϕ)is  H±=β2(1+Kr 2)p2 r 2+L2 2r2+G±1 r1+Kr 2,(2.1) where Lis the angular momentum and L2can be considered as an angular Hamiltonian Hθϕin the variables (θ, ϕ), having the form L2=Hθϕ=p2 θ+p2 ϕ sin2θ(2.2) being K, Greal constants and β=m/na rational number. We can also consider p2 ϕas a free Hamiltonian Hϕin the variables (ϕ, pϕ), defined on the unit circle. The angular variables have the usual range: 0 <θ<πand 0 <ϕ<2π. The radial variable has different ranges depending on K. If K≥0, then 0 <r<∞, while if K<0, rmust be restricted to the finite interval 0 <r<1/√−K. In his paper [1] Perlick considered only the negative sign in front of the square root of the potential, that is  H−, in order to have bounded motions. However, we will show that for the case K<0 both Hamiltonians  H±should be taken into account in order to have a well defined Hamiltonian on a three dimensional sphere. The metric associated to this Hamiltonian is [3] ds2=dr2 β2(1+Kr 2)+r2(dθ2+sin2θdϕ2). (2.3) Another point of view on the Hamiltonians (2.1) is to look at them not as defined in a curved space but as position dependent mass Hamiltonians in a flat space [32,33]. Since {L2,  H±} =0, L2is a constant of motion and takes the positive values L2=2. Here, {·, ·} stands for the Poisson brackets (PBs) that for the functions f(r, θ, φ), g(r, θ, ϕ)in spherical coordinates are defined by {f,g}=∂f ∂r ∂g ∂pr−∂f ∂pr ∂g ∂r+∂f ∂θ ∂g ∂pθ−∂f ∂pθ ∂g ∂θ +∂f ∂ϕ ∂g ∂pϕ−∂f ∂pϕ ∂g ∂ϕ.(2.4) For the sake of simplicity, we also choose G =0, so when we replace L2by its constant value 2, we can rewrite the initial Hamiltonian (2.1) as an effective Hamiltonian  H± rin the variable r:  H± r=β2(1+Kr 2)p2 r 2+2 2r2±1 r1+Kr 2.(2.5) Since {pϕ,  H±} =0, pϕis another constant of motion: pϕ=z= const. In the same way, after replacing Hϕby zin (2.2), we have an effective Hamiltonian Hθin θ: Hθ=p2 θ+2 z sin2θ.(2.6) Notice that Hθis singular at the angles θ=0 and θ=π, so the trajectories lie between these two values (i.e. the North and South poles cannot be reached unless z=0). Once fixed the values of , z, the turning points for θare given by the solutions of 2= 2 z/sin2θ. In order to take into account different signs and values of K, we will make use of the κ-dependent cosine and sine functions defined by Cκ(u)≡⎧ ⎨ ⎩ cos√κuκ>0 1κ=0 cosh√−κuκ<0 , Sκ(u)≡⎧ ⎪ ⎨ ⎪ ⎩ 1 √κsin√κuκ>0 uκ=0 1 √−κsinh√−κuκ<0 .(2.7) The κ-tangent is defined by Tκ(u)≡Sκ(u) Cκ(u). Some relations among these κ-functions are [35,36]: C2 κ(u)+κS2 κ(u)=1,d duCκ(u)=−κSκ(u), d duSκ(u)=Cκ(u). (2.8) Now, let us introduce the parameter κinstead of Kand make the following change of canonical variables, K=−κ,r=Sκ(ξ), pr=pξ Cκ(ξ) ,(2.9) where the range of ξis as follows: (a) for κ≤0, 0 <ξ<∞; (b) for κ>0, we have two natural choices: 0 <ξ<π/(2√κ)or ¸S. Kuru et al. / Physics Letters A 381 (2017) 3355–3363 3357 π/(2√κ) <ξ<π/√κ). This change of coordinates can be interpreted in the following way. The variables (ξ, θ, ϕ)can be considered as the angular coordinates of the points of a (pseudo) sphere in R4. When κ<0they parametrize the points of one sheet of a three dimensional (3D) hyperboloid; when κ=0the coordinates (ξ, θ, ϕ)represent the spherical coordinates of the points of R3; finally, for κ>0these variables with 0 <ξ<π/(2√κ)parametrize the points of one half (the north hemisphere) of a deformed 3D sphere, and the values π/(2√κ) <ξ<π/√κgive the points of the south hemisphere. In the case κ<0, this choice of coordinates is not important since the north and south hemispheres of a hyperboloid are not connected and can be taken independently without any problem, but in the case κ>0this parametrization of the points of a sphere is quite relevant. Next, we introduce a Hamiltonian Hin the variables (ξ, θ, ϕ) defined as follows for any value of κ: H(ξ, θ, ϕ)=β2p2 ξ 2+L2 2 1 S2 κ(ξ) −1 Tκ(ξ) ,(2.10) and the corresponding effective Hamiltonian Hξby Hξ=β2p2 ξ 2+2 2 1 S2 κ(ξ) −1 Tκ(ξ) =β2p2 ξ 2+Veff(ξ) . (2.11) We will see the relation of this Hamiltonian with the Perlick Hamiltonians  H±for different values of κ. •κ≤0. In this case, both variables ξand rvary in the positive semi-line (0, ∞). If we change the variables in (2.1) according to (2.9) it is easy to check that H(ξ, θ, ϕ)= H−(r,θ,ϕ). (2.12) •κ>0. Here, there are two complementary intervals for the range of variable ξas mentioned above. If we change the variables in (2.1) according to (2.9), we get H(ξ, θ, ϕ)= H−(r,θ,ϕ), 0<ξ< π 2√κ,0<r<1 √κ,  H+(r,θ,ϕ), π 2√κ<ξ< π √κ,1 √κ>r>0. (2.13) Therefore, in this case we have a unique Hamiltonian Hwell defined for all the points of the sphere 0 <ξ<π/√κ, such that on the north hemisphere is described by  H−and on the south by  H+. When these points are parametrized in terms of (r, θ, ϕ)variables, each hemisphere gives rise to the two distinct Hamiltonians  H±. This is very important since in general the trajectories on the sphere are not restricted to the upper or the lower hemispheres, and a full description of all of them should be done by means of H. Henceforth, we will use the following notation. For all the values of κwe will take the Hamiltonian H(ξ, θ, ϕ)as given in (2.10) or the effective Hamiltonian Hξ(ξ) in (2.11). The ranges of the variable ξare taken according to the previous discussion depending on the values of κ, in particular for κ>0the interval is 0 <ξ< π √κ. 3. Constants of motion from factorization properties In this model, we already have three independent constants of motion: H, Hθϕ, Hϕwith constant values given by E, 2, 2 z. Our aim is to find additional constants of motion by means of the factorization method in order to show the maximal superintegrability of the system. A first pair of constants X±will be obtained from the effective Hamiltonians Hξand Hθ, and a second pair Y±from the next two effective Hamiltonians, Hθand Hϕ. 3.1. The constants of motion X± The first set of constants X±are constructed in terms of the shift functions of Hξand the ladder functions of Hθwhich are obtained as follows [18]. •Shift functions of Hξ The Hamiltonian (2.11) can be factorized as Hξ=B+B−+λξ,(3.1) where B±=1 √2∓iβpξ+ Tκ(ξ) −1 , λξ=−1 21 2−κ2.(3.2) The shift functions B±are complex conjugate of each other and they satisfy the following PBs together with the Hamiltonian Hξ {B−,B+}=iβ Sκ2(ξ) ,{Hξ,B±}=±iβ Sκ2(ξ) B±. (3.3) The second PB of (3.3) implies that {H,B±}=±iβHθϕ Sκ2(ξ) B±.(3.4) From the factorization given in (3.1) or the effective potential (2.11), we conclude that the energy Eof the total Hamiltonian Hξfor bounded motions must satisfy the following inequalities depending on κ: κ<0,−√|κ|>E≥−1 21 2+|κ|2, κ=0,0>E≥− 1 22, κ>0,∞>E≥−1 21 2−κ2. (3.5) In Fig. 1, it is shown the effective potential Veff(ξ) given by (2.11) together with the conditions on the energy (3.5) for different values of κ. For all these three cases, bounded motions have two turning points in the variable ξgiven by E=Veff(ξ). For E≥−√|κ|the trajectory will be unbounded with only one turning point for κ≤0. •Ladder functions of Hθ In order to find the ladder functions for the angular Hamiltonian Hθ, defined by (2.6), first we multiply it by sin2θand after rearranging, we get −2 z=p2 θsin2θ−Hθsin2θ. (3.6) Now, we can factorize the right hand side of this equality in terms of complex conjugate ladder functions −2 z=A+A−+λθ,(3.7) where A±=∓isin θpθ+Hθcosθ, λ θ=−Hθ.(3.8) These ladder functions A±together with the Hamiltonian Hθ satisfy the following PBs {A−,A+}=2iHθ,{Hθ,A±}=±2iHθA±.(3.9) 3358 ¸S. Kuru et al. / Physics Letters A 381 (2017) 3355–3363 Fig. 1. Plot of the effective potential Veff(ξ) for κ=−1(left), κ=0(center) and κ=1(right) together with the limiting values of the energy given by the inequalities (3.5) for  =0.5. The range of κfor κ≤0is 0 <ξ<∞and for κ>0it is 0 <ξ<π/√κ. Therefore, the PB with His {H,A±}=∓iHθϕ Sκ2(ξ) A±.(3.10) Notice that the factorization (3.7) implies the following inequality: 2≥2 z,(3.11) which is reasonable in view of the physical interpretation of  (the total angular momentum) and z(the component of the angular momentum in the z-direction). •Construction of X± From the PBs (3.4) and (3.10), taking into account that β= m/n, we obtain the constants of motion X±of the Perlick system type I in the following form {H,X±}=0,X±=(A±)m(B∓)n.(3.12) We can denote the values of these constants as follows, X±=qxe±iαx,0≤qx<∞,−π≤αx<π.(3.13) The absolute value qxis directly obtained from the factorization properties of A±and B±: qx=|X±|=2−2 zm/2E+1 21 2−κ2n/2 .(3.14) By means of these constants of motion, the relation between the variables ξ(or r) and θalong the trajectories can be found from (3.13) (and the relative frequencies as we will see later). However, when  =z, this relation breaks, because in this case (2.6) and (3.8) entail pθ=0 and θ=π/2. In other words, if  =z, the motion takes place in the horizontal plane z=0. This particular case will be considered in Section 4. In principle, the constants of motion X±given by (3.12) are not polynomial in the momentum functions, since A±and B±depend on which is a square root (see (2.6)). However, if we expand the powers mand nin (3.12), then the real and imaginary parts of these functions will give rise to polynomial constants of motion in the momenta [22]. 3.2. The constants of motion Y± This new set of constants of motion is derived from the shift functions of Hθand the ladder functions of Hϕ. They are obtained in a similar way as the previous set. •Shift functions of Hθ The angular Hamiltonian defined by (2.6) is factorized as Hθ=p2 θ+2 z sin2θ=C+C−+λ,(3.15) where C±=∓ip θ+zcotθ, λ =2 z.(3.16) These factor functions C±together with the Hamiltonian Hθ satisfy the following PBs {C−,C+}=2iz sin2θ,{Hθ,C±}=±2iz sin2θC±. (3.17) The second PB of (3.17) implies that {Hθϕ,C±}=±2iHϕ sin2θC±.(3.18) •Ladder functions of Hϕ The Hamiltonian Hϕ=p2 ϕis factorized as Hϕ=D+D−,D±=Hϕe∓iϕ.(3.19) These ladder functions D±together with the Hamiltonian Hϕ satisfy the following PBs {D−,D+}=2iHϕ,{Hϕ,D±}=±2iHϕD±.(3.20) Then, the PB with the Hamiltonian Hθϕis {Hθϕ,D±}=±2iHϕ sin2θD±.(3.21) •The building of Y± The Hamiltonian Hθϕdefined by (2.6), besides Hϕ, has the constants of motion Y±that now can be found with the help of the relations (3.18) and (3.21): {Hθϕ,Y±}=0,Y±=(C±)(D∓). (3.22) The geometric meaning of these constants of motion can be appreciated by rewriting them in the form Y±=−Lz(Lx±iLy), (3.23) where Lx, Lyand Lzare the components of the angular momentum in the x, yand zdirection, respectively: Lx=−sin ϕpθ−cot θcosϕpϕ, Ly=cosϕpθ−cotθsin ϕpϕ,Lz=pϕ.(3.24) If we denote the values of these constants of motion by Y±=qye±iαy,0≤qy<∞,π≤αy<π,(3.25) with qy=|Y±|=z(2−2 z)1/2,(3.26) while the angle αycoincides with the azimuthal angle of the angular momentum vector L=(Lx, Ly, Lz). The value of this ¸S. Kuru et al. / Physics Letters A 381 (2017) 3355–3363 3359 Fig. 2. Plot of the trajectories of β=1 for the values E=−1(left),E=−6(right),κ=−1, =0.25 and z=0.1. constant of motion fixes the relation of the angles θand ϕ along a trajectory. For example, for y=0, αy=π, the relation (3.25) gives the following expression for θ(ϕ)(or ϕ(θ)) cotθ=−x zcosϕ.(3.27) However, as mentioned before, this relation also breaks when  =z, since in this case x=0 and therefore θ=π/2. Note that, from expression (3.23), it is clearly seen that Y±are polynomial in the momentum variables. In this subsection, we have arrived at an obvious result: the Hamiltonian L2has the symmetries Lzand Y±or equivalently Lx, Lyand Lz. Nevertheless, the basis Lz, Y±will be the most adequate to write the symmetry algebra. 3.3. Trajectories of the Perlick system type I In summary, once fixed the values of the constants of motion E, , z, the new constants of motion X±and Y±determine the relation between the coordinates r–θand θ–ϕ, respectively. In this way, one can get all the trajectories of the system. Now, in order to characterize each trajectory we will use the properties of the ladder and shift functions. The ladder functions, B±of (3.2), and the shift functions, A± of (3.8), can be expressed as B±(ξ, pξ)=E+1 21 2−κ21/2e±ib(ξ,pξ), A±(θ, pθ)=2−2 z1/2e±ia(θ,pθ), (3.28) where b(ξ, pξ)and a(θ, pθ)are real phase functions that depend also on the constants of motion E, and z. The effective Hamiltonian Hξgiven in (2.11) depends of the variables (ξ, pξ). When the energy Esatisfies the restrictions (3.5) the motion is periodic between two turning points. The variables (θ, pθ)are described by the effective Hamiltonian Hθgiven in (2.6). For z=0, the motion of these variables will be periodic, the range of θis determined by its corresponding turning points. As a consequence, the functions b(ξ, pξ)and a(θ, pθ)will also be periodic. Now, taking into account the constants of motion X±as given in (3.13) the phases of A±, B±and X±are related as follows ma(θ, pθ)−nb(ξ, pξ)=αx,ξ 1≤ξ≤ξ2,θ 1≤θ≤θ2. (3.29) This equation fixes the relation of the variables (ξ, pξ)and (θ, pθ) along the motion. Therefore, if we differentiate (3.29) with respect to time, we get m˙ a(θ, pθ)−n˙ b(ξ, pξ)=0.(3.30) This implies that the frequencies ωξand ωθare related by mωθ−nωξ=0.(3.31) In the same way, we can write the functions C±and D±in the form C±(θ, pθ)=(2−2 z)1/2e±ic(θ,pθ), D±(ϕ,pϕ)=ze±id(ϕ,pϕ), (3.32) where c(θ, pθ)and d(ϕ, pϕ)are real phase functions. Due to the constants of motion Y±these variables are related by c(θ, pθ)−d(ϕ,pϕ)=αy,θ 1≤θ≤θ2,−π≤ϕ<π. (3.33) As the motion of the (θ, pθ)variables and the (ϕ, pϕ)is periodic, differentiating (3.33) with respect to time, we obtain ωθ−ωϕ=0.(3.34) Hence, the frequencies of the angular variables θand ϕare equal (except for the case  =z). In conclusion, when the energy Esatisfies the restrictions (3.5), the motion is bounded, and the frequencies of the three variables are related by (3.31) and (3.34). This implies that the bounded motion is periodic and the trajectories are closed. In conclusion, we have checked in this case that the Bertrand’s theorem is satisfied. In Fig. 2 and Fig. 3 some examples of the trajectories for different values of the energies Eand of the parameter βcorresponding to bounded and unbounded motions are shown for the case κ=−1. The trajectories of the initial Hamiltonian (2.1) are plotted in a three dimensional space (rsinθcosϕ, rsin θsinϕ, rcos θ), where (r, ϕ, θ) are the spherical coordinates in R3. 3.4. Algebra of the constants of motion for the Perlick system type I As we have seen in the previous sections, we have obtained seven constants of motion for the three dimensional Perlick system: H, Hθϕ, Hϕ, X±, Y±. Only five of these constants are independent, for example we can choose: H, Hθϕ, Hϕ, X+, Y+. Therefore, the Perlick system type I for a rational β=m/nis a maximally superintegrable system. It turns out that, all the aforementioned seven constants are useful to express in a simple form the algebraic structure defined in terms of Poisson brackets: 3360 ¸S. Kuru et al. / Physics Letters A 381 (2017) 3355–3363 Fig. 3. Plot of the trajectories of β=2(left),β=3(right)forthevaluesE=−6, κ=−1, =0.25 and z=0.1. {H,·}=0,{Hθϕ,Y±}=0, {Hθϕ,X±}=±2imHθϕX±,{Hϕ,Y±}=∓2iHϕY±, {Hϕ,X±}=0,{Y+,Y−}=2iHϕHθϕ−2Hϕ, {X+,X−}=imn(Hθϕ−Hϕ)m(Hξ+1 2(1 Hθϕ−κHθϕ))n−1 ×(κHθϕ+1 H3/2 θϕ )−2im2(Hθϕ−Hϕ)m−1 ×(Hξ+1 2(1 Hθϕ−κHθϕ))nHθϕ, {X±,Y±}=∓ im Hθϕ+Hϕ X±Y±, {X±,Y∓}=∓ im Hθϕ−Hϕ X±Y∓.(3.35) In this way, we have found the algebra of the constants of motion for any two coprime integers mand nand any value of the constant κ. The corresponding polynomial algebras can be found as in [22]. The complex constants of motion X±and Y±allow to get real constants: YR=Re(Y±), YI=Im(Y±), XR=Re(X±), XI=Im(X±), which will close a real algebra. Besides, for β=1, κ=0, the complex constants of motion X±can be expressed in terms of the angular momentum L =(Lx, Ly, Lz)and Runge–Lenz A= (Ax, Ay, Az)vectors, X±=Re(X±)±iIm(X±)=Az √2±i(L×A)z √2,(3.36) where A=p×L−ˆ r.(3.37) Here, we have used the expressions of r, p, Lin spherical coordinates. Notice that in the context of the analysis of the Perlick system, a generalization of the Runge–Lenz vector has been proposed in ref. [4]. In the next section we will consider the motion in a plane corresponding to special case θ=π/2. 4. Special case of the Perlick system type I In this section, we will study the motion of the Perlick system type I in a plane, in this way we want to simplify the relevant constants of motion and the expression of the trajectories determined by such constants. Let us take θ=π/2; with this election the motion is limited to x–yplane and the angular momentum along the z-axis is equal to the total angular momentum: L =(0, 0, Lz)( =z). The corresponding Hamiltonian in the remaining r, ϕvariables is  H±=β2(1+Kr 2)p2 r 2+pϕ2 2r2±1 r1+Kr 2.(4.1) Now, we have two trivial constants of motion: the total energy E and z. Following the same procedure as in Sections 2, 3, with the change of the variables (r, pr)by (ξ, pξ), we get the Hamiltonian H(ξ, θ, ϕ)=β2p2 ξ 2+pϕ2 2 1 S2 κ(ξ) −1 Tκ(ξ) ,(4.2) with the same range of the variable ξas specified in Section 2. In this case the effective Hamiltonian Hξis given by Hξ=β2p2 ξ 2+z2 2 1 S2 κ(ξ) −1 Tκ(ξ) =β2p2 ξ 2+Veff(ξ) . (4.3) It is easy to find two additional constants of motion Z±for the Hamiltonian (4.2): {H,Z±}=0,Z±=(D±)m(B∓)n,(4.4) where B±=1 √2∓iβpξ+z Tκ(ξ) −1 z,D±=Hϕe∓iϕ, (4.5) and Hξ=B+B−+λξ,λ ξ=−1 21 2 z−κ2 z,(4.6) Hϕ=p2 ϕ=D+D−.(4.7) These constants of motion Z±have complex values denoted by Z±=qze±iϕz,(4.8) where −π≤ϕz<πand 0 ≤qz<∞. The modulus qzhas a value determined by the other constants of motion Eand z, qz=|Z±|=m zE+1 21 2 z−κ2 zn/2 .(4.9) ¸S. Kuru et al. / Physics Letters A 381 (2017) 3355–3363 3361 Fig. 4. Plot of the trajectories of β=1(left), β=2(center), β=3(right) for the values E=−8.03 (minimum energy of the potential), E=−6, E=−1, E=4. For β=1, respectively. Here we have chosen κ=−1,  =0.25 and z=0.1. The functions B±and D±can be written as B±(ξ, pξ)=E+1 21 2 z−κ2 z1/2 e±ib(ξ,pξ), D±(ϕ,pϕ)=ze±id(ϕ,pϕ), (4.10) where b(ξ, pξ)and d(ϕ, pϕ)are the corresponding real phase functions. When the energy Esatisfies the restriction (3.5), the bounded motion of the variables (ξ, pξ)is periodic. Besides, the motion of the variables (ϕ, pϕ)is also periodic, due to the angular character of ϕ. Then, substituting in (4.8) we get md(ϕ,pϕ)−nb(ξ, pξ)=ϕz,ξ 1≤ξ≤ξ2,−π≤ϕ<π. (4.11) Hence, after differentiating with respect to the time we obtain the relation between the frequencies of the two variables: mωϕ−nωξ=0.(4.12) Let us write the equation of the constant of motion Z±, (ze∓iϕ)m(∓iβ √2pξ+z √2 1 Tκ(ξ) −1 z√2)n=qze±iϕz.(4.13) Taking the real and imaginary parts of this equality, we get two equations z √2 1 Tκ(ξ) −1 z√2=qz m z1/n cos1 nϕz+m nϕ,(4.14) −β √2pξ=qz m z1/n sin1 nϕz+m nϕ.(4.15) From (4.14), we obtain the relation between ξand ϕalong the trajectories [4]: cos1 nϕz+m nϕ= 2 z Tκ(ξ) −1 2E2 z+1−κ4 z .(4.16) Relation (4.16) becomes a well known conic section equation for the values κ=0, β=m/n =1 and ϕz=0: α ξ=1+εcosϕ,(4.17) where ε2=2 E2 z+1 (eccentricity) and α=2 z(semi-latus rectum) [34]. In this case, the problem is reduced to the Kepler–Coulomb system in the Euclidean plane. If κ=−1(κ=1) and β=1, ϕz=0, then we get the conic section equation in the hyperboloid (sphere) [12]: κ=−1,(Hyperbolic) α tanhξ=1+ε2+α2cos ϕ,(4.18) κ=1,(Spherical) α tanξ=1+ε2−α2cos ϕ.(4.19) So, we may say that the equation (4.16) can be considered as a generalized conic section equation. Taking into account the relation (4.18) for κ=−1 and the relation (4.19) for κ=1different examples of trajectories (ξ cos ϕ, ξsin ϕ), where (ξ, ϕ)are polar coordinates, are plotted in Figs. 4–7. In these figures there are represented bounded and unbounded trajectories, according to the values of the energy E. In the case of bounded motion, due to the above relation of the frequencies (4.12) the motion must be periodic and the trajectories are closed. In the special case where κ=0 and β=m/n =1, the constants of motion Z±can also be expressed in terms of the Runge–Lenz vector A=(Ax, Ay, 0), Z±=Re(Z±)±iIm(Z±)=Ax √2∓iAy √2.(4.20) For the other cases (κ=±1), the constants of motion Z±can be written in terms of a type of generalized Runge–Lenz vector [4]. Note that, the constants H, Hϕ, Z±close a similar algebra as in the general case. 5. Conclusions and remarks In this paper, we have started an algebraic approach to the study of Perlick’s systems, the family of maximally superintegrable systems discovered in 1992, that represent the most general extension of the classical Bertrand systems to curved (though conformally flat) N-dimensional (Riemannian) manifolds. We would summarize our main new findings as follows. On one hand, we have given a thorough description of the algebraic and geometrical properties of the classical Perlick system of type I by means of the factorization approach: we have identified both the shift and the ladder functions for the radial as well as for the angular Hamiltonian. Henceforth, we have been able to identify the full set of constants of motion and their related Poisson algebra, and unveiled the intimate connection with the geometric features of the orbits provided by superintegrability. On the other hand, we have been able to single out the role played by the two parameters that characterize the system: the real parameter K, entering both in the metrics and in the potential, is related with the compact or noncompact nature of the manifold where the motion takes place; the rational parameter β, which does not appear in the potential, is in turn responsible for what we would call “the complexity” of the trajectories, namely their winding number. These features emerge 3362 ¸S. Kuru et al. / Physics Letters A 381 (2017) 3355–3363 Fig. 5. Plot of the trajectories of β=1/2(left), β=1/3(right) for the values E=−8.03 (minimum energy of the potential), E=−6, E=−1, E=4, corresponding to circular, elliptic, parabolic and hyperbolic orbits, respectively. Here we have chosen κ=−1,  =0.25 and z=0.1. Fig. 6. Plot of the trajectories of β=1(left), β=2(center), β=3(right) for the values E=−7.96875 (minimum energy of the potential), E=−3, E=0, E=1. Here we have chosen κ=1, z=0.25. Fig. 7. Plot of the trajectories of β=1/2(left), β=1/3(right) for the values E=−7.96875 (minimum energy of the potential), E=−3, E=0, E=1. Here we have chosen κ=1, z=0.25. ¸S. Kuru et al. / Physics Letters A 381 (2017) 3355–3363 3363 in a perspicuous manner from the graphic representations of the (effective) potential as well as of the orbits, reported in a number of figures for different values of Kand β. The further steps to be accomplished are clear. 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