Results in Physics 19 (2020) 103406 Contents lists available at ScienceDirect Results in Physics journal homepage: www.elsevier.com/locate/rinp The dynamics of the relativistic Kepler problem Elbaz I. Abouelmagd a, *, Juan Luis García Guirao b, Jaume Llibre c a Celestial Mechanics and Space Dynamics Research Group (CMSDRG), Astronomy Department, National Research Institute of Astronomy and Geophysics (NRIAG), Helwan – 11421, Cairo, Egypt b Departamento de Matemática Aplicada y Estadística, Universidad Politécnica de Cartagena, Hospital de Marina, 30203 Cartagena, Región de Murcia, Spain c Departament de Matemàtiques, Universitat Autònoma de Barcelona, Bellaterra, 08193 Barcelona, Catalonia, Spain A R T I C L E I N F O A B S T R A C T Keywords: We deal with the Hamiltonian system (HS) associated to the Hamiltonian in polar coordinates Kepler problem ( ) p2 Global dynamics H = 12 p2r + r2ϕ − 1r − 2r∊2 , where ∊ is a small parameter. This Hamiltonain comes from the correction given by the Lyapunov special relativity to the motion of the two-body problem, or by the first order correction to the two-body problem Stability coming from the general relativity. This Hamiltonian system is completely integrable with the angular mo­ mentum C and the Hamiltonian H. We have two objectives. First we describe the global dynamics of the Hamiltonian system (HS) in the following sense. Let Sh and Sc are the subset of the phase space where H = h and C = c, respectively. Since C and H are first integrals, the sets Sc , Sh and Shc = Sh ∩ Sc are invariant by the action of the flow of the Hamiltonian system (HS). We determine the global dynamics on those sets when the values of h and c vary. Second recently Tudoran (2017) provided a criterion which detects when a non-degenerate equilibrium point of a completely integrable system is Lyapunov stable. Every equilibrium point q of the completely integrable Hamiltonian system (HS) is degenerate and has zero angular momentum, so the mentioned criterion cannot be applied to it. But we will show that this criterion is also satisfied when it is applied to the Hamiltonian system (HS) restricted to zero angular momentum. 1. Introduction gravity obeys the law of inverse square distance between two bodies. Thus for example, the motion of two stars around each other, the planets In celestial mechanics, the Kepler problem is a special dynamical moving surrounding the Sun, a satellite orbiting a planet, and many system coming from the two-body problem. In this model, two objects other examples of orbital motion. The Kepler problem has also a sig­ move under their mutual Newtonian gravitational force [1]. This nificant relevance in the study of the motion of two charged particles, attractive force varies in size as the inverse square of the separation because Coulomb’s law of electrostatics follows the law of inverse distance between them and it is proportional to the product of the square distance too. For Example the hydrogen atom, muonium and masses of the two bodies. This system is used to find the position and the positronium, which have played serious roles as models of dynamical velocity vectors of the two bodies at specified time. Using the laws of systems to test physical theories and measuring constants of nature, see classical mechanics, the solution can be provided as a Kepler orbit [3–5]. through finding the six orbital elements. This problem is called the The importance of the Kepler problem is not only related to its varied Kepler problem in honor of the German astronomer Johannes Kepler, applications in different fields of science, but rather to its use in devel­ after he proposed Kepler’s laws of the motion of the planets and illus­ oping some mathematical methods. Thus, this problem has been used to trated the kinds of forces which can provide orbits obeying those laws develop new methods in classical mechanics, like Hamiltonian me­ [2]. chanics, the Hamilton-Jacobi equation, Lagrangian mechanics and The Kepler problem appears in various fields, some of these are action-angle coordinates. Furthermore, the Kepler and simple harmonic beyond the physics, which have been studied by Kepler himself. This oscillator problems are two of the most fundamental problems in clas­ problem is important in celestial mechanics because the Newtonian sical mechanics. They are also the only integrable dynamical systems * Corresponding author. E-mail addresses: elbaz.abouelmagd@nriag.sci.eg (E.I. Abouelmagd), juan.garcia@upct.es (J.L.G. Guirao), jllibre@mat.uab.cat (J. Llibre). https://doi.org/10.1016/j.rinp.2020.103406 Received 1 May 2020; Received in revised form 23 August 2020; Accepted 11 September 2020 Available online 19 September 2020 2211-3797/© 2020 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). E.I. Abouelmagd et al. Results in Physics 19 (2020) 103406 which have closed orbits for open sets of possible initial conditions. The Kepler problem is an unperturbed version form of the two-body {q(ϕ) = (r, ϕ, pϕ , pr ) = ( − ∊, ϕ, 0, 0) : ϕ ∈ S1 }. problem, which must be extended or generalized to include to additional The Hamiltonian dynamical system (2) is perfectly integrable within forces as perturbations, that represent realistic natural phenomena or to frame the of Liouville-Arnold because it has two independent first in­ obtain precise and accuracy results. Within the frame of anisotropic tegrals the Hamiltonian H and the angular momentum pϕ in involution. perturbation in [6], the authors have showed that the anisotropic Kepler For more details on perfectly integrable Hamiltonian systems see problem is equivalent to two massless particles which move in a plane or [30,31]. on perpendicular lines and which interact according to the Newton’s law The objective of this paper is double. Our first objective is to describe of gravitation. They have also proved that the generalized issue of the global dynamics of the Hamiltonian system (2) in the following anisotropic Kepler problem and anisotropic two centres problem are sense. Let Sh , Sc are the subset of the phase space, H = h and pϕ = c non-integrable systems. For an analysis of the Kepler problem, specific where h and c are the constants of integration. Due to the fact that H and to radial periodic perturbation of a central force field, it has been proved pϕ are first integrals, the sets Sh , Sc and Shc = Sh ∩ Sc are invariant under the existence of rotating periodic solutions nearly circular orbits [7]. the flow of the Hamiltonian system (2). We determine the global dy­ Implicit Function Theorem is used to find such solutions. namics on those sets when h and c vary. Moreover, we describe the The solution of the Kepler problem allowed researchers to investi­ foliation of the phase space by the invariant sets Sh , and the foliation of gate the unperturbed and the perturbed Lagrange’s planetary motion. Sh by the invariant sets Shc . See Section 2. Thus, these motions were clarified entirely by the classical mechanics In fact, this first objective is a particular case of the general cases and the Newton’s law of universal gravitation [8,9]. More recently, the studied inside the Manev Hamiltonian H with arbitrary values of a and perturbed Kepler problem have been investigated by [10,11]. The per­ b, see [32]. However, from that work, it is not easy to obtain the global turbed models are not limited to Kepler models but also include the dynamics for the particular case a = − 1 and b = − ∊/2, corresponding to restricted three-body problem, which can be reduced to the perturbed our Hamiltonian (1). We cover this case in this work. Kepler problem in some cases [12–14]. Second, we shall show that every equilibrium point q(ϕ) of the The Manev Hamiltonian is Hamiltonian system (2) restricted to zero angular momentum is Lya­ ( ) punov stable. However, our interest in this Lyapunov stability rises in 1 2 p2ϕ a b H = pr + 2 + + 2 , showing that the criterion provided recently by Tudoran in [33] which 2 r r r detects when a non-degenerate equilibrium point of a completely inte­ where a and b are arbitrary constants. This Hamiltonian describes the grable system is Lyapunov stable, also can be extended to the degenerate motion of a two-body problem, which is governed by the potential a/r + equilibrium points of the completely integrable system (2) restricted to b/r2 , where r is the separation distance between the two bodies and (pr , zero angular momentum, where the equilibrium points of system (2) live, see Section 3. pϕ ) are the momenta in polar coordinates. The analysis of the motion in the two-body problems has a long 2. On the global dynamics history. Just after the Newton’s work on the two-body gravitational problem, some discrepancies appeared between the theoretical motions 2.1. Critical points and critical values of the pericenters of the planets and the observed ones. Consequently, some doubts on the accuracy of the inverse square law of Newton for We will consider the notation and terminology of [32], as conse­ gravitation. Motivated scientists to construct alternative gravitational quence, we suppose that Sh and Sc are the subset of the phase space models and corrections trying to reconcile these discrepancies. In fact, where H = h and pϕ . Science H and pϕ are first integrals, the sets Sh , Sc Newton was the first to consider what we now call the Manev systems, and Shc = Sh ∩ Sc are invariant under the flow of the Hamiltonian system see the book I, section IX, proposition XLIV, theorem XIV and Corollary 2 (2), i.e. if an orbit of the Hamiltonian system has a point on the set Shc of the Principia. then the whole orbit is contained in this set. Several authors tried to find or construct an appropriate model with the features of the Newtonian one but with the convenient corrections, Let r ∈ R+ = (0, ∞), ϕ ∈ S1 , (pr , pϕ ) ∈ R2 and ℵ = R+ × S1 × R2 , to make closer theory and observation. where There were several pre- and post-relativistic attempts to get such Sh = {(r, ϕ, pr , pϕ ) ∈ ℵ : H(r, ϕ, pr , pϕ ) = h}, modified models. This is the case of the Manev Hamiltonian introduced Sc = {(r, ϕ, pr , pϕ ) ∈ ℵ : pϕ = c}. by Manev in [15–18]. See for more information on the two-body prob­ lem, for instance, [19–23]. In this case the set of critical points of H is The correction given by the special relativity to the motion of the C = {(r, ϕ, pr , pϕ ) ∈ ℵ : r + ∊ = 0, ϕ ∈ S1 } two-body problem, or by the first order correction to this problem coming from the general relativity is since r > 0, then ( ) { } 1 2 p2ϕ 1 ∊ ∅ if ∊⩾0, H= pr + 2 − − , (1) C = 2 r r 2r2 − ∊ if ∊ < 0. where |∊|≪1 is a small parameter, for details see [24–28]. Note that, hence the critical value of H is 1/2∊ if ∊ < 0 when ∊ = 0, we have the rotating Kepler problem, for more information on this last problem see [29]. 2.2. Hill Regions Note that the Hamiltonian H is a particular case of the Manev Hamiltonian H when a = − 1 and b = − ∊/2. Let ℵ and R = R+ × S1 be the phase space and the configuration The Hamiltonian system defined by the Hamiltonian H is space of the Hamiltonian system (2), and let Γ : ℵ→R be the projection from ℵ to R . Then for each h belongs to real set R, the regions of motion pϕ p2ϕ − ∊ 1 ṙ = pr , ϕ̇ = 2 , p˙r = − 2, p˙ϕ = 0. (2) Rh (Hill regions) of Sh are defined by Γ(Sh ) = Rh , for more details see r r3 r [34,32], hence For a fixed ∊ < 0 small, this Hamiltonian system has the circle of equilibria 2 E.I. Abouelmagd et al. Results in Physics 19 (2020) 103406 Fig. 1. When h < 0 : (a) The surface g − 1 (h). (b) Manifold Sh /S1 . 1 ∊ Rh = {(r, ϕ) ∈ R : − − ⩽h} p2ϕ 1 ( )} r 2r2 ℵ(r,ϕ) = {(r, ϕ, pr , pϕ ) ∈ ℵ : p2r + = 2hr2 + 2r + ∊ . r2 4r2 = {r ∈ R+ : 2hr2 + 2r + ∊⩾0} × S1 . For every point (r, ϕ) ∈ Rh the set ℵ(r,ϕ) is an ellipse, a point or the Therefore, if h < 0 then Rh is homeomorphic to empty set if the point (r, ϕ) belongs to the interior of the Hill region Rh , 1 to the boundary of Rh , or does not belong to Rh , respectively. Hence ∅ if ∊ < , every energy level Sh of the planar relativistic Kepler problem is ho­ 2h { 1 } 1 meomorphic, from (3) and (6), to − × S1 if ∊ = , h 2h 1 [ √̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅ √̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅ ] (3) ∅ if ∊ < 2h , 1− 1 − 2h∊ 1 + 1 − 2h∊ 1 − ,− × S1 if < ∊ < 0, 1 h h 2h S1 if ∊ = , ( √̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅ ] 2h 1 + 1 − 2h∊ 0, − × S1 if ∊⩾0. 1 h S2 × S1 if < ∊ < 0, 2h If h = 0 then Rh is homeomorphic to (S2 × S1 )⧹S1 if 0⩽∊, [ ∊ ) − , ∞ × S1 if ∊ < 0, if h < 0; and from ()()()(4)–(6) to 2 (4) + 1 R ×S if ∊⩾0. (S2 × S1 )⧹S1 if ∊ < 0, R+ × S1 × S1 if ∊⩾0, If h > 0 then Rh is homeomorphic to [√̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅ ) if h⩾0. 1 − 2h∊ − 1 , ∞ × S1 if ∊ < 0, h (5) 2.4. The sets Sc R+ × S 1 if ∊⩾0. Since Sc = {(r,ϕ,pr ,pϕ ) ∈ ℵ : pϕ = c}, we get that Sc is homeomorphic 2.3. The sets Sh to R+ × S1 × R for all c ∈ R. Now we determine the topology of the invariant energy levels Sh by 2.5. The foliation of Sh by Shc using the fact: Sh = {(r, ϕ, pr , pϕ ) ∈ ℵ : g(r, pr , pϕ ) = h} We can evaluate the invariant set Shc from knowing the set {g− 1 (h)} ≈ {g− 1 (h)} × S1 , and Shc = Sh ∩ Sc where = Sh ∩ {pϕ = c} ) ( ) = ({g− 1 (h)} ∩ {pϕ = c} × S1 1 2 p2ϕ 1 ∊ g(r, pr , pϕ ) = pr + 2 − − , 2 r r 2r2 Hence the foliation of Sh by Shc can be described when h varies If h is a regular momentum value of the map g : R+ × R2 ⟶R and the through the following cases: set of points {g− 1 (h)} ∕ = ∅, then the set {g− 1 (h)} is a surface in R+ × R2 . Case 1: h < 0. Then the surface g− 1 (h) is the topological plane R2 of Hence the intersection of {g− 1 (h)} with {r = r0 = constant}, is. either an Fig. 1(a). The curves γhc = {g− 1 (h)} ∩ {pϕ = c} for each |c|⩽c1 = √̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅ ellipse if r10 + 2r∊2 + h > 0, or a point if r10 + 2r∊2 + h = 0, or the empty set if (2∊h − 1)/(2h) are homeomorphic to: 0 0 1 + 2r∊2 + h < 0. r0 0 • one component homeomorphic to R if 0⩽c⩽c2 , From the definition of the energy levels Sh we obtain √̅̅̅ • one component homeomorphic to S1 if c2 = ∊ < |c| < c1 , and ⋃ Sh = ℵ(r,ϕ) (6) • one component homeomorphic to a point if |c| = c1 . (r,ϕ)∈Rh The manifold Sh is homeomorphic to a solid torus without its where boundary. Hence we can find the dynamics on Sh by rotating Fig. 1(b) 3 E.I. Abouelmagd et al. Results in Physics 19 (2020) 103406 Fig. 2. When h⩾0 and ∊⩾0 : (a) The surface g − 1 (h). (b) Manifold Sh /S1 . Fig. 3. When h⩾0 and ∊ < 0 : (a) The surface g− 1 (h). (b) Manifold Sh /S1 . around the e-axis. From this figure we have. neighborhood U of q, there exists an open neighborhood V⊆U of q where ϒt (p) ∈ U for any p ∈ V and any t⩾0. An equilibrium state which is not • one periodic orbit (topologically a circle) Shc for |c| = c1 , Lyapunov stable is called unstable. • one two-dimensional torus Shc for c2 < |c| < c1 , and For ∊ < 0 sufficiently small we restrict the dynamics of the Hamil­ • one cylinder Shc for 0 < |c| < c2 , tonian system (2) to the space pϕ = 0, where we have the circle of equilibria q(ϕ) for ϕ ∈ S1 . Then we have the differential system which foliate Sh . 1 ∊ Case 2: h⩾0 and ∊⩾0. Then g− 1 (h) is homeomorphic to a cylinder R × ṙ = pr , ϕ̇ = 0, p˙r = − − . (7) 1 r2 r3 S see Fig. 2(a), and the curves γ hc are formed by. This differential system is completely integrable because it has the • two components each of them homeomorphic to R if c = 0, and two functionally independent first integrals • one component homeomorphic to R if c⩾0. 1 1 ∊ C1 = ϕ and C2 = p2r − − . 2 r 2r2 The manifold Sh is homeomorphic to a solid torus without its boundary and without its central circular axis. Hence we can find the According with [33] for our system (7) a non-degenerate equilibrium dynamics on Sh by rotating Fig. 2(b) around the e axis. From this figure point is a point which satisfies that the determinant of the Hessian of the we obtain one cylinder Shc for every c ∈ R⧹{0}, and two cylinders Shc for function C2 is non-zero at that equilibrium, see Definition 3.3 of [33]. every c = 0, which foliate Sh . Then every equilibrium point ( − ∊, ϕ, 0) of system (7) is degenerate, Case 3: h⩾0 and ∊ < 0. Now g− 1 (h) is homeomorphic to a plane R2 because the determinant of the Hessian of the function C2 at this equi­ see Fig. 3(a), and the curves γ hc are homeomorphic to R. librium is zero. Indeed, the mentioned Hessian is The manifold Sh is homeomorphic to a solid torus without its ⎛ 0 0 1 ⃒⃒ ⎞⃒ boundary. The dynamics of Sh can be obtained by rotating Fig. 3(b) ⎜ ⎟⃒ ⎜ ⎟⃒ around the e axis. From this figure the foliation of Sh is done by the ⎜ 0 0 0⎟⃒ ⎜ ⎟⃒ , (8) cylinders Shc for every c ∈ R⧹{0}. Figs. ()1–3 already have been ⎜ ⎝1 ⎟⃒ ⎠⃒ appeared in [32] in a more general context. 0 0 ⃒ ⃒ ∊3 (r,ϕ,pr )=(− ∊,ϕ,0) 3. On the Lyapunov stability of the equilibria and clearly its determinant is zero. In [33] to a non-degenerate equilibria q is associated a real number Let ϒt (p) be the solution of a differential system (S) such that ϒt (p) = I(q) as follows. If P(x) is the characteristic polynomial defined by the p, i.e. ϒt (p) is the flow defined by the system (S). An equilibrium point q linear part of the complete integrable differential system at q, then of the differential system (S) is called Lyapunov stable if for each open 4 E.I. Abouelmagd et al. Results in Physics 19 (2020) 103406 Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgements The first and second authors are partially supported by Fundación Séneca (Spain), grant 20783/PI/18, and Ministerio de Ciencia, Innovación y Universidades (Spain), grant PGC2018-097198-B-I00. Also, the third author is partially supported by the Ministerio de Econ­ omía, Industria y Competitividad, Agencia Estatal de Investigación grant MTM2016-77278-P (FEDER) and MDM-2014-0445, the Agència de Gestió d’Ajuts Universitaris i de Recerca grant 2017SGR1617, and the Fig. 4. The graphic of the potential V(r) = − 1/r − ∊/r2 for − ∊ = 1/10. H2020 European Research Council grant MSCA-RISE-2017-777911. References P(x) = (− x)n− 2 (x2 +I(q)) where n is the dimension of the differential system, see Theorem 5.4 of [33]. 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