This is a preprint of: “The dynamics of the relativistic Kepler problem”, Elbaz I. Abouelmagd, Juan Luis Garcia Guirao, Jaume Llibre, Results in Physics, vol. 19, 103406, 2020. DOI: [10.1016/j.rinp.2020.103406] THE DYNAMICS OF THE RELATIVISTIC KEPLER PROBLEM ELBAZ I. ABOUELMAGD1 , JUAN LUIS GARCÍA GUIRAO2 AND JAUME LLIBRE3 Abstract. We deal with the Hamiltonian system (HS) provided by the correction given by the special relativity to the motion of the two–body problem, or by the first order correction to this problem coming from the general relativity. This Hamiltonian system is completely integrable with the angular momentum 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 Ih and Ic are the subset of the phase space where H = h and C = c. Since C and H are first integrals, the sets Ic , Ih and Ihc = Ih ∩ Ic are invariant by the action of the flow of the Hamiltonian system (HS). We determine the global dynamics on those sets when h and c vary. Second, recently Tudoran in [21] 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 an- gular 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 The Manev Hamiltonian is   1 p2 a b H= p2r + 2θ + + 2, 2 r r r where a and b are arbitrary constant. This Hamiltonian describes the motion of a two–body problem, which is governed by the potential a/r + b/r2 , where r is the separation distance between the two bodies. The analysis of the motion of the two–body problems has a long story. Just after the Newton’s work on the two–body gravitational problem, some discrepancies appeared between the theoretical motions of the pericenters of the planets and the observed ones. Consequently some doubts on the accuracy of the inverse square law of Newton for gravitation motivated the scientists to construct alternative gravitational models and corrections trying to reconcile these discrepancies. In fact Newton was the first in considering what we call now the Manev systems, see the book I, section IX, proposition XLIV, theorem XIV and corollary 2 of the Principia. Several authors tried to find or construct a appropriate model with the features of the Newtonian one but with the convenient corrections in order that the orbits of planets being close to the observe ones in the solar system. There were several pre– and post–relativistic 2010 Mathematics Subject Classification. Primary: 70F07, 70F15. Key words and phrases. Relativistic Kepler problem, global dynamics, Lyapunov stability. 1 2 E. I. ABOUELMAGD, J. L. G. GUIRAO AND J. LLIBRE attempts to get such modified models. This is the case of the Manev Hamiltonian introduced by Manev in [14, 15, 16, 17]. See for more information on the two–body problem for instance [1, 2, 3, 4, 13]. The correction given by the special relativity to the motion of the two–body problem, or by the first order correction to this problem coming from the general relativity is   1 2 p2θ 1 ε H= pr + 2 − − 2 , 2 r r 2r where ε > 0 is small, for details see [7, 8, 10, 19, 20]. Note that when ε = 0 we have the rotating Kepler problem, for more information on this last problem see [18]. Note that the Hamiltonian H is a particular case of the Manev Hamiltonian H when a = −1 and b = −ε/2. The Hamiltonian system defined by the Hamiltonian H is ṙ = pr , pθ θ̇ = 2 , r (1) p2θ − ε 1 p˙r = − , r3 r2 p˙θ = 0. For a fixed ε < 0 small this Hamiltonian system has the circle of equilibria {q(θ) = (r, θ, pθ , pr ) = (−ε, θ, 0, 0) : θ ∈ S1 }. The Hamiltonian dynamical system (1) is perfectly integrable within frame of Liouville– Arnold because it has two independent first integrals the Hamiltonian H and the angular momentum pθ in involution, for more details on perfectly integrable Hamiltonian systems see [5, 6]. The objective of this paper is double. Our first objective is to describe the global dynamics of the Hamiltonian system (1) in the following sense. Let Ih , Ic are the subset of the phase space, H = h and pθ = c where h c are the constants of integration. Due to the fact that H = h and pθ = c are first integrals, the sets Ih , Ic and Ihc = Ih ∩ Ic are invariant under the flow of the Hamiltonian system (1). We determine the global dynamics on those sets when h and c vary. Moreover, we describe the foliation of the phase space by the invariant sets Ih , and the foliation of Ih by the invariant sets Ihc . See section 2. In fact this first objective was studied inside the Manev Hamiltonian H with arbitrary values of a and b, see [11], but from there it is not easy to obtain the global dynamics for the particular case a = −1 and b = −ε/2. We do this work here. Second we shall show that every equilibrium point q(θ) of the Hamiltonian system (1) restricted to zero angular momentum is Lyapunov stable. But our interest in this Lyapunov stability becomes for showing that the criterion provided recently by Tudoran in [21] which detects when a non–degenerate equilibrium point of a completely integrable system is Lya- punov stable, also can be extended to the degenerate equilibrium points of the completely integrable system (1) restricted to zero angular momentum where the equilibrium points of system (1) live, see section 3. THE DYNAMICS OF THE RELATIVISTIC KEPLER PROBLEM 3 2. On the global dynamics 2.1. Critical points and critical values. We will consider the notation and terminology of [11], as consequence, we suppose that Ih and Ic are the subset of the phase space where H = h and pθ = c. Science H and pθ are first integrals, the sets Ih , Ic and Ihc = Ih ∩ Ic are invariant under the flow of the Hamiltonian system (1), i.e. if an orbit of the Hamiltonian system has a point on the set Ihc then the whole orbit is contained in this set. Let r ∈ R+ = (0, ∞), θ ∈ S1 , (pr , pθ ) ∈ R2 and E = R+ × S1 × R2 , where Ih = {(r, θ, pr , pθ ) ∈ E : H(r, θ, pr , pθ ) = h}, Ic = {(r, θ, pr , pθ ) ∈ E : pθ = c}. In this case the set of critical points of H is C = {(r, θ, pr , pθ ) ∈ E : r + ε = 0, θ ∈ S1 } since r > 0, then ( ) ∅ if ε ≥ 0, C= −ε if ε < 0. hence the critical value of H is 1/2ε if ε < 0 2.2. Hill Regions. Let E and R = R+ × S1 be the phase space and the configuration space of the Hamiltonian system (1), and let Γ : E → R be the natural projection from E to R. Then for each h belongs to real set R, the regions of motion Rh (Hill regions) of Ih are defined by Γ(Ih ) = Rh , for more details see [9, 11], hence 1 ε Rh = {(r, θ) ∈ R : − − 2 ≤ h} r 2r = {r ∈ R : 2hr + 2r + ε ≥ 0} × S1 . + 2 Therefore, if h < 0 then Rh is homeomorphic to 1 ∅ if ε < , 2h   1 1 − × S1 if ε = , h 2h (2)  √ √  1− 1 − 2hε 1 + 1 − 2hε 1 − ,− × S1 if < ε < 0, h h 2h  √  1 + 1 − 2hε 0, − × S1 if ε ≥ 0. h If h = 0 then Rh is homeomorphic to h ε  − , ∞ × S1 if ε < 0, (3) 2 R+ × S1 if ε ≥ 0. 4 E. I. ABOUELMAGD, J. L. G. GUIRAO AND J. LLIBRE If h > 0 then Rh is homeomorphic to √  1 − 2hε − 1 , ∞ × S1 if ε < 0, (4) h R+ × S1 if ε ≥ 0. 2.3. The sets Ih . Now we determine the topology of the invariant energy levels Ih by using the fact: Ih = {(r, θ, pr , pθ ) ∈ E : g(r, pr , pθ ) = h} ≈ {g −1 (h)} × S1 , where   1 2 p2θ 1 ε g(r, pr , pθ ) = pr + 2 − − 2 , 2 r r 2r If h is a regular momentum value of the map g : R × R2 −→ R and the set of points + 6 ∅, then the set {g −1 (h)} is a surface in R+ × R2 . Hence the intersection of {g −1 (h)} = −1 {g (h)} with {r = r0 = constant}, is 1 ε either an ellipse if + 2 + h > 0, r0 2r0 1 ε or a point if + + h = 0, r0 2r02 1 ε or the empty set if + 2 + h < 0. r0 2r0 ¿From the definition of the energy levels Ih we obtain [ (5) Ih = E(r,θ) (r,θ)∈Rh where p2θ 1  E(r,θ) = {(r, θ, pr , pθ ) ∈ E : p2r +2 = 2 2hr2 + 2r + ε }. r 4r For every point (r, θ) ∈ Rh the set E(r,θ) is an ellipse, a point or the empty set if the point (r, θ) belongs to the interior of the Hill region Rh , to the boundary of Rh , or does not belong to Rh , respectively. Therefore every energy level Ih of the planar relativistic Kepler problem is homeomorphic, from (2) and (6), to 1 ∅ if ε < , 2h 1 S1 if ε = , 2h 1 S2 × S1 if < ε < 0, 2h (S2 × S1 ) \ S1 if 0 ≤ ε, if h < 0; and from (3), (4) and (6) to (S2 × S1 ) \ S1 if ε < 0, R+ × S1 × S1 if ε ≥ 0, THE DYNAMICS OF THE RELATIVISTIC KEPLER PROBLEM 5 if h ≥ 0. 2.4. The sets Ic . Since Ic = {(r, θ, pr , pθ ) ∈ E : pθ = c}, we get that Ic is homeomorphic to R+ × S1 × R for all c ∈ R. 2.5. The foliation of Ih by Ihc . We can evaluate the invariant set Ihc from knowing the set {g −1 (h)} and Ihc = Ih ∩ Ic (6) = Ih ∩ {pθ = c} = ({g −1 (h)} ∩ {pθ = c}) × S1 Hence the foliation of Ih by Ihc can be described when h varies through the following cases: topological plane R2 of Fig. 1(a). The curves Case 1: h < 0. Then the surface g −1 (h) is the p −1 γhc = {g (h)} ∩ {pθ = c} for each |c| ≤ c1 = (2εh − 1)/(2h) are homeomorphic to: • one component homeomorphic to R if 0 ≤ c √≤ c2 , • one component homeomorphic to S1 if c2 = ε < |c| < c1 , and • one component homeomorphic to a point if |c| = c1 . Figure 1. When h < 0: (a) The surface g −1 (h). (b) Manifold Ih /S1 . The manifold Ih is homeomorphic to a solid torus without its boundary. Hence we can find the dynamics on Ih by rotating Fig. 1(b) around the e−axis. ¿From this figure we have • one periodic orbit (topologically a circle) Ihc for |c| = c1 , • one two-dimensional torus Ihc for c2 < |c| < c1 , and • one cylinder Ihc for 0 < |c| < c2 , which foliate Ih . Case 2: h ≥ 0 and ε ≥ 0. Then g −1 (h) is homeomorphic to a cylinder R × S1 see Fig. 2(a), and the curves γhc are formed by • two components each of them homeomorphic to R if c = 0, and • one component homeomorphic to R if c ≥ 0. 6 E. I. ABOUELMAGD, J. L. G. GUIRAO AND J. LLIBRE Figure 2. When h ≥ 0 and ε ≥ 0: (a) The surface g −1 (h). (b) Manifold Ih /S1 . The manifold Ih is homeomorphic to a solid torus without its boundary and without its central circular axis. Hence we can find the dynamics on Ih by rotating Fig. 2(b) around the e axis. From this figure we obtain one cylinder Ihc for every c ∈ R \ {0}, and two cylinders Ihc for every c = 0, which foliate Ih . Case 3: h ≥ 0 and ε < 0. Now g −1 (h) is homeomorphic to a plane R2 see Fig. 3(a), and the curves γhc are homeomorphic to R. Figure 3. When h ≥ 0 and ε < 0: (a) The surface g −1 (h). (b) Manifold Ih /S1 . The manifold Ih is homeomorphic to a solid torus without its boundary. The dynamics of Ih can be obtained by rotating Fig. 3(b) around the e axis. From this figure the foliation of Ih is done by the cylinders Ihc for every c ∈ R \ {0}. 3. On the Lyapunov stability of the equilibria Let φt (p) be the solution of a differential system (S) such that φt (p) = p, i.e. φt (p) is the flow defined by the system (S). An equilibrium point q of the differential system (S) is called Lyapunov stable if for every open neighborhood U of q, there exists an open neighborhood THE DYNAMICS OF THE RELATIVISTIC KEPLER PROBLEM 7 V ⊆ U of q where φt (p) ∈ U for any p ∈ V and any t ≥ 0. An equilibrium state which is not Lyapunov stable is called unstable. For ε < 0 sufficiently small we restrict the dynamics of the Hamiltonian system (1) to the space pθ = 0, where we have the circle of equilibria q(θ) for θ ∈ S1 . Then we have the differential system ṙ = pr , (7) θ̇ = 0, 1 ε p˙r = − 2 − 3. r r This differential system is completely integrable because it has the two functionally inde- pendent first integrals 1 1 ε C1 = θ and C2 = p2r − − 2 . 2 r 2r According with [21] for our system (7) a non-degenerate equilibrium point is a point which satisfies that the determinant of the Hessian of the function C2 is non–zero at that equilibrium, see Definition 3.3 of [21]. Then every equilibrium point (−ε, θ, 0) of system (7) is degenerate, because the determinant of the Hessian of the function C2 at this equilibrium is zero. Indeed, the mentioned Hessian is   0 0 1    0 0 0 (8)   , 1  0 0 ε3 (r,θ,pr )=(−ε,θ,0) and clearly its determinant is zero. In [21] to a non-degenerate equilibria q is associated a real number I(q) as follows. If P (x) is the characteristic polynomial defined by the linear part of the complete integrable differential system at q, then P (x) = (−x)n−2 (x2 + I(q)) where n is the dimension of the differential system, see Theorem 5.4 of [21]. In Theorem 5.4 of [21] it is also shown that a non-degenerate equilibrium point q is unstable if I(q) < 0, and in Theorem 5.6 it is proved that a non-degenerate equilibrium point q is Lyapunov stable if I(q) > 0. From (8) the characteristic polynomial of the linear part of system (7) at the equilibrium is (−ε, θ, 0) is   2 1 λ λ − 3 . ε So applying the Turoban criterium (which in fact we cannot apply because the equilibrium (−ε, θ, 0) is degenerate) we obtain that 1 (9) I(−ε, θ, 0) = − > 0. ε3 because ε < 0, and by the mentioned criterium the equilibrium (−ε, θ, 0) would be Lyapunov stable. 8 E. I. ABOUELMAGD, J. L. G. GUIRAO AND J. LLIBRE Now we shall prove that really the equilibrium point (−ε, θ, 0) is Lyapunov stable for all θ ∈ S1 . Consequently we have proved that the criterion of Tudoran also works for the degenerate equilibrium points of the completely integrable system (7). We note that we cannot apply the so-called “Arnol’d stability test” to the equilibrium (−ε, θ, 0) because the second condition of that test does not hold, see the statement of this test in Theorem 5.5 of [21]. From the differential system (7) it is clear that the variable θ is Figure 4. The graphic of the potential V (r) = −1/r − ε/r2 for −ε = 1/10. constant for any solution. So for studying the Lyapunov stability at an equilibrium (−ε, θ, 0) of system (7) we can restrict to study it at the equilibrium (−ε, 0) of the Hamiltonian system with one degree of freedom ṙ = pr , (10) 1 ε p˙r = − 2 − 3 . r r with Hamiltonian C2 . Since the potential V (r) = −1/r − ε/r2 of system (10) has the graphic of the Figure 4, and the minimum of this graphic takes place at r = −ε, recall that ε < 0 is fixed and small, it follows that the equilibrium point (−ε, 0) of system (10) is a center, i.e. all the orbits in a convenient neighborhood of it are periodic with the exception of the equilibrium point (see [12] for more information). Hence given any neighborhood U of (−ε, 0) in the plane (r, pr ) there is another neighborhood V ⊆ U of (−ε, 0) formed by sufficiently small periodic orbits surrounding the point (−ε, 0), and consequently contained in U . Hence the equilibrium point (−ε, 0) is Lyapunov stable for system (10), and consequently for the system (7). In summary, we have proved that the degenerate equilibrium points of the completely integrable system (7) satisfies the Tudoran criterion (see (9)) and are Lyapunov stable. THE DYNAMICS OF THE RELATIVISTIC KEPLER PROBLEM 9 Acknowledgements The third author is partially supported by the Ministerio de Economı́a, Industria y Com- petitividad, 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 H2020 European Research Council grant MSCA-RISE-2017-777911. References [1] E.I. Abouelmagd, J.L.G. Guirao and J. Llibre, Periodic Orbits of the Planar Anisotropic Kepler Problem, International Journal of Bifurcation and Chaos, 27(3), (2017),1750039 (6 Pages). [2] E.I. Abouelmagd, S.M. Elshaboury and H.H. 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Tudoran, A stability criterion for non-degenerate equilibrium states of completely integrable systems, J. Differential Equations 263 (10) (2017), 6731–6763. 10 E. I. ABOUELMAGD, J. L. G. GUIRAO AND J. LLIBRE 1 Nonlinear Analysis and Applied Mathematics Research Group (NAAM), Mathematics Department, Faculty of science, King Abdulaziz University, Jed- dah, Saudi Arabia 2 Celestial Mechanics Unit, Astronomy Department, National Research Institute of Astronomy and Geophysics (NRIAG). Helwan–11421, Cairo, Egypt. E-mail address: eabouelmagd@gmail.com or elbaz.abouelmagd@nriag.sci.eg 2 Departamento de Matemática Aplicada y Estadı́stica. Universidad Politécnica de Carta- gena, Hospital de Marina, 30203-Cartagena, Región de Murcia, Spain. E-mail address: juan.garcia@upct.es 3 Departament de Matemàtiques. Universitat Autònoma de Barcelona, Bellaterra, 08193- Barcelona, Catalonia, Spain E-mail address: jllibre@mat.uab.cat