This is a preprint of: “Periodic orbits of the two fixed centers problem with a variational gravi- tational field”, Fabao Gao, Jaume Llibre, Celestial Mech. Dynam. Astronom., vol. 132(6-7), art. no. 39, 2020. DOI: [10.1007/s10569-020-09979-z] PERIODIC ORBITS OF THE TWO FIXED CENTERS PROBLEM WITH A VARIATIONAL GRAVITATIONAL FIELD FABAO GAO1,2 , JAUME LLIBRE2 Abstract. Within a given range of energy levels the two fixed centers problem under a variational gravitational field admits pe- riodic orbits bifurcating from the Kepler problem. The analytical expressions of these periodic orbits are given when the mass pa- rameter of the system is sufficiently small. 1. Introduction and statement of the main result It is well known that due to the non-integratable feature of the re- stricted three-body problem. Scientists have not yet obtained analyti- cal expressions of its general solutions, and that its periodic orbits have extremely important applications in practical space missions. This fact which has attracted a large number of mathematicians and astronomers to carry out research on the periodic orbits of the restricted three-body problem (see [1] and the references therein). The extensive research covered three categories: qualitative analysis (see [1]-[4] and so on), analytical calculation (see [5]-[7] ), and numerical simulation (see [8]- [13]). For the case of the planar circular restricted three-body problem, Zotos [14] investigated the problem with two equivalent masses with strong gravitational field, which was controlled by power of the grav- itational potential. They revealed the influences of the gravitational potential power on the nature of orbits. Based on the continuation method, Llibre and Paşca [15] proved that the circular and the elliptic symmetric periodic orbits of the planar rotating Kepler problem can be continued into periodic orbits of the planar collision restricted three- body problem. The method was also applied by Llibre and Makhlouf Key words and phrases. three-body problem, periodic orbit, averaging theory, vari- ational gravitational field. 1 2 FABAO GAO1,2 , JAUME LLIBRE2 [16] to provide sufficient conditions of periodic solutions of a fourth- order differential system. For large value of eccentricity e, Abouel- magd et al. [17] found that the anisotropic Kepler problem with small anisotropy has two periodic orbits in every negative energy level bi- furcating from elliptic orbits of the Kepler problem by using averaging theory. In addition, they also presented the approximate analytic ex- pressions of the continued periodic orbits. Recently, for a sufficiently small parameter and each value of eccentricity e ∈ (0, 1), Llibre and Yuan [18] continued elliptic periodic orbits of the Kepler problem to the the planar anisotropic Manev problem and of two perturbations of the hydrogen atom problem. As for the analytical calculation of periodic orbits of three-body problem, there were already scientists conducting research on it as early as a few decades ago. In 1973, Farquhar and Kamel [5] developed an approximation for periodic orbit, spurring a spate of investigation into the computation of periodic orbits. Richardson [6], [7] constructed the classical periodic ‘halo’ orbits seven years later, he presented a third-order analytical solution for halo-type periodic motion about the collinear points of the circular restricted problem based on the method of successive approximations and a technique similar to the Lindstedt- Poincaré method. Mainly motivated by Zotos [14] and Llibre [16] , this paper focuses on the question of what kind of periodic orbits of the Kepler problem can be continued to two fixed centers with the a variational gravitational field, and presenting the analytical expression of periodic orbits. 1.1. Equations of Motion. The equations of motion of a particle in the two fixed centers problem with a variational gravitational field can be written as ∂H ∂H (1) qi0 = , p0i = − , for i = 1, 2, ∂pi ∂qi where 1 2  1−µ µ (2) H= p1 + p22 − − p, 2 r1 r2 p p and r1 = (q1 + µ)2 + q22 , r2 = (q1 − 1 + µ)2 + q22 . Here the prime represents derivative with respect to the time t and µ (0 < µ  1) denotes the mass parameter of the two masses, one of mass 1 − µ fixed at (−µ, 0) and the other of mass µ fixed at (1 − µ, 0). The parameter p represents the power of the gravitational potential. When 3 p is negative, then the interaction is variationaler for short distances and stronger for large distances. Zotos in [14] studied the restricted circular three-body problem when p > 1. Here we shall study the two fixed centers problem with the p = −2. Our main result is the following: Theorem 1. For the mass parameter µ > 0 sufficiently small and for every value h ∈ (−1/2, 0), the elliptic periodic solution with eccentricity e = −2h of the Kepler problem can be continued to the two fixed centers problem with variational gravitational field p = −2. 2. Proof of theorem Introducing the McGehee coordinates (r, θ, v, u), of which r and θ are the radius and the angle in polar coordinate, v and u are the scaled radial and angular components of velocity, respectively, as follows (q1 , q2 ) = r(cos θ, sin θ), r−1/2 v = (p1 , p2 ) · (cos θ, sin θ), r−1/2 u = (p1 , p2 ) · (− sin θ, cos θ). Then the Hamiltonian of system (2) with p = −2 becomes 1 2 1 H= (p1 + p22 ) − p 2 2 q1 + q22 (3) ! q1 + q12 + q22 +µ −1 + 2q1 − q12 − q22 + 3/2 + O(µ2 ), (q12 + q22 ) and we have r0 = r−1/2 v, θ0 = r−3/2 u,   0 −3/2 2 v2 v = r u + −1 (4) 2 +µ r−5/2 [r (1 + 2r3 ) + 2 (1 − r3 ) cos θ] + O(µ2 ),   0 −3/2 1 u = r − uv + µ r−5/2 (1 + 2r3 ) sin θ + O(µ2 ). 2 4 FABAO GAO1,2 , JAUME LLIBRE2 Note that r = 0 corresponds to the collision singularity of equations (4). Let dt/dτ = r5/2 , hence equations (4) can be rewritten as ṙ = r2 v, θ̇ = ru,   2 v2 v̇ = r u + −1 (5) 2 + µ [r (1 + 2r3 ) + 2 (1 − r3 ) cos θ] + O(µ2 ),   1 u̇ = r − uv + µ (1 + 2r3 ) sin θ + O(µ2 ), 2 where the dot denotes derivative with respect to the new time τ . Taking into account that the Hamiltonian system (3) is reduced to the Kepler problem when µ = 0, we will study what periodic orbits of the Kepler problem can be continued to a given negative energy level of the two fixed centers problem with variational gravitational field p = −2. For a given energy level H = h we have   1 2 2  3 (1 + 2r3 ) (6) u +v −1+µ 1−r−r + cos θ = rh. 2 r Isolating r = r(θ, v, u, h) from equation (6) in power series of the small mass parameter µ we obtain  (7) r = r(θ, v, u, h) = R0 + µR1 + O µ2 , where 1 2 R0 = (u + v 2 − 2) , 2h 1 h 3 2 2 2 2 2 3 i R1 = 8h − 4h (u + v − 2) − (u + v − 2) 8h4 3 1 4h3 + (u2 + v 2 − 2) + 3 . 2h u2 + v 2 − 2 5 Therefore substituting equation (7) into equations (5) and taking θ as the new independent variable, equations (5) become " 3 dv 2u2 + v 2 − 2 1 (u2 + v 2 − 2) = +µ + dθ 2u u 4h3 u # 3 8h3 − (u2 + v 2 − 2) (8) + cos θ + O (µ2 ) , 2h2 (u2 + v 2 − 2) u 3 du v 4h3 + (u2 + v 2 − 2) = − +µ sin θ + O (µ2 ) . dθ 2 2h2 (u2 + v 2 − 2) u If µ = 0 equations (8) will be reduced to the following the unper- turbed system dv 2u2 + v 2 − 2 = , dθ 2u (9) du v =− , dθ 2 which admits the general solution e sin(θ − θ0 ) v(θ; e, θ0 ) = p , 1 + e cos(θ − θ0 ) p u(θ; e, θ0 ) = 1 + e cos(θ − θ0 ), where θ0 ∈ [0, 2π) and e are the argument of pericenter and the ec- centricity of the planar Kepler problem, respectively. The eccentricity e = 0 corresponds to circular periodic solutions, and e ∈ (0, 1) indi- cates that the corresponding periodic solutions are elliptical ones. In order to apply the averaging theory described in the Appendix to equations (8), we do the following transformations v   v(θ; e, θ0 ) e  x= u , x(θ; z, 0) = u(θ; e, θ0 ) , z= θ0 , F01  F11  F0 = F02 , F1 = F12 , 6 FABAO GAO1,2 , JAUME LLIBRE2 where 2u2 + v 2 − 2 F01 = , 2u v F02 =− , 2 3 3 1 (u2 + v 2 − 2) 8h3 − (u2 + v 2 − 2) F11 = + + cos θ, u 4h3 u 2h2 (u2 + v 2 − 2) u 3 4h3 + (u2 + v 2 − 2) F12 = sin θ. 2h2 (u2 + v 2 − 2) u Now we calculate the averaged function F(z) in equation (15) in the Appendix. Let (10) (G1 (θ; e, θ0 ), G2 (θ; e, θ0 )) = Mz−1 (θ, z) F1 (θ, x(θ; z, 0)), where ! y1 (θ; e, θ0 ) y2 (θ; e, θ0 ) Mz−1 (θ, z) = , y3 (θ; e, θ0 ) y4 (θ; e, θ0 ) denotes the inverse of the fundamental matrix of equations (9). Here (1 + e cos(θ − θ0 ))1/2 y1 (θ; e, θ0 ) = [2 cos θ(1 + e cos θ0 ) 2(1 + e cos θ0 )3/2 + e sin θ sin θ0 ] , 1 y2 (θ; e, θ0 ) = − 8(1 + e cos(θ − θ0 ))1/2 (1 + e cos θ0 )3/2   · 2 8 + e2 cos θ + 8e cos cos θ0 sin θ + 2(4 cos θ0 + 3e cos 2θ0 ) sin 2θ + 16e(2 + cos θ) sin2 (θ/2) sin θ0  + 6e2 sin2 θ sin 2θ0 , (1 + e cos(θ − θ0 ))1/2 y3 (θ; e, θ0 ) = sin θ, 2(1 + e cos θ0 )1/2 4 cos θ + e(cos(2θ − θ0 ) + 3 cos θ0 ) y4 (θ; e, θ0 ) = . 4(1 + e cos(θ − θ0 ))1/2 (1 + e cos θ0 )1/2 7 Hence G1 (θ; e, θ0 ) " 3 1 (−1 + e2 ) = 4+ 8(1 + e cos θ0 )3/2 h3 (1 + e cos(θ − θ0 ))3 16h cos θ(1 + e cos(θ − θ0 )) + −1 + e2   (−1 + e2 )3 · 1− 3 8h (1 + e cos(θ − θ0 ))3 · (2 cos θ(1 + e cos θ0 ) + e sin θ sin θ0 )   2h sin θ (−1 + e2 )3 + 1+ 3 −1 + e2 4h (1 + e cos(θ − θ0 ))3 2  · −2 8 + e cos θ + 8e cos θ0 sin θ − e(4 cos θ0 + 3e cos 2θ0 ) sin 2θ  2 !# (11) θ − 16e(2 + cos θ) sin sin θ0 − 6e2 sin2 θ sin 2θ0 , 2 G2 (θ; e, θ0 ) " 3 sin θ (−1 + e2 ) = 4+ 3 8(1 + e cos θ0 )1/2 h (1 + e cos(θ − θ0 ))3 16h cos θ(1 + e cos(θ − θ0 )) + −1 + e2   (−1 + e2 )3 · 1− 3 8h (1 + e cos(θ − θ0 ))3 4h(4 cos θ + e(cos(2θ − θ0 ) + 3 cos θ0 )) + −1 + e2   (−1 + e2 )3 · 1+ 3 . 4h (1 + e cos(θ − θ0 ))3 Substituting equations (11) into equation (15) in the Appendix and computing the corresponding integrals we obtain (12) F(z) = (f1 (e, θ0 ), f2 (e, θ0 )), 8 FABAO GAO1,2 , JAUME LLIBRE2 where √ 3 1 − e2 f1 (e, θ0 ) = [3e2 − 8h 32h3 (1 + e cos θ0 )3/2 (13) −4e(−1 + 2h) cos θ0 + e2 cos 2θ0 ] , √ 3e 1 − e2 sin θ0 f2 (e, θ0 ) = √ . 16h3 1 + e cos θ0 The function F(z) in equation (12) is the averaged function of system (8). According with the averaging theory (see the Appendix) we must compute the simple zeros of the system f1 (e, θ0 ) = 0, f2 (e, θ0 ) = 0. Here we must consider e 6= 0, otherwise f2 (e, θ0 ) ≡ 0, and then the averaging theory does not provide any information. From the equation f2 (e, θ0 ) = 0 we get θ01 = 0, θ02 = π. Case I: If θ01 = 0, we have e = 1 or e = 2h from the first equation of (13). However, both values of e should be discarded because e ∈ (0, 1) and h < 0. Case II: If θ02 = π, we have e = 1 or e = −2h from the first equation of (13), and e = 1 should also be discarded because it does not provide a periodic solution of the Kepler problem. Thus e = −2h with the h ∈ (−1/2, 0). Note that the Jacobian ∂g1 ∂g1 ∂e ∂θ0 9(2h − 1) J= ∂g2 ∂g2 = 6= 0. 64h5 ∂e ∂θ0 (e, θ0 )=(−2h, π) Therefore if h ∈ (−1/2, 0), we find that the periodic solution   2h sin θ √ (v(θ; −2h, π), u(θ; −2h, π)) = √ , 1 + 2h cos θ , 1 + 2h cos θ can be continued to a periodic solution of the two fixed center problem with variational gravitational field p = −2 when the mass parameter µ is sufficiently small. This completes the proof of Theorem 1. 9 Appendix We consider the problem of the bifurcation of T -periodic solutions from the following system (14) ẋ = F0 (t, x) + µ F1 (t, x) + µ2 F2 (t, x, µ), with µ = 0 to µ 6= 0 sufficiently small. Here F0 , F1 : R × Ω → Rn , F2 : R × Ω × (−µ0 , µ0 ) → Rn are C 2 functions and T -periodic with respect to variable t, and Ω is an open subset of Rn . We assume that there is a submanifold V of periodic solutions with the same period of the unperturbed system ẋ = F0 (t, x). Let x(t, z, µ) be the solution of this unperturbed system such that x(0, z, µ) = z. Then its linearization system along a periodic x(t, z, 0) can be writ- ten as ẏ = Dx F0 (t, x(t, z, 0))y, and the fundamental matrix can be denoted by Mz (t). Supposing that there exists an open set V with Cl(V ) ⊂ Ω such that x(t, z, 0) is T -periodic for each z ∈ Cl(V ) and x(0, z, 0)) = z. Here Cl(V ) denotes the closure of V in Rn . Then we have the following result (please see corollary 1 of [19] for an easy proof): If there is an open and bounded set V with Cl(V ) ⊂ Ω such that x(t, z, 0) is T -periodic for each z ∈ Cl(V ). Consider the function F : Cl(V ) → Rn , i.e., Z 1 T −1 (15) F(z) = Mz (θ, z) F1 (θ, x(θ; z, 0)) dθ. T 0 Assume that there exists a ∈ V such that F(a) = 0 and det((dF/dx)(a)) 6= 0, then there is a T -periodic solution φ(t, µ) of system (14) such that φ(0, µ) → a as µ → 0. Acknowledgments The first author gratefully acknowledges the support of the Na- tional Natural Science Foundation of China (NSFC) through grant Nos.11672259 and 11571301, the China Scholarship Council through grant No.201908320086, the Ministry of Land and Resources Research of China in the Public Interest through grant No.201411007. The second author gratefully acknowledges the support of the Min- isterio de Economı́a, Industria y Competitividad, Agencia Estatal de Investigación grants MTM2016-77278-P (FEDER) and MDM-2014- 0445, the Agència de Gestió d’Ajuts Universitaris i de Recerca grant 10 FABAO GAO1,2 , JAUME LLIBRE2 2017SGR1617, and the H2020 European Research Council grant MSCA- RISE-2017-777911. References [1] Musielak Z. E., Quarles B., The three-body problem, Reports on Progress in Physics 77: 065901, 2014. [2] Gómez G., Ollé M., Second-species solutions in the circular and elliptic re- stricted three-body problem I. existence and asymptotic approximation, Ce- lestial Mechanics and Dynamical Astronomy 52: 107-146, 1991. [3] Koon W.S., Lo M.W., Marsden J.E., Ross S.D., Dynamical systems, the three- body problem and space mission design, Marsden Books, ISBN 978-0-615- 24095-4, 2011. [4] Gao F.B., Zhang W., A study on periodic solutions for the circular restricted three-body problem, The Astronomical Journal 148: 116, 2014. [5] Farquhar R.W., Kamel A.A., Quasi-periodic orbits about the translunar libra- tion point, Celestial Mechanics 7: 458-473, 1973. [6] Richardson D.L., Halo orbit formulation for the ISEE-3 mission, Journal of Guidance and Control 3: 543-548, 1980. [7] Richardson D.L., Analytical construction of periodic orbits about the collinear points, Celestial Mechanics 22: 241-253, 1980. [8] Hénon M., Generating families in the restricted three-body problem, Springer- Verlag, 1997. [9] Moore C., Braids in classical dynamics, Physical Review Letters 70: 3675-3679, 1993. [10] Chenciner A., Montgomery R., A remarkable periodic solution of the three- body problem in the case of equal masses, Annals of Mathematics 152: 881- 901, 2000. [11] Simó C., Dynamical properties of the figure eight solution of the three-body problem, “Celestial Mechanics, dedicated to Donald Saari for his 60th Birth- day”, A. Chenciner, R. Cushman, C. Robinson, Z.J. Xia ed., Contemporary Mathematics 292: 209-228, 2002. [12] Šuvakov M., Dmitrašinović V., Three classes of Newtonian three-body planar periodic orbits, Physical Review Letters 110: 114301, 2013. [13] Li, X.M., Jing, Y.P. and Liao, S.J., Over a thousand new periodic orbits of a planar three-body system with unequal masses, Publications of the Astronom- ical Society of Japan 70: 64, 2018. [14] Zotos E.E., Investigating the planar circular restricted three-body problem with strong gravitational field, Meccanica 52: 1995-2021, 2017. [15] Llibre J., Paşca D., Periodic orbits of the planar collision restricted 3-body problem, Celestial Mechanics and Dynamical Astronomy 96: 19-29, 2006. [16] Llibre J., Makhlouf A., On the limit cycles for a class of fourth-order differential equations, Journal of Physics. A: Mathematical and Theoretical 45: 055214, 2012. [17] Abouelmagd E.I., Llibre J., Garcı́a Guirao J.L., Periodic orbits of the planar anisotropic Kepler problem, International Journal of Bifurcation and Chaos 27: 1750039, 2017. 11 [18] Llibre J., Yuan P.F., Periodic orbits of the planar anisotropic Manev problem and of the perturbed hydrogen atom problem, Qualitative Theory of Dynamical Systems: 1-18, 2019. [19] Buică A., Françoise J.P., Llibre J., Periodic solutions of nonlinear periodic dif- ferential systems with a small parameter, Communications on Pure & Applied Analysis 6: 103-111, 2007. 1 School of Mathematical Science, Yangzhou University, Yangzhou 225002, China E-mail: gaofabao@sina.com (Fabao Gao) 2 Departament de Matemàtiques, Universitat Autònoma de Barcelona, Bellaterra 08193, Barcelona, Catalonia, Spain E-mail: jllibre@mat.uab.cat (Jaume Llibre)