Periodic orbits of the two fixed centers problem with a variational gravitational field
aa r X i v : . [ n li n . C D ] A ug PERIODIC ORBITS OF THE TWO FIXED CENTERS PROBLEMWITH A VARIATIONAL GRAVITATIONAL FIELD
FABAO GAO , AND JAUME LLIBRE Abstract.
We prove the existence of periodic orbits of the two fixed centers prob-lem bifurcating from the Kepler problem. We provide the analytical expressions ofthese periodic orbits when the mass parameter of the system is sufficiently small. Introduction
The non-integrability of the restricted three-body problem prevents to obtain theanalytical expressions of its general solutions. The periodic orbits of this problemhave extremely important applications in practical space missions. This fact hasattracted a large number of mathematicians and astronomers to carry out researchon the periodic behavior of the classical restricted three-body problem (see [16] andthe references therein). The extensive research covered three categories: qualitativeanalysis (see [5], [6], [10], [16], and so on), analytical calculation (see [4], [17], [18]),and numerical simulation (see [3], [9], [11], [15], [19], [20]).
Key words and phrases. three-body problem, periodic orbit, averaging theory, variational gravita-tional field. , AND JAUME LLIBRE For the planar circular restricted three-body problem, Zotos [22] investigated theproblem with two equivalent masses with strong gravitational field, which was con-trolled by the power p of gravitational potential. He revealed the great influence ofthe power p on the nature of orbits. For the planar rotating Kepler problem, Llibreand Pa¸sca [13] proved that some of the symmetric periodic orbits can be continued tothe case of the restricted three-body problem colliding on a plane by using a contin-uation method. The method was also applied by Llibre and Makhlouf [12] to providesufficient conditions for periodic orbits of a fourth-order differential system. For largevalues of the eccentricity, Abouelmagd et al. [1] found that the anisotropic Keplerproblem with small anisotropy has two periodic orbits in every negative energy levelbifurcating from elliptic orbits of the Kepler problem by using averaging theory. Inaddition, they also presented the approximate analytic expressions of the continuedperiodic orbits. Recently, for each eccentricity and a sufficiently small parameter,Llibre and Yuan [14] continued elliptic periodic orbits of the Kepler problem to ahydrogen atom problem and an anisotropic Manev problem, respectively.Some relevant works on the analytical calculation of the periodic orbits of the circu-lar restricted three-body problem have been done by Farquhar and Kamel [4]. Theyproposed an approximation method for computing periodic orbits, which motivateda plenty of research for computing periodic orbits of that problem. Based on theLindstedt-Poincar´e-like method and a successive approximation method, Richardson [17], [18] constructed the classical periodic halo-type orbits of the circular restrictedproblem several years later. He gave a third-order analytical solution that was lateron used in the orbital design.Mainly motivated by Zotos [22] and Llibre [12] we shall use the fact that whenthe masses of two primaries differ greatly, i.e. the mass parameter of the two bodycenters problem is very small, then the two fixed centers problem becomes closeto Kepler problem. In this paper we study which Keplerian periodic orbits can becontinued, using the averaging theory, to the fixed-center problem with a variationalgravitational field. Moreover we provide an analytical estimation of these periodicorbits.The two body problem when one of the primaries is very big and the other issmall, is very close to the restricted three-body problem with one of the primariesvery big and the other small; these two problems have many interesting applicationsin astronomy, and in particular in the Solar system, see for instance [7, 8, 21].2. Equations of Motion
The equations of motion of a particle in the two fixed centers problem with avariational gravitational field can be written as(1) q ′ k = ∂ H ∂p k , p ′ k = − ∂ H ∂q k , for k = 1 or 2 , FABAO GAO , AND JAUME LLIBRE where(2) H = p + p − − µr − µr p , and r = p ( q + µ ) + q , r = p ( q − µ ) + q . Here the prime denotes deriv-ative with respect to the time t , and µ (0 < µ ≪
1) denotes the mass parameter ofthe two masses, one of mass 1 − µ fixed at ( − µ,
0) and the other of mass µ fixed at(1 − µ, p denotes the power of the gravitational potential. When p is negative, the interaction is variational for different distances. Zotos in [22] studiedthe restricted circular three-body problem when p >
1. Here we shall study the twofixed centers problem with p = − Theorem 1.
For the mass parameter µ > sufficiently small and for every value H ∈ ( − / , , the periodic solution of the Kepler problem with elliptic eccentric-ity e = − H can be continued to the two fixed centers problem with variationalgravitational field p = − . Proof of theorem
Introducing the so-called McGehee coordinate system and denoting it with ( r, θ, v, u ),where r and θ are the radius and the angle in polar coordinates, v and u are the scaled components of velocity in the radial and angular directions, respectively. Moreprecisely ( q , q ) = r (cos θ, sin θ ) ,r − / v = ( p , p ) · (cos θ, sin θ ) ,r − / u = ( p , p ) · ( − sin θ, cos θ ) . Then the Hamiltonian of system (2) with p = − H = p + p − p q + q + µ − q − q − q + q + q + q ( q + q ) / ! + O ( µ ) , and we have(4) r ′ = r − / v,θ ′ = r − / u,v ′ = r − / (cid:18) u + v − (cid:19) + µ r − / [ r (1 + 2 r ) + 2 (1 − r ) cos θ ] + O ( µ ) ,u ′ = r − / (cid:18) − uv (cid:19) + µ r − / (1 + 2 r ) sin θ + O ( µ ) . FABAO GAO , AND JAUME LLIBRE Note that r = 0 corresponds to the collision singularity of equations (4). Per-forming the change d t/ d τ = r / in the independent variable, equations (4) can berewritten as(5) d r d τ = r v, d θ d τ = ru, d v d τ = r (cid:18) u + v − (cid:19) + µ [ r (1 + 2 r ) + 2 (1 − r ) cos θ ] + O ( µ ) , d u d τ = r (cid:18) − uv (cid:19) + µ (1 + 2 r ) sin θ + O ( µ ) . Consider that when µ = 0, the Hamiltonian system (3) is reduced to the Keplerproblem. We shall study which periodic orbits of the Kepler problem can be contin-ued to the two fixed centers problem with variational gravitational field p = − H = H we have(6) u + v − µ (cid:20) − r − r + (1 + 2 r ) r cos θ (cid:21) = r H . Isolating the radius r from equation (6) and performing a series expansion withrespect to the small mass parameter µ we get(7) r = r ( θ, v, u, H ) = R + µR + O (cid:0) µ (cid:1) , where R = 12 H ( u + v − ,R = 18 H h H − H ( u + v − − ( u + v − i + 12 H H + ( u + v − u + v − . Therefore substituting equation (7) into equations (5) and changing the independentvariable τ by the variable θ , equations (5) become(8) d v d θ = 2 u + v − u + µ " u + ( u + v − H u + 8 H − ( u + v − H ( u + v − u cos θ + O ( µ ) , d u d θ = − v µ H + ( u + v − H ( u + v − u sin θ + O ( µ ) . FABAO GAO , AND JAUME LLIBRE If µ = 0 equations (8) will be reduced to the following unperturbed system(9) d v d θ = 2 u + v − u , d u d θ = − v , which admits the general solution v ( θ ; e, θ ) = e sin( θ − θ ) p e cos( θ − θ ) ,u ( θ ; e, θ ) = p e cos( θ − θ ) , where θ ∈ [0 , π ) and e are the argument of pericenter and the eccentricity ofthe Kepler problem, respectively. The value e = 0 corresponds to circular periodicsolutions, and e ∈ (0 ,
1) indicates that the corresponding periodic solutions areelliptical ones.In order to apply the averaging theory summarized in the Appendix to equations(8), we do the following transformations x = (cid:0) vu (cid:1) , x ( θ ; z ,
0) = (cid:0) v ( θ ; e, θ ) u ( θ ; e, θ ) (cid:1) , z = (cid:0) eθ (cid:1) , F = (cid:0) F F (cid:1) , F = (cid:0) F F (cid:1) , where F = 2 u + v − u ,F = − v ,F = 1 u + ( u + v − H u + 8 H − ( u + v − H ( u + v − u cos θ,F = 4 H + ( u + v − H ( u + v − u sin θ. Now we calculate the averaged function F ( z ), see equation (15) in the Appendix.Let(10) ( G ( θ ; e, θ ) , G ( θ ; e, θ )) = M − z ( θ, z ) F ( θ, x ( θ ; z , , where M − z ( θ, z ) = y ( θ ; e, θ ) y ( θ ; e, θ ) y ( θ ; e, θ ) y ( θ ; e, θ ) , , AND JAUME LLIBRE denotes the inverse of the fundamental matrix of equations (9) with y ( θ ; e, θ ) = (1 + e cos( θ − θ )) / e cos θ ) / [2 cos θ (1 + e cos θ )+ e sin θ sin θ ] ,y ( θ ; e, θ ) = − e cos( θ − θ )) / (1 + e cos θ ) / · (cid:2) (cid:0) e cos θ + 8 e cos cos θ (cid:1) sin θ + 2(4 cos θ + 3 e cos 2 θ ) sin 2 θ + 16 e (2 + cos θ ) sin ( θ/
2) sin θ + 6 e sin θ sin 2 θ (cid:3) ,y ( θ ; e, θ ) = (1 + e cos( θ − θ )) / e cos θ ) / sin θ,y ( θ ; e, θ ) = 4 cos θ + e (cos(2 θ − θ ) + 3 cos θ )4(1 + e cos( θ − θ )) / (1 + e cos θ ) / . Hence, we obtain G ( θ ; e, θ )= 18(1 + e cos θ ) / " − e ) H (1 + e cos( θ − θ )) + 16 H cos θ (1 + e cos( θ − θ )) − e · (cid:18) − ( − e ) H (1 + e cos( θ − θ )) (cid:19)(cid:19) · (2 cos θ (1 + e cos θ ) + e sin θ sin θ )+ 2 H sin θ − e (cid:18) − e ) H (1 + e cos( θ − θ )) (cid:19) · (cid:0) − (cid:0) e cos θ + 8 e cos θ (cid:1) sin θ − e (4 cos θ + 3 e cos 2 θ ) sin 2 θ − e (2 + cos θ ) sin (cid:18) θ (cid:19) sin θ − e sin θ sin 2 θ ! ,G ( θ ; e, θ )= sin θ e cos θ ) / " − e ) H (1 + e cos( θ − θ )) + 16 H cos θ (1 + e cos( θ − θ )) − e · (cid:18) − ( − e ) H (1 + e cos( θ − θ )) (cid:19)(cid:19) + 4 H (4 cos θ + e (cos(2 θ − θ ) + 3 cos θ )) − e · (cid:18) − e ) H (1 + e cos( θ − θ )) (cid:19)(cid:21) . (11) , AND JAUME LLIBRE Substituting equations (11) into equation (15) of the Appendix and computing thecorresponding integrals we obtain(12) F ( z ) = ( f ( e, θ ) , f ( e, θ )) , where(13) f ( e, θ ) = 3 √ − e H (1 + e cos θ ) / [3 e − H − e ( − H ) cos θ + e cos 2 θ ] ,f ( e, θ ) = 3 e √ − e sin θ H √ e cos θ . The function F ( z ) in equation (12) is the averaged function of system (8). Accord-ing to the averaging theory (see the Appendix) we must compute the simple zerosof the system f ( e, θ ) = 0 , f ( e, θ ) = 0. Here we must consider e = 0, otherwise f ( e, θ ) ≡
0, and then we cannot obtain any information based on the averagingtheory. From the equation f ( e, θ ) = 0 we get θ = 0 , θ = π. Case I: If θ = 0, we have e = 1 or e = 2 H from the first equation of (13).However, both values of e should be discarded because e ∈ (0 ,
1) and H < Case II: If θ = π , we have e = 1 or e = − H from the first equation of (13),and e = 1 is excluded because this case does not generate periodic solutions. Thus e = − H with the H ∈ ( − / , J = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∂g ∂e ∂g ∂θ ∂g ∂e ∂g ∂θ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ( e, θ )=( − H , π ) = 9(2 H − H = 0 . Therefore if H ∈ ( − / , v ( θ ; − H , π ) , u ( θ ; − H , π )) = (cid:18) H sin θ √ H cos θ , p H cos θ (cid:19) , can be continued to the periodic solution of the two fixed centers problem withvariational gravitational field p = − µ is sufficientlysmall. For the energy H , defined in (6), we plotted in Figure 1 the continuedperiodic orbits at the energy levels H = − / , − /
8, and − /
16, respectively. Asit is shown in Figure 1 the size of the periodic orbits decreases when the energyincreases.This completes the proof of Theorem 1. , AND JAUME LLIBRE Figure 1.
Periodic orbits at the energy level H = − / , − /
8, and − /
16, respectively.
Appendix
Consider the differential system(14) ˙ y = K ( t, y ) + µ K ( t, y ) + µ K ( t, y , µ ) , with µ = 0 to µ = 0 small enough. The C functions K , K : R × Ω → R n , K : R × Ω × ( − µ , µ ) → R n are T -periodic with respect to variable t , and Ω isdefined as an open subset of R n .We assume that there is a submanifold W of periodic solutions with the sameperiod of the unperturbed system ˙ y = K ( t, y ). Let y ( t, x , µ ) be the solutionof this system such that y (0 , x , µ ) = x . Then the linearized equation of theoriginal unperturbed system along a periodic solution y ( t, x ,
0) can be written as˙ z = D y K ( t, y ( t, x , z , and the fundamental matrix can be denoted by M x ( t ). Suppose that there is an open set W with Cl ( W ) ⊂ Ω such that y ( t, x ,
0) is ω -periodic for each x ∈ Cl ( W ) and y (0 , x ,
0) = x . Here Cl ( W ) denotes the closureof W in R n . Then we have the following proposition (see Corollary 1 of [2] for aneasy proof): Proposition 1 . Let W be an open and bounded set such that Cl ( W ) ⊂ Ω andfor each x ∈ Cl ( W ) the solution y ( t, x , satisfying y (0 , x ,
0) = x is T -periodic.Define the function P : Cl ( W ) → R n as (15) P ( x ) = 1 ω Z ω M − x ( θ, x ) K ( θ, y ( θ ; x , θ. Assume that there exists c ∈ W such that P ( c ) = and that det((d P / d y )( c )) = 0 .Then system (14) admits an ω -periodic solution ψ ( t, µ ) such that ψ (0 , µ ) → c for µ → . Acknowledgments
We thank the reviewers and the editor-in-chief for their comments which helpedus to improve the presentation of this paper.The first author gratefully acknowledges the support of the National Natural Sci-ence Foundation of China (NSFC) through grant No.11672259, the China ScholarshipCouncil through grant No.20190832 0086. , AND JAUME LLIBRE The second author gratefully acknowledges the support of the Ministerio de Econom´ıa,Industria y Competitividad, Agencia Estatal de Investigaci´on grants MTM2016-77278-P (FEDER), the Ag`encia de Gesti´o d’Ajuts Universitaris i de Recerca grant2017SGR1617, and the H2020 European Research Council grant MSCA-RISE-2017-777911.
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Meccanica : 1995-2021, 2017. School of Mathematical Science, Yangzhou University, Yangzhou 225002, China