dumbbell dynamics

DYNAMICS OF A DUMBBELL SATELLITE UNDER THE ZONAL HARMONIC EFFECT OF AN OBLATE BODY ELBAZ I. ABOUELMAGD1;2, JUAN L.G. GUIRAO3AND JUAN A. VERA4 Abstract. The aim of the present paper is to study the dynamics of a dumbbell satellite moving in a gravity eld generated by an oblate body considering the eect of the zonal harmonic parameter. We prove that the pass trajectory of the mass center of the system is periodic and dierent from the classical one when the eect of the zonal harmonic parameter is non zero. Moreover, we complete the classical theory show- ing that the equations of motion in the satellite approximation can be reduced to Beletsky's equation when the zonal harmonic parameter is zero. The main tool for proving these results is the Lindstedt{Poincare's technique. 1.Introduction From the end of the sixth decade of the last century, a part of the math- ematical community, has directed its attention to the study the so called dumbbell body or satellite in central gravity, see for instance Mor an [21], Schechter [25], Brereton and Modi [11], Beletsky [10, 9], Maciejewski et al. [20]; Kirchgraber et al. [18], Krupa et al. [19], Elipe et al. [15], Burov and Dugain [12] or Nakanishi et al. [23]. Recall that a dumbbell body is a quite simple structure composed by two masses connected by a massless rod. It is assume that this object is moving around a planet whose gravity eld is approximated by the eld of the attracting center. In general, the distance between the two points masses is considered to be much smaller that the distance between the satellite's center of mass and the attracting center of mass. Thus, it is common to neglect the in uence of the attitude dynamics on the motion of the center of mass and treat it as an unperturbed Keplerian one. Rodnikov [24] studied equilibrium positions of a weight on a cable xed to a dumbbell{shaped space station moving along a circular geocentric orbit. This model is composed by two masses coupled by a weightless rod, while the cable is weightless and non-stretched. The equations of motion are stated when the motion is produced in a single plane and the center of mass of the Key words and phrases. Dumbbell satellite, Lindstedt{Poincare's technique, Zonal har- monic parameter, Beletsky's equation. 2010 Mathematics Subject Classication. Primary: 70E17, 70E20, 70E40. Secondary: 37C27. 1arXiv:1408.1214v1 [astro-ph.EP] 6 Aug 2014 2 E.I. ABOUELMAGD, J.L.G. GUIRAO, J.A. VERA system moves along a circular geocentric orbit. Moreover, the equilibrium congurations of the system are obtained and the Lyapunov stability of congurations for two situations, rst when the station is composed of equal masses, second when masses at the ends of the station are dierent are analyzed. For the \dumbbells{load" system with two unilateral connections, all rel- ative equilibria on the circular Keplerian orbit were established by Munitsina [22]. Recall that a relative equilibria of the system is a point of the phase space giving an evolution which is a one{parameter orbit of the action of the symmetry group of the system. These results were interpreted for studying the relative equilibria for which both connections are stretched in geometri- cal terms. The necessary and sucient conditions for stability of the relative equilibria were stated. Celletti and Sidorenko [14] investigated the dumbbell satellite's attitude dynamics, when the center of mass moves on a Keplerian trajectory. They found a stable relative equilibrium position in the case of circular orbits which disappears as far as elliptic trajectories are considered. They replaced the equilibrium position by planar periodic motions and they proved this motion is unstable with respect to out-of-plane perturbations. They also gave some numerical evidences of the existence of stable spatial periodic motions. Burov et al. [13] considered the motion of a dumbbell{shaped body in an attractive Newtonian central eld. They used the Poincare's theory to determine the conditions for the existence of families of system periodic motions depending on the arising small parameter and passing into some stable radial steady{state motion of the unperturbed problem as the small parameter tends to zero. They also proved that, each of the radial relative equilibria generates one family of such periodic motions, for suciently small parameter values. Furthermore, they studied the stability of the obtained periodic solutions in the linear approximation as well as these solutions were calculated up to terms of the rst order in the small parameter. Guirao et al. [17] gave sucient conditions for the existence of periodic solutions of the perturbed attitude dynamics of a rigid dumbbell satellite in a circular orbit. The statement of our main results is the following. Theorem 1. Consider a dumbbell satellite moving in a gravity eld gener- ated by an oblate body considering the eect of the zonal harmonic parameter A. The pass trajectory of the mass center of the system is periodic and dif- ferent from the classical one. If Ais equal to zero our solution coincides with the elliptical classical one. Finally, considering the motion in the satellite approximation we complete the classical theory, stating the following result. DYNAMICS OF A DUMBBELL SATELLITE... 3 Theorem 2. The equations of motion in the satellite approximation can be reduced to Beletsky's equation when Ais equal to zero. Note that Theorem 1 generalizes Celletti and Sidorenko [14], Burov and Dugain [12] and Nakanishi et al. [23] due to oblateness parameter. The structure of the paper is as follows. In Section 2 we present the model description, the potential, the kinetic energy and the Lagrangian function of the system. In Section 3 we present the morphology of the equations of motion and the equation of the mass center of the system. In Sections 4 and 5 we respectively provide proof of Theorems 1 and 2. We remark that whenJ2= 0 is clear that the dynamics occurs on a plane, however when the coecient J2is considered the eects of the gravitational potential are not the same for planes with dierent inclinations and a natural question is if there is an invariant plane for the dynamics. The answer of this fact is positive and it will be a key point in the proofs of our main result. In the Appendix we provide a proof of this property. 2.Model description 2.1.Hypothesis. We assume that the dumbbell satellite is formed by mass- less rod of length lwith to masses m1andm2placed at its ends. Let consider cthe center of mass of the two masses moving in a gravity eld generated by an oblate body whose mass mhaving mass center located at 0 where the distance between 0 and cisrandrl. Let us consider the orbital reference frame cxy with origin at the dumb- bell's center, and the polar coordinates of the center are ( r;). While the rotation of the satellite relative to ray ocwill be determined by an angle . Furthermore we denote the reduced mass by and the sum of the two masses bymswhere=m1m2=msandms=m1+m2, see Figure 1 for details. Now, we assume that riis the position vector of miwith respect to 0. Moreover, let the vector nidenotes the position vector of miwith respect to the center of mass of the dumbbell satellite, i2f1;2g. Therefore, the magnitudes of the position vectors riare controlled by (1) r2 i=r2+n2 i+ 2(1)2inircos  where (2) ni=m3il=ms: 2.2.The potential of the model. From the potential theory, the gravi- tational potential (any object has axial symmetry m0) experienced by the satellitemwill be controlled by (see Murray and Dermott [16]) 4 E.I. ABOUELMAGD, J.L.G. GUIRAO, J.A. VERA Figure 1. The dumbbell satellite model (3) V=Gm 0m r0" 1+1X n=2JnR r0 pn(sin())# where: (1)Gis the universal constant; m0is the mass of the oblate object and mis the mass of the satellite; (2)Ris the mean radius of the oblate object; (3)Jnis a dimensionless coecient that characterizes the size of non{ spherical components of the potential; (4)r0is the distance between m0andm; (5)pnsin() are the Legendre polynomials of degree n; (6)denotes the latitude of the satellite. If the two bodies move in the same plane, then = 0 and equation (3) can be written as: (4) V=Gm 0m r0" 1+1X n=2JnR r0 pn(0)# ; where p2n(0) =(1)n2n! 22n(n!)2,p2n+1(0) = 0. In the present model we shall consider the planar motion, for more details on it see the Appendix, and take the eect of the zonal harmonic up to J2, hence equation (4) can be rewritten as DYNAMICS OF A DUMBBELL SATELLITE... 5 V0=Gm 0m(1 r0+J2R2 2r3 0); see [1, 2, 3, 4, 5, 6, 7, 8] for more details. If we assume that Rrepresent the unit of distance, m0is also the unit mass and denote J2byA. We have that the potential experienced by the massesm1andm2areV1andV2such that (5) V1=Gm 1(1 r1+A 2r3 1); (6) V2=Gm 2(1 r2+A 2r3 2): Therefore the total potential Vcan be written as (7) V=k(m1 r1+m2 r2+A(m1 2r3 1+m2 2r3 2)); wherek=Gdenotes the gravity parameter associated to the oblate body. 2.3.The kinetic energy of the model. Let the vectors e1ande2be an orthogonal set of unitary vectors with e1corresponding to the direction from 0 toc. Consideriandjbe another orthogonal set of unitary vectors such that iis a vector in the direction of xaxis. Consequently the vectors of the locationsriand associates velocities viof massesmican be written as ri=r+ni; vi=dri dt; where r=r(cosi+ sinj); ni= (1)ini(cos e1+ sin e2); ei= (1)i cos + i i+ sin + i j : Therefore, after some calculations, we obtain (8) v2 i=8 < :_r2+r2_2+n2 i(_+_)2 2(1)i_rni(_+_)sin +2(1)irni_(_+_)cos9 = ;: Since the kinetic energy of the dumbbell satellite system is 6 E.I. ABOUELMAGD, J.L.G. GUIRAO, J.A. VERA (9) T=1 22X i=1miv2 i: Substituting equation (8) into (9), the kinetic energy can be written in the form T=Ts+Tr: where (10) Ts=1 2ms( _r2+r2_2): Tr=1 2l2(_+_)2: Hence (11) T=1 2ms( _r2+r2_2) +1 2l2(_+_)2: 2.4.The Lagrangian function of the model. Since the Lagrange's func- tion is dened by L=TVfrom equations (7) and (11) we get (12)L=1 2ms( _r2+r2_2) +1 2l2(_+_)2+k(m1 r1+m2 r2+A(m1 2r3 1+m2 2r3 2)): Therefore, the equations of motion will be governed by (13)d dt(@L @_)@L @= 0; 2fr;;g: 3.Equation of motion 3.1.Equations of motion for the general case. Substituting equation (12) into (13) when 2fr;;gthe equations of motion can be written in the following form (14) (msr2+l2)_+l2_ =p=Q; ms(rr(pl2_ msr2+l2)2) =k8 < :(m1(rncos) r3 1+m2(r+(ln)cos) r3 2) +3 2A(m1(rncos) r5 1+m2(r+(ln)cos) r5 2)9 = ;;  +2 _r(l2_p) r(msr2+l2)=k(msr2+l2) sin  mslr((1 r3 11 r3 2) +3 2A(1 r5 11 r5 2)); whereQis constant and n1=n,n2=lnwhilepis a constant expresses the angular momentum conservation. DYNAMICS OF A DUMBBELL SATELLITE... 7 3.2.Dumbbell's center of motion. Since (r;) is the coordinate of the dumbbell's center, therefore the kinetic energy Tsis given by equation (10), while the potential of the center of mass Vsis given by Vs=Gms(1 r+A 2r3): Consequently, the Lagrange function Lsof the center of mass can be represented in the form (15) Ls=1 2ms( _r2+r2_2) +Gms(1 r+A 2r3): Substituting equation (15) into (13) with L=Lsand2fr;gand taking account that the equations of motion can be written on the form d dt(@Ls @_r)@Ls @r= 0; d dt(@Ls @_)@Ls @= 0; we state that the motion of dumbbell's center will be controlled by (16)rr_2=k(1 r2+3A 2r4); msr2_=Forr2_=h; 1 2_r2k r6kA r3=E; whereFis a constant, his the angular momentum which is constant too, that can be evaluated by the initial conditions and Eis the preservation of the total energy for the dumbbell's center. Let ber=1 u, consequently (17)d2u d2+u=k h2(1 +3 2Au2): (18)_=hu2; 1 2h2du d2 ku6kAu3=E: It is worth mentioning that the system of equations (17) does not represent only dumbbell's center motion, it represents too the motion of two{body problem under the eect of the zonal harmonic motion which can be reduced to the motion of the classical case when A= 0. Therefore our results on the dumbbell's center of motion can be applied it to the motion of two{body problem. Now let us go back to dumbbell's center motion in which we can be assumed that this motion follows a Kepler's type orbit when the eect of 8 E.I. ABOUELMAGD, J.L.G. GUIRAO, J.A. VERA oblateness parameter is absent ( A= 0). Consequently the solution can be written as r0=h2=k (1 +e0cos) wherer0=1 u0,e0is the orbit eccentricity such that 0 e0<1, in the framework of elliptic orbits and is a true anomaly of the center of mass. When= 0; u0=1 rp=k h2(1 +e0); rp=a(1e0) is the pericenter (periapsis) and ais a semi{major axis, see Figures 2 and 3. Now we look for solutions in the form u(;) under the condition 0 < 1. SinceA=J2andJ22[1103;1106] for the most of celestial bodies then we can replace by A. In addition, this solution must hold the initial conditions u(0;A) =1 rp; Du(0;A) = 0: Therefore, we search for straight forward expansion of an asymptotes solution as a tends to zero in the following form (19) u(;A) =u0() +Au1() +o(A2): The eect of the zonal harmonic of J2is taking account, but the perturba- tion due to J2is of order about 103of the unperturbed main term ( m1=r1) or (m2=r2). While all other coecients of zonal harmonic are about 106 or less. Therefore, it is sucient from practical point of view, we take the expansion in equation (19) up to A. On the other hand, o(A2) represents the eect of the zonal harmonic J4while our potential does not contain the zonal harmonic J4. Consequently we truncate the expansion in equation (19) up to the linear term A. In this case the leading{order perturbation equations are Du0+u0=k h2; Du1+u1=3 2k h2u2 0: Under the conditions u0(0) =1 rp,Du0= 0; u1(0) = 0 and Du1= 0, hence the solution is governed by u0=k h2(1 +e0cos) and u1=3k3 2h6[1 +1 2e2 0(1 +1 3e2 0) cos+e0sin1 6e2 0cos 2]: DYNAMICS OF A DUMBBELL SATELLITE... 9 Therefore, the general expression of the dumbbell's center motion up to o(A) will be governed by (20) u(;A) =u0() +Au1(): 4.Proof of Theorem 1 Since equation (20) represents a solution which contains a secular term that grows in . As a result, the expansion is not uniformly valid in and breaks down when =o(A), furthermore Au1is no longer a small correction ofu0. But convergent series approximation of the periodic solution can be determined by the continuation method known as the Lindstedt{Poincare's technique. Since equation (17) is a second order dierential equation, it describes a dynamical system in which A is a small parameter. Consequently if A= 0 the system will be reduced to a harmonic oscillator which has a solution with period T= 2=! 0where!0= 1. The continuation method enables us to construct a periodic solution for A6= 0. If we consider that the angular velocity changes due to the non{ linear terms, the asymptotic solution u(; A) and the angular velocity !of the dynamical system can be expanded as (21)u(;A) =u0() +Au1() +A2u2() +::: != 1 +A!1+A2!2+::: To construct a uniformly valid solution, we will introduce a stretched variable=!, therefore (22)d d=!d d; d2 d2=!2d2 d2: Substituting equations (22) into (17) we obtain (23) !2d2u d2+u=k h2(1 +3 2Au2): Now, under the following conditions u(0;A) =1 rp; u(0;A) = 0; u(+ 2;A) =u(;A); we insert the series expansion (21) into (23) and equating terms of the same order inAwith keeping the terms up to rst order of A, we obtain the following: 10 E.I. ABOUELMAGD, J.L.G. GUIRAO, J.A. VERA The coecient of A0gives a homogeneous equation in the form d2u0 d2+u0=k h2 whereu0(0) =1 rp,du0(0) d= 0 andu0(+ 2;A) =u0(;A) with a solution (24) u0g() =k h2(1 +e0cos); being (25) r=h2=k 1 +e0cos: The coecient of A gives a non{homogeneous equation in the form (26)d2u1 d2+u1=a1+a2cos+a3cos 2 whereu1(0) = 0 ,du1(0) d= 0 and a1=3k3 2h6(1 +1 2e2 0); a2=3e0k3 h6(1 +2!1h4 3k2); a3=3e2 0k3 4h6 with a particular solution u1(+ 2;A) =u0(;A) u1p=a1+1 2a2cos+1 2a2sin1 3a3cos 2: This solution contain a secular term a2sin=2, to avoid this term and the solution becomes periodic we have to equate it coecient by zero, hence !1=3k2 2h4: Therefore the general solution of equation (26) is controlled by (27)u1g=3k3 2h6(1 +1 2e2 0)k3 2h6(3 +e2 0) cose2 0k3 4h6cos 2: Substituting equations (24) and (27) into (21), the general solution of equation (23) becomes u=k h2(1 +k1)[1 + (e0k2 1 +k1) cos+k3 1 +k1cos 2] DYNAMICS OF A DUMBBELL SATELLITE... 11 where k1=3Ak2 4h4(2 +e2 0); k2=Ak2 2h4(3 +e2 0); k3=Ak2e2 0 4h4; = (13Ak2 2h4): Therefore (28) r=h2=k (1 +ecos+ecos 2); with k=k(1 +k1); e= (e0k2 1 +k1); e=k3 1 +k1: In short, it is clear that the trajectory of the mass center diers from that as- sumed by Celletti and Sidorenko [14], Burov and Dugain [12] and Nakanishi et al. [23] due to oblateness parameter. Although, this solution is periodic. While this trajectory is the same as their solutions when the eect of oblate- ness is ignored. Since e0<1 andA<< 1 as a result Ae2 0<<1 is very small. Therefore, if we neglect all terms that include Ae2 0, the equation (28) will be reduced to (29) r=h2=k (1 +ecos); k=k(1 +3Ak2 2h4); e=e03Ak2 2h4(1 +e0): This means that the trajectory of the mass center is elliptic as the classical case with the decreasing of the elliptical parameter and the eccentricity, ending the proof. Remark 1. Taking account the oblateness eect we have proved that the solution is periodic, see equation (28). While for the small value of the parameter A, we have elliptical solutions as in the classical case with the decreasing in the elliptical parameter, see (29). Thus, there is no discontinuity and the solution varies smoothly as A approaches zero. 12 E.I. ABOUELMAGD, J.L.G. GUIRAO, J.A. VERA Figure 2. Variation in the trajectory of dumbbell's center whene0= 0:3 for dierent values of zonal harmonic param- eter. Figure 3. Variation in the trajectory of dumbbell's center whene0= 0:8 for dierent values of zonal harmonic param- eter. DYNAMICS OF A DUMBBELL SATELLITE... 13 Note 1. Figures 2 and 3 represent the changes in the trajectory of the dumbbell's center corresponding to the changes in the eccentricity and in the zonal harmonic parameter, here we have considered that kandhare equal to 1. We denote the curves representing the classical case (the eect of zonal harmonic is switched o) by (CC). If the eect of zonal harmonic is consider ignoring all terms with coecients Ae2 0the curves will be denoted by (EC). Finally, by (PC) we denote the general trajectory. We observe that all trajectories are quasi{elliptical and the decreasing in the ellipse parameters is very small for small values of classical eccentricity e0and the zonal harmonic parameter J2, see some cases of Figure 2. While for some relative large values of the classical eccentricity e0the decreasing in the ellipse parameter is observed especially when the parameter of zonal harmonic is assigned by big value. 5.Proof of Theorem 2 We shall start by the deduction of the equations of motion in satellite approximation. Indeed, substituting equations (1) and (2) into (7), the approximation of the potential energy can be written as (30) V=k(ms(1 r+A 2r3) +l2 2r3(3 cos21)): In this potential we neglect all terms that contain coecients (1 =r) with power four or more, since lr. Therefore the Lagrangian function becomes (31)L=1 2ms( _r2+r2_2) +1 2l2(_+_)2 +k(ms(1 r+A 2r3) +l2 2r3(3 cos21)): Substituting equation (31) into (13), the approximation equations of mo- tion are (32)ms(rr_2) =k(ms(1 r2+3A 2r4) +3l2 2r4(3 cos21)); (msr2+l2)_+l2_ =p; l2(+) =3kl2 r3cos  sin : Now replacing the independent variable twith the starched variable  where=!andr2_=htherefore, it is possible to write _ = () such that (33) ( ) =!k2 h3(1 +ecos+ecos 2)2: 14 E.I. ABOUELMAGD, J.L.G. GUIRAO, J.A. VERA Hence (34)d dt= d d; d2 dt2= 2d2 d2+ 0d d; where (
)0meansd d. Inserting equations (34) into (32) and using equation (33), we obtain (35) (1+ecos+ecos 2)002(esin+2esin 2)(1 !+0)+3k !2kcos  sin  = 0 : Since=!, we can rewrite equation (35) in the form (36)(1 +ecos!+ecos 2!)d2 d22!(esin!+ 2esin 2!)(1 +d d) +3k kcos  sin  = 0 where != 13Ak2 2h4; e=Ak2e2 0 4h4; and k=k[1 +3Ak2 4h4(2 +e2 0)]: Now, for nishing only remark that equation (36) can be reduced to Belet- sky's equation, see [9] for more details, if we assume the oblateness eect is not consider (i.e., A= 0) obtaining the relation (1 +ecos)d2 d22esind d+ 3 cos  sin  = 2 esin; which ends the proof. Appendix Let us introduce now an inertial reference frame I(O;E1,E2,E3). The coordinates of a generic vector in this reference system are denoted by x= (x;y;z )I. Recall that we are considering a dumbbell formed by two material points M1, of massm1andM2of massm2rigidly connected by a segment of constant length land negligible mass mutually attracted by DYNAMICS OF A DUMBBELL SATELLITE... 15 a gravitational potential due to nearly spherical body M. Recall that the potential is given by V(x) =GM kxk 11X n=2JnR kxkn Pnz kxk! withGis the gravitational constant, Mthe mass of the body, and Rthe equatorial radius. Pn(u) is the Legendre polynomial of degree nand argu- mentu, and theJnare constant coecients characterizing the potential of the bodyM:We can dene a rotating frame R(G;e1,e2,e3), withGbe the center of masses of the dumbbell, such that the unitary vector e3is directed along the dumbbell towards the point M2ande1,e2are two orthonormal vectors, perpendicular to e3. In this frame, the principal moments of inertia (I1;I2;I3) of the dumbbell are I1=I2=l2; I3= 0 with =m1m2 ms wherems=m1+m2and l1=n=m2l ms; l 2=ln=m1l ms the distances from M1andM2toG: The attitude of the dumbbell is given by two angles, namely nutation  and precession . The coordinates of points M1andM2in the space frame Sare M1l1(sin  sin;sin  cos;cos)S M2l2(sin  sin;sin  cos;cos)S: The coordinates of Grespect to the inertial frame I;using cylindrical coordinates are G(rcos;rsin;z)I: Using the Koenig's Theorem, the Lagrangian of the dumbbell is L(P;VP) equal to ms 2 dr dt2 +r2d dt2 +dz dt2! +l2 2 d dt2 +d dt2 sin2! U(P) with ( P;VP) = r;z;; ;;dr dt;dz dt;d dt;d dt;d dt ;and U(P) =V(xM1) +V(xM2): The coordinates of xM1andxM2are xM1(rcosl1sin  sin;rsin+l1sin  cos;zl1cos)I xM2(rcos+l2sin  sin;rsinl2sin  cos;z+l2cos)I 16 E.I. ABOUELMAGD, J.L.G. GUIRAO, J.A. VERA and kxM1k2=r2+z2+l2 12l1(zcos+rsinsin ()) kxM2k2=r2+z2+l2 2+ 2l2(zcos+rsinsin ()): The potential of the system has the following expression U(r;z;;) =GMm 1 kxM1k 1P1 n=2Jn R kxM1kn Pn zl1cos kxM1k + GMm 2 kxM2k 1P1 n=2Jn R kxM2kn Pn z+l2cos kxM2k A.1 Hamiltonian expressions. From the expressions of the kinetic energy and the potential, we can derive the Hamiltonian H(P;TVP) =1 2ms P2 r+P2  r2+P2 z +1 2l2P2  sin2+P2  +U(r;z;;) with (P;TVP) = (r;z;; ;;Pr;P;Pz;P;P): The angles and  appear only as the dierence  , we can reduce the order of the Hamiltonian by means of the canonical transformation (;;P;P)!(; ;P P;P ): The new Hamiltonian is (A:1)H=1 2ms P2 r+(P P)2 r2+P2 z +1 2l2 P2 sin2+P2 ! +U(r;z; ; ): The variable is cyclic and the momentum Pis a constant of the motion. The Hamiltonian itself is another integral. A.2 Equations of the motion. The Hamiltonian equations of the motion are (A:2:1)dr dt=Pr ;dPr dt=(P P)2 r3@U @r; dz dt=Pz ;dPz dt=@U @z; d dt=P P r2+P l2sin2;dP dt=@U @ ; d dt=P l2;dP dt=P2 cos l2sin3@U @: Theorem. The equations (A.2.1) has an invariant manifold given by z 0;Pz0;=2 andP0: DYNAMICS OF A DUMBBELL SATELLITE... 17 Proof. Using the equations dz dt=Pz ;dPz dt=@U @z; d dt=P l2;dP dt=P2 cos l2sin3@U @: the result is immediate. The Hamiltonian (A.1) restricted to the invariant manifold is H=1 2ms P2 r+P2 l2+(P P)2 r2! +U1(r; ): with (A:2:2)U1(r; ) =GM" m1p r2+l2 12l1rsin +m2p r2+l2 2+ 2l2rsin ! + R2J20 B@m1p r2+l2 12l1rsin 3+m2p r2+l2 2+ 2l2rsin 31 CA+O(J4)3 75 Ifr>>l andR= 1,M= 1,k=G,J2=Awe obtain (A:2:3)U1(r; ) =k ms1 r+A 2r3 +l2 2r3 3 cos2 1
 : A.3 The Lagrangian function. The Lagrangian are L r;; ;dr dt;d dt;d dt =ms 2 dr dt2 +r2d dt2! +l2 2d(+ ) dt2 U1(r; ) and the second order equations of the motion are given by (A:3)ms d2r dt2rd dt2! =@U1 @r; l2d2 dt2+d2 dt2 =@U1 @ ; msr2d dt+l2d dt+d dt = constant: It is clear that if we replace the symbols byand by  the equation (A.2.3) is the same of equation (30). Also the system of equations (A.3) becomes into the system of equations (14) when U1is represented by equa- tion (A.2.3) and it is the same of equation (32) when U1is represented by equation (A.3). 18 E.I. ABOUELMAGD, J.L.G. GUIRAO, J.A. VERA Acknowledgements This work has been partially supported by MICINN/FEDER grant num- ber MTM2011{22587. References [1]E.I. Abouelmagd, S.M. El{Shaboury ,Periodic orbits under combined eects of oblateness and radiation in the restricted problem of three bodies , Astrophys Space Sci. 341(2012), 331{341 [2]E.I. Abouelmagd ,Existence and stability of triangular points in the restricted three{ body problem with numerical applications , Astrophys Space Sci. 342(2012), 45{53 [3]E.I. Abouelmagd ,Stability of the triangular points under combined eects of radia- tion and oblateness in the restricted three-body problem , Earth Moon and Planets 110 (2013), 143{155 [4]E.I. Abouelmagd ,Existence and stability of triangular points in the restricted three{ body problem with numerical applications , Astrophys Space Sci. 342(2012), 45{53 [5]E.I. Abouelmagd ,The eect of photogravitational force and oblateness in the per- turbed restricted three-body problem , Astrophys Space Sci. 346(2013), 321{332 [6]E.I. Abouelmagd, M.A. Sharaf ,The motion around the libration points in the restricted three-body problem with the eect of radiation and oblateness , Astrophys Space Sci. 344(2013), 321{332 [7]E.I. Abouelmagd, H.M. Asiri, M.A. Sharaf ,The eect of oblateness in the per- turbed restricted three-body problem , Meccanica 48(2013), 2479{2490 [8]E.I. Abouelmagd, M.E. Awad, E.M.A. Elzayat, I.A. Abbas ,Reduction the secular solution to periodic solution in the generalized restricted three-body problem , Astrophys Space Sci. 350(2014), 495{505 [9]V.V. Beletsky ,Motion of an articial satellite about its center of mass , Israel pro- gram for scientic translations. Jerusalem (1966) [10]V.V. Beletsky and D.V. Pankova ,Connected bodies in the orbit as dynamic bil- liard, Regular Chaot. Dynam. 1(1996), 87{103 [11]R.C. Brereton and V.J. Modi ,On the stability of planar librations of a dumbbell satellite in an elliptic orbit , Aeronaut. J. 70(1966), 1098{1102 [12]A. Burov and A. Dugain ,Planar oscillations a vibrating dumbbell{like body in a central eld of forces , Aeronaut. J. 49(4) (2011), 353{359 [13]A. Burov, I.I. Kosenko and H. Troger ,On periodic motions of an orbital dumbbell{shaped body with a cabin elevator , Mechanics of Solids 47(3) (2012), 269{284 [14]A. Celletti and V. Sidorenko ,Some properties of the dumbbell satellite attitude , Celestial Mech. Dyn. Astr. 101(2008), 105{126 [15]A. Elipe, M. Palacios and H. Pretka{Ziomek ,Equilibria of the two{body problem with rigid dumbbell satellite , Chaos Soliton & Fractals 35(2008), 830{842 [16]C.D. Murray and S.F. Dermott ,Solar system dynamics , Cambridge University Press, (1999) [17]J.L.G. Guirao, J.A. Vera and B.A. Wade ,On the periodic solutions of a rigid dumbbell satellite in a circular orbit , Astrophys Space Sci. 346(2013), 437{442 [18]U. Kirchgraber, U. Manz and D. Stoffer ,Rigorous proof of chaotic behavior in a dumbbell satellite model , J. Math. Anal. Appl. 251(2000), 897{911 [19]M. Krupa, A. Steindl and I.I. Troger ,Stability of relative equilibria. Part II: Dumbbell satellites , Meccanica 35(2001), 353{371 [20]A.J. Maciejewski ,About attitude motion of a satellite with time dependent mo- ments of inertia , Articial Satellites 17(2-3) (1982), 49{60 [21]J.P. Mor an,Eects of plane librations on the orbital motion of a dumbbell satellite , ARS Journal 31(8) (1961), 1089{1096 DYNAMICS OF A DUMBBELL SATELLITE... 19 [22]M.A. Munitsina ,Relative equilibrium on the circular Keplerian orbit of the \Dumbbells-Load" system with unilateral connections , Automation and Remote Con- trol68(9) (2007), 1476{1481 [23]K. Nakanishi, H. Kojima and T. Watanabe ,Trajectories of in plane periodic solutions tethered satellite system projected on van der Pol planes , Acta Astronautica 68(2011), 1024{1030 [24]A.V. Rodnikov ,Equilibrium positions of a weight on a cable xed to a dumbbell{ shaped space station moving along a circular geocentric orbit , Cosmic Research 44 (2006), 58{68 [25]H.B. Schechter ,Dumbbell librations in elliptic orbits , AIAA Journal 2(1964), 1000{1004 1Mathematics Department, Faculty of Science and Arts (Khulais), King Abdulaziz, University, Jeddah, Saudi Arabia 2Nonlinear Analysis and Applied Mathematics Research Group (NAAM) Department of Mathematics, King Abdulaziz University, Jeddah, Saudi Arabia E-mail address :[email protected], [email protected] 3Departamento de Matem atica Aplicada y Estad stica. Universidad Polit ecnica de Cartagena, Hospital de Marina, 30203{Cartagena, Regi on de Murcia, Spain.{ Corresponding Author{ E-mail address :[email protected] 4Centro Universitario de la Defensa. Academia General del Aire. Uni- versidad Polit ecnica de Cartagena, 30720-Santiago de la Ribera, Regi on de Murcia, Spain. E-mail address :[email protected]