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DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS SERIES B Volume, Number, pp. ON THE STABILITY OF THE LAGRANGIAN HOMOGRAPHIC SOLUTIONS IN A CURVED THREE-BODY PROBLEM ON S Regina Martínez Departament de Matemàtiques Universitat Autònoma de Barcelona Bellaterra, Barcelona, Spain Carles Simó Departament de Matemàtica Aplicada i Anàlisi Universitat de Barcelona Gran Via 585, 87 Barcelona, Spain (Communicated by) Abstract. The problem of three bodies with equal masses in S is known to have Lagrangian homographic orbits. We study the linear stability and also a practical (or effective) stability of these orbits on the unit sphere. A nuestro buen amigo Ernesto Lacomba, en recuerdo de años pasados y con nuestros mejores deseos para los venideros. 1. Introduction. The classical (Euclidean) N-body problem in R or R 3 was modified by Bolyai and Lobachevsky near years ago to consider the case of curved spaces. Since then it has received sporadically some attention, but the interest has been renewed in the last years with several contributions. We refer to the nice paper [1] for historical details and basic results. In particular that paper proves the existence of several kinds of Lagrangian-like homographic orbits and classifies all the possible solutions of this type. In [1] both the Lagrangian and Eulerian homographic solutions are studied and the curvature κ is kept as a parameter. As in the classical case one can scale the masses of the three bodies so that the sum of the masses is equal to 1 or, if the masses are equal, they can be all set to 1. But the curvature κ cannot be scaled. Hence, it is an essential parameter in the problem. But in what follows we shall consider just the motion on the sphere of radius 1. The methodology used in present paper can be extended to arbitrary positive values of κ, but the results will be different. Our main point is that not only the existence of periodic solutions (either in a fixed or rotating frame) is relevant, but mainly its local stability properties. Another question that we face is that even orbits starting close to unstable homographic Mathematics Subject Classification. Primary: ; Secondary:. Key words and phrases. curved 3-body problem, stability of solutions, homographic orbits, practical stability. 1 MARTÍNEZ AND SIMÓ solutions can move, during a very long time, in a vicinity of moderate size of these orbits due to the existence of many invariant nearby tori and its sticky properties. We consider the motion of three point particles of masses equal to 1 moving on S embedded in R 3. Let q i, i = 1,, 3 denote the position of the i-th mass. Hence, if we denote as (, ) the scalar product, we have (q i, q i ) = 1. The force function which extends from the plane to S is U(q) = 1 i j 3 (q i, q j ) [1 (q i, q j ) ] 1/. The equations of motion are a particular case of the equations (8) in [1] and read as 3 q j (q i, q j )q i q i = [1 (q i, q j ) ] ( q i, q 3/ i )q i, i = 1,, 3, (1) j=1,j i where denotes differentiation with respect to the time t. In contrast with the Euclidean case that system no longer has the centre of mass integrals, but it keeps the energy and angular momentum integrals. Concretely if T = 1 3 i=1 ( q i, q i ) and H = T U and c = 3 i=1 q i q i then H and c are first integrals. Due to the invariance of the equations in (1) under the action of SO(3) one can always assume that c points in the direction of the positive z axis if c. The modulus of c will be denoted simply as c. A Lagrangian-like solution is a solution of (1) in which the three masses form an equilateral triangle for all t. In [1] it is proved (Theorem 4) that these solutions are only possible with three equal masses. This is why we restrict our attention to this case. Furthermore the possible Lagrangian-like solutions, in any sphere of arbitrary radius, belong to different classes, as given by Theorem in [1]. We present these results in a slightly modified version, adapted to the unit sphere. This will be obtained from the phase portrait of the Hamiltonian H z to be introduced in Section 3 (see Figure 3). So, the Lagrangian-like solutions belong to one of the following types. 1) If c = then there are a) Fixed points at the equator. b) Homothetic orbits, analogous to the homothetic orbits in the planar case, which are ejected from the north pole and return to collision with the north pole after reaching a minimal value z m of z which is positive. Each one of the masses moves along a meridian. Symmetrical orbits exist exchanging north pole by south pole and being now z m, negative, the maximal value of z. c) Homothetic orbits ejected from the north pole and going to collision at the south pole and vice versa. d) Finally we find orbits ejected from the north pole and going asymptotically to the equator when t +. They spiral towards the equator. Also the orbits symmetrical of these one by time reversal and the corresponding ones in the southern hemisphere. ) If c 8/ 3 then there are a) A relative equilibrium solution with the bodies at the equator, b) Relative equilibrium solutions, with the bodies moving in a parallel, which depends on the modulus of c with constant angular velocity. Thet can be seen as fixed points in a rotating frame. It is analogous to the planar STABILITY OF HOMOGRAPHIC 3-BODY ORBITS IN S 3 case. There are two such solutions, one in the northern and one in the southern hemisphere. c) Homographic solutions, such that the bodies rotate with non-constant angular velocity and the distance between them is changing. There are three kinds of these orbits: the ones confined to the northern hemisphere with z min z z max 1, the symmetrical ones confined to the southern hemisphere, and the orbits which visit, in a symmetric way, both hemispheres crossing the equator and ranging in 1 z max z z max 1. d) Finally there are separatrix like orbits, either in one hemisphere or the other, which reach an extreme value of z, depending on the modulus of c, and go spiraling asymptotically to the equator for t ±, approaching the relative equilibrium solution at the equator (with some phase shift). 3) If c 8/ 3, only the relative equilibrium at the equator, like in a), and the homographic solutions which cross the equator like in c), subsist. Full details and some illustrations will be given in Section 3. We shall study the case b) and the homographic solutions confined to the northern hemisphere of the case c). We have found relevant differences with the planar case. For the relative equilibrium solutions in the planar three-body problem, according to the classical results (see e.g. [7]), the Siegel exponents for these solutions give instability. For the relative equilibria in b) of the three-body problem in S, we show that there are ranges of the angular momentum for which these orbits are linearly stable. This is the contents of Section. In the homographic case c) a first difference with the planar problem is that the orbits are quasiperiodic in a fixed frame (unless the ratio of the frequencies is rational, which is a zero measure case), while in the planar case they are periodic, with the three bodies moving in ellipses. Furthermore, results in [5] (already announced in [4]) show that the Lagrangian-like homographic orbits for equal masses are totally hyperbolic. In the present case there are open sets of energy and momentum for which these orbits are stable. These properties are studied in Sections 3 and 4. Finally Section 5 is devoted to study the dynamics of homographic orbits by direct numerical simulation. A very rough escape criterion is set up, to decide that, definitely, an orbit starting close to a homographic solution has gone away from it. We found that orbits starting very close to some unstable homographic orbit can remain relatively close to them for very long time. Even if there is no linear stability one can consider that they have a kind of practical or effective stability. Furthermore, for the orbits displaying linear stability one can not prove, in general, the non-linear stability. Even if KAM theorem can be applied, in the case of three or more degrees of freedom there is no way to prevent from the existence of Arnol d diffusion, but the possible escape is extremely slow. The idea is similar to what happens around totally elliptic fixed points, as studied in [] for the Lagrangian solutions of the RTBP. In the present case a reference orbit (a homographic one) can be unstable, but many invariant tori can be close, with sticky properties (as follows using standard tools of averaging) and, hence, the departure from the vicinity of them is a very slow process. This study is completed with a plot of the rate of escape. 4 MARTÍNEZ AND SIMÓ As a final comment we consider the essential number of eigenvalues to be computed. In principle the system has 6 degrees of freedom and hence one has to consider 1 eigenvalues. Due to the energy (and time shift) we can skip two of them. As we can assume the components c x and c y equal to zero one can skip two additional eigenvalues. Finally the fact that c z is constant and the node elimination allow to reduce to 6 essential eigenvalues. The reduction process can be seen like the one in the Euclidean problem, as described, e.g., in [8]. Anyway, for most of the analytical and numerical computations we shall not use reduction, but rather the non-relevant eigenvalues are skipped at the end of the computations. In this way the equations to integrate are kept easy and this is also used as an additional check.. Stability of the relative equilibria. The system (1) has a Lagrangian-like periodic solution in which the three bodies move on a parallel, forming an equilateral triangle for all time, and rotating with angular velocity ω. Let r (, 1) be the radius of the parallel. Then the vertical coordinate z = ± 1 r remains constant. For concreteness we introduce the coordinates q i = (x i, y i, z i ) T. Just asking that the values of z i remain constant it is immediate to obtain ω = (4/(1r 9r 4 ) 3/ ) 1/. It is clear that, when the radius tends to zero, the dynamics approaches the planar motion, but with present variables the value of ω in these periodic solutions tends to. As a first step it is convenient to introduce scaled x, y variables and a new time τ defined as follows x i = rx i, y i = ry i, t = r 3/ τ. () In this way, when r tends to zero the equations tend to the planar ones plus a perturbation which is O(r ). Let denote d/dτ. Then the angular frequency becomes Ω = (4/(1 9r ) 3/ ) 1/, bounded and bounded away from zero for the full range r (, 1). Next step is the introduction of a rotating frame. We define new variables ξ i, η i and the rotation R(θ), θ = Ωτ, as ( Xi Y i ) = R ( ξi η i ), R(θ) = ( cos(θ) sin(θ) sin(θ) cos(θ) ). (3) The equations of motion become (the indices i, j ranging in {1,, 3} and keeping in mind that r is a constant parameter here) ( ) [( ) ( )] ( ) ξ i ξj ξi η i (q η i, q j ) r ξi h j η i, i η i where = Ω ( ξi η i ) +Ω ( η i ξ i ) + g 3/ i,j j i g i,j = ρ i + ρ j p i,j zi,j r (ρ i ρ j + p i,j), ρ i = ξ i + η i, p i,j = ξ i ξ j + η i η j, z i,j = (1 r ρ i )(1 r ρ j), (q i, q j ) = z i,j + r p i,j, h i = Ω ρ i + Ω(ξ i η i η i ξ i) + (ξ i) + (η i) + r (ξ i ξ i + η i η i) /(1 r ρ i ). Our present goal is to prove the following result. Theorem 1. Consider the Lagrangian-like periodic orbits of three equal masses moving on S. Let the motion take place on a parallel with z = 1 r. Then the orbits are linearly stable (or totally elliptic) for r (r 1, r ) (r 3, 1) and linearly unstable for r (, r 1 ) (r, r 3 ). In the unstable domains the local behaviour around (4) STABILITY OF HOMOGRAPHIC 3-BODY ORBITS IN S 5 the orbits consists of two elliptic planes and a complex saddle. The bifurcations at r k, k = 1,, 3 are Hamiltonian-Hopf bifurcations. Remark 1. As it will be shown along the proof, the values of r k at which the bifurcations are produced can be obtained from the zeros which belong to (, 1), of the polynomial P 1 (R) = 7R R R R 4716R + 648, where R denotes r. Approximate values are r 1 = , r = , r 3 = Proof of Theorem 1. It is easy to check that ξ 1 =1, η 1 =, ξ = 1/, η = 3/, ξ 3 =ξ, η 3 = η, ξ 1 =η 1 =ξ =η =ξ =η = (5) is a fixed point in the rotating frame. We are interested on the eigenvalues of the differential of the vector field (4), written as a system of 1 first order equations, at that point. The structure of the differential of the vector field at the fixed points is easily seen to be of the form ( ) I Df =. A B Some tedious but elementary computations give A = Ω Â being (4 18r )Â the matrix whose components â i,j are given by â 1,1 =7r 4 1r +44, â 1, =â,1 =â 4,5 =â 6,3 =, â 1,3 =â 1,5 = 18r 4 +7r 1, â 1,4 = â 1,6 =(6r 4 11r +6) 3, â, = 4r +, â,3 = â,5 =( 9r +6) 3, â,4 =â,6 =3r +, â 3,1 =â 5,1 = 9r 4 +4r 1, â 3, = â 5, =(3r 4 1r +6) 3, â 3,3 =â 5,5 =18r 4 48r +6, â 3,4 =â 4,3 = â 5,6 = â 6,5 =( 18r 4 +4r 6) 3, â 3,5 =â 5,3 = 6r +8, â 3,6 = â 5,4 =(6r 4 r ) 3, â 4,1 = â 6,1 =(9r 4 1r +6) 3, â 4, =â 6, = 9r 4 +6r +, â 4,4 =â 6,6 =54r 4 96r +38, â 4,6 =â 6,4 = 18r 4 +36r 16, and r B =Ω ˆB, r being ˆB 3 = r + 3r r 3. r 3 r + 3r r 3 We are interested in the characteristic multipliers of the periodic orbit, that is in λ = exp(ζt ), where T = π/ω is the period and ζ an eigenvalue of Df. Let us introduce µ such that ζ = Ωµ and then λ = exp(πµ). In this way we can skip the factor in Ω in the characteristic polynomial, and the equation for µ becomes p(µ) := det(â + µ ˆB µ I) =. 6 MARTÍNEZ AND SIMÓ Let now R denote r and M = µ. The characteristics polynomial, after multiplying by (4 3R) 4 to skip denominators, dividing by M(M + 1) to take out some eigenvalues associated to first integrals and removing a factor depending on R, nonzero in [, 1], turns out to be P (M)= (7R 3 18R + 144R 64)M 4 +(81R 3 34R + 43R 19)M 3 +( 81R R 3 168R + 11R 336)M +(54R 6 7R 5 9R R R + 181R 35)M +(135R 6 45R R 4 R 3 164R + 888R 144). We recall that the condition for the linear stability of the relative equilibria is that the solutions M of the previous equation be real and negative. In Figure 1 we plot the real zeros of P (M) as a function of r. The values of r 1, r and r 3 given in Remark 1, when the number of real negative zeros changes from to 4 or vice versa are shown by short vertical lines Figure 1. The evolution of the real zeros of P (M) as a function of r. We consider first the limit cases, r =, 1 (R =, 1) for P (M). If R = the roots are M = 1 (double) and M = 1/ ± i. Last ones give µ = ±1/ ± i, the well known Siegel exponents for equal masses in the planar case, see [7]. If R = 1 the roots are M = and M = 1 (triple). We have to consider first what happens for R small and for R 1 but close to 1. For R small the roots obtained by continuation of the Siegel exponents are simple and only move slightly. To analyze the behaviour of the other roots we set M = 1 + N. After simplification P (M) becomes 1R + 3RN 4N + O( (N, R) 3 ) and a standard Newton polygon argument shows that the roots have expansions of the form N = R + O(R ), N = 5R/4 + O(R ). Hence, M continues to be real and close to 1. For R close to 1 we introduce R = 1+S and expanding P (M) around S =, M = we obtain M+1S+O( (M, S) ). Hence the root M = for S = has the form M = 1S + O(S ) and keeps being negative for S near zero. For the triple root we set M = 1+N and then we obtain 144S +16NS+N 3 +O(S 3, S N, SN, N 4 ). Again a Newton polygon argument gives roots of the form N = 9S + O(S ), N = ±4 S + O(S 3/ ). Therefore, for S near zero (i.e., R 1) the roots M keep being real and close to 1. STABILITY OF HOMOGRAPHIC 3-BODY ORBITS IN S 7 Changes on the stability of the Lagrangian-like periodic orbits can be produced when a couple of conjugate imaginary roots µ become real (i.e., M crosses ) or when two couples of conjugate imaginary roots collide and move outside the imaginary axis (i.e., M has a negative double root which moves to the complex), the so-called Hamiltonian-Hopf bifurcation. Setting M = in P (M) = we find that the possible values for R are R 1, = (1 ± 1)/15. These are double zeros, beyond the trivial zeros R = 1, R = /3. Let now write R = R 1, + S. The expansions of P (M) around M =, S = give, in both cases (i.e., for R = R 1 and R = R ), expressions of the form αm + βs + O(M, MS, S 3 ) with α , β . Hence, the value of M is not crossing zero. This can be seen in Figure 1. Finally it remains to check negative double roots for M. To this end we compute the resultant of P (M) and dp (M)/dM as a polynomial in R which factorizes as R (R 1) 3 (3R 4) 1 (3R 1) P 1 (R)P (R), where P 1 and P are irreducible polynomials of degree 5 P 1 (R) = 7R R R R 4716R + 648, P (R) = 486R 5 133R R 3 484R + 51R 468. The roots of the first three factors are irrelevant in the range of interest. The double factors 3R 1, P (R) give rise to double zeros, but there are no complex roots for M because the roots return to the reals. This happens at values of R equal to 1/3 and approximately equal to , , (other zeros of P being complex). But these concrete values are not relevant. Hence, the only possible bifurcations are associated to the zeros of P 1. As P 1 (1) , P 1 () one should have one root with R 1 and another with R . A simple computation gives three roots R 1, R, R 3 (, 1) which correspond to the radii r 1, r, r 3 given in Remark 1. There are other values of r for which the corresponding values of µ are of the form k i/, k Z and, therefore, corresponding to eigenvalues of the monodromy matrix of the periodic orbit equal to ±1. But these values of µ are simple and cannot give rise to bifurcations. This ends the proof of Theorem 1. The zeros of P (M) as a function of R = r have been computed numerically and from them the eigenvalues λ 1,..., λ 8, being λ j+4 = λ 1 j for j = 1,..., 4 are obtained. We recall that the eigenvalues already known to be associated to first integrals have been skipped. On the other hand we have computed all the characterictic multipliers by direct numerical integration of the equations (1). As expected four of them are equal to 1. For the remaining ones the results agree with the ones obtained from the roots of P (M) within the expected numerical accuracy. Moreover, as said at the Introduction, we can expect a further reduction. Indeed, it has been checked that the eigenvalues λ 1,..., λ 8 are not independent, beyond the conditions imposed by the symplectic character. In any case, for the full range of r (, 1) one finds three indices j 1, j, j 3, the differences between them being always different from 4, such that λ j1 λ j λ j3 = 1. This is the effect of the first integrals not taken into account until this point. It is immediate to check also that the same holds by taking the eigenvalues as exp(πµ j ) where µ j = M j and M j are the roots of the polynomial P (M) for any value of R [, 1]. 8 MARTÍNEZ AND SIMÓ 3. Preliminaries for the homographic solutions. Now we pass to the homographic orbits, keeping the constant curvature κ = 1 and scaling again time and momentum to normalise m = 1 for the three bodies. We look for homographic solutions in the form x i = r(t) cos(θ(t)+(i 1)π/3), y i = r(t) sin(θ(t)+(i 1)π/3), i = 1,, 3, (6) for some scalar functions r(t), θ(t). We recall that q i = (x i, y i, z i ), i = 1,, 3. To study the homographic solutions we introduce Q i = (x i, y i ), i = 1, and a rotating and pulsating reference system as Q i = r(t)r(θ(t))u i, U i = (ξ i, η i ) T, i = 1,, 3 where R(θ) is the rotation defined in (3). We remark that if we take r(t) as constant, and θ(t) = Ωr 3/ t, we recover the change of variables introduced in Section. However, to study homographic solutions now, we preserve the initial time t. The equations in the new variables b
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