The reduction on the linear stability of elliptic Euler-Moulton solutions of the n-body problem to those of 3-body problems
aa r X i v : . [ m a t h . D S ] J u l The reduction on the linear stability of elliptic Euler-Moultonsolutions of the n -body problem to those of 3-body problems Qinglong Zhou ∗ Yiming Long † , School of MathematicsShandong University, Jinan 250100, Shandong, China Chern Institute of Mathematics and LPMCNankai University, Tianjin 300071, ChinaAugust 28, 2018
Abstract
In this paper, we consider the elliptic collinear solutions of the classical n -body problem, where the n bodies always stay on a straight line, and each of them moves on its own elliptic orbit with the sameeccentricity. Such a motion is called an elliptic Euler-Moulton collinear solution. Here we prove thatthe corresponding linearized Hamiltonian system at such an elliptic Euler-Moulton collinear solutionof n -bodies splits into ( n −
1) independent linear Hamiltonian systems, the first one is the linearizedHamiltonian system of the Kepler 2-body problem at Kepler elliptic orbit, and each of the other ( n − n -body problem is reduced to those of the corresponding elliptic Euler collinear solutions of the 3-bodyproblems, which for example then can be further understood using numerical results of Martin´ez, Sam`aand Sim´o in [13] and [14] on 3-body Euler solutions in 2004-2006. As an example, we carry out thedetailed derivation of the linear stability for an elliptic Euler-Moulton solution of the 4-body problemwith two small masses in the middle. Keywords: n -body problem, elliptic Euler-Moulton collinear solution, reduction, linear stability. AMS Subject Classification : 70F10, 70H14, 34C25.
When one considers a system of n bodies including the Earth, the Moon and ( n −
2) space stations in themiddle, one tries to find places for these space stations so that they can be easily put there and easily takenaway. When n =
3, by the linear stability study it is well-known that such a middle place should be theEuler point, because at such a point the essential part of the linearized Hamiltonian system possesses twopairs of Floquet multipliers with suitable masses and eccentricity, one of which is elliptic and the other ishyperbolic. This paper is devoted to study the problem for general n ≥
3, and in fact here we prove that thestudy on such an n -body problem can be reduced to those of ( n −
2) related 3-body problems. ∗ Partially supported by NSFC (No.11501330, No.11425105) of China. E-mail:[email protected] † Partially supported by NSFC (No. 11131004), LPMC of MOE of China, Nankai University, and BAICIT at Capital NormalUniversity. E-mail: [email protected] e ∈ [0 , n positive masses, there exists a unique collinear central configuration of n -bodies. After them in general,for the classical n -body problem we call a solution elliptic Euler-Moulton homographic motion of n -bodies( EEM for short below), if the n bodies always form a collinear central configuration and each body travelsalong a specific Keplerian elliptic orbit about the center of mass of the system with the same eccentricity.Specially when e =
0, the n bodies run circularly around the center of mass with the same angular velocity,which are called Euler-Moulton relative equilibria traditionally.Given n positive masses m = ( m , . . . , m n ) ∈ ( R + ) n on n points q = ( q , . . . , q n ) ∈ ( R ) n respectively.According to Newton’s gravitation law, their motion is governed by the system, m i ¨ q i = ∂ U ( q ) ∂ q i , for i = , , . . . , n , (1.1)where U ( q ) = P ≤ i < j ≤ n m i m j | q i − q j | is the potential function and | · | denotes the norm of vectors in R .Let ˆ X : = q = ( q , q , . . . , q n ) ∈ ( R ) n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n X i = m i q i = , q i , q j , ∀ i , j . Then critical points of the action functional A ( q ) = Z π n X i = m i | ˙ q i ( t ) | + U ( q ( t )) dt defined on the space W , ( R / π Z , ˆ X ) correspond to 2 π -periodic solutions of the system (1.1) one-to-one.To transform (1.1) to a Hamiltonian system, we let p = ( p , . . . , p n ) with p i = m i ˙ q i ∈ R for 1 ≤ i ≤ n andobtain ˙ p i = − ∂ H ∂ q i , ˙ q i = ∂ H ∂ p i , for i = , , . . . , n , (1.2)where the Hamiltonian function is given by H ( p , q ) = n X i = | p i | m i − U ( q ) . (1.3)It is well-known (cf. [13], [14]) that the linear stability of an EEM solution of the 3-body problem withmasses m = ( m , m , m ) ∈ ( R + ) is determined by the eccentricity e ∈ [0 ,
1) and the mass parameter β = m (3 x + x + + m x ( x + x + x + m [( x + ( x + − x ] , (1.4)where x is the unique positive solution of the Euler quintic polynomial equation( m + m ) x + (3 m + m ) x + (3 m + m ) x − (3 m + m ) x − (3 m + m ) x − ( m + m ) = , (1.5)and the three bodies form a central configuration of m , which are denoted by q = q = ( x α, T and q = ((1 + x ) α, T with α = | q − q | > x α = | q − q | .In this paper we prove that the linear stability problem of the EEM in the n -body case for every integer n ≥ n −
2) related EEM of 3-body cases. More precisely,2ased on the central configuration coordinate method of K. Meyer and D. Schmidt in [16], we reduce thelinear stability of the n -body EEM to two parts symplectically, one of which is the same as that of the Keplersolutions, and the other is a 4( n − n -bodies. Then we prove that this essential part is thesum of ( n −
2) independent linear Hamiltonian systems, each of which is the essential part of the linearizedHamiltonian system of some EEM of a related 3-body problem.To describe our main reduction result more precisely, given positive masses m = ( m , m , . . . , m n ) ∈ ( R + ) n , let a = ( a , . . . , a n ) be the unique n -body collinear central configuration of m with a i = ( a ix , T for1 ≤ i ≤ n which satisfies a ix < a jx if i < j . Without lose of generality, we normalize the masses by n X i = m i = , (1.6)and normalize the positions a i with 1 ≤ i ≤ n by n X i = m i a i = , and n X i = m i a i = I ( a ) = . (1.7)Moreover, we define µ = U ( a ) = X ≤ i < j ≤ n m i m j | a i − a j | , (1.8)and ˜ M = diag( m , . . . , m n ) . (1.9)Let B = ( B i j ) = U ′′ ( a ) be the Hessian of U ( q ) at the collinear central configuration q = a which is an n × n symmetric matrix given by B i j = m i m j | a i − a j | , if i , j , ≤ i , j ≤ n , (1.10) B i j = − X ≤ j ≤ nj , i m i m j | a i − a j | , if i = j , ≤ i ≤ n . (1.11)We let D = µ I n + ˜ M − B . (1.12)Then the following lemma is crucial for our study, whose proof is due to C. Conley according to F.Pacella ([21], 1987) and R. Moeckel ([17] of 1990 as well as [18] of 1994). For reader’s conveniences, asketch of this proof will be given in the Appendix of this paper below following [21], [17] and [18]. Lemma 1.1
The n × n matrix D possesses a simple eigenvalue λ = µ > and a second eigenvalue λ = .The other n − eigenvalues of D besides µ and this are non-positive. Consequently they satisfy λ > λ = ≥ λ ≥ · · · ≥ λ n . (1.13)Then we define β i = − λ i + µ ≥ , ∀ ≤ i ≤ n − . (1.14)Based on these β i s, our main result of this paper is the following3 heorem 1.2 In the planar n-body problem with given masses m = ( m , m , . . . , m n ) ∈ ( R + ) n , denote theEEM with eccentricity e ∈ [0 , for m by q m , e ( t ) = ( q ( t ) , q ( t ) , . . . , q n ( t )) . Then the linearized Hamiltoniansystem at q m , e is reduced into the sum of ( n − independent Hamiltonian systems, the first one is thelinearized system of the Kepler -body problem at the corresponding Kepler orbit, and the i-th part of theother ( n − parts with ≤ i ≤ n − is the essential part of the linearized Hamiltonian system of some EEMof a -body problem with the original eccentricity e and the mass parameter β i given by (1.14) instead ofthat β given by (1.4). Remark 1.3 (i) J. Liouville first observed in [10] of 1842 that the Moon stays always on the straight linepassing through the centers of the Sun and the Earth and on the opposite side of the Sun with respect to theEarth, i.e., the Moon always enlightens the Earth during the nights, is impossible due to the instability ofsuch a configuration. According to R. Moeckel (cf. p.300, [18]) of 1994, the stability analysis of collinearrelative equilibria can be attributed to M. Andoyer [1] in 1906 and M. Meyer [15] in 1933. Subsequentstudies on the linear stability of EEMs can be found in [16] of K. Meyer and D. Schmidt in 2005, [12], [13]and [14] of R. Mart´ınez, A. Sam`a and C. Sim´o in 2004-2006, and the recent preprints [26] of Q. Zhou andY. Long, and [6] of X. Hu and Y. Ou. Researches on Lagrangian equilateral triangle elliptic solutions (cf.[9]) and related topics were done by M. Gascheau ([4], 1843), E. Routh ([23], 1875), J. Danby ([2], 1964),R. Moeckel ([19], 1995), G. Roberts ([22], 2002), X. Hu and S. Sun ([7], 2010), and X. Hu, Y. Long and S.Sun ([5], 2014).(ii) Based on our above reduction theorems, the numerical results obtained by R. Mart´ınez, A. Sam`a andC. Sim´o in [13] and [14] for 3-body Euler solutions can be applied to get the linear stability of the n -bodyelliptic Euler-Moulton collinear solutions using our formula of β i s in (1.14) for any positive integer n ≥ β and β in (1.14) are calculated explicitly there, and hence their linear stability can bedetermined numerically using results in [13] and [14] of 2004-2006 for example. It is interesting to see thatwhen the masses of the two middle particles tend to 0, the e ff ect of both of them does not disappear. In theAppendix, a sketch of the proof of Lemma 1.1 is given. n -body problem to ( n − collinear -bodyproblems In their paper [16] of 2005, K. Meyer and D. Schmidt introduced the central configuration coordinates fora class of periodic solutions of the n -body problem. Our study on the EEM solutions of n -bodies is basedupon their method. Here the key point is that we found the reduction of the linear stability of the n -bodyEEM problem to those of ( n −
2) three body problems. This reduction needs more techniques for the n bodycase.As in Section 1, for the given masses m = ( m , m , . . . , m n ) ∈ ( R + ) n satisfying (1.6), suppose the n particles are all on the x -axis with a = ( a x , T , a = ( a x , T , . . . , a n = ( a nx , T satisfying a ix < a jx if i < j . In this section we always denote by a = ( a , . . . , a n ) the unique collinear central configuration for the4ass m determined by [20]. Using normalization and notations (1.6)-(1.9), we have n X j = , j , i m j ( a jx − a ix ) | a jx − a ix | = U ( a )2 I ( a ) a ix = µ a ix . (2.1)Based on the matrix B of (1.10)-(1.11), besides D we further define˜ D = µ I n + ˜ M − / B ˜ M − / = ˜ M / D ˜ M − / . (2.2)where µ is given by (1.8).Since ˜ D is symmetric, all its eigenvalues are real, which are denoted by λ = µ , λ = λ , . . . , λ n withcorresponding eigenvectors ˜ v = ˜ M / v , ˜ v = ˜ M / v , ˜ v , . . . , ˜ v n . Moreover, we can suppose that ˜ v , ˜ v , . . . ,˜ v n form an orthonormal basis of R n .Letting v i = ˜ M − / ˜ v i for 3 ≤ i ≤ n , we have Dv i = ˜ M − / ˜ D ˜ M / ( ˜ M − / ˜ v i ) = ˜ M − / ˜ D ˜ v i = ˜ M − / λ i ˜ v i = λ i v i . Thus v i is the eigenvector of D belonging to its eigenvalue λ i . Moreover, by the orthonormal basis propertyof ˜ v , ˜ v , . . . , ˜ v n , we have v Ti ˜ Mv j = ˜ v Ti ˜ v j = δ ji , ∀ ≤ i , j ≤ n . (2.3)Denote the eigenvector v i belonging to the eigenvalue λ i of the matrix D by v Ti = ( b i , b i , . . . , b ni ) for3 ≤ i ≤ n , i.e., D ( b k , b k , . . . , b nk ) T = λ k ( b k , b k , . . . , b nk ) T , ≤ k ≤ n . (2.4)Then it yields µ b ik − n X j = , j , i m j ( b ik − b jk ) | a i − a j | = λ k b ik , ≤ i ≤ n . (2.5)Let F ik = n X j = , j , i m i m j ( b ik − b jk ) | a i − a j | , ≤ i ≤ n , ≤ k ≤ n , (2.6)then we have F ik = ( µ − λ k ) m i b ik . (2.7)Moreover, we have n X i = F ik b ik = n X i = ( µ − λ k ) m i b ik = µ − λ k = µ (1 + β k − ) , (2.8)where in the last equality, we used (1.14).Now as in p.263 of [16], we define P = p p . . . p n , Q = q q . . . q n , Y = GZW . . . W n − , X = gzw . . . w n − , (2.9)where p i , q i with i = , , . . . , n , G , Z , W i with 1 ≤ i ≤ n − g , z , and w i with 1 ≤ i ≤ n − R . We make the symplectic coordinate change P = A − T Y , Q = AX , (2.10)5here the matrix A is constructed as in the proof of Proposition 2.1 in [16]. More precisely, the matrix A ∈ GL ( R n ) is given by A = I A B . . . B n I A B . . . B n . . . . . . . . . . . . . . . I A n B n . . . B nn , (2.11)where each A i is a 2 × A i = ( a i , Ja i ) = (cid:18) a ix a ix (cid:19) = a ix I . (2.12)Let B ki = (cid:18) b ki b ki (cid:19) = b ki I . (2.13)Then A T MA = I n holds (cf. (13) in p.263 of [16]).As in Theorem 2.1 on pp.261-262, setting G = g = n -body problem in the new variablesbecomes H ( Z , W , . . . , W n − , z , w , . . . , w n − ) = K ( Z , W , . . . , W n − ) − U ( z , w , . . . , w n − ) (2.14)where the kinetic energy satisfies K =
12 ( | Z | + | W | + . . . + | W n − | ) , (2.15)and the potential function satisfies U ( z , w , . . . , w n − ) = X ≤ i < j ≤ n U i j ( z , w , . . . , w n − ) , (2.16)with U i j ( z , w , . . . , w n − ) = m i m j d i j ( z , w , . . . , w n − ) , (2.17) d i j ( z , w , . . . , w n − ) = | ( A i − A j ) z + n X k = ( B ik − B jk ) w k − | = | ( a ix − a jx ) z + n X k = ( b ik − b jk ) w k − | , (2.18)where we have used (2.12) and (2.13). Recall that each Z , W i , z , w i with 1 ≤ i ≤ n − R .Here z = z ( t ) is the Kepler elliptic orbit given through the true anomaly θ = θ ( t ), r ( θ ( t )) = | z ( t ) | = p + e cos θ ( t ) , (2.19)where p = a (1 − e ) and a > Proposition 2.1
There exists a symplectic coordinate change ξ = ( Z , W , . . . , W n − , z , w , . . . , w n − ) T ¯ ξ = ( ¯ Z , ¯ W , . . . , ¯ W n − , ¯ z , ¯ w , . . . , ¯ w n − ) T , (2.20)6 uch that using the true anomaly θ as the variable the resulting Hamiltonian function of the n-body problemis given by H ( θ, ¯ Z , ¯ W , . . . , ¯ W n − , ¯ z , ¯ w , . . . , ¯ w n − ) =
12 ( | ¯ Z | + n − X k = | ¯ W k | ) + (¯ z · J ¯ Z + n − X k = ¯ w k · J ¯ W k ) + p − r ( θ )2 p ( | ¯ z | + n − X k = | ¯ w k | ) − r ( θ ) σ U (¯ z , ¯ w , . . . , ¯ w n − ) , (2.21) where J = (cid:18) −
11 0 (cid:19) , r ( θ ) = p + e cos θ , µ is given by (1.8), σ = ( µ p ) − / and p is given in (2.19). Remark 2.2
Proposition 2.1 is a modified version of Lemma 3.1 of [16] in our case of n -bodies. As pointedout in Section 11 of [11], in the 3-body case, the σ in (2.23) given by σ = p β in the original computationon line 9 of p.273 in [16] is incorrect, and should be corrected to σ = ( µ p ) − / . Note also that in the line11 of p.273 in [16], the stationary solution (0 , , , , , , , T is not correct too and should be corrected to(0 , σ, , , σ, , , T as in [11], and in general it may not be possible to have σ = Proof of Proposition 2.1.
Because of reasons mentioned in this remark, for reader’s conveniences, wegive the complete details of the proof of this proposition below.Following the proof of Lemma 3.1 of [16], we carry the coordinate changes in four steps.
Step 1.
Rotating coordinates via the matrix R ( θ ( t )) in time t. We change first the coordinates ξ toˆ ξ = ( ˆ Z , ˆ W , . . . , ˆ W n − , ˆ z , . . . , ˆ w , ˆ w n − ) T ∈ ( R ) n − , (2.22)which rotates with the speed of the true anomaly. The transformation matrix is given by the rotation matrix R ( θ ) = (cid:18) cos θ − sin θ sin θ cos θ (cid:19) . The generating function of this transformation is given byˆ F ( t , Z , W , . . . , W n − , ˆ z , ˆ w , . . . , ˆ w n − ) = − Z · R ( θ )ˆ z − n − X i = W i · R ( θ ) ˆ w i , (2.23)and for 1 ≤ i ≤ n − z = − ∂ ˆ F ∂ Z = R ( θ )ˆ z , ˆ Z = − ∂ ˆ F ∂ ˆ z = R ( θ ) T Z , (2.24) w i = − ∂ ˆ F ∂ W i = R ( θ ) ˆ w i , ˆ W i = − ∂ ˆ F ∂ ˆ w i = R ( θ ) T W i . (2.25)Writing ˙ R ( θ ( t )) = ddt R ( θ ( t )), and noting that R ( θ ) T = R ( θ ) − and ˙ R ( θ ) = ˙ θ JR ( θ ) we obtain the functionˆ F t ≡ ∂ ˆ F ∂ t = − Z · ˙ R ( θ )ˆ z − n − X i = W i · ˙ R ( θ ) ˆ w i = − ˆ Z · R ( θ ) T ˙ R ( θ )ˆ z − n − X i = ˆ W i · R ( θ ) T ˙ R ( θ ) ˆ w i = − ˙ θ ˆ Z · R ( θ ) T JR ( θ )ˆ z + n − X i = ˆ W i · R ( θ ) T JR ( θ ) ˆ w i = − ˙ θ ˆ Z · J ˆ z + n − X i = ˆ W i · J ˆ w i . A i s and (2.13) of B ki s, we obtain A i R ( θ ) = R ( θ ) A i , B ki R ( θ ) = R ( θ ) B ki , ∀ ≤ i ≤ n − . (2.26)By (2.16), this then implies U ( z , w , . . . , w n − ) = X ≤ i < j ≤ m i m j | ( A i − A j ) z + P n − i = ( B i − B j ) w i | = X ≤ i < j ≤ m i m j | ( A i − A j ) R ( θ )ˆ z + P n − i = ( B i − B j ) R ( θ ) ˆ w i | = X ≤ i < j ≤ m i m j | ( A i − A j )ˆ z + P n − i = ( B i − B j ) ˆ w i | = U (ˆ z , ˆ w , . . . , ˆ w n − ) , (2.27)by the orthogonality of R ( θ ). Because θ = θ ( t ) depends on t , by adding the function ∂ ˆ F ∂ t to the Hamiltonianfunction H in (2.14), as in Line 5 in p.272 of [16], we obtain the Hamiltonian function ˆ H in the newcoordinates: ˆ H ( t , ˆ Z , ˆ W , . . . , ˆ W n − , ˆ z , ˆ w , . . . , ˆ w n − ) = H ( Z , W , . . . , W n − , z , w , . . . , w n − ) + ˆ F t =
12 ( | ˆ Z | + n − X i = | ˆ W i | ) + (ˆ z · J ˆ Z + n − X i = ˆ w i · J ˆ W i )˙ θ − U (ˆ z , ˆ w , . . . , ˆ w n − ) , (2.28)where the variables of H are functions of θ , ˆ Z , ˆ W , . . . , ˆ W n − , ˆ z , ˆ w , . . . , ˆ w n − given by (2.24)-(2.25). Step 2.
Dilating coordinates via the polar radius r = | z ( t ) | . We change the coordinates ˆ ξ to ˜ ξ = ( ˜ Z , ˜ W , . . . , ˜ W n − , ˜ z , ˜ w , . . . , ˜ w n − ) which dilate with r = | z ( t ) | givenby (2.19). The position coordinates are transformed byˆ z = r ˜ z , ˆ w i = r ˜ w i , ∀ ≤ i ≤ n − . (2.29)It is natural to scale the momenta by 1 / r to get ˆ Z = ˜ Z / r and ˆ W i = ˜ W i / r . But it turns out that the newtransformation with 1 ≤ i ≤ n − Z = r ˜ Z + ˙ r ˜ z , ˆ W i = r ˜ W i + ˙ r ˜ w i (2.30)makes the resulting Hamiltonian function simpler. This transformation is generated by the function˜ F ( t , ˜ Z , ˜ W , . . . , ˜ W n − , ˆ z , ˆ w , . . . , ˆ w n − ) = r ( ˜ Z · ˆ z + n − X i = ˜ W i · ˆ w i ) + ˙ r r ( | ˆ z | + n − X i = | ˆ w i | ) , (2.31)and is given by ˜ z = ∂ ˜ F ∂ ˜ Z = r ˆ z , ˆ Z = ∂ ˜ F ∂ ˆ z = r ˜ Z + ˙ rr ˆ z = r ˜ Z + ˙ r ˜ z , ˜ w i = ∂ ˜ F ∂ ˜ W i = r ˆ z , ˆ W i = ∂ ˜ F ∂ ˆ w i = r ˜ W i + ˙ rr ˆ w i = r ˜ W i + ˙ r ˜ w i , with ∂ ˜ F ∂ t = − ˙ rr ( ˜ Z · ˆ z + n − X i = ˜ W i · ˆ w i ) + ¨ rr − ˙ r r ( | ˆ z | + n − X i = | ˆ w i | ) = − ˙ rr ( ˜ Z · ˜ z + n − X i = ˜ W i · ˜ w i ) + ¨ rr − ˙ r | ˜ z | + n − X i = | ˜ w i | ) , (2.32)8y (2.30).In this case, as in the last two lines on p.272 of [16], the Hamiltonian function ˆ H in (2.28) becomes thenew Hamiltonian function ˜ H in the new coordinates:˜ H ( t , ˜ Z , ˜ W , . . . , ˜ W n − , ˜ z , ˜ w , . . . , ˜ w n − ) ≡ ˆ H ( t , ˆ Z , ˆ W , . . . , ˆ W n − , ˆ z , ˆ w , . . . , ˆ w n − )) + ˜ F t = r ( | ˜ Z | + n − X i = | ˜ W i | ) + ˙ rr ( ˜ Z · ˜ z + n − X i = ˜ W i · ˜ w i ) + ˙ r | ˜ z | + n − X i = | ˜ w i | ) + (˜ z · J ˜ Z + n − X i = ˜ w i · J ˜ W i )˙ θ − U ( r ˜ z , r ˜ w i , . . . , r ˜ w n − ) + F t = r ( | ˜ Z | + n − X i = | ˜ W i | ) + r ¨ r | ˜ z | + n − X i = | ˜ w i | ) + (˜ z · J ˜ Z + n − X i = ˜ w i · J ˜ W i )˙ θ − r U (˜ z , ˜ w , . . . , ˜ w n − ) . (2.33) Step 3.
Coordinates via the true anomaly θ as the independent variable. Here we want to use the true anomaly θ ∈ [0 , π ] as an independent variable instead of t ∈ [0 , T ] tosimplify the study. This is achieved by dividing the Hamiltonian function ˜ H in (2.33) by ˙ θ . Assuming˙ θ ( t ) > t ∈ [0 , T ], for ˜ ξ ∈ W , ( R / ( T Z ) , R ) we consider the action functional corresponding to theHamiltonian system: f ( ˜ ξ ) = Z T ( 12 ˙˜ ξ ( t ) · J ˜ ξ ( t ) − ˜ H ( t , ˜ ξ ( t ))) dt = Z π
12 ˙˜ ξ ( t ( θ ))˙ θ ( t ) · J ˜ ξ ( t ) − ˜ H ( t , ˜ ξ ( t ( θ )))˙ θ ( t ) d θ = Z π
12 ˜ ξ ′ ( θ ) · J ˜ ξ ( θ ) − ˜ H ( θ, ˜ ξ ( θ )) ! d θ. Here we used ˜ ξ ′ ( θ ) to denote the derivative of ˜ ξ ( θ ) with respect to the variable θ . But in the following weshall still write ˙˜ ξ ( θ ) for the derivative with respect to θ instead of ˜ ξ ′ ( θ ) for notational simplicity.It is well known that the elliptic Kepler orbit (2.19) satisfies r ( t ) ˙ θ ( t ) = √ µ p = q µ a (1 − e ) = σ with σ = ( µ p ) / . Note that a = µ / ( T / π ) / with T being the minimal period of the orbit (2.19), we have σ = ( µ a (1 − e )) / = µ / ( T π ) / (1 − e ) / ∈ (0 , µ / ( T π ) / ]depending on e , when the mass µ and the period T are fixed. Note that similarly we have p = σ /µ dependson e too. Note that the function r satisfies¨ r = µ pr − µ r = µ pr − r ! . Therefore we get the Hamiltonian function ˜ H in the new coordinates:˜ H ( θ, ˜ Z , ˜ W , . . . , ˜ W n − , ˜ z , ˜ w , . . . , ˜ w n − ) ≡ θ ˜ H ( t , ˜ Z , ˜ W , . . . , ˜ W n − , ˜ z , ˜ w , . . . , ˜ w n − )9 r ( t )˙ θ ( t ) ( | ˜ Z | + n − X i = | ˜ W i | ) + r ( t )¨ r ( t )2˙ θ ( t ) ( | ˜ z | + n − X i = | ˜ w i | ) + (˜ z · J ˜ Z + n − X i = ˜ w i · J ˜ W i ) − r ( t )˙ θ ( t ) U (˜ z , ˜ w , . . . , ˜ w n − ) = σ ( | ˜ Z | + n − X i = | ˜ W i | ) + (˜ z · J ˜ Z + n − X i = ˜ w i · J ˜ W i ) + µ ( p − r ( θ ))2 σ ( | ˜ z | + n − X i = | ˜ w i | ) − r ( θ ) σ U (˜ z , ˜ w , . . . , ˜ w n − ) , (2.34)where r ( θ ) = p / (1 + e cos θ ). Note that now the minimal period T of the elliptic solution ˜ z = ˜ z ( θ ) becomes2 π in the new coordinates in terms of true anomaly θ as an independent variable. Step 4.
Coordinates via the dilation of σ = ( p µ ) / . The last transformation is the dilation( ˜ Z , ˜ W , . . . , ˜ W n − , ˜ z , ˜ w , . . . , ˜ w n − ) ( σ ¯ Z , σ ¯ W , . . . , σ ¯ W n − , σ − ¯ z , σ − ¯ w , . . . , σ − ¯ w n − ) . (2.35)This transformation is symplectic and independent of the true anomaly θ . Thus the Hamiltonian function ˜ H in (2.34) becomes a new Hamiltonian function: H ( θ, ¯ Z , ¯ W , . . . , ¯ W n − , ¯ z , ¯ w , . . . , ¯ w n − ) ≡ ˜ H ( θ, σ ¯ Z , σ ¯ W , . . . , σ ¯ W n − , σ − ¯ z , σ − ¯ w , . . . , σ − ¯ w n − ) =
12 ( | ¯ Z | + n − X i = | ¯ W i | ) + (¯ z · J ¯ Z + n − X i = ¯ w i · J ¯ W i ) + p − r p ( | ¯ z | + n − X i = | ¯ w i | ) − r σ U (¯ z , ¯ w , . . . , ¯ w n − ) , (2.36)where one σ is factored out from U ( σ − ¯ z , σ − ¯ w , . . . , σ − ¯ w n − ).The proof is complete.Motivated by ideas in Sections 2 and 3 of [16], we now derive the linearized Hamiltonian system at suchan EEM solution of n -bodies, where σ = ( µ p ) − / is important. Theorem 2.3
Using notations in (2.9), the EEM solution ( P ( t ) , Q ( t )) T in time t of the system (1.2) withQ ( t ) = ( r ( t ) R ( θ ( t )) a , r ( t ) R ( θ ( t )) a , . . . , r ( t ) R ( θ ( t )) a n ) T , P ( t ) = M ˙ Q ( t ) , (2.37) where we denote by M = diag( m , m , . . . , m n , m n ) , is transformed to the new solution ( Y ( θ ) , X ( θ )) T in thetrue anomaly θ as the new variable with G = g = for the original Hamiltonian function H of (2.21), whichis given by Y ( θ ) = ¯ Z ( θ )¯ W ( θ ) . . . ¯ W n − ( θ ) = σ. . .. . . , X ( θ ) = ¯ z ( θ )¯ w ( θ ) . . . ¯ w n − ( θ ) = σ . . .. . . . (2.38) Moreover, the linearized Hamiltonian system at the EEM solution ξ ≡ ( Y ( θ ) , X ( θ )) T = (0 , σ, . . . , . . . , , | {z } n − , σ, , . . . , . . . , , | {z } n − ) T ∈ R n − epending on the true anomaly θ with respect to the Hamiltonian function H of (2.21) is given by ˙ ζ ( θ ) = JB ( θ ) ζ ( θ ) , (2.39) with B ( θ ) = H ′′ ( θ, ¯ Z , ¯ W , . . . , ¯ W n − , ¯ z , ¯ w , . . . , ¯ w n − ) | ¯ ξ = ξ = I O . . . O − J O . . .
OO I . . . O O − J . . . O . . . . . . . . . . . . . . . . . . . . . . . . O . . . O I O . . . O − JJ O . . .
O H ¯ z ¯ z ( θ, ξ ) O . . . OO J . . .
O O H ¯ w ¯ w ( θ, ξ ) . . . O . . . . . . . . . . . . . . . . . . . . . . . . O . . . O J O . . .
O H ¯ w n − ¯ w n − ( θ, ξ ) , (2.40) andH ¯ z ¯ z ( θ, ξ ) = (cid:18) − − e cos θ + e cos θ
00 1 (cid:19) , H ¯ w i ¯ w i ( θ, ξ ) = − β i + − e cos θ + e cos θ β i + + e cos θ + e cos θ ! , for ≤ i ≤ n − , (2.41) where each β i with ≤ i ≤ n − is given by (1.14), and H ′′ is the Hessian Matrix of H with respect to itsvariables ¯ Z, ¯ W , . . . , ¯ W n − , ¯ z, ¯ w , . . . , ¯ w n − . The corresponding quadratic Hamiltonian function is given byH ( θ, ¯ Z , ¯ W , . . . , ¯ W n − , ¯ z , ¯ w , . . . , ¯ w n − )12 | ¯ Z | + ¯ Z · J ¯ z + H ¯ z ¯ z ( θ, ξ ) | ¯ z | + n − X i = | ¯ W i | + ¯ W i · J ¯ w i + H ¯ w i ¯ w i ( θ, ξ ) | ¯ w i | ! . (2.42) Proof.
In this proof, we generalize the computations in [26] for the EEM of the 3-body case to the n -body case here. For reader’s conveniences, we given all details here. We only need to compute H ¯ z ¯ z ( θ, ξ ), H ¯ z ¯ w i ( θ, ξ ) and H ¯ w i ¯ w j ( θ, ξ ) for 1 ≤ i , j ≤ n − H in (2.21). By (2.21), we have H z = JZ + p − rp z − r σ U z ( z , w , . . . , w n − ) , H w i = JW i + p − rp w i − r σ U w i ( z , w , . . . , w n − ) , and H zz = p − rp I − r σ U zz ( z , w , . . . , w n − ) , H zw l = H w l z = − r σ U zw l ( z , w , . . . , w n − ) , for l = , . . . , n − , H w l w l = p − rp I − r σ U w l w l ( z , w , . . . , w n − ) , for i = l , . . . , n − , H w l w s = H w s w l = − r σ U w l w s ( z , w , . . . , w n − ) , for l , s = , . . . , n − , l , s , (2.43)where all the items above are 2 × H x and H xy the derivative of H with respectto x , and the second derivative of H with respect to x and then y respectively for x and y ∈ R .By (2.17) for U i j with 1 ≤ i < j ≤ n and 1 ≤ l ≤ n −
2, we obtain ∂ U i j ∂ z ( z , w , . . . , w n − ) = − m i m j ( a ix − a jx ) | ( a ix − a jx ) z + P nk = ( b ik − b jk ) w k − | ( a ix − a jx ) z + n X k = ( b ik − b jk ) w k − ,∂ U i j ∂ w l ( z , w , . . . , w n − ) = − m i m j ( b i , l + − b j , l + ) | ( a ix − a jx ) z + P nk = ( b ik − b jk ) w k − | ( a ix − a jx ) z + n X k = ( b ik − b jk ) w k − , ∂ U i j ∂ z ( z , w , . . . , w n − ) = − m i m j ( a ix − a jx ) | ( a ix − a jx ) z + P nk = ( b ik − b jk ) w k − | I + m i m j ( a ix − a jx ) | ( a ix − a jx ) z + P nk = ( b ik − b jk ) w k − | · ( a ix − a jx ) z + n X k = ( b ik − b jk ) w k − ( a ix − a jx ) z + n X k = ( b ik − b jk ) w k − T ,∂ U i j ∂ z ∂ w l ( z , w , . . . , w n − ) = − m i m j ( a ix − a jx )( b i , l + − b j , l + ) | ( a ix − a jx ) z + P nk = ( b ik − b jk ) w k − | I + m i m j ( a ix − a jx )( b i , l + − b j , l + ) | ( a ix − a jx ) z + P nk = ( b ik − b jk ) w k − | · ( a ix − a jx ) z + n X k = ( b ik − b jk ) w k − ( a ix − a jx ) z + n X k = ( b ik − b jk ) w k − T ,∂ U i j ∂ w l ( z , w , . . . , w n − ) = − m i m j ( b i , l + − b j , l + ) | ( a ix − a jx ) z + P nk = ( b ik − b jk ) w k − | I + m i m j ( b i , l + − b j , l + ) | ( a ix − a jx ) z + P nk = ( b ik − b jk ) w k − | · ( a ix − a jx ) z + n X k = ( b ik − b jk ) w k − ( a ix − a jx ) z + n X k = ( b ik − b jk ) w k − T . Set K = (cid:18) − (cid:19) , K = (cid:18) (cid:19) . Now evaluating the corresponding functions at the special solution (0 , σ, . . . , , | {z } n − , σ, , . . . , , | {z } n − ) T ∈ R n − of (2.38) with z = ( σ, T , w l = (0 , T for 1 ≤ l ≤ n −
2, and summing them up, we obtain ∂ U ∂ z (cid:12)(cid:12)(cid:12) ξ = X ≤ i < j ≤ n ∂ U i j ∂ z (cid:12)(cid:12)(cid:12) ξ = X ≤ i < j ≤ n − m i m j ( a ix − a jx ) | ( a ix − a jx ) σ | I + m i m j ( a ix − a jx ) | ( a ix − a jx ) σ | ( a ix − a jx ) σ K = σ X ≤ i < j ≤ m i m j | a ix − a jx | K = µσ K , (2.44) ∂ U ∂ w l (cid:12)(cid:12)(cid:12) ξ = X ≤ i < j ≤ n ∂ U i j ∂ w l (cid:12)(cid:12)(cid:12) ξ = X ≤ i < j ≤ n − m i m j ( b i , l + − b j , l + ) | ( a ix − a jx ) σ | I + m i m j ( b i , l + − b j , l + ) | ( a ix − a jx ) σ | ( a ix − a jx ) σ K σ X ≤ i < j ≤ n m i m j ( b i , l + − b j , l + ) | a ix − a jx | K = σ X ≤ i < j ≤ n m i m j ( b i , l + − b j , l + ) b i , l + | a ix − a jx | − X ≤ i < j ≤ n m i m j ( b i , l + − b j , l + ) b j , l + | a ix − a jx | K = σ X ≤ i < j ≤ n m i m j ( b i , l + − b j , l + ) b i , l + | a ix − a jx | + X ≤ j < i ≤ n m i m j ( b i , l + − b j , l + ) b i , l + | a ix − a jx | K = σ n X i = b i , l + n X j = , j , i m i m j ( b i , l + − b j , l + ) | a ix − a jx | K = σ n X i = b i , l + F i , l + K = µ (1 + β l ) σ K , (2.45)where the last equality of the first formula follows from (1.8), and the last equality of the second formulafollows from the definition (2.8). Similarly, we have ∂ U ∂ z ∂ w l (cid:12)(cid:12)(cid:12) ξ = X ≤ i < j ≤ n ∂ U i j ∂ z ∂ w l (cid:12)(cid:12)(cid:12) ξ = X ≤ i < j ≤ − m i m j ( a ix − a jx )( b i , l + − b j , l + ) | ( a ix − a jx ) σ | I + m i m j ( a ix − a jx )( b i , l + − b j , l + ) | ( a ix − a jx ) σ | ( a ix − a jx ) σ K ! = X ≤ i < j ≤ n m i m j ( b i , l + − b j , l + ) · sign ( a ix − a jx ) | ( a ix − a jx ) | K σ = X ≤ i < j ≤ n m i m j ( b i , l + − b j , l + ) · ( − | ( a ix − a jx ) | K σ = − X ≤ i < j ≤ n m i m j b i , l + | ( a ix − a jx ) | + X ≤ i < j ≤ n m i m j b j , l + | ( a ix − a jx ) | K σ = − n X i = b i , l + n X j = i + m i m j | ( a ix − a jx ) | + n X j = b j , l + j − X i = m i m j | ( a ix − a jx ) | K σ = − n X i = b i , l + n X j = i + m i m j | ( a ix − a jx ) | + n X i = b i , l + i − X j = m i m j | ( a ix − a jx ) | K σ = n X i = b i , l + n X j = i + m i m j ( a ix − a jx ) | ( a ix − a jx ) | + n X i = b i , l + i − X j = m i m j ( a ix − a jx ) | ( a ix − a jx ) | K σ = − n X i = b i , l + n X j = , j , i m i m j ( a ix − a jx ) | ( a ix − a jx ) | K σ − n X i = b i , l + µ m i a ix K σ = O , (2.46)where in the fourth and fourth last equality, we used the ascending order of a ix , ≤ i ≤ n , in the second lastequation, we used (2.1), and in the last equality, we used (2.3). Moreover, for l , s , we have ∂ U ∂ w l ∂ w s (cid:12)(cid:12)(cid:12) ξ = X ≤ i < j ≤ n ∂ U i j ∂ w l ∂ w s (cid:12)(cid:12)(cid:12) ξ = X ≤ i < j ≤ n − m i m j ( b i , l + − b j , l + )( b i , s + − b j , s + ) | ( a ix − a jx ) σ | I + m i m j ( b i , l + − b j , l + )( b i , s + − b j , s + ) | ( a ix − a jx ) σ | ( a ix − a jx ) σ K ! = σ X ≤ i < j ≤ n m i m j ( b i , l + − b j , l + )( b i , s + − b j , s + ) | a ix − a jx | K = σ X ≤ i < j ≤ n m i m j b i , l + ( b i , s + − b j , s + ) | a ix − a jx | − X ≤ i < j ≤ n m i m j b j , l + ( b i , s + − b j , s + ) | a ix − a jx | K = σ X ≤ i < j ≤ n m i m j b i , l + ( b i , s + − b j , s + ) | a ix − a jx | + X ≤ j < i ≤ n m j m i b i , l + ( b i , s + − b j , s + ) | a jx − a ix | K = σ n X i = b i , l + n X j = , j , i m i m j ( b i , s + − b j , s + ) | a ix − a jx | K = σ ( n X i = b i , l + F i , s + ) K = σ ( n X i = b i , l + ( µ − λ s + ) m i b i , s + ) K = O , (2.47)where in the third last equality, we used (2.6), and in the last equality of (2.47), we used (2.3) and (2.4).By (2.44), (2.45), (2.46) and (2.43), we have H zz | ξ = p − rp I − r µσ K = I − rp I − r µ p µ K = I − rp ( I + K ) = (cid:18) − − e cos θ + e cos θ
00 1 (cid:19) , H zw l | ξ = − r σ ∂ U ∂ z ∂ w l | ξ = O , for 1 ≤ l ≤ n − , H w l w s | ξ = − r σ ∂ U ∂ w l ∂ w s | ξ = O , for 1 ≤ l , s ≤ n − , l , s , H w l w l | ξ = p − rp I − r (1 + β l ) µσ K = I − rp I − r (1 + β l ) µ p µ K = I − rp ( I + (1 + β l ) K ) = − β l + − e cos θ + e cos θ β l + + e cos θ + e cos θ ! , for 1 ≤ l ≤ n − . (2.48)Thus the proof is complete. 14 emark 2.4 (i) When we set n = β is precisely the mass parameter β defined by(1.4), and the corresponding linearized Hamiltonian system at the EEM q m , e ( t ) is given by z ′ = J − − − β − + e cos( t )1 + e cos( t )
01 0 0 β + + e cos( t )1 + e cos( t ) z . (2.49)Note that this system was derived in [13] and [26] too.(ii) The Hamiltonian equation of the i -th part of the other ( n −
2) parts with 1 ≤ i ≤ n − z ′ = J − − − β i − + e cos( t )1 + e cos( t )
01 0 0 β i + + e cos( t )1 + e cos( t ) z . (2.50)Also, β coincides with β c in Table 2 of [13] when α = Proof of Theorem 1.2.
Note that by Theorem 2.3, specially (2.39)-(2.41), we obtain that the matrix H ¯ z ¯ z ( θ, ξ ) together with the first identity matrix I in the diagonal of the matrix B ( θ ) in (2.40) yield a 4-dimensional Hamiltonian system corresponding to the Kepler 2-body problem, and each matrix H ¯ w i ¯ w i ( θ, ξ )together with the ( i + I in the diagonal of the matrix B ( θ ) in (2.40) yield a 4-dimensionalHamiltonian system (2.50) with β i given by (1.14), which corresponds to the linear system (2.49) of the Euler3-body problem with β replaced by β i for 1 ≤ i ≤ n −
2. Therefore Theorem 1.2 holds. -body problem with two small masses in the middle We now consider the linear stability of special collinear central configurations in the four body problem withtwo small masses in the middle. A typical example is the EEM orbit of the 4-bodies, the Earth, the Moonand two space stations in the middle as mentioned at the beginning of this paper with n =
4. We try to givean analytical way following which one can numerically find out the best elliptic-hyperbolic positions for thetwo space stations using results in [13] and [14]. Specially, for the four masses we fix m = m ∈ (0 , m = ǫ , m = τǫ , m = − m − ( τ + ǫ with τ > < ǫ < − m τ + . They satisfy m + m + m + m = . (3.1)Suppose q , q , q and q are four points on the x -axis in R , and form a central configuration. Usingnotations similar to those in [26], we set q = (0 , T , q = ( x α, T , q = ( y α, T , q = ( α, T , (3.2)where α = α ǫ,τ = | q − q | , x = x ǫ,τ , y = y ǫ,τ satisfy 0 < x < y <
1. Then the center of mass of the fourparticles is q c = m q + m q + m q + m q = ([ m x + m y + m ] α, T = ([(1 − m ) − (1 + τ − x − τ y ) ǫ ] α, T , (3.3)where (3.1) is used to get the last equality. 15or i =
1, 2, 3 and 4, let a i = q i − q c , and denote by a ix and a iy the x and y -coordinates of a i respectively.Then we have a x = [ m − + (1 + τ − x − τ y ) ǫ ] α, a x = [ m + x − + (1 + τ − x − τ y ) ǫ ] α, (3.4) a x = [ m + y − + (1 + τ − x − τ y ) ǫ ] α, a x = [ m + (1 + τ − x − τ y ) ǫ ] α, (3.5)and a iy = , for i = , , , . (3.6)Next we study properties of this central configuration. Step 1.
Computations on α and x , y. Scaling α by setting P i = m i | a i | =
1, we obtain1 α = P i = m i | a i | α = m [ m − + (1 + τ − x − τ y ) ǫ ] + ǫ [ m + x − + (1 + τ − x − τ y ) ǫ ] + τǫ [ m + y − + (1 + τ − x − τ y ) ǫ ] + (1 − m − ( τ + ǫ )[ m + (1 + τ − x − τ y ) ǫ ] = m (1 − m ) + m (1 − m )( x + τ y − − τ ) ǫ + m ( x + τ y − − τ ) ǫ + ǫ [( x − + ( m + (1 + τ − x − τ y ) ǫ ) + x − m + (1 + τ − x − τ y ) ǫ )] + τǫ [( y − + ( m + (1 + τ − x − τ y ) ǫ ) + y − m + (1 + τ − x − τ y ) ǫ )] + (1 − m )[ m + m (1 + τ − x − τ y ) + (1 + τ − x − τ y ) ǫ ] − ( τ + ǫ ( m + (1 + τ − x − τ y ) ǫ ) = m (1 − m ) + (1 + τ − x − τ y ) ǫ + ǫ [( x − + τ ( y − ] + ǫ m (1 + τ )( m + (1 + τ − x − τ y ) ǫ ) − ǫ (1 + τ − x − τ y )( m + (1 + τ − x − τ y ) ǫ ) = m (1 − m ) + [(1 − x ) + τ (1 − y ) + m (1 + τ ) − m (1 + τ − x − τ y )] ǫ + [2 m (1 + τ )(1 + τ − x − τ y ) − (1 + τ − x − τ y ) ] ǫ + m (1 + τ )(1 + τ − x − τ y ) ǫ . (3.7)Moreover, let α = lim ǫ → α = [ m (1 − m )] − , (3.8)and q c , = lim ǫ → q c = (1 − m ) α , (3.9)and hence a x , = lim ǫ → a x = − (1 − m ) α , (3.10) a x , = lim ǫ → a x = m α . (3.11)The potential µ is given by µ = µ ǫ,τ = X ≤ i < j ≤ m i m j | a i − a j | , (3.12)and by Lemma 3 of [8], we have µ = lim ǫ → µ = m (1 − m ) α = α − . (3.13)In the following, we will use the subscript 0 to denote the limit value of the parameters when ǫ → emma 3.1 When ǫ → , a and a must converge to the same point a ∗ . Moreover, a , , a ∗ , a , is thecentral configuration of the restricted 3-body problem with given masses ˜ m = m , ˜ m = , ˜ m = − m whichthe small mass lies in the segment between the other two masses. Proof. If a and a do not converge to the same point when ǫ →
0, there is a sequence { ǫ n } ∞ n = convergentto 0 such that a ∗ x ≡ lim n →∞ ( a x − a x ) , . (3.14)Up to a subsequence of { ǫ n } ∞ n = , and we denote it still by { ǫ n } ∞ n = , we havelim n →∞ a x = a ∗ x . (3.15)Then lim n →∞ a x = lim n →∞ a x + lim n →∞ ( a x − a x ) = a ∗ x + a ∗ x . (3.16)Because a , a , a and a form a central configuration, for the two middle points we have m ( a − a ) | a − a | + m ( a − a ) | a − a | + m ( a − a ) | a − a | = µ a , (3.17) m ( a − a ) | a − a | + m ( a − a ) | a − a | + m ( a − a ) | a − a | = µ a . (3.18)Let ǫ = ǫ n , n ∈ N , and n → ∞ , together with (3.5), (3.9), (3.10), (3.13) and (3.14), (3.17) and (3.18) become m ( a ∗ x − a x , ) − − m ( a ∗ x − a x , ) = µ a ∗ x , (3.19) m ( a ∗ x + a ∗ x − a x , ) − − m ( a ∗ x + a ∗ x − a x , ) = µ ( a ∗ x + a ∗ x ) . (3.20)We define f ( t ) = m ( t − a x , ) − − m ( t − a x , ) − µ t , for t ∈ ( a x , , a x , ) . (3.21)Then f is a strictly decreasing function satisfyinglim t → a x , f ( t ) = + ∞ , and lim t → a x , f ( t ) = −∞ . (3.22)Thus there is a unique zero point of f in [ a x , , a x , ], which we denote by a ∗ x . Here (3.22) yields a x , < a ∗ x < a x , .Now (3.19) and (3.20) yield two zero points a ∗ x and a ∗ x + a ∗ x of f in [ a x , , a x , ] respectively, whichthen yields a contradiction. Therefore, we have proved lim ǫ → ( a x − a x ) = ǫ → a x = a ∗ x . If not, there is a sequence { ˜ ǫ n } ∞ n = converges to 0, such thatlim n →∞ a x = ˜ a ∗ , a ∗ . Then lim n →∞ a x = ˜ a ∗ .Now adding m m + m times (3.17) to m m + m times (3.18) yields m τ + a − a | a − a | + ττ + a − a | a − a | ! + m τ + a − a | a − a | + ττ + a − a | a − a | ! = µ a + τ a + τ , (3.23)Let ǫ = ˜ ǫ n , n ∈ N , and n → ∞ , together with (3.6), (3.10), (3.11) and (3.13), (3.23) becomes m (˜ a ∗ − a x , ) − − m (˜ a ∗ − a x , ) = µ ˜ a ∗ , (3.24)17hen using also the property of unique zero point of f ( x ), we obtain a contradiction. Thus we must havelim ǫ → a x = lim ǫ → a x = a ∗ x .By direct computations, we can check that a , = ( a x , , T , a ∗ = ( a ∗ x , T and a , = ( a x , , T forma collinear central configuration with given masses ˜ m = m , ˜ m = m = − m . The uniqueness isobtained by these three given ordered masses as in [20].By Lemma 3.1, we can suppose lim ǫ → x = lim ǫ → y = x , (3.25)and hence a x , = lim ǫ → a x = ( m + x − α , (3.26) a x , = lim ǫ → a x = ( m + x − α . (3.27)Note that a , , a , and a , form a central configuration with given masses ˜ m = m , ˜ m = m = − m .Then ˜ x = x − x is the unique positive root of Euler’s quintic polynomial equation (cf. p. 276 of [25] and p.29of [11]): (1 − m ) ˜ x + (3 − m ) ˜ x + (3 − m ) ˜ x − m ˜ x − m ˜ x − m = . (3.28)Thus x satisfies: x − (3 − m ) x + (3 − m ) x − mx + mx − m = . (3.29)Next we derive the equations satisfied by x = x ( ǫ ) and y = y ( ǫ ). Because a , a , a and a form a centralconfiguration, we have ǫ x α + τǫ y α + − m − (1 + τ ) ǫα = µ [1 − m − (1 + τ − x − τ y ) ǫ ] α, (3.30) − mx α + τǫ ( y − x ) α + − m − (1 + τ ) ǫ (1 − x ) α = µ [1 − m − x − (1 + τ − x − τ y ) ǫ ] α, (3.31) − my α − ǫ ( y − x ) α + − m − (1 + τ ) ǫ (1 − y ) α = µ [1 − m − y − (1 + τ − x − τ y ) ǫ ] α, (3.32) − m α − ǫ (1 − x ) α − τǫ (1 − y ) α = − µ [ m + (1 + τ − x − τ y ) ǫ ] α. (3.33)From (3.30) and (3.31), we have0 = " ǫ x + τǫ y + − m − (1 + τ ) ǫ [1 − m − x − (1 + τ − x − τ y ) ǫ ] − " − mx + τǫ ( y − x ) + − m − (1 + τ ) ǫ (1 − x ) [1 − m − (1 + τ − x − τ y ) ǫ ] = (1 − m )(1 − m − x ) − " − mx + − m (1 − x ) (1 − m ) + ǫ " − (1 − m )(1 + τ − x − τ y ) + (1 − m − x )( 1 x + τ y − − τ ) − (1 − m )( τ ( y − x ) − + τ (1 − x ) ) + ( − mx + − m (1 − x ) )(1 + τ − x − τ y ) + ǫ " ( 1 x + τ y − − τ )(1 + τ − x − τ y ) + ( τ ( y − x ) − + τ (1 − x ) )(1 + τ − x − τ y ) = − (1 − m ) x − (3 − m ) x + (3 − m ) x − mx + mx − mx (1 − x ) ǫ " ( m − − mx + − m (1 − x ) )(1 + τ − x − τ y ) + (1 − m − x )( 1 x + τ y − − τ ) − (1 − m )( τ ( y − x ) − + τ (1 − x ) ) + ǫ " x + τ y + τ ( y − x ) − + τ (1 − x ) − − τ (1 + τ − x − τ y ) . (3.34)We denote the right hand side of (3.34) by g ǫ ( x , y ), then x (1 − x ) y ( y − x ) g ǫ ( x , y ) is a binary polynomialin x , y . Similarly, from (3.30) and (3.32), we have h ǫ ( x , y ) = , (3.35)where h ǫ ( x , y ) = − (1 − m ) y − (3 − m ) y + (3 − m ) y − my + my − my (1 − y ) + ǫ " ( m − − my + − m (1 − y ) )(1 + τ − x − τ y ) + (1 − m − y )( 1 x + τ y − − τ ) + (1 − m )( 1( y − x ) + + τ (1 − y ) ) + ǫ " x + τ y − y − x ) − + τ (1 − y ) − − τ (1 + τ − x − τ y ) . (3.36)Therefore, x and y can be solved out from g ǫ ( x , y ) = h ǫ ( x , y ) = ǫ → g ǫ ( x , y ) = h ǫ ( x , y ) = ǫ → x (1 − x ) y ( y − x ) g ǫ ( x , y ) = − (1 − m ) y ( y − x ) [ x − (3 − m ) x + (3 − m ) x − mx + mx − m ] , (3.37)lim ǫ → y (1 − y ) x ( x − y ) h ǫ ( x , y ) = − (1 − m ) x ( x − y ) [ y − (3 − m ) y + (3 − m ) y − my + my − m ] . (3.38)Here in (3.37) and (3.38) we have the same polynomial again as that in the left hand side of (3.29). Step 2.
Computations on β i s. Now in our case, D is given by D = µ − α [ ǫ x + τǫ y + − m − (1 + τ ) ǫ ] , ǫ x α , τǫ y α , − m − (1 + τ ) ǫα mx α , µ − α [ mx + τǫ ( y − x ) + − m − (1 + τ ) ǫ (1 − x ) ] , τǫ ( y − x ) α , − m − (1 + τ ) ǫ (1 − x ) α my α , ǫ ( y − x ) α , µ − α [ my + ǫ ( y − x ) + − m − (1 + τ ) ǫ (1 − y ) ] , − m − (1 + τ ) ǫ (1 − y ) α m α , ǫ (1 − x ) α , τǫ (1 − y ) α , µ − α [ m + ǫ (1 − x ) + τǫ (1 − y ) ] . (3.39)Recall that the other two eigenvalues of D are λ and λ , then we havedet( D − λ I ) = − λ ( µ − λ )( λ − λ )( λ − λ ) = λ − ( µ + λ + λ ) λ + ( λ λ + µ ( λ + λ )) λ − λ λ µλ (3.40)On the other hand det( D − λ I ) = λ − ( trD ) λ + ( X i , j = , i < j det E i j ) λ + . . . , (3.41)19here E i j is the principal minor when deleting all the rows and columns except for i and j . Then we have µ + λ + λ = trD = X i = D ii = X i = µ − X j = , j , i D i j = µ − X i , j = , i , j D i j = µ − α " m + ǫ x + m + τǫ y + − (1 + τ ) ǫ + (1 + τ ) ǫ ( y − x ) + − m − τǫ (1 − x ) + − m − ǫ (1 − y ) , (3.42) λ λ + µ ( λ + λ ) = X i , j = , i < j det E i j = X i , j = , i < j (cid:16) D ii D j j − D i j D ji (cid:17) = X i , j = , i < j D ii D j j − X i , j = , i < j D i j D ji = (cid:16)P i = D ii (cid:17) − P i = D ii − X i , j = , i < j D i j D ji =
12 ( trD ) − X i = µ − X s = , s , i D is − X i , j = , i < j D i j D ji =
12 ( trD ) − µ + µ X i , j = , i , j D i j − X i = X s = , s , i D is − X i , j = , i < j D i j D ji (3.43)Let δ = µ X i , j = , i , j D i j = µα " m + ǫ x + m + τǫ y + − (1 + τ ) ǫ + (1 + τ ) ǫ ( y − x ) + − m − τǫ (1 − x ) + − m − ǫ (1 − y ) , (3.44)then we have λ + λ = trD − µ = µ − δµ − µ = − (2 δ − µ, (3.45) λ λ = − µ ( λ + λ ) +
12 ( trD ) − µ + µ X i , j = , i , j D i j − X i = X s = , s , i D is − X i , j = , i < j D i j D ji = (2 δ − µ +
12 (4 − δ ) µ − µ + δµ − α ǫ x + τǫ y + − m − (1 + τ ) ǫ ! mx + τǫ ( y − x ) + − m − (1 + τ ) ǫ (1 − x ) ! + my + ǫ ( y − x ) + − m − (1 + τ ) ǫ (1 − y ) ! + m + ǫ (1 − x ) + τǫ (1 − y ) ! + m ǫ x + m τǫ y + m (1 − m − (1 + τ ) ǫ ) + τǫ ( y − x ) + ǫ (1 − m − (1 + τ ) ǫ )(1 − x ) + τǫ (1 − m − (1 + τ ) ǫ )(1 − y ) ! = µ " (2 δ − δ + − µ α ǫ x + τǫ y + − m − (1 + τ ) ǫ ! + mx + τǫ ( y − x ) + − m − (1 + τ ) ǫ (1 − x ) ! + my + ǫ ( y − x ) + − m − (1 + τ ) ǫ (1 − y ) ! + m + ǫ (1 − x ) + τǫ (1 − y ) ! + m ǫ x + m τǫ y + m (1 − m − (1 + τ ) ǫ ) + τǫ ( y − x ) + ǫ (1 − m − (1 + τ ) ǫ )(1 − x ) + τǫ (1 − m − (1 + τ ) ǫ )(1 − y ) ! . (3.46)Moreover, we have ∆ = ( λ + λ ) − λ λ = µ − δ + δ − + µ α ǫ x + τǫ y + − m − (1 + τ ) ǫ ! + mx + τǫ ( y − x ) + − m − (1 + τ ) ǫ (1 − x ) ! + my + ǫ ( y − x ) + − m − (1 + τ ) ǫ (1 − y ) ! + m + ǫ (1 − x ) + τǫ (1 − y ) ! + m ǫ x + m τǫ y + m (1 − m − (1 + τ ) ǫ ) + τǫ ( y − x ) + ǫ (1 − m − (1 + τ ) ǫ )(1 − x ) + τǫ (1 − m − (1 + τ ) ǫ )(1 − y ) ! (3.47)Letting ˜ ∆ = ∆ µ , and note that λ ≥ λ are real numbers, we have λ = − δ + + p ˜ ∆ ! µ, (3.48) λ = − δ + − p ˜ ∆ ! µ. (3.49)Therefore, we obtain β = − λ µ = δ − − p ˜ ∆ , β = − λ µ = δ − + p ˜ ∆ . (3.50)Then using the numerical results by R. Mart´ınez, A. Sam`a and C. Sim´o in [13] and [14], we can obtain thestability pattern of our four body problem. Step 3.
Computations on the limit case.
We need to compute the mass parameter of the restricted three-body problem of given masses ˜ m = m ,˜ m =
0, and ˜ m = − m . By (A.3) of [14], β (they use β c there) is given by β c = − + α a α + [1 + ( ρ + ] " ( ρ +
2) ( ρ + m + m ρ α + + ( ρ + m ρ + m ( ρ + + m − m ρ ( ρ + α + , ρ and a is given by (A.2) of [14].Note that, when in our case α =
1, (A.2) of [14] is just the Euler’s quintic equation, then together with ρ = x − x of (3.28), we have β = − + ( ρ + + ( ρ + " ( ρ +
2) ( ρ + m ρ + (1 − m )( ρ + + (1 − m ) − m ρ ( ρ + = − + ( x − x + + ( x − x + ( x − x +
2) ( x − x + m ( x − x ) + (1 − m )( x − x + + (1 − m ) − m x − x ( x − x + = − + − x )[(1 − x ) + m (2 − x )(1 − x ) x + (1 − m ) 1(1 − x ) + (1 − x ) (1 − x − m ) = − + − x )[(1 − x ) + m (2 − x )(1 − x ) x − m (1 − x ) x + − x )[(1 − x ) + " (1 − m ) 1(1 − x ) + (1 − m ) + − x )[(1 − x ) + m (1 − x ) x − (1 − m ) + (1 − x ) (1 − x − m ) = − + mx + − m (1 − x ) + − x )[(1 − x ) + − x + (3 − m ) x − (3 − m ) x + mx − mx + mx . = − + mx + − m (1 − x ) , (3.51)where in the last equality, we used (3.29).Following pp.171 in [24], for q = ( q x , q y ) T ∈ R , we define V ( q ) = m | a , − q | + − m | a , − q | + α − | q | (3.52)where α − is an extra parameter because Z. Xia fixed λ = λ = α − . Thenwe have ∂ V ∂ q x = − m | a , − q | − − m | a , − q | + α + " m ( − (1 − m ) α − q x ) | a , − q | + (1 − m )( m α − q x ) | a , − q | , (3.53) ∂ V ∂ q x ∂ q y = − " m ( − (1 − m ) α − q x ) q y | a , − q | + (1 − m )( m α − q x ) q y | a , − q | (3.54) ∂ V ∂ q x = − m | a , − q | − − m | a , − q | + α + mq y | a , − q | + (1 − m ) q y | a , − q | . (3.55)Therefore ∂ V ∂ q x (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) q = ( x α , T = − mx α − − m (1 − x ) α + α + mx α x α + (1 − m )(1 − x ) α (1 − x ) α = α mx α + − m (1 − x ) α + α = (2 β + α − , (3.56)22 V ∂ q x ∂ q y (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) q = ( x α , T = , (3.57) ∂ V ∂ q x (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) q = ( x α , T = − mx α − − m (1 − x ) α + α = − βα − , (3.58)and hence D V ( q ) | q = ( x α , T = (cid:18) (2 β + α − − βα − (cid:19) . (3.59)By the Case (ii) in p.173 of [24], we havelim ǫ → a − a ( m + m ) = lim ǫ → r ′ = ± [(2 β + α − ] − , (3.60)and hence lim ǫ → m | a − a | = + τ lim ǫ → m + m | a − a | = (2 β + α − + τ = (2 β + µ + τ , (3.61)lim ǫ → m | a − a | = τ + τ lim ǫ → m + m | a − a | = τ (2 β + α − + τ = τ (2 β + µ + τ . (3.62)Note that m = ǫ, m = τǫ and lim ǫ → | a i − a j | , i < j , ( i , j ) , (2 , D = lim ǫ → D = m µ − m ) µ mx µ [ − β − τ (2 β + + τ ] µ τ (2 β + + τ µ − mx µ mx µ β + + τ µ [ − β − β + + τ ] µ − mx µ m µ − m ) µ , (3.63)where we have used (3.13), (3.51), (3.61) and (3.62). Then the characteristic polynomial of D is given bydet( D − λ I ) = − λ ( µ − λ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) [ − β − τ (2 β + + τ ] µ − λ τ (2 β + + τ µ β + + τ µ [ − β − β + + τ ] µ − λ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = λ ( λ − µ )( λ + βµ )( λ + β + µ ) . (3.64)Then all eigenvalues of D are given by λ , = µ , λ , = , λ , = − βµ , λ , = − β + µ , (3.65)and hence by (2.4), we have β , = − λ , µ = β, (3.66) β , = − λ , µ = β + . (3.67)From (3.65), the four eigenvalues of D are di ff erent, then for ǫ > ǫ → λ i = λ i , , ≤ i ≤ . (3.68)Thus, we also have lim ǫ → β i = β i , , ≤ i ≤ . (3.69)23herefore, the linear stability problem of the limiting case of our four-body problem when letting ǫ → β , and the other has mass parameter 3( β + Example 3.2
Computations on the actual case of the Earth-Moon-two space stations system.
We denote by ESSM system the short hand notation for the Earth-two space stations-Moon system.From https: // en.wikipedia.org / wiki / Earth and https: // en.wikipedia.org / wiki / Moon, one can find that the massof Earth is E = . × kg, the mass of the Moon is M = . × kg, the distance between theEarth and the Moon is d = e ≈ . m = EE + M ≈ . . (3.70)For two space stations in the line segment between the Earth and the Moon, as their masses tends to 0 theirlimit position x given by (3.25) is determined by (3.29) and m . When m is given by (3.70), by a numericalcomputation, we have x ≈ . d S M = d × (1 − x ) ≈ β for the EEM of the 3-body problem is given by β = − + mx + − m (1 − x ) ≈ . . (3.72)Thus the linear stability property of the ESSM system is determined by the eccentricity e ≈ . β = β ≈ . , β = β + ≈ . . (3.73)On the other hand, by (1.5)-(1.8) of [26], we haveˆ β ≈ . , ˆ β ≈ . , ˆ β ≈ . , ˆ β ≈ . , (3.74)where ˆ β n and ˆ β n + , n ∈ N are the parameter values when the resonances of the linearized system appear.Indeed, ˆ β n is the n -th value such that γ β, (2 π ) has eigenvalue 1, and ˆ β n + is the n -th value such that γ β, (2 π )has eigenvalue −
1. Here γ β, (2 π ) is the end matrix at time t = π of the fundamental solution of the linearizedHamiltonian system (2.49) at the Euler solution EEM q m , e with e = β < β < ˆ β , ˆ β < β < ˆ β . (3.75)Since the eccentricity e ≈ . e =
0. Then by Theorem 1.5 of [26], the linear stability pattern of the ESSMsystem is R ( θ ) ⋄ D (2) ⋄ R ( θ ) ⋄ D (2) (3.76)for some θ and θ ∈ (0 , π ). Here for θ ∈ R and λ ∈ R \ { , ± } we denote the elliptic and hyperbolic matricesby R ( θ ) = (cid:18) cos θ − sin θ sin θ cos θ (cid:19) , D ( λ ) = (cid:18) λ λ − (cid:19) , respectively. 24 Appendix: A sketch of the proof of Lemma 1.1.
For reader’s conveniences, following [17] of R. Moeckel (cf. also [21], [18]), next we sketch the ideas ofthe proof of Lemma 1.1 due to C. Conley.
A sketch of the proof of Lemma 1.1.
Note first that both the matrices D in (1.12) and ˜ D in (2.2) possessthe same eigenvalues by the definition of ˜ D . Because ˜ D is symmetric, all its eigenvalues are real, and thenso does D , although it may not be symmetric in general.Note that 2 B ( a ) = U ′′ ( a ) is the Hessian of U ( q ) at the collinear central configuration q = a , and U ′′ ( a ) + U ( a ) ˜ M is the Hessian of U | S with S being the hypersurface determined by (1.7). By the homogeneityof U , we obtain that D has the first eigenvalue λ = µ = U ( a ) with the eigenvector v = (1 , , . . . , T , i.e.,( Dv ) i = µ holds for 1 ≤ i ≤ n .From the definition (2.1) of a as a central configuration, we obtain that D has the second eigenvalue λ = v = ( a x , a x , . . . , a nx ) T . More precisely for 1 ≤ i ≤ n by (2.1) we have( Dv ) i = ( µ − n X j = , j , i m j | a i − a j | ) a ix + n X j = , j , i m j a jx | a i − a j | = µ a ix + n X j = , j , i m j ( a jx − a ix ) | a jx − a ix | = . Note that by (1.6)-(1.7), the vectors v and v form an ˜ M -orthonormal sub-basis, i.e., they satisfy v T ˜ Mv = v T ˜ Mv =
0, and v T ˜ Mv =
1. Denote all the other eigenvalues of D by λ , . . . , λ n . Nextgoal is to show that the other ( n −
2) eigenvalues of D are non-positive.Following [17], this is equivalent to showing that all the eigenvalues of D are non-positive when werestricted to the subspace spanned by vectors orthogonal to ˜ Mv , and observing that this is equivalent toshowing that in the flow on the space of lines through the origin determined by the following linear systemon u , ˙ u = M − B ( a ) u , (4.1)the line determined by v is an attractor.Let K = u = ( u , u , . . . , u n ) T (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n X i = m i u i = , u ≤ u ≤ . . . ≤ u n . Then for any u ∈ K , we have u ⊥ ˜ Mv . Moreover, we have rv ∈ K for any r ∈ R . We will show that,around the line in K which is carried strictly inside itself by the flow defined by (4.1) except for the origin.Note that the boundary ∂ K of K consists of points where one or more equalities hold. However, exceptfor the origin, at least one strict inequality must hold, otherwise u = k (1 , , . . . , T ∈ K and hence k = u i = u i + = · · · = u j < u j + , ≤ i < j < n , or u i − = u i = · · · = u j < u j + , < i < j ≤ n . The di ff erential equation (4.1) becomes˙ u i = X k , i m k r ik ( u k − u i ) , ˙ u j = X k , j m k r jk ( u k − u j ) , r mk = | a mx − a kx | for m = i , j . Since u i = u j we get˙ u j − ˙ u i = X k , i , j m k ( u k − u j ) r jk − r ik . Every term in this sum is non-negative, since(i) if k < i , ( u k − u i ) ≤ r jk − r ik < i ≤ k ≤ j , u k − u i = k > j , ( u k − u i ) ≥ r jk − r ik > u i with 1 ≤ i ≤ n are equal. Thus ˙ u j − ˙ u i > K as required.Now we consider the central configurations in R . Let S = q = ( q , q , . . . , q n ) T , q i ∈ R (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n X i = m i q i = , n X i = m i q i = , q i , q j if i , j , C = ( q = ( q , q , . . . , q n ) T ∈ S (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) q i ∈ R × { } × { } , ∀ ≤ i ≤ n ) , E = ( q = ( q , q , . . . , q n ) T ∈ S (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) q i ∈ { } × R , ∀ ≤ i ≤ n ) , ˜ C = n q = ( q , q , . . . , q n ) T ∈ S | q is collinear along some line o . Then C ⊂ ˜ C holds and ˜ C is the orbit of C under SO(3).Now on S , the central configuration equation is F ( q ) = ˜ M − U ′ ( q ) + U ( q ) q = , where U ′ ( q ) denotes the gradient of U with respect to q = ( q , . . . , q n ). Then when we consider the gradientflow of the system ˙ q = F ( q ) , (4.2)a central configuration is a fixed point of this flow. Note that C , ˜ C and E are invariant sub-manifolds underthe gradient flow of (4.2). For the central configuration q = ( q , , q , , . . . , q n , ) with q i , = ( a ix , , T , wehave F ′ ( q ) | C = − M − B + µ I n , (4.3) F ′ ( q ) | E = diag( ˜ M − B , ˜ M − B ) + µ I n . (4.4)Note that in the first Corollary on p.507 of [17], R. Moeckel proved that any orbits near ˜ C are attracted to˜ C by the gradient flow of (4.2). Therefore it yields that F ′ ( q ) | E in (4.4) is non-negative definite as required.In fact, using notations in [17], an explicit neighborhood U = { q ∈ S | Θ ( q ) ≤ π } of ˜ C in S can be definedsuch that the orbits of the gradient flow of (4.2) in U get more and more collinear.Here following [17] the function Θ ( q , L ) measures the approximate collinearity of a configuration q ∈ S and a line L in R is defined by Θ ( q , L ) = max i , j ∠ ( L , q i − q j ) , where ∠ ( L , q i − q j ) denotes the acute angle between L and q i − q j . Θ ( q , L ) vanishes if and only if q iscollinear along a line parallel to L . Then let Θ ( q ) = min L Θ ( q , L ) , q is collinear.Note that in U , Θ ( q ) is strictly decreasing along orbits q = q ( t ) of the gradient flow of (4.2), and itsu ffi ces to prove Θ ( q ( t )) < Θ ( q (0)) , ∀ t > . (4.5)Now we refer readers to pp.504-505 of [17] on the details of the proof of (4.5). Acknowledgements.
The authors would like to thank sincerely the anonymous editor for informing us andhelps on finding the paper of J. Liouville, and valuable comments. They thank sincerely also the anonymousreferees on their careful reading and helpful comments on the manuscript of this paper.
References [1] M. Andoyer, Sur les solutiones periodiques voisines des position d’equilibre relatif dans ie problemedes n corps. Bull. Astron. 23, (1906) 129-146.[2] J. Danby, The stability of the triangular Lagrangian point in the general problem of three bodies.
Astron.J.
69. (1964) 294-296.[3] L. Euler, De motu restilineo trium corporum se mutus attrahentium.
Novi Comm. Acad. Sci. Imp. Petrop.
11. (1767) 144-151.[4] M. Gascheau, Examen d’une classe d’´equations di ff ´erentielles et application `a un cas particulier duprobl`eme des trois corps. Comptes Rend. Acad. Sciences.
16. (1843) 393-394.[5] X. Hu, Y. Long, S. Sun, Linear stability of elliptic Euler solutions of the classical planar three-bodyproblem via index theory.
Arch. Ration. Mech. Anal. n -body problem. http: // arxiv.org / pdf / (2015). Comm. Math. Phys. to appear.[7] X. Hu, S. Sun, Morse index and stability of elliptic Lagrangian solutions in the planar three-body prob-lem.
Advances in Math. http: // arxiv.org / abs / (2015).[9] J. Lagrange, Essai sur le probl`eme des trois corps. Chapitre II. Œuvres Tome 6, Gauthier-Villars, Paris.(1772) 272-292.[10] J. Liouville, Sur un cas particulier du probl`eme des trois corps. J. Math. Pures Appl.
7. (1842) 110-113.[11] Y. Long, Lectures on Celestial Mechanics and Variational Methods.
Preprint. ff erential equations. Hasselt, 2003, eds,Dumortier, Broer, Mawhin, Vanderbauwhede and Lunel, World Scientific, (2004) 1005-1010.[13] R. Mart´ınez, A. Sam`a, C. Sim´o, Stability diagram for 4D linear periodic systems with applications tohomographic solutions. J. Di ff . Equa. R . Applications. J. Di ff . Equa. J. Di ff . Equa. Math. Z.
205 (1990) 499-517.[18] R. Moekel, Celestial Mechanics (especially central configurations). http: // / ∼ rmoeckel / notes / CMNotes.pdf . 1994.[19] R. Moekel, Linear stability analysis of some symmetrical classes of relative equilibria. H. S. Dumas etal. (eds.), Hamiltonian Dynamical Systems, Springer, New York. (1995) 291-317.[20] F. Moulton, The straight line solutions of the n -body problem. Ann. of Math . II Ser. 12 (1910) 1-17.[21] F. Pacella, Central configurations and the equivariant Morse theory.
Arch. Ration. Mech. Anal.
J.Di ff . Equa. Proc. LondonMath. Soc.
6. (1875) 86-97.[24] Z. Xia, Central Configurations with Many Small Masses.
J. Di ff . Equa.