Linear Stability of Elliptic Rhombus Solutions of the Planar Four-body Problem
aa r X i v : . [ m a t h . D S ] J un Linear Stability of Elliptic Rhombus Solutions of the PlanarFour-body Problem
Bowen Liu ∗ Chern Institute of MathematicsNankai University, Tianjin 300071, ChinaJune 7, 2018
Abstract
In this paper, we study the linear stability of the elliptic rhombus homographic solutions inthe classical planar four-body problem which depends on the shape parameter u ∈ (1 / √ , √ and eccentricity e ∈ [0 , . By an analytical result obtained in the study of the linear stabilityof elliptic Lagrangian solutions, we prove that the linearized Poincare map of elliptic rhombussolution possesses at least two pairs of hyperbolic eigenvalues, when ( u, e ) ∈ ( u , /u ) × [0 , or ( u, e ) ∈ (cid:0) [1 / √ , u ) ∪ (1 /u , √ (cid:1) × [0 , ˆ f ( ) − / ) where u ≈ . and ˆ f ( ) − / ≈ . .By a numerical result obtained in the study of the elliptic Lagrangian solutions, we analyticallyprove that the elliptic rhombus solution is hyperbolic, i.e., it possesses four pairs of hyperboliceigenvalues, when ( u, e ) ∈ [1 / √ , √ × [0 , . Key words:
Linear stability, Morse Index, Maslov-type ω -index, hyperbolic region, ellipticrhombus solution, planar four-body problem. Running title:
Linear Stability of Elliptic Rhombus Solution.
We consider the classical planar four-body problem in celestial mechanics. Denote by q , q , q , q ∈ R the position vectors of four particles with masses m , m , m , m > m i ¨ q i = ∂U∂q i , i = 1 , , , , (1.1) ∗ Partially supported by NSFC (No. 11131004, No. 11671215). Email: [email protected] U ( q ) = U ( q , q , q , q ) = P ≤ i
3. In 2007, Perez-Chavela and Santoprete in [19] proved that if the configuration is convex and m = m = m , m = m = 1, the central configuration must be a rhombus and this central configuration isunique. In 2008, Albouy, Fu ans Sun in [1] studied the symmetry of the four body problem of thecentral configuration and they proved that for four particles forming a convex quadrilateral centralconfiguration, the central configuration is symmetric with respect to the diagonal if and only if twoparticles on the opposite sides of the diagonal possess equal masses.In 2005, Meyer and Schmidt in [16] decomposed the fundamental solution of the elliptic La-grangian orbit into two parts symplectically using central configuration coordinates. They obtainedresults on stability by normal form theory for small enough eccentricity e ≥
0. In 2010-2014, Hu,Long and Sun introduced a Maslov-type index method and operator theory of the studying thestability in elliptic Lagrangian solutions of the planar three-body problem in [2] and [6]. In [2],the authors analytically proved the stability bifurcation diagram of the elliptic Lagrangian solu-tionsv in the parameter rectangle ( β, e ) ∈ [0 , × [0 , m = m = m = m = 1 is hyperbolic for any eccentricity e . In 2017, Mansur, Offin and Lewisin [14] proved the instability of the constrained elliptic rhombus solution in reduced space by theminimizing property of the action functional and assuming the nondegeneracy of variational prob-lem, i.e., the linearized Poincare map which is the ending point of the fundamental solution of thelinearized problem possesses at least one pair of hyperbolic eigenvalues. Especially, when e = 0, by[18], they obtained instability in the reduce space, i.e., the linearized Poincare map possesses onepair hyperbolic eigenvalues. In this paper, without the assumption on nondegeneracy, we obtainthe fundamental solution at the end point possesses at least two pairs of hyperbolic eigenvalueswhich yields the instability by the analytical method. By the numerical results on linear stabilityof elliptic Lagrangian solution, we obtain that the eigenvalues of the linearized Poincare map of theessential part are all hyperbolic.Furthermore, in 2017, Zhou and Long applied the Maslov-type index theory on the Euler-Moulton solutions. They reduced the elliptic Euler-Moulton solutions of the N -body problem3o those 3-body problem in [21] by the central configuration coordinate and obtained the linearstability of the elliptic Euler solution of the 3-body problem by the Maslov-type indices in [20].In this paper, we use the technique introduced by Meyer and Schmidt in [16] to reduce thesystem to three independent Hamiltonian systems of γ ( t ), γ u,e ( t ) and η u,e ( t ). The Hamiltoniansystem of γ ( t ) is fully studied in [6]. For the rest two Hamiltonian systems γ u,e ( t ) and η u,e ( t ), weanalyze the ω -Maslov type indices of γ u,e ( t ) and η u,e ( t ) and the ω -Morse indices of the correspondingoperators.Before stating our results, we need the following results on the positivity of certain operatorsobtained in the studies of the linear stability of the elliptic Lagrangian solutions in [4] and [15]. Lemma 1.1. (i) By the analytical result of Theorem 1.8 of [4], the operator A ( β, e ) defined by(2.126) are positive definite for any ω -boundary condition with zero nullity where ω ∈ U and ( β, e ) ∈{ }× [0 , ˆ f ( ) − / ) or ( β, e ) ∈ { β }× [0 , ˆ f ( β ) − / ) where β is given by (4.27), ˆ f ( ) − / ≈ . and ˆ f ( β ) − / ≈ . can be obtained by Theorem 1.8 of [4].(ii) By the numerical result in section 7 of [15], for ( β, e ) ∈ { } × [0 , or ( β, e ) ∈ { β } × [0 , ,the operator A ( β, e ) is positive definite for any ω -boundary condition with zero nullity where ω ∈ U . By the analytical and numerical results of the elliptical Lagrangian solutions in Lemma 1.1, weanalytically obtain the linear stability of the elliptic rhombus solutions.
Theorem 1.2. (i) By (i) of Lemma 1.1, when ( u, e ) ∈ ( u , /u ) × [0 , or ( u, e ) ∈ (cid:0) (1 / √ , u ) ∪ (1 /u , √ (cid:1) × [0 , ˆ f ( ) − / ) where u ≈ . is given by (4.36), the linearizedPoincare map, which is the end pint γ (2 π ) of the fundamental solution of the linearized Hamil-tonian system, possesses at least two pairs of hyperbolic eigenvalues, i.e., at least two pairs ofeigenvalues are not on U .(ii) By (ii) of Lemma 1.1, for ( u, e ) ∈ [1 / √ , √ × [0 , , γ (2 π ) possesses four pairs of hyper-bolic eigenvalues, i.e, all the eigenvalues of the essential parts are hyperbolic. This paper is organized as follows. In Section 2, we introduce the ω -Maslov-type indices and ω -Morse indices, and reduce the linearized Hamiltonian system to three subsystems. In Section 3, westudy the linear stability along the three boundary segments of the rectangle ( u, e ) ∈ [1 / √ , √ × [0 , u, e ) ∈ [1 / √ , √ × [0 ,
1) andprove the Theorem 1.2. 4
Preliminaries ω -Maslov-Type Indices and ω -Morse Indices Let ( R n , Ω) be the standard symplectic vector space with coordinates ( x , ..., x n , y , ..., y n ) andthe symplectic form Ω = P ni =1 dx i ∧ dy i . Let J = ( − I n I n ) be the standard symplectic matrix,where I n is the identity matrix on R n . Given any two 2 m k × m k matrices of square block form M k = ( A k B k C k D k ) with k = 1 ,
2, the symplectic sum of M and M is defined (cf. [8] and [10]) by thefollowing 2( m + m ) × m + m ) matrix M ⋄ M : M ⋄ M = A B A B C D C D . For any two paths γ j ∈ P τ (2 n j ) with j = 0 and 1, let γ ⋄ γ ( t ) = γ ( t ) ⋄ γ ( t ) for all t ∈ [0 , τ ].It is well known that that the fundamental solution γ ( t ) of the linear Hamiltonian system withthe continuous symmetric periodic coefficients is a path in the symplectic matrix group Sp(2 n )starting from the identity. In the Lagrangian case, when n = 2, the Maslov-type index i ω ( γ )is defined by the usual homotopy intersection number about the hypersurface Sp(2 n ) = { M ∈ Sp(2 n ) | D ω ( M ) = 0 } where D ω ( M ) = ( − n − ω n det( M − ωI n ). And the nullity is defined by ν ω ( M ) = dim C ker C ( γ (2 π ) − ωI n ). Please refer to [8, 9, 10] for more details on this index theoryof symplectic matrix paths and periodic solutions of Hamiltonian system.For T >
0, suppose x is a critical point of the functional F ( x ) = Z T L ( t, x, ˙ x ) dt, ∀ x ∈ W , ( R /T Z , R n ) , where L ∈ C (( R /T Z ) × R n , R ) and satisfies the Legendrian convexity condition L p,p ( t, x, p ) > x satisfies the corresponding Euler-Lagrangian equation: ddt L p ( t, x, ˙ x ) − L x ( t, x, ˙ x ) = 0 , (2.1) x (0) = x ( T ) , ˙ x (0) = ˙ x ( T ) . (2.2)For such an extremal loop, define P ( t ) = L p,p ( t, x ( t ) , ˙ x ( t )) , Q ( t ) = L x,p ( t, x ( t ) , ˙ x ( t )) , R ( t ) = L x,x ( t, x ( t ) , ˙ x ( t )) . (2.3)Note that F ′′ ( x ) = − ddt ( P ddt + Q ) + Q T ddt + R. (2.4)5or ω ∈ U , set D ( ω, T ) = { y ∈ W , ([0 , T ] , C n ) | y ( T ) = ωy (0) } . (2.5)We define the ω -Morse index φ ω ( x ) of x to be the dimension of the largest negative definite subspaceof h F ′′ ( x ) y , y i , for all y , y ∈ D ( ω, T ), where h· , ·i is the inner product in L . For ω ∈ U , we alsoset D ( ω, T ) = { y ∈ W , ([0 , T ] , C n ) | y ( T ) = ωy (0) , ˙ y ( T ) = ω ˙ y (0) } . (2.6)Then F ′′ ( x ) is a self-adjoint operator on L ([0 , T ] , R n ) with domain D ( ω, T ). We also define ν ω ( x ) = dim ker( F ′′ ( x )) . In general, for a self-adjoint operator A on the Hilbert space H , we set ν ( A ) = dim ker( A ) anddenote by φ ( A ) its Morse index which is the maximum dimension of the negative definite subspaceof the symmetric form h A · , ·i . Note that the Morse index of A is equal to the total multiplicity ofthe negative eigenvalues of A .On the other hand, ˜ x ( t ) = ( ∂L/∂ ˙ x ( t ) , x ( t )) T is the solution of the corresponding Hamiltoniansystem of (2.1)-(2.2), and its fundamental solution γ ( t ) is given by˙ γ ( t ) = J B ( t ) γ ( t ) , (2.7) γ (0) = I n , (2.8)with B ( t ) = P − ( t ) − P − ( t ) Q ( t ) − Q ( t ) T P − ( t ) Q ( t ) T P − ( t ) Q ( t ) − R ( t ) ! . (2.9) Lemma 2.3. ([10], p.172) For the ω -Morse index φ ω ( x ) and nullity ν ω ( x ) of the solution x = x ( t ) and the ω -Maslov-type index i ω ( γ ) and nullity ν ω ( γ ) of the symplectic path γ correspondingto ˜ x , for any ω ∈ U we have φ ω ( x ) = i ω ( γ ) , ν ω ( x ) = ν ω ( γ ) . (2.10)A generalization of the above lemma to arbitrary boundary conditions is given in [5]. For moreinformation on these topics, readers may refer to [10]. In 2005, Meyer and Schmidt gave the essential part of the fundamental solution of the ellipticLagrangian orbit (cf. p. 275 of [16]). Readers may also refer to [12]. Note that X i =1 m i a i = 0 and X i =1 m i | a i | = 1 . (2.11)6e define M = diag { m I, m I, m I, m I } , ˜ J = diag { J , J , J , J } and J is the standard 2 × U ( q ) at the central configuration a and obtain B ij | q = a ≡ ∂ U∂q i ∂q j (cid:12)(cid:12)(cid:12)(cid:12) q = a = m i m j | a i − a j | (cid:18) I − a j − a i )( a j − a i ) T | a i − a j | (cid:19) , (2.12)and B ii | q = a ≡ ∂ U∂q i (cid:12)(cid:12)(cid:12)(cid:12) q = a = n X j = i m i m j | a i − a j | (cid:18) − I + 3( a i − a j )( a i − a j ) T | a i − a j | (cid:19) . (2.13)By the symmetry of the configuration, we have that a − a = a − a and a − a = a − a . Theseyield that B = B = B = B = α m (1 + u ) / u − u u − u ! , (2.14) B = B = B = B = α m (1 + u ) / u − − u − u − u ! , (2.15) B = B = α m u − ! , (2.16) B = B = α − ! . (2.17)Note that B ii = − P j = i B ij . These yield that B = B = 2 α m (1 + u ) / − u
00 2 u − ! + α m u − ! ; (2.18) B = B = 2 α m (1 + u ) / − u
00 2 u − ! + α − ! . (2.19)As the in p. 263 of [16], Section 11.2 of [12], we define P = p p p p , Q = q q q q , Y = GZW W , X = gzw w , (2.20)where p i , q i , i = 1 , , , G , Z , W , W , g , z , w , w are all column vectors in R . We makethe symplectic coordinate change P = A − T Y, Q = AX, (2.21)7here the matrix A is constructed as in the proof of Proposition 2.1 in [16]. Concretely, the matrix A ∈ GL ( R ) is given by A = I A A A I A A A I A A A I A A A , (2.22)satisfying that ˜ J A = A ˜ J , A T M A = I. (2.23)Note that (2.23) is equivalent to A ij J = J A ij , X i =1 A Tij
M A ik = δ kj I . (2.24) A i is given by A = uα J , A = 1 α I , A = − uα J , A = − α I . (2.25)Readers may verify that P i =1 m i A i = 0 and P i =1 m i A Ti A i = I hold. We define A i s by A = A = − √ m + 2 m I , A = A = r m m + 2 I . (2.26)Readers may verify that P i =1 m i A i = 0, P i =1 m i A Ti A i = 0 and P i =1 m i A Ti A i = I hold. Wedefine A i s by A = − √ mα I , A = − u √ mα J , A = 1 √ mα I , A = u √ mα J . (2.27)Readers may verify that P i =1 m i A i = 0, P i =1 m i A Ti A i = 0, P i =1 m i A Ti A i = 0 and P i =1 m i A Ti A i = I hold. Above all, we have the matrix A satisfying (2.24) which is A = I A A A I A A A I A A A I A A A = − uα − √ m +2 − √ mα
00 1 uα − √ m +2 − √ mα α q m m +2 u √ mα α q m m +2 − u √ mα
01 0 0 uα − √ m +2 √ mα
00 1 − uα − √ m +2 √ mα − α q m m +2 − u √ mα − α q m m +2 u √ mα . (2.28)8n following discussion, we also need to name each column of A by defining A = ( c , c , ..., c ) where c i s are column vectors.Under the change of (2.21), we have the kinetic energy K = 12 ( | G | + | Z | + | W | + | W | ) , (2.29)and the potential function U ( z, w , w ) = X n = 4. Proposition 2.4.
There exists a symplectic coordinate change ξ = ( Z, W , W , z, w , w ) T ¯ ξ = ( ¯ Z, ¯ W , ¯ W , ¯ z, ¯ w , ¯ w ) T (2.32) such that using the true anomaly θ as the variable the resulting Hamiltonian function of the n -bodyproblem is given by H ( θ, ¯ Z, ¯ W , ¯ W , ¯ z, ¯ w , ¯ w ) = 12 | ¯ Z | + X k =3 | ¯ W k | ! + (¯ z · J ¯ Z + X k =3 ¯ w k · J ¯ W k )+ p − r ( θ )2 p | ¯ z | + X k =3 | ¯ w k | ! − r ( θ ) σ U (¯ z, ¯ w , ¯ w ) , (2.33) where r ( θ ) = p e cos θ , µ is given by (1.9), σ = ( µp ) / and p is given in (2.31). The proof of this proposition can be found in pp. 271-275 of [16] and pp.403-407 of [21]. Weomit it here.
Proposition 2.5.
Using the notations in (2.20), elliptic rhombus solution ( P ( t ) , Q ( t )) T of thesystem (1.4) with Q ( t ) = ( r ( t ) R ( θ ( t )) a , r ( t ) R ( θ ( t )) a , r ( t ) R ( θ ( t )) a , r ( t ) R ( θ ( t )) a ) T , P ( t ) = M ˙ Q ( t ) (2.34)9 n time t with the matrix M = diag { m I , m I , m I , m I } , is transformed to the new solution ( Y ( θ ) , X ( θ )) T in the variable true anomaly θ with G = g = 0 with respect to the original Hamilto-nian function H of (2.33), which is given by Y ( θ ) = ¯ Z ( θ )¯ W ( θ )¯ W ( θ ) = σ , X ( θ ) = ¯ z ( θ )¯ w ( θ )¯ w ( θ ) = σ , (2.35) Moreover, the linearized Hamiltonian system is given at the elliptic rhombus solution ξ ≡ ( Y ( θ ) , X ( θ )) T = (0 , σ, , , , , σ, , , , , ∈ R (2.36) depending on the true anomaly θ with respect to the Hamiltonian function H defined in (2.33) isgiven by ˙ γ ( θ ) = J B ( θ ) γ ( θ ) (2.37) with B ( θ ) is given by B ( θ ) = H ′′ ( θ, ¯ Z, ¯ W , ¯ W , ¯ z, ¯ w , ¯ w ) | ¯ ξ = ξ = I O O − J O OO I O O − J OO O I O O − JJ O O H zz ( θ, ξ ) O OO J O O H w w ( θ, ξ ) OO O J O O H w w ( θ, ξ ) (2.38) and H zz ( θ, ξ ) is given by H zz ( θ, ξ ) = − − e cos θ e cos θ
00 1 ! , (2.39) H w w ( θ, ξ ) is given by H w w ( θ, ξ ) = (cid:18) −
11 + e cos θ (cid:19) I − m + 1) α µ (1 + e cos θ )(1 + u ) / − u
00 2 u − ! , (2.40)10 w w ( θ, ξ ) is given by H w w ( θ, ξ ) = (cid:16) −
11 + e cos θ (cid:17) I − αµ (1 + e cos θ ) × u ) / m u + (6 m − m − u + 2 00 − m u + (2 m − m + 2) u − ! + (cid:18) mu m u (cid:19) − !! , (2.41) where H ′′ is the Hession matrix of H with respect to its variables ¯ Z , ¯ W , ¯ W ¯ z , ¯ w and ¯ w . Thecorresponding quadric Hamiltonian function is given by H ( θ, ¯ Z, ¯ W , ¯ W , ¯ z, ¯ w , ¯ w ) = 12 | ¯ Z | + ¯ Z · J ¯ z + 12 H ¯ z ¯ z ( θ, ξ )¯ z · ¯ z + X i =3 (cid:16) | ¯ W i | + ¯ W i · J ¯ w i + 12 H ¯ w i ¯ w i ( θ, ξ ) ¯ w i · ¯ w i (cid:17) . (2.42) Proof.
The proof is similar to those of Proposition 11.11 and Proposition 11.13 of [12]. Readermay also refer to a similar proof in pp.404-407 in [21]. We only focus on the H ¯ z ¯ z ( θ, ξ ), H ¯ z ¯ w ( θ, ξ ), H ¯ z ¯ w ( θ, ξ ), H ¯ w ¯ w ( θ, ξ ), H ¯ w ¯ w ( θ, ξ ), H ¯ w ¯ w ( θ, ξ ).For simplicity , we omit all the upper bars on the variables of H in (2.33) in this proof. Notethat we have transformed ( x , x , x , x ) to ( g, z, w , w ) by Q = AX . By this transformation, wehave the linearized system is given by H zz = p − rp I − rσ U zz ( z, w , w ) ,H zw l = H w l z = − rσ U zw l ( z, w , w ) , for l = 3 , H w l w l = p − rp I − rσ U w l w l ( z, w , w ) , for l = 3 , H w l w s = H w s w l = − rσ U w l w s ( z, w , w ) , for l, s = 3 , , l = s. (2.43)Then we have B ( θ ) = H ′′ ( θ, ¯ Z, ¯ W , ¯ W , ¯ z, ¯ w , ¯ w ) | ¯ ξ = ξ = I O O − J O OO I O − J OO O I O O − JJ O O H zz ( θ, ξ ) H zw ( θ, ξ ) H zw ( θ, ξ ) O J O H zw ( θ, ξ ) H w w ( θ, ξ ) H w w ( θ, ξ ) O O J H w z ( θ, ξ ) H w w ( θ, ξ ) H w w ( θ, ξ ) . (2.44)11e define Φ ij and Ψ ij ( k ) byΦ ij = A i − A j = ( a i − a j , J ( a i − a j )); (2.45)Ψ ij ( k ) = A ik − A jk = ( a ik − a jk , J ( a ik − a jk )) , (2.46)where a i = ( a i , a i ) and A ij = a ij, − a ij, a ij, a ij, ! . (2.47)Then the potential U ( x ) can be written as U ( z, x ) = X ≤ i 00 2 u − ! + α m u ( m + 1) − ! , (2.57) A T B A = α m ( m + 1)(1 + u ) / − u 00 2 u − ! + α m m + 1) − ! , (2.58)13 T B A = α m u ( m + 1) − ! , A T B A = α m m + 1) − ! , (2.59) A T B A = A T B A = − α m (2 m + 2)(1 + u ) / u − u u − u ! , (2.60) A T B A = A T B A = − α m (2 m + 2)(1 + u ) / u − − u − u − u ! . (2.61)Then we have that σ ∂ U∂w (cid:12)(cid:12)(cid:12)(cid:12) ξ = 2( A T B A + A T B A + A T B A + A T B A + A T B A + A T B A + A T B A + A T B A )= 2( m + 1) α (1 + u ) / − u 00 2 u − ! . (2.62)Next, we consider ∂ U∂w ∂w which satisfies ∂ U∂w ∂w (cid:12)(cid:12)(cid:12)(cid:12) ξ = 1 σ X i =1 4 X j =1 A Ti B ij A j . (2.63)Since (2.14-2.19)and (2.27), we have that A T B A = − A T B A , A T B A = − A T B A , (2.64) A T B A = − A T B A , A T B A = − A T B A , (2.65) A T B A = − A T B A , A T B A = − A T B A , (2.66) A T B A = − A T B A , A T B A = − A T B A . (2.67)We can rearrange the order of P i =1 P j =1 A Ti B ij A j and obtain that σ ∂ U ( X ) ∂w ∂w (cid:12)(cid:12)(cid:12)(cid:12) ξ = ( A T B A + A T B A ) + ( A T B A + A T B A )+( A T B A + A T B A ) + ( A T B A + A T B A )+( A T B A + A T B A ) + ( A T B A + A T B A )+( A T B A + A T B A ) + ( A T B A + A T B A )= 0 , (2.68)14here the last equality holds because every bracket is zero by (2.64-2.67).Next, we consider ∂ U∂w (cid:12)(cid:12)(cid:12) ξ which satisfies ∂ U∂w (cid:12)(cid:12)(cid:12)(cid:12) ξ = 1 σ X i =1 4 X j =1 A Ti B ij A j . (2.69)Since (2.14-2.19) and (2.26-2.27), we have that A T B A = A T B A , A T B A = A T B A , (2.70) A T B A = A T B A , A T B A = A T B A , (2.71) A T B A = A T B A , A T B A = A T B A , (2.72) A T B A = A T B A , A T B A = A T B A . (2.73)Then we only need to calculate the left hand of each equation (2.70-2.73). A T B A = 2 α (1 + u ) / − u 00 2 u − ! + αm u − ! , (2.74) A T B A = 2 αm u (1 + u ) / u − − u ! + αmu − ! , (2.75) A T B A = αm u − ! , A T B A = αmu − ! , (2.76) A T B A = αum (1 + u ) / u − u − u − u ! , (2.77) A T B A = αmu (1 + u ) / u − u − u − u ! , (2.78) A T B A = αmu (1 + u ) / u u − u − − u ! , (2.79) A T B A = αmu (1 + u ) / u u − u − − u ! . (2.80)15y (2.74-2.80), we have that ∂ U∂w (cid:12)(cid:12)(cid:12)(cid:12) ξ = 1 σ X i =1 4 X j =1 A Ti B ij A j = 4 ασ u ) / m u + (6 m − m − u + 2 00 − m u + (2 m − m + 2) u − ! + (cid:18) mu m u (cid:19) − !! . (2.81)Then, ∂ U∂z∂w s (cid:12)(cid:12)(cid:12) ξ is obtained by following computations. ∂ U∂z∂w s (cid:12)(cid:12)(cid:12)(cid:12) ξ = X ≤ i 00 1 ! , (2.89) H w w ( θ, ξ ) is given by H w w ( θ, ξ ) = p − rp I − rσ U w w ( z, w , w )= (cid:18) − 11 + e cos θ (cid:19) I − m + 1) α µ (1 + e cos θ )(1 + u ) / − u 00 2 u − ! , (2.90)and H w w ( θ, ξ ) is given by H w w ( θ, ξ ) = p − rp I − rσ U w w ( z, w , w )= (cid:18) − 11 + e cos θ (cid:19) I − αµ (1 + e cos θ ) × u ) / m u + (6 m − m − u + 2 00 − m u + (2 m − m + 2) u − ! + (cid:18) mu m u (cid:19) − !! . (2.91)Then this theorem holds.Then Hamiltonian system (2.33) can be decomposed to three independent Hamiltonian systems.The first one is the Kepler 2-body problem at the corresponding Kepler orbit which is given by γ ′ = J B γ = J − − − − e cos θ e cos θ 01 0 0 1 γ . (2.92)17ccording to Proposition 3.6. of [6], p. 1012 of [2] and (3.4- 3.5) of [20], we have that i ω ( γ ) = , if ω = 1 , , if ω ∈ U \ { } , ν ω ( γ ) = , if ω = 1 , , if ω ∈ U \ { } . (2.93)In the following, we only need to discuss the linear stability of the rest of two linearized Hamil-tonian systems γ ′ u,e = J B γ u,e = J I − JJ H w w ( u, e ) ! γ u,e , (2.94) η ′ u,e = J B η u,e = J I − JJ H w w ( u, e ) ! η u,e , (2.95)where ( u, e ) ∈ (1 / √ , √ × [0 , u ∈ (1 / √ , √ 3) we define ϕ ( u ) = 1 + 2( m + 1) α (2 − u ) µ (1 + u ) / , (2.96) ϕ ( u ) = 1 + 2( m + 1) α (2 u − µ (1 + u ) / , (2.97) ψ ( u ) = 1 + 4 αµ (cid:18) m u + (6 m − m − u + 2(1 + u ) / − mu − m u (cid:19) , (2.98) ψ ( u ) = 1 + 4 αµ (cid:18) − m u + (2 m − m + 2) u − u ) / + mu m u (cid:19) . (2.99)In following discussion, we will write ϕ i and ψ i instead of ϕ i ( u ) and ψ i ( u ) when it does not causeany confusion in the context. Note that ϕ i and ψ i are both smooth functions of u on the interval1 / √ < u < √ m , µ and α are smooth functions of u on that interval. Furthermore, ϕ i and ψ i , for i = 1 , 2, all converge when u tends to 1 / √ √ u →√ ϕ ( u ) = lim u → / √ ϕ ( u ) = lim u →√ ψ ( u ) = lim u → / √ ψ ( u ) = 34 , (2.100)lim u → / √ ϕ ( u ) = lim u →√ ϕ ( u ) = lim u →√ ψ ( u ) = lim u → / √ ψ ( u ) = 94 . (2.101)Then we extend the domain of u to [1 / √ , √ / √ ≤ u ≤ √ ϕ ( u ) = ϕ (1 /u ) , ψ ( u ) = ψ (1 /u ) , ψ ( u ) = ψ (1 /u ) . (2.102)18e define K u,e ( t ) and T u,e ( t ) by K u,e ( t ) ≡ 11 + e cos t ϕ ϕ ! , (2.103) T u,e ( t ) ≡ 11 + e cos t ψ ψ ! . (2.104)Therefore, H w w ( t ) and H w w ( t ) can be respectively written as H w w ( t ) = I − K u,e ( t ) = I − 11 + e cos t ϕ ϕ ! , (2.105) H w w ( t ) = I − T u,e ( t ) = I − 11 + e cos t ψ ψ ! . (2.106) Proposition 2.6. For any given ( u, e ) ∈ [1 / √ , √ × [0 , , the ω -Maslov-type indices andnullities of γ u,e ( t ) and η u,e ( t ) satisfying that for any ω ∈ U , i ω ( γ u,e ) = i ω ( γ /u,e ) , i ω ( η u,e ) = i ω ( η /u,e ) , (2.107) ν ω ( γ u,e ) = ν ω ( γ /u,e ) , ν ω ( η u,e ) = ν ω ( η /u,e ) . (2.108) Proof. Note that J − B ( u, e ) J where J = diag( J , J ) satisfies J − B ( u, e ) J = B (1 /u, e ) , (2.109)where the equality holds because of ϕ ( u ) = ϕ (1 /u ). Next we consider following systemdd t γ /u,e ( t ) = J B (1 /u, e ) γ /u,e ( t ) = J J − B ( u, e ) J γ /u,e ( t ) = J − J B ( u, e ) J γ /u,e ( t ) , (2.110)where the third equality holds because of J − J = J J − . Therefore, the fundamental solution γ /u,e ( t ) and γ u,e ( t ) satisfy γ /u,e ( t ) = J − γ u,e ( t ) J . (2.111)Then we have that for any ω ∈ U and ( u, e ) ∈ [1 / √ , √ × [0 , i ω ( γ u,e ) = i ω ( γ /u,e ) , ν ω ( γ u,e ) = ν ω ( γ /u,e ) . (2.112)Note that ψ ( u ) = ψ (1 /u ) and ψ ( u ) = ψ (1 /u ). We have that T u,e ( t ) = T /u,e ( t ), and then η u,e ( t ) = η /u,e ( t ). Therefore, we have that i ω ( η u,e ) = i ω ( η /u,e ) , ν ω ( η u,e ) = ν ω ( η /u,e ) . (2.113)Therefore, this proposition holds. 19 .3 A modification on the path γ u,e ( t ) According to the discussion of [2], we can transform the Lagrangian system to a simpler linearoperator corresponding to a second order Hamiltonian system with the same linear stability as γ u,e (2 π ) and η u,e (2 π ), using R ( t ) and R = diag( R ( t ) , R ( t )) as in [2], we let ξ u,e ( t ) = R ( t ) γ u,e ( t ) , ∀ θ ∈ [0 , π ] , ( u, e ) ∈ [1 / √ , √ × [0 , , (2.114)and ζ u,e ( t ) = R ( t ) η u,e ( t ) , ∀ θ ∈ [0 , π ] , ( u, e ) ∈ [1 / √ , √ × [0 , . (2.115)One can show by direct computations thatdd t ξ u,e ( t ) = J I R ( t )( I − K u,e ( t )) R ( t ) T ! ξ u,e ( t ) , (2.116)dd t ζ u,e ( t ) = J I R ( t )( I − T u,e ( t )) R ( t ) T ! ζ u,e ( t ) , (2.117)where K u,e ( t ) is given by (2.103) and T u,e ( t ) is given by (2.104). Note that R (0) = R (2 π ) = I ,so γ u,e (2 π ) = ξ u,e (2 π ) and η u,e (2 π ) = ζ u,e (2 π ) hold. Then the linear stabilities of the systems aredetermined by the same matrix and thus are precisely the same.By (2.114) and (2.115) the symplectic paths γ u,e and ξ u,e are homotopic to each other via thehomotopy h ( s, t ) = R ( st ) γ u,e ( t ) for ( s, t ) ∈ [0 , × [0 , π ]. Because R ( s ) γ u,e (2 π ) for s ∈ [0 , 1] isa loop in Sp(4) which is homotopic to the constant loop γ u,e (2 π ), h ( · , π ) is contractible in Sp(4).Therefore by the proof of Lemma 5.2.2 on p.117 of [12], the homotopy between γ u,e and ξ u,e canbe modified to fix the end point γ u,e (2 π ) for all s ∈ [0 , u, e ) ∈ [1 / √ , √ × [0 , i ω ( ξ u,e ) = i ω ( γ u,e ) , ν ω ( ξ u,e ) = ν ω ( γ u,e ) , ∀ ω ∈ U . (2.118)Similarly, we have that for ( u, e ) ∈ [1 / √ , √ × [0 , i ω ( ζ u,e ) = i ω ( η u,e ) , ν ω ( ζ u,e ) = ν ω ( η u,e ) , ∀ ω ∈ U . (2.119)Note that the first order linear Hamiltonian systems (2.116) and (2.117) correspond to thefollowing second order linear Hamiltonian systems receptively¨ x ( t ) = − x ( t ) + R ( t ) K u,e ( t ) R ( t ) T x ( t ) , (2.120)20nd ¨ x ( t ) = − x ( t ) + R ( t ) T u,e ( t ) R ( t ) T x ( t ) . (2.121)For ( u, e ) ∈ [1 / √ , √ × [0 , D ( ω, π ) corresponding to (2.120) and (2.121) are given by A ( u, e ) = − d d t I − I + R ( t ) K u,e ( t ) R ( t ) T , (2.122)and B ( u, e ) = − d d t I − I + R ( t ) T u,e ( t ) R ( t ) T , (2.123)where K u,e ( t ) and T u,e ( t ) are defined by (2.103-2.104) and D ( ω, π ) is given by (2.5). By directcomputations, we have that A ( u, e ) = − d d t I − I + 12(1 + e cos t ) (( ϕ + ϕ ) I + ( ϕ − ϕ ) S ( t )) , (2.124) B ( u, e ) = − d d t I − I + 12(1 + e cos t ) (( ψ + ψ ) I + ( ψ − ψ ) S ( t )) , (2.125)where S ( t ) = ( cos 2 t sin 2 t sin 2 t − cos 2 t ). In [2], the authors defined a operator A ( β, e ) is given by A ( β, e ) = − d d t I − I + 12(1 + e cos t ) (3 I + p − βS ( t )) . (2.126)We will use this operator A ( β, e ) in Section 3 and Section 4.The operators A ( u, e ) and B ( u, e ) are both self-adjoint and depend on the parameters u and e . By p. 172 of [10], we have for any ( u, e ) ∈ [1 / √ , √ × [0 , φ ω ( A ( u, e )) and φ ω ( B ( u, e )) and nullities which are ν ω ( A ( u, e )) and ν ω ( B ( u, e )) on the domain D ( ω, π ) satisfy φ ω ( A u,e ) = i ω ( ξ u,e ) , ν ω ( A u,e ) = ν ω ( ξ u,e ) , ∀ ω ∈ U , (2.127)and φ ω ( B u,e ) = i ω ( η u,e ) , ν ω ( B u,e ) = ν ω ( η u,e ) , ∀ ω ∈ U . (2.128)In the rest of this paper, we shall use both of the paths γ u,e and ξ u,e to study the linear stabilityof γ u,e (2 π ) = ξ u,e (2 π ) and use both of the paths ζ u,e and η u,e to study the linear stability of ζ u,e (2 π ) = η u,e (2 π ). Because of (2.118) and (2.119), in many cases and proofs below, we shall notdistinguish these two paths. 21 Stability on the Three Boundary Segments of the Rectangle [1 / √ , √ × [0 , [1 / √ , √ × { } If e = 0 which means that the orbits of four bodies are circles, H w w ( t ) and H w w ( t ) are given by H w w ( t ) = I − K u,e ( t ) = I − ϕ ϕ ! , (3.1) H w w ( t ) = I − T u,e ( t ) = I − ψ ψ ! , (3.2)where ϕ i s and ψ i s are given by (2.96- 2.99). The system of γ u, is given by γ ′ u, = J B γ u, = J − − − ϕ 01 0 0 1 − ϕ γ u, , (3.3)and the system of η u, is given by η ′ u, = J B η u, = J − − − ψ 01 0 0 1 − ψ η u, . (3.4) Theorem 3.1. For any given ( u, e ) ∈ [1 / √ , √ × { } and ω ∈ U , all the eigenvalues ofmatrices γ u, (2 π ) and η u, (2 π ) are all hyperbolic, i.e., all the eigenvalues are not on U , i ω ( γ u, ) = φ ω ( A ( u, , ν ω ( γ u, ) = ν ω ( A ( u, , (3.5) i ω ( η u, ) = φ ω ( B ( u, , ν ω ( η u, ) = ν ω ( B ( u, . (3.6) Therefore, the operators A ( u, and B ( u, are positive definite on the space ¯ D ( ω, π ) with zeronullity. Proof. The characteristic polynomial det( J B − λI ) of J B is given by p ( λ ) = λ + (4 − ϕ − ϕ ) λ + ϕ ϕ (3.7)22he roots of p ( λ ) are all pure imaginary if and only if4 − ϕ − ϕ > , (3.8) ϕ ϕ > , (3.9)(4 − ϕ − ϕ ) − ϕ ϕ ≥ , (3.10)hold at the same time. Note that4 − ϕ − ϕ = 2 − m + 1) α µ (1 + u ) / ( u + 1) , (3.11)and ϕ ϕ = 4( m + 1) α µ (1 + u ) (2 − u )(2 u − 1) + 2( m + 1) α µ (1 + u ) / ( u + 1) + 1 . (3.12)Note that the denominator of dd u (4 − ϕ − ϕ ) is positive and the numerator of dd u (4 − ϕ − ϕ ) isa polynomial on Z [ u, √ u ] of degree 20. Note that dd u (4 − ϕ − ϕ ) | u =1 = 0. By the numericalcomputations with the step length √ − / √ , u = 1 is the only root of dd u (4 − ϕ − ϕ ) = 0 in theinterval [1 / √ , √ dd u (4 − ϕ − ϕ ) | u =0 . ≈ − . dd u (4 − ϕ − ϕ ) | u = √ = √ , wehave that dd u (4 − ϕ − ϕ ) < / √ , 1] and dd u (4 − ϕ − ϕ ) > , √ / √ ≤ u ≤ √ 3, by (2.102),( − √ − ϕ (1) − ϕ (1) ≤ − ϕ − ϕ ≤ − ϕ (cid:18) √ (cid:19) − ϕ (cid:18) √ (cid:19) = 1 . (3.13)The denominator of dd u ( ϕ ϕ ) is positive and the numerator of dd u ( ϕ ϕ ) is a polynomial on Z [ u, √ u ] of degree 39. Note that dd u ( ϕ ϕ ) | u =1 = 0. By the numerical computations withthe step length √ − / √ , u = 1 is the only root of dd u ( ϕ ϕ ) = 0 in the interval [1 / √ , √ dd u ( ϕ ϕ ) | u =0 . ≈ . dd u ( ϕ ϕ ) | u = √ = − √ , we have dd u ( ϕ ϕ ) > / √ , 1] and dd u ( ϕ ϕ ) < , √ / √ ≤ u ≤ √ 3, by(2.102), 2716 = ϕ (cid:18) √ (cid:19) ϕ (cid:18) √ (cid:19) ≤ ϕ ϕ ≤ ϕ (1) ϕ (1) = 233 − √ . (3.14)Then(4 − ϕ − ϕ ) − ϕ ϕ ≤ (cid:18) − ϕ (cid:18) √ (cid:19) − ϕ (cid:18) √ (cid:19)(cid:19) − ϕ (cid:18) √ (cid:19) ϕ (cid:18) √ (cid:19) = − . (3.15)Let ¯ λ = λ and we have that¯ p (¯ λ ) = ¯ λ + (4 − ϕ − ϕ )¯ λ + ϕ ϕ . (3.16)23herefore, we have that the two roots of ¯ p ( λ ) is given by¯ λ = r e iθ = 12 (cid:0) (4 − ϕ − ϕ ) + p (4 − ϕ − ϕ ) − ϕ ϕ (cid:1) , (3.17)¯ λ = r e − iθ = 12 (cid:0) (4 − ϕ − ϕ ) − p (4 − ϕ − ϕ ) − ϕ ϕ (cid:1) , (3.18)where r = (4 − ϕ − ϕ ) − ϕ ϕ and θ = π because (4 − ϕ − ϕ ) − ϕ ϕ < / √ ≤ u ≤ √ p ( λ ), which are λ = √ r e iθ , λ = √ r e iθ + π , λ = √ r e − iθ , λ = √ r e − iθ + π , (3.19)are complex numbers with non-zero real parts because θ = π . This yields that γ u, (2 π ) is hyper-bolic and for any ω ∈ U and u ∈ [1 / √ , √ i ω ( γ u, ) = 0 , ν ω ( γ u, ) = 0 . (3.20)By (2.127), for any ω ∈ U and u ∈ [1 / √ , √ A ( u, 0) is non-degenerate and φ ω ( A ( u, , ν ω ( A ( u, . (3.21)The characteristic polynomial det( J B − λI ) of J B is given by p ( λ ) = λ + (4 − ψ − ψ ) λ + ψ ψ . (3.22)Note that 4 − ψ − ψ = 2 − αµ (cid:18) m u + 1(1 + u ) / + mu m u (cid:19) = 1 , (3.23)and ψ ψ = 16 α µ (cid:18) m u + (6 m − m − u + 2(1 + u ) / − mu − m u (cid:19) × (cid:18) − m u + (2 m − m + 2) u − u ) / + mu m u (cid:19) + 2 (3.24)where the last equality of (3.23) is obtained by the symbolic computations of Mathematica. Theroots of p ( λ ) are all pure imaginary if and only if4 − ψ − ψ = 1 > , (3.25) ψ ψ > , (3.26)(4 − ψ − ψ ) − ψ ψ = 1 − ψ ψ ≥ , (3.27)hold at the same time. 24ote that the denominator of dd u ( ψ ψ ) is positive and the numerator of dd u ( ψ ψ ) is a polyno-mial on Z [ u, √ u ] of degree 35. Note that dd u ( ψ ψ ) | u =1 = 0. Since dd u ( ψ ψ ) | u =1 / √ . ≈ . dd u ( ψ ψ ) | u =0 . ≈ − . dd u ( ψ ψ ) | u =1 . ≈ . dd u ( ψ ψ ) | u = √ = − √ ,there exists at least two more roots of dd u ( ψ ψ ) = 0 in the interval [1 / √ , √ 3] except u = 1. By thenumerical computations, u = ¯ u ≈ . u = 1 and u = 1 / ¯ u are three roots of dd u ( ψ ψ ) = 0in the interval [1 / √ , √ dd u ( ψ ψ ) > / √ , ¯ u ) ∪ (1 , / ¯ u ),and dd u ( ψ ψ ) < u , ∪ (1 / ¯ u , √ / √ ≤ u ≤ √ ψ (cid:18) √ (cid:19) ψ (cid:18) √ (cid:19) ≤ ψ ψ ≤ ψ (¯ u ) ψ (¯ u ) = 2 . , (3.28)and − . − ψ (¯ u ) ψ (¯ u ) ≤ − ψ ψ ≤ − ψ (cid:18) √ (cid:19) ψ (cid:18) √ (cid:19) = − . (3.29)Let ˜ λ = λ and we have that ¯ p (˜ λ ) = ˜ λ + ˜ λ + ψ ψ . (3.30)Therefore, we have that the two roots of ¯ p (˜ λ ) is given by˜ λ = ˜ r e i ˜ θ = 12 (cid:0) − p − ψ ψ (cid:1) , (3.31)˜ λ = ˜ r e − i ˜ θ = 12 (cid:0) − − p − ψ ψ (cid:1) , (3.32)where ˜ r = √ − ψ ψ and θ = π by 1 − ψ ψ < / √ ≤ u ≤ √ 3. Therefore, we havethe four roots of p ( λ ) are given by λ = p ˜ r e i ˜ θ , λ = p ˜ r e i ˜ θ + π , λ = p ˜ r e − i ˜ θ , λ = p ˜ r e − i ˜ θ + π , (3.33)which are complex numbers with non-zero real parts because ¯ θ = π . Therefore, the roots of p ( λ )have non-zero real part. This yields that η u, (2 π ) is hyperbolic, i.e., i ω ( η u, ) = 0 , ν ω ( η u, ) = 0 . (3.34)By (2.128), we have that for any ω ∈ U the operator is non-degenerate and φ ω ( B ( u, , ν ω ( B ( u, , (3.35)when u ∈ [1 / √ , √ .2 The segment { } × [0 , This case has been discussed in [3]. Here we paraphrase their results in our notations. When u = 1,we have that m = 1, α = 2, µ = 4 √ ϕ (1) = 1 + 2 √ √ , ϕ (1) = 1 + 2 √ √ , ψ (1) = 1 + 4 √ − √ , ψ (1) = 1 + 2 − √ √ . (3.36)Therefore, we have the operator A (1 , e ) and B (1 , e ) are given by A (1 , e ) = − d d t I − I + 4 √ √ e cos t ) I , (3.37) B (1 , e ) = − d d t I − I + 32(1 + e cos t ) I + 6 √ − √ e cos t ) S ( t ) . (3.38)By Proposition 2 of [3] and √ √ > 1, they obtain following results. Theorem 3.2. (cf. Theorem 2 of [3]) For any ω ∈ U and e ∈ [0 , , the operators A (1 , e ) and B (1 , e ) are positive definite on D ( ω, π ) with zero nullity, i.e., i ω ( γ ,e ) = φ ω ( A (1 , e )) = 0 , ν ω ( γ ,e ) = ν ω ( A (1 , e )) = 0; (3.39) i ω ( η ,e ) = φ ω ( B (1 , e )) = 0 , ν ω ( η ,e ) = ν ω ( B (1 , e )) = 0 . (3.40) Therefore, all the eigenvalues of γ ,e (2 π ) and η ,e (2 π ) are hyperbolic, i.e., all the eigenvalues arenot on U . {√ } × [0 , and { / √ } × [0 , In this section, we consider the linear stability of the system when ( u, e ) ∈ {√ } × [0 , ∪ { / √ } × [0 , u = √ 3, by (2.100-2.101), (2.105) and (2.106), we have that H w w ( √ , e ) = I − e cos θ ) ! , (3.41) H w w ( √ , e ) = I − e cos θ ) ! . (3.42)Note that H w w ( √ , e ) = H w w ( √ , e ). When u = 1 / √ 3, we have that H w w (1 / √ , e ) = I − e cos θ ) ! , (3.43) H w w (1 / √ , e ) = I − e cos θ ) ! . (3.44)26 heorem 3.3. (i) By (i) of Lemma 1.1, when ( u, e ) ∈ { / √ } × [0 , ˆ f ( ) − / ) or ( u, e ) ∈{√ } × [0 , ˆ f ( ) − / ) , for any ω ∈ U , the operators A ( u, e ) and B ( u, e ) are positive definite withzero nullity on the space ¯ D ( ω, π ) , i.e., φ ω ( A ( u, e )) = i ω ( γ u,e ) = 0 , ν ω ( A ( u, e )) = ν ω ( γ u,e ) = 0 , (3.45) φ ω ( B ( u, e )) = i ω ( η u,e ) = 0 , ν ω ( B ( u, e )) = ν ω ( η u,e ) = 0 . (3.46) Then all eigenvalues of the matrices γ u,e (2 π ) and η u,e (2 π ) are both hyperbolic, i.e., all the eigen-values are not on U , when ( u, e ) ∈ { / √ } × [0 , ˆ f ( ) − / ) or ( u, e ) ∈ {√ } × [0 , ˆ f ( ) − / ) .(ii) By (ii) of Lemma 1.1, when ( u, e ) ∈ { / √ } × [0 , or ( u, e ) ∈ {√ } × [0 , , the results of(i) hold. Proof. Since for all e ∈ [0 , H w w ( √ , e )( t ) = H w w (1 / √ , e )( t ) = H w w ( √ , e )( t ) , wehave that γ √ ,e ( t ) = η √ ,e ( t ) = η / √ ,e ( t ). This yields that for any ω ∈ U , i ω ( γ √ ,e ) = i ω ( η √ ,e ) = i ω ( η / √ ,e ) , (3.47) ν ω ( γ √ ,e ) = ν ω ( η √ ,e ) = ν ω ( η / √ ,e ) . (3.48)By Proposition 2.6, we have that i ω ( γ / √ ,e ) = i ω ( γ √ ,e ) = i ω ( η √ ,e ) = i ω ( η / √ ,e ) , (3.49) ν ω ( γ / √ ,e ) = ν ω ( γ √ ,e ) = ν ω ( η √ ,e ) = ν ω ( η / √ ,e ) . (3.50)For the system of γ / √ ,e ( t ), by (2.124), the corresponding operator is given by A (1 / √ , e ) = − d d t I − I + 12(1 + e cos t ) (3 I + 32 S ( t )) . (3.51)By the definition of A ( β, e ) in (2.126), when β = , we have that A (1 / √ , e ) = A ( 274 , e ) . (3.52)By (i) of Lemma 1.1, when β = , 0 ≤ e < ˆ f ( ) − / ≈ . A ( , e ) is positive operatorwith zero nullity on any ω boundary condition where ω ∈ U and ˆ f ( β ) is given by (1.22) of [4], i.e., φ ω ( A ( 274 , e )) = 0 , ν ω ( A ( 274 , e )) = 0 . (3.53)By (3.53), (3.49-3.50), and i ω ( γ / √ ,e ) = φ ω ( A (1 / √ , e )) = φ ω ( A ( 274 , e )) , (3.54) ν ω ( γ / √ ,e ) = ν ω ( A (1 / √ , e ) = ν ω ( A ( 274 , e )) , (3.55)27e have that (i) of this theorem holds.By (ii) of Lemma 1.1, we have that for any e ∈ [0 , 1) and any ω ∈ U , φ ω ( A ( 274 , e )) = 0 , ν ω ( A ( 274 , e )) = 0 . (3.56)By (3.56), we have (ii) of this theorem holds. [1 / √ , √ × [0 , By direct computations, the denominator of ϕ ( u ) − ϕ ( u ) is negative on the interval [1 / √ , √ ϕ ( u ) − ϕ ( u ) is a polynomial on Z [ u, √ u ]. Furthermore, u = 1 is theonly root of ϕ ( u ) − ϕ ( u ) = 0 , and ϕ ( u ) − ϕ ( u ) > , when 1 / √ ≤ u < , (4.1) ϕ ( u ) − ϕ ( u ) < , when 1 < u ≤ √ . (4.2)By (2.124), we define¯ A ( u, e ) = A (1 ,e ) ϕ − ϕ + S ( t )2(1+ e cos t ) , when 1 / √ ≤ u < , A (1 ,e ) ϕ − ϕ − S ( t )2(1+ e cos t ) , when 1 < u ≤ √ . (4.3)Then when 1 / √ ≤ u < A ( u, e ) can be written as A ( u, e ) = ( ϕ − ϕ ) (cid:18) A (1 , e ) ϕ − ϕ + S ( t )2(1 + e cos t ) (cid:19) = ( ϕ − ϕ ) ¯ A ( u, e ) , (4.4)and when 1 < u ≤ √ A ( u, e ) can be written as A ( u, e ) = ( ϕ − ϕ ) (cid:18) A (1 , e ) ϕ − ϕ − S ( t )2(1 + e cos t ) (cid:19) = ( ϕ − ϕ ) ¯ A ( u, e ) . (4.5)By (4.1-4.2) and (4.4-4.5), we have that φ ω ( A ( u, e )) = φ ω ( ¯ A ( u, e )) , ν ω ( A ( u, e )) = ν ω ( ¯ A ( u, e )) . (4.6)By direct computations, the denominator of d( ϕ − ϕ )d u is positive on the interval [1 / √ , √ 3] andthe numerator of d( ϕ − ϕ )d u is a polynomial on Z [ u, √ u ] of degree 24. Note that d( ϕ − ϕ )d u | u =0 . ≈− . d( ϕ − ϕ )d u | u =1 = −√ and d( ϕ − ϕ )d u | u = √ = − √ . Then there exist at least two rootsof d( ϕ − ϕ )d u = 0 in the interval [1 / √ , √ u ≈ . u = 1 /u are only two roots of d( ϕ − ϕ )d u = 0 in the interval [1 / √ , √ ϕ − ϕ )d u < , when 1 / √ < u < u , (4.7)d( ϕ − ϕ )d u > , when u < u < , (4.8)and d( ϕ − ϕ )d u < , when 1 < u < u , (4.9)d( ϕ − ϕ )d u > . when u < u < √ . (4.10) Lemma 4.1. (i) For each fixed e ∈ [0 , , the operator ¯ A ( u, e ) is increasing with respect to u ∈ ( u , ∪ ( u , √ and is decreasing with respect to u ∈ (1 / √ , u ) ∪ (1 , u ) where u and u aretwo roots of ∂ ( ϕ − ϕ ) ∂u = 0 when u ∈ [1 / √ , √ . Especially, ∂∂u ¯ A ( u, e ) | u = u = A (1 ,e )( ϕ − ϕ ) ∂ ( ϕ − ϕ ) ∂u , when / √ < u < A (1 ,e )( ϕ − ϕ ) ∂ ( ϕ − ϕ ) ∂u , when < u < √ . (4.11) for u ∈ [1 / √ , √ is positive definite operator when u ∈ ( u , ∪ ( u , √ and is negative definiteoperator when u ∈ [1 / √ , u ) ∪ (1 , u ) .(ii) For every eigenvalue λ u = 0 of ¯ A ( u , e ) with ω ∈ U for some ( u , e ) ∈ [1 / √ , √ × [0 , ,there hold dd u λ u | u = u > , when u ∈ ( u , ∪ ( u , √ , (4.12) and dd u λ u | u = u < u ∈ [1 / √ , u ) ∪ (1 , u ) . (4.13) Proof. By (3.39), A (1 ,e )( ϕ − ϕ ) and A (1 ,e )( ϕ − ϕ ) are always a positive definite operators on D ( ω, π )for any ω ∈ U . Then the first claim of this lemma is proved.Let x = x ( t ) with unit norm such that¯ A ( u , e ) x = 0 . (4.14)Fix e . Then ¯ A ( u, e ) is an analytic path of strictly increasing self-adjoint operators with respectto u when u ∈ ( u , ∪ ( u , √ 3] and is an analytic path of strictly decreasing self-adjoint operatorswith respect to u when u ∈ [1 / √ , u ) ∪ (1 , u ). 29ollowing Kato ([7], p.120 and p.386), we can choose a smooth path of unit norm eigenvectors x u with x u = x belonging to a smooth path of real eigenvalues λ u of the self-adjoint operator¯ A ( u, e ) on D ( ω, π ) such that for small enough | u − u | , we have¯ A ( u, e ) x u = λ u x u , (4.15)where λ u = 0. Taking inner product with x u on both sides of (4.15) and then differentiating itwith respect to u at u , we get ∂∂u λ u | u = u = h ∂∂u ¯ A ( u, e ) x u , x u i| u = u + 2 h ¯ A ( u, e ) x u , ∂∂u x u i| u = u = h ∂∂u ¯ A ( u , e ) x , x i = ϕ − ϕ ) ∂ ( ϕ − ϕ ) ∂u hA (1 , e ) x , x i , when 1 / √ < u < ϕ − ϕ ) ∂ ( ϕ − ϕ ) ∂u hA (1 , e ) x , x i , when 1 < u < √ . (4.16)where the second equality follows from (4.15), the last equality follows from the definition of ¯ A ( u, e ).By (4.7 - 4.10) and the the non-negative definiteness of A (1 , e ), we have thatdd u λ u | u = u > , when u ∈ ( u , ∪ ( u , √ , (4.17)and dd u λ u | u = u < u ∈ (1 / √ , u ) ∪ (1 , u ) . (4.18)Thus, this lemma holds. Corollary 4.2. For every fixed e ∈ [0 , and ω ∈ U , the index function φ ω ( A ( u, e )) , andconsequently i ω ( γ u,e ) , is non-decreasing as u increases from u to and from u to √ ; and theyare non-increasing as u increases from / √ to u and from to u . Especially, the index function φ ω ( A ( u, e )) satisfies φ ω ( A ( u, e )) ≥ φ ω ( A ( u , e )) , when u ∈ (1 / √ , , (4.19) φ ω ( A ( u, e )) ≥ φ ω ( A ( u , e )) , when u ∈ [1 , √ . (4.20) Proof. For u ≤ u ′ < u ′′ < e ∈ [0 , u increases from u ′ to u ′′ , it is possiblethat negative eigenvalues of ¯ A ( u ′ , e ) pass through 0 and become positive ones of ¯ A ( u ′′ , e ), but itis impossible that positive eigenvalues of ¯ A ( u ′ , e ) pass through 0 and become negative by (ii) ofLemma 4.1. Similar arguments also hold when u in the intervals ( u , √ / √ , u ) and (1 , u ).Therefore the first and the second claims hold. 30ext we consider Morse index and nullity of A ( u, e ) when u = u and u = u . Lemma 4.3. (i) By (i) of Lemma 1.1, for any ω boundary condition, when e ∈ [0 , ˆ f ( β ) − / ) ,both the operators A ( u , e ) and A ( u , e ) are non-degenerate positive operators with zero nullity, i.e., φ ω ( A ( u , e )) = φ ω ( A ( u , e )) = 0 , ν ω ( A ( u , e )) = ν ω ( A ( u , e )) = 0 . (4.21) (ii) By (ii) of Lemma 1.1, when e ∈ [0 , , the results of (i) hold. Proof. By u = 1 /u and Proposition 2.6, we have that i ω ( γ u ,e ) = i ω ( γ u ,e ) , ν ω ( γ u ,e ) = ν ω ( γ u ,e ) . (4.22)Then φ ω ( A ( u , e )) = φ ω ( A ( u , e )) , ν ω ( A ( u , e )) = ν ω ( A ( u , e )) . (4.23)We only need to consider the case of u = u . By the direct computations, we have that ϕ ( u ) + ϕ ( u ) ≈ . , ϕ ( u ) − ϕ ( u ) ≈ . . (4.24)The operator A ( u , e ) is given by A ( u , e ) = − d d t I − I + 12(1 + e cos t ) (( ϕ ( u ) + ϕ ( u )) I + ( ϕ ( u ) − ϕ ( u )) S ( t )) . (4.25)Since ϕ ( u ) + ϕ ( u ) > I e cos t ) is a positive operator on D ( ω, π ), we have A ( u , e ) > − d d t I − I + 12(1 + e cos t ) (3 I + ( ϕ ( u ) − ϕ ( u )) S ( t )) . (4.26)Note that there exists a β = 9 − ( ϕ ( u ) − ϕ ( u )) ≈ − (1 . = 6 . A ( β , e ) = − d d t I − I + 12(1 + e cos t ) (3 I + p − β S ( t )) . (4.27)where A ( β, e ) is defined by (2.126). Then we have that for any ω boundary condition A ( u , e ) > A ( β , e ) . (4.28)By (i) of Lemma 1.1, when β = β and 0 ≤ e < ˆ f ( β ) − / ≈ . A ( β , e ) is positive operatorwith zero nullity on any ω boundary condition where ω ∈ U and ˆ f ( β ) is given by (1.22) of [4].Then for e ∈ [0 , ˆ f ( β ) − / ) and ω ∈ U , φ ω ( A ( u , e )) = 0 , ν ω ( A ( u , e )) = 0 , ∀ ω ∈ U . (4.29)31y (4.23), we obtain (i) of this lemma.By (ii) of Lemma 1.1 and (4.28), we have that for any e ∈ [0 , 1) and ω ∈ U , A ( u , e ) is alsopositive definite with zero nullity, i.e., φ ω ( A ( u , e )) = 0 , ν ω ( A ( u , e )) = 0 , ∀ ω ∈ U . (4.30)Again, by (4.23), we obtain (ii) of this lemma. Theorem 4.4. (i) By (i) of Lemma 1.1, for any ( u, e ) ∈ [1 / √ , √ × [0 , ˆ f ( β ) − / ) and ω ∈ U , A ( u, e ) is a positive definite operator with zero nullity on the space D ( ω, π ) , i.e., i ω ( γ u,e ) = φ ω ( A ( u, e )) = 0 , ν ω ( γ u,e ) = ν ω ( A ( u, e )) = 0 . (4.31) Then all the eigenvalues of the matrix γ u,e (2 π ) are hyperbolic, i.e., all the eigenvalues are not on U . (ii) By (ii) of Lemma 1.1, for any ( u, e ) ∈ [1 / √ , √ × [0 , , the results of (i) hold. Proof. By Lemma 4.1, Corollary 4.2 and Lemma 4.3, we have that for any given e ∈ [0 , φ ω ( A ( u, e )) ≥ φ ω ( A ( u , e )) > , when u ∈ (1 / √ , , (4.32) φ ω ( A ( u, e )) ≥ φ ω ( A ( u , e )) > , when u ∈ [1 , √ . (4.33)By (i) of Lemma 4.3, we have that for any ( u, e ) ∈ [1 / √ , √ × [0 , ˆ f ( β ) − / ), φ ω ( A ( u, e )) = 0 , ν ω ( A ( u, e )) = 0 . (4.34)Since i ω ( γ u,e ) = φ ω ( A ( u, e )) , ν ω ( γ u ,e ) = ν ω ( A ( u, e )) , (4.35)we have (i) of the theorem holds.By (4.32-4.33), (ii) of Lemma 4.3 and (4.35), we have (ii) of this theorem holds. Remark 4.5. By the discussion in Section 3.2, we have that A (1 , e ) are positive definiteoperator for e ∈ [0 , . Then there exists a u ∗ ∈ (1 / √ , such that when ( u, e ) ∈ ( u ∗ , /u ∗ ) × [0 , , A ( u, e ) is a positive definite operator with zero nullity and the matrix γ u,e (2 π ) is hyperbolic. Next we consider the operator B ( u, e ) and the symplectic path η u,e ( t ). Since ψ i ( u ) = ψ i (1 /u )for i = 1 , B ( u, e ) = B (1 /u, e ) and η u,e ( t ) = η /u,e ( t ) for ( u, e ) ∈ [1 / √ , √ × [0 , B ( u, e ) and η u,e ( t ) in the domain ( u, e ) ∈ [1 / √ , × [0 , ψ ( u ) − ψ ( u ) is positive on the interval [1 / √ , √ ψ ( u ) − ψ ( u ) is a polynomial on Z [ u, √ u ] of degree 12. Note that ψ ( u ) − ψ ( u ) | u =1 = − √ and ψ ( u ) − ψ ( u ) | u =1 / √ = ψ ( u ) − ψ ( u ) | u = √ = − . Then thereexists at least one root of ψ ( u ) − ψ ( u ) = 0 in the interval [1 / √ , u = u ≈ . ψ ( u ) − ψ ( u ) = 0 in the interval [1 / √ , ψ ( u ) − ψ ( u ) < , when 1 / √ ≤ u < u , (4.36) ψ ( u ) − ψ ( u ) > , when u < u ≤ . (4.37)When u = u , we have that ψ ( u ) + ψ ( u ) = 3 , ψ ( u ) − ψ ( u ) = 0 . (4.38)The operator B ( u , e ) is given by B ( u , e ) = − d d t I − I + 32(1 + e cos t ) . (4.39)By the definition of A ( β, e ) in (2.126), we have that B ( u , e ) = A (9 , e ) (4.40)By Corollary 4.3 of [2], we have that A (9 , e ) is a positive definite operator with zero nullity for any ω boundary condition. So is B ( u , e ). We define the operator ¯ B ( u, e ) by¯ B ( u, e ) = B ( u ,e ) ψ − ψ − S ( t )2(1+ e cos t ) , when 1 / √ < u < u , B ( u ,e ) ψ − ψ + S ( t )2(1+ e cos t ) , when u < u < . (4.41)By the definition of B ( u, e ) in (2.125), when 1 / √ < u < u , B ( u, e ) can be written as B ( u, e ) = ( ψ − ψ ) (cid:18) B ( u , e ) ψ − ψ − S ( t )2(1 + e cos t ) (cid:19) = ( ψ − ψ ) ¯ B ( u, e ) , (4.42)and when u < u < B ( u, e ) can be written as B ( u, e ) = ( ψ − ψ ) (cid:18) B ( u , e ) ψ − ψ + S ( t )2(1 + e cos t ) (cid:19) = ( ψ − ψ ) ¯ B ( u, e ) . (4.43) Lemma 4.6. (i) For each fixed e ∈ [0 , , the operator ¯ B ( u, e ) is increasing when u ∈ [1 / √ , u ) and is decreasing when u ∈ ( u , . Especially, ∂∂u ¯ B ( u, e ) | u = u = B ( u ,e )( ψ − ψ ) ∂ ( ψ − ψ ) ∂u , when / √ ≤ u < u , B ( u ,e )( ψ − ψ ) ∂ ( ψ − ψ ) ∂u , when u < u ≤ . (4.44)33 s positive definite operator when u ∈ [1 / √ , u ) and is negative definite operator when u ∈ ( u , .(ii) For every eigenvalue λ u = 0 of ¯ B ( u , e ) with ω ∈ U for some ( u , e ) ∈ (1 / √ , × [0 , ,there hold dd u λ u | u = u > , when u ∈ (1 / √ , u ) , (4.45) and dd u λ u | u = u < u ∈ ( u , . (4.46) Proof. Note that B ( u ,e )( ψ − ψ ) is always a positive definite operator on D ( ω, π ). By direct com-putations, the denominator of d( ψ − ψ )d u is positive on the interval [1 / √ , √ 3] and the numerator of d( ψ − ψ )d u is a polynomial on Z [ u, √ u ] of degree 22. Note that d( ψ − ψ )d u | u =1 = 0. Furthermore, d( ψ − ψ )d u | u =0 . ≈ . d( ψ − ψ )d u | u = √ = − √ . By the numerical computations withthe step length √ − / √ , u = 1 is the only one root of d( ψ − ψ )d u = 0 in the interval [1 / √ , √ / √ ≤ u < u , d( ψ − ψ )d u > 0; when u < u ≤ d( ψ − ψ )d u < 0. Therefore,the eigenvalues of ¯ B ( u, e ) are not decreasing when 1 / √ ≤ u < u and the eigenvalues of ¯ B ( u, e )are not increasing when u < u ≤ 1. By the proof of Lemma 4.1, this lemma can be proved. Corollary 4.7. For every fixed e ∈ [0 , and ω ∈ U , the index function φ ω ( B ( u, e )) andconsequently i ω ( η u,e ) are non-decreasing as u increases from / √ to u and are non-increasing as u increase from u to . Especially, the index function φ ω ( B ( u, e )) satisfies φ ω ( B ( u, e )) ≥ φ ω ( B (1 / √ , e )) , when u ∈ [1 / √ , u ) ∪ (1 /u , √ , (4.47) φ ω ( B ( u, e )) ≥ φ ω ( B (1 , e )) , when u ∈ [ u , /u ] . (4.48)The proof of Corollary 4.7 is similar as the proof of Corollary 4.2. We omit it here. Theorem 4.8. (i) By (i) of Lemma 1.1, for any ( u, e ) ∈ [1 / √ , u ) × [0 , ˆ f ( ) − / ) , ( u, e ) ∈ (1 /u , √ × [0 , ˆ f ( ) − / ) , or ( u, e ) ∈ [ u , /u ] × [0 , , the operator B ( u, e ) is positive definitewith zero nullity on the space D (2 π, ω ) , i.e., i ω ( η u,e ) = φ ω ( B ( u, e )) = 0 , ν ω ( η u,e ) = ν ω ( B ( u, e )) = 0 . (4.49) Then all the eigenvalues of the matrix η u,e (2 π ) are hyperbolic, i.e., all the eigenvalues are not on U . (ii) By (ii) of Lemma 1.1, when ( u, e ) ∈ [1 / √ , √ × [0 , , the results of (i) hold. Since the proof of Theorem 4.8 is similar as the one of Theorem 4.4, we sketch the proof here.34 ketch of proof. By (4.47) and (i) of Theorem 3.3, we have that (4.49) holds when ( u, e ) ∈ [1 / √ , u ) × [0 , ˆ f ( ) − / ) and ( u, e ) ∈ (1 /u , √ × [0 , ˆ f ( ) − / ). By (4.48) and Theorem 3.2,(4.49) holds when ( u, e ) ∈ [ u , /u ] × [0 , u, e ) ∈ [1 / √ , √ × [0 , Proof of Theorem 1.2. Note that the fundamental solution of the linearized Hamiltonian system γ (2 π ) satisfies γ (2 π ) = γ (2 π ) ⋄ γ u,e (2 π ) ⋄ η u,e (2 π ). By (i) of Theorem 4.8 , η u,e (2 π ) possesses twopairs of hyperbolic eigenvalues when ( u, e ) ∈ [ u , /u ] × [0 , f ( ) − / > ˆ f ( β ) − / , i.e., γ u,e (2 π ) possesses two pairs of hyperbolic eigenvalues when( u, e ) ∈ (cid:0) (1 / √ , √ (cid:1) × [0 , ˆ f ( β ) − / ) and η u,e (2 π ) possesses two pairs of hyperbolic eigenvalueswhen ( u, e ) ∈ (cid:0) (1 / √ , u ) ∪ ( u , √ (cid:1) × [0 , ˆ f ( ) − / ), we have that γ u,e (2 π ) ⋄ η u,e (2 π ) possessesat least two pair of hyperbolic eigenvalues when ( u, e ) ∈ (cid:0) (1 / √ , u ) ∪ ( u , √ (cid:1) × [0 , ˆ f ( ) − / ).Then (i) of Theorem 1.2 holds.By (ii) of Theorem 4.4 and (ii) of Theorem 4.8, γ (2 π ) possesses four pair of eigenvalues when( u, e ) ∈ [1 / √ , √ × [0 , Acknowledgment. This paper is a part of my Ph.D. thesis. 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