Existence of the Broucke periodic orbit and its linear stability

J. Math. Anal. Appl. 389 (2012) 656–664

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Existence of the Broucke periodic orbit and its linear stability ✩ Duokui Yan School of Mathematics and System Sciences, Beihang University, Beijing 100191, China

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i n f o

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Article history: Received 20 October 2011 Available online 13 December 2011 Submitted by W. Sarlet

In this paper, we study the existence and linear stability of the Broucke periodic orbit in the planar three-body problem. In each period of this orbit, there are two binary collisions (or BC for short) between the outer bodies, while the inner body reaches its minimum or maximum at the time of each BC. A surprising simple existence proof of this orbit is given. The initial condition of this orbit is shown to be a supremum of some well-chosen set. The linear stability is then analyzed by Roberts’ symmetry reduction method. It is shown that the Broucke periodic orbit with equal masses is linearly stable. © 2011 Elsevier Inc. All rights reserved.

Keywords: Celestial mechanics Broucke periodic orbit Three-body problem Periodic solution with singularity Regularization

1. Introduction The planar three-body problem considers a system of three points with masses m1 , m2 , m3 in the same plane attracting each other only by Newtonian gravitational law. In this work, we study a special symmetric periodic orbit with masses 1, m, 1, which is called the Broucke periodic orbit later. Numerically in 1979, Broucke [2] found an amazing orbit with singularities in the equal mass planar three-body problem. In each period of this Broucke periodic orbit, there are two binary collisions (BC) between the two outer bodies, while the inner body reaches its minimum or maximum at the time of each BC. Actually, there are two other famous periodic orbits in the planar three-body problem: the Schubart orbit [18] and the figure-eight orbit [3,11]. Although these orbits were discovered and studied numerically decades ago, the mathematical research in their existences and linear stabilities has not had much progress until recently. In 1993, Moore found the figure-eight orbit numerically. Recently in 2000, Chenciner and Montgomery [3] gave a variational proof to the existence of this periodic orbit. They made a major breakthrough by considering minimization in some symmetric loop space. In 2007, Roberts [17] introduced a symmetry reduction method and gave a computer-assisted proof for the linear stability of figure-eight orbit. In the same year, Kapela and Simó [9] gave a different computer-assisted proof. Later in 2009, Hu and Sun [8] studied the stability from the point of view of index. They introduced the iteration theory of Maslov-type index [12] and built up some stability criteria for symmetric periodic orbits. Different from the figure-eight orbit, the existences of Schubart orbit and Broucke periodic orbit have more difficulties due to the collision singularity. In 2008, Venturelli [23] and Moeckel [10] proved the existence of the Schubart orbit where the outer masses are equal and the inner mass is arbitrary. In 2010, Shibayama [19] extended the variational approach to the case of arbitrary masses. Further, he claimed that the same variational approach works for the Broucke periodic orbit. For the importance and more properties of Schubart orbit and related topics, we refer to the works of Hénon [4,5], Hietarinta and Mikkola [6,7], Saito and Tanikawa [20], Sweatman [21,22] and Ouyang et al. [13–16]. ✩

Supported by NNSFC (No. 11101221). E-mail address: [email protected].

0022-247X/$ – see front matter doi:10.1016/j.jmaa.2011.12.024

©

2011 Elsevier Inc. All rights reserved.

D. Yan / J. Math. Anal. Appl. 389 (2012) 656–664

657

Fig. 1. The mass configuration.

Fig. 2. Broucke periodic orbit.

In this paper, we give a simple and direct existence proof to the Broucke periodic orbit. The initial condition of the differential system can be reduced to one parameter. To find the right initial value for the periodic orbit, we introduce a well-chosen set such that the supremum of this set exactly corresponds to the initial condition of the periodic orbit. In the second part of this paper, we study the linear stability of this orbit. A careful study shows that its symmetry group is isomorphic to D 2 . By applying Roberts’ symmetric reduction method, we prove that the Broucke periodic orbit with equal masses is linearly stable. The setup of this problem is introduced first. Without loss of generality, we assume the three bodies lying on the x-axis at the initial time t = 0. As in Fig. 1, the outer bodies have the same masses 1 and they locate symmetrically on both sides of mass m initially. Also, they have the same initial velocity which is perpendicular to x-axis. We number the three bodies by 1, 2, 3 from left to right. In each period of the Broucke periodic orbit, there are two binary collisions between the outer two bodies 1 and 3. At t = 0, the three masses locate in a line and have only vertical velocities. At t = t 0 , the outer two bodies collide and the inner body reaches its maximum. For t ∈ [t 0 , 2t 0 ], t ∈ [2t 0 , 3t 0 ] and t ∈ [3t 0 , 4t 0 ], the motion follows by symmetry. Fig. 2 shows a picture of the motion described above. At t = 0, we set the coordinate of the three bodies to be (−1, 0), (0, 0) and (1, 0) separately. In general, let body 3 have coordinate (x1 , x2 ) and body 1 have coordinate (−x1 , x2 ). We assume the center of mass is fixed at the origin, then the coordinate of body 2 is (0, −2x2 /m). As in Fig. 2, we set the initial velocity of body 2 to be positive. Then the equations of motion are

x¨1 = − x¨2 = −

mx1

[x21 + (1 + m2 )2 x22 ]3/2 (m + 2)x2 [x21 + (1 +

2 2 2 3/2 ) x2 ] m

−

1 4x21

(1.1)

,

(1.2)

,

with initial conditions

x1 (0) = 1,

x2 (0) = 0,

x˙ 1 (0) = 0,

x˙ 2 (0) = − v ,

(1.3)

where v > 0. This paper is organized as follows. In Section 2, the Hamiltonian of this system is regularized by a Levi–Civita type transformation and an appropriate scaling of time, which is adapted from Aarseth and Zare [1]. In Section 3, we consider

658


the regularized Hamiltonian and show the existence of the Broucke periodic orbit. The symmetry of this orbit with m = 1 is shown to be isomorphic to D 2 in Section 4 and then Roberts’s symmetry reduction method [17] is applied to study the linear stability. It turns out that the Broucke periodic orbit with equal masses is linearly stable. 2. Regularization The Hamiltonian of the system (1.1) and (1.2) in terms of {x1 , x2 , w 1 , w 2 } is

H=

1

w 21 +

4

w 22 1+

−

2 m

1 2x1

−

2m x21

+ (1 +

(2.1)

, 2 2 2 ) x2 m

2 where w 1 = 2x˙ 1 and w 2 = 2(1 + m )˙x2 . Since body 3 has coordinate (x1 , x2 ) and it always locates on the right side of the inner body by our setting, it follows that x1 0 at any time. There are two possible types of collision singularities for the system: binary collision and total collision. A binary collision between body 1 and 3 occurs if and only if x1 (t 1 ) = 0 and x2 (t 1 ) = 0 for some t 1 . A total collision occurs if x1 (t 2 ) = x2 (t 2 ) = 0 for some t 2 . Next we introduce a new transformation to regularize binary collisions. The Hamiltonian (2.1) is regularized by the Aarseth–Zare [1] method. Let Q 12 = x1 , P 1 = 2Q 1 w 1 . A new time variable s is introduced and it satisfies dt /ds = x1 . Then the regularized Hamiltonian Γ = x1 ( H − E ) has the form

Γ = x1 ( H − E ) =

1 16

P 12 +

1 4(1 +

2 ) m

Q 12 w 22 −

1 2

−

2m Q 12

[ Q 14

+ (1 +

2 2 2 1/2 ) x2 ] m

− E Q 12 ,

(2.2)

where E is the total energy of the Hamiltonian H . This regularized Hamiltonian gives the following differential equations:

Q 1 = x2 =

1

(2.3)

P 1,

8

m 2m + 4

P 1 = −

Q 12 w 2 ,

m 2m + 4

w 2 = −

(2.4)

w 22 Q 1 +

2 2 2 ) x2 m 2 2 2 3/2 ) x2 ] m

4m Q 1 (1 +

[ Q 14 + (1 +

+ 2E Q 1 ,

(2.5)

2 2 ) x2 m , 2 2 2 3/2 ) x2 ] m

2m Q 12 (1 +

[ Q 14 + (1 +

(2.6)

with initial conditions

Q 1 (0) = 1,

x2 (0) = 0,

P 1 (0) = 0,

w 2 (0) = −2 1 +

2 m

v,

(2.7)

where derivatives are with respect to s, and E is the total energy. Actually by the initial condition (1.3) at t = 0 and the formula of H (2.1), the total energy E is

E = E (0) = 1 +

2 m

v 2 − 2m −

1 2

.

3. Existence of the Broucke periodic orbit In this section, we show the existence of the Broucke periodic orbit. We first introduce a definition. Definition 3.1. We define sc to be the first time when Q 1 (sc ) = 0 and sm to be the first time when w 2 (sm ) = 0. If body 2 never has zero velocity, we define sm = +∞. Remark. When v = 0, the system has a total collision. In this case, sm = 0 and sc > 0. When v > 0, body 2 reaches its local maximum at s = sm and x2 (s) < 0 for s ∈ (0, sm ]. Note that at s = 0, the initial point is away from collision. Therefore, for any v 0, there is no total collision for s ∈ [0, sm ]. Lemma 3.2. sc = sc ( v ) is continuous in v for any v > 0. sm = sm ( v ) is continuous whenever Q 1 (sm , v ) = 0 and v > 0.


659

Proof. We first show sc is continuous with respect to v. In order to apply the implicit function theorem for Q 1 = Q 1 (sc , v ) = 0, we need to show that (∂ Q 1 /∂ s)(sc , v ) = 0. By the differential equation (2.3),

∂Q1 1 (sc , v ) = Q 1 (sc , v ) = P 1 (sc ). ∂s 8 Note that Γ = 0, at s = sc , we have

0=

1 16

P 1 (sc )2 −

√

1 2

.

Then P 1 (sc ) = ±2 2. Hence, ∂∂Qs1 (sc , v ) = 0. By the implicit function theorem, sc is a continuous function of v. Similarly, we consider w 2 = w 2 (sm , v ) = 0. From the differential equation (2.6), 2 2 2m Q 2 (1 + m ) x2 ∂ w2 (sm , v ) = w 2 (sm , v ) = − 4 1 . 2 2 ∂s [ Q 1 + (1 + m ) x22 ]3/2

When v > 0, x2 (sm , v ) < 0. Since Q 1 (sm , v ) = 0, it implies ∂∂ws2 (sm , v ) = 0. Then sm is also a continuous function of v when v > 0 and Q 1 (sm , v ) = 0. 2 Next, we show there exists some initial velocity v = v 1 > 0, such that sc ( v 1 ) < sm ( v 1 ). Intuitively, when v is big enough, there will be a bunch of collisions between body 1 and 3 before body 2 reaches its maximum. That is, sc < sm . The following lemma shows this fact analytically. Lemma 3.3. There exist a v 1 > 0, such that sc ( v 1 ) < sm ( v 1 ). Furthermore, sc ( v ) < sm ( v ) for any v v 1 . Proof. Consider the function P 1 Q 1 + 2x2 w 2 . By the differential equations (2.3) to (2.6),

( P 1 Q 1 + 2x2 w 2 ) = where E = Choose

1 8

P 12 + Q 12

(1 + m2 ) v 2 − 2m − 12 is the m(4m+3) , such that v1 = 2m+4

m 2m + 4

w 22 + 2E ,

total energy. E = 1. It implies that ( P 1 Q 1 + 2x2 w 2 ) 0 and the equality holds if and only if

P 1 = Q 1 = 0. Hence for any s1 ∈ (0, sc ( v 1 )],

s1 P 1 (s1 ) Q 1 (s1 ) + 2x2 (s1 ) w 2 (s1 ) = ( P 1 Q 1 + 2x2 w 2 ) ds > 0.

(3.1)

0

If s1 ∈ (0, sc ( v 1 )), body 1 and 3 haven’t had their first binary collision. It follows that x1 (s1 ) > 0, x2 (s1 ) < 0, w 1 (s1 ) = 2x˙ 1 (s1 ) < 0 for s1 ∈ (0, sc ( v 1 )). Then Q 1 (s1 ) > 0 and P 1 (s1 ) = 2Q 1 w 1 < 0. Note that x2 (s1 ) < 0, inequality (3.1) implies that w 2 (s1 ) < 0. If s1 = sc , Q 1 (s1 ) = 0. Then inequality (3.1) implies x2 (sc ( v 1 )) w 2 (sc ( v 1 )) < 0. By continuity, x2 (sc ( v 1 )) < 0, then w 2 (sc ( v 1 )) < 0. Hence,

w 2 ( s ) < 0,

for any s ∈ 0, sc ( v 1 ) .

That is, body 2’s velocity −mw+2 (2s) is positive for any s ∈ [0, sc ( v 1 )]. So body 2 hasn’t reached its first maximum yet at s = sc ( v 1 ). Therefore, sc ( v 1 ) < sm ( v 1 ). To prove the second part of this lemma, we note that for any v v 1 , inequality (3.1) always holds. A similar argument shows that sc ( v ) < sm ( v ) holds for any v v 1 . 2 In order to find the right initial condition v which generates the Broucke periodic orbit, we define

v 0 = sup v 0 sc ( v ) − sm ( v ) > 0 . Note that when v = 0, sc (0) > sm (0) = 0. It follows that 0 ∈ { v 0 | sc ( v ) − sm ( v ) > 0}. So the set { v 0 | sc ( v ) − sm ( v ) > 0} is not empty. By Lemma 3.3, this set has an upper bound v 1 . Then v 0 is well defined and v 1 v 0 0. Next we want to show that v 0 = 0 and it is exactly the initial condition we want. Remark. For any v ∈ { v 0 | sc ( v ) − sm ( v ) > 0}, body 1 and 3 haven’t had the first collision at time s = sm . It follows that Q 1 (sm ( v ), v ) > 0. By Lemma 3.2, both sc and sm are continuous for v ∈ { v 0 | sc ( v ) − sm ( v ) > 0} and v = 0. Lemma 3.4. v 0 = 0 and sc ( v 0 ) − sm ( v 0 ) = 0.

660


Proof. We first show that v 0 = 0 by contradiction. Suppose that v 0 = 0, then sm = 0 < sc . By continuity of these functions, there exists v c > 0 such that sm ( v c ) < sc ( v c ). But on the other hand we know that sm ( v ) > sc ( v ) for all v > 0, which is a contradiction. Hence, v 0 = 0. Next we show sc ( v 0 ) − sm ( v 0 ) = 0. If sc ( v 0 ) − sm ( v 0 ) > 0, since v 0 > 0, by continuity, there exists some ε > 0, such that sc ( v ) − sm ( v ) > 0 for any v ∈ [ v 0 − ε , v 0 + ε ]. It contradicts with the definition of v 0 ! If sc ( v 0 ) − sm ( v 0 ) < 0, we can get a contradiction similarly. Therefore, sc ( v 0 ) − sm ( v 0 ) = 0. 2 Remark. By the above lemma, we know that at v = v 0 , there exists a time sc = sm ≡ s0 , such that

Q 1 ( s 0 ) = 0,

w 2 ( s 0 ) = 0,

and

Q 1 (s) < 0,

w 2 (s) > 0 for any s ∈ (0, s0 ).

Note that x2 (s0 ) < x2 (0) = 0. Then for any s ∈ [0, s0 ], Q 14 + x22 > 0. Therefore, there is no total collision for s ∈ [0, s0 ]. It follows that the differential system (2.3) to (2.6) has analytic solutions for s ∈ [0, s0 ]. Theorem 3.5. When v = v 0 , the solution of the differential system (2.3) to (2.6) with initial condition (2.7) is the Broucke periodic orbit. Proof. The proof follows by the uniqueness of solution of ODE system. At time s = 0, the three bodies lie on x-axis and have the initial condition

Q 1 (0) = 1,

x2 (0) = 0,

P 1 (0) = 0,

w 2 (0) = −2 1 +

2

v 0.

m

At time s = s0 , body 2 reaches its maximum and body 1 and 3 has a collision,

Q 1 ( s 0 ) = 0,

x2 (s0 ) = − A 0 ,

√

P 1 (s0 ) = −2 2,

w 2 ( s 0 ) = 0,

where A 0 > 0. Compare the motion for s ∈ [0, s0 ] with the motion for s ∈ [s0 , 2s0 ]. The orbit from s = s0 to s = 2s0 and the orbit from s = 0 to s = s0 satisfy the following properties:

Q 1 (2s0 − s) = − Q 1 (s),

x2 (2s0 − s) = x2 (s),

P 1 (2s0 − s) = P 1 (s),

w 2 (2s0 − s) = − w 2 (s),

where 0 s s0 . By the uniqueness of solution of the regularized Hamiltonian system, the trajectory from s = s0 to s = 2s0 can be generated by the trajectory from s = 0 to s = s0 by symmetry. And at the time s = 2s0 when these three bodies lie on the same line again,

Q 1 (2s0 ) = −1,

x2 (2s0 ) = 0,

P 1 (2s0 ) = 0,

w 2 (2s0 ) = 2 1 +

2

v 0.

m

By symmetry and uniqueness again, at time s = 3s0 ,

Q 1 (3s0 ) = 0,

x2 (3s0 ) = A 0 ,

√

P 1 (3s0 ) = 2 2,

w 2 (3s0 ) = 0.

At time s = 4s0 ,

Q 1 (4s0 ) = 1,

x2 (4s0 ) = 0,

P 1 (4s0 ) = 0,

w 2 (4s0 ) = −2 1 +

2 m

v 0,

which coincides with the initial conditions at s = 0. Therefore when v = v 0 , the solution is a periodic orbit with period 4s0 and it is exactly the Broucke periodic orbit.

2

4. Linear stability In this section, we apply Roberts’ symmetry reduction method [17] to study the linear stability of the Broucke periodic orbit with m = 1. From Broucke’s work [2], the initial condition for the Broucke periodic orbit with m = 1 is

Q 1 (0) = 1,

x2 (0) = 0,

P 1 (0) = 0,

w 2 (0) = −3.5406.

In each period of this Broucke periodic orbit, the inner body keeps moving up and down and it reaches its minimum or maximum when the outer bodies collide.


661

Let z(t ) be a periodic solution with period T of the Hamiltonian system with Hamiltonian H , namely

z˙ (t ) = J H t , z(t ) ,

(4.1)

z(0) = z( T ), 0 I where J = − I 0n and I n is the identity matrix on Rn . The associated fundamental solution γ ≡ γz (t ) satisfies n

(4.2)

γ˙ (t ) = J H t , z(t ) γ (t ),

(4.3)

γ (0) = I 2n .

(4.4)

γ is a path in the set of symplectic group Sp(2n) = M ∈ G L (2n) M T J M = J ,

It is well known that

and γ ( T ) is called the Monodromy matrix, or the linear Poincaré map. Solution z is called linear stable if γ ( T )k is bounded for all k ∈ N and spectral stable (or elliptic) if all the eigenvalues of γ ( T ) are on U , the unit circle in the complex plane C . 4.1. Symmetry of the Broucke periodic orbit Let z(s) = ( Q 1 (s), x2 (s), P 1 (s), w 2 (s)) T be the periodic solution of the Hamiltonian Γ (2.2) corresponding to the Broucke periodic orbit. To consider the linear stability of it, we first show that the Broucke periodic orbit has a symmetry group isomorphic to D 2 . Lemma 4.1.

z(s + T /2) = S 1 z(s), where

⎡

−1

⎢ 0

S1 = ⎢ ⎣

0 0

z( T − s) = R 1 z(s),

⎤

0 0 0 −1 0 0 ⎥ ⎥, 0 −1 0 ⎦ 0 0 −1

⎡

1 ⎢0 ⎢ R1 = ⎣ 0 0

(4.5)

⎤

0 0 0 −1 0 0 ⎥ ⎥. 0 −1 0 ⎦ 0 0 1

Proof. The proof follows by the uniqueness of solution of the ODE system. Consider the regularized Hamiltonian Γ and its corresponding differential system (2.3) to (2.6). Note that

T − Q 1 ( T /2 + s), −x2 ( T /2 + s), − P 1 ( T /2 + s), − w 2 ( T /2 + s) = S 1 z(s + T /2) = S 1−1 z(s + T /2) = − z(s + T /2).

It is easy to check that − z(s + T /2) satisfies the differential system (2.3) to (2.6). Also, the initial condition is S 1 z( T /2) = −z( T /2) = z(0). Therefore, by uniqueness we obtain S 1−1 z(s + T /2) = z(s), i.e. z(s + T /2) = S 1 z(s). Similarly we have z ( T − s ) = R 1 z ( s ). 2 Remark. The matrix S 1 and R 1 satisfy S 12 = R 21 = ( S 1 R 1 )2 = I . So the Broucke periodic orbit has a symmetry group isomorphic to the dihedral group D 2 . 4.2. A good basis Let S = − S 1 R 1 = R 1 . Then

z(−s + T ) = R 1 z(s) = S z(s),

z(−s + T /2) = − S z(s).

By the symmetry reduction in Roberts’ work [17], the Monodromy matrix

γ ( T ) satisfies

2 γ ( T ) = S γ ( T /4)−1 S γ ( T /4) . Let D 1 = γ ( T /4)−1 S γ ( T /4), then

γ ( T ) = ( S D 1 )2 , where S 2 = D 21 = I . We have reduced the stability analysis to the first quarter of the periodic orbit. Let Y (s) be the fundamental matrix solution to the linearized equation (4.3) with arbitrary initial conditions Y (0) = Y 0 . Then the matrix solution Y (s) satisfies

662


Y (s) = γ (s)Y 0 . Consequently, the Monodromy matrix To study the eigenvalues of

γ ( T ) = Y ( T )Y 0−1 . Note that Y 0−1 Y ( T ) is similar to γ ( T ) = Y ( T )Y 0−1 .

γ ( T ), it is equivalent to consider the eigenvalues of Y 0−1 Y ( T ).

Let B = Y ( T /4). By Roberts’ work [17], the matrix Y 0−1 Y ( T ) satisfies

Y 0−1 Y ( T ) =

Y 0−1 S Y 0 B −1 S B

2

.

Then the question of stability reduces to showing that the eigenvalues of

W = Y 0−1 S Y 0 B −1 S B are on the unit circle. An appropriate choice of Y 0 will simplify the factor Y 0−1 S Y 0 in W . Set

Λ=

I 0 0 −I

.

Lemma 4.2. There exists Y 0 such that 1. Y 0 is orthogonal and symplectic, and 2. Y 0−1 S Y 0 = Λ. Proof. By the differential equations (2.3) to (2.6) and initial conditions (2.7) of Q 1 , x2 , P 1 and w 2 , we can find out

z (0) = 0 a b

0

T

,

where a = − v 0 = −0.5901, b = −5, and c = of Y 0 to be

1

col3 (Y 0 ) = z (0)/ z (0) =

c

√

a2 + b2 . Let coli (Y 0 ) denote the i-th column of Y 0 . Choose the third column

0 a b

0

T

,

and define

col1 (Y 0 ) = J · col3 (Y 0 ) =

1 c

b

T

0 0 −a

.

Choose col4 (Y 0 ) such that it is orthogonal to col3 (Y 0 ), and it is in the eigenspace of S with respect to its eigenvalue −1, which is

span

0

1

0

0

T T . , 0 0 1 0

We define

col4 (Y 0 ) =

1 c

0 −b

a

0

T

and

col2 (Y 0 ) = J · col4 (Y 0 ) = Then the matrix

⎡

1 c

a

0

0 b

T

.

⎤

b 1⎢ 0 ⎢ Y0 = ⎣ 0 c −a

a 0 0 0 a −b ⎥ ⎥ 0 b a ⎦ b 0 0

is both symplectic and orthogonal and it satisfies Y 0−1 S Y 0 = Λ. Theorem 4.3. Let D = B −1 S B, Λ = Y 0−1 S Y 0 =

1 2

W + W −1 =

KT 0

0 K

.

Furthermore, the first column of K is [−1 0] T .

I 0 0 −I

2

and W = Λ D. There exists a 2 × 2 matrix K such that

(4.6)


Proof. Since W = Λ D where Λ = Y 0−1 S Y 0 =

KT 0 0 K

663

, D = B −1 S B, and Λ2 = D 2 = I , it follows that

W −1 = D Λ. Because B is a symplectic matrix, a short computation using the formula for the inverse of a symplectic matrix shows that D has the form

KT −L2

L1 −K

for 2 × 2 matrices K , L 1 , L 2 . It follows that

W =

I 0 0 −I

and

W

−1

=

KT −L2

L1 −K

Hence,

1 2

KT −L2

W + W −1 =

L1 −K

I 0 0 −I

KT 0

0 K

=

KT L2

=

L1 K

KT −L2

,

−L1

K

.

.

Next we show that the first column of K is [−1 0] T . By Roberts’ work [17], S Y 0 B −1 S B = Y ( T /2), where B = Y ( T /4). Since Y 0−1 = Y 0T , it follows that

W = Y 0−1 S Y 0 B −1 S B = Y 0−1 Y ( T /2) = Y 0T Y ( T /2). Set v = Y 0−1 z (0). By the choice of the matrix Y 0 ,

⎡

⎤

0 ⎢ 0 ⎥ −1 T ⎥ v = Y 0 z (0) = Y 0 z (0) = ⎢ ⎣ z (0) ⎦ = z (0) e 3 . 0 Because z (s) is a solution to the linearized equation ξ˙ = J D 2 Γ ( z(s))ξ and z (0) = Y (0)Y 0−1 z (0), then z (s) = Y (s)Y 0−1 z (0) = Y (s) v for all s. Hence,

W v = Y 0T Y ( T /2) v = Y 0T z ( T /2).

(4.7)

Since z(s) satisfies z(s + T /2) = − z(s) for all s, it follows that

z (s) = − z (s + T /2). Setting s = 0 in this gives z (0) = − z ( T /2), and consequently that

Y 0T z ( T /2) = −Y 0T z (0) = −Y 0−1 z (0) = − v .

(4.8)

Eqs. (4.7) and (4.8) now combine to show that W v = − v. Hence −1 is an eigenvalue of W and e 3 is an eigenvector for W corresponding to this eigenvalue. Then the first column of K is as claimed. 2 By the formula for the inverse of a symplectic matrix and the definition of D, the matrix K has the form

K=

−1 c 1T ( S J c 4 ) 0

c 2T ( S J c 4 )

,

where c i is the i-th column of B = Y ( T /4). The next result is adopted from Roberts’ work [17]. It tells us the linear stability of the Broucke periodic orbit is equivalent to show that the eigenvalues of K are real and have absolute value smaller than or equal to 1. Lemma 4.4. For a symplectic matrix W , suppose there is a matrix K such that

1 2

W +W

−1

=

KT 0

0 K

.

Then W is stable if and only if all of the eigenvalues of K are real and have absolute value smaller than or equal to 1. Therefore, the linear stability problem reduces to the comparison of an absolute value |c 2T ( S J c 4 )| and 1, where S = diag{1, −1, −1, 1}, and c i is the i-th column of B = Y ( T /4).

664


5. Numerical calculations The initial conditions for the Broucke periodic orbit are

Q 1 (0) = 1,

x2 (0) = 0,

P 1 (0) = 0,

w 2 (0) = −3.5406

with a period T satisfying T /4 = 3.556, and energy E ≈ −1.4553. Using MATLAB and a Runge–Kutta–Fehlberg algorithm, we computed the columns of the matrix Y ( T /4) with an absolute error tolerance of 5 × 10−3 . From this we got

c 2T ( S J c 4 ) = 0.9884. For the Broucke periodic orbit, the rigorous estimate of the eigenvalue c 2T ( S J c 4 ) of K and its distinctiveness from the eigenvalue 1 of K combine with Lemma 4.4 to give the following stability result. Theorem 5.1. The Broucke periodic orbit in the collinear equal mass three-body problem is linearly stable. The symmetry reductions used to compute the eigenvalues over just one-fourth of the period and the rigorous estimate of c 2T ( S J c 4 ) showing that it is clearly between −1 and 1, assures the linear stability of the Broucke periodic orbit. Acknowledgments The author is pleased to acknowledge several valuable conversations with Prof. Yiming Long and Prof. Tiancheng Ouyang on these and related topics. He is also indebted to all the referees for reading the original manuscript and suggesting improvements.

References [1] S.J. Aarseth, K. Zare, A regularization of the three-body problem, Celestial Mech. 10 (1974) 185–205. [2] R. Broucke, On the isosceles triangle configuration in the planar general three body problem, Astron. Astrophys. 73 (1979) 303–313. [3] A. Chenciner, R. Montgomery, A remarkable periodic solution of the three-body problem in the case of equal masses, Ann. of Math. 152 (2000) 881–901. [4] M. Hénon, A family of periodic orbits of the planar three-body problem, and their stability, Celestial Mech. Dynam. Astronom. 13 (1976) 267–285. [5] M. Hénon, Stability and interplay motions, Celestial Mech. Dynam. Astronom. 15 (1977) 243–261. [6] J. Hietarinta, S. Mikkola, Chaos in the one-dimensional gravitational three-body problem, Chaos 3 (1993) 183–203. [7] S. Mikkola, J. Hietarinta, A numerical investigation of the one-dimensional Newtonian three-body problem, III, Celestial Mech. Dynam. Astronom. 51 (1991) 379–394. [8] X. Hu, S. Sun, Index and stability of symmetric periodic orbits in Hamiltonian system with application to figure-eight orbit, Comm. Math. Phys. 290 (2009) 737–777. [9] T. Kapela, C. Simó, Computer assisted proofs for nonsymmetric planar choreographies and for stability of the eight, Nonlinearity 20 (2007) 1241–1255. [10] R. Moeckel, A topological existence proof for the Schubart orbits in the collinear three-body problem, Discrete Contin. Dyn. Syst. Ser. B 10 (2008) 609–620. [11] C. Moore, Braids in classical dynamics, Phys. Rev. Lett. 70 (1993) 3675–3679. [12] Y. Long, Index Theory for Symplectic Paths with Applications, Birkhäuser, Basel, Boston, Berlin, 2002. [13] T. Ouyang, D. Yan, Periodic solutions with alternating singularities in the collinear four-body problem, Celestial Mech. Dynam. Astronom. 109 (2011) 229–239. [14] T. Ouyang, S. Simmons, D. Yan, Periodic solutions with singularities in two dimensions in the n-body problem, preprint, arXiv:0811.0227. [15] L. Bakker, T. Ouyang, G. Roberts, D. Yan, S. Simmons, Linear stability for some symmetric periodic simultaneous binary collision orbits in the four-body problem, Celestial Mech. Dynam. Astronom. 108 (2010) 147–164. [16] L. Bakker, T. Ouyang, D. Yan, S. Simmons, Existence and stability of symmetric periodic simultaneous binary collision orbits in the planar pairwise symmetric four-body problem, Celestial Mech. Dynam. Astronom. 110 (2011) 271–290. [17] G. Roberts, Linear stability analysis of the figure-eight orbit in the three-body problem, Ergodic Theory Dynam. Systems 27 (2007) 1947–1963. [18] J. Schubart, Numerische aufsuchung periodischer Lösungen im dreikörperproblem, Astron. Nachr. 283 (1956) 17–22. [19] M. Shibayama, Minimizing periodic orbits with regularizable collisions in the n-body problem, Arch. Ration. Mech. Anal. 199 (2010) 821–841. [20] M. Saito, K. Tanikawa, The rectilinear three-body problem using symbol sequence, II. Role of the periodic orbits, Celestial Mech. Dynam. Astronom. 103 (2009) 191–207. [21] W. Sweatman, Symmetrical one-dimensional four-body problem, Celestial Mech. Dynam. Astronom. 82 (2002) 179–201. [22] W. Sweatman, A family of symmetrical Schubart-like interplay orbits and their stability in the one-dimensional four-body problem, Celestial Mech. Dynam. Astronom. 94 (2006) 37–65. [23] A. Venturelli, A variational proof for the existence of Von Schubart’s orbit, Discrete Contin. Dyn. Syst. Ser. B 10 (2008) 699–717.