NEW PERIODIC ORBITS IN THE PLANAR EQUAL-MASS THREE-BODY PROBLEM

. It is known that there exist two sets of nontrivial periodic orbits in the planar equal-mass three-body problem: retrograde orbit and prograde orbit. By introducing topological constraints to a two-point free boundary value problem, we show that there exists a new set of periodic orbits for a small interval of rotation angle θ .


1.
Introduction. The planar three-body problem studies the motion of three masses m 1 , m 2 , m 3 moving in a fixed plane satisfying Newton's law of gravitation: where q i = (q ix , q iy ), a row vector in the xy plane, is the position of m i , q =   q 1 q 2 q 3   is a 3 × 2 matrix and the kinetic energy K and the potential energy U are given as follows: It is known that in the planar equal-mass three-body problem, there exist two families of orbits [1,7]: retrograde orbit and prograde orbit. "A retrograde orbit of the planar three-body problem is a relative periodic solution of the Newtonian equation (1) with two adjacent masses revolving around each other in one direction while their mass center revolves around the third mass in the other direction. The orbit is said to be prograde if both revolutions follow the same direction." [3] Actually, Chen [2,3] introduced a level estimate argument and showed the existence of such two families of orbits for a general mass set M = [m 1 , m 2 , m 3 ] = [1, 1, m].
The purpose of this paper is to show the existence of a new set of orbits in the planar equal-mass three-body problem. A sample picture of these orbits is given in Fig. 2. We first introduce some notations before stating our main results.
Let the masses be m 1 = m 2 = m 3 = 1. We consider a two-point free boundary value problem with topological constraints. Let where a 1 ≥ 0, a 2 ≥ 0, b 1 ≥ 0, b 2 ∈ R, and R(θ) = cos(θ) sin(θ) − sin(θ) cos(θ) . The two configuration sets are defined as follows: where Q s and Q e are defined by the matrices in (2). Geometrically, the configuration Q S is on a horizontal line with order q 2x (0) ≤ q 1x (0) ≤ q 3x (0). The configuration Q E is an isosceles triangle with q 1 as its vertex, and the symmetry axis of each Q e in (3) is a counterclockwise θ rotation of the x−axis. Furthermore, in the configuration Q e · R(−θ), body 2 is always above the x−axis. Pictures of the two configurations Q S and Q E are shown in Fig. 1 respectively. Figure 1. The configurations Q S and Q E are shown, where blue dots represent q 1 , red dots represent q 2 and black dots represent q 3 . In Q S , three masses are on the x−axis with an order q 2x ≤ q 1x ≤ q 3x . In Q E , three masses form an isosceles triangle, whose symmetry axis is a counterclockwise θ rotation of the x−axis. q 1 is on the symmetry axis and q 2 is above the symmetry axis in Q E .

Remark 1.
In the equal-mass three-body problem, there is a remarkable periodic orbit, figure-eight orbit, which has been shown to exist [4] by Chenciner and Montgomery in 2000. Indeed, one twelveth of the figure-eight orbit can be characterized as an action minimizer connecting two specific boundary sets: a horizontal Euler configuration set and an isosceles configuration set

NEW PERIODIC ORBITS IN THE PLANAR THREE-BODY PROBLEM 2189
However, the two boundaries Q S1 and Q E1 do NOT satisfy our boundary conditions: Q S and Q E in (3). Actually in the figure-eight orbit [4], the vertex of the isosceles triangle in Q E1 corresponds to the first body of the Euler configuration set Q S1 , which lies either on the left or on the right end of a horizontal line in Q S1 . While in our case, the vertex of the isosceles triangle in Q E corresponds to the second body of the collinear configuration set Q S (as in Fig. 1), which lies in the middle of a horizontal line in Q S . Furthermore, the angular momentum of the figure-eight orbit is 0, but the angular momenta in our case turn out to be nonzero. Therefore, the orbits in this paper are different from the well-known figure-eight orbit.
Without loss of generality, we assume the center of mass to be at the origin. That is, q ∈ χ, where Let a = (a 1 , a 2 , b 1 , b 2 ). We define the set Λ to be Given θ = π/10 and a ∈ Λ, the position matrices Q s and Q e in (2) are fixed. We set P (Q s , Q e ) to be the path set in the Soblev space connecting the two fixed ends Q s and Q e : It is known that there exists an action minimizer P connecting the two fixed ends, which satisfies In general, the minimizer P is not a part of a periodic solution. In order to find a periodic or quasi-periodic solution, we consider the following free boundary value problem: For θ = π/10, standard results imply the existence of an action minimizer P 0 ∈ H 1 ([0, 1], χ), which minimizes A in (6). However, P 0 may not be a solution of the Newtonian equation (1) since it could have collision singularities. The main difficulty in this paper is to exclude possible collisions in P 0 . By Marchal [9] and Chenciner [5], P 0 is collision-free in (0, 1). To prove P 0 is a solution of the Newtonian equation, we only need to exclude possible boundary collisions in P 0 . Besides that, we also need to show that P 0 is nontrivial, which means that it does not coincide with a relative equilibrium. By applying Chen's level estimate argument [2,3,6] and introducing a new test path, we can show that Theorem 1.1. For θ = π/10, there exists a nontrivial and collision-free minimizing path P 0 ≡ P 0 (t ∈ [0, 1]), which satisfies The minimizer P 0 can be extended to a periodic orbit.
The proof of Theorem 1.1 can be found in Theorem 4 and Theorem 5.1.
where B = 1 0 0 −1 and R(π/5) is the rotation matrix defined in (2). It implies that this orbit has a symmetry group D 10 . However, P 0 is not the action minimizer under the symmetry constraint D 10 since it is different from the retrograde orbit. In fact, P 0 can be seen as a local action minimizer under the D 10 symmetry group constraint. A picture in Fig. 2 shows P 0 and its periodic extension.  A picture of the periodic orbit extended by the action minimizer P 0 with θ = π/10. The three dots represent the starting configuration Q S , and the three crosses represent the ending configuration Q E of P 0 . In the graph, the red curve is the trajectory of body 2, the blue curve is for body 1 and the black curve is for body 3.
Actually, the argument in Theorem 1.1 can be extended to the case when θ is close to π/10. Recall that the two boundary sets Q S and Q E in (2) are where with R(θ) defined in (2).
By the definition of Q S and Q E in (8), the action minimizer corresponding to a given θ could be a part of an Euler orbit, which is a relative equilibrium. Besides the difficulty of excluding collisions in the action minimizer, another challenge is to show that this minimizer is not a part of a relative equilibrium. By carefully choosing the test paths, the following theorem holds, while its proof is partially computer assisted and can be found in Theorem 6.1. Rough numerical simulations indicate that these solutions are dynamically unstable. It will be very interesting if one can understand their stabilities mathematically. Theorem 1.2. For each θ ∈ [0.084π, 0.183π], there exists a nontrivial and collisionfree minimizing path P 0, θ ≡ P 0, θ (t ∈ [0, 1]) connecting the two configuration sets Q S and Q E in (3), and it can be extended to a periodic or quasi-periodic orbit.
The paper is organized as follows. Section 2 shows the coercivity of the action functional under our topological constraints. Section 3 defines the test path for θ = π/10, while θ represents the angle between the symmetry axis of Q E and the x−axis as shown in Fig. 1. In Section 4, a lower bound of the action A(P 0 ) is given if the minimizer P 0 has some collision singularities. Section 5 extends the minimizer P 0 to a periodic orbit for θ = π/10. In the end, Section 6 applys the argument in Sections 3 and 4 to an interval of θ and proves the existence of periodic or quasi-periodic orbits for θ ∈ [0.084π, 0.183π].
2. Topological constraint and coercivity. In this section, we introduce a general coercivity result and apply it to our case, which implies the existence of a minimizer P 0 satisfying Furthermore, if one of the boundaries of P 0 has no collision, we show that it can be extended.
where q i ∈ R d (i = 1, 2, . . . , N, d = 1, 2, or 3) are row vectors, and Q s , Q e ∈ χ. The variational argument is a two-step minimizing procedure. First, we consider a fixed-end boundary value problem, which is also known as the Bolza problem. For given values of a 1 , . . . a k and b 1 , . . . b s , the two matrices Q s and Q e are fixed. There exists an action minimizer P, which satisfies where K(q(t)) is the standard kinetic energy, U (q(t)) is the standard potential energy, and P(Q s , Q e ) is defined as follows: If one wants P to be a part of a periodic solution, the boundaries must be special. Hence, we introduce another minimizing procedure. Instead of fixing the boundaries, we free several parameters on the boundaries q(0) = Q s and q(1) = Q e . The Lagrangian action functional is then minimized over these parameters. The resulting minimizing path may be extended to a periodic or quasi-periodic solution. A general coercivity theorem [11] is introduced here to show the existence of the minimizer in a subset of H 1 ([0, 1], χ). This coercivity result is standard, while its proof basically follows by Arzelá-Ascoli theorem and can be found in [8,11]. Similar coercivity results can also be found in [2].
S. Assume that both Q s α ∈ R k and Q e β ∈ R s are linear spaces and their intersection satisfies: Then there exist a path sequence {P n l } and a minimizer P 0 in H 1 ([0, 1], χ), such that for each n l , . . , a kn l ) and β n l = (b 1n l , . . . , b sn l ). For t ∈ [0, 1], P n l (t) converges to P 0 (t) uniformly. In particular, As its application, we show that the two configurations Q s and Q e defined in (2) satisfy the assumptions in Theorem 2.1. Let a = (a 1 , By Theorem 2.1, there exists an action minimzier P 0 ∈ H 1 ([0, 1], χ) and a vector a 0 ∈ Λ, such that Let q(t) (t ∈ [0, 1]) be the position matrix path of P 0 . We then show that if one of the ends q(0) or q(1) of P 0 is collision-free, the path P 0 = P 0 ([0, 1]) can be extended.
Proof. The proof mainly follows by the first variation formulas. If q(0) = Q s ( a 0 ) has no collision, by the first variation formulas, the velocities q i (0) (i = 1, 2, 3) must satisfyq

NEW PERIODIC ORBITS IN THE PLANAR THREE-BODY PROBLEM 2193
Then the extension of q(t) (t ∈ [0, 1)) can be defined as follows where B = 1 0 0 −1 . It is easy to check that q(t) in (11) is smoothly connected at t = 0. Hence, by the uniqueness of solution of initial value problem in an ODE system, If q(1) = Q e ( a 0 ) has no collision, by the first variation formulas, the velocities q i (1) (i = 1, 2, 3) satisfẏ where B is as in (11) and R(2θ) is as in (2). Then the extension of q(t) (t ∈ [0, 1]) can be defined as follows It is clear that q(t) in (12) is The proof is complete.
3. An upper bound of A (P 0 ). Let where Q S and Q E are given by (8). By choosing a test path in P(Q S , Q E ) and calculating its action, we can find an upper bound of A (P 0 ) in the case when θ = π/10. This test path is basically a continuous and piecewise linear function, which can be seen as a linear approximation of the minimizing path P 0 . Let be the position matrix of the minimizer P 0 = P 0,π/10 . Approximated values of q i,k = q i (k/10) (i = 1, 2, 3; k = 0, 1, 2, . . . , 10) are given as follows: . This path is defined by connecting the adjacent points q i,k and q i,(k+1) , (i = 1, 2, 3, k = 0, 1, . . . , 9) with straight lines, where the positionsq i (t) (i = 1, 2, 3) satisfỹ It follows that the test path P test is a continuous and piecewise smooth function.
It implies that the action of P test is an upper bound of the minimum action A (P 0 ), that is The following lemma shows that the minimum action value A (P 0 ) is less than 3.964.

Exclusion of collisions.
In this section, we show that The proof of this theorem follows by Lemma 4.3, Corollary 1 and Lemma 4.5. Note that, by the works of Marchal and Chenciner, the action minimizer P 0 has no singularity in (0, 1). The upper bound 3.964 of A (P 0 ) in Section 3 will be used to exclude the possible collisions on the two boundaries q(0) and q(1) in P 0 = P 0 ([0, 1]). We assume that in the minimizing path P 0 , q(0) and q(1) are where θ = π/10, a 10 ≥ 0, a 20 ≥ 0, b 10 ≥ 0 and b 20 ∈ R. The lower bound estimate of collision paths is based on the following result, which is introduced by Chen [2, 3]. Give any θ ∈ (0, π], T > 0, consider the following path spaces: The symbol < ·, · > stands for the standard scalar product in R 2 and | · | represents the standard norm in R 2 . Define the Keplerian action functional I µ,α,T : Note that the center of mass is assumed to be at the origin, it follows that the action A can be rewritten as Let 4.1. Exclusion of triple collisions. The following result implies that whenever there is a triple collision in P 0 , its action must be greater than 3.964. Therefore, P 0 has no triple collision.
Proof. If P 0 has a triple collision, by Lemma 4.2, we have Hence, the lower bound of action for paths with triple collision is about 6.6927, which is greater than 3.964. By Lemma 3.1, P 0 must have no triple collision. The proof is complete.

Exclusion of binary collisions at
where b 1 ≥ 0, b 2 ∈ R, and R(θ) = cos(θ) sin(θ) − sin(θ) cos(θ) . Let θ = π/10. To show P 0 has no collision at t = 1, we only need to exclude the binary collision between body 2 and body 3. We will give a lower bound for the action of orbits with binary collision between body 2 and body 3 at t = 1: q 2 (1) = q 3 (1). Proof. By assumption, bodies 2 and 3 collide at t = 1. By Lemma 4.2, In the minimizing path P 0 , the position matrix is given by with b 10 ≥ 0. Note that we assume bodies 2 and 3 collide at t = 1, it implies that b 10 = 0. If meanwhile b 20 = 0, it is a total collision. By Lemma 4.3, it is not possible for P 0 . It follows that b 10 = 0 and b 20 = 0.
Next, we find a lower bound of A 12 + A 13 for P 0 with a binary collision between  Proof. Note that bodies 2 and 3 will not collide at t = 0 or t = 1. It is clear that in P 0 , the vector − − → q 3 q 2 = q 2 − q 3 rotates at least 2π/5. By Lemma 4.2, In order to find a good lower bound for the collision path, we will apply the extension formulas in Lemma 2.2 of Section 2. By Corollary 1, there is no singularity on q(1) of P 0 . It follows that the orbit can be extended to the time period t ∈ [0, 2]. And the action value for t ∈ [0, 2] is 2A(P 0 ). If at t = 0, bodies 1 and 2 collide, that is a 10 = 0. By the extension formula (12) in Lemma 2.2, it implies that at t = 2, bodies 1 and 3 collide. Similarly, the binary collision between bodies 1 and 3 at t = 0 implies a binary collision between bodies 1 and 2 at t = 2. Therefore, by Lemma 4.2, It follows that A(P 0 ) ≥ 4.0219 > 3.964. By Lemma 3.1, P 0 has no collision singularity at t = 0. The proof is complete. 4.4. P 0 is nontrivial. An orbit is called nontrivial if it does not coincide with a relative equilibrium in the N-body problem. At the end of Section 4, we show that P 0 is not a part of a relative equilibrium. By the definition of Q S and Q E in (8) and (9), the only possible relative equilibrium is an Euler orbit. In this case, the isosceles configuration in Q E degenerates to an Euler configuration. And from a straight line in Q S to an Euler configuration in Q E , this Euler orbit rotates 2π/5. It follows that the corresponding action of this part of the Euler orbit is By Lemma 3.1, P 0 is nontrivial.

5.
Extension to a periodic orbit. In Section 4, we show that the minimizing path P 0 = P 0 ([0, 1]) is collision-free. The extension of P 0 is shown by the first variation formulas and the uniqueness of solution of initial value problem in an ODE system.
Proof. By Lemma 2.2, P 0 = P 0 ([0, 1]) can be extended to t ∈ [0, 2]. The extension is defined as follows where B = 1 0 0 −1 and R(2θ) is the rotation matrix defined in (2). It is clear that for t ∈ [0, 2], the path P 0 ([0, 2]) in (28) is smooth. Note that by the first variation formulas, at t = 0, the velociteq i (0) of the minimizing path P 0 satisfẏ Then one can extend the path P 0 ([0, 2]) to t ∈ [0, 4]: It is easy to check that the extension in (29) is smooth. Furthermore, at t = 4, By the uniqueness of solution of initial value problem in an ODE system, the position 4]) can be extended smoothly to any t ∈ R: Note that θ = π/10. By taking n = 5, we have q(t + 20) = q(t). Hence, q(t) is periodic. The proof is complete.
Remark 3. The periodic orbit (Fig. 2) extended by P 0 ≡ P 0,π/10 has a nonzero angular momentum J: The initial condition of this orbit is By running the above initial condition in the ode solver ODE45 of Matlab, this orbit looks unstable numerically since it breaks the periodic shape when t = 40. However, we have not investigated its stability rigorously.
6. Properties of the action minimizer for θ ∈ [0.084π, 0.183π]. In this last section, we will follow the idea in Section 4 and show that the action minimizer P 0 = P 0, θ is nontrivial and collision-free for θ ∈ [0.084π, 0.183π].  (3), and it can be extended to a periodic or quasi-periodic orbit.
For each given θ 0 , let q(t) =   q 1 q 2 q 3   (t) be the position matrix path of the minimizer   (t) be the position matrix path of P test,θ . We can then define a test path P test,θ by connecting the following 11 points: In fact,q(t) satisfies where i = 0, 1, . . . , 9. It is easy to check thatq(0) ∈ Q S andq(1) ∈ Q E , where Q S and Q E are the boundary configuration sets defined in (3). Once the values of q i 10 (i = 0, 1, . . . , 10) in P 0,θ0 are given, the action of the test path A test = A (P test,θ ) can be calculated accurately as in the proof of Lemma 3.1. For readers' convenience, the data of the 7 test paths and the corresponding figures of action values are given in Appendix A.
Note that for a given set of 11 interpolation points, formula (16) implies that the action of the text path A test = A(P test,θ ) is a smooth function with respect to θ. In the last step, we compare the value of the two smooth functions: A test = A(P test,θ ) and min {f 2 (θ), g(θ)} in different intervals of θ as in Fig. 3. To do so, we calculate the value of the two functions for θ ∈ [0.084π, 0.183π] with a step π × 10 −6 .
In order to compare the two functions for every θ, a linear interpolation method is introduced. Actually, we use straight lines to replace the real curves of the two functions in each step. The error of this linear interpolation is 1 8 (π × 10 −6 ) 2 ∆, where ∆ is the maximum of the second derivative of the corresponding function. For θ ∈ [0.084π, 0.183π], it turns out that ∆ ≤ 20 π for both functions. It implies that the error is bounded by By (33) and (34), it follows that Numerically, for θ ∈ [0.084π, 0.183π], the minimum value of min{f 2 (θ), g(θ)}−A test in Fig. 3 is 3.08 × 10 −4 > 7.85 × 10 −12 . Therefore, for each given θ ∈ [0.084π, 0.183π], the action of the test path A test satisfies A test < min {f 2 (θ), g(θ)} .
It follows that the action minimizer P 0,θ is nontrivial and collision-free when θ ∈ [0.084π, 0.183π]. The proof is complete. In summary, this paper studies a new set of periodic orbits in the planar equalmass three-body problem. For each given θ ∈ [0.084π, 0.183π], the orbit can be characterized as a local action minimizer connecting a collinear configuration and an isosceles configuration. Topological constraints are introduced on the two boundary configurations as in Fig. 1, and the orbit generated by the corresponding minimizer is different from the retrograde orbit, the prograde orbit in [3] and the well-known figure-eight orbit [4].
The topological constraints introduced in this paper are simple and have clear geometric meanings, which can be extended to study periodic orbits in four-body or five-body problem. used in this paper is created by Tiancheng Ouyang. We are grateful for his endless help and support. We thank all the referees for carefully checking the paper and appreciate their helpful comments. D. Yan is partially supported by NSFC (No. 11432001) and the China Scholarship Council. Part of this work is done when D. Yan is visiting Brigham Young University. He really appreciates the support from the department of mathematics at BYU.
Appendix A. Data of the 7 test paths.