VARIATIONAL PROOF OF THE EXISTENCE OF BRAKE ORBITS IN THE PLANAR 2 -CENTER PROBLEM

. The restricted three-body problem is an important subject that deals with significant issues referring to scientific fields of celestial mechanics, such as analyzing asteroid movement behavior and orbit designing for space probes. The 2-center problem is its simplified model. The goal of this paper is to show the existence of brake orbits, which means orbits whose velocities are zero at some times, under some particular conditions in the 2 -center problem by using variational methods.


Introduction & main theorem.
The n-center problem is given by the following ODEs:q where a k ∈ R d is a constant vector. A solution q(t) of (1) is called a brake orbit if there are real numbers T 1 and T 2 (T 2 > T 1 ) such thaṫ q(T 1 ) =q(T 2 ) = 0 (2) and q(t) is not a stationary solution. A brake orbit is a periodic orbit with period 2(T 2 − T 1 ). The fact is shown in Section 2. The 2-center problem is a simplified model of the restricted three-body problem [7]. The 2-center problem is integrable, but its first integrals are complicated(for further details, see [1]). We can not immediately know what types of periodic solutions exist.
For various Lagrange systems, it has been researched for a long time to find periodic solutions with variational methods. In the n-center problem, it is shown that there exist periodic orbits that move around one or several primaries ( [8], [10]). The brake orbits we prove to exist in this paper do not wind around particles.
Brake orbits are a special type of periodic orbits. Chen [3] proved that brake orbits exist in the planar isosceles three-body problem using collision manifold. In [5], Moeckel, Montgomery and Venturelli show the existence of brake orbits using variational methods with respect to the Jacobi-Maupertuis functional. The Lagrangian actional functional have not been used to find brake orbits.
In this paper, we will show that brake orbits exist in the planar 2-center problem by minimizing the Lagrangian action functional. We can set m 1 = 1 and a 1 = −a 2 = (1, 0) without loss of generality for the planar 2-center problem as stated in Section 3. More precisely, we shall prove the following theorem: then a 4T -periodic brake orbit q(t)(= (q 1 (t), q 2 (t))) exists in the planar 2-center problem. The orbit is orthogonal to the x-axis at t = 0 and has zero velocity at t = T . The orbit q(t) satisfies (q 1 (t), q 2 (t)) = (q 1 (−t), −q 2 (−t)). Here, the set D is defined by  This paper is organized as follows. Section 2 contains some of well-known facts about brake orbits and variational methods. In Section 3, we introduce the variational settings in the planar 2-center problem and set the boundary condition. In Section 4, we complete the proof of Theorem 1.1 by eliminating the possibility that minimizer is a equilibrium solution or a collision path. In Appendix, we extend the theorem to a domain larger than D. 2.1. Brake orbits. Consider ordinary differential equations: Definition 2.1 (Reversible). Let R be an involuntary liniear map from R n to R n , i.e. R 2 = E n . If (3) satisfies F R + RF = 0, then (3) is said to be reversible with respect to R.
With a simple calculation, we get the following proposition: . See [6] for more detailed explanation for reversible systems.
Consider the following Lagrangian: The differential equations of the Lagrangian system are reversible with respect to In this case, the fixed space is Fix(R) = {(q, 0) | q ∈ R n }.
The n-center problem is a Lagrangian system with form (4), Corollary 1. In the n-center problem, if a solution q satisfies (2), then it is a 2(T 2 − T 1 )-periodic orbit.

2.2.
Existence of the minimizer. Let C A,B,T be the set of C 2 curves in an open set D ⊂ R n connecting from A to B : where A, B ⊂ D are affine spaces. The action functional for (4) is defined by: The following is well-known.

Proposition 3. Let L be a Lagrangian of the form (4) and A be the action functional. If q ∈ C A,B,T is a critical value of A, then q(t) satisfies the Euler-Lagrange
We take The norm is defined by In general, action functionals for potential systems are weakly lower semicontinuous ( [4]).

Lemma 2.3 ([9]). Assume that A is weakly lower semi-continuous. If
A| Ω is coercive, then there exists a minimizer q * of A in the weak closureΩ of Ω.

Lemma 2.4. Define Ω by
If A is a bounded set, then A| Ω is coercive.
Proof. Here we prove this lemma, but similar proofs have appeared in some other settings (see for example [2]).
For any q ∈ Ω, we take By the Cauchy-Schwarz inequality, By letting ξ = sup q∈A |q|, holds. Since 3. Variational setting for the 2-center problem. We consider the planar 2center problem i.e. take n = 2 and d = 2 in (1). We fix masses and positions of the primaries as follows: • Fix the position of the primaries at a 1 and a 2 .
We can assume the above setting without loss of generality for the 2-center problem, because for any a 1 , a 2 ∈ R 2 , m 1 > 0 and m 2 > 0, it can be reduced the above case with appropriate transformation and scaling.
We define its action functional by problem is equivalent to the variational problem: We fix a positive number T and search for a brake orbit q(t) = (q 1 (t), q 2 (t)) satisfying In order to obtain such brake orbits, we take a class of curves as follows: From Lemma 2.3 and 2.4, (6) has a minimizer in the weak closureΩ of Ω. Let q * (t) = (q * 1 (t), q * 2 (t)) be a minimizer. If q * is neither a trivial solution nor a collision solution, it is a quarter (fundamental) part of a brake orbit from Proposition 2 and 3 (See figure 2). In fact, the system has a reversibility with respect to: By collorary 1 if q(t) = (q 1 (t), q 2 (t)) is a solution, then so is q(t) = (q 1 (−t), −q 2 (−t)). Thus, we get the entire trajectory of a 4T -periodic brake orbit like figure 3. 4.1. Estimate of equilibrium point. Let q eq denote an equilibrium point of (6), i.e. 1 |q eq − a| 3 (q eq − a) + m |q eq + a| 3 (q eq + a) = 0. From a simple calculation, q eq is determined by: and the value of the action functional at q eq is We will obtain a condition under which the equilibrium point is not the minimizer by estimating the second variation. The second variation A ′′ (q)(δ) is given by where q ∈ H 1 ([0, T ], R 2 ) and δ ∈ H 1 ([0, T ], R 2 ). (For details, see [11].) If there exists δ such that A ′′ (q)(δ) is negative, then q is not the minimizer of (6). Since we obtain We substitute the second variation of q eq for (9) is negative if From this, the following lemma is proved.
is not a minimizer of A(q).

YUIKA KAJIHARA AND MISTURU SHIBAYAMA
The value of action functional at q col is This inequality becomes an equality if and only if θ(t) is identically zero. We can obtain the similar estimate in the case that q col collides with m 2 , and it is no less than the former one since m ≥ 1. It follows that the collision path moves on the x-axis like Figure 4.
We will call the solution of Lemma 4.2 a collision-ejection solution of the 2-center problem and represent it by q col (t) = (q col (t), 0). By Lemma 4.2, we consider only a collision-ejection solution to get a lower bound estimate for the value of the action functional for any collision path.

Lemma 4.5. For any q col in collision solutions,
Proof. From Lemma 12, we have

Test path vs. collision path.
Lemma 4.6. If f (m, T, c) ≥ 0, the collision path q col is not a minimizer.
Proof. We take a test path: Clearly g(m, T ) >g(T ) holds, so we obtaing(T ) ≥ A(q eq ), i.e. if , then q col is not a minimizer and if T > α(m), then q eq is not a minimizer. If there exists T such that α(m) < T < β(m), then For 1 ≤ m < 3.1164778, (16) holds.
Appendix A. . In this section, we will reconsider estimate of (6) of collisions. For all λ ∈ (0, 1), let and By [4], we get the following estimate of (17): To estimate (18), we will use a comparison of (18) and a part of the linear Kepler orbit.
We fix H and assume −m/2 < H < 0. Let Q(t) denote a collision-ejection solution with respect to (18) satisfying Q(t 0 ) = 2,Q(T + t 0 ) = 0 and Thus we obtain Gordon [4] gives Proof. By the definition of T (x, H), we get The proof is completed.
In the same way as the proof of Lemma 4.6, ifḡ(m, λ, y) − A(q c ) ≥ 0, then q col is not a minimizer. From the above discussion, we show: Theorem A.2. If (m, T ) ∈ D ′ , then 4T -periodic brake orbits q(t)(= (q 1 (t), q 2 (t))) satisfying the same condition of Theorem 1.1 exists in the planar 2-center problem.