Some remarks on symmetric periodic orbits in the restricted three-body problem

The planar circular restricted three body problem (PCRTBP) is symmetric with respect to the line of masses and there is a corresponding anti-symplectic involution on the cotangent bundle of the 2-sphere in the regularized PCRTBP. Recently it was shown that each bounded component of an energy hypersurface with low energy for the regularized PCRTBP is fiberwise starshaped. This enable us to define a Lagrangian Rabinowitz Floer homology which is related to periodic orbits symmetric for the anti-symplectic involution in the regularized PCRTBP and hence to symmetric periodic orbits in the unregularized problem. In this paper we compute of this homology and discuss about symmetric periodic orbits.


Introduction
The project to apply holomorphic curve techniques to the restricted three body problem just began, see [AFvKP12,AFFHvK11,AFFvK12,CFvK11]. In particular, Albers-Frauenfeldervan Koert-Paternain [AFvKP12] proved that each bounded component of the regularized energy hypersurface is a fiberwise starshaped hypersurface in T * S 2 for energy less than the first critical value. As they mentioned this opens up the possibility of applying holomorphic curve techniques. In this paper we compute a related Lagrangian Rabinowitz Floer homology and using this computation discuss about symmetric periodic orbits which we will introduce below.
We refer to two massive primaries as the earth and the moon and to the other body with negligible mass as the satellite. The configuration space is R 2 \ {q E , q M } and the phase space is given by T * (R 2 \ {q E , q M }) = (R 2 \ {q E , q M }) × R 2 . Here q E and q M lying on the real line are the positions of the earth and the moon respectively. The Hamiltonian for the planar circular restricted three body problem (PCRTBP) is given by where µ ∈ (0, 1) is the normalized mass of the moon. The energy hypersurface H −1 (c) with energy c ∈ R below the first critical value H(L 1 ) is composed of three connected components. Following [AFvKP12], we denote by Σ E c resp. Σ M c the bounded component close to the earth resp. to the moon. Since these components are noncompact due to collisions, we compactify each of them into Σ E c and Σ M c by Moser regularization. Since the discussions in this paper go through for both Σ E c and Σ M c with c < H(L 1 ), we call them Σ for convenience. The regularized phase space is the cotangent bundle of S 2 and Σ is diffeomorphic to the unit cotangent bundle of S 2 . We denote the Hamiltonian function corresponding to H via Moser regularization by Q ∈ C ∞ (T * S 2 ). More details can be found in Section 2.
An interesting feature of the PCRTBP is that there is an involution such that the problem is symmetic with respect to this involution. More precisely, there exists an anti-symplectic involution R on T * R 2 given by R(q 1 , q 2 , p 1 , p 2 ) = (q 1 , −q 2 , −p 1 , p 2 ) such that H • R = H. Thus the Hamiltonian vector field of H is invariant under R. Through Moser regularization, R induces the anti-symplectic involution R = I • T * ρ : T * S 2 −→ T * S 2 where I : T * S 2 → T * S 2 given by I(ξ, η) = (ξ, −η) and ρ is the reflection on S 2 about a great circle. The fixed locus of R is the conormal bundle of the great circle. In the present paper we are concerned with a periodic orbit of prescribed energy which is carried into itself by R, i.e. (x, 2T ) satisfying which we refer to a symmetric periodic orbit. Here X Q is the Hamiltonian vector field of Q. As mentioned above, Σ is shown to be a fiberwise starshaped hypersurface (tight RP 3 ) in T * S 2 and thus Fix R ∩ Σ is diffeomorphic to the disjoint union of two circles L + and L − (legendrian knots). We note that every symmetric periodic orbit (x, 2T ) intersects with L + ∪ L − exactly twice at time 0 and T (after an appropriate time shift). It is an interesting question whether a symmetric periodic orbit intersects with both circles or only one of them.
Definition 1.1. A symmetric periodic orbit on Σ is called of type I if it intersects with both L + and L − . Otherwise, we call it of type II.
For explicit computations, let us consider the case Σ = Σ M c with c < H(L 1 ). We embed the cotangent bundle of S 2 in R 6 as below.
Then the inverse process of Moser regularization gives a correspondence where the moon is located at q M = (q M 1 , q M 2 ) = (−(−1 − µ), 0) for µ ∈ (0, 1). Via this map, the anti-symplectic involution R on T * R 2 corresponds to the anti-symplectic involution R on T * S 2 defined by As mentioned, R can be regarded as the composition of two involutions where ρ is the reflection on S 2 about the great circle L = {ξ ∈ S 2 ⊂ R 3 | ξ 1 = 0}. Then the fixed locus of R is the conormal bundle of L.
Since Σ is a fiberwise starshaped hypersurface, Fix R ∩ Σ is composed of two circles is a positive/negative function. Let π : T * R 2 → R 2 be the footpoint projection map. Then the regions of L M + and L M − in the configuration space of PCRTBP are as below. Suppose that a symmetric periodic orbit (x, 2T ) does not pass through the north pole of S 2 (i.e. does not collide with the moon). Then there is a periodic solution ((q x (t), p x (t)), 2T ) of the Hamiltonian system of (1.1) corresponding to (x, 2T ). Since (q x (t), p x (t)) passes through Fix R at time 0 and T anḋ q x (t) cuts the q 1 -axis at right angle at time 0 and T . The figures 1.2 and 1.3 describes the geometric motions of symmetric periodic orbits in the configuration space Figure 1.2. We note that the Birkhoff retrograde orbit [Bir15] which looks like X 1 is of type I. On the other hand, if (x, 2T ) is of type II, (q x Figure 1.3. We doubt whether there is a symmetric periodic orbit which does not surround the primary like X 3 . But we expect Type II symmetric periodic orbits like X 4 mostly exist in the PCRTBP for arbitrary c < H(L 1 ) and µ ∈ (0, 1). Indeed, when µ = 0 (the rotating Kepler problem), there always exist such Type II symmetric periodic orbits for every energy below the first critical value: A symmetric periodic orbit which is a k-fold covered ellipse in an l-fold covered coordinate system (defined in [AFFvK12]) is of type II whenever k + l is odd. Then the perturbation method based on 0 1 00 00 11 11 Figure 1.3. Type II symmetric periodic orbits the implicit function theorem, for instance [Are63,Bar65], ensures survival of them at least for small µ ≈ 0. Since Σ is fiberwise starshaped it is a graph of f ∈ C ∞ ( over the unit cotangent bundle S * S 2 . We note that the Hamiltonian vector field X Q on Σ can be lifted to the Reeb vector field on a starshaped hypersurface in R 4 with respect to the contact form α := 1 2 (x 1 dy 1 − y 1 dx 1 + x 2 dy 2 − y 2 dx 2 ), see e.g. [HP08]. We denote such a double cover of Σ by S ⊂ R 4 and the covering map by Π : S → Σ. We refer to S dynamically convex if the Conley-Zehnder index (defined in Section 3) of every periodic Reeb orbit is greater than or equal to 3. It was proved that a strictly convex hypersurface is R 4 is dynamically convex, see [HWZ98,Theorem 3.4] or [Lon02,Chapter 15]. This enables us to check the convexity for given mass ratio µ ∈ (0, 1) and energy c < H(L 1 ) using computer. Moreover it turns out that S is dynamically convex in some cases [AFFHvK11,AFFvK12] and it is believed that S is dynamically convex for all µ ∈ (0, 1) and energy c < H(L 1 ).
Theorem A. Suppose that Σ is nondegenerate and S is dynamically convex.
(A1) There exist at least two symmetric periodic orbits on Σ.
(A2) If there are precisely two periodic orbits on Σ, both are symmetric periodic orbits of type I. (A3) There exist infinitely many periodic orbits on Σ if a type II symmetric periodic exists. Aforementioned we meet the assumption on the existence of a type II symmetric periodic orbit in some cases. The nondegeneracy condition will be defined in Section 3 but we expect that this can be removed in the theorem. The assertions (A2) and (A3) are immediate consequences of (A1) and a theorem in [HWZ98], see Remark 1.3. Indeed if there is a type II symmetric periodic orbit (x, 2T ), there are two distinct periodic Reeb orbits (x 1 , 2T ) and (x 2 , 2T ) on S such that π(x 1 ) = π(x 2 ) = x. But there is another periodic Reeb orbit on S due to (A1) and thus the theorem of [HWZ98] guarantees the existence of infinitely many periodic orbits on S and hence on Σ as well.
It is worth mentioning that making use of an idea behind of the proof of (A1), we can find multiple brake orbits (see Remark 1.2) on dynamically convex hypersurfaces in R 2n , see [AKM12]. We think that there are a couple of ways to prove (A1). In this paper (A1) will be proved by using the Lagrangian Rabinowitz Floer homology computation (1.3) which will be carried out in Theorem 3.6. We hope that this homology computation will provide more information rather than (A1).
Remark 1.2. We note that S is a centrally symmetric hypersurface, i.e. S = −S in R 4 and that there is an anti-symplectic involution R on R 4 given by such that Ψ * α = α, the result on R-symmetric periodic orbits on S can be inferred from the result on periodic orbits symmetric with respect to It is worth remarking that N -symmetric periodic orbits are well known as brake orbits in classical mechanics which has a rich history. In particular, we can employ a theorem of [LZZ06] to prove the assertion (A1) when S is strictly convex. We have not checked whether their theorem is applicable to the dynamically convex case. We close the remark by pointing out that centrally symmetric brake orbits resp. centrally asymmetric brake orbits on Ψ(S) correspond to Type I symmetric periodic orbits resp. Type II symmetric periodic orbits in the regularized PCRTBP.
Remark 1.3. In order for (A2) and (A3) we observe dynamics on the ellipsoid which is a typical example of a strictly convex hypersurface in (C 2 , i 2 (dz 1 ∧ dz 1 + dz 2 ∧ dz 2 )). The Reeb flow on the ellipsoid E 4 (r 1 , r 2 ) is given by The minimal periods of z 1 (t) and z 2 (t) are T 1 = πr 1 and T 2 = πr 2 respectively. Thus if r 1 /r 2 / ∈ Q, there are precisely two periodic orbits ( √ r 1 e 2ti/r 1 , 0) and (0, √ r 2 e 2ti/r 2 ). In contrast, if r 1 /r 2 ∈ Q, all orbits are periodic with the minimal period T = lcm(p, q)T 1 /p = lcm(p, q)T 2 /q where p, q ∈ N satisfy p/q = r 1 /r 2 . This shows that the ellipsoid possesses either two or infinitely many periodic orbits. In fact this dichotomy remains true for a wider class of 3-dimensional starshaped hypersurfaces: dynamically convex starshaped hypersurfaces [HWZ98]; see also [HWZ03].
Further discussions. We close this introductory section with some expectable applications.
1. For c ∈ (H(L 1 ), H(L 2 )) energy between the first critical value and the second critical value, the regularized energy hypersurface Σ Thus it may be an interesting question to ask what is the Lagrangian Rabinowitz Floer homology (1.4) for energy level c ∈ (H(L 1 ), H(L 2 )) although we can define it only for c slight above H(L 1 ) at the present, see [AFvKP12]. It is conceivable that the computation of (1.4) can be expressed in terms of the computation (1.3) as in the wrapped Floer homology case [Iri10]; see also the symplectic homology and (periodic) Rabinowitz Floer homology case [Cie02, CFO10, AF12].
2. Since the computation (1.3) only uses the fact that Σ is a nondegenerate fiberwise starshaped hypersurface in T * S 2 and the proof of Theorem A continues to hold whenever Σ is dynamically convex, the theorem may be applicable to the Hill's lunar problem [Hil78]. The Hill's Hamiltonian H Hill : Interestingly, the Hill's Hamiltonian carries an additional involution R ′ defined by as well as R. That is, there are two anti-symplectic involutions R and R ′ on T * R 2 such that H Hill = H Hill • R = H Hill • R ′ . Via the regularization process as in the PCRTBP case, R corresponds to an anti-symplectic involution R = I • T * ρ and R ′ corresponds to an anti-symplectic involution  If Σ H c is dynamically convex, we can apply Theorem A. This assumption is not groundless due to [AFFHvK11] and for small energies this is actually true and can be checked by straightforward computations. If this is the case, there exist two R-symmetric periodic orbits and two R ′ -symmetric periodic orbits on Σ H c . But R-symmetric periodic orbits may coincide with R ′ -symmetric periodic orbits after time shift (i.e. geometrically the same). Therefore if there are precisely two periodic orbits on Σ H c , both have to be doubly symmetric. The existence of a doubly symmetric periodic orbit is known thanks to Birkhoff [Bir15] again.

The regularized restricted three body problem
Though the content of this section can be found in [AFvKP12], we briefly review the regularized PCRTBP to make this paper self-contained. As the name of the PCRTBP indicates, we assume that the moon and the earth rotate in a circular trajectory with center at the center of masses and that the satellite is massless and moves on the plane where the moon and the earth rotate. Let m E be the mass of the earth and m M be the mass of the moon. We denote the normalized mass of m M by µ, i.e.
In the rotating coordinate system, the earth and the moon are located at respectively and the phase space is given by The positions of the earth and the moon are removed to avoid collisions. The Hamiltonian for the satellite H : The Hamiltonian H carries exactly five critical points (L 1 , L 2 , L 3 , L 4 , L 5 ) called Lagrange points. We may assume that The Hamiltonian H is invariant under the anti-symplectic involution which preserves three (colinear) Lagrange points and interchanges two (equilateral) Lagrange points. We denote by the footpoint projection map π : T * R 2 → R 2 . Then the Hill's region where the satellite with energy c moves is π (H −1 (c)). The region π(H −1 (c)) for c < H(L 1 ) is composed of two bounded regions and one unbounded region. We abbreviate the bounded regions by K E c and K M c such that q E ∈ cl(K E c ) and q M ∈ cl(K M c ). Likewise H −1 (c) consists of two bounded components and one unbounded component. We denote by Σ E c resp. Σ M c the bounded component corresponding to K E c resp. K M c . In [Mos70], Moser regularized an energy hypersurfaces of the Kepler problem with negative energy into the unit tangent bundle of S 2 . The PCRTBP (in the rotating coordinate system) can also be regularized in a similar way, see [AFvKP12,Section 6]. In what follows we briefly outline the regularization process for Σ M c which is the bounded component close to the moon. We first introduce an independent variable s = dt |q − q M | and define the Hamiltonian K(q, p) by Here H −1 (c) is the energy hypersurface to be regularized. One can easily check that the Hamiltonian flow of K at energy level 0 with time parameter s corresponds to the Hamiltonian flow of H at energy level c ∈ R with time parameter t. We set p = −x, q − q M = y and perform the inverse of the stereographic projection. Here the stereographic projection where ξ = (ξ 0 , ξ 1 , ξ 2 ) ∈ S 2 ⊂ R 3 and η = (η 0 , η 1 , η 2 ) ∈ T * ξ S 2 , i.e. ξ 0 η 0 + ξ 1 η 1 + ξ 2 η 2 = 0. Then we obtain the Hamiltonian function K • S on T * S 2 . Since K • S is not smooth at the zero section, we consider instead Q : T * S 2 → R, Q := 1 2 |η| 2 (K + µ) 2 .
Then one can readily check that Hamiltonian vector fields on Σ M c are lifted to those of Q −1 (µ 2 /2) =: Σ

Rabinowitz Floer homology and Proof of Theorem A
A simple observation shows that the problem (1.2) can be interpreted as the boundary value problem for the Lagrangian submanifold Fix R. Indeed if (x, T ) solves (3.1) then so does x R (t) := Rx(T − t) and we obtain a symmetric periodic orbit As mentioned in the introduction, Σ can be both Σ E c and Σ M c for c < H(L 1 ). Our appoach to this boundary value problem is Lagrangian Rabinowitz Floer homology studied by Merry [Mer10,Mer11], which is the Lagrangian intersection theoretic version of Rabinowitz Floer homology introduced by Cieliebak-Frauenfelder [CF09]. The existence of a symmetric periodic orbit is by now a standard application of Rabinowitz Floer homology theory due to the computation of Lagrangian Rabinowitz Floer homology group which will be carried out in Theorem 3.6. Moreover using this computation we shall prove Theorem A.

Construction of Lagrangian Rabinowitz Floer homology.
We briefly introduce Lagrangian Rabinowitz Floer homology and refer to [Mer10,Mer11] for further details. We also refer the reader to [Frau04, Appendix A] for Morse-Bott homology and Floer's celebrated papers [Flo88a,Flo88b]. Let M be a closed n-dimensional manifold and Q be a closed d-dimensional submanifold in M . We denote by T * M the cotangent bundle of M and by N * Q the conormal bundle of Q. We note that N * Q is an exact Lagrangian submanifold in (T * N, dλ) where λ is the Liouville 1-form. We denote by Let H ∈ C ∞ (T * M ) such that H −1 (0) is a smooth fiberwise starshaped hypersurface. Then since H −1 (0) splits T * M into one bounded component and one unbounded component, we can modify H to be constant near infinity. For notational convenience we write again H for the modified Hamiltonian function. Then the Rabinowitz action functional A H : where X H is the Hamiltonian vector field associated to H defined implicitly by i X H ω = dH. Thus if (x, η) is a nontrivial critical point of A H , i.e. η = 0, (x η , η) where x η (t) := x(t/η) solves We choose an ω-compatible almost complex structure J and define a metric m J on P N * Q T * M × R by   The standard arguments in Floer theory proves that following nontrivial facts. For a generic almost complex structure J and a generic Riemannian metric g,

Computation of Lagrangian Rabinowitz Floer homology.
Making use of the Abbondandolo-Schwartz short exact sequence in [AS09], Merry proved the following theorem in [Mer10, Theorem B] (see also Remark 7.7 and Remark 12.6 in [Mer11]). We should mention that he proved more general statements.
Theorem 3.1. Let Q and M be closed manifolds with d ≤ n/2. Then where the path space P Q M is defined below.
Remark 3.2. Although in proving the above theorem one has to use a Hamiltonian function defining Σ which has quadratic growth [AS09,Mer10] or linear growth [CFO10] near infinity, the resulting Floer homology coincides with the Rabinowitz Floer homology defined in the previous subsection, see [ In what follows we compute the singular homology groups in Theorem 3.1 in a special case. Let Z be a connected topological space and Y be a connected subspace. We denote by ΩZ the based loop space of Z. We further abbreviate relative path spaces by for z, z ′ ∈ Z, P z,z ′ Z := γ ∈ C 0 ([0, 1], Z) | (γ(0), γ(1)) = (z, z ′ ) , Here we deal with continuous paths but the homotopy types of above path spaces do not change even if we consider W 1,2 -, or C ∞ -paths instead. Suppose that Y is contractible to z ∈ Z in Z; that is, there exists a continuous map F : Proposition 3.3. Let Y be contractible to z ∈ Z in Z as above. Then we have the following homotopy equivalences: Proof. We define for each y ∈ Y , γ y a path in Z by In particular, γ y (0) = z and γ y (1) = y. We set γ y (t) := γ y (1 − t), γ y r (t) := γ y (rt), r ∈ [0, 1]. We define a map Φ which will give the desired homotopy equivalence. Here we abbreviate # for the concatenation operation for paths.
The map Ψ below will be a homotopical inverse of Φ.
Here we consider ΩZ as a loop space of Z with the base point z ∈ Z. In order to show that Ψ • Φ is homotopic to the identity, we construct a homotopy Performing some reparametrizations on G at time r = 0, we deduce In a similar vein, using the homotopy (w, y, r) −→ (γ y r #γ y r #w, y) such that R(w, y, 0) = w(3t), y we obtain after some reparametrizations as before, This proves P z,Y Z ≃ ΩZ × Y and thus the second equivalence is proved. The first equivalence P Y Z ≃ P z,Y Z × Y follows analogously.
Corollary 3.4. Let S 1 be an embedded circle in S 2 . Then we have In particular, we compute Remark 3.5. In an alternative way, one can directly compute the singular homology of P S 1 S 2 by means of the Leray-Serre spectral sequence. We consider the evaluation map ev 1 : P z,S 1 S 2 → S 1 defined by ev 1 (u) = u(1). Then we have a fibration We note that the spectral sequence for this fibration degenerates at the second page for dimension reasons, i.e. E ∞ = E 2 . Even though S 1 is not simply-connected, the E 2 -page has a simple formula. Since S 1 is contractible in S 2 , the above fibration has trivial monodromy π 1 (S 1 ) → Aut H n (ΩS 2 ) , ℓ → Id Hn(ΩS 2 ) , ∀ℓ ∈ π 1 (S 1 ), n ∈ N ∪ {0}, and thus E 2 Therefore we have The exactly same arguments go through for a fibration where ev 0 is the evaluation map at time zero. Therefore we derive Therefore Theorem 3.1 and Proposition 3.3 result in the following.
Theorem 3.6. Let Q and M be as above and Q be contractible to a point in M .
In particular if S 1 is an embedded circle in S 2 , 3.3. Robbin-Salamon index.
We denote by L(R 2n ) the Grassmanian manifold of all Lagrangian subspaces in (R 2n , ω 0 = dx ∧ dy). Let V ∈ L(R 2n ) and Λ : [0, T ] → L(R 2n ). We choose W ∈ L(R 2n ) a Lagrangian complement of Λ(t) for t ∈ [0, T ]. For v ∈ Λ(t) and small ǫ we can find a unique w(ǫ) ∈ W such that v + w(ǫ) ∈ Λ(t + ǫ). The crossing form at time t ∈ [0, T ] is defined by It is independent of the choice of W . A crossing time t ∈ [0, T ], i.e. Λ(t) ∩ V = {0}, is said to be regular if Γ(Λ, V, t) is nondegenerate. Since regular crossings are isolated, the number of crossings for a regular path which has only regular crossings is finite. Thus for a regular path Λ(t) ∈ L(R 2n ) and V ∈ L(R 2n ), the Robbin-Salamon index [RS93] can be defined as below. where ∆ is the diagonal of R n × R n . Returning to the regularized PCRTBP, let (x, T ) be a solution of (3.1). Then (x R , T ) and (x m , mT ) for m ∈ N defined by x R (t) := Rx(T − t), x 2k := x R # · · · #x#x R #x 2k , x 2k+1 := x# · · · #x#x R #x 2k+1 solve (3.1) as well. Now we associate the Robbin-Salamon index to each solution of (3.1). We first symplectically trivialize the hyperplane field ker λ Σ (contact structure) by a pair of global sections where n Σ = (n ξ ∂ ξ , n η ∂ η ) is the outward pointing normal vector field on Σ. We denote the induced global symplectic trivialization by We note that Φ is a vertical preserving symplectization trivialization and maps T R to the reflection about the X 2 -axis, i.e.
where T v T * S 2 := ker T π be the vertical subbundle of T T * M and We abbreviate the fixed locus of R by V := Fix R = R × (0).
Lemma 3.7. If (x, T ) is a solution of (3.1) for a periodic solution (x 2 , 2T ), see [Lon02]. In consequence, we obtain the following proposition which plays crucial roles in proving Theorem A.