Non-Collision singularities in the Planar two-Center-two-Body problem
NNON-COLLISION SINGULARITIES IN THE PLANARTWO-CENTER-TWO-BODY PROBLEM
JINXIN XUE AND DMITRY DOLGOPYAT
Abstract.
In this paper, we study a restricted four-body problem called planartwo-center-two-body problem. In the plane, we have two fixed centers Q and Q of masses 1, and two moving bodies Q and Q of masses µ (cid:28)
1. They interactvia Newtonian potential. Q is captured by Q , and Q travels back and forthbetween two centers. Based on a model of Gerver, we prove that there is a Cantorset of initial conditions which lead to solutions of the Hamiltonian system whosevelocities are accelerated to infinity within finite time avoiding all earlier collisions.This problem is a simplified model for the planar four-body problem case of thePainlev´e conjecture. Contents
1. Introduction 31.1. Statement of the main result 31.2. Motivations. 41.3. Extension to the four-body problem 51.4. Plan of the paper 62. Proof of the main theorem 62.1. Idea of the proof 62.2. Main ingredients 62.3. Gerver map 92.4. Asymptotic analysis, local map 112.5. Asymptotic analysis, global map 122.6. Admissible surfaces 132.7. Construction of the singular orbit 153. Hyperbolicity of the Poincar´e map 173.1. Construction of invariant cones 173.2. Expanding directions of the global map 193.3. Checking transversality 244. C estimates for global map 264.1. Equations of motion in Delaunay coordinates 26 Date : February 5, 2018. a r X i v : . [ m a t h . D S ] J u l JINXIN XUE AND DMITRY DOLGOPYAT a priori bounds 284.3. Proof of Lemma 4.1 314.4. Proof of Lemma 2.5 325. Derivatives of the Poincar´e map 356. Variational equations 366.1. Estimates of the coefficients 366.2. Estimates of the solutions 427. Boundary contributions and the proof of Proposition 3.7 447.1. Dependence of (cid:96) on variables ( X, Y ) 447.2. Asymptotics of matrices ( I ) , ( III ) , ( V ) from the Proposition 3.7 458. Switching foci 468.1. From the right to the left 468.2. From the left to the right 489. Approaching close encounter 4810. C estimate for the local map 5010.1. Justifying Gerver’s asymptotics 5110.2. Proof of Lemma 2.4 and 2.6 5411. Consequences of C estimates 5511.1. Avoiding collisions 5511.2. Choosing angular momentum 5712. Derivative of the local map 5812.1. Justifying the asymptotics 5812.2. Proof of the Lemma 3.9 6712.3. Proof of the Lemma 3.10 69Appendix A. Delaunay coordinates 73A.1. Elliptic motion 73A.2. Hyperbolic motion 74A.3. Large (cid:96) asymptotics: auxiliary results 77A.4. First order derivatives 78A.5. Second order derivatives 79Appendix B. Gerver’s mechanism 81B.1. Gerver’s result in [G2] 81B.2. Numerical information for a particularly chosen ε = 1 / ONCOLLISION SINGULARITIES IN THE 2-CENTER-2-BODY PROBLEM 3
References 861.
Introduction
Statement of the main result.
We study a two-center two-body problem.Consider two fixed centers Q and Q of masses m = m = 1 located at distance χ from each other and two small particles Q and Q of masses m = m = µ (cid:28) Q i sinteract with each other via Newtonian potential. If we choose coordinates so that Q is at (0 ,
0) and Q is at ( − χ,
0) then the Hamiltonian of this system can be written as(1.1) H = | P | µ + | P | µ − µ | Q | − µ | Q − ( − χ, | − µ | Q | − µ | Q − ( − χ, | − µ | Q − Q | . We assume that the total energy of the system is zero.We want to study singular solutions of this system, that is, the solutions whichcan not be continued for all positive times. We will exhibit a rich variety of singularsolutions. Fix ε < χ. Let ω = { ω j } ∞ j =1 be a sequence of 3s and 4s. Definition 1.1.
We say that ( Q ( t ) , Q ( t )) is a singular solution with symbolicsequence ω if there exists a positive increasing sequence { t j } ∞ j =0 such that • t ∗ = lim j →∞ t j < ∞ . • | Q ( t j ) − Q | ≤ ε , | Q ( t j ) − Q | ≤ ε . • For t ∈ [ t j − , t j ] , | Q − ω j ( t ) − Q | ≤ ε and { Q ω j ( t ) } t ∈ [ t j − ,t j ] leaves the ε neighborhood of Q , winds around Q exactly once then reenters the ε neigh-borhood of Q . • lim sup t ↑ t ∗ | ˙ Q i ( t ) | → ∞ for i = 3 , . During the time interval [ t j − , t j ] we refer to Q ω j as the traveling particle and to Q − ω j as the captured particle. Thus ω j prescribes which particle is the travelerduring the j trip. The phrase that the traveler winds around Q exactly once meansthat the angle from Q to the traveler changes by 2 π + O (1 /χ ) . We denote by Σ ω the set of initial conditions of singular orbits with symbolicsequence ω . Note that if ω contains only finitely many 3s then there is a collision of Q and Q at time t ∗ . If ω contains only finitely many 4s then there is a collision of Q and Q at time t ∗ . Otherwise at we have a collisionless singularity at t ∗ . Theorem 1.
There exists µ ∗ (cid:28) such that for µ < µ ∗ the set Σ ω (cid:54) = ∅ . Moreover there is an open set U on the zero energy level and a foliation of U bytwo-dimensional surfaces such that for any leaf S of our foliation Σ ω ∩ S is a Cantorset. Remark 1.2.
By rescaling space and time variables we can assume that χ (cid:29) . Inthe proof we shall make this assumption and set ε = 2 . JINXIN XUE AND DMITRY DOLGOPYAT
Remark 1.3.
It follows from the proof that the Cantor set described in Theorem 1can be chosen to depend continuously on S. In other words Σ ω contains a set whichis locally a product of a five dimensional disc and a Cantor set. The fact that on eachsurface we have a Cantor set follows from the fact that we have a freedom of choosinghow many rotations the captured particle makes during j -th trip. Remark 1.4.
The construction presented in this paper also works for small nonzeroenergies. Namely, it is sufficient that the total energy is much smaller than the kineticenergies of the individual particles. The assumption that the total energy is zero ismade to simplify notation since then the energies of Q and Q have the same absolutevalues. Remark 1.5.
One can ask if Theorem 1 holds for other choices of masses. The factthat the masses of the fixed centers Q and Q are the same is not essential and ismade only for convenience. The assumption that Q and Q are light is importantsince it allows us to treat their interaction as a perturbation except during the closeencounters of Q and Q . The fact that the masses of Q and Q are equal allows us touse an explicit periodic solution of a certain limiting map (Gerver map) which is foundin [G2] . It seems likely that the conclusion of Theorem 1 is valid if m = µ, m = cµ where c is a fixed constant close to 1 and µ is sufficiently small but we do not have aproof of that. Motivations.
Non-collision singularities in N-body problem.
Our work is motivated by thefollowing fundamental problem in celestial mechanics.
Describe the set of initial con-ditions of the Newtonian N-body problem leading to global solutions.
The complimentto this set splits into the initial conditions leading to the collision and non-collisionsingularities.It is clear that the set of initial conditions leading to collisions is non-empty for all
N >
Conjecture 1.
The set of non-collision singularities is non-empty for all
N > . Conjecture 2.
The set of non-collision singularities has zero measure for all
N > . Conjecture 1 probably goes back to Poincar´e who was motivated by King Oscar IIprize problem about analytic representation of collisionless solutions of the N -bodyproblem. It was explicitly mentioned in Painlev´e’s lectures [Pa] where the authorproved that for N = 3 there are no non-collision singularities. Soon after Painlev´e,von Zeipel showed that if the system of N bodies has a non-collision singularity thensome particle should fly off to infinity in finite time. Thus non-collision singularitiesseem quite counterintuitive. However in [MM] Mather and McGehee constructed asystem of four bodies on the line where the particles go to infinity in finite time after aninfinite number of binary collisions (it was known since the work of Sundman [Su] thatbinary collisions can be regularized so that the solutions can be extended beyond the ONCOLLISION SINGULARITIES IN THE 2-CENTER-2-BODY PROBLEM 5 collisions). Since Mather-McGehee example had collisions it did not solve Conjecture1 but it made it plausible. Conjecture 1 was proved independently by Xia [X] for thespacial five-body problem and by Gerver [G1] for a planar 3 N body problem where N is sufficiently large. The problem still remained open for N = 4 and for small N in the planar case. However in [G2] (see also [G3]) Gerver sketched a scenario whichmay lead to a non-collision singularity in the planar four-body problem. Gerver hasnot published the details of his construction due to a large amount of computationsinvolved (it suffices to mention that even technically simpler large N case took 68pages in [G1]). The goal of this paper is to realize Gerver’s scenario in the simplifiedsetting of two-center-two-body problem.Conjecture 2 is mentioned by several authors, see e.g. [Sim, Sa3, K]. It is knownthat the set of initial conditions leading to the collisions has zero measure [Sa1] andthat the same is true for non-collisions singularities if N = 4 . To obtain the completesolution of this conjecture one needs to understand better of the structure of thenon-collision singularities and our paper is one step in this direction.1.2.2.
Well-posedness in other systems.
Recently the question of global well-posednessin PDE attracted a lot of attention motivated in part by the Clay Prize problemabout well-posedness of the Navier-Stokes equation. One approach to constructinga blowup solutions for PDEs is to find a fixed point of a suitable renormalizationscheme and to prove the convergence towards this fixed point (see e.g. [LS]). Thesame scheme is also used to analyze two-center-two-body problem and so we hope thatthe techniques developed in this paper can be useful in constructing singular solutionsin more complicated systems.1.2.3.
Poincar´e’s second species solution.
In his book [Po], Poincar´e claimed the exis-tence of the so-called second species solution in three-body problem, which are periodicorbits converging to collision chains as µ →
0. The concept of second species solutionwas generalized to the non-periodic case. In recent years significant progress was madein understanding second species solutions in both restricted [BM, FNS] and full [BN]three-body problem. However the understanding of general second species solutionsgenerated by infinite aperiodic collision chains is still incomplete. Our result can beconsidered as a generalized version of second species solution. All masses are positiveand there are infinitely many close encounters. Therefore the techniques developed inthis paper can be useful in the study of the second species solutions.1.3.
Extension to the four-body problem.
Consider the same setting as in ourmain result but suppose that Q and Q are also free (not fixed). Then we canexpect that during each encounter light particle transfers a fixed proportion of theirenergy and momentum to the heavy particle. The exponential growth of energy andmomentum would cause Q and Q to go to infinity in finite time leading to a non-collision singularity.Unfortunately a proof of this involves a significant amount of additional compu-tations due to higher dimensionality of the full four-body problem. A good news isthat similarly to the problem at hand, the Poincar´e map of the full four-body problem JINXIN XUE AND DMITRY DOLGOPYAT will have only two strongly expanding directions whose origin could be understood bylooking at our two-center-two-body problem. The other directions will be dominatedby the most expanding ones. This allows our strategy to extend to the full four-bodyproblem leading to the complete solution of the Painlev´e conjecture. However, due tothe length of the arguments, the details are presented in a separate paper [Xu].1.4.
Plan of the paper.
The paper is organized as follows. Section 2 and 3 consti-tute the framework of the proof. In Section 2 we give a proof of the main Theorem 1based on a careful study of the hyperbolicity of the Poincar´e map. In Section 3, wesummarize all calculations needed in the proof of the hyperbolicity. All the later sec-tions provide calculations needed in Sections 2 and 3. We define the local map to studythe local interaction between Q and Q and global map to cover the time intervalwhen Q is traveling between Q and Q . Sections 4, 6, 7 and 8 are devoted to theglobal map, while Sections 9,10, and 12 study local map. Relatively short Sections5 and 11 contain some technical results pertaining to both local and global maps.Finally, we have two appendices. Appendix A contains an introduction to the Delau-nay coordinates for Kepler motion, which are used extensively in our calculations. InAppendix B, we summarize the information about Gerver’s model from [G2].2. Proof of the main theorem
Idea of the proof.
The proof of the Theorem 1 is based on studying the hy-perbolicity of the Poincar´e map. Our system has four degrees of freedom. We pickthe zero energy surface and then consider a Poincar´e section. The resulting Poincar´emap is six dimensional. In turns out that for orbits of interest (that is, the orbitswhere the captured particle rotates around Q and the traveler moves back and forthbetween Q and Q ) there is an invariant cone field which consists of vectors closeto a certain two dimensional subspace such that all vectors in the cone are stronglyexpanding. This expansion comes from the combination of shearing (there are longstretches when the motion of the light particles is well approximated by the Keplermotion and so the derivatives are almost upper triangular) and twisting caused bythe close encounters between Q and Q and between Q and Q . We restrict ourattention to a two dimensional surface whose tangent space belong to the invariantcone and construct on such a surface a Cantor set of singular orbits as follows. Thetwo parameters coming from the two dimensionality of the surface will be used tocontrol the phase of the close encounter between the particles and their relative dis-tance. The strong expansion will be used to ensure that the choices made at the nextstep will have a little effect on the parameters at the previous steps. This Cantor setconstruction based on the instability of near colliding orbits is also among the keyingredients of the singular orbit constructions in [MM] and [X].2.2.
Main ingredients.
In this section we present the main steps in proving Theorem1. In Subsection 2.3 we describe a simplified model for constructing singular solutionsgiven by Gerver [G2]. This model is based on the following simplifying assumptions:
ONCOLLISION SINGULARITIES IN THE 2-CENTER-2-BODY PROBLEM 7 • µ = 0 , χ = ∞ so that Q (resp. Q ) moves on a standard ellipse (resp.hyperbola). • The particles Q , Q do not interact except during a close encounter. • Velocity exchange during close encounters can be modeled by an elastic colli-sion. • The action of Q on light particles can be ignored except that during theclose encounters of the traveler particle with Q the angular momentum of thetraveler with respect to Q can be changed arbitrarily.The main conclusion of [G2] is that the energy of the captured particle can be increasedby a fixed factor while keeping the shape of its orbit unchanged. Gerver designs a twostep procedure with collisions having the following properties: • The incoming and outgoing asymptotes of the traveler are horizontal. • The major axis of the captured particle remains vertical. • After two steps of collisions, the elliptic orbit of the captured particle has thesame eccentricity but smaller semimajor axis compared with the elliptic orbitbefore the first collision (see Fig 1 and 2).For quantitative information, see Appendix B.Since the shape is unchanged after the two trips described above the procedure canbe repeated. Then the kinetic energies of the particles grow exponentially and so thetime needed for j -th trip is exponentially small. Thus the particles can make infinitelymany trips in finite time leading to a singularity. Our goal therefore is to get rid ofthe above mentioned simplifying assumptions. Figure 1.
Angular momentum transferIn Subsection 2.4 we study near collision of the light particles. This assumptionthat velocity exchange can be modeled by elastic collision is not very restrictive since
JINXIN XUE AND DMITRY DOLGOPYAT
Figure 2.
Energy transferboth energy and momentum are conserved during the exchange and any exchange ofvelocities conserving energy and momentum amounts to rotating the relative velocityby some angle and so it can be effected by an elastic collision. In Subsection 2.5 westate a result saying that away from the close encounters the interaction between thelight particles as well as the action of Q on the particle which is captured by Q canindeed be disregarded. In Subsection 2.6 we study the Poincar´e map correspondingto one trip of the traveller particle around Q . After some technical preparations wepresent the main result of that section–Lemma 2.10 which says that after this tripthe angular momentum of the traveler particle indeed can change in an arbitrary way.Finally in Subsection 2.7 we show how to combine the above ingredients to constructa Cantor set of singular orbits.In (1.1), we make the change of variables P i = µv i , i = 3 , µ . This rescaling changes the symplectic form by a conformal factor butdoes not change the Hamiltonian equations. The rescaled Hamiltonian, still denotedby H has the following form(2.1) H = | v | | v | − | Q | − | Q + ( χ, | − | Q | − | Q + ( χ, | − µ | Q − Q | . We have v i = ˙ Q i and we use x, y to denote the components of Q , Q i = ( x i , y i ) , i = 3 , v, Q ) ∈ R orin Delaunay coordinates ( L, (cid:96), G, g ). The symplectic transformation between the twocoordinates is given explicitly in Appendix A. The geometric meanings of the Delaunayvariables are as follows. For elliptic motion, L is the length of the semi major axis, LG is the length of the semi minor axis, and g is the argument of periapsis (direction).These three variables characterize the shape of the ellipse. The variable (cid:96) calledmean anomaly indicates the position of the moving body on the ellipse. For Kepler ONCOLLISION SINGULARITIES IN THE 2-CENTER-2-BODY PROBLEM 9 hyperbolic motion, Delaunay coordinates can also be introduced and have similarmeanings. See Appendix A for more details. In the following we use subscript 3 , Q or Q .2.3. Gerver map.
Following [G2], we discuss in this section the limit case µ =0 , χ = ∞ . We assume that Q has elliptic motion and Q has hyperbolic motion withrespect to the focus Q . Since µ = 0 , Q and Q do not interact unless they haveexact collision. Since we assume that Q just comes from the interaction from Q located at ( −∞ ,
0) and the new traveler particle is going to interact with Q in thefuture, the slope of incoming asymptote θ − of Q and that of the outgoing asymptote¯ θ + of the traveler particle should satisfy θ − = 0 , ¯ θ + = π .The Kepler motions of Q and Q has three first integrals E i , G i and g i where E i denotes the energy, G i denotes the angular momentum and g i denotes the argumentof periapsis. Since the total energy of the system is zero we have E = − E . Notethat(2.2) E := − L = | v | − | Q | . It turns out convenient to use eccentricities(2.3) e i = (cid:113) G i E i instead of G i since the proof of Theorem 1 involves a renormalization transformationand e i are scaling invariant. The Gerver map describes the parameters of the ellipticorbit change during the interaction of Q and Q . The orbits of Q and Q intersectin two points. We pick one of them. We label the intersection points in the reversechronological order with respect to the motion of Q . (This labeling is done so that thefirst intersection point is used at the first step of the Gerver’s construction and thesecond point is used at the second step of the Gerver construction, see Figures 1 and2.). Thus we use a discrete parameter j ∈ { , } to describe which intersection pointis selected.Since Q and Q only interact when they are at the same point the only effect of theinteraction is to change their velocities. Any such change which satisfies energy andmomentum conservation can be described by an elastic collision. That is, velocitiesbefore and after the collision are related by(2.4) v +3 = v − + v − (cid:12)(cid:12)(cid:12)(cid:12) v − − v − (cid:12)(cid:12)(cid:12)(cid:12) n ( α ) , v +4 = v − + v − − (cid:12)(cid:12)(cid:12)(cid:12) v − − v − (cid:12)(cid:12)(cid:12)(cid:12) n ( α ) , where n ( α ) is a unit vector making angle α with v − − v − . With this in mind we proceed to define the Gerver map G e ,j,ω ( E , e , g ) . Thismap depends on two discrete parameters j ∈ { , } and ω ∈ { , } . The role of j hasbeen explained above, and ω will tell us which particle will be the traveler after thecollision.To define G we assume that Q moves along the hyperbolic orbit with parameters( − E , e , g ) where g is fixed by requiring that the incoming asymptote of Q ishorizontal. We assume that Q and Q arrive to the j -th intersection point of their orbit simultaneously. At this point their velocities are changed by (2.4). After that theparticle proceed to move independently. Thus Q moves on an orbit with parameters( ¯ E , ¯ e , ¯ g ), and Q moves on an orbit with parameters ( ¯ E , ¯ e , ¯ g ) . If ω = 4, we choose α in (2.4) so that after the exchange Q moves on hyperbolicorbit and ¯ θ +4 = π and let G e ,j, ( E , e , g ) = ( ¯ E , ¯ e , ¯ g ) . If ω = 3 we choose α in (2.4) so that after the exchange Q moves on hyperbolic orbitand ¯ θ +3 = π and let G e ,j, ( E , e , g ) = ( ¯ E , ¯ e , ¯ g ) . Remark 2.1.
If the index j is used to define to the Gerver map then we refer to j -thintersection point of the orbits of Q and Q as Gerver collision point . We refer toAppendix B for the coordinates of Gerver’s collision points. It is important in Gerver’smodel that if Q and Q have a close encounter near the Gerver point then they donot have another close encounter before the next trip of the traveller particle. Thisfact is proven in [G2] . For the reader’s convenience we reproduce Gerver’s argumentin Section 11.1.In the following, to fix our notation, we always call the captured particle Q and thetraveler Q . Below we denote the ideal orbit parameters in Gerver’s paper [G2] of Q and Q before the first (respectively second) collision with * (respectively **). Thus, forexample, G ∗∗ will denote the angular momentum of Q before the second collision.Moreover, the actual values after the first (respectively, after the second) collisions aredenoted with a bar or double bar .Note G has a skew product form¯ e = f e ( e , g , e ) , ¯ g = f g ( e , g , e ) , ¯ E = E f E ( e , g , e ) . This skew product structure will be crucial in the proof of Theorem 1 since it willallow us to iterate G so that E grows exponentially while e and g remains almostunchanged.The following fact plays a key role in constructing singular solutions. Lemma 2.2 ([G2]) . Assume that the total energy of the Q , Q , Q system is zero. (a) For E ∗ = , g ∗ = π and for any e ∗ ∈ (0 , √ ) , there exist e ∗ , e ∗∗ , λ > suchthat ( e , g , E ) ∗∗ = G e ∗ , , ( e , g , E ) ∗ , ( e , − g , λ E ) ∗ = G e ∗∗ , , ( e , g , E ) ∗∗ , where E ∗∗ = E ∗ = , g ∗∗ = g ∗ = π and e ∗∗ = (cid:112) − e ∗ . (b) There is a constant ¯ δ such that if ( e , g , E ) lie in a ¯ δ neighborhood of ( e ∗ , g ∗ , E ∗ ) , then there exist smooth functions e (cid:48) ( e , g ) , e (cid:48)(cid:48) ( e , g ) , and λ ( e , g , E ) suchthat e (cid:48) ( e ∗ , g ∗ ) = e ∗ , e (cid:48)(cid:48) ( e ∗ , g ∗ ) = e ∗∗ , λ ( e ∗ , g ∗ , E ∗ ) = λ , ONCOLLISION SINGULARITIES IN THE 2-CENTER-2-BODY PROBLEM 11 (¯ e , ¯ g , ¯ E ) = G e (cid:48) ( e ,g ) , , ( e , g , E ) , ( e ∗ , − g ∗ , λ ( e , g , E ) E ∗ ) = G e (cid:48)(cid:48) ( e ,g ) , , (cid:0) ¯ e , ¯ g , ¯ E (cid:1) . In Section 12.3, we will give a set of equations (equations (12.30)-(12.38)) whosesolutions give the map G , and the smoothness of e (cid:48) , e (cid:48)(cid:48) follows from the implicitfunction theorem. We remark that e (cid:48) , e (cid:48)(cid:48) do not depend on E since e , e , g arerescaling invariant, and we can always rescale E to E ∗ . Part (a) allows us to increaseenergy after two collisions without changing the shape of the orbit in the limit case µ = 0 , χ = ∞ . Part (b) allows us to fight against the perturbation coming from thefact that µ > χ < ∞ . Lemma 2.2 is a slight restatement of the main result of[G2]. Namely part (a) is proven in Sections 3 and 4 of [G2] and part (b) is stated inSection 5 of [G2] (see equations (5-10)–(5-13)). The proof of part (b) proceeds by aroutine numerical computation. For the reader’s convenience we review the proof ofLemma 2.2 in Appendix B explaining how the numerics is done.
Remark 2.3.
We try to minimize the use of numerics in our work. The use of numer-ics is always preceded by mathematical derivations. Readers can see that the numericsin this paper can also be done without using computer. We prefer to use the computersince computers are more reliable than humans when doing routine computations.
Asymptotic analysis, local map.
Starting from this section, we work on theHamiltonian system (1.1). We assume that the two centers are at distance χ (cid:29) Q , Q have positive masses 0 < µ (cid:28)
1. We will see below that χ grows exponentially to infinity under iterates due to the renormalization, so we alwaysassume 1 /χ (cid:28) µ (cid:28) Q and Q can be approximated by Kepler motions at least for a short time interval if theyare away from collisions. We use the Delaunay coordinates ( L, (cid:96), G, g ) , (elliptic for3 and hyperbolic for 4) to describe the motions of Q and Q when Q and Q arein a O χ →∞ (1) neighborhood of Q . We assume Q is captured by Q . Namely, theenergy E of Q is negative where the energy (2.2) is the sum of the kinetic energyand the potential energy relative to Q . The system has four degrees of freedom.By restricting to the zeroth energy level and picking a Poincar´e section, we get a sixdimensional space as our phase space on which the Poincar´e map is defined. ThePoincar´e section is chosen as { x = − , ˙ x > } . We choose the orbit parameters as( E , (cid:96) , e , g , e , g ) ∈ R × T which are obtained from the Delaunay variables using(2.2)–(2.3). The energy E of Q is eliminated using energy conservation and (cid:96) istreated as the new time, which is also eliminated by considering the Poincar´e mapinstead of flow.We consider initial conditions in the following sets. We denote K := max † = ∗ , ∗∗ (cid:107) d G e † , , ( e , g , E ) † (cid:107) + 1 , K (cid:48) := max † = ∗ , ∗∗ (cid:107) d ( e (cid:48) , e (cid:48)(cid:48) )( e , g ) † (cid:107) + 1 . Given δ < ¯ δ/ ( KK (cid:48) ) where ¯ δ is in Lemma 2.2, consider open sets in the phase space(zero energy level and the Poincar´e section { x = − , ˙ x > } ) defined by U ( δ ) = (cid:26)(cid:12)(cid:12)(cid:12)(cid:12) E − (cid:18) − (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) , | e − e ∗ | , | g − g ∗ | , | θ − | < δ, | e − e ∗ | < K (cid:48) δ (cid:27) , U ( δ ) = (cid:8) | E − E ∗∗ | , | e − e ∗∗ | , | g − g ∗∗ | , | θ − | < Kδ, | e − e ∗∗ | < KK (cid:48) δ (cid:9) . In both U ( δ ) and U ( δ ), the angle (cid:96) can take any value in T . Throughout the paper, we reserve the notations
K, K (cid:48) , δ, ¯ δ . We let particles move until one of the particles moving on hyperbolic orbit reachesthe surface { x = − , ˙ x < } . We measure the final orbit parameters ( ¯ E , ¯ (cid:96) , ¯ e , ¯ g , ¯ e , ¯ g ).We call the mapping moving initial positions of the particles to their final positionsthe local map L . In Fig. 3 of Section 3.2 the local map is to the right of the section { x = − } . We are only interested in those initial conditions in U j ( δ ) , j = 1 , Q and Q , since otherwise Q moves on one slightlyperturbed hyperbola with non-horizontal outgoing asymptote and will escape fromthe system (Sublemma 4.9). To select these initial conditions of interest, we imposeone more boundary condition. Lemma 2.4.
Fix any constant C > and j ∈ { , } . Suppose that the initial orbitparameters ( E , (cid:96) , e , g , e , g ) are chosen in U j ( δ ) , such that the orbit passes througha δ neighborhood of the j -th Gerver’s collision point, and the traveler particle ( s ) satisfy | θ − | ≤ C µ and | ¯ θ +4 − π | ≤ C µ . Then the following asymptotics holds uniformly ( ¯ E , ¯ e , ¯ g ) = G e ,j, ( E , e , g ) + o (1) , as 1 /χ (cid:28) µ → . Thus the condition that the orbit parameters of Q (in particular ¯ θ +4 ) change signif-icantly forces Q and Q to have a closer encounter. The lemma tells us that Gervermap is a good C approximation of the local map L for the real case 0 < /χ (cid:28) µ (cid:28) Asymptotic analysis, global map.
As before we assume that the two centersare at distance χ (cid:29) . Fix a large constant C . We assume that initially Q moveson an elliptic orbit, Q moves on hyperbolic orbit and { x (0) = − , ˙ x (0) < } . We assume that | y (0) | < C and that, after moving around Q , Q hits the surface { x = − , ˙ x > } so that | y | < C . We call the mapping moving initial positions ofthe particles to their final positions the (pre) global map G . In Section 2.6 we willslightly modify the definition of the global map but it will not change the essentialfeatures discussed here. In Fig. 3 from Section 3.2, the global map is to the left ofthe section { x = − } . We let ( E , (cid:96) , e , g , e , g ) denote the initial orbit parametersmeasured in the section { x = − , ˙ x < } and ( ¯ E , ¯ (cid:96) , ¯ e , ¯ g , ¯ e , ¯ g ) denote the finalorbit parameters measured in the section { x = − , ˙ x > } . Lemma 2.5.
Assume that | y | < C holds both at initial and final moments andassume that we have initially | E − E † | , | e − e † | , | g − g † | < δ where † = ∗ or ∗∗ and ( E † , e † , g † ) are defined in Lemma 2.2. Then there exists C such that uniformlyin χ, µ we have the following estimates (a) | ¯ E − E | ≤ C µ, | ¯ e − e | ≤ C µ, | ¯ g − g | ≤ C µ. (b) | θ +4 − π | ≤ C µ, | ¯ θ − | ≤ C µ. (c) The flow time between the initial and final moments bounded by C χ . ONCOLLISION SINGULARITIES IN THE 2-CENTER-2-BODY PROBLEM 13
The proof of this lemma is given in Section 4. Notice that in the above two lemmas,we control the orbit parameters E , e , g , θ , but we do not talk about (cid:96) , e (recallthat g can be solved from θ , L , G ). Most of the work of the paper is devoted toshowing that there are two strongly expanding directions of the Poincar´e map whichenable us to prescribe (cid:96) , e arbitrarily.We also need the following fact which says that Q if initially captured by Q willalways be captured. Lemma 2.6.
Let C be as in Lemma 2.5. Suppose the initial orbit parameters x =( E , (cid:96) , e , g , e , g ) ∈ U j ( δ ) and the image G ◦ L ( x ) has | y | ≤ C . Then there areconstants µ , χ , D such that for µ ≤ µ and χ ≥ χ we have | Q ( t ) | ≤ − D for all t up to the time needed to define G ◦ L . The proof of this lemma is also given in Section 10.2.6.
Admissible surfaces.
Given a sequence ω we need to construct orbits havingsingularity with symbolic sequence ω . We will study the Poincar´e map P = G ◦ L to the surface { x = − , ˙ x > } . It isa composition of the local and global maps defined in the previous sections.We will also need the renormalization map R defined as follows. In Cartesian co-ordinates, we partition our six dimensional section { x = − , ˙ x > } into coordinatecubes of size 1 / √ χ . We next evaluate E at the center of each cube and denote itsvalue by − λ/
2, where λ > δ -close to λ in Lemma 2.2. The locally constant map R amounts to zooming in the configuration Q i = ( x i , y i ) , i = 3 , , by multiplying by λ and slowing down the velocity v i , i = 3 , √ λ. In addition wereflect the coordinates along the x axis. In Cartesian coordinates, the renormalizationtakes the form(2.5) R (( v i,x , v i,y ) , ( x i , y i ) , H, t ) = (cid:18) ( v i,x , − v i,y ) λ / , λ ( x i , − y i ) , Hλ , λ / t (cid:19) , i = 3 , . Since the renormalization R sends the section { x = − } to { x /λ = − } , we pushforward each cube along the flow to the section { x = − /λ, ˙ x > } . We include thepiece of orbits from the section { x = − , ˙ x > } to { x = − /λ, ˙ x > } to theglobal map and apply the R to the section { x = − /λ, ˙ x > } . This is then followedby a reflection. We have R ( { x = − /λ, ˙ x > } ) = { x = − , ˙ x > } , and R ( E , (cid:96) , e , g , e , g ) = ( E /λ, (cid:96) , e , − g , e , − g ) , where minus signs are the effect of the reflection.Note that the rescaling changes (for the orbits of interest, increases) the distancebetween the fixed centers by sending χ to λχ . Observe that at each step we have thefreedom of choosing the centers of the cubes. We describe how this choice is made inSection 3. In the following we give a proof of the main theorem based on the threelemmas, whose proofs are in the next section.We need to define cone fields K on T U ( R × T ) and K on T U ( R × T ) . Fix asmall constant η. Definition 2.7.
Let K to be the set of vectors which make an angle less than a smallnumber η with span ( d R w , ˜ w ) , and K to be the set of vectors which make an angleless than η with span ( w , ˜ w ) , where ˜ w = ∂∂(cid:96) and w j = ∂e ∂G ∂∂e − L L + G ∂∂g , j = 1 , . Lemma 2.8.
There is a constant c > such that for all x ∈ U ( δ ) satisfying P ( x ) ∈ U ( δ ) , and for all x ∈ U ( δ ) satisfying R ◦ P ( x ) ∈ U ( δ ) , (a) d P ( K ) ⊂ K , d ( R ◦ P )( K ) ⊂ K . (b) If v ∈ K , then (cid:107) d P ( v ) (cid:107) ≥ cχ (cid:107) v (cid:107) .If v ∈ K , then (cid:107) d ( R ◦ P )( v ) (cid:107) ≥ cχ (cid:107) v (cid:107) . We call a two dimensional C surface S ⊂ U ( δ ) (respectively S ⊂ U ( δ )) admis-sible if T S ⊂ K (respectively T S ⊂ K ). Then item (a) of Lemma 2.8 implies thatthe image of admissible surface is also admissible. More precisely, if S is admissibleand P ( S ) ∩ U ( δ ) (cid:54) = ∅ , then T U ( δ ) P ( S ) ⊂ K . A similar statement holds for thehigher iterates.From the explicit construction of the cones we get the following lemma. Lemma 2.9. (a)
The vector ˜ w = ∂∂(cid:96) is in K i . (b) For any plane Π in K i the projection map π e ,(cid:96) = ( de , d(cid:96) ) : Π → R isone-to-one. In other words ( e , (cid:96) ) can be used as coordinates on admissiblesurfaces. Using the invariance of the cone fields, we can reduce the six dimensional Poincar´emap to a two dimensional map defined on a cylinder. The reduction is done as follows.We introduce the following cylinder sets C ( δ ) = ( e ∗ − K (cid:48) δ, e ∗ + K (cid:48) δ ) × T , C ( δ ) = ( e ∗∗ − KK (cid:48) δ, e ∗∗ + KK (cid:48) δ ) × T . By Lemma 2.9, each piece of admissible surface S in U j ( δ ) is a graph of a function S of the variables ( e , (cid:96) ) ∈ C j ( δ ) . Hence P ( S ( e , (cid:96) )) becomes a function of twovariables ( e , (cid:96) ). However, P ( S ( · , · )) is well defined only on subsets of small measurein C j ( δ ) , since for most points ( e , (cid:96) ) ∈ C j ( δ ) the points S ( e , (cid:96) ) have orbits forwhich Q escapes from the system. The next lemma shows that certain open set V can always be found in C j ( δ ) on which P ( S ( · , · )) is defined and has large image wherewe call an admissible surface S large if π e ,(cid:96) S contains C j ( δ ) . In particular, given e ∈ ( e ∗ − K (cid:48) δ, e ∗ + K (cid:48) δ ) or ( e ∗∗ − KK (cid:48) δ, e ∗∗ + KK (cid:48) δ ), we can prescribe (cid:96) arbitrarily.Since the part of P ( S ) consisting of points which land on U ( δ ) or U ( δ ) is alsoadmissible by Lemma 2.8, we can apply Lemma 2.9 again to project the image to the( e , (cid:96) ) cylinder. Therefore we introduce the notation Q := π e ,(cid:96) P ( S ( · , · )) , Q := π e ,(cid:96) R ◦ P ( S ( · , · )) , whenever they are defined. Q j is a map from a subset of C j ( δ ) to C − j ( δ ) , j = 1 , Lemma 2.10.
For any < δ ≤ ¯ δ/ ( KK (cid:48) ) , we have the following. ONCOLLISION SINGULARITIES IN THE 2-CENTER-2-BODY PROBLEM 15 (a)
Given a large admissible surface S ⊂ U ( δ ) and ˜ e ∈ ( e ∗ − K (cid:48) δ, e ∗ + K (cid:48) δ ) thereexists ˜ (cid:96) such that P ( S (˜ e , ˜ (cid:96) )) ∈ U ( δ ) . Moreover if | ˜ e − e ∗ | < K (cid:48) δ − /χ ,then there is a neighborhood V (˜ e ) ⊂ C ( δ ) of (˜ e , ˜ (cid:96) ) such that Q maps V surjectively to C ( δ ) . (b) Given a large admissible surface S ⊂ U ( δ ) and ˜ e ∈ ( e ∗∗ − KK (cid:48) δ, e ∗∗ + KK (cid:48) δ ) there exists ˜ (cid:96) such that R ◦ P ( S (˜ e , ˜ (cid:96) )) ∈ U ( δ ) . Moreover if | ˜ e − e ∗∗ |
Construction of the singular orbit.
Fix a number ε which is much smallerthan δ but is much larger than both µ and 1 /χ. Pick (ˆ e , ˆ g ) so that | ˆ e − e ∗ | ≤ δ , | ˆ g − g ∗ | ≤ δ . Let S be an admissible surface such that the diameter of S is much larger than 1 /χ and such that on S we have | e − ˆ e | < ε, | g − ˆ g | < ε. For example, we can pick a point x ∈ U ( δ ) and let ˆ w be a vector in K ( x ) such that ∂∂(cid:96) ( ˆ w ) = 0 . Then let S = { ( E , (cid:96) , e , g , e , g )( x ) + a ˆ w + (0 , b, , , ,
0) where | a | ≤ ε/ ¯ K, b ∈ T } and ¯ K is a large constant.We wish to construct a singular orbit in S . We define S j inductively so that S j is a component of P ( S j − ) ∩ U ( δ ) if j is odd and S j is a component of ( R ◦P )( S j − ) ∩ U ( δ ) if j is even (we shall show below that such components exist). Let x = lim j →∞ ( RP ) − j S j . We claim that x has singular orbit.We define t = 0 and let t j be the time of x ’s 2 j -th visit to the section { x = − x > } . Since the global map gives only O ( µ ) small oscillation to E = | v | − | Q | by Lemma 2.5, and the local map is approximated by the Gerver map by Lemma 2.4,we apply Lemma 2.2 to get the unscaled energy of Q satisfies − E ( t j ) ≥ ( λ − ˜ δ ) j/ where ˜ δ → δ → , µ → . For the local map part in the rescaled system, bypart (c) of Lemma 2.9, Q and Q stay away from collision. By the continuity ofthe flow there is an upper bound τ of the flow time defining the local map for thoseinitial values satisfying the assumption of Lemma 2.4. Therefore without doing therescalings, during the j -th trip the time spent during the local map part is boundedfrom above by τ / ( λ − ˜ δ ) j/ using (2.5). For the global map part, we note that, by (2.1), the velocity of Q during the trip j is | v ( t j ) | > (cid:112) | E ( t j ) | ≥ ( λ − ˜ δ ) j/ . According to the definition of the renormalization R , the rescaled distance between Q and Q is χ j = | E ( t j ) | χ , where χ = | Q − Q | is the distance in the systemwithout rescalings, and using part (c) of Lemma 2.5, we have that without rescalingthe time defining the global map during the j -th trip is less than χ j / | E ( j ) | / ≤ const .χ ( λ − ˜ δ ) − j/ . Therefore combining the above analysis for the local and global maps, we have | t j +1 − t j | ≤ const .χ ( λ − ˜ δ ) − j/ and so t ∗ = lim j →∞ t j < ∞ as needed. It is also clear from the estimate of − E ( t j )and | v ( t j ) | that lim sup t → t ∗ | v i ( t ) | = lim sup t → t ∗ | ˙ Q i ( t ) | = ∞ , i = 3 , j we can find a component of P ( S j ) inside U ( δ )and a component of ( R ◦ P ( S j +1 )) inside U ( δ ) . We proceed inductively. So we assume that the statement holds for j (cid:48) < j and thatthere exist (ˆ e ,j , ˆ g ,j ) such that on S j we have(2.6) | e − ˆ e ,j | ≤ ε, | g − ˆ g ,j | ≤ ε. Note that due to rescaling defined in subsection 2.6 we have that on S j (cid:12)(cid:12)(cid:12)(cid:12) E − (cid:12)(cid:12)(cid:12)(cid:12) = O ( µ ) . Since S j is admissible it is a graph of a map S j : C ( δ ) → R × T . Let(2.7) S j +1 = P ( S j ( V ( e (cid:48) (ˆ e ,j , ˆ g ,j )))) . We claim that S j +1 is a large admissible surface in U ( δ ) . Indeed, by Lemma 2.5(b) θ − = O ( µ ) on S j +1 . Also e on S j +1 satisfies | e − e ∗∗ | ≤ KK (cid:48) δ since Q maps V ( e (cid:48) (ˆ e ,j , ˆ g ,j )) onto C ( δ ) . Therefore we have the required control on the orbit pa-rameters of Q . Next, Lemmas 2.4 and 2.5 show that on S j +1 we have | e − e ∗∗ | ≤ Kε, | g − g ∗∗ | ≤ Kε and | E − E ∗∗ | ≤ Kε.
Thus S j +1 ⊂ U ( δ ) and by Lemma 2.8, S j +1 is admissible. In fact, it is a largeadmissible surface due to Lemma 2.9(a).In addition, since S j +1 ⊂ U ( δ ) it follows that P : S j → S j +1 is strongly ex-panding. We claim that this implies that the oscillations of e and g of S j +1 are lessthan ε if µ is small enough. Namely, by Lemma 2.8(b) the preimage of S j +1 has size O (1 /χ ) . Hence e and g have oscillations of size O (1 /χ ) on S j V ( e (cid:48) (ˆ e ,j , ˆ g ,j )) whileLemmas 2.4 and 2.5 show that the oscillations do not increase much after applicationof local and global maps. Thus there are numbers ˜ e ,j and ˜ g ,j such that on S j +1 | e − ˜ e ,j | ≤ ε, | g − ˜ g ,j | ≤ ε. Since S j +1 is admissible, it is a graph of a map S j +1 : C ( δ ) → R × T . Let(2.8) S j +2 = R ◦ P ( S j +1 ( V ( e (cid:48)(cid:48) (ˆ e ,j , ˆ g ,j )))) . ONCOLLISION SINGULARITIES IN THE 2-CENTER-2-BODY PROBLEM 17
The same argument as for S j +1 shows that S j +2 is a large admissible surface in U ( δ )and that (2.6) holds on S j +2 (with j replaced by j + 1). The only caveat is that thesurfaces S j are not smooth but only piecewise smooth since the rescaling map R isdiscontinuous. However we can use the freedom to choose the appropriate partition inthe definition of R to ensure that R is continuous on the preimage of V ( e (cid:48) (ˆ e ,j , ˆ g ,j ))so that S j V ( e (cid:48) (ˆ e ,j , ˆ g ,j )) is a smooth surface.This completes the construction of a singular orbit. Remark 2.11.
In fact we do not need to use exactly e (cid:48) (ˆ e ,j , ˆ g ,j ) and e (cid:48)(cid:48) (ˆ e ,j , ˆ g ,j ) in (2.7) and (2.8) . Namely any V ( e † ) and V ( e ‡ ) would do provided that (cid:12)(cid:12)(cid:12) e † − e (cid:48) (ˆ e ,j , ˆ g ,j ) (cid:12)(cid:12)(cid:12) < ε, (cid:12)(cid:12)(cid:12) e ‡ − e (cid:48)(cid:48) (ˆ e ,j , ˆ g ,j ) (cid:12)(cid:12)(cid:12) < ε. Different choices of e † and e ‡ allow us obtain different orbits. Since such freedomexists at each step of our construction we have a Cantor set of singular orbits with agiven symbolic sequence ω . Hyperbolicity of the Poincar´e map
Construction of invariant cones.
Here we derive Lemma 2.8, 2.9 and 2.10dealing with the asymptotics of the derivative of local and global maps.
Lemma 3.1.
Fix j ∈ { , } meaning the first or second collision. ( a ) Let ˜ θ be a small constant. Consider x ∈ U j ( δ ) satisfying (1) the orbit with initial value x passes through a δ neighborhood of the j -thGerver’s collision point. (2) | θ − ( x ) | ≤ C µ where C is as in Lemma 2.4. (3) y = L ( x ) ∈ { x = − , ˙ x < } satisfies | ¯ θ +4 ( y ) − π | ≤ ˜ θ. Then there exist continuous functions u j ( x , ¯ θ +4 ) , l j ( x ) and B j ( x , ¯ θ +4 ) such that d L ( x ) = 1 µ ( u j ( x , ¯ θ +4 ) + o (1)) ⊗ ( l j ( x ) + o (1)) + B j ( x , ¯ θ +4 ) + o (1) , as 1 /χ (cid:28) µ → . ( b ) Moreover there exist a linear functional ˆ l j , a vector ˆ u j and a matrix ˆ B j withbounded norms, such that if we take further limits δ → and ˜ θ → , we have l j ( x ) → ˆ l j , u j ( x , ¯ θ +4 ) → ˆ u j , B j ( x , ¯ θ +4 ) → ˆ B j . This lemma is proven in Section 12.
Lemma 3.2.
Fix j ∈ { , } meaning the first or second collision.Let x ∈ { x = − , ˙ x < } and y = G ( x ) ∈ { x = − , ˙ x > } be such that | y ( x ) | ≤ C , | y ( y ) | ≤ C where C is as in Lemma 2.5. Then (a) there exist continuous linear functionals ¯ l j ( x ) and ¯¯ l j ( x ) and vectorfields ¯ u j ( y ) and ¯¯ u j ( y ) , such that as /χ (cid:28) µ → d G ( x ) = χ (¯ u j ( y ) + o (1)) ⊗ (cid:0) ¯ l j ( x ) + o (1) (cid:1) + χ (¯¯ u j ( y ) + o (1)) ⊗ (cid:16) ¯¯ l j ( x ) + o (1) (cid:17) + O ( µχ ) . (b) If x ∈ U j ( δ ) satisfies G ◦ L ( x ) ∈ U − j ( δ ) for j = 1 or R◦ G ◦ L ( x ) ∈ U − j ( δ ) for j = 2 , and the orbit with initial value x passes through a δ neighborhood of the j -th Gerver’s collision point, then there exist vector w j and linear functionals ˆ¯ l j , ˆ¯¯ l j such that for δ → , we have ¯ l j ( x ) → ˆ¯ l j , ¯¯ l j ( x ) → ˆ¯¯ l j , span (¯ u j ( y ) , ¯¯ u j ( y )) → span ( w j , ˜ w ) . (c) Finally if we define in Delaunay coordinates ˆ¯ l = (cid:18) G /L L + G , , , , − L + G , − L (cid:19) , ˆ¯¯ l = (1 , , , , , ,w = (cid:18) , , , , , L L + G (cid:19) T , ˜ w = (0 , , , , , T , (3.1) then ˆ¯ l j and ˆ¯¯ l j are obtained from ˆ¯ l and ˆ¯¯ l respectively by evaluating G , L atGerver’s collision point immediately after the j -th collision, and w j is obtainedfrom w by evaluating G , L at Gerver’s collision point immediately before the (3 − j ) -th collision. Remark 3.3.
We remark that the w j , j = 1 , in Definition 2.7 is the same as the w j here, but written in different coordinates. This lemma is proven in Section 3.2.
Lemma 3.4.
The following non degeneracy conditions are satisfied for E ∗ = − / , e ∗ =1 / , g ∗ = π/ . (a1) span (ˆ u , B (ˆ l ( ˜ w ) d R w − ˆ l ( d R w ) ˜ w )) is transversal to Ker (ˆ¯ l ) ∩ Ker (ˆ¯¯ l ) . (a2) de ( span ( d R w , d R ˜ w )) (cid:54) = 0 . (b1) span (ˆ u , B (ˆ l ( ˜ w ) w − ˆ l ( w ) ˜ w )) is transversal to Ker (ˆ¯ l ) ∩ Ker (ˆ¯¯ l ) . (b2) de ( w ) (cid:54) = 0 . This lemma is proven in Section 3.3.
Proof of Lemma 2.8.
Consider for example the case where x ∈ U ( δ ) . We claim thatif δ, µ are small enough then d L (span( w , ˜ w )) is transversal to Ker¯ l ∩ Ker¯¯ l . Indeedtake Γ such that l (Γ) = 0 . If Γ = aw + ˜ a ˜ w then a l ( w ) + ˜ a l ( ˜ w ) = 0 . It followsthat the direction of Γ is close to the direction of ˆΓ = ˆ l ( ˜ w ) w − ˆ l ( w ) ˜ w. Next take˜Γ = bw + ˜ b ˜ w where b l ( w ) + ˜ b l ( ˜ w ) (cid:54) = 0 . Then the direction of d L ˜Γ is close to ˆ u andthe direction of d L (Γ) is close to B (ˆΓ) so our claim follows.Thus for any plane Π close to span( w , ˜ w ) we have that d L (Π) is transversal toKer¯ l ∩ Ker¯¯ l . Take any Y ∈ K . Then either Y and w are linearly independentor Y and ˜ w are linearly independent. Hence d L (span( Y, w )) or d L (span( Y, ˜ w )) istransversal to Ker¯ l ∩ Ker¯¯ l . Accordingly either ¯ l ( d L ( Y )) (cid:54) = 0 or ¯¯ l ( d L ( Y )) (cid:54) = 0 . If¯ l ( d L ( Y )) (cid:54) = 0 then the direction of d ( G ◦ L )( Y ) is close to ¯ u . If ¯ l ( d L ( Y )) = 0 thenthe direction of d ( G ◦ L )( Y ) is close to ¯¯ u . In either case d ( R G ◦ L )( Y ) ∈ K and ONCOLLISION SINGULARITIES IN THE 2-CENTER-2-BODY PROBLEM 19 (cid:107) d ( G ◦ L )( Y ) (cid:107) ≥ cχ (cid:107) Y (cid:107) . This completes the proof in the case x ∈ U ( δ ) . The casewhere x ∈ U ( δ ) is similar. (cid:3) To prove Lemma 2.10 we need two auxiliary results.
Sublemma 3.5.
In the notation and setting of part ( a ) of Lemma 2.10, given ˜ e thereexists ˜ (cid:96) such that P ( S (˜ e , ˜ (cid:96) )) ∈ U ( δ ) . There is a corresponding statement to part ( b ) of Lemma 2.10. The proof of this sublemma is postponed to Section 11.2.
Sublemma 3.6.
Let F be a map on R which fixes the origin and such that if |F ( z ) | Without the loss of generality we may assume that a = ( r, . Let V ( z ) be thedirection field defined by the condition that the direction of d F ( V ( z )) is parallel to(1 , . Let γ ( t ) be the integral curve of V passing through the origin and parameterizedby the arclength. Then F ( γ ( t )) has form ( σ ( t ) , 0) where σ (0) = 0 and | ˙ σ ( t ) | ≥ ¯ χ aslong as | σ | < R. Now the statement follows easily. (cid:3) Proof of Lemma 2.10. ( a ) We claim that it suffices to show that for each (¯ e , ¯ (cid:96) ) suchthat | ¯ e − e ∗∗ | < √ δ there exist (ˆ e , ˆ (cid:96) ) such that(3.2) Q (ˆ e , ˆ (cid:96) ) = (¯ e , ¯ (cid:96) ) . Indeed in that case Sublemma 4.9 from Section 4.3 says that the outgoing asymp-tote is almost horizontal. Therefore by Lemma 2.4 our orbit has ( E , e , g ) close to G ˜ e , , ( E (ˆ e , ˆ (cid:96) ) , e (ˆ e , ˆ (cid:96) ) , g (ˆ e , ˆ (cid:96) )) . Next Lemma 2.5 shows that after the applica-tion of G , ( E , e , g ) change little and θ − becomes O ( µ ) so that P ( S (ˆ e , ˆ (cid:96) )) ∈ U ( δ ) . We will now prove (3.2). Due to Lemma 2.8 we can apply Sublemma 3.6 to thecovering map ˜ Q : R → R with ¯ χ = cχ obtaining (3.2). This completes the proof ofpart (a).Part (b) is similar to part (a).Part (c) follows from Lemma 10.2 proven in Section 10. (cid:3) Expanding directions of the global map. Estimating the derivative of theglobal map is the longest part of the paper. It occupies Sections 5–8.It will be convenient to use the Delaunay coordinates ( L , (cid:96) , G , g ) for Q and( G , g ) for Q . Delaunay coordinates are action-angle coordinates for the Keplerproblem. We collect some facts about the Delaunay coordinates in Appendix A.We divide the plane into several pieces by lines x = − x = − χ . Those linescut the orbit of Q into 4 pieces: • { x = − , ˙ x < } → (cid:8) x = − χ , ˙ x < (cid:9) . We call this piece ( I ). • (cid:8) x = − χ , ˙ x < (cid:9) → (cid:8) x = − χ , ˙ x > (cid:9) turning around Q . We call it( III ). • (cid:8) x = − χ , ˙ x > (cid:9) → { x = − , ˙ x > } . We call it ( V ) • { x = − , ˙ x > } → { x = − , ˙ x < } turning around Q .We composition of the first three pieces constitutes the global map. The last piecedefines the local map. See Fig 3. Notice that when we define R in Section 2.6, afterthe second collision in Gerver’s construction, the global map sends { x = − , ˙ x < } to { x = − /λ, ˙ x > } . Then R sends { x = − /λ, ˙ x > } to { x = − , ˙ x > } before applying local map. So without leading to confusion, when we are talkingabout sections after the second collision, we always talk about R ◦ G so that thesection { x = − , ˙ x < } is sent to { x = − , ˙ x > } . Figure 3. Poincar´e sectionsThe line x = − χ is convenient because if Q is moving to the right of the line x = − χ , its motion can be treated as a hyperbolic motion focused at Q withperturbation caused by Q and Q . If Q is moving to the left of this line, its motioncan be treated as a hyperbolic motion focused at Q perturbed by Q and Q .Since we use different guiding centers to the left and right of the line of x = − χ wewill need to change variables when Q hits this line. This will give rise to two morematrices for the derivative of the global map: ( II ) will correspond to the change ofcoordinates from right to left and ( IV ) will correspond for the change of coordinatesfrom left to right. Thus d G = ( V )( IV )( III )( II )( I ) . In turn, each of the matrices ( II )and ( IV ) will be products of three matrices corresponding to changing one variableat a time. Thus we will have ( II ) = [( iii )( ii )]( i ) and ( IV ) = ( iii (cid:48) )[( ii (cid:48) )( i (cid:48) )].The asymptotics of the above mentioned matrices is presented in the two proposi-tions below.To refer to a certain subblock of a matrix ( (cid:93) ), we use the following convention:( (cid:93) ) = (cid:20) ( (cid:93) ) ( (cid:93) ) ( (cid:93) ) ( (cid:93) ) (cid:21) . Thus ( (cid:93) ) is a 4 × (cid:93) ) is a 2 × i, j ) − th entry of a matrix ( (cid:93) ) (in the Delaunay coordinates mentioned above) we use ( (cid:93) )( i, j ).For example, ( I )(1 , 3) means the derivative of L with respect to G when the orbitmoves between sections { x = − } and (cid:8) x = − χ (cid:9) . ONCOLLISION SINGULARITIES IN THE 2-CENTER-2-BODY PROBLEM 21 Proposition 3.7. Under the assumptions of Lemma 3.2 the matrices introduced abovesatisfy the following estimates. ( I ) = Id + O ( µ ) O ( µ ) × O ( µ ) × O ( χ ) O ( µχ ) × O ( µχ ) × O ( µ ) × O ( µ ) × O ( µ ) × O (1) × O ( µ ) × O (1) × , ( i ) = × × × ˜ G R /k R ˜ L k R ˜ L + ˜ G R + O ( χ ) O ( χ ) × − k R ˜ L + ˜ G R + O ( χ ) − k R ˜ L + O ( χ ) , [( iii )( ii )] = × × O (1 /χ ) O (1 /χ ) × − χO (1 /χ ) O (1 /χ ) × L + O (1 /χ ) − χ ˜ L + O (1) , ( III ) = Id + O (1 /χ ) O (1 /χ ) × O ( µ/χ ) × O ( χ ) O (1 /χ ) × O (1) × O (1 /χ ) × O (1 /χ ) × O ( µ/χ ) × O ( µ ) × O ( µ/χ ) × O (1) × , [( ii (cid:48) )( i (cid:48) )] = × × O (1) O (1 /χ ) × χ ˆ L + O (1) χ ˆ L + O (1) O (1 /χ ) O (1 /χ ) × L + O (1 /χ ) L + O (1 /χ ) , ( iii (cid:48) ) = × × × − ˆ G R / ( k R )( k R ˆ L + G R ) + O ( χ ) O ( χ ) × k R ˆ L k R ˆ L + ˆ G R + O ( χ ) k R ˆ L + O ( χ ) , ( V ) = Id + O ( µχ ) O ( µ ) × O ( µ ) × O ( χ ) O ( µ ) × O (1) × O ( µχ ) × O ( µ ) × O ( µ ) × O ( µχ ) × O ( µ ) × O (1) × . where k R = 1 + µ, ˜ L , ˜ G are the initial values of G of L , G and ˆ L , ˆ G are the finalvalues of G of L , G . Moreover, the matrix of the renormalization map R has theform diag {√ λ, , −√ λ, − , −√ λ, − } , where the constant λ is the dilation rate definedin Section 2.6 and the “ − ” appears due to the reflection. Proposition 3.8. (a) The O ( χ ) entries in the matrices ( I ) , ( III ) , ( V ) are c I χ, c III χ, c V χ ,where c I , c III , c V (cid:54) = 0 and have the same sign. (b) The O (1) blocks in Proposition 3.7 can be written as a continuous function of x and y plus an error which vanishes in the limit µ → , χ → ∞ . Moreoverthe O (1) blocks have the following limits for orbits of interest. ( I ) = − ˜ L L + ˜ G ) − ˜ L L L + ˜ G ) ˜ L L + ˜ G ) , ( III ) = (cid:20) − L L (cid:21) , ( V ) = / L ˆ L + ˆ G − / L / L (ˆ L + ˆ G ) − / L ˆ L + ˆ G . In addition for map (I) we have(( I )(5 , , ( I )(6 , T = (cid:32) ˜ G ˜ L 2( ˜ L + ˜ G ) , − ˜ G ˜ L 2( ˜ L + ˜ G ) (cid:33) T . where tilde, hat have the same meanings as in the previous proposition. The estimates of ( I ) , ( III ) , ( V ) from Proposition 3.7 are proven in Sections 4–7. The estimates of ( II ) , ( IV ) are given in Section 8. Proposition 3.8 is proven inSection 6.2. Now we prove Lemma 3.2 based on the Proposition 3.8. Proof of Lemma 3.2. For the matrices ( I ) , ( III ) , ( V ), we separate the (2 , 1) entry fromthe matrices to get the following decompositions into a tensor part and a remainder.(3.3) ( I ) = c I χ ¯ u ⊗ ¯ l + R I , ( III ) = c III χ ¯ u ⊗ ¯ l + R III , ( V ) = c V χ ¯ u ⊗ ¯ l + R III where ¯ u = (0 , , , , , l = (1 , , , , , 0) and the tensor term picks out the O ( χ )entry in each matrix. For the matrices [( iii )( ii )] and [( ii (cid:48) )( i (cid:48) )], we separate the leadingterms of the 44 blocks to get the following decompositions[( iii )( ii )] = χu II ⊗ l II + R II , [( ii (cid:48) )( i (cid:48) )] = χu IV ⊗ l IV + R IV where u II = (cid:18) , , , , , L (cid:19) , l II = (cid:18) , , , , χ , − (cid:19) u IV = (cid:18) , , , , , χ (cid:19) , l IV = (cid:18) , , , , L , L (cid:19) . Notice l IV · ¯ u = ¯ l · u II = 0 . Multiplying ( ii (cid:48) )( i (cid:48) )( III )( iii )( ii ), we get( ii (cid:48) )( i (cid:48) )( III )( iii )( ii ) = ( χu IV ⊗ l IV + R IV )( III )( χu II ⊗ l II + R II )= χ u IV ⊗ l IV ( III ) u II ⊗ l II + χR IV ( III ) u II ⊗ l II + χu IV ⊗ l IV ( III ) R II + R IV ( III ) R II = χ u IV ⊗ l IV R III u II ⊗ l II + χR IV R III u II ⊗ l II + χu IV ⊗ l IV R III R II + R IV ( III ) R II We define c = l IV R III u II and v = R IV R III u II , v (cid:48) = l IV R III R II . ONCOLLISION SINGULARITIES IN THE 2-CENTER-2-BODY PROBLEM 23 Continuing the computation we get(3.4)= cχ u IV ⊗ l II + χv ⊗ l II + χu IV ⊗ v (cid:48) + R IV ( III ) R II = cχ (cid:18) u IV + vcχ (cid:19) ⊗ (cid:18) l II + v (cid:48) cχ (cid:19) − c v ⊗ v (cid:48) + c III χR IV ¯ u ⊗ ¯ lR II + R IV R III R II where(a) c = (cid:18) × , L , L (cid:19) (cid:18)(cid:20) id 00 ( III ) (cid:21) + (cid:20) O (1 /χ ) O (1) O ( µ ) O ( µ ) (cid:21)(cid:19) (cid:18) × , , L (cid:19) T → (cid:18) L , L (cid:19) ( III ) (cid:18) , L (cid:19) T = 2 L (cid:54) = 0 . (b) v = O (cid:16) µχ , , µχ , µχ , , χ (cid:17) T and v (cid:48) = O (cid:16) χ , µχ , µχ , µχ , χ , (cid:17) .(c) R IV ¯ u = ¯ u + O (0 × , /χ , /χ ) and ¯ lR II = ¯ l. (d) The remainder R IV R III R II is explicitly computed= (cid:20) id 00 0 (cid:21) + O χ ( χ ) × µχ µχ χ ) × χ χ ) × ( χ ) × ( µχ ) × ( µχ ) × µχ ) × χ χ ( µχ ) × χ χ .We next multiply ( i )( I ) from the right and ( V )( iii (cid:48) ) from the left to ( ii (cid:48) )( i (cid:48) )( III )( iii )( ii )to get ( V )( IV )( III )( II )( I ) = ( c I + c III + c V ) χ ¯ u ⊗ ¯ l + cχ ¯¯ u ⊗ ¯¯ l + O ( µχ )where(1) we have the following limit using Proposition 3.8¯¯ u = ( V )( iii (cid:48) ) (cid:18) u IV + vcχ (cid:19) → w + const . ¯ u, ¯¯ l = (cid:18) l II + v (cid:48) cχ (cid:19) ( i )( I ) → ˆ¯ l as 1 /χ (cid:28) µ → 0. In fact ¯¯ u is essentially the fifth column of ( iii (cid:48) ) and ¯¯ l isessentially the sixth row of ( i ).(2) using item (b) above, we have ( V )( iii (cid:48) ) v ⊗ v (cid:48) ( i )( I ) = O (1). The estimate( V )( iii (cid:48) ) v = O (1) essentially picks out the second, fifth and sixth columns of( V ), the estimate v (cid:48) ( i )( I ) = O (1) essentially picks out the first, fifth and sixthrows of ( I ), and the O ( χ ) entries in ( I ) and ( V ) are suppressed by the smallentries of v (cid:48) (the second entry) and v (the first entry) respectively.(3) using item (c) above, we have( V )( iii (cid:48) )( c III χR IV ¯ u ⊗ ¯ lR II )( i )( I ) = c III χ ¯ u ⊗ ¯ l + O (1) . (4) using the decomposition of ( I ) and ( V ) in (3.3), we can verify that c V χ ¯ u ⊗ ¯ l ( iii (cid:48) ) R IV R III R II ( i ) c I χ ¯ u ⊗ ¯ l = ( c I + c V ) χ ¯ u ⊗ ¯ l + O (1) . Here the O (1) estimate of the remainder comes from the O (1 /χ ) estimate ofthe (1 , 2) entry of the matrix R IV R III R II , which in turn comes from the sameestimate of the (1 , 2) entry of ( III ) . (5) All the remaining terms in ( V )( iii (cid:48) ) R IV R III R II ( i )( I ) other than item (4)above are absorbed into O ( µχ ). (Note that the special structure of the ma-trices is important for the estimate. In particular, though the first column of( V ) and the second row of ( I ) are large, the first row and second column in R IV R III R II are small.) (cid:3) Checking transversality. In Lemma 3.1 and 3.2 when we take limits ˜ θ, δ, µ, /χ → 0, the dynamics in the phase space reduces to Gerver’s case. The limiting vectorsˆ l j , ˆ¯ l j , ˆ¯¯ l j and ˆ u j , w, ˜ w can be computed explicitly and evaluated at Gerver’s collisionpoints. In the following lemmas we consider Gerver’s orbits with the choice of E ∗ = − and e ∗ = . All the other orbit parameters are determined by E ∗ , e ∗ as shown in Ap-pendix B.2.The O (1 /µ ) part of d L in Lemma 3.1 satisfies the following estimates. Lemma 3.9. The asymptotics ˆ l j and ˆ u j of the vectors l j , u j in the O (1 /µ ) part ofthe matrix d L satisfy the following: (a) ˆ l j · ˜ w (cid:54) = 0 , ˆ l j · w − j (cid:54) = 0 , ˆ¯ l j · ˆ u j (cid:54) = 0 ,j = 1 , meaning the first or the second collision. (b) If Q and Q switch roles after the collisions, the vectors ˆ u and ˆ u get a “ − ”sign. To check the nondegeneracy condition, it is enough to know the following. Lemma 3.10. Let x ∈ U j ( δ ) and | ¯ θ +4 − π | < ˜ θ (cid:28) be as in Lemma 3.1. If we take thedirectional derivative at x of the local map along a direction Γ ∈ span { ¯ u − j , ¯¯ u − j } ⊂ T x U j ( δ ) , such that ¯ l j · ( d L Γ) = 0 , j = 1 , , then lim /χ (cid:28) µ → ∂E +3 ∂ Γ is a continuous function of both x and ¯ θ +4 , where E +3 ( respectively ¯ θ +4 ) is the energy of Q ( respectively outgoing asymptote of Q ) after the close en-counter with Q . If we take further limits δ → and ˜ θ → , we have lim δ, ˜ θ → lim /χ (cid:28) µ → ∂E +3 ∂ Γ (cid:54) = 0 . The proofs of the two lemmas are postponed to Section 12. Now we can check thenondegeneracy condition. Proof of Lemma 3.4. We prove (b1) and (b2). The proofs of (a1) and (a2) are similarand are left to the reader. ONCOLLISION SINGULARITIES IN THE 2-CENTER-2-BODY PROBLEM 25 To check (b2), de we differentiate e = (cid:112) G /L ) to get de = 1 e (cid:18) G L dG − G L dL (cid:19) . Thus (3.1) gives de w = G L (cid:54) = 0 as claimed.Next we check (b1) which is equivalent to the following condition(3.5) det (cid:32) ˆ¯ l (ˆ u ) ˆ¯ l ( ˆ B Γ (cid:48) ))ˆ¯¯ l (ˆ u ) ˆ¯¯ l ( ˆ B Γ (cid:48) ) (cid:33) (cid:54) = 0 . where Γ (cid:48) = ˆ l ( ˜ w ) w − ˆ l ( w ) ˜ w. The vector Γ (cid:48) (cid:54) = 0 due to part (a) of Lemma 3.9.Let Γ be a vector satisfying ˆ¯ l · ( d L Γ) = 0 and chosen as follows. d L Γ is a vectorin span { ˆ u i , ˆ B i Γ (cid:48) i } , so it can be represented as d L Γ i = b ˆ u + b (cid:48) ˆ B Γ (cid:48) . Thus we can take b = − ˆ¯ l · ˆ B Γ (cid:48) and b (cid:48) = ˆ¯ l (ˆ u ) to ensure that d L Γ i ∈ Ker ˆ¯ l . Note that we have b (cid:48) (cid:54) = 0by part (a) of Lemma 3.9. Hencedet (cid:32) ˆ¯ l (ˆ u ) ˆ¯ l ( ˆ B Γ (cid:48) )ˆ¯¯ l ( u ) ˆ¯¯ l ( ˆ B Γ (cid:48) ) (cid:33) = 1 b (cid:48) det (cid:32) ˆ¯ l (ˆ u ) ˆ¯ l ( d L Γ)ˆ¯¯ l (ˆ u ) ˆ¯¯ l ( d L Γ) (cid:33) = ˆ¯¯ l ( d L Γ)where the last equality holds since ˆ¯ l ( d L Γ) = 0 . By (3.1) ˆ¯¯ l i = (1 , , , , , l ( d L Γ) = ∂E +3 ∂ Γ and so (b2) follows from Lemma 3.10. (cid:3) Remark 3.11. Let us describe the physical and geometrical meanings of the vectors ¯ l , ¯¯ l , ¯ u , ¯¯ u , l , u and the results in this section. (1) The structure of d L shows that a significant change of the behavior of theoutgoing orbit parameters occurs when we vary the orbit parameters in thedirection of l , which is actually varying the closest distance (called impactparameter) between Q and Q (see Section 12, especially Corollary 12.1).The vector w in d G shows that after the global map, the variable G getssignificant change as asserted by Lemma 2.10. So ˆ l i · w − i (cid:54) = 0 in Lemma 3.9means that by changing G after the global map, we can change the impactparameter and hence change the outgoing orbit parameters after the local mapsignificantly. Similarly we see ˆ l i · ˜ w (cid:54) = 0 means the same outcome by varying (cid:96) instead of G . (2) The result ˆ¯ l i · ˆ u i (cid:54) = 0 in Lemma 3.9 means that by changing the outgoing orbitparameter of the local map in ˆ u direction, which is in turn changed significantlyby changing the impact parameter in the local map, we can change the finalorbit parameter of the global map in the ¯ u direction significantly. The vector ˆ¯ l has clear physical meaning. If we differentiate the outgoing asymptote θ +4 = g +4 + arctan G +4 L +4 , where + means after close encounter of Q and Q , we get dθ +4 = L +4 ˆ¯ l . (3) Lemma 3.10 means that if we vary the incoming orbit parameter of the localmap in the direction Γ such that there is no significant change of the outgoing parameters of the local map in certain direction, then the energy (and, hence,semimajor axis) of the ellipse after Q , Q interaction will change accordingly.One may think this as varying the incoming orbit parameter while holding theoutgoing asymptotes unchanged. The change of energy means the change ofperiods of the ellipses according to Kepler’s law. Ellipses with different periodswill accumulate huge phase difference during one return time O ( χ ) of Q . Thisis the mechanism that we use to fine tune the phase of Q such that Q comesto the correct phase to interact with Q . Since the phase is defined up to π ,we get a Cantor set as initial condition of singular orbits. C estimates for global map Equations of motion in Delaunay coordinates. We use Delaunay variablesto describe the motion of Q and Q (for reader’s convenience we collect the basic prop-erties of Delaunay variables in Appendix A). We have eight variables ( L , (cid:96) , G , g )and ( L , (cid:96) , G , g ). We consider the Hamiltonian (2.1).When Q is moving to the left of the section { x = − χ/ } , we consider the motionof Q as elliptic motion with focus at Q , and Q as hyperbolic motion with focusat Q , perturbed by other interactions. We can write the Hamiltonian in terms ofDelaunay variables as H L = − L + 12 L − | Q | − | Q − ( − χ, | − µ | Q − Q | . When Q is moving to the right of the section { x = − χ/ } , we consider the motion of Q as an elliptic motion with focus at Q , and that of Q as a hyperbolic motion withfocus at Q attracted by the pair Q , Q which has mass 1+ µ plus a perturbation. For | Q | ≥ O is in the sense | Q | → ∞ ,µ | Q − Q | = µ | Q | + µQ · Q | Q | + O (cid:18) µ | Q | (cid:19) . Hence the Hamiltonian takes form H = v v − | Q | − µ | Q | − | Q − ( − χ, | − | Q − ( − χ, | − µQ · Q | Q | + O (cid:18) µ | Q | (cid:19) . In terms of the Delaunay variables we have(4.1) H R = − L + (1 + µ ) L − | Q + ( χ, | − | Q + ( χ, | − µQ · Q | Q | + O (cid:18) µ | Q | (cid:19) . We shall use the following notation. The coefficients of L in the Hamiltonian willbe called k L = 1 and k R = 1 + µ. The terms in the Hamiltonian containing Q will bedenoted by(4.2) V R = − | Q + ( χ, | − µQ · Q | Q | + O (cid:18) µ | Q | (cid:19) , and V L = − | Q | − µ | Q − Q | . Here subscripts L and R mean that the corresponding expressions are used when Q is to the left (respectively to the right) of the line Q = − χ . Likewise for the terms ONCOLLISION SINGULARITIES IN THE 2-CENTER-2-BODY PROBLEM 27 containing Q we define(4.3) U R = − | Q + ( χ, | − µQ · Q | Q | + O (cid:18) µ | Q | (cid:19) , U L = − | Q − ( − χ, | − µ | Q − Q | . The use of subscripts R, L here is the same as above. Let us write down the fullHamiltonian equations with the subscripts R and L suppressed.(4.4) ˙ L = − ∂Q ∂(cid:96) · ∂U∂Q , ˙ (cid:96) = 1 L + ∂Q ∂L · ∂U∂Q , ˙ G = − ∂Q ∂g · ∂U∂Q , ˙ g = ∂Q ∂G · ∂U∂Q , ˙ L = − ∂Q ∂(cid:96) · ∂V∂Q , ˙ (cid:96) = − k L + ∂Q ∂L · ∂V∂Q , ˙ G = − ∂Q ∂g · ∂V∂Q , ˙ g = ∂Q ∂G · ∂V∂Q . Next we use the energy conservation to eliminate L . Setting H = 0, we have(4.5) L k R = k R L · (cid:18) − L (cid:18) | Q + ( χ, | + 1 | Q + ( χ, | + µQ · Q | Q | + O (cid:18) µ | Q | (cid:19) + O (1 /χ ) (cid:19)(cid:19) := k R L + W R ,L k L = k L L (cid:18) − L (cid:18) | Q + ( χ, | + 1 | Q | − µ | Q − Q | + O (1 /χ ) (cid:19)(cid:19) : = k L L + W L . We use (cid:96) as the independent variable. Dividing (4.4) by ˙ (cid:96) and using (4.5) toeliminate L we obtain(4.6) dL d(cid:96) = ( kL + W ) ∂Q ∂(cid:96) · ∂U∂Q (cid:18) kL + W ) ∂Q ∂L · ∂V∂Q (cid:19) d(cid:96) d(cid:96) = − ( kL + W )( 1 L + ∂Q ∂L · ∂U∂Q ) (cid:18) kL + W ) ∂Q ∂L · ∂V∂Q (cid:19) dG d(cid:96) = ( kL + W ) ∂Q ∂g · ∂U∂Q (cid:18) kL + W ) ∂Q ∂L · ∂V∂Q (cid:19) dg d(cid:96) = − ( kL + W ) ∂Q ∂G · ∂U∂Q (cid:18) kL + W ) ∂Q ∂L · ∂V∂Q (cid:19) dG d(cid:96) = ( kL + W ) ∂Q ∂g · ∂V∂Q (cid:18) kL + W ) ∂Q ∂L · ∂V∂Q (cid:19) dg d(cid:96) = − ( kL + W ) ∂Q ∂G · ∂V∂Q (cid:18) kL + W ) ∂Q ∂L · ∂V∂Q (cid:19) + O (cid:18) µ | Q | + 1 /χ (cid:19) . We shall use the following notation: X = ( L , (cid:96) , G , g ) , Y = ( G , g ) . a priori bounds. In this section, we give some estimates that will be used toestimate the derivatives of the global map in later sections.4.2.1. Estimates of positions. We have the following estimates for the positions. Lemma 4.1. Given C and D > there exists C (cid:48) such that if (4.7) | Q | < − D, | Q y | < C then (a) we have (4.8) (cid:12)(cid:12)(cid:12)(cid:12) ∂Q ∂X (cid:12)(cid:12)(cid:12)(cid:12) < C (cid:48) ;(b) when Q is moving to the right of the section { x = − χ/ } we have (4.9) | Q ( (cid:96) ) | (cid:40) ≥ , if | (cid:96) ∗ | ≤ | (cid:96) | ≤ C ∈ (cid:2) , (cid:3) L ( (cid:96) ∗ ) | (cid:96) | , if | (cid:96) | ≥ C, where (cid:96) ∗ is the value of (cid:96) restricted on x = − when Q is moving to the left of the section { x = − χ/ } , we have (4.10) | Q ( (cid:96) ) − Q | ≤ L ( (cid:96) ∗ ) | (cid:96) | + C (cid:48) for some constant C (cid:48) where (cid:96) ∗ is the value of (cid:96) on the section { x = − χ/ } . This lemma justifies the following intuitive facts. Since Q and Q are away fromclose encounter, the motion of Q is almost Kepler elliptic motion hence we get item(a). The motion of Q is a perturbed Kepler hyperbolic motion for both the leftand the right case, hence for most of the time Q as a function of the time (cid:96) isalmost linear (item (b)). To give the complete proof we have to use the Hamiltonianequations. See Section 4.3. The next several lemmas relies on the conclusion of thislemma. Lemma 4.2. If inequalities (4.7) , (4.9) , (4.10) are valid and in addition (4.11) 1 /C ≤ | L | , | L | ≤ C, | G | , | G | < C, then we have ∂Q ∂(cid:96) = O (1) , ∂Q ∂ ( L , G , g ) = O ( (cid:96) ) , ∂Q ∂g · Q = 0 and ∂Q ∂G · Q = O ( (cid:96) ) as | (cid:96) | → ∞ .Proof. This follows directly from Lemma A.3 in Appendix A.4. (cid:3) ONCOLLISION SINGULARITIES IN THE 2-CENTER-2-BODY PROBLEM 29 Estimates of potentials. Lemma 4.3. Under the assumptions of Lemma 4.2 we have the following estimatesfor the potentials U, V, W as /χ (cid:28) µ → : (a) When Q is moving to the right of the section { x = − χ/ } , we have V R , U R , W R = O (cid:18) χ + µ(cid:96) + 1 (cid:19) . (b) When Q is moving to the left of the section { x = − χ/ } , we have V L , U L , W L = O (cid:18) χ (cid:19) . Proof. This follows directly from equations (4.2), (4.3) and (4.5) and (4.9) in Lemma4.1. For part (a), the estimate O ( χ ) comes from | Q , +( χ, | in the potentials V R , U R , W R and the estimate O ( µ(cid:96) +1 ) comes from the term µQ · Q | Q | since Q moves away from Q almost linearly in (cid:96) according to (4.9). Our choice of the section { x = − } excludesthe collision between Q and Q . So we put µ(cid:96) +1 to stress the fact that the denomina-tor is bounded away from zero. We do the same thing in the following proofs withoutmentioning it any more. (cid:3) Estimates of gradients of potentials. In this section, we estimate the gradientsof the potentials U, V , which appears in the Hamiltonian equations. Lemma 4.4. Under the assumptions of Lemma 4.2 we have the following estimatesfor the gradients of the potentials U, V as /χ (cid:28) µ → ∂U R ∂Q , ∂Q ∂ ( G , g ) ∂V R ∂Q = O (cid:18) χ + µ(cid:96) + 1 (cid:19) , ∂V R ∂Q = O (cid:18) χ + µ | (cid:96) | + 1 (cid:19) ,∂U L ∂Q = O (cid:18) χ (cid:19) , ∂V L ∂Q = O (cid:18) χ (cid:19) , ∂Q ∂ ( G , g ) ∂V L ∂Q = O (cid:18) χ (cid:19) . Proof. The estimates for the ∂∂Q , terms are straightforward. Indeed, we only needto use the fact (cid:12)(cid:12)(cid:12) ddx | x | k (cid:12)(cid:12)(cid:12) = k | x | k +1 together with the estimates in Lemma 4.1.The estimates of all ∂∂ ( G ,g ) terms are similar. We consider for instance ∂Q ∂G ∂V R ∂Q . We have(4.13) ∂Q ∂G ∂V R ∂Q = ∂Q ∂G Q + ( χ, | Q + ( χ, | + O (cid:18) µ (cid:12)(cid:12)(cid:12)(cid:12) ∂Q ∂G (cid:12)(cid:12)(cid:12)(cid:12) | Q | − (cid:19) . The second term here is O ( µ/ ( (cid:96) + 1)) due to (4.9) and Lemma A.3(a). To handle thefirst term let ∂Q ∂G = ( a , b ) , Q = ( x, y ) . Note that equations (A.3), (A.4), (4.7), (4.9),and (4.11) show that x, (cid:96) are all comparable in the sense that the ratios betweenany two of these qualities are bounded from above and below. On the other handLemma A.3(a) tells us that a x + b y = O ( (cid:96) ) . Since b y = O ( b ) = O ( (cid:96) ) we conclude that a x = O ( (cid:96) ) and thus a = O (1) . Thus the first term in (4.13) is ∂Q ∂G · Q + a χ | Q +( χ, | . Thenumerator here is O ( χ ) while the denominator is at least ( χ/ . This completes theestimate of ∂Q ∂G ∂V R ∂Q . Other derivatives are similar. (cid:3) Plugging the above estimates into (4.6) we obtain the following estimate of theHamiltonian equations. Lemma 4.5. Under the assumptions of Lemma 4.2 we have the following estimateson the RHS of (4.6) as /χ (cid:28) µ → . (a) When − χ ≤ x ≤ − we have dL d(cid:96) , dG d(cid:96) , dg d(cid:96) , dG d(cid:96) , dg d(cid:96) = O (cid:18) χ + µ(cid:96) + 1 (cid:19) , d(cid:96) d(cid:96) = − O ( µ ) . (b) When Q is moving to the left of the section { x = − χ/ } , we have dL d(cid:96) , dG d(cid:96) , dg d(cid:96) , dG d(cid:96) , dg d(cid:96) = O (cid:18) χ (cid:19) , d(cid:96) d(cid:96) = − O (cid:18) χ (cid:19) . Proof. The proof is simply an application of Lemma 4.4. We only remark that in theleft case, the orbit is very close to collision and the hyperbolic Delaunay coordinatesbecomes singular. We use Lemma A.1 to show that the derivatives of the Cartesiancoordinates with respect to L , G , g are bounded. Moreover, since we treat (cid:96) as thenew time, we never take the (cid:96) derivative in the RHS of the Hamiltonian equations,hence the dependence on (cid:96) is continuous. (cid:3) In Section 6 we will need the following bounds on the second derivatives to estimatethe variational equations. Lemma 4.6. Under the assumptions of Lemma 4.2 we have the following estimatesfor the second derivatives. (4.14) ∂ U R ∂Q = O (cid:18) χ + µ(cid:96) + 1 (cid:19) , ∂ V R ∂Q = O (cid:18) χ + µ(cid:96) + 1 (cid:19) ,∂ ( U R , V R ) ∂Q ∂Q = O (cid:18) µ | (cid:96) | + 1 (cid:19) ,∂ U L ∂Q = O (cid:18) χ (cid:19) , ∂ V L ∂Q = O (cid:18) χ (cid:19) , ∂ ( U L , V L ) ∂Q ∂Q = O (cid:18) χ (cid:19) . We omit the proof since it is again a direct computation. ONCOLLISION SINGULARITIES IN THE 2-CENTER-2-BODY PROBLEM 31 Proof of Lemma 4.1. Proof of Lemma 4.1. Let τ be the maximal time interval such that(4.15) 34 | L ( (cid:96) ∗ ) | ≤ | L | ≤ | L ( (cid:96) ∗ ) | , | G i ( (cid:96) ∗ ) | ≤ | G i ( (cid:96) ) | ≤ | G i ( (cid:96) ∗ ) | , i = 3 , , on [0 , τ ] where (cid:96) ∗ is the value (cid:96) restricted on { x = − } . (4.15) implies that e = (cid:112) G /L is bounded. We always have we have | Q | ≥ Q is to the left ofthe section { x = − } . Therefore (4.5) implies that L = L + O ( µ ) in the right caseand L = L + O (1 /χ ) in the left case. Now formula (A.3) and Lemma A.2 allow usreplace sinh u, cosh u by (1 + o (1)) (cid:96) e as | (cid:96) | → ∞ . (4.16) | Q | = L (cid:113) L (cosh u − e ) + G sinh u = L (cid:113) L (cid:0) cosh u − e cosh u + e (cid:1) + ( L e − L ) sinh u = L (cid:113) − e cosh u + e + e sinh u = L ( e cosh u − t ≤ min( τ, ¯ τ ) where ¯ τ is the first time then x reaches − χ . Thus for t ≤ min( τ, ¯ τ ) the assumptions of Lemma 4.5 are satisfied and hence(4.17) dL d(cid:96) , dG d(cid:96) , dG d(cid:96) = O (cid:18) χ + µ | Q − Q | (cid:19) (note that to prove the estimates in Lemma 4.5 in the right case we do not need theassumption (4.10)). If we integrate (4.17) w.r.t. (cid:96) on the interval of size O ( χ ) wefind that the oscillations of L , G , G are O ( µ ) . Therefore ¯ τ < τ and we obtain theestimates of (4.9) up to the time ¯ τ . The analysis of the cases when Q is to the left of the section { x = − χ/ } andthen it travels back from { x = − χ/ } to { x = − } is similar once we establish thebounds on the angular momentum at the beginning of the corresponding pieces of theorbit. Let us show, for example, that at the moment when the orbit hits { x = − χ } for the first time, the angular momentum of Q w.r.t. Q is O (1) . Indeed we havealready established that G R = − χv y − yv x = O (1) . Also (4.15) shows that v = O (1)and so (4.7) implies that yv x = O (1) . Accordingly χv y = − G R − yv y = O (1)and hence G L = G R + χv y = O (1) as claimed. The argument for the second timethe orbit hits { x = − χ } is the same. This completes the proof of part (b).To show part (a), we notice ∂Q ∂X depends on (cid:96) , g periodically according to equation(A.1). So part ( a ) follows since we have already obtained bounds on L and G . (cid:3) The next lemma gives more information about the Q part of the orbit thanLemma 4.1. It justifies the assumptions of Lemma A.3. Lemma 4.7. Under the hypothesis of Lemma 4.2, we have as /χ (cid:28) µ → : (a) when Q is moving to the right of the section { x = − χ/ } , we have tan g = − sign( u ) G L + O (cid:18) µ | (cid:96) | + 1 + 1 χ (cid:19) . (b) when Q is moving to the left of the section { x = − χ/ } , then G , g = O (1 /χ ) . Proof. We prove part (b) first. From equation (A.5) we see that if (cid:96) is of order χ and y = O (1) then G cos g + sign( u ) L sin g = O (1 /χ ) . Integrating the estimates ofLemma 4.5(b) we see that during the time x ≤ − χ/ G = G ∗ + O (1 /χ ) , L = L ∗ + O (1 /χ ) , g = g ∗ + O (1 /χ )where ( L ∗ , G ∗ , g ∗ ) are the orbit parameters of Q then it first hits { x = − χ/ } . Itfollows that both G ∗ cos g ∗ + L ∗ sin g ∗ = O (1 /χ ) , and G ∗ cos g ∗ − L ∗ sin g ∗ = O (1 /χ ) . Since L ∗ is not too small this is only possible if G ∗ = O (1 /χ ) , g ∗ = O (1 /χ ) . Now part(b) follows from (4.18).The proof of part (a) is similar. Consider for example the case when Q moves tothe right. Now (4.18) has to be replaced by(4.19) ( G , L , g ) = ( G ∗ , L ∗ , g ∗ ) + O (cid:18) µ | (cid:96) | + 1 + 1 χ (cid:19) , (since we use part (a) of Lemma 4.5 rather than part (b)). As before we have G ∗ cos g ∗ − L ∗ sin g ∗ = O (1 /χ ) . Since cos g ∗ can not be too small (since otherwise G ∗ cos g ∗ − L ∗ sin g ∗ ≈ L ∗ sin g wouldnot be small) we can divide the last equation by L ∗ cos g ∗ to gettan g ∗ = − G ∗ L ∗ + O (cid:18) χ (cid:19) . Now part (a) follows from (4.19). (cid:3) Proof of Lemma 2.5. We begin by demonstrating that the orbits satisfyingthe conditions of Lemma 2.5 satisfy the assumptions of Lemma 4.5. Lemma 4.8. (a) Given D, C there exist constants ˆ C, µ such that for µ ≤ µ the following holds. Consider a time interval [0 , T ] and an orbit satisfying thefollowing conditions (i) x ( t ) ∈ ( − χ − , − for t ∈ (0 , T ) , x (0) = − , x ( T ) = − χ. (ii) y (0) ≤ C, y ( T ) ≤ C. (iii) At time , Q moves on an elliptic orbit which is completely contained in { x ≥ − (2 − D ) } . Then | y ( t ) | ≤ ˆ C for all t ∈ [0 , T ] . (b) The result of part (a) remains valid if (i) is replaced by (˜i) x ( t ) < − for t ∈ (0 , T ) , x (0) = x ( T ) = − . ONCOLLISION SINGULARITIES IN THE 2-CENTER-2-BODY PROBLEM 33 Proof. To prove part (a) we first establish a preliminary estimate showing that Q travels roughly in the direction of Q . Sublemma 4.9. Given ˜ θ > there exists µ , χ such that the following holds for µ ≤ µ , χ > χ . If the outgoing asymptote satisfies (4.20) | π − θ +4 (0) | > ˜ θ then Q escapes from the two center system.Proof. We consider the case θ +4 (0) < π − ˜ θ, the other case is similar. If we disregardthe influence of Q and Q then Q would move on a hyperbolic orbit and its velocitywould approach ( (cid:112) E (0) cos θ +4 (0) , (cid:112) E (0) sin θ +4 (0)) . Accordingly given R we canfind ¯ t, µ such that uniformly over all orbits satisfying (i)-(iii) and θ +4 (0) < π − ˜ θ wehave for µ ≤ µ y (¯ t ) > R, v y (¯ t ) > . (cid:112) E (0) sin ˜ θ. Let ˜ t = inf { t > ¯ t : v y < √ E (0)2 sin ˜ θ } . We shall show that ˜ t = ∞ which implies thesublemma since for t ∈ [¯ t, ˜ t ] we have(4.21) y ( t ) > R + (˜ t − ¯ t ) √ E θ. To see that ˜ t = ∞ note that (4.21) implies that | ˙ v y | ≤ R + (˜ t − ¯ t ) √ E sin ˜ θ ) and so | v y (˜ t ) − v y (¯ t ) | ≤ (cid:90) ∞ ds ( R + s √ E sin ˜ θ ) = 2 R √ E sin ˜ θ . Hence if R is sufficiently large we have v y (˜ t ) ≥ √ E sin ˜ θ which is only possible if˜ t = ∞ . (cid:3) We now consider the case | π − θ +4 | < ˜ θ. Arguing as above we see that given R, wecan find for µ small enough a time ¯ t such that x (¯ t ) < − R, v x (¯ t ) < − . (cid:112) E (0) cos ˜ θ. Let ˆ t be the first time after ¯ t such that x = − ( χ − R ) . Arguing as in Sublemma 4.9we see that for t ∈ [¯ t, ˆ t ] we have | v x | ≥ √ E (0)2 cos ˜ θ. Hence the force from Q and Q is O (1 /t ) and the force from Q is O (1 / (ˆ t − t ) ) . Accordingly v remains O (1) so theenergy of Q remains bounded. Next if | y (ˆ t ) | > R then the argument of Sublemma4.9 shows that y ( T ) > R/ R > C. Accordingly we have for t ∈ [ˆ t, T ] that E = O (1) , y = O (1) and | G L (ˆ t ) | = O (1). We point out that the O (1)’shere are as χ → ∞ and might depend on R . It remains to show that | y ( t ) | < ˆ C for t ∈ [¯ t, ˆ t ] . To this end let t ∗ be the first time when x = − χ . We first get E = O (1) for t ∈ [ t ∗ , ˆ t ] since by arguing as in the Sublemma we get the oscillation of v is bounded.Next, we have that G L ( t ∗ ) = O (1) since ˙ G L = O (1 /χ ),(this estimate of ˙ G L = ¨ v × x does not need any assumption on G L .) On other other hand, we have G R ( t ∗ ) = O (1) by integrating the equation ˙ G = O (1 /χ ) with initial condition G R (0) = O (1) providedby the assumption of the lemma. Therefore χv y ( t ∗ ) = G R − G L = O (1) and so v y ( t ∗ ) = O (1 /χ ) . Since G L ( t ∗ ) = (cid:16) χ v y − y v x (cid:17) ( t ∗ ) we have y ( t ∗ ) = O (1) . Nextfor t ∈ [ t ∗ , ˆ t ] we have y ( t ) = y ( t ∗ ) + v y ( t ∗ )( t − t ∗ ) + (cid:90) tt ∗ (cid:90) ut ∗ ¨ y ( s ) dsdu. Note that ¨ y ( s ) = O (cid:18) y | Q − Q | (cid:19) = O (cid:18) y (ˆ t − s + R ) (cid:19) . Combining the last two estimates we get | y ( t ) | ≤ C + C sup s {| y ( s ) |} (cid:90) tt ∗ (cid:90) ut ∗ dsdu (ˆ t − s + R ) ≤ C + C (cid:18) R + 1 χ (cid:19) sup s | y ( s ) | . Here C might depend on R through the estimates of y ( t ∗ ) , v y ( t ∗ ) but C does not.We choose R large enough to get that | y | is bounded on [ t ∗ , ˆ t ]. The argument for [¯ t, t ∗ ]is the same except that the force from Q is O (cid:16) µy | Q | (cid:17) . This completes the proof ofpart (a).To prove part (b) we note that if | y (ˆ t ) | > R then Q escapes by the argument ofSublemma 4.9. Hence | y (ˆ t ) | < R . This implies (via already established part (a) ofthe lemma) that y is uniformly bounded on [0 , ˆ t ] . The argument for [ˆ t, T ] is the samewith the roles of Q and Q interchanged. (cid:3) Proof of Lemma 2.5. Initially we have 1 /C ≤ | L | ≤ C, | G | , | G | ≤ C for someconstant C > 1. We assume (4.15) from time 0 to some time τ . Due to the previouslemma, we can use Lemma 4.5 to get the estimates on the time interval [0 , τ ] dL d(cid:96) , dG d(cid:96) , dg d(cid:96) , dG d(cid:96) , dg d(cid:96) = O (cid:18) χ + µ(cid:96) + 1 (cid:19) . We integrate the equations to get O ( µ ) oscillations of L , G , G so that τ can beextended to as large as χ . For part (a) of Lemma 2.5, we integrate the equations of dL d(cid:96) , dG d(cid:96) , dg d(cid:96) , over time of order χ as Q first moves away from Q and then comesback. Therefore we get O (cid:18) (cid:90) χ (cid:20) µ(cid:96) + 1 + 1 χ (cid:21) d(cid:96) (cid:19) = O ( µ )estimate for the change of L , G and g proving part (a).Part (b) of Lemma 2.5 follows from Lemma 4.7.For part (c), applying the bounds 1 /C ≤ | L | ≤ C, | G | , | G | ≤ C to equation(4.5), we get 1 /C (cid:48) < | L | < C (cid:48) for some constant C (cid:48) > 1. Next, when restricted tothe section { x = − χ/ } , we set in (A.5) q = − χ/ q = χ/ 2. We useLemma 4.7 In both the left and the right cases to get | (cid:96) | = O ( χ ) restricted to thesection { x = − χ/ } . Next use the ˙ (cid:96) equation in the Hamiltonian equation (4.4)to get | ˙ (cid:96) | > c > c . Therefore for each piece of orbit I, III, V , ONCOLLISION SINGULARITIES IN THE 2-CENTER-2-BODY PROBLEM 35 it takes time | t | = O ( χ ) to get | (cid:96) | = O ( χ ). Adding the time for the three piecestogether, we get that the total time defining the global map is O ( χ ). (cid:3) Derivatives of the Poincar´e map In computing C asymptotics of both local and global maps we will need formulasfor the derivatives of Poincar´e maps between two sections. Here we give the formulasfor such derivatives for the later reference.Recall our use of notations. X denotes Q part of our system and Y denotes Q part. Thus(5.1) X = ( L , (cid:96) , G , g ) , Y = ( G , g ) . ( X, Y ) i will denote the orbit parameters at the initial section and ( X, Y ) f will denotethe orbit parameters at the final section. Likewise we denote by (cid:96) i the initial “time”when Q crosses some section, and by (cid:96) f final “time” when Q arrives at the next.We abbreviate the RHS of (4.6)) as X (cid:48) = U , Y (cid:48) = V . Here (cid:48) is the derivative w.r.t. (cid:96) . We also denote Z = ( X, Y ) and W = ( U , V ) tosimplify the notations further.Suppose that we want to compute the derivative of the Poincar´e map between thesections S i and S f . Assume that on S i we have (cid:96) = (cid:96) i ( Z i ) and on S f we have (cid:96) = (cid:96) f ( Z f ) . We want to compute the derivative D of the Poincar´e map along theorbit starting from ( Z i ∗ , (cid:96) i ∗ ) and ending at ( Z f ∗ , (cid:96) f ∗ ) . We have D = dF dF dF where F is the Poincar´e map between S i and { (cid:96) = (cid:96) i ∗ } , F is the flow map between thetimes (cid:96) i ∗ and (cid:96) f ∗ , and F is the Poincar´e map between { (cid:96) = (cid:96) f ∗ } and S f . We have F = Φ( Z i , (cid:96) ( Z i ) , (cid:96) i ∗ ) where Φ( Z, a, b ) denotes the flow map starting from Z at time a and ending at time b. Since ∂ Φ ∂Z ( Z i ∗ , (cid:96) i ∗ , (cid:96) i ∗ ) = Id , ∂ Φ ∂a = −W we have dF = Id − W ( (cid:96) i ) ⊗ D(cid:96) i DZ i . Inverting the time we get dF = (cid:32) Id − W ( (cid:96) f ) ⊗ D(cid:96) f DZ f (cid:33) − . Finally dF = DZ ( (cid:96) f ∗ ) DZ ( (cid:96) i ∗ ) is just the fundamental solution of the variational equationbetween the times (cid:96) i ∗ and (cid:96) f ∗ . Thus we get(5.2) D = (cid:32) Id − W ( (cid:96) f ) ⊗ D(cid:96) f DZ f (cid:33) − DZ ( (cid:96) f ) DZ ( (cid:96) i ) (cid:18) Id − W ( (cid:96) i ) ⊗ D(cid:96) i DZ i (cid:19) . Here the term DZ ( (cid:96) f ) DZ ( (cid:96) i ) is the fundamental solution to the variational equation fromtime (cid:96) i to (cid:96) f . It does not give us the correct derivative of the Poincar´e map since thePoincar´e sections are not defined by (cid:96) i,f =constant (equal time) but by { x = − χ/ } (equal space). As a result, different orbits may take different time to travel from onesection to the next. The two other terms in D corresponding to dF , dF are usedto go from equal space section to equal time section and vice versa, which we call boundary contributions. Variational equations The next step in the proof is the C analysis of the global map. It occupies sections6-8. We shall work under the assumptions of Lemma 3.2. In particular we will usethe estimates of Section 4 and Appendix A.The plan of the proof of Proposition 3.7 is the following. Matrices ( I ), ( III ) and( V ) are treated in Sections 6 and 7. Namely, in Sections 6 we study the variationalequation while in Section 7 we estimate the boundary contributions. Finally in Section8 we compute matrices ( II ) and ( IV ) which describe the change of variables betweenthe Delaunay coordinates with different centers which are used to the left and to theright of the line x = − χ . Estimates of the coefficients.Lemma 6.1. We have the following estimates for the RHS of the variational equationunder the assumption of Lemma 4.2. (a) When Q is moving to the right of the section { x = − χ/ } , we have ∂ U R ∂X ∂ U R ∂Y∂ V R ∂X ∂ V R ∂Y = O χ ( χ ) × ( χ ) × χ ( χ ) × ( χ ) × ( χ ) × ( χ ) × ( χ ) × ( χ ) × ( χ ) × ( χ ) × + O (cid:18) µ | Q | (cid:19) In addition we have for ξ = | Q | χ = | Q − Q | χ ∈ (0 , / ∂ V ∂Y = 1 χ ξ (1 − ξ ) − L sign( ˙ x )( G + L ) L − L ( G + L ) L sign( ˙ x )( G + L ) + O (cid:18) µχ + µ | Q | (cid:19) ,∂ V ∂L = 1 χ ξ (1 − ξ ) (cid:18) G L sign( ˙ x )( L + G ) , G L ( L + G ) (cid:19) T + O (cid:18) µχ + µ | Q | (cid:19) . ONCOLLISION SINGULARITIES IN THE 2-CENTER-2-BODY PROBLEM 37 (b) When Q is moving to the left of the section x = − χ/ , we have O ∂ U L ∂X ∂ U L ∂Y∂ V L ∂X ∂ V L ∂Y = χ ( χ ) × ( µχ ) × χ ( χ ) × ( χ ) × ( χ ) × ( χ ) × ( µχ ) × ( χ ) × ( µχ ) × ( χ ) × In addition we have for ξ = | Q − Q | χ ∈ (0 , / ∂ V ∂Y = − χ ξ (1 − ξ ) (cid:20) L sign( ˙ x ) L − L − L sign( ˙ x ) (cid:21) + O (cid:18) µχ (cid:19) . Proof. Before going the the calculations, we remark that the variable (cid:96) is treated asthe new time hence we do not take partial derivatives with respect to it when derivingthe variational equations. We need only C dependence on (cid:96) in the RHS of boththe Hamiltonian equation and the variational equation, which is satisfied even if whenthe orbits come close to collision. We need to use Lemma A.1 when taking G , L partial derivatives for small (cid:96) in the left case to show that the first and second orderderivatives of Q with respect to G , L are always bounded.(a) We estimate the four blocks of the derivative matrix separately. • We begin with ∂ U R ∂X part. We consider first the partial derivatives of (cid:96) since it is the largest component of U . Opening the brackets in the second line of (4.6) we get(6.1) d(cid:96) d(cid:96) = − k + 1 L W + kL ∂Q ∂L · ∂U∂Q + k L ∂Q ∂L · ∂V∂Q +2 kW ∂Q ∂L · ∂V∂Q + O (cid:18) χ + µ | Q | (cid:19) . Note that by (4.5)(6.2) W R = k R L (cid:18) | Q + ( χ, | + 1 | Q + ( χ, | + µQ · Q | Q | (cid:19) + O (cid:18) µ | Q | (cid:19) = O (cid:18) χ + µ | Q | (cid:19) Observe that the RHS of (6.1) depends on L in three ways. First, in contains severalterms of the form L m . Second, Q depends on L via (A.2). Third, Q depends on L via (A.5) and L depends on L via (4.5). In particular we need to consider thecontribution to ∂∂L d(cid:96) d(cid:96) coming from ∂L ∂L ∂∂L = ∂L ∂L ∂Q ∂L ∂∂Q . By Lemma A.3 and equation (4.9) we have ∂Q ∂L = O ( | Q | ) . Therefore the main contri-bution to (2,1) entry is O (cid:16) χ + µ | Q | (cid:17) and it comes from ∂W R ∂Q ∂Q ∂L ∂L ∂L , W R ∂∂L L and ∂L ∂L ∂∂L (cid:16) k L ∂Q ∂L · ∂V∂Q (cid:17) . For the (2 , , (2 , , (2 , entries, the computations are similar. We need to act ∂∂(cid:96) , ∂∂G , ∂∂g on (6.1). (4.5) and (6.2) show that the contributioncoming from ∂L ∂ ( (cid:96) ,G ,g ) is O (cid:16) χ + µ | Q | (cid:17) . It remains to consider the contributioncoming from ∂Q ∂ ( (cid:96) ,G ,g ) ∂∂Q . Now the bound for (2 , , (2 , 3) and (2 , 4) entries followsdirectly from Lemmas 4.1, 4.3, 4.4, and 4.6. The entries ( i, j ) , i ∈ { , , } , j ∈ { , , } are done together. These involve second order derivatives with respect to (cid:96) , G , g . The estimate O ( µ | Q | ) in the statement comes from the term µQ · Q | Q | in U R . For the term | Q +( χ, | = O (1 /χ ) in U R , each Q derivative amounts to improve the estimate by multiplying1 /χ . Here we need to take two Q derivatives. Moreover, ∂Q ∂ ( G ,g ,(cid:96) ) , ∂ Q ∂ ( G ,g ,(cid:96) ) = O (1)due to the periodicity. So we get the estimate in the statement. We point out that theimprovement compared to the first column and second row in this block is becausethat we do not take L partial derivative. Next, consider (1 , entry. We need to estimate ∂∂L (cid:18) ( kL + W ) ∂Q ∂(cid:96) · ∂U∂Q (cid:18) kL + W ) ∂Q ∂L · ∂V∂Q (cid:19)(cid:19) . Using the Leibniz rule we see that the leading term comes from ∂∂L (cid:16) kL ∂Q ∂(cid:96) · ∂U∂Q (cid:17) and it is of order O (cid:16) χ + µ | Q | (cid:17) . The estimates for other entries of the ∂ U R ∂X part aresimilar to the (1 , 1) entry. This completes the analysis of ∂ U R ∂X . • Next, we consider ∂ V R ∂Y . Using the Leibniz rule again we see that the main contribution to the derivativesof V comes from differentiating (cid:34) L ∂Q ∂g · ∂V∂Q − L ∂Q ∂G · ∂V∂Q (cid:35) Consider the (5 , entry. The main contribution to this entry comes from ∂∂G (cid:18) L ∂Q ∂g · ∂V∂Q (cid:19) = L (cid:18) ∂ Q ∂G ∂g · ∂V∂Q + ∂Q ∂g · ∂ V∂Q · ∂Q ∂G (cid:19) . By Lemmas 4.4 and 4.6 the first term is | Q | · O (cid:16) χ + µ | Q | (cid:17) = O (cid:16) χ + µ | Q | (cid:17) and thesecond term is | Q | · O (cid:16) χ + µ | Q | (cid:17) = O (cid:16) χ + µ | Q | (cid:17) . This gives the desired upperbound of the (5 , 5) entry. Notice that O (1 /χ ) term comes from L ∂∂G (cid:16) ∂Q ∂g · ∂ ˜ V∂Q (cid:17) where ˜ V = − | Q +( χ, | . Thus we need to find the asymptotics of(6.3) L ∂∂G (cid:32) ∂Q ∂g · ( Q + ( χ, | Q + ( χ, | (cid:33) . Let ∂Q ∂g = ( a , b ) . Arguing in the same way as in the estimation of (4.13) we seethat a = O (1) . Accordingly the numerator in (6.3) is O ( χ ) so if we differentiate the ONCOLLISION SINGULARITIES IN THE 2-CENTER-2-BODY PROBLEM 39 denominator of (6.3) the resulting fraction will be of order O ( χ ) O ( χ − ) = O ( χ − ) . Hence O (1 /χ ) term comes from L ∂∂G (cid:16) ∂Q ∂g · ( Q +( χ, (cid:17) | Q +( χ, | . The numerator here equals to ∂∂G (cid:18) ∂Q ∂g · Q (cid:19) + ∂ Q ∂G ∂g · ( χ, . The first term vanishes due to Lemma A.3(a) so the main contribution comes fromthe second term. Using Lemma A.5 we see that (5 , 5) entry equals to L L (cid:112) L + G χ sinh u | Q + ( χ, | + O (cid:18) µχ + µ | Q | (cid:19) . Recall that L = L (1 + o (1)) (due to (4.5)) and sinh u = sign( u ) | (cid:96) | L √ L + G (due to(A.4)). Since Lemma 4.1 implies that | Q | = | (cid:96) | /L (1 + o (1)) we obtain that O (1 /χ )-term in (5 , 5) is asymptotic to L sign( u ) L + G χ | Q | ( χ −| Q | ) . Since u and ˙ x have opposite signswe obtain the asymptotics of O (1 /χ )-term claimed in part (a) of the Lemma 6.1. Theanalysis of other entries of ∂ V R ∂Y is similar. • Next, consider the ∂ U R ∂Y term. The analysis of (2 , 5) entry is similar to the analysis of (2 , 2) entry except that ∂∂G (cid:16) k L ∂Q ∂L ∂V∂Q (cid:17) contains the term k L ∂ Q ∂L ∂G ∂V∂Q which is of order O (1 /χ ) dueto Lemmas 4.6 and A.5 and this term provides the leading contribution for large t. The analysis of (2 , 6) is similar to (2 , . The estimate of the remaining entries of ∂ U R ∂Y is similar to the analysis of (1 , 1) entry. • Thus to complete the proof of (a) it remains to consider ∂ V R ∂X . We beginwith (5 , 1) entry. We need to act by ∂∂L + ∂L ∂L ∂∂L on( kL + W ) ∂Q ∂g · ∂V∂Q (cid:18) kL + W ) ∂Q ∂L · ∂V∂Q (cid:19) . The leading term for the estimate of (5 , 1) comes from (cid:18) ∂∂L + ∂L ∂L ∂∂L (cid:19) (cid:18) ∂Q ∂g · ∂V∂Q (cid:19) = ∂L ∂L ∂∂L (cid:18) ∂Q ∂g · ∂V∂Q (cid:19) + O (cid:18) χ + µ | Q | (cid:19) = O (cid:18) χ + µ | Q | (cid:19) . Observe that O (1 /χ ) term here comes from ∂∂L (cid:16) ∂Q ∂g · ∂V∂Q (cid:17) which can be analyzedin the same way as (5 , 5) term. The analysis of (6 , 1) is the same as of (5 , . The (5 , 2) entry is equal to (cid:16) ∂∂(cid:96) + ∂L ∂(cid:96) ∂∂L (cid:17) (cid:104)(cid:16) ∂Q ∂g · ∂V∂Q (cid:17) Γ (cid:105) whereΓ = kL + W + k L ∂Q ∂L · ∂V∂Q + 2 kL W ∂Q ∂L · ∂V∂Q + W ∂Q ∂L · ∂V∂Q . Now the estimate of the (5 , 2) entry follows from the following estimatesΓ = O (1) , (cid:18) ∂Q ∂g · ∂V∂Q (cid:19) = O (cid:18) χ + µ | Q | (cid:19) , (cid:18) ∂∂(cid:96) + ∂L ∂(cid:96) ∂∂L (cid:19) (cid:18) ∂Q ∂g · ∂V∂Q (cid:19) = ∂Q ∂g · ∂∂(cid:96) ∂V∂Q + ∂L ∂(cid:96) ∂L (cid:18) ∂Q ∂g · ∂V∂Q (cid:19) = O (cid:18) µ | Q | + (cid:18) χ + µ | Q | (cid:19) (cid:18) χ + µ | Q | (cid:19)(cid:19) = O (cid:18) χ + µ | Q | (cid:19) , and (cid:18) ∂∂(cid:96) + ∂L ∂(cid:96) ∂∂L (cid:19) Γ = O (cid:18) χ + µ | Q | (cid:19) . The remaining entries of ∂ V ∂X are similar to the (5 , 2) entry. This completes the proofof part (a).(b) • The estimate of ∂ V L ∂Y and ∂ U L ∂X are the same as in part (a). However,now | Q | is of order χ so O ( µ/ | Q | ) is dominated by other terms. In addition tocompute the leading part we need to use part (c) Lemma A.5 rather than part (b).Moreover, in order to be able to use the formulas of that Lemma we need to shift theorigin to Q . Therefore the coordinates of Q become ( χ, ∂ V L ∂Y = L ∂ Q ∂G∂g · ( − χ, | Q − ( χ, | ∂ Q ∂g · ( − χ, | Q − ( χ, | − ∂ Q ∂G · ( − χ, | Q − ( χ, | − ∂ Q ∂G∂g · ( − χ, | Q − ( χ, | + O (cid:18) µχ (cid:19) . Now the asymptotic expression of ∂ V L ∂Y follows directly from Lemma A.5(c). We pointout that the “ − ” sign in front of the matrices of ∂V∂Y and ∂V∂L comes from the fact thatthe new time (cid:96) that we are using satisfies d(cid:96) dt = − L + o (1) as µ → , χ → ∞ . • Next, we consider the ∂ U L ∂Y term. First consider (1 , . We need to find G derivative of (cid:20) ∂Q ∂(cid:96) · ∂U∂Q (cid:21) ( kL + W ) (cid:18) kL + W ) ∂Q ∂L · ∂V∂Q (cid:19) . Differentiating the first factor we get using Lemma 4.6(6.5) ∂∂G (cid:18) ∂Q ∂(cid:96) · ∂U∂Q (cid:19) = ∂Q ∂(cid:96) · ∂ U∂Q ∂Q ∂Q ∂G = O (cid:18) µχ (cid:19) . When we differentiate the product of the remaining factors then the main contributioncomes from(6.6) ∂∂G (cid:18) ∂Q ∂L · ∂V∂Q (cid:19) = ∂ Q ∂L ∂G · ∂V∂Q + ∂Q ∂L · ∂∂G (cid:18) ∂V∂Q (cid:19) . To bound the last expression we use Lemma A.5. Namely, the second derivative ∂ Q ∂G ∂L = O (1) + (cid:96) (0 , , is almost vertical and ∂V L ∂Q = Q | Q | + µ ( Q − Q ) | Q − Q | is almost ONCOLLISION SINGULARITIES IN THE 2-CENTER-2-BODY PROBLEM 41 horizontal. This shows that ∂ Q ∂G ∂L · ∂V∂Q = χ . The main contribution to the secondsummand in (6.6) comes from ∂∂G (cid:16) ∇ (cid:16) Q (cid:17)(cid:17) . Using Lemma A.3, we get ∂Q ∂L · ∂∂G (cid:18) ∇ (cid:18) Q (cid:19)(cid:19) = ( (cid:96) (1 , O (1)) (cid:18) − Id | Q | + 3 Q ⊗ Q | Q | (cid:19) ( (cid:96) (0 , O (1)) = 1 χ . Since ∂Q ∂(cid:96) · ∂U∂Q = O (1 /χ ) we get the required estimate for (1 , 5) entry.The estimates of other ∂ U L ∂Y terms are similar to the estimate of (1 , 5) entry, exceptfor (2 , 5) and (2 , 6) entries which are different because d(cid:96) d(cid:96) is larger than the othercoordinates of U . Now consider (2 , entry. We need to compute(6.7) − ∂∂G (cid:18) ( kL + W )( 1 L + ∂Q ∂L · ∂U∂Q ) (cid:18) kL + W ) ∂Q ∂L · ∂V∂Q (cid:19)(cid:19) = − ∂∂G (cid:18) k + 1 L W + kL ∂Q ∂L · ∂U∂Q + k L ∂Q ∂L · ∂V∂Q + 2 kW ∂Q ∂L · ∂V∂Q + 1 χ (cid:19) = 0 + 1 χ + µχ + 1 χ + 1 χ + 0 = O (cid:18) χ (cid:19) where the analysis of the leading terms is similar to (6.5), (6.6). • Finally, we consider ∂ V L ∂X . We begin with (5 , entry. We need to compute (cid:20) ∂∂L + ∂L ∂L ∂∂L (cid:21) (cid:18)(cid:18) ∂Q ∂g · ∂V∂Q (cid:19) Γ (cid:19) where Γ = kL + W + k L ∂Q ∂L · ∂V∂Q + 2 kL W ∂Q ∂L · ∂V∂Q + W ∂Q ∂L · ∂V∂Q . The main contribution to (cid:104) ∂∂L + ∂L ∂L ∂∂L (cid:105) (cid:16) ∂Q ∂g · ∂V∂Q (cid:17) comes from ∂L ∂L ∂∂L (cid:18) ∂Q ∂g · ∂V∂Q (cid:19) = ∂L ∂L ∂ Q ∂L ∂g · ∂V∂Q + ∂L ∂L ∂Q ∂g · ∂ V∂Q ∂Q ∂L . The two summands above can be estimated by O (1 /χ ) by the argument used tobound (6.6). Next a direct calculation shows thatΓ = O (1) , (cid:20) ∂∂L + ∂L ∂L ∂∂L (cid:21) Γ = O (1)while (cid:16) ∂Q ∂g · ∂V∂Q (cid:17) = O (1 /χ ) by Lemma 4.4. This gives the required bound for the(5 , 1) entry. The bound for the (6 , 1) entry is similar. Next, consider (5 , . It equals to (cid:20) ∂∂(cid:96) + ∂L ∂(cid:96) ∂∂L (cid:21) (cid:18)(cid:18) ∂Q ∂g · ∂V∂Q (cid:19) Γ (cid:19) . The main contribution to (cid:104) ∂∂(cid:96) + ∂L ∂(cid:96) ∂∂L (cid:105) (cid:16) ∂Q ∂g · ∂V∂Q (cid:17) comes from ∂∂(cid:96) (cid:16) ∂Q ∂g · ∇ (cid:16) µ | Q − Q | (cid:17)(cid:17) = O (cid:16) µχ (cid:17) . On the other hand the main contribution to (cid:104) ∂∂(cid:96) + ∂L ∂(cid:96) ∂∂L (cid:105) Γ comes from ∂W∂(cid:96) = O (cid:16) χ (cid:17) . Combining this with C bounds men-tioned used in the analysis of (5 , 1) we obtain the required estimate on the (5 , 2) entry.The remaining entries of ∂ V L ∂X are similar to (5 , . (cid:3) Estimates of the solutions. We integrate the variational equations to get the ∂ ( X,Y )( (cid:96) f ) ∂ ( X,Y )( (cid:96) i ) in equation (5.2). Lemma 6.2. Under the hypothesis of Lemma 4.2 the following estimates are valid as /χ (cid:28) µ → For maps ( I ) and ( V ) , (6.8) ∂ ( X, Y )( (cid:96) f ) ∂ ( X, Y )( (cid:96) i ) = Id + O µ ( µ ) × ( µ ) × µ ) × (1) × ( µ ) × ( µ ) × ( µ ) × (1) × ( µ ) × (1) × . (b) For map ( III ) , (6.9) ∂ ( X, Y )( (cid:96) f ) ∂ ( X, Y )( (cid:96) i ) = Id + O χ ( χ ) × ( µχ ) × χ ) × ( χ ) × ( χ ) × ( χ ) × ( µχ ) × ( χ ) × ( µχ ) × (1) × . (c) ∂Y ( (cid:96) f ) ∂Y ( (cid:96) i ) and ∂Y∂L have the same asymptotics as item (b) of Proposition 3.8. Parts (a) and (b) of this lemma claim that we can integrate the estimates of Lemma6.1 over (cid:96) -interval of size O ( χ ). Proof. We use the following convention. For two matrices M , M , by M ≤ M , wemean the inequality for each corresponding matrix entries. Similarly, the notation | M | means to take the absolute value in each entry of M . We use the following versionof Gronwall inequality, which can be proven by either comparing the series obtainedfrom Picard iterations (see below) or by applying standard comparison theorem forthe ODEs. Lemma 6.3. Consider two linear systems X (cid:48) = M ( t ) X and X (cid:48) = M ( t ) X . Sup-pose that | M ( t ) | ≤ M ( t ) . Then the corresponding fundamental solutions satisfy com-ponentwise inequalities | Φ ( t ) | ≤ Φ ( t ) for all t ≥ . Let us consider part (b) first, which is easier since the estimate (b) in Lemma6.1 does not depend on (cid:96) . Consider ODE system X (cid:48) ( t ) = KX where K is the ONCOLLISION SINGULARITIES IN THE 2-CENTER-2-BODY PROBLEM 43 matrix in (b) of Lemma 6.1. It can be verified by straightforward computation that( Kχ ) ≤ CKχ where C is a constant independent of χ. thus ( Kχ ) n ≤ C n Kχ , so weget X ( χ ) ≤ Id + e C Kχ . Now part (b) follows from Lemma 6.3.Next, we work on part (a). After a rescaling (cid:96) = χt/ t ∈ (0 , 1) we compare thevariational equation with the ODE(6.10) X (cid:48) = K ( t ) X where(6.11) K ( t ) := cAχ + cµχ ( χt/ + 1 is an upper bound for χ times the estimate of Lemma 6.1(a), c is a large positiveconstant, A is the constant matrix and is the matrix whose entries are all 1’s. Wecan verify in the same way as the proof of part (b) that(6.12) χ A ≤ CχA and | A | ≤ Cχ By Lemma 6.3, it is enough to show that the upper bound of the fundamentalsolution X (1) of (6.10) is given by the estimate in part (a).Solving (6.10) by Picard iteration we get X ( t ) = Id + (cid:90) t K ( s ) X ( s ) ds = Id + (cid:90) t K ( s ) ds + (cid:90) t K ( s ) (cid:90) s K ( s ) ds ds + · · · . The terms which do not contain µ sum to e χA which is Id + O ( χA ) by the sameargument as in part (a). We claim that the remaining terms sum to O ( µ ) . To thisend let k ( t ) = ˜ C max(1 , χχ t +1 ) . By (6.11) and (6.12) the contribution of the termscontaining µ is less that µY (1) where Y is the fundamental solution of ˙ Y = k ( t ) Y. Since (cid:82) k ( t ) dt = O (1) we have Y ( t ) = O (1) as claimed.To prove part (c) we need to find the asymptotics of V . Consider map (I) first. V satisfies V (cid:48) = ∂ V ∂Y V . By already established part (a) V = O (1) so the above equationcan be rewritten as V (cid:48) = ξL χ (1 − ξ ) A V + O (cid:18) µ(cid:96) + 1 + µχ (cid:19) . where A = (cid:34) L ( G + L ) L − L ( G + L ) − L ( G + L ) (cid:35) . Now Gronwall Lemma gives V ≈ ˜ V where ˜ V is the fundamental solution of ˜ V (cid:48) = ξL χ (1 − ξ ) A ˜ V . Using ξ as the independent variablewe get d ˜ V dξ = − ξ (1 − ξ ) A ˜ V . Note that ξ ( (cid:96) i ) = o (1) , ξ ( (cid:96) f ) = + o (1) . Making a furthertime change dτ = ξdξ (1 − ξ ) we obtain the constant coefficient linear equation d ˜ V dτ = − A ˜ V . Observe that Tr( A ) =det( A ) = 0 and so A = 0 . Therefore(6.13) ˜ V ( σ, τ ) = Id − ( τ − σ ) A. Since τ = ξ − ξ ) we have τ (0) = 0 , τ (cid:0) (cid:1) = . Plugging this into (6.13) we get theclaimed asymptotics for map (I). The analysis of map (V) is similar. To analyze map(III) we split ∂Y ( (cid:96) f ) ∂Y ( (cid:96) i ) = ∂Y ( (cid:96) f ) ∂Y ( (cid:96) m ) ∂Y ( (cid:96) m ) ∂Y ( (cid:96) i )where (cid:96) m = (cid:96) i + (cid:96) f . Using the argument presented above we obtain ∂Y ( (cid:96) m ) ∂Y ( (cid:96) i ) = − L L , ∂Y ( (cid:96) f ) ∂Y ( (cid:96) m ) = − L L . Multiplying the above matrices we obtain the required asymptotics for map (III).Next using the same argument as in analysis of ∂Y ( (cid:96) f ) ∂Y ( (cid:96) i ) we obtain ∂Y∂L ≈ W where W (cid:48) = ξL χ (1 − ξ ) (cid:34) A W + (cid:18) − GL ( L + G ) , GL ( L + G ) (cid:19) T (cid:35) . In terms of the new time this equation reads d W dτ = − (cid:34) A W + (cid:18) − GL ( L + G ) , GL ( L + G ) (cid:19) T (cid:35) . Solving this equation using (6.13) and initial condition (0 , T , we obtain the asymp-totics of ∂Y∂L . (cid:3) Boundary contributions and the proof of Proposition 3.7 According to (5.2) we need to work out the boundary contributions in order tocomplete the proof of Proposition 3.7.7.1. Dependence of (cid:96) on variables ( X, Y ) . To use the formula (5.2) we need towork out ( U , V )( (cid:96) i ) ⊗ ∂(cid:96) i ∂ ( X,Y ) i and ( U , V )( (cid:96) f ) ⊗ ∂(cid:96) f ∂ ( X,Y ) f . Consider x component of Q (see equation (A.5)). x = − cos g ( L sinh u − e ) + sin g ( L G cosh u ) . For fixed x = − χ/ − 2, we can solve for (cid:96) as a function of L , G , g . From thecalculations in the Appendix A.2, Lemma A.3, and the implicit function theorem, weget for the section x = − χ/ , (cid:18) ∂(cid:96) ∂L , ∂(cid:96) ∂G , ∂(cid:96) ∂g (cid:19) (cid:12)(cid:12)(cid:12) x = − χ/ = ( O ( χ ) , O (1) , O (1)) , for the section x = − , (cid:18) ∂(cid:96) ∂L , ∂(cid:96) ∂G , ∂(cid:96) ∂g (cid:19) (cid:12)(cid:12)(cid:12) x = − = ( O (1) , O (1) , O (1)) . ONCOLLISION SINGULARITIES IN THE 2-CENTER-2-BODY PROBLEM 45 Explicitly, the O ( χ ) term is(7.1) ∂(cid:96) ∂L = − ∂x ∂L / ∂x ∂(cid:96) = sinh u (2 (cid:112) L + G )sign( u ) L + O (1)using Lemma A.3 and 4.7. This shows that the O ( χ ) term has always positive co-efficient. Using equation (4.5) which relates L to L , we obtain for the section { x = − χ/ } ,(7.2) ∂(cid:96) ∂ ( X, Y ) (cid:12)(cid:12)(cid:12) x = − χ/ = ( O ( χ ) , O (1 /χ ) , O (1 /χ ) , O (1 /χ ) , O (1) , O (1)) , ( U , V ) (cid:12)(cid:12)(cid:12) x = − χ/ = ( O (1 /χ ) , − O (1 /χ ) , O (1 /χ ) × ) T , For the section { x = − } ,(7.3) ∂(cid:96) ∂L (cid:12)(cid:12)(cid:12) x = − = ( O (1) , O ( µ ) , O ( µ ) , O ( µ ) , O (1) , O (1)) , ( U , V ) (cid:12)(cid:12)(cid:12) x = − = (0 , − , , , , T + O ( µ ) . The matrix ( U , V ) ⊗ ∂(cid:96) ∂ ( X,Y ) (cid:12)(cid:12)(cid:12) x = − χ/ has rank 1 and the only nonzero eigenvalue is O (1 /χ ), and ( U , V ) ⊗ ∂(cid:96) ∂ ( X,Y ) (cid:12)(cid:12)(cid:12) x = − has rank 1 and the only nonzero eigenvalue is O ( µ ). So the inversion appearing in (5.2) is valid.7.2. Asymptotics of matrices ( I ) , ( III ) , ( V ) from the Proposition 3.7. Herewe complete the computations of matrices (I), (III) and (V). The boundary contribution to ( I ) . In this case, (cid:96) i stands for the section { x = − } and (cid:96) f stands for the section { x = − χ/ } . So we use equation (7.3) to form( U , V )( (cid:96) i ) ⊗ ∂(cid:96) i ∂ ( X,Y ) i in equation (5.2) and equation (7.2) to form ( U , V )( (cid:96) f ) ⊗ ∂(cid:96) f ∂ ( X,Y ) f .We have(7.4) (cid:18) Id − ( U , V )( (cid:96) f ) ⊗ ∂(cid:96) i ∂ ( X, Y ) i (cid:19) − = Id + ∞ (cid:88) k =1 (cid:18) ( U , V )( (cid:96) f ) ⊗ ∂(cid:96) i ∂ ( X, Y ) i (cid:19) k = Id + (cid:18) ( U , V )( (cid:96) f ) ⊗ ∂(cid:96) i ∂ ( X, Y ) i (cid:19) ∞ (cid:88) k =0 (cid:18) ∂(cid:96) i ∂ ( X, Y ) i · ( U , V )( (cid:96) f ) (cid:19) k = Id + (cid:18) ( U , V )( (cid:96) f ) ⊗ ∂(cid:96) i ∂ ( X, Y ) i (cid:19) (1 + O (1 /χ )) . Now we use equation (5.2) and Lemma 6.2 to obtain the asymptotics of the matrix( I ) stated in Proposition 3.7. The boundary contribution to ( III )This time we use equation (7.2) to form both ( U , V )( (cid:96) i ) ⊗ ∂(cid:96) i ∂ ( X,Y ) i and ( U , V )( (cid:96) f ) ⊗ ∂(cid:96) f ∂ ( X,Y ) f in equation (5.2). The matrix (cid:18) Id − ( U , V )( (cid:96) f ) ⊗ ∂(cid:96) f ∂ ( X,Y ) f (cid:19) − has the same form as (7.4). Now we useequation (5.2) and Lemma 6.2 to obtain the asymptotics of the matrix ( III ) statedin Proposition 3.7. The boundary contribution to ( V )This time we use equation (7.2) to form ( U , V )( (cid:96) i ) ⊗ ∂(cid:96) i ∂ ( X,Y ) i and equation (7.3) toform ( U , V )( (cid:96) f ) ⊗ ∂(cid:96) f ∂ ( X,Y ) f in equation (5.2).The matrix (cid:32) Id − ( U , V )( (cid:96) f ) ⊗ ∂(cid:96) f ∂ ( X, Y ) f (cid:33) − = Id − ( U , V )( (cid:96) f ) ⊗ ∂(cid:96) f ∂ ( X, Y ) f (1 + O ( µ )).Now we use equation (5.2) and Lemma 6.2 to obtain the asymptotics of the matrix( V ) stated in Proposition 3.7.Now we are ready to finish the proof of Proposition 3.8. Proof of Proposition 3.8. The matrices ( I ) , ( III ) , ( V ) are obtained by multiplying thesolution to the variational equations (Lemma 6.2) and the boundary contributionsaccording to (5.2). By explicit calculation it can be verified that the O ( χ ) terms,i.e. the (2 , 1) entries of the ( I ) , ( III ) , ( V ) come from the O ( χ ) term in the boundarycontribution, i.e. the d(cid:96) dL term, which is always positive (see (7.1) in Section 7.1). Thisfinishes the proof of part (a). Again explicit calculation shows that the estimate ofpart (b) comes mainly from the solution to the variational equation (Lemma 6.2). (cid:3) Switching foci Recall that we treat the motion of Q as a Kepler motion focused at Q when it ismoving to the right of the section { x = − χ/ } and treat it as a Kepler motion focusedat Q when it is moving to the left of the section { x = − χ/ } . Therefore, we need tomake a change of coordinates when Q crosses the section { x = − χ/ } . These aredescribed by the matrices ( II ) and ( IV ). Under this coordinate change the Q part ofthe Delaunay variables does not change. The change of G is given by the differenceof angular momentums w.r.t. different reference points ( Q or Q ). To handle it weintroduce an auxiliary variable v y -the y component of the velocity of Q . Relating g with respect to the different reference points to v y we complete the computation.8.1. From the right to the left. We have( II ) = ∂ ( L , (cid:96) , G , g , G L , g L ) ∂ ( L , (cid:96) , G , g , G R , g R ) (cid:12)(cid:12)(cid:12) x = − χ/ = ( iii )( ii )( i )where matrices ( i ) , ( ii ) and ( iii ) correspond to the following coordinate changes re-stricted to the section { x = − χ/ } .( G, g ) R ( i ) −→ ( G, v y ) R ( ii ) −→ ( G, v y ) L ( iii ) −→ ( G, g ) L . ONCOLLISION SINGULARITIES IN THE 2-CENTER-2-BODY PROBLEM 47 Computation of matrices (i) and (iii)(ii) in Proposition 3.7. ( i ) is given by the rela-tion v y = L R sinh u R sin g R + G R L R cos g R cosh u R − e R cosh u R , L R = k R L − W R L . where last relation follows from (4.5). Recall that by Lemma 4.7 g R = − arctan G R L R + O (1 /χ ) . In addition (8.1) below and the fact that G R and G L are O (1) implies v y = O ( χ ) . Now the asymptotics of (i) is obtained by a direct computation. We compute dv y dL theother derivatives are similar but easier. We have dv y dL = dv y dL R ∂L R ∂L . The second termis k R + O (1 /χ ) . On the other hand dv y dL = ∂∂L R (cid:16) L R sinh u R sin g R + G R L R cos g R cosh u R (cid:17) − e R cosh u R + v y ∂e R ∂L R cosh u R − e R cosh u R + ∂v y ∂(cid:96) R ∂(cid:96) R ∂L R . The main contribution comes from the first term which equals G R L R ( L R + G R ) + O (1 /χ ) . The second term is O (1 /χ ) since v R = O (1 /χ ) . Next rewriting v y = L R tanh u R sin g R + G R L R cos g R (1 / cosh u R ) − e R we see ∂v y ∂(cid:96) R ∂(cid:96) R ∂L R = O (1 /χ ) × O ( χ ) = O (1 /χ ) since ∂(cid:96) R ∂L R = O ( χ ) by (7.2).( ii ) is given by(8.1) G L = G R − χv y , which comes from the simple relation v × ( Q − Q ) = v × Q − v × Q . Here G R and v y are independent variables so the computation of the derivative of (ii) isstraightforward.To compute the derivative of ( iii ) we use the relation from (A.6) v y = − L L sinh u L sin g L − G L L L cos g L cosh u L − e L cosh u L where u L < 0. Arguing the same way as for (i) and using the fact that by Lemma 4.7, G L , g L = O (1 /χ ), − sinh u L , cosh u L (cid:39) (cid:96) L e L we obtain δv y = δG L k R L − δg L k R L + HOT. Hence δg L = δG L k R L − k R L δv y + HOT = δG R − χδv y k R L + HOT completing the proof of the lemma. (cid:3) From the left to the right. At this step we need to compute( IV ) = ∂ ( L , (cid:96) , G , g , G R , g R ) ∂ ( L , (cid:96) , G , g , G L , g L ) (cid:12)(cid:12)(cid:12) x = − χ/ = ( iii (cid:48) )( ii (cid:48) )( i (cid:48) ) . where the matrices ( iii (cid:48) ) , ( ii (cid:48) ) and ( i (cid:48) ) correspond to the following changes of variablesrestricted to the section { x = − χ/ } .( G, g ) L ( i (cid:48) ) −→ ( G, v y ) L ( ii (cid:48) ) −→ ( G, v y ) R ( iii (cid:48) ) −→ ( G, g ) R . Computation of matrices ( iii (cid:48) ) and ( ii (cid:48) )( i (cid:48) ) in Proposition 3.7. ( i (cid:48) ) is given by v y = − L L sinh u L sin g L − G L L L cos g L cosh u L − e L cosh u L < . Here u L > G L , g L = O (1 /χ ).( ii (cid:48) ) is given by G R = G L + χv yL . Now the analysis is similar to Subsection 8.1. In particular the main contribution to[( ii (cid:48) )( i (cid:48) )] comes from ∂ ( G R , v y ) ∂ ( G L , g L ) = ∂ ( G R , v y ) ∂ ( G L , v y ) ∂ ( G L , v y ) ∂ ( G L , g L ) = (cid:20) χ (cid:21) (cid:34) L + O (cid:16) χ (cid:17) L + O (cid:16) χ (cid:17) (cid:35) . The analysis of (43) part is similar.( iii (cid:48) ) is given by G R = G R , v y = L R sinh u R sin g R + G R L R cos g R cosh u R − e R cosh u R < . Here u R < 0, and by Lemma 4.7, tan g R = G R L R + O (1 /χ ) . To get the asymptoticsof the derivative we first show that similarly to Subsection 8.1, we have dv y = (cid:18) − G R L ( k R L + G R ) , , , , k R L + G R , k R L (cid:19) + O (cid:18) χ , χ , χ , χ , χ , χ (cid:19) and then take the inverse. (cid:3) Approaching close encounter In this paper we choose to separate local and global maps by section { x = − } . We could have used instead { x = − } , or { x = − } . Our first goal is to showthat the arbitrariness of this choice does not change the asymptotics of derivative ofthe local map (we have already seen in Sections 6.2 and 7 that it does not in changethe asymptotics of the derivative of the global map).We choose the section {| Q − Q | = µ κ } , / < κ < / 2. Outside the section theorbits are treated as perturbed Kepler motions and inside the section the orbits aretreated as two body scattering. We shall estimate the errors of this approximation.We break the orbit into three pieces: from { x = − , ˙ x > } to {| Q − − Q − | = µ κ } , ONCOLLISION SINGULARITIES IN THE 2-CENTER-2-BODY PROBLEM 49 from {| Q − − Q − | = µ κ } to {| Q +3 − Q +4 | = µ κ } and from {| Q +3 − Q +4 | = µ κ } to { x = − , ˙ x < } . Here and below, we use the following convention. Convention: A variable with superscript − (reap. + ) means its value measuredon the section | Q − Q | = µ κ before (resp. after) Q , Q coming to close encounter. In this section we consider the two pieces of orbit outside the section {| Q − Q | = µ κ } . We use Hamiltonian (2.1). Then we convert the Cartesian coordinates to Delau-nay coordinates. The resulting Hamiltonian is(9.1) H = − L + 12 L − | Q + ( χ, | − | Q + ( χ, | − µ | Q − Q | . The difference with the Hamiltonian (4.1) is that we do not do the Taylor expansionto the potential − µ | Q − Q | .The next lemma and the remark after it tell us that we can neglect those two pieces. Lemma 9.1. Consider the orbits satisfying the conditions of Lemma 3.1. For thepieces of orbit from x = − , ˙ x > to | Q − − Q − | = µ κ and from | Q +3 − Q +4 | = µ κ to x = − , ˙ x > , the derivative matrices have the following form in Delaunaycoordinates ∂ ( X, Y ) − ∂ ( X, Y ) | x = − , ∂ ( X, Y ) | x = − ∂ ( X, Y ) + = × O (1) 2 O (1) × × × Id + O (cid:18) µ − κ + 1 χ (cid:19) . Proof. The proof follows the plan in Section 5. We first consider the integrationof the variational equation. We treat the orbit as Kepler motions perturbed by Q and interaction between Q and Q . Consider first the perturbation coming fromthe interaction of Q and Q . The contribution of this interaction to the variationalequation is of order µ | Q − Q | . If we integrate the variational equation along an orbitsuch that | Q − Q | goes from − µ κ , then the contribution has the order(9.2) O (cid:18)(cid:90) µ κ − µ | t | dt (cid:19) = O ( µ − κ ) . Similar consideration shows that the perturbation from Q is O (1 /χ ).On the other hand absence of perturbation, all Delaunay variables except (cid:96) areconstants of motion. The (2 , 1) entry is also o (1) following from the same estimateas the (2 , 1) entry of the matrix in Lemma 6.1. After integrating over time O (1), thesolutions to the variational equations have the formId + O ( µ − κ + 1 /χ ) . Next we compute the boundary contributions. The analysis is the same as Section 7.The derivative is given by formula (5.2). We need to work out ( U , V )( (cid:96) i ) ⊗ ∂(cid:96) i ∂ ( X,Y ) i and ( U , V )( (cid:96) f ) ⊗ ∂(cid:96) f ∂ ( X,Y ) f . In both cases we have( U , V ) = (0 , , , , , 0) + O ( µ − κ ) . For the section { x = − } , we use (7.3). For the section {| Q − Q | = µ κ } , we have(9.3) ∂(cid:96) ∂ ( X, Y ) = − (cid:18) ∂ | Q − Q | ∂(cid:96) (cid:19) − ∂ | Q − Q | ∂ ( X, Y ) = − ( Q − Q ) · ∂ ( Q − Q ) ∂ ( X,Y ) ( Q − Q ) · ∂ ( Q − Q ) ∂(cid:96) We will prove in Lemma 10.2(c) below that the angle formed by Q − Q and v − v is O (cid:0) µ − κ (cid:1) (the proof of Lemma 10.2 does not rely on section 9). Thus in (9.3) wecan replace Q − Q by v − v making O (cid:0) µ − κ (cid:1) error. Hence ∂(cid:96) ∂ ( X, Y ) = ( v − v ) · ∂ ( Q − Q ) ∂ ( X,Y ) ( v − v ) · ∂Q ∂(cid:96) + O ( µ − κ ) , Note that ∂Q ∂(cid:96) is parallel to v . Using the information about v and v from AppendixB.1 we see that (cid:104) v , v (cid:105) (cid:54) = (cid:104) v , v (cid:105) . Therefore the denominator in (9.3) is bounded awayfrom zero and so ∂(cid:96) ∂ ( X, Y ) = ( O (1) , O (1) , O (1) , O (1) , O (1) , O (1)) . We also need to make sure the second component ∂(cid:96) ∂(cid:96) is not close to 1, so that Id − ( U , V )( (cid:96) f ) ⊗ ∂(cid:96) f ∂ ( X,Y ) f is invertible when | Q − Q | = µ κ serves as the final section. Infact, due to (4.6), ∂(cid:96) ∂(cid:96) (cid:39) − 1. Using formula (5.2), we get the asymptotics stated inthe lemma. (cid:3) Remark 9.2. Using the explicit value of the vectors ˆ¯ l , ˆ¯ l , w, ˜ w in equations (3.1),we find that in the limit µ → , χ → ∞ (cid:18) ∂ ( X, Y ) − ∂ ( X, Y ) | x = − (cid:19) span { w, ˜ w } = span { w, ˜ w } and ˆ¯ l (cid:18) ∂ ( X, Y ) | x = − ∂ ( X, Y ) + (cid:19) = ˆ¯ l , ˆ¯ l (cid:18) ∂ ( X, Y ) | x = − ∂ ( X, Y ) + (cid:19) = ˆ¯ l This tells us that we can neglect the derivative matrices corresponding to the piecesof orbit from x = − , ˙ x > to | Q − − Q − | = µ κ and from | Q +3 − Q +4 | = µ κ to x = − , ˙ x > . We thus can identify d L with ∂ ( L , (cid:96) , G , g , G , g ) + ∂ ( L , (cid:96) , G , g , G , g ) − + O ( µ − κ ) where ( L , (cid:96) , G , g , G , g ) ± denote the Delaunay variables measured on the section {| Q ± − Q ± | = µ κ } . C estimate for the local map In Sections 10 and 12 we consider the piece of orbit from | Q − − Q − | = µ κ to | Q +3 − Q +4 | = µ κ . Because of Remark 9.2, we simply write d L for the derivative forthis piece. ONCOLLISION SINGULARITIES IN THE 2-CENTER-2-BODY PROBLEM 51 Justifying Gerver’s asymptotics. It is convenient to use the coordinates ofrelative motion and the motion of mass center. We define(10.1) v ± = v ± v , Q ± = Q ± Q . Here ”-” refers to the relative motion and ”+” refers to the center of mass motion.To study the relative motion, we make the following rescaling:(10.2) q − := Q − /µ, τ := t/µ and v − remains unchanged.In this way, we zoom in the picture of Q and Q by a factor 1 /µ .Then we have the following lemma. Lemma 10.1. Inside the sphere | Q − Q | = µ κ , / < κ < / , as µ → , (a) the equation governing the motion of the center of mass is a Kepler motionfocused at Q perturbed by O ( µ κ ) , (10.3) ˙ Q + = v + , ˙ v + = − Q + | Q + | + O ( µ κ ) . (b) In the rescaled variables, the equation governing the relative motion is a Keplermotion focused at the origin perturbed by O ( µ κ ) , (10.4) dq − dτ = v − , dv − dτ = q − | q − | + O ( µ κ ) . Proof. Note that (10.1) preserves the symplectic form. dv ∧ dQ + dv ∧ dQ = dv − ∧ dQ − + dv + ∧ dQ + , The Hamiltonian becomes(10.5) H = | v − | − µ | Q − | + | v + | − | Q + + Q − | − | Q + − Q − |− | Q + + Q − + ( χ, | − | Q + − Q − + ( χ, | = | v − | − µ | Q − | + | v + | − | Q + | + | Q − | | Q + | − | Q + · Q − | | Q + | + O ( µ κ ) + O (1 /χ ) , where the O ( µ κ ) includes the | Q − | and higher order terms. In the following, wedrop O (1 /χ ) terms since 1 /χ (cid:28) µ. So the Hamiltonian equations for the motion ofthe mass center part are ˙ Q + = v + , ˙ v + = − Q + | Q + | + O ( µ κ )proving part (a) of the lemma.Next, we study the relative motion. From equation (10.5), we get the equations ofmotion for the center of mass˙ Q − = v − , ˙ v − = − µQ − | Q − | − Q − | Q + | + 3 | Q + · Q − | Q + | Q + | + O ( µ κ ) , as µ → 0, where O ( µ κ ) includes quadratic and higher order terms of | Q − | . Aftermaking the rescaling according to (10.2) the equations for the relative motion partbecome (cid:3) (10.6) dq − dτ = v − , dv − dτ = q − | q − | + µ q − | Q + | − µ | Q + · q − | Q + | Q + | + O ( µ κ ) . Lemma 10.1 implies the following C estimate. Lemma 10.2. (a) We have the following equations for orbit crossing the section {| Q − Q | = µ κ } , / < κ < / and µ → , (10.7) v +3 = 12 R ( α )( v − − v − ) + 12 ( v − + v − ) + O ( µ (1 − κ ) / + µ κ − ) ,v +4 = − R ( α )( v − − v − ) + 12 ( v − + v − ) + O ( µ (1 − κ ) / + µ κ − ) ,Q +3 + Q +4 = Q − + Q − + O ( µ k ) , | Q − − Q − | = 2 µ κ , | Q +3 − Q +4 | = 2 µ κ , where R ( α ) = (cid:20) cos α − sin α sin α cos α (cid:21) , (10.8) α = π + 2 arctan (cid:18) G in µ L in (cid:19) , and 14 L in = v − − µ | Q − | , G in = 2 v − × Q − . (b) We have /c ≤ L in ≤ c for some constant c > . If α is bounded away from and π by an angle independent of µ then G in = O ( µ ) and the closest distancebetween Q and Q is bounded away from zero by δµ and from above by µ/δ for some δ > independent of µ . (c) Also if α is bounded away from and π by an angle independent of µ, thenthe angle formed by Q − and v − is O ( µ − κ ) . (d) The time interval during which the orbit stays in the sphere | Q − | = 2 µ κ is ∆ t = µ ∆ τ = O ( µ κ ) . Proof. In the proof, we omit the subscript in standing for the variables inside thesphere | Q − | = 2 µ κ without leading to confusion.The idea of the proof is to treat the relative motion as a perturbation of Keplermotion and then approximate the relative velocities by their asymptotic values for theKepler motion.Fix a small number δ . Below we derive several estimates valid for the first δ unitsof time the orbit spends in the set | Q − | ≤ µ k . We then show that ∆ t (cid:28) δ . It willbe convenient to measure time from the orbit enters the set | Q − | < µ k . Using the formula in the Appendix A.1, we decompose the Hamiltonian (10.5) as H = H rel + h ( Q + , v + ) where H rel = µ L + | Q − | | Q + | − | Q + · Q − | | Q + | + O ( µ κ ) , as µ → , and h depends only on Q + and v + . ONCOLLISION SINGULARITIES IN THE 2-CENTER-2-BODY PROBLEM 53 Note that H is preserved and ˙ h = O (1) which implies that Lµ is O (1) and moreoverthat ratio does not change much for t ∈ [0 , δ ] . Using the identity µ L = v − − µ | Q − | we see that initially Lµ is uniformly bounded from below for the orbits from Lemma2.4. Thus there is a constant δ such that for t ∈ [0 , δ ] we have δ µ ≤ L ( t ) ≤ µδ . Expressing the Cartesian variables via Delaunay variables (c.f. equation (A.3) inSection A.2) we have up to a rotation by g (10.9) q = 1 µ L (cosh u − e ) , q = 1 µ LG sinh u,O ( µ κ ) = | Q − | = (cid:112) | q | + | q | = L µ ( e cosh u − , following from the same calculation as (4.16) with (cid:96) and u related by u − e sinh u = (cid:96). This gives(10.10) (cid:96) = O ( µ κ − ) . Next(10.11)˙ (cid:96) = − ∂H∂L = − µ L − ∂H rel ∂Q − ∂Q − ∂L = − µ L + O ( µ κ ) O ( µ κ − ) = − µ L + O ( µ κ − ) . Since the leading term here is at least δ µ while (cid:96) = O ( µ κ − ) we obtain part (d) ofthe lemma. In particular the estimates derived above are valid for the time the orbitsspend in | Q − | ≤ µ κ . Next, without using any control on G (using the inequality (cid:12)(cid:12) ∂e∂G (cid:12)(cid:12) = L G/Le ≤ L ), we have(10.12) ˙ G = ∂H∂Q − ∂Q − ∂g = O ( | Q − | ) = O ( µ κ ) , ˙ L = ∂H∂Q − ∂Q − ∂(cid:96) = O ( µ κ +1 ) , (10.13) ˙ g = ∂H∂Q − ∂Q − ∂G = O ( µ κ ) O ( µ κ − ) = O ( µ κ − ) . Integrating over time ∆ t = O ( µ κ ) we get the oscillation of g and arctan GL are O ( µ κ − ) . We are now ready to derive the first two equations of (10.7). It is enough to show v + − = R ( α ) v −− + O ( µ (1 − κ ) / + µ κ − ) where α = 2 arctan GL is the angle formed by thetwo asymptotes of the Kepler hyperbolic motion. We first have | v + − | = | v −− | + O ( µ κ )using the total energy conservation. It remains to show the expression of α . Let usdenote till the end of the proof φ = arctan GL , γ = (1 / − κ . Recall (see (A.3)) that for v − = ( p , p ) , (10.14) p = ˜ p cos g + ˜ p sin g, p = − ˜ p sin g + ˜ p cos g where˜ p = µL sinh u − e cosh u , ˜ p = µGL cosh u − e cosh u . Consider two cases. (I) G ≤ µ κ + γ . In this case on the boundary of the sphere | Q − | = 2 µ κ we have (cid:96) > δ µ − γ for some constant δ . Thus p p = µGL cosh u cos g + µL sinh u sin g − µGL cosh u sin g + µL sinh u cos g = GL ± tan g ± − GL tan g + O ( e − | u | ) = tan( g ± φ ) + O ( µ γ ) . where the plus sign is taken if u > u < . Since arctanis globally Lipschitz, this completes the proof in case (I) by choosing α = 2 φ .(II) G > µ κ + γ . In this case GL (cid:29) p p (or p p ) changeslittle during the time the orbit is inside the sphere. Consider first the case where | g − | > π so sin g is bounded from below. Then p p = cot g + O ( µ − ( κ + γ ) )proving the claim of part (a) in that case. The case | g − | ≤ π is similar but we needto consider p p . This completes the proof in case (II).Combining equation (10.3) and Lemma 10.1(c) we obtain(10.15) Q ++ = Q − + + O ( µ κ ) . We also have Q + − = Q −− + O ( µ κ ) due to to the definition of the sections {| Q ±− | = 2 µ κ } .This proves the last two equation in (10.7). Plugging (10.15) into (10.3) we see that v ++ = v − + + O ( µ κ ) . This completes the proof of part (a).The first claim of part (b) has already been established. The estimate of G followsfrom the formula for α. The estimate of the closest distance follows from the fact thatif α is bounded away from 0 and π then the Q − orbit of Q − ( t ) is a small perturbationof Kepler motion and for Kepler motion the closest distance is of order G. We integratethe ˙ G equation (10.12) over time O ( µ κ ) to get the total variation ∆ G is at most µ κ ,which is much smaller than µ . So G is bounded away from 0 by a quantity of order O ( µ ).Finally part (c) follows since we know G = µ κ | v − | sin (cid:93) ( v − , Q − ) = O ( µ ) . (cid:3) Proof of Lemma 2.4 and 2.6. With the help of Lemma 10.2, we are readyto prove Lemma 2.4 and 2.6. Proof of Lemma 2.4. Since we assume the outgoing asymptote ¯ θ + is close to π , weget that the orbit under consideration has to intersect the section | Q − Q | = µ κ andalso achieve | Q − Q | = O ( µ ) Lemma 10.2. With the same initial E , e , g , e , wedetermine a solution of the Gerver’s map. It follows from (9.1) that the equationsof motion outside the section | Q − Q | = µ κ is a O ( µ − κ ) perturbation of theKepler motion. We get that the v − , , Q − , at collision in Gerver’s case is close tothose values measured on the section | Q − Q | = µ κ in the µ > | Q − Q | = µ κ is not singular. Letting µ = 0 in the first two equations of (10.7) ONCOLLISION SINGULARITIES IN THE 2-CENTER-2-BODY PROBLEM 55 we obtain the equations of elastic collisions. Namely, both the kinetic energy andmomentum conservations hold | v +3 | + | v +4 | = | v − | + | v − | , v +3 + v +4 = v − + v − . On the other hand, the Gerver’s map G in Lemma 2.4 is also defined through elasticcollisions. If we could show that the rotation angle α in the µ > v +3 , , Q +3 , are closein both cases. We then complete the proof using the fact that the orbit outside | Q − Q | = µ κ is a small perturbation of the Kepler motion after running the orbittill the section { x = − } . By converting v +4 , Q +4 into Delaunay coordinates, we canexpress the outgoing asymptote ¯ θ + as a function of v +4 , Q +4 therefore a function of α, v − , v − , Q − , Q − using (10.7) where µ = 0 corresponds to Gerver’s case. To comparethe angle α , it is enough to show that the outgoing asymptote ¯ θ + as a function of α has non degenerate derivative so that we can apply the implicit function theorem tosolve α as a function of ¯ θ + and the initial conditions. In fact we have d ¯ θ + dα = ¯ l · u upto a multiplicative non vanishing factor c , which is non vanishing due to Lemma 3.9.Here the vectors ¯ l and u are in Lemma 3.1 and 3.2 with subscripts omitted. See item(2) of Remark 3.11 for the derivation of dθ + = c ¯ l and Corollary 12.1 for ∂∂α = u . Sothe assumption | ¯ θ + − π | ≤ ˜ θ implies that α in (10.7) is ˜ θ -close to its value in Gerver’scase. (cid:3) Proof of Lemma 2.6. We follow the same argument as in the proof of Lemma 2.4 toget that the orbit of Q is a small deformation of Gerver’s Q ellipse. So we onlyneed to prove this lemma in Gerver’s setting. Since the Q ellipse has semimajor 1in Gerver’s case, the distance from the apogee to the focus is strictly less than 2.Therefore we can find some D > | Q | ≤ − D in the Gerver case. Nextwe know from the Sublemma 4.9 and its proof that Q moves away almost linearly(the oscillation of v is small). We then integrate the dL d(cid:96) equation to get that theoscillation of L is O ( µ ) . (cid:3) Consequences of C estimates Here we obtain corollaries C estimates for the local and global maps. Namely, insubsection 11.1 we show that the orbits we construct are collision free. In subsec-tion 11.2 we show that the angular momentum can be prescribed freely during theconsecutive iterations of the inductive scheme, that is, we prove Sublemma 3.5.11.1. Avoiding collisions. Here we exclude the possibility of collisions. The possiblecollisions may occur for the pair Q , Q and the pair Q , Q . The fact that there isno collision between Q and Q is a consequence of the following result. Lemma 11.1. If an orbit satisfies the conditions of Lemma 4.1 and there is a collisionbetween Q and Q then we have ¯ G + G = O ( µ ) where G and ¯ G denote the angularmoment of Q before and after the application of the global map respectively. Proof. We write the equations of motion as Y (cid:48) = V , where Y = ( L , G , g ; G , g )and V is the RHS of the Hamiltonian equations (4.4).We run the orbit coming to a collision backward so that we can compare it tothe orbit exiting collision. We can still use the hyperbolic Delaunay coordinates toestimate the variational equation for collisional orbits as explained at the beginning ofthe proof of Lemma 6.1. We shall use the subscript in to refer to the orbit coming tocollision with time direction reversed the subscript out for the orbit exiting collision.We have( Y in − Y out ) (cid:48) = O (cid:18)(cid:13)(cid:13)(cid:13)(cid:13) ∂ V ∂ Y (cid:13)(cid:13)(cid:13)(cid:13)(cid:19) ( Y in − Y out ) + O (cid:18) µ | Q − Q | (cid:19) where the last term comes from the µ | Q − Q | term in the potential V L . We integratethis estimate for (cid:96) starting from the collision and ending when the outgoing orbithits the section { x = − χ/ } . The initial condition is Y in − Y out = 0 since L , G , g assume the same values before and after the Q - Q collision. Next, (cid:13)(cid:13) ∂ V ∂ Y (cid:13)(cid:13) = O (cid:16) χ (cid:17) (this is proven in Lemma 6.1(b)). Now the estimates (cid:90) (cid:96) f (cid:96) i ∂ V ∂ Y d(cid:96) = O (1) , (cid:90) (cid:96) f (cid:96) i O (cid:18) µ | Q − Q | (cid:19) d(cid:96) = O ( µ/χ )and the Gronwall Lemma imply that(11.1) Y in ( (cid:96) f ) − Y out ( (cid:96) f ) = O ( µ/χ ) . Next we estimate the angular momentum of Q w.r.t. Q . We have(11.2) G R = G L + v × ( − χ, 0) = G L + v y χ, where v y is the y component of the velocity of Q at the time the orbit hits thesection { x = − χ/ } . Using the equation (A.5) in the Appendix A.2 and Lemma 4.7we see that for the orbits of interest v y = kL ( L sin g − G cos g ) + O (cid:18) χ (cid:19) . Now (11.1) shows that v y,in − v y,out = O ( µ/χ ), where we need to use (4.5) to get thatthe difference of L is also O ( µ/χ ) from other variables when restricted to the section { x = − χ/ } . Hence (11.2) implies that G R,in − G R,out = O ( µ ) . Finally the proof ofLemma 4.1 shows that the angular momentum of Q with respect to Q changes by O ( µ ) during the time the orbits moves from the section { x = − χ/ } to the section { x = − } . (cid:3) Now we exclude the possibility of collisions between Q and Q . Note that Q and Q have two potential collision points corresponding to two intersections of the ellipseof Q and the branch of the hyperbola utilized by Q . See Fig 1 and 2 in Section 2.3.Now it follows from Lemma 10.2(b) that Q and Q do not collide near the intersectionwhere they have the close encounter. We need also to rule out the collision near thesecond intersection point. This was done by Gerver in [G2]. Namely he shows thatthe time for Q and Q to move from one crossing point to the other are different. As ONCOLLISION SINGULARITIES IN THE 2-CENTER-2-BODY PROBLEM 57 a result, if Q and Q come to the correct intersection points nearly simultaneously,they do not collide at the wrong points. To see that the travel times are differentrecall that by second Kepler’s law the area swiped by the moving body in unit time isa constant for the two-body problem. In terms of Delaunay coordinates, this fact isgiven by the equation ˙ (cid:96) = ± L where − is for hyperbolic motion and + for elliptic. Inour case, we have L ≈ L when µ (cid:28) , χ (cid:29) . Therefore in order to collide Q and Q must swipe nearly the same area within the unit time. We see from Fig 1 and Fig2, the area swiped by Q is a proper subset of that by Q between the two crossingpoints. Therefore the travel time for Q is shorter.11.2. Choosing angular momentum. Proof of the Sublemma 3.5. The idea is to apply the strong expansion of the Poincar´emap in a neighborhood of the collisional orbit studied in Lemma 11.1. Notice Delaunaycoordinates regularize double collisions and our estimate of d G holds also for collisionalorbits. Step 1. We first show that there is a collisional orbit as (cid:96) varies. The proof ofLemma 11.1 shows that Q nearly returns back to its initial position. Sublemma 4.9shows that if after the application of the local map we have θ +4 (0) = π − ˜ θ then theorbit hits the line x = − χ so that its y coordinate is a large positive number andif θ +4 (0) = π + ˜ θ then the orbit hits the line x = − χ so that its y coordinate is alarge negative number. Therefore due to the Intermediate Value Theorem it sufficesto show that our surface S j , j = 1 , , contains points x , x such that θ +4 ( x ) = π − ˜ θ,θ +4 ( x ) = π + ˜ θ. We have the expression θ +4 = g +4 − arctan G +4 L +4 . By direct calculationwe find dθ + = L +4 ˆ¯ l (see also item (2) of Remark 3.11). Since T S j ⊂ K j and the cone K j is centered at the plane span { ¯ u − j , ¯¯ u − j } . Note that ¯¯ u − j → ˜ w = ∂∂(cid:96) . We getusing Lemma 3.9 dθ + · ( d L ¯¯ u − j ) = L +4 ˆ¯ l j · (cid:18) µ (ˆ u j (ˆ l j ˜ w ) + o (1)) + O (1) (cid:19) = c j ( x ) /µ, c j ( x j ) (cid:54) = 0 . So it is enough to vary (cid:96) in a O ( µ ) neighborhood of a point whose outgoing asymptotessatisfies the assumption of Lemma 3.1. We choose ˜ θ (cid:28) µ suchthat the assumption of Lemma 3.1 and Sublemma 4.9 is satisfied. Step 2. We show that there exists (cid:96) such that ¯ e ( P ( S ( (cid:96) , ˜ e ))) is close to e ∗∗ . Wefix ˜ e then P ( S ( · , ˜ e )) becomes a function of one variable (cid:96) . Suppose the collisionalorbit in Step 1 occurs at (cid:96) = ˆ (cid:96) . As we vary (cid:96) , the same calculation as in Step 1 givesˆ¯ l j · ( d L ¯¯ u − j ) = ¯ c j ( x ) /µ, ¯ c j ( x j ) (cid:54) = 0 and that ¯ u j contains nonzero ∂/∂e component.Therefore the projection of P = G ◦ L to the e component, i.e. ¯ e ( (cid:96) , ˜ e ) as a functionof (cid:96) is strongly expanding with derivative bounded from below by ¯ cχ µ provided thatthe assumptions of Lemma 4.1 are satisfied (for the orbits of interest this will alwaysbe the case according to Lemma 4.8). Considering the map ¯ e ( (cid:96) , ˜ e ) is not injective,we study ¯ G ( (cid:96) , ˜ e ) instead of ¯ e ( (cid:96) , ˜ e ) using the relation e = (cid:112) G/L ) . Wehave the same strong expansion for ¯ G ( (cid:96) , ˜ e ) since our estimates of the d L , d G are done using G instead of e . Thus it follows from the strong expansion of the map¯ G ( (cid:96) , ˜ e ) that a R -neighborhood of G ∗∗ (corresponding to e ∗∗ ) is covered if (cid:96) variesin a Rµ ¯ cχ -neighborhood of ˆ (cid:96) . Taking R large we can ensure that ¯ G changes from alarge negative number to a large positive number. Then we use the intermediate valuetheorem to find e such that | ¯ G − G ∗∗ | < KK (cid:48) δ, hence | ¯ e − e ∗∗ | < KK (cid:48) δ. Step 3. We show that for the orbit just constructed P ( S (˜ (cid:96) , ˜ e )) ∈ U ( δ ) . ByLemma 2.5, we get θ +4 = O ( µ ). Therefore by Lemma 2.4 L (˜ e , (cid:96) ) has ( E , e , g )close to G ˜ e , , ( E (˜ e , (cid:96) ) , e (˜ e , ˜ (cid:96) ) , g (˜ e , ˜ (cid:96) )) . It follows that | E − E ∗∗ | < KK (cid:48) δ/ , | e − e ∗∗ | < KK (cid:48) δ/ , | g − g ∗∗ | < KK (cid:48) δ/ . Next Lemma 2.5 shows that after the application of G , ( E , e , g ) change little and θ − becomes O ( µ ) . (cid:3) Derivative of the local map Justifying the asymptotics. Here we give the proof of Lemma 3.1. Our goalis to show that the main contribution to the derivative comes from differentiating themain term in Lemma 10.2. Proof of Lemma 3.1. Since the transformation from Delaunay to Cartesian variablesis symplectic and the norms of the transformation matrices are independent of µ ,it is sufficient to prove the lemma in terms of Cartesian coordinates. To go to thecoordinates system used in Lemma 3.1, we only need to multiply the Cartesian de-rivative matrix by O (1) matrices, namely, by ∂ ( L ,(cid:96) ,G ,g ,G ,g ) + ∂ ( Q ,v ,Q ,v ) + on the left and by ∂ ( Q ,v ,Q ,v ) − ∂ ( L ,(cid:96) ,G ,g ,G ,g ) − on the right. This does not change the form of the d L stated inLemma 3.1.As before we use the formula (5.2). We need to consider the integration of thevariational equations and also the boundary contribution. Recall that the subscripts − and + mean relative motion and mass center motionrespectively, and the superscripts − and + mean incoming and outgoing respectively.In the following, we are most interested in the relative motion, so we drop the subscript − of Q − , v − , L − , G − , g − for simplicity and without leading to confusion. Step 1, the Hamiltonian equations, the variational equations and theboundary contributions. It is convenient to use the variable L = L/µ . Lemma 10.2 gives that 1 /c < L < c and µ/c ≤ G ≤ cµ for some c > α is bounded away from 0 and π . We also have g, Q + , v + , v = O (1) and Q = O ( µ κ ). From the Hamiltonian (10.5),we have ˙ (cid:96) = − µ L + O ( µ κ ) (see (10.11)). Using (cid:96) as the time variable we get from(10.5) that the equations of motion take the following form (recall that (cid:96) = O ( µ κ − )due to (10.10)): ONCOLLISION SINGULARITIES IN THE 2-CENTER-2-BODY PROBLEM 59 (12.1) ∂ L ∂(cid:96) = − L ∂H∂(cid:96) (cid:18) − L ∂H∂ L + . . . (cid:19) (1 + O ( µ κ +1 )) = O ( µ κ ) ,∂G∂(cid:96) = − µ L ∂H∂g (cid:18) − L ∂H∂ L + . . . (cid:19) (1 + O ( µ κ +1 )) = O ( µ κ ) ,∂g∂(cid:96) = 2 µ L ∂H∂G (cid:18) − L ∂H∂ L + . . . (cid:19) (1 + O ( µ κ +1 )) = O ( µ κ ) ,dQ + d(cid:96) = − v + µ L )(1 + O ( µ κ +1 )) = O ( µ ) dv + d(cid:96) = (cid:18) Q + | Q + | + O ( µ κ ) (cid:19) (2 µ L )(1 + O ( µ κ +1 )) = O ( µ ) . where . . . denote the higher order terms. The estimates of the last two equations aresimple. In the first three equations, the main contribution in H is coming from | Q | and | Q + · Q | , both of which are O ( µ κ ). We have the estimate (cid:12)(cid:12)(cid:12)(cid:12)(cid:18) ∂∂ L , ∂∂(cid:96) , ∂∂G , ∂∂g (cid:19) Q (cid:12)(cid:12)(cid:12)(cid:12) = O ( µ κ , µ, µ κ − , µ κ )using (10.9) for Q = ( q , q ) up to a rotation by g . In fact, the ∂∂(cid:96) amounts todividing by the scale of (cid:96) , i.e. µ − κ . The derivatives ∂∂ L , ∂∂g do not change the orderof magnitude. Finally since G = O ( µ ), the ∂∂G amounts to dividing by µ. Next weanalyze the variational equations. This estimate is much easier than that of the globalmap part. The same rules as those used to obtain (12.1) apply here.(12.2) dd(cid:96) δ L δGδgδQ + δv + = O µ κ µ κ µ κ µ κ µ κ µ κ µ κ µ κ µ κ µ κ − µ κ µ κ µ µ κ +1 µ κ +2 µ κ +2 µµ µ κ +1 µ κ +2 µ δ L δGδgδQ + δv + . We need to integrate this equation over time µ κ − . As we did in the proof of Lemma6.2, we have Gronwall inequality for linear systems (Lemma 6.3). Recall also that the“ ≤ ” for matrices means “ ≤ ” entry-wise.Thus we compare the solution to the variational equation with a constant linearODE of the form X (cid:48) = AX. Its solution has form X ( µ κ − ) = (cid:80) ∞ n =0 ( Aµ κ − ) n n ! . We willshow that(12.3) ( Aµ κ − ) ≤ C (( Aµ κ − ) + ( Aµ κ − ) ) . Then we have( Aµ κ − ) n ≤ C n (( Aµ κ − ) + ( Aµ κ − ) ) , C n = C (1 + C ) n . Hence X ( µ κ − ) ≤ Id + C (( Aµ κ − ) + ( Aµ κ − ) ). We next integrate the variationalequations over time O ( µ κ − ) to get the estimate of its fundamental solution. Fromnow on, we fix κ = 2 / ∈ (1 / , / . We get the following bound for the fundamentalsolution of the variational equation in the case of κ = 2 / 5, in which case (12.3) holds and so two steps of Picard iteration are enough(12.4) Id + O µ κ µ κ − µ κ µ κ µ κ µ κ µ κ − µ κ µ κ µ κ µ κ − µ κ − µ κ − µ κ − µ κ − µ κ µ κ − µ κ µ κ µ κ µ κ µ κ − µ κ µ κ µ κ . This calculation can either be done by hand or using computer.Next, we compute the boundary contribution using the formula (5.2). In terms ofthe Delaunay variables inside the sphere | Q | = 2 µ κ , we have(12.5) ∂(cid:96)∂ ( L , G, g, Q + , v + ) = − (cid:18) ∂ | Q | ∂(cid:96) (cid:19) − ∂ | Q | ∂ ( L , G, g, Q + , v + ) = ( O ( µ κ − ) , O ( µ κ − ) , , , . Indeed, due to (10.9) we have ∂ | Q | ∂g = 0, ∂ | Q | ∂(cid:96) = O ( µ ), ∂ | Q | ∂ L = O ( µ κ ) and ∂ | Q | ∂G = O ( µ κ − ) . Combining this with (12.1) we get(12.6) (cid:18) ∂∂(cid:96) ( L , G, g, Q + , v + ) (cid:19) ⊗ ∂(cid:96)∂ ( L , G, g, Q + , v + )= O ( µ κ , µ κ , µ κ , µ, µ ) ⊗ O ( µ κ − , µ κ − , , , . Step 2, the analysis of the relative motion part. The structure of d L comes mainly from the relative motion part, on which we nowfocus. We neglect the Q + , v + part and will study them in the last step. Substep 2.1, the strategy. Using (5.2) we obtain the derivative matrix(12.7) ∂ ( L , G, g ) + ∂ ( L , G, g ) − = Id + O µ κ µ κ − µ κ µ κ − µ κ − µ κ − − × Id + O µ κ µ κ − µ κ µ κ µ κ − µ κ µ κ − µ κ − µ κ − Id − O µ κ µ κ − µ κ µ κ − µ κ − µ κ − = Id + O µ κ µ κ − µ κ µ κ µ κ − µ κ µ κ − µ κ − µ κ − := Id + P. For the position variables q , we are only interested in the angle Θ := arctan (cid:16) q q (cid:17) since the length | ( q , q ) | = 2 µ κ is fixed when restricted on the sphere.We split the derivative matrix as follows:(12.8) ∂ (Θ , v ) + ∂ (Θ , v ) − = ∂ (Θ , v ) + ∂ ( L , G, g ) + ∂ ( L , G, g ) + ∂ ( L , G, g ) − ∂ ( L , G, g ) − ∂ (Θ , v ) − = ONCOLLISION SINGULARITIES IN THE 2-CENTER-2-BODY PROBLEM 61 ∂ (Θ , v ) + ∂ ( L , G, g ) + ∂ ( L , G, g ) − ∂ (Θ , v ) − + ∂ (Θ , v ) + ∂ ( L , G, g ) + P ∂ ( L , G, g ) − ∂ (Θ , v ) − = I + II. In the following, we prove Claim: (12.9) I = 1 µ O (1) × ⊗ ∂G − ∂ (Θ , v ) − + O (1) , II = 1 µ O ( µ κ − ) × ⊗ ∂G − ∂ (Θ , v ) − + O ( µ κ − ) . We will give the expressions of O (1) terms explicitly. Substep 2.2, the estimate of I in the splitting (12.8) . Using equations (10.9) and (10.14) we obtain(12.10) ∂ (Θ , v ) + ∂ ( L , G, g ) + = O µ − µ − µ − . Next, we consider the first term in (12.8).(12.11) I = ∂ (Θ , v ) + ∂ L + ⊗ ∂ L − ∂ (Θ , v ) − + ∂ (Θ , v ) + ∂G + ⊗ ∂G − ∂ (Θ , v ) − + ∂ (Θ , v ) + ∂g + ⊗ ∂g − ∂ (Θ , v ) − . Using the expressions L = v − µ | Q | , G = v × Q = | v | · | Q | sin (cid:93) ( v, Q ), we see that(12.12) ∂ L − ∂ (Θ , v ) − = O (1) , ∂G − ∂ (Θ , v ) − = ( O ( µ κ ) , O ( µ κ )) . It only remains to get the estimate of ∂g − ∂ (Θ ,v ) − . Next, we claim that(12.13) ∂g − ∂ (Θ , v ) − = (cid:20) ∂∂G − arctan (cid:18) G − µ L (cid:19)(cid:21) ∂G − ∂ (Θ , v ) − + O (1) = O (1 /µ ) ∂G − ∂ (Θ , v ) − + O (1) . We use the fact p p = sin g sinh u ± Gµ L cos g cosh u cos g sinh u ∓ Gµ L sin g cosh u = tan g ± Gµ L ∓ Gµ L tan g + e − | u | E ( G/µ L , g, u ) , where E is a smooth function satisfying ∂E∂g = O (1) as (cid:96) → ∞ . Therefore we get g = arctan (cid:18) p p − e − | u | E ( G/µ L , g ) (cid:19) ∓ arctan Gµ L as (cid:96) → ∞ . We choose the + when considering the incoming orbit parameters. Thus ∂g∂ (Θ , v ) (cid:16) O ( e − | u | ) (cid:17) = ∂ arctan p p ∂ (Θ , v ) + ∂ arctan Gµ L ∂ L ∂ L ∂ (Θ , v )+ (cid:32) ∂ arctan Gµ L ∂G + O ( e − | u | /µ ) (cid:33) ∂G∂ (Θ , v ) + O ( e − | u | )proving (12.13). Plugging (12.10), (12.12) and (12.13) back to (12.11) we get the estimate of I in(12.9). More explicitly, I = µ U ⊗ ∂G − ∂ (Θ ,v ) − + B , where (12.14) U = (cid:32) µ ∂ (Θ , v ) + ∂G + + µ ∂ arctan G − µ L − ∂G − ∂ (Θ , v ) + ∂g + + O ( e − | u | ) (cid:33) B = ∂ (Θ , v ) + ∂ L + ⊗ ∂ L − ∂ (Θ , v ) − + ∂ (Θ , v ) + ∂g + ⊗ ∂ arctan p − p − ∂ (Θ , v ) + ∂ arctan G − µ L − ∂ L − ∂ L − ∂ (Θ , v ) + O ( e − | u | ) . Substep 2.3, the estimate of II in the splitting (12.8) . Now we study the second term in (12.8)(12.15) II = O µ − µ − µ − · O µ κ µ κ − µ κ µ κ µ κ − µ κ µ κ − µ κ − µ κ − ∂ ( L , G, g ) − ∂ (Θ , v ) − = O µ κ − µ κ − µ κ − µ κ − µ κ − µ κ − µ κ − µ κ − µ κ − ∂ ( L , G, g ) − ∂ (Θ , v ) − = µ κ − (cid:20) O (1) × ⊗ ∂ L − ∂ (Θ , v ) − + O ( µ − ) × ⊗ ∂G − ∂ (Θ , v ) − + O (1) × ⊗ ∂g − ∂ (Θ , v ) − (cid:21) where we use that µ κ < µ κ − and µ κ − < µ κ − since κ < / 2. The first summandin (12.15) is O ( µ κ − ). Applying (12.13), we get the estimate of II in (12.9). Substep 2.4, going from Θ to Q . We use the variable Θ for the relative position Q and we have ∂G − ∂ (Θ ,v ) − = O ( µ κ ). Toobtain ∂ ( Q,v ) + ∂ ( Q,v ) − , we use that Q = 2 µ κ (cos Θ , sin Θ) = ( x, y ) , Θ = arctan yx . So we have the estimate ∂Q + ∂ ( L ,G,g ) + = O ( µ κ ) ∂ Θ + ∂ ( L ,G,g ) + = O ( µ κ − ). To get ∂ − ∂Q − , we usethe transformation from polar coordinates to Cartesian, ∂ − ∂Q − = ∂ − ∂ ( r, Θ) − ∂ ( r, Θ) − ∂Q − , where r = | Q − | = 2 µ κ . Therefore we have ∂r − ∂Q − = 0 , ∂ − ∂Q − = µ κ ∂ − ∂ Θ − ( − sin Θ − , cos Θ − ) . So we have the estimate ∂G − ∂Q − = O (1), and ∂ L − ∂Q − = ∂ L − ∂ Θ − = 0 since in the expression L = v − µ | Q | , the angle Θ plays no role. Finally, we have ∂∂Q −− arctan p − p − = 0.Applying these estimates to the concrete expressions of U , B , and (12.15) for the O ( µ κ − ) remainder, so we get(12.16) ∂ ( Q, v ) + ∂ ( Q, v ) − = 1 µ ( O ( µ κ ) × , O (1) × ) ⊗ ( O (1) × , O ( µ κ ) × ) + O (1) × . In particular, the estimate of B + O ( µ κ − ) = O (1) instead of O ( µ − κ ) is due to ∂ L − ∂Q − = ∂ L − ∂ Θ − = 0 and ∂p −− ∂Q − = ∂p −− ∂ Θ − = 0 . ONCOLLISION SINGULARITIES IN THE 2-CENTER-2-BODY PROBLEM 63 Step 3, the contribution from the motion of the mass center. Substep 3.1, the decomposition. Consider the following decomposition(12.17) D := ∂ (Θ , v, Q + , v + ) + ∂ (Θ , v, Q + , v + ) − = ∂ (Θ , v ; Q + , v + ) + ∂ ( L , G, g ; Q + , v + ) + ∂ ( L , G, g ; Q + , v + ) + ∂ ( L , G, g ; Q + , v + )( (cid:96) f ) ∂ ( L , G, g ; Q + , v + )( (cid:96) f ) ∂ ( L , G, g ; Q + , v + )( (cid:96) i ) ∂ ( L , G, g ; Q + , v + )( (cid:96) i ) ∂ ( L , G, g ; Q + , v + ) − ∂ ( L , G, g ; Q + , v + ) − ∂ (Θ , v ; Q + , v + ) − := (cid:20) M 00 Id (cid:21) (cid:20) A B Id (cid:21) (cid:20) C DE F (cid:21) (cid:20) A (cid:48) B (cid:48) Id (cid:21) (cid:20) N 00 Id (cid:21) = (cid:20) M ACA (cid:48) N + M ADB (cid:48) N M AD ( BC + E ) A (cid:48) N + ( BD + F ) B (cid:48) N BD + F (cid:21) . Each of the above matrix is 7 × Substep 3.2, the estimate of each block. The matrix M = ∂ (Θ ,v ) + ∂ ( L ,G,g ) + is given by (12.10) and N = ∂ ( L ,G,g ) − ∂ (Θ ,v ) − is given by (12.10),(12.12), (12.13) M = O µ − µ − µ − , N = O (1) × ∂G − ∂ (Θ ,v ) − O ( µ ) ∂G − ∂ (Θ ,v ) − + O (1) .C, D, E, F form the matrix (12.4), the fundamental solution of the variational equa-tion, (cid:18) C DE F (cid:19) = Id + O µ κ µ κ − µ κ ( µ κ ) × ( µ κ ) × µ κ µ κ − µ κ ( µ κ ) × ( µ κ ) × µ κ − µ κ − µ κ − ( µ κ − ) × ( µ κ − ) × ( µ κ ) × ( µ κ − ) × ( µ κ ) × ( µ κ ) × ( µ κ ) × ( µ κ ) × ( µ κ − ) × ( µ κ ) × ( µ κ ) × ( µ κ ) × .A, B, A (cid:48) , B (cid:48) are given by (12.6), boundary contributions, (cid:20) A B Id (cid:21) , (cid:20) A (cid:48) B (cid:48) Id (cid:21) = Id + O ( µ κ , µ κ , µ κ ; µ × ) ⊗ O ( µ κ − , µ κ − , 0; 0 × ) . Substep 3.3, the estimate of the first block M ACA (cid:48) N + M ADB (cid:48) N in D . By (12.7) ACA (cid:48) = Id + P = Id + O µ κ µ κ − µ κ µ κ µ κ − µ κ µ κ − µ κ − µ κ − (Recall that (12.7) is the part of ∂ ( L ,G,g ) + ∂ ( L ,G,g ) − without considering the motion of the masscenter), and by (12.9) and (12.14)(12.18) M ACA (cid:48) N = M (Id + P ) N = 1 µ (cid:0) U + O (cid:0) µ κ − (cid:1)(cid:1) ⊗ ∂G − ∂ (Θ , v ) − + B + O (cid:0) µ κ − (cid:1) . Indeed, using the notation of (12.8), we have I = M N and II = M P N . The estimatesof I and II are given in (12.9).Next we claim that(12.19) M ADB (cid:48) N = O (cid:0) µ κ − (cid:1) ∂G − ∂ (Θ , v ) − + O (cid:0) µ κ − (cid:1) so it can be absorbed into the errror terms of (12.18). To this end we split N = N + N ,A = Id + A where N = × ∂G − ∂ (Θ ,v ) − O ( µ ) ∂G − ∂ (Θ ,v ) − , N = O (1) × × O (1) × ,A = O ( µ κ , µ κ , µ κ ) ⊗ O ( µ κ − , µ κ − , . Thus M ADB (cid:48) N = M DB (cid:48) N + M A DB (cid:48) N. Let us work on the first term. A direct computation shows that DB (cid:48) = O µ κ µ κ − µ κ µ κ − µ κ − µ κ − , and M DB (cid:48) = O (cid:0) µ κ − × , µ κ − × , × (cid:1) . Now it is easy to see that M DB (cid:48) N can beabsorbed into the first term in (12.19) and M DB (cid:48) N can be absorbed into the secondterm. The key is that N has rank one and the second row of N is zero. The analysisof M A DB (cid:48) N is even easier since a direct computation shows that DB (cid:48) dominates A DB (cid:48) componentwise. This proves (12.19) and shows that M ACA (cid:48) N + M ADB (cid:48) N has the same asymptotics as (12.18). Substep 3.4, estimate of the remaining blocks in D . The following estimates are obtained by a direct computation BD + F = O ( µ × ) ⊗ O ( µ κ − , µ κ − , O ( µ κ ) × ( µ κ ) × ( µ κ ) × ( µ κ ) × ( µ κ − ) × ( µ κ − ) × + Id + O (cid:18) ( µ κ ) × ( µ κ ) × ( µ κ ) × ( µ κ ) × (cid:19) = Id + O ( µ κ ) × .BC + E = O ( µ × ) ⊗ O ( µ κ − , µ κ − , O µ κ µ κ − µ κ µ κ µ κ − µ κ µ κ − µ κ − µ κ − + (cid:0) ( µ κ ) × , ( µ κ − ) × , ( µ κ ) × (cid:1) = O (cid:0) ( µ κ ) × , ( µ κ − ) × , ( µ κ − ) × (cid:1) . Accordingly using (12.12) and (12.13) for N , and arguing the same way as on substep3.3 we get(12.20) ( BC + E ) A (cid:48) N + ( BD + F ) B (cid:48) N = 1 µ [ O ( µ κ )] × ⊗ ∂G − ∂ (Θ , v ) −− + O ( µ κ ) . Finally, we have M AD = [ O ( µ κ − )] × . Substep 3.5, completing the asymptotics of D . ONCOLLISION SINGULARITIES IN THE 2-CENTER-2-BODY PROBLEM 65 Substeps 3.1–3.4 above can be summarized as follows(12.21) D =1 µ ( U + O ( µ κ − ); O ( µ κ ) × ) ⊗ (cid:32) ∂G − ∂ (Θ , v ) −− ; 0 × (cid:33) + (cid:18) B 00 Id (cid:19) + O (cid:0) µ κ − (cid:1) . Finally, when we use the coordinates ( Q − , v − ) instead of (Θ − , v − ) as we did in Sub-step 2.4, we get U + O ( µ κ − ) = O ( µ κ × , × ) and B + O (cid:0) µ κ − (cid:1) = O (1), and ∂G − ∂ ( Q,v ) −− = O (1 × , µ κ × ) in terms of the coordinates ( Q − , v − , Q + , v + ). Hence, simi-larly to (12.16), we get ∂ ( Q − , v − , Q + , v + ) + ∂ ( Q − , v − , Q + , v + ) − = 1 µ O ( µ κ × , × ; O ( µ κ ) × ) ⊗ (cid:0) × , µ κ × ; 0 × (cid:1) + O (1) . This is the structure of d L stated in the lemma.It remains to obtain the asymptotics of the leading terms in Lemma 3.1. Belowwe use the Delaunay variables ( L , (cid:96) , G , g , G , g ) ± as the orbit parameters outside the sphere | Q − | = 2 µ κ and add a subscript in to the Delaunay variables inside thesphere. We relate C estimates of Lemma 10.2 to the C estimates obtained above.Namely consider the following equation which is obtained by discarding the o (1) errorsin (10.7)(12.22) Q + − = 0 , v + − = R ( α ) v −− , Q ++ = Q − + , v ++ = v − + , where α is given by (10.8). We have the following corollary saying that d L can beobtained by taking derivative directly in (12.22). Corollary 12.1. Under the assumption of Lemma 3.1, the derivative of the local maphas the following form (12.23) d L = 1 µ (ˆ u j + O ( µ κ )) ⊗ l j + ˆ B j + O ( µ κ − ) , where ˆ u j , l j and ˆ B j are computed from (12.22) and the variables are evaluated at the j -th Gerver’s collision point, j = 1 , . In particular, (12.24)ˆ u j = ∂ ( L , (cid:96) , G , g , G , g ) + ∂ ( Q , v , Q , v ) + ∂ ( Q , v , Q , v ) + ∂ ( Q − , v − , Q + , v + ) + ∂ ( Q − , v − , Q + , v + ) + ∂α (cid:18) µ ∂α∂G in (cid:19) , l j = ∂G in ∂ ( Q − , v − , Q + , v + ) − ∂ ( Q − , v − , Q + , v + ) − ∂ ( Q , v , Q , v ) − ∂ ( Q , v , Q , v ) − ∂ ( L , (cid:96) , G , g , G , g ) − . As /χ (cid:28) µ → , we have that l j is a continuous function of ( L , (cid:96) , G , g , G , g ) − ,and ˆ u j is a continuous function of both ( L , (cid:96) , G , g , G , g ) − and α .Proof. We begin by computing the rank 1 terms in the expression for D . To get(12.24) we need to multiply the vector by ∂ ( L ,(cid:96) ,G ,g ,G ,g ) + ∂ ( Q ,v ,Q ,v ) + ∂ ( Q ,v ,Q ,v ) + ∂ ( Q − ,v − ,Q + ,v + ) + and thelinear functional by ∂ ( Q − ,v − ,Q + ,v + ) − ∂ ( Q ,v ,Q ,v ) − ∂ ( Q ,v ,Q ,v ) − ∂ ( L ,(cid:96) ,G ,g ,G ,g ) − . For the map (12.22) we have ∂ ( Q + , v + ) + ∂ ( Q + , v + ) − = Id , ∂ ( Q + , v + ) + ∂ ( Q − , v − ) − = ∂ ( Q − , v − ) + ∂ ( Q + , v + ) − = 0 , ∂ ( Q + , v + ) + ∂α = ∂G in ∂ ( Q + , v + ) − = 0which agrees with the corresponding blocks in (12.21) up to an o (1) error as µ → ∂ ( Q − ,v − ) + ∂ ( Q − ,v − ) − .Now the expression for l j follows from (12.18).Differentiating (12.22) we get ∂ ( Q − ,v − ) + ∂α = (cid:18) , ∂v + − ∂α (cid:19) . Thus to get the expression ofˆ u in (12.24), it is enough to show (cf. (12.14)) that for the map (12.22) we have(12.25) ∂v + − ∂α (cid:18) ∂α∂G in (cid:19) = (cid:32) ∂v + − ∂G + + ∂ arctan G − µ L − ∂G − ∂v + − ∂g + (cid:33) , G in = G − . Write v + − = V ( G + , µ L , g + ) where G + and g + depend on G − as follows. First, G + = G − . Second, (A.3) givesarctan (cid:18) v +2 v +1 (cid:19) ∼ g + − arctan (cid:18) G + µ L (cid:19) , arctan (cid:18) v − v − (cid:19) ∼ g − − arctan (cid:18) G − µ L (cid:19) , arctan (cid:18) v +2 v +1 (cid:19) ∼ arctan (cid:18) v − v − (cid:19) + α where ∼ means that the difference between the LHS and the RHS is O (cid:0) e − u (cid:1) . Thus g + ∼ g − + α and so ∂v + − ∂G − = ∂ V ∂G + + ∂ V ∂g + ∂g + ∂G − ∼ ∂ V ∂G + + ∂ V ∂g + ∂α∂G − proving (12.25).To complete the proof of the corollary we have to show that the formula for ˆ B isobtained by taking the derivatives of (12.22) with respect to variables different from G − . This is done by comparing (12.24) with (12.14) similarly to the derivation of(12.24). (cid:3) It remains to show that the RHS of (12.24) has the dependence on x , θ +4 required byLemma 3.1. To this end we note that the variable α can be solved using the implicitfunction theorem as a function of the outgoing asymptote ¯ θ +4 in the limit µ → (cid:3) The next corollary says that the small remainders in (10.7) is also C small if thederivative is taken along a correct direction, i.e. the direction with small change of G − in . ONCOLLISION SINGULARITIES IN THE 2-CENTER-2-BODY PROBLEM 67 Corollary 12.2. Let γ ( s ) : ( − ε, ε ) → R be a C curve such that Γ = γ (cid:48) (0) = O (1) and ∂G − in ◦ γ (0) ∂s = ∂G − in ∂ Γ = O ( µ ) then when taking derivative with respect to s in equations | v +3 | + | v +4 | = | v − | + | v − | + o (1) ,v +3 + v +4 = v − + v − + o (1) ,Q +3 + Q +4 = Q − + Q − + o (1) , obtained from equation (10.7) , the o (1) terms are small in the C sense as µ → .Proof. For the motion of the mass center, it follows from Corollary 12.1 that ∂ ( Q + , v + ) + ∂ ( Q − , v − , Q + , v + ) − = 1 µ ∂ ( Q + , v + ) + ∂α ⊗ l + (0 × , Id × ) + o (1) . We already obtained that ∂ ( Q + ,v + ) + ∂α = O ( µ κ ) (see equation (12.21)). Due to Corol-lary 12.1 our assumption that ∂G − in ∂s = O ( µ ) implies that(12.26) l · Γ = O ( µ )which suppresses the 1 /µ term. This proves the last two identities of the corollary.To derive the first equation it is enough to show dds ( | v + − | − | v −− | ) = o (1) since wealready have the required estimate for the velocity of the mass center. We use thefact that RHS (10.5) is the same in incoming and outgoing variables (superscripts +and − respectively). In (10.5), the terms involving only Q + , v + are handled using theresult of the previous paragraph. The term − µ | Q − | vanishes when taking derivativesince | Q − | = 2 µ κ is constant. All the remaining terms have Q − to the power 2 orhigher. We have ∂Q −− ∂s = O (1) since Γ = O (1). We also have ∂Q + − ∂s = O (1) due to(12.26). Therefore after taking the s derivative, any term involving Q − is of order O ( µ κ ). This completes the proof of the energy conservation part. (cid:3) Proof of the Lemma 3.9. In this section we work out the O (1 /µ ) term in thelocal map. Proof. The proof is relies on a numerical computation. Before collision, l = ∂G in ∂ − . According to Corollary 12.1 we can differentiate theasymptotic expression of Lemma 10.2. We have (cid:16) ∂G in ∂G − , ∂G in ∂g − (cid:17) = − ( v − − v − ) × (cid:18) ∂∂G − , ∂∂g − (cid:19) Q − ( v − − v − ) × (cid:18) ∂Q ∂(cid:96) − (cid:19) · (cid:18) ∂(cid:96) − ∂G − , ∂(cid:96) − ∂g − (cid:19) + O ( µ κ + µ − κ ) , where O ( µ κ ) comes from (cid:16) ∂∂ − ( v − − v − ) (cid:17) × ( Q − Q ) and O ( µ − κ ) comes from ∂Q ∂L − ∂L − ∂ − where L is solved from the Hamiltonian (9.1) H = 0 . We need to eliminate (cid:96) using the relation | Q − Q | = µ κ . (cid:18) ∂(cid:96) − ∂G − , ∂(cid:96) − ∂g − (cid:19) = − (cid:18) ∂ | Q − Q | ∂(cid:96) − (cid:19) − (cid:18) ∂ | Q − Q | ∂G − , ∂ | Q − Q | ∂g − (cid:19) = − ( Q − Q ) · (cid:16) ∂Q ∂G − , ∂Q ∂g − (cid:17) ( Q − Q ) · ∂Q ∂(cid:96) − = − ( v − − v − ) · (cid:16) ∂Q ∂G − , ∂Q ∂g − (cid:17) ( v − − v − ) · ∂Q ∂(cid:96) − + O ( µ − κ ) . Here we replaced Q − − Q − by v − − v − using the fact that the two vectors form anangle of order O ( µ − κ ) by Lemma 10.2(c). Therefore (cid:18) ∂G in ∂G − , ∂G in ∂g − (cid:19) = − ( v − − v − ) × (cid:18) ∂∂G − , ∂∂g − (cid:19) Q +( v − − v − ) × ∂Q ∂(cid:96) − ( v − − v − ) · (cid:16) ∂Q ∂G − , ∂Q ∂g − (cid:17) ( v − − v − ) · ∂Q ∂(cid:96) − + O ( µ κ + µ − κ ) . Similarly, we get ∂G in ∂(cid:96) − = ( v − − v − ) × ∂Q ∂(cid:96) − + ( v − − v − ) × ∂Q ∂(cid:96) − ( v − − v − ) · ∂Q ∂(cid:96) − ( v − − v − ) · ∂Q ∂(cid:96) − + O ( µ κ + µ − κ ) . We use Mathematica and the data in the Appendix B.2 to work out ∂G in ∂ − . Theresults are : for the first collision, ˆ l = [ ∗ , − . , ∗ , ∗ , . , − . , and for the secondcollision: ˆ l = [ ∗ , − . , ∗ , ∗ , . , − . l i · w − i (cid:54) = 0 andˆ l i · ˜ w (cid:54) = 0 for i = 1 , After collision, ˆ u = ∂ − ∂α . In equation (10.7), we let µ → 0. Recall (5.1). Applyingthe implicit function theorem to (10.7) with µ = 0 we obtain (cid:18) ∂ ( Q +3 , v +3 , Q +4 , v +4 ) ∂ ( X + , Y + ) + ∂ ( Q +3 , v +3 , Q +4 , v +4 ) ∂(cid:96) +4 ⊗ ∂(cid:96) +4 ∂ ( X + , Y + ) (cid:19) · ∂ ( X + , Y + ) ∂α = 12 (cid:16) , , R (cid:16) π α (cid:17) ( v − − v − ) , , , − R (cid:16) π α (cid:17) ( v − − v − ) (cid:17) T = 12 (cid:16) , , R (cid:16) π (cid:17) ( v +3 − v +4 ) , , , − R (cid:16) π (cid:17) ( v +3 − v +4 ) (cid:17) T . where R ( π/ α ) = dR ( α ) dα and ∂(cid:96) +4 ∂ ( X + ,Y + ) is given by (9.3). Again we use Mathematica to work out the ∂ − ∂α . The results are: for the first collision ˆ u = [ − . , ∗ , ∗ , ∗ , − . , − . u = [ − . , ∗ , ∗ , ∗ , . , − . l i · ˆ u i (cid:54) = 0 for i = 1 , , v +3 − v +4 ) → − ( v +3 − v +4 ). So we only need to send ˆ u → − ˆ u . (cid:3) ONCOLLISION SINGULARITIES IN THE 2-CENTER-2-BODY PROBLEM 69 Proof of the Lemma 3.10. In this section, we prove Lemma 3.10, whichguarantees the non degeneracy condition Lemma 3.4 (see the proof of Lemma 3.4).Since we have already obtained l and u in d L and ¯ l , ¯¯ l , ¯ u , ¯¯ u in d G , one way to proveLemma 3.4 is to work out the matrix B explicitly using Corollary 12.1 on a computer.In that case, the current section is not necessary. However, in this section, we use adifferent approach, which simplifies the computation and has several advantages. Thefirst advantage is that this treatment has clear physical and geometrical meaning.Second, we use the same way to control the shape of the ellipse in Appendix B.3.Third, this method gives us a way to deal with the singular limit d L as µ → d L = 1 µ u j ⊗ l j + B + O ( µ κ ) , d G = χ ¯ u j ⊗ ¯ l j + χ ¯¯ u j ⊗ ¯¯ l j + O ( µ χ ) , where j = 1 , o (1)errors into the vectors. Moreover, in the limit 1 /χ (cid:28) µ → δ → θ → span { ¯ u j , ¯¯ u j } → span { w j , ˜ w } , l j → ˆ l j , ¯ l j → ˆ¯ l j , ¯¯ l j → ˆ¯¯ l j , j = 1 , . We first prove an abstract lemma that reduces the study of the local map of the µ > µ = 0 case. It shows that we can find a direction in span { ¯ u , ¯¯ u } , along whichthe directional derivative of d L is not singular. Lemma 12.3. Consider x ∈ U j ( δ ) , j=1,2 and | ¯ θ +4 − π | < ˜ θ as in Lemma 3.1. Supposethe vector ˜Γ µ ∈ span { ¯ u − j , ¯¯ u − j } ⊂ T x U j ( δ ) satisfies ¯ l j ( d L ˜Γ µ ) = 0 and (cid:107) ˜Γ µ (cid:107) ∞ = 1 . Then we have (a) l j (˜Γ µ ) = O ( µ ) as µ → , (b) the limits lim µ → ˜Γ µ and lim µ → d L ˜Γ µ exist, and lim µ → ˜Γ µ is continuous in x and lim µ → d L ˜Γ µ is continuous in x and ¯ θ +4 , (c) ˆ¯ l j ( lim δ, ˜ θ → lim µ → d L ˜Γ µ ) = 0 .Proof. Denote Γ (cid:48) µ = l j (¯ u − j )¯¯ u − j − l j (¯¯ u − j )¯ u − j ∈ Ker l j and let v µ be a vector inspan(¯ u − j , ¯¯ u − j ) such that v µ → v as µ → l j ( v µ ) = 1 . Suppose that˜Γ µ = a µ v µ + b µ Γ (cid:48) µ then(12.27) d L (˜Γ µ ) = a µ µ l j ( v µ ) u j + a µ B j ( v µ ) + b µ B j Γ (cid:48) µ + o (1) . So ¯ l j ( d L (˜Γ µ )) = 0 implies that(12.28) a µ = − µ b µ ¯ l j ( B j Γ (cid:48) µ ) + o (1) l j ( v µ )¯ l j ( u j ) + µ ¯ l j B j ( v µ ) . The denominator is not zero since l j ( v µ ) = 1 and ¯ l j ( u j ) (cid:54) = 0 using Lemma 3.9.Therefore a µ = O ( µ ) . Hence ˜Γ µ = b µ Γ (cid:48) µ + O ( µ ) and l j (˜Γ µ ) = O ( µ ) . The continuous dependence on variables in part (b) follows from part (a) of Lemma 3.1 and 3.2. Nowthe remaining statements of the lemma follow from equations (12.27) and (12.28). (cid:3) To compute the numerical values it is more convenient for us to work with polarcoordinates. We need the following quantities. Definition 12.4. • ψ : polar angle, related to u by tan ψ = (cid:113) e − e tan u forellipse. We choose the positive y axis as the axis ψ = 0 . E : energy; e : eccen-tricity; G : angular momentum, g : argument of periapsis. • The subscripts , stand for Q or Q . The superscript ± refers to before orafter collision. Recall that all quantities are evaluated on the sphere | Q − Q | = µ κ . Recall the formula r = G − e cos ψ for conic sections in which the perigee lies on theaxis ψ = π . In our case we have(12.29) r ± = ( G ± ) − e ± sin( ψ ± + g ± ) + o (1) ,r ± = ( G ± ) − e ± sin( ψ ± − g ± ) + o (1) .o (1) terms are small when µ → χ (cid:29) /µ ). Lemma 12.5. Under the assumptions of Corollary 12.2 we have dr +3 ds = dr +4 ds + o (1) , dr − ds = dr − ds + o (1) , dψ +3 ds = dψ +4 ds + o (1) , dψ − ds = dψ − ds + o (1) . Moreover in (12.29) the o (1) terms are also C small when taking the s derivative.Proof. To prove the statement about (12.29), we use the Hamiltonian (2.1). The r , obey the Hamiltonian system (2.1). The estimate (9.2) shows the − µ | Q − Q | gives smallperturbation to the variational equations. The two O (1 /χ ) terms in (2.1) are alsosmall. This shows that the perturbations to Kepler motion is C small.Next we consider the derivatives ∂r ± , ∂s . We consider first the case of “ − ”. From thecondition | (cid:126)r − (cid:126)r | = µ κ , for the Poincar´e section we get( (cid:126)r − (cid:126)r ) · dds ( (cid:126)r − (cid:126)r ) = 0 . This implies ( (cid:126)r − (cid:126)r ) ⊥ dds ( (cid:126)r − (cid:126)r ).We also know the angular momentum for the relative motion is G in = ( ˙ (cid:126)r − ˙ (cid:126)r ) × ( (cid:126)r − (cid:126)r ) = O ( µ ) , which implies ˙ (cid:126)r − ˙ (cid:126)r is almost parallel to (cid:126)r − (cid:126)r . The condition ∂G − in ∂s = O ( µ ) reads (cid:18) dds ( ˙ (cid:126)r − ˙ (cid:126)r ) (cid:19) × ( (cid:126)r − (cid:126)r ) + ( ˙ (cid:126)r − ˙ (cid:126)r ) × (cid:18) dds ( (cid:126)r − (cid:126)r ) (cid:19) = O ( µ ) . ONCOLLISION SINGULARITIES IN THE 2-CENTER-2-BODY PROBLEM 71 Since the first term is O ( µ κ ) due to our choice of the Poincare section we see that( ˙ (cid:126)r − ˙ (cid:126)r ) × (cid:18) dds ( (cid:126)r − (cid:126)r ) (cid:19) = o (1) . Since dds ( (cid:126)r − (cid:126)r ) is almost perpendicular to ( ˙ (cid:126)r − ˙ (cid:126)r ) by the analysis presented abovewe get dds ( (cid:126)r − (cid:126)r ) = o (1). Taking the radial and angular part of this vector identityand using that r = r + o (1) , ψ = ψ + o (1) we get ” − ” part of the lemma.To repeat the above argument for “+” variables, we first need to establish ∂G + in ∂s = O ( µ ) . Indeed, using equations (12.7) and (12.17) we get ∂G + in ∂ψ = ∂G + in ∂ ( L , G in , g, Q + , v + ) − ∂ ( L , G in , g, Q + , v + ) − ∂ψ = O ( µ κ , , µ κ , µ κ × , µ κ × ) · O (1 , µ, , × , × ) = O ( µ ) . It remains to show (cid:16) dds ( ˙ (cid:126)r − ˙ (cid:126)r ) (cid:17) = O (1) in the “ + ” case. Since we know it is truein the “-” case, the “+” case follows, because the directional derivative of the localmap d L Γ is bounded due to our choice of Γ. (cid:3) We are now ready to describe the computation of Lemma 3.10. The reader maynotice that the computations in the proofs of Lemmas 3.10 and 2.2 are quite similar.Note however that Lemma 3.10 describes the subleading term for the derivative of thelocal map. By contrast the leading term can not be understood in terms of the Gervermap since it comes from the possibility of varying the closest distance between Q and Q and this distance is assumed to be zero in Gerver’s model.We will use the following set of equations which follows from (12.22).(12.30) E +3 + E +4 = E − + E − , (12.31) G +3 + G +4 = G − + G − , (12.32) e +3 G +3 cos( ψ +3 + g +3 )+ e +4 G +4 cos( ψ − − g − ) = e − G − cos( ψ − + g − )+ e − G − cos( ψ − − g − ) , (12.33) ( G +3 ) − e +3 sin( ψ +3 + g +3 ) = ( G − ) − e − sin( ψ − + g − ) , (12.34) ψ +3 = ψ − , (12.35) ( G +3 ) − e +3 sin( ψ +3 + g +3 ) = ( G +4 ) − e +4 sin( ψ +4 − g +4 ) , (12.36) ( G − ) − e − sin( ψ − + g − ) = ( G − ) − e − sin( ψ − − g − ) , (12.37) ψ − = ψ − , (12.38) ψ +4 = ψ +3 . In the above equations we have dropped o (1) terms for brevity. We would like toemphasize that the above approximations hold not only in C sense but also in C sense when we take the derivatives along the directions satisfying the conditions ofCorollary 12.2. (12.30) is the approximate conservation of the energy, (12.31) is theapproximate conservation of the angular momentum and (12.32) follows from theapproximate momentum conservation (see the derivation of (B.2) in Appendix B.3).The possibility of differentiating these equations is justified in Corollary 12.2. Theremaining equations reflect the fact that Q ± and Q ± are all close to each other. Thepossibility of differentiating these equations is justified by Lemma 12.5.We set the total energy to be zero. So we get E ± = − E ± . This eliminates E ± .Then we also eliminate ψ ± by setting them to be equal ψ ± . Proof of the Lemma 3.10. Lemma 12.3 and Corollary 12.1 show that the assumptionof Lemma 3.10 implies that the direction Γ along which we take the directional deriv-ative satisfies ∂G in ∂ Γ = O ( µ ). So we can directly take derivatives in equations (12.30)-(12.36). Recall that we need to compute dE +3 ( d L Γ) where Γ ∈ Ker l j ∩ span { w − j , ˜ w } .(3.1) tells us that in in Delaunay coordinates we have(12.39) ˜ w = (0 , , , , , , w = (0 , , , , , a ) where a = − L − ( L − ) + ( G − ) . The formula tan ψ = (cid:113) e − e tan u which relates ψ to (cid:96) through u shows that (12.39)also holds if we use ( L , ψ , G , g , G , g ) as coordinates. Hence Γ has the form(0 , , , , c, ca ). To find the constant c we use (12.36).Note that the expression dE +3 ( d L Γ) does not involve dψ +3 . Therefore we can elim-inate ψ +3 from consideration by setting ψ +3 = ψ − = ψ (see (12.34)). Let L de-note the projection of our map to ( L , G , g , G , g ) variables. Thus we need tofind dE +3 ( d L Γ) . To this end write the remaining equations ((12.31), (12.32), (12.33),and (12.35)) formally as F ( Z + , Z − ) = 0, where in Z + = ( E +3 , G +3 , g +3 , G +4 , g +4 ) and Z − = ( E − , ψ, G − , g − , G − , g − ).We have ∂ F ∂Z + d L Γ + ∂ F ∂Z − Γ = 0 . However, ∂ F ∂Z + is not invertible since F involves only four equations of F while Z + has 5 variables. To resolve this problem we use that by definition of Γ we have¯ l · ∂Z + ∂ψ = 0, where ¯ l = (cid:16) G +4 /L +4 ( L +4 ) +( G +4 ) , , , , − L +4 ) +( G +4 ) , L +4 (cid:17) by (3.1). Thus we get (cid:34) ¯ l ∂ F ∂Z + (cid:35) d L Γ = − (cid:34) ∂ F ∂Z − Γ (cid:35) ONCOLLISION SINGULARITIES IN THE 2-CENTER-2-BODY PROBLEM 73 and so d L Γ = − (cid:34) ¯ l ∂ F ∂Z + (cid:35) − (cid:34) ∂ F ∂Z − Γ (cid:35) . We use computer to complete the computation. We only need the entry ∂E +3 ∂ψ to proveLemma 3.10. It turns out this number is 1 . 855 for the first collision and − . 608 forthe second collision. Neither is zero as needed. (cid:3) Appendix A. Delaunay coordinates A.1. Elliptic motion. The material of this section could be found in [Al]. Considerthe two-body problem with Hamiltonian H ( P, Q ) = | P | m − k | Q | , ( P, Q ) ∈ R . This system is integrable in the Liouville-Arnold sense when H < 0. So we can intro-duce the action-angle variables ( L, (cid:96), G, g ) in which the Hamiltonian can be writtenas H ( L, (cid:96), G, g ) = − mk L , ( L, (cid:96), G, g ) ∈ T ∗ T . The Hamiltonian equations are˙ L = ˙ G = ˙ g = 0 , ˙ (cid:96) = mk L . We introduce the following notation E -energy, M -angular momentum, e -eccentricity, a -semimajor axis, b -semiminor axis. Then we have the following relations which ex-plain the physical and geometrical meaning of the Delaunay coordinates. a = L mk , b = LGmk , E = − k a , M = G, e = (cid:115) − (cid:18) GL (cid:19) . Moreover, g is the argument of periapsis and (cid:96) is called the mean anomaly, and (cid:96) canbe related to the polar angle ψ through the equationstan ψ (cid:114) e − e · tan u , u − e sin u = (cid:96). We also have the Kepler’s law a T = π ) which relates the semimajor axis a and theperiod T of the ellipse.Denoting particle’s position by ( q , q ) and its momentum ( p , p ) we have thefollowing formulas in case g = 0 . (cid:40) q = a (cos u − e ) ,q = a √ − e sin u, p = −√ mka − / sin u − e cos u ,p = √ mka − / √ − e cos u − e cos u , where u and l are related by u − e sin u = (cid:96) . Expressing e and a in terms of Delaunay coordinates we obtain the following(A.1) q = L mk (cid:32) cos u − (cid:114) − G L (cid:33) , q = LGmk sin u.p = − mkL sin u − (cid:114) − G L cos u , p = mkL G cos u − (cid:114) − G L cos u . Here g does not enter because the argument of perihelion is chosen to be zero. Ingeneral case, we need to rotate the ( q , q ) and ( p , p ) using the matrix (cid:20) cos g − sin g sin g cos g (cid:21) . Notice that the equation (A.1) describes an ellipse with one focus at the originand the other focus on the negative x -axis. We want to be consistent with [G2], i.e.we want g = π/ y -axis (see Appendix B.2). Therefore we rotatethe picture clockwise. So we use the Delaunay coordinates which are related to theCartesian ones through the equation(A.2) q = 1 mk (cid:32) L (cid:32) cos u − (cid:114) − G L (cid:33) cos g + LG sin u sin g (cid:33) ,q = 1 mk (cid:32) − L (cid:32) cos u − (cid:114) − G L (cid:33) sin g + LG sin u cos g (cid:33) . A.2. Hyperbolic motion. The above formulas can also be used to describe hyper-bolic motion, where we need to replace “sin → sinh , cos → cosh”(c.f.[Al, F]). Namely,we have for g = 0(A.3) q = L mk (cid:32) cosh u − (cid:114) G L (cid:33) , q = LGmk sinh u,p = − mkL sinh u − (cid:113) G L cosh u , p = − mkL G cosh u − (cid:113) G L cosh u . where u and l are related by(A.4) u − e sinh u = (cid:96), where e = (cid:115) (cid:18) GL (cid:19) . This hyperbola is symmetric w.r.t. the x -axis, opens to the right and the particlemoves counterclockwise on it when u increases ( (cid:96) decreases) in the case when theangular momentum G = p × q < 0. The angle g is defined to be the angle measuredfrom the positive x -axis to the symmetric axis. There are two such angles that differby π depending on the orientation of the symmetric axis. This π difference disappearsin the symplectic form and the Hamiltonian equation, so it does not matter whichangle to choose. ONCOLLISION SINGULARITIES IN THE 2-CENTER-2-BODY PROBLEM 75 When the particle moves to the right of x = − χ line we have a hyperbola openingto the left and the particle moves counter-clockwise. To get the picture studied in[G1], we rotate (A.3) by π + g. In this case, we choose g to be the angle measuredfrom the positive x -axis to the symmetric axis pointing to the perigee. Thus we have(A.5) q = 1 mk (cid:0) − cos gL (cosh u − e ) + sin gLG sinh u (cid:1) ,q = 1 mk (cid:0) − sin gL (cosh u − e ) − cos gLG sinh u (cid:1) ,P = mk − e cosh u (cid:18) L sinh u cos g − GL sin g cosh u, L sinh u sin g + GL cos g cosh u (cid:19) . If the incoming asymptote is horizontal, (see the arrows in Figure 1 for “incoming”and “outgoing”), then the particle comes from the left, and as u tends to −∞ , the y -coordinate is bounded and x -coordinate is negative. In this case we have tan g = GL , g ∈ ( − π/ , . We use u < u tends to + ∞ , the y -coordinate is bounded and x -coordinate is negative. In thiscase we have tan g = − GL , g ∈ (0 , π/ u > g = − sign( u ) GL with G < , L > Q is moving to the left of the section { x = − χ/ } , we treat themotion as hyperbolic motion focused at Q . We move the origin to Q . The hyperbolaopens to the right. The particle Q moves on the hyperbola counterclockwise withnegative angular momentum G , we then rotate by angle g and g is the angle measuredfrom the positive x -axis to the symmetric axis pointing to the opening of the hyperbola.The orbit has the following parametrization(A.6) q = 1 mk (cid:0) cos gL (cosh u − e ) − sin gLG sinh u (cid:1) ,q = 1 mk (sin gL (cosh u − e ) + cos gLG sinh u ) ,P = mk − e cosh u (cid:18) − L sinh u cos g + GL sin g cosh u, − L sinh u sin g − GL cos g cosh u (cid:19) . In the left case the orbits we consider have G is close to zero, i.e. the system is closeto the double collision. In this case, the hyperbolic Delaunay coordinates are singularwhen (cid:96) is close to zero. Indeed when we set e = 1 in (A.4), we find (cid:96) = u + h.o.t. Hence u as a function (cid:96) in a neighborhood of 0 is only C but not C . One can verifythat for G = 0 and (cid:96) (cid:54) = 0 the hyperbolic Delaunay coordinates still give a symplectictransformation, so we only have singular behavior when G and (cid:96) are both close tozero. To control this singular behavior, we need the following estimates. Lemma A.1. In the hyperbolic Delaunay coordinates, as G → , u → and L beingclose to , we have the following estimates of the first order derivatives (cid:12)(cid:12)(cid:12)(cid:12) ∂u∂G (cid:12)(cid:12)(cid:12)(cid:12) ≤ , (cid:12)(cid:12)(cid:12)(cid:12) ∂u∂L (cid:12)(cid:12)(cid:12)(cid:12) ≤ | G | and the second order derivatives (cid:12)(cid:12)(cid:12)(cid:12) ∂Q∂u ∂ u∂G (cid:12)(cid:12)(cid:12)(cid:12) ≤ , (cid:12)(cid:12)(cid:12)(cid:12) ∂Q∂u ∂ u∂L (cid:12)(cid:12)(cid:12)(cid:12) ≤ G , (cid:12)(cid:12)(cid:12)(cid:12) ∂Q∂u ∂ u∂G∂L (cid:12)(cid:12)(cid:12)(cid:12) ≤ | G | . Proof. For the first order derivatives, it follows from (A.4) that ∂u∂G − e cosh u ∂u∂G = sinh u ∂e∂G . We have ∂e∂G = GeL and ∂e∂L = − G eL . Hence we get for small G and u (cid:12)(cid:12)(cid:12)(cid:12) ∂u∂G (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) sinh u ∂e∂G − e cosh u (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∼ (cid:12)(cid:12)(cid:12)(cid:12) uGG + u (cid:12)(cid:12)(cid:12)(cid:12) ≤ . To get ∂u∂L , we replace G by L in the above expression we get (cid:12)(cid:12)(cid:12)(cid:12) ∂u∂L (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) sinh u ∂e∂L − e cosh u (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∼ (cid:12)(cid:12)(cid:12)(cid:12) uG G + u (cid:12)(cid:12)(cid:12)(cid:12) ≤ G. Next, we work on second order derivatives. We have ∂ u∂G − ∂e∂G ∂u∂G cosh u − e sinh u (cid:18) ∂u∂G (cid:19) − ∂ e∂G sinh u − e cosh u ∂ u∂G = 0which gives ∂ u∂G = 11 − e cosh u (cid:32) ∂e∂G ∂u∂G cosh u + e sinh u (cid:18) ∂u∂G (cid:19) + ∂ e∂G sinh u (cid:33) ∼ G + uG + u for small u and G by substituting sinh u ∼ u, cosh u ∼ , ∂e∂G ∼ G and ∂u∂G ∼ 1. Onthe other hand, we have ∂Q∂u = ∂∂u ( L cosh u, LG sinh u ) = ( L sinh u, LG cos u ) ∼ ( u, G ) , where we choose g = 0 in Q since a rotation by g does not change the Euclidean norm.When we consider ∂Q∂u ∂ u∂G , we get (cid:12)(cid:12)(cid:12)(cid:12) ∂Q∂u ∂ u∂G (cid:12)(cid:12)(cid:12)(cid:12) ≤ ( | u | + | G | ) u + G ≤ ∂Q∂u ∂ u∂L , we need to replace in the expression of ∂ u∂G everywhere G by L ,which gives us the estimate ∂ u∂L ∼ G + uG + u G . To get ∂Q∂u ∂ u∂L∂G , we have ∂ u∂L∂G = 11 − e cosh u (cid:18)(cid:18) ∂e∂L ∂u∂G + ∂e∂G ∂u∂L (cid:19) cosh u − e sinh u ∂u∂G ∂u∂L − ∂ e∂L∂G sinh u (cid:19) which is estimated as G G + uG + u . This completes the proof. (cid:3) ONCOLLISION SINGULARITIES IN THE 2-CENTER-2-BODY PROBLEM 77 A.3. Large (cid:96) asymptotics: auxiliary results. In the remaining part of Appen-dix A we obtain estimates onthe first and second order derivatives of Q w.r.t. thehyperbolic Delaunay variables ( L, (cid:96), G, g ) which are needed in our proof. The nextlemma allows us to simplify the computations. Since the hyperbolic motion approachesa linear motion, this lemma shows that, we can replace u by ln( ∓ (cid:96)/e ) when takingfirst and second order derivatives. Lemma A.2. Let u be the function of (cid:96), G and L given by (A.4) . Then we canapproximate u by ln( ∓ (cid:96)/e ) in the following sense. u ∓ ln ∓ (cid:96)e = O (ln | (cid:96) | /(cid:96) ) , ∂u∂(cid:96) = ± /(cid:96) + O (1 /(cid:96) ) , (cid:18) ∂∂L , ∂∂G (cid:19) ( u ± ln e ) = O (1 / | (cid:96) | ) , (cid:18) ∂∂L , ∂∂G (cid:19) ( u ± ln e ) = O (1 / | (cid:96) | ) , Here the first sign is taken if u > and the second sign is taken then u < . Theestimates above are uniform as long as | G | ≤ K, /K ≤ L ≤ K, (cid:96) > (cid:96) and the impliedconstants in O ( · ) depend only on K and (cid:96) . Proof. We see from formula (A.4) that sinh u (cid:39) cosh u = − (cid:96)e + O (ln | (cid:96) | ) when u > u (cid:39) − cosh u (cid:39) − (cid:96)e + O (ln | (cid:96) | ) when u < | u | large enough. This proves C estimate.Now we consider the first order derivatives. We assume that u > (cid:96) we get ∂u∂(cid:96) − e cosh u ∂u∂(cid:96) = 1 , ∂u∂(cid:96) = 1 /(cid:96) + O (1 /(cid:96) ) . Next, we differentiate (A.4) with respect to L to obtain ∂u∂L − ∂e∂L sinh u − e cosh u ∂u∂L = 0 . Therefore, ∂u∂L = sinh u − e cosh u ∂e∂L = − e ∂e∂L + O ( e −| u | ) = − ∂∂L ln( e ) + O (1 / | (cid:96) | ) . The same argument holds for ∂∂G . This proves C part of the Lemma.Now we consider second order derivatives. We take ∂ ∂L as example. Combining ∂ u∂L − ∂ e∂L sinh u − u ∂e∂L ∂u∂L − e cosh u ∂ u∂L − e sinh u (cid:18) ∂u∂L (cid:19) = 0 . with C estimate proven above we get ∂ u∂L = − e ∂ e∂L − ∂ee∂L ∂u∂L + (cid:18) ∂u∂L (cid:19) + O (cid:18) (cid:96) (cid:19) = − e ∂ e∂L + (cid:18) e ∂e∂L (cid:19) + O (cid:18) (cid:96) (cid:19) = ∂ ∂L ln e + O (cid:18) (cid:96) (cid:19) . This concludes the C part of the lemma. (cid:3) In the estimate of the derivatives presented in the next two subsections we shalloften use the following facts. Let f = ln e. Then(A.7) f G = GL + G , f L = − G L ( L + G ) , (A.8) ( f ) GG = L − G ( L + G ) , f LG = − GL ( L + G ) . A.4. First order derivatives. In the following computations, we assume for sim-plicity that m = k = 1 . To get the general case we only need to divide positions by mk. Lemma A.3. Under the same conditions as in Lemma A.2 we have the followingresult for the first order derivatives (a) (cid:12)(cid:12)(cid:12)(cid:12) ∂Q∂(cid:96) (cid:12)(cid:12)(cid:12)(cid:12) = O (1) , (cid:12)(cid:12)(cid:12)(cid:12) ∂Q∂ ( L, G, g ) (cid:12)(cid:12)(cid:12)(cid:12) = O ( (cid:96) ) , ∂Q∂g · Q = 0 ,∂Q∂G · Q = O C ( L,G,g ) ( (cid:96) ) . (b) If in addition we have (cid:12)(cid:12) g + sign( u ) arctan GL (cid:12)(cid:12) ≤ C/ | (cid:96) | then we have the follow-ing bounds for (A.5) ∂Q∂G = − L sinh u √ L + G (0 , 1) + O (1) , ∂Q∂L = sinh u (cid:18) − (cid:112) L + G , GL √ L + G (cid:19) + O (1) . (c) If in addition to the conditions of Lemma A.2 we have G, g = O (1 /χ ) and (cid:96) = O ( χ ) , then we have the following bounds for (A.6) ∂Q∂G = sinh u (0 , L ) + O (1) , ∂Q∂L = sinh u (2 L, 0) + O (1) . Remark A.4. The assumptions of the lemma and the next lemma hold in our situa-tion due to Lemma 4.7.Proof. We write the position variables in (A.5) as q = ( L cosh u, LG sinh u ) − L e (1 , 0) = cosh uL ( L, G sign( u )) + O (1)and Q is obtained by rotating q by angle π + g in case ( b ) and by angle g in case (c). ONCOLLISION SINGULARITIES IN THE 2-CENTER-2-BODY PROBLEM 79 Using Lemma A.2, we obtain ∂q∂G = − sign( u ) · f G ( L sinh u, LG cosh u ) + L sinh u (0 , 1) + O (1)= GL + G ( − L , − sign( u ) LG ) cosh u + sign( u ) L cosh u (0 , 1) + O (1)= L cosh uL + G ( − G, sign( u ) L ) + O (1) ,∂q∂L = − sign( u ) · f L ( L sinh u, LG cosh u ) + (2 L cosh u, G sinh u ) + O (1)= − G L ( L + G ) ( − L , − sign( u ) LG ) cosh u + (2 L, G sign( u )) cosh u + O (1)= ( L, 0) cosh u + L + 2 G L + G ( L, sign( u ) G ) cosh u + O (1) . Now the estimates on ∂Q∂G and ∂Q∂L follow since, by (A.4), cosh u and sinh u are O ( (cid:96) ) . The estimates on ∂Q∂g follow since Q is obtained from q by a rotatation. Also ∂q∂l = − sign( u ) ∂u∂l (cid:0) L sinhu, LG cosh u (cid:1) so the estimate of ∂Q∂g follows from Lemma A.2.To prove the last estimate of part (a) we observe that Q · ∂Q∂G = q · ∂q∂G = cosh uL ( L, G sign u ) · L L + G ( − G, sign( u ) L ) + O ( (cid:96) ) = O ( (cid:96) ) . Next, we work on (b). First consider g = − sign( u ) arctan GL , G < 0. Then ∂Q∂G is arotation of ∂q∂G by π + g . We see from above that ∂q∂G is a vector with polar anglesign( u ) arctan L − G = sign( u )( π − arctan − GL ). So after rotating by angle g + π , finallywe get that ∂Q∂G has polar angle π + sign( u ) π = − sign( u ) π , we get ∂Q∂G = − sinh u L √ L + G (0 , 1) + O (1) . When g is in a 1 / | (cid:96) | neighborhood of − sign( u ) arctan GL , we get the same estimate byabsorbing the error into O (1). By the same argument, we get that ∂Q∂L = (cid:18) − (cid:112) L + G cosh u, sinh u LG √ L + G (cid:19) + O (1) . Part (c) follows directly from the formulas for ∂q∂G , ∂q∂L , since both g and arctan GL are O (1 /χ ) . (cid:3) A.5. Second order derivatives. The following bounds of the second order deriva-tives are used in estimations of the variational equation. Lemma A.5. We have the following information for the second order derivatives of Q w.r.t. the Delaunay variables. (a) Under the conditions of Lemma A.3(a) we have ∂ Q∂g = − Q, ∂ Q∂g∂G ⊥ ∂Q∂G , (cid:18) ∂∂G , ∂∂g (cid:19) (cid:18) ∂ | Q | ∂g (cid:19) = (0 , ,∂ Q∂G = O ( (cid:96) ) , ∂ Q∂L = O ( (cid:96) ) , ∂ Q∂G∂L = O ( (cid:96) ) . (b) Under the conditions of Lemma A.3(b) we have we have ∂ Q∂G = L ( L + G ) / ( L cosh u, G sinh u ) + O (1) ,∂ Q∂g∂G = (cid:18) L sinh u √ L + G , (cid:19) + O (1) ,∂ Q∂g∂L = (cid:18) − GL sinh u √ L + G , − (cid:112) L + G cosh u (cid:19) + O (1) ,∂ Q∂G∂L = − L ( L + G ) / (cid:0) LG cosh u, ( L + 3 G ) sinh u (cid:1) + O (1) . (c) Under the conditions of Lemma A.3(c) we have ∂ Q∂G = − cosh u (1 , 0) + O (1) , ∂ Q∂g∂G = − L sinh u (1 , 0) + O (1) ,∂ Q∂g∂L = L sinh u (0 , 2) + O (1) , ∂ Q∂G∂L = cosh u (0 , 1) + O (1) . Proof. The estimates of ∂ Q∂G , ∂ Q∂L , and ∂ Q∂G∂L follows from similar estimates on thederivatives of q. The estimates on the second derivatives of q follow by straightforwarddifferentiation of (A.3) using Lemma A.3. The other estimates of part (a) follow since Q depends on g via a rotation.Next we prove parts (b) and (c). Again we first work on q then rotate by g + π for(b) and by g for (c), ∂ q∂G = (cid:18)(cid:18) L L + G (cid:19) G cosh u + L sinh uu G L + G (cid:19) ( − G, sign( u ) L ) + L cosh uL + G ( − , 0) + O (1)= cosh u (cid:18) − L G ( L + G ) (cid:19) ( − G, sign( u ) L ) + L cosh uL + G ( − , 0) + O (1) ∂ q∂L∂G = (cid:18)(cid:18) L L + G (cid:19) L cosh u + L sinh uu L L + G (cid:19) ( − G, sign( u ) L ) + L cosh uL + G (0 , sign( u )) + O (1)= cosh u (cid:18) LG ( L + G ) (cid:19) ( − G, sign( u ) L ) + L cosh uL + G (0 , sign( u )) + O (1) ONCOLLISION SINGULARITIES IN THE 2-CENTER-2-BODY PROBLEM 81 After rotating by angle π + g with g = − sign( u ) · arctan GL , we get ∂ Q∂G = sinh u L G ( L + G ) / (0 , 1) + L cosh u ( L + G ) / ( L, − sign( u ) G ) + O (1)= L ( L + G ) / ( L cosh u, G sinh u ) + O (1) ∂ Q∂L∂G = sinh u − LG ( L + G ) / (0 , 1) + L sinh u ( L + G ) / ( − sign( u ) G, − L ) + O (1)= − L ( L + G ) / ( LG cosh u, ( L + 3 G ) sinh u ) + O (1) . This gives the estimates on ∂ Q∂G and ∂ Q∂L∂G in part (b). The estimates of part (c)are similar. The estimates of ∂ Q∂L∂g and ∂ Q∂G∂g follow easily from parts (b) and (c) ofLemma 3.2. (cid:3) Appendix B. Gerver’s mechanism B.1. Gerver’s result in [G2] . We summarize the result of [G2] in the followingtable. Recall that the Gerver scenario deals with the limiting case χ → ∞ , µ → Q disappears at infinity and there is no interaction between Q and Q . Hence both particles perform Kepler motions. The shape of each Kepler orbitis characterized by energy, angular momentum and the argument of periapsis. InGerver’s scenario, the incoming and outgoing asymptotes of the hyperbola are alwayshorizontal and the semimajor of the ellipse is always vertical. So we only need todescribe on the energy and angular momentum.1st collision @( − ε ε , ε + ε ) 2nd collision @( ε , Q Q Q Q energy − 12 12 − → − ε ε → ε ε angular momentum ε → − ε p → − p − ε √ ε eccentricity ε → ε ε → ε semimajor 1 − → (cid:16) ε ε (cid:17) → − ε ε semiminor ε → ε p → p ε → ε ε √ ε → √ ε Here p , = − Y ± (cid:112) Y + 4( X + R )2 , R = (cid:112) X + Y . and ( X, Y ) stands for the point where collision occurs (the parenthesis after @ in thetable). We will call the two points the Gerver’s collision points.In the above table ε is a free parameter and ε = (cid:112) − ε . At the collision points, the velocities of the particles are the following. For the first collision, v − = (cid:18) − ε ε ε + 1 , − ε ε ε + 1 (cid:19) , v − = (cid:18) − YRp , Rp (cid:19) .v +3 = (cid:18) ε ε ε + 1 , ε ε ε + 1 (cid:19) , v +4 = (cid:18) − YRp , − Rp (cid:19) . For the second collision, v − = (cid:18) − ε ε , − ε (cid:19) , v − = (cid:32) , √ ε (cid:33) , v +3 = (cid:18) , − ε (cid:19) , v +4 = (cid:32) − ε ε , √ ε (cid:33) . B.2. Numerical information for a particularly chosen ε = 1 / . For the firstcollision e : → √ .We want to figure out the Delaunay coordinates ( L, u, G, g ) for both Q and Q . (Herewe replace (cid:96) by u for convenience.) The first collision point is( X, Y ) = ( − ε ε , ε + ε ) = (cid:32) − √ , √ (cid:33) . Before collision( L, u, G, g ) − = (cid:32) , − π , √ , π/ (cid:33) , ( L, u, G, g ) − = (1 , . , − p , − arctan p ) ,v − = (cid:18) − √ , − √ (cid:19) (cid:39) − (0 . , . ,v − = (cid:32) − √ √ p , √ p (cid:33) (cid:39) ( − . , . , where p = − Y + (cid:112) Y + 4( X + R )2 = − ( ε + ε ) + √ ε ε . . After collision( L, u, G, g ) +3 = (cid:18) , π , − , π/ (cid:19) , ( L, u, G, g ) +4 = (1 , . , p , − arctan p ) ,v +3 = (cid:32) √ , √ √ (cid:33) (cid:39) (0 . , . ,v +4 = (cid:32) − √ √ p , − √ p (cid:33) (cid:39) ( − . , . p = − Y − (cid:112) Y + 4( X + R )2 = − ( ε + ε ) − √ ε ε − . . ONCOLLISION SINGULARITIES IN THE 2-CENTER-2-BODY PROBLEM 83 For the second collision e : √ → .The collision point is ( X, Y ) = ( ε , 0) = (cid:18) , (cid:19) .Before collision( L, u, G, g ) − = (cid:18) , − π , − , π/ (cid:19) , ( L, u, G, g ) − = (cid:32) , . , −√ / , − arctan √ (cid:33) ,v − = (cid:16) −√ , − (cid:17) , v − = (cid:16) , √ (cid:17) . After collision( L, u, G, g ) +3 = (cid:18) √ , π , − , − π (cid:19) , ( L, u, G, g ) +4 = (cid:32) √ , − . , − √ , arctan √ (cid:33) ,v +3 = (1 , − , v +4 = (cid:16) −√ , √ (cid:17) . B.3. Control the shape of the ellipse. As it was mentioned before Lemma 2.2was stated by Gerver in [G2]. There is a detailed proof of part ( a ) of our Lemma2.2 in [G2]. However since no details of the proof of part ( b ) were given in [G2] wego other main steps here for the reader’s convenience even though computations arequite straightforward. Proof of Lemma 2.2. Recall that Gerver’s map depends on a free parameter e (orequivalently G ). In the computations below however it is more convenient to use thepolar angle ψ of the intersection point as the free parameter. It is easy to see that as G changes from large negative to large positive value the point of intersection coversthe whole orbit of Q so it can be used as the free parameter. Our goal is to show thatby changing the angles ψ and ψ of the first and second collision we can prescribethe values of ¯¯ e and ¯¯ g arbitrarily. Due to the Implicit Function Theorem it sufficesto show that det (cid:34) ∂ ¯¯ e ∂ψ ∂ ¯¯ g ∂ψ ∂ ¯¯ e ∂ψ ∂ ¯¯ g ∂ψ (cid:35) (cid:54) = 0 . To this end we use the following set of equations(B.1) G +3 + G +4 = G − + G − , (B.2) e +3 G +3 cos( ψ + g +3 ) + e +4 G +4 cos( ψ − g − ) = e − G − cos( ψ + g − ) + e − G − cos( ψ − g − ) , (B.3) ( G +3 ) − e +3 sin( ψ + g +3 ) = ( G − ) − e − sin( ψ + g − ) , (B.4) ( G +3 ) − e +3 sin( ψ + g +3 ) = ( G +4 ) − e +4 sin( ψ − g +4 ) , (B.5) g +4 = arctan G +4 L +4 . Here e , e and L are functions of the other variables according to the formulas ofAppendix A.(B.1)–(B.5) are obtained as follows. (B.1) is the angular momentum conservation,(B.3) means that the position of Q does not change during the collision, (B.4) meansthat Q and Q are at the same point immediately after the collision and (B.5) saysthat after the collision the outgoing asymptote of Q is horizontal.It remains to derive (B.2). Represent the position vector as (cid:126)r = r ˆ e r . Then thevelocity is ˙ (cid:126)r = ˙ r ˆ e r + r ˙ ψ ˆ e ψ . The momentum conservation gives( ˙ (cid:126)r ) − + ( ˙ (cid:126)r ) − = ( ˙ (cid:126)r ) + + ( ˙ (cid:126)r ) + . Taking the angular component of the velocity we get(B.6) r − ˙ ψ − + r − ˙ ψ − = r +3 ˙ ψ +3 + r +4 ˙ ψ +4 . In our notation the polar representation of the ellipse takes form r = G − e sin( ψ + g ) . Differentiating this equation we obtain the following relation for the radial componentof the Kepler motion˙ r = G (1 − e sin( ψ + g )) e cos( ψ + g ) ˙ ψ = r G e cos( ψ + g ) Gr = eG cos( ψ + g ) . Plugging this into (B.6) we obtain (B.2).We can write (B.1)–(B.5) in the form F ( Z − , ˜ Z, Z + ) = 0where Z − = ( E − , G − , g − , ψ ), Z + = ( E +3 , G +3 , g +3 , G +4 , g +4 ) , and ˜ Z = ( G − , g − ) areconsidered as functions Z − . By the Implicit Function Theorem we have ∂Z + ∂Z − = − (cid:18) ∂ F ∂Z + (cid:19) − (cid:32) ∂ F ∂Z − + ∂ F ∂ ˜ Z ∂ ˜ Z∂Z − (cid:33) . Thus to complete the computation we need to know ∂ ˜ Z∂Z − . In order to compute thisexpression we use the equations(B.7) g − = − arctan G − L − which means that the incoming asymptote of Q is horizontal and(B.8) ( G − ) − e − sin( ψ + g − ) = ( G − ) − e − sin( ψ − g − ) , which means that Q and Q are at the same place immediately before the collision.Writing these equations as I ( Z − , ˜ Z ) = 0 we get by the Implicit Function Theorem ∂ ˜ Z∂Z − = − (cid:18) ∂ I ∂ ˜ Z (cid:19) − ∂ I ∂Z − ONCOLLISION SINGULARITIES IN THE 2-CENTER-2-BODY PROBLEM 85 so that the required derivative equals to(B.9) ∂Z + ∂Z − = − (cid:18) ∂ F ∂Z + (cid:19) − (cid:32) ∂ F ∂Z − − ∂ F ∂ ˜ Z (cid:18) ∂ I ∂ ˜ Z (cid:19) − ∂ I ∂Z − (cid:33) . Combining (B.9) with the formula de = − G E dG + G dE (cid:112) − G E which follows from the relation e = (cid:112) − G E we obtain the two entries ∂ ¯¯ e ∂ψ = − . ∂ ¯¯ g ∂ψ = 0 . . The meanings of these two entries are the changes of the eccentricity and argumentof periapsis after the second collision if we vary the phase of the second collision.We need more work to figure out the two entries ∂ ¯¯ e ∂ψ and ∂ ¯¯ g ∂ψ , which are the changesof the eccentricity and argument of periapsis after the second collision if we vary thephase of the first collision. We describe the computation of the first entry, the secondone is similar. We use the relation ∂ ¯¯ e ∂ψ = ∂ ¯¯ e ∂ ¯ E +3 ∂ ¯ E +3 ∂ψ + ∂ ¯¯ e ∂ ¯ G +3 ∂ ¯ G +3 ∂ψ + ∂ ¯¯ e ∂ ¯ g +3 ∂ ¯ g +3 ∂ψ . Now (cid:16) ∂ ¯ E +3 ∂ψ , ∂ ¯ G +3 ∂ψ , ∂ ¯ g +3 ∂ψ (cid:17) is computed using (B.9) and the data for the first collision.Noticing that the quantities E , G , g after the first collision are the same as thosebefore the second collision, we replace (cid:16) ∂ ¯¯ e ∂ ¯ E +3 , ∂ ¯¯ e ∂ ¯ G +3 , ∂ ¯¯ e ∂ ¯ g +3 (cid:17) by (cid:18) ∂ ¯¯ e ∂ ¯¯ E − , ∂ ¯¯ e ∂ ¯¯ G − , ∂ ¯¯ e ∂ ¯¯ g − (cid:19) andcompute it using (B.9) and the data for the second collision. It turns out that theresulting matrix is ∂ ¯¯ e ∂ψ ∂ ¯¯ g ∂ψ ∂ ¯¯ e ∂ψ ∂ ¯¯ g ∂ψ = (cid:20) . . − . (cid:21) , which is obviously nondegenerate. (cid:3) Acknowledgement The authors would like to thank Prof. John Mather many illuminating discussions,and Vadim Kaloshin for introducing us to the problem. We would also like to thankJ. Fejoz for pointing to us that the hyperbolic Delaunay coordinates are singular nearcollision, which is handled in Lemma A.1. This research was supported by the NSFgrant DMS 1101635. References [A] Arnold, Vladimir Igorevich. Mathematical methods of classical mechanics. Vol. 60. SpringerScience & Business Media, 1989..[Al] Albouy, Alain. Lectures on the two-body problem. Classical and celestial mechanics (Recife,1993/1999) (2002): 63-116.[BM] Bolotin, Sergey, and Robert Sinclair MacKay. Nonplanar second species periodic and chaotictrajectories for the circular restricted three-body problem. Celestial Mechanics and DynamicalAstronomy 94.4 (2006): 433-449.[BN] Bolotin, Sergey, and Piero Negrini. Variational approach to second species periodic solutionsof Poincar´e of the 3 body problem. arXiv preprint arXiv:1104.2288 (2011).[F] Floria, L. A simple derivation of the hyperbolic Delaunay variables. The Astronomical Journal110 (1995): 940.[FNS] Font, Joaquim, Ana Nunes, and Carles Sim´o. Consecutive quasi-collisions in the planar cir-cular RTBP. Nonlinearity 15.1 (2002): 115.[G1] Gerver, Joseph L. The existence of pseudocollisions in the plane. Journal of Differential Equa-tions 89.1 (1991): 1-68.[G2] Gerver, Joseph L. Noncollision Singularities: Do Four Bodies Suffice!. Experimental Mathe-matics 12.2 (2003): 187-198.[G3] J. Gerver, Noncollision singularities in the n-body problem ∼ knill/seminars/intr/index.html.[LL] L. Landau, Lifschitz, Mechanics . Third Edition: Volume 1 (Course of Theoretical Physics).[LS] Li, D., and Ya G. Sinai. Blowups of complex-valued solutions for some hydrodynamic models. Regular and Chaotic Dynamics 15.4-5 (2010): 521-531.[MM] Mather, J. N., and Richard McGehee. Solutions of the collinear four body problem whichbecome unbounded in finite time. Dynamical systems, theory and applications. Springer BerlinHeidelberg, 1975. 573-597.[Pa] P. Painlev´e, Le¸cons sur la th´eorie analytique des ´equations diff´erentielles , Hermann, Paris,1897.[Po] H. Poincar´e, New methods of celestial mechanics. Translated from the French. History ofModern Physics and Astronomy, 13. American Institute of Physics, New York, 1993.[Sa1] Saari, Donald Gene. Improbability of collisions in Newtonian gravitational systems. Transac-tions of the American Mathematical Society (1971): 267-271.[Sa2] Saari, Donald G. A global existence theorem for the four-body problem of Newtonian mechanics. Journal of Differential Equations 26.1 (1977): 80-111.[Sa3] Saari, Donald G. Collisions, rings, and other Newtonian N-body problems. AMC 10 (2005):12.[Sim] B. Simon, Fifteen problems in mathematical physics. Perspectives in mathematics, 423–454,Birkhauser, Basel, 1984.[Su] K. F. Sundman, Nouvelles recherches sur le probl`eme des trois corps , Acta. Soc. Sci. Fennicae (1909), 3–27.[X] Xia, Zhihong. The existence of noncollision singularities in Newtonian systems. Annals ofmathematics (1992): 411-468.[Xu] Xue, Jinxin. Noncollision Singularities in a Planar Four-body Problem. arXiv preprintarXiv:1409.0048 (2014). University of Chicago, Chicago, IL, 60637 E-mail address : [email protected] University of Maryland, College Park, MD, 20740 E-mail address ::