collective branch regularization of simultaneous binary collisions in the 3d n-body problem

Collective branch regularization of simultaneous binary collisions in the 3D N -bodyproblemMohamed Sami ElBialy Citation: Journal of Mathematical Physics 50, 052702 (2009); doi: 10.1063/1.3119002 View online: http://dx.doi.org/10.1063/1.3119002 View Table of Contents: http://scitation.aip.org/content/aip/journal/jmp/50/5?ver=pdfcov Published by the AIP Publishing Articles you may be interested in On the periodic orbits and the integrability of the regularized Hill lunar problem J. Math. Phys. 52, 082701 (2011); 10.1063/1.3618280 Some insights from total collapse in the N -body problem Am. J. Phys. 76, 1034 (2008); 10.1119/1.2970051 On the Elliptic Restricted ThreeBody Problem AIP Conf. Proc. 1043, 203 (2008); 10.1063/1.2993640 Numerical experiments of the planar equal-mass three-body problem: The effects of rotation Chaos 17, 033105 (2007); 10.1063/1.2753126 Implementing an efficient collisionless N -body code on the Cray T3D Comput. Phys. 11, 378 (1997); 10.1063/1.168608

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Collective branch regularization of simultaneous binarycollisions in the 3D N-body problem

Mohamed Sami ElBialya�

Department of Mathematics, University of Toledo, Toledo, Ohio 43606, USA

�Received 23 September 2008; accepted 13 March 2009; published online 5 May 2009�

In this work we study simultaneous binary collision �SBC� singularities of M�N /2 binaries in the three dimensional classical gravitational N-body problem. Weshow the following: �1� In the generalized Kustaanheimo–Stiefel variables, thetotality of SBC orbits, the totality of simultaneous binary collisions �SBE� orbits,and the collision singularity itself together form a real analytic submanifold whichwe call the collision-ejection manifold. �2� We use the collision-ejection manifoldto show geometrically, without writing down any power series, that SBC solutionscan be collectively analytically continued. That is, all SBC orbits, not just a singleorbit, can be written as a convergent power series in s= t1/3 with coefficients thatdepend real analytically on initial conditions that lie in a real analytic submanifold.�3� There are two important ingredients in our work. �i� We use the intrinsic ener-gies and properly rescaled intrinsic angular momenta of the binaries as variables inorder to reduce the order of the singularity and to parametrize �distinguish betweendifferent� collision orbits that constitute the stable manifolds of the rest points thatappear on the collision manifold in the McGehee coordinates. �ii� We use what wecall the Kustaanheimo–Stiefel-projective transformation near a SBC singularity toresolve the singularity and isolate collision and ejection orbits from nearby near-collision and near-ejection orbits. We will see that quaternionic multiplication andquaternionic projective spaces are not suitable. © 2009 American Institute ofPhysics.�DOI: 10.1063/1.3119002�

I. INTRODUCTION

In 1920 Levi-Civita23 introduced what later came to be known as the Levi-Civita map �LCM�to regularize the perturbed planar Kepler problem. The LCM is the double cover x=z2 with R2

identified with C. Levi-Civita tried to generalize his method to the three dimensional problemwithout any success. “This may be the reason his ingenious method is not described in most of thetextbooks of celestial mechanics” �Ref. 43, p. 23�.

In 1964–1965 Kustaanheimo and Stiefel21,22,43 used their expertise in the theory of spinorsand topology to introduce what is now known as the Kustaanheimo-Stiefel map �KSM� to regu-larize the perturbed three-dimensional �3D� Kepler problem. The KSM is given by � :U*→X*,��u�ªL�u�u, where U*, X*, and L�u� are given in Sec. I E The restriction of the KSM to the unitsphere S3 is nothing but the Hopf �it so happened that Stiefel was a student of Hopf� fibrationH3,2 :S3→S2. For a thorough presentation of the Hopf fibration see Hopf14,15 and Steenrod�Ref.41. Sec. 20� Abraham and Marsden �Ref. 1 p. 722� presented the Hopf fibration in connection withcentral configurations in celestial mechanics.

At this point it was apparent why Levi-Civita had not been able to generalize his method tothe 3D problem: There are only three Hopf fibrations, namely, Hi,j :Si→Sj, �i , j�= �1,1�, �3, 2�, and

a�Electronic mail: [email protected].

JOURNAL OF MATHEMATICAL PHYSICS 50, 052702 �2009�

50, 052702-10022-2488/2009/50�5�/052702/50/$25.00 © 2009 American Institute of Physics


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http://dx.doi.org/10.1063/1.3119002

http://dx.doi.org/10.1063/1.3119002

http://dx.doi.org/10.1063/1.3119002

�7, 4� �Ref. 41 Sec. 20�. The LCM restricted to the unit circle S1 is the Hopf fibration H1,1 whichis nothing more than the standard double cover of the unit circle.

The three integers 1, 3, and 7 bring to mind two equivalent fundamental theorems.

• The only parallelizable �an n-dimensional manifold is parallelizable if and only if it has nnonvanishing continuous vector fields� spheres are S1, S3, and S7.6,18

• The celebrated theorem by Hurwitz17 which states that the only normed real division alge-bras that satisfy �this property is also known as the composition law; algebras that satisfy itare called composition algebras �Ref. 2 p. 67�� xy�= �x��y� are the real, complex, quater-nionic, and Cayley algebras �the algebra of octonions� �Ref. 41 Sec. 20.7.

These statements are weaker than the Hopf fibration statement. The reason is that although theLCM z�z2 gives us the Hopf map H1,1 :S1→S1, squaring quaternions �q��q�ªq2� does notgive us the Hopf fibration H3,2 :S3→S2. In fact, ��S3�=S3 not S2. Moreover, fixing a quaterniona�H, the dimension of the solution set of the quaternionic equation q2=a is not constant �in thisequation we identify R3 with quaternions whose fourth components vanish�: The set �−1�1� con-sists of two points while �−1�−1� is a 2-sphere. See Sec. I N On the other hand the solution set of��u�=a ,u�U* ,a�R3 is always a circle which one might call a Hopf fiber or a Hopf circle.

There is a more geometrical interpretation to KS multiplication rule, namely,

L�u�q = q1u + q2I2u + q3I3u + q4I4u ,

where the matrices I2, I3, and I4 are given by �B5�. The multiplication rules for �I2 , I3 , I4� show thatthey define a hypercomplex structure on U. On the other hand if quaternion is represented by amatrix Q�u�, then

Q�u�q = q1u + q2I2u + q3I3u − q4I4u .

The three matrices �I2 , I3 ,−I4� do not give rise to a hypercomplex structure on U.These observations show that the KS map �Hopf fibration� u�L�u�u is the appropriate gen-

eralization of the squaring map z�z2 and not the quaternionic squaring q�q2. They also showthat it is solving the equation ��u�=a rather than the equation ��q�=a that will allow us togeneralize the square root map from two dimensions to four. “And any attempt to substitute thetheory of KS-matrices by the more popular theory of the quaternion matrices leads, therefore, tofailure or at least a very unwieldy formalism,” Stiefel and Scheifele �Ref. 43 p. 286�. �From nowon U stands for R4 with KS multiplication u�v=L�u�v and H stands for R4 with quaternionmultiplication which we represent in the matrix form as u1u2�Q�u1�u2, where Q�u� is given by�B3�.�

In fact, in Ref. 12 we define what we call KS cylindrical coordinates on U* with a twodimensional axis of rotation. We use the KS cylindrical coordinates to define the full square rootmap on an S1-cover of R3 given by �R3�S1� /, where is an equivalence relation on �x1�−

�S1�S1, and �x1�−�S1 is the blown-up negative x1-axis in R3. The equivalence relation is�x1 ,� ,��x1� ,�� ,��, if and only if x1=x1��0 and �+�=��+��.

When we study SBC singularities in the collinear and coplanar cases, we utilize some form ofa projective transformation in order to determine the relative direction of any two binaries so thatwe can resolve �blow up� the singularity, determine the limiting configuration of the relativedirections of the different binaries, and separate near-collision (near-ejection) orbits from colli-sion (ejection) orbits. To achieve that end we use the real projective space RPk in the collinearproblem9 and the complex projective space CPk in the planar problem11 For example, in the planarproblem with the identification R2�C made, the direction of z2 relative to z1 is determined by thestandard projective coordinate �=z2 /z1= z1z2 / �z1�2. In U*, using the KS matrix L�u� we can definethe direction of u2 relative to u1 to be �= �u1�−2L�u1�{u2. We use this operation to define what wecall the KS projective space UPk. The reason we can do that with the KS map and not withquaternion multiplication is that the KS map defines an S1-principal bundle �U* ,X* ,� ,S1�.

052702-2 Mohamed Sami ElBialy J. Math. Phys. 50, 052702 �2009�


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A. The different types of regularization

In celestial mechanics the question of regularizing collision singularities is studied at threedifferent levels.

At one level one addresses the question of branch regularization (or analytic continuation) ofa single collision orbit: A single collision orbit is said to be branch regularizable if and only if wecan write it as a convergent power series in = t1/3. In this case the collision orbit is analyticallycontinued through complex time to an ejection orbit. �An ejection orbit is a collision orbit inreversed time.� We call these two orbits together a collision-ejection orbit �CE orbit�.

At another level one considers the question of block regularization where one studies thesection map from the totality of collision and near-collision orbits to the totality of ejection andnear-ejection orbits. If such a map exists and is Ck one says that the singularity is Ck blockregularizable. �The terms branch regularization and block regularization were introduced byMcGehee.28 Easton5 refered to block regularization as regularization by surgery.�

In between lies what we call the problem of collective analytic continuation �CAC�, or col-lective branch regularization of simultaneous binary collisions �SBC�. Although it is a problem inanalysis, our approach to CAC is closer to block regularization because it is geometric in natureand it does not use any power series techniques. We exploit the geometric nature of the singularity,which is no longer isolated, to blow it up and separate CE orbits from near-collision and near-ejection orbits. To do that we introduce in this work two sets of variables: the generalized KS(GKS) variables and the projective KS variables. Then we we show that in the projective KSvariables, in the 3D problem, the totality of CE orbits, together with the singularity itself, forms areal analytic stable manifold WL of a real analytic normally hyperbolic submanifold of rest points.Then we project down to the GKS variables and show that in the GKS variables, the totality of CEorbits, together with the singularity itself, form a real analytic submanifold WK. �We used thegeneralized and projective LC variables in the collinear and planar problem.9,11�

The existence of the two real analytic manifolds WL and WK allows us to demonstrategeometrically that in the physical variables, the totality of CE orbits (not including near-collisionand near-ejection orbits) can be written as one convergent power series in t1/3 with coefficientsthat depend real analytically on initial conditions that lie in a real analytic submanifold. This iswhat we mean when we say that the SBC singularity can be collectively branch regularized oranalytically continued. Since all the transformations we use are real analytic away from thesingularity, this shows that each of the totality of collision orbits and the totality of ejection orbitsforms a real analytic submanifold in physical space.

B. Analytic continuation „branch regularization…

�1� In 1906 Sundman44 showed that any single solution of the three-body problem that ends ina binary collision as t↗0 can be written as a convergent power series in a new time = t1/3. Wintner49 demonstrated the assertion for a single binary collision in the n-body prob-lem. Since the vector field of the n-body problem is reversible, Sundman’s theorem meansthat such a solution is analytically continued in the complex t-plane to an ejection solutionwhich ends in a binary collision as t↘0.

�2� We will refer to branch regularization (analytic continuation) of a single binary collision asSundman’s problem or Sundman’s question.

�3� In 1941 Siegel35 investigated triple collisions in the three-body problem and showed that theset of triple-collision solutions that are branch regularizable have measure zero. In fact, heshowed that any triple-collision solution that ends in an equilateral central configuration canbe written as a convergent power series in 1= t2/3, 2= t�, and 3= t�, where � and � are realbut not necessarily rationales. If the orbit ends in a collinear central configuration it can bewritten as a convergent power series in 1= t2/3 and 2= t�. See Ref. 36 �pp. 88 �39� and�40��. Thus, only those solutions for which � and � are rational and have odd denominatorscan be analytically continued. He also showed that the totality of triple-collision solutions inthe three-body problem form a C submanifold.

052702-3 Collective branch regularization J. Math. Phys. 50, 052702 �2009�


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�4� In 1978 Simó37 studied triple collisions when � and � in Sigel’s work are rational with odddenominators.

�5� In 1968 Sperling39 studied Sundman’s problem in complex space and time: The position ofa particle is given by x= �x1 ,x2 ,x3��C3 and time ��C= �c�t��to� t� t*��C, where C is arectifiable curve in the complex plane. The collision occurs as t↗ t*. Sperling replaced thenon-negative distance in the potential by

r ª xx xy ª x12y1

2 + x22y2

2 + x32y3

2, x,y � C3.

In this case x=0⇒r=0 but =0⇏x=0. Sperling showed the following �Ref. 39 �Theorem 4,p. 313�, which we state in a slightly modified notation: “A binary collision takes place at analgebraic branch point t* in the complex t plane.”

• If →0 but x⇏0 “then the branch point is of the first order, and in a neighborhood of t*the coordinates x , . . . are meromorphic functions of �t− t*�1/2 with finite principal parts att*.”

• If x→0, and hence r→0, “then the branch point is of the second order, and in a neigh-borhood of” t* the coordinates x , . . . are meromorphic functions of �t− t*�1/3 with finiteprincipal parts at t*.”

Sperling used a rather technical definition for a binary collision given in �Ref. 39 pp. 309�.

�6� In 1970 Sperling ��Ref. 40 Sec. 411� settled Sundman’s question for SBC singularities in realspace and showed that a single SBC solution in the n-body problem is branch regularizable.Sperling relied on several results by Horn16 from 1896 to 1997 that state that for systems ofthe form x�= f�x�+g�x ,�, where f�x� is real analytic and g�x ,� is real analytic for small��; if a solution x��→0 as →0, then x�� is real analytic for sufficiently small .

�7� In 1984 Saari �Ref. 34 Theorem 5.1� also showed that a single SBC solution in the n-bodyproblem is branch regularizable. He did not rely on Horn’s work. After introducing atime-dependent change in variables, he used standard power series techniques which as usualculminated with the use of the method of majorant to prove conversion.

�8� The works of Sundmann, Siegel, Sperling, and Saari on branch regularization of a singleorbit were demonstrated �either directly or by relying on assertions that had been demon-strated� using power series techniques that are suitable for investigating a single solution toa system of ordinary differential equations. The dependence of the coefficients of the powerseries on nearby collision orbits was not studied. It was not even clear from their workwhether one can find a real analytic submanifold of initial conditions that end in SBCs northat the totality of SBC orbits forms a real analytic submanifold. We demonstrated theseassertions for the collinear problem in Ref. 9 and for the planar problem in Ref. 11.

C. Binary collisions

There are several investigations of binary collisions that are geometric in nature.

�1� In the planar problem a single binary collision can be real analytically block regularized in ageometric way via the Levi-Civita transformation �LCT�.23 Because the time rescaling of theLCT is �x�� t1/3, we can easily show that this singularity is branch regularizable withoutrelying on power series techniques.

�2� In 1970 Moser32 regularized the Kepler problem in Rn by constructing a correspondencebetween the flow of the n-dimensional Kepler problem on a surface of fixed negative energyand the geodesic flow on the unit tangent bundle of the n-sphere with a point missing. Themissing point and the unit �n−1�-sphere attached to it correspond to the collisions of theKepler problem.

�3� In 1982 Kummer20 showed that “the KS transformation owes its existence to the localisomorphism of SO�2,4� and SU�2,2�.” Kummer also showed that there is “an intimaterelationship between the Moser regularization32 and the KS regularization22 of the 3D Ke-pler problem.” He gave explicit formulas that link the two methods of regularization. That



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raises an interesting question: Can we use Moser’s method to regularize SBC singularities inany dimension? In R4 it might be cumbersome but in principal, one can use the formulas thatKummer provided to translate the current work to Moser’s variables. But in other dimensions�including dimension 3 without lifting the vector field to R4 via the KS map�, it is not clearhow one can define the relative orientations of two vectors. In other words it is not clear howto divide one vector by another as we do in R2 and R4.

�4� In 1981 McGehee28 investigated geometrically branch regularization of binary collisions ofclassical particle systems with nongravitational interactions.

�5� In 1992 Simó and Lacomba38 showed that a SBC singularity is Co block regularizable.�6� In 1993 �Ref. 9� we showed that a SBC singularity in the collinear N-body problem is C1

block regularizable.�7� In 1996 �Ref. 11� we showed that SBC singularities in the planar problem are collectively

branch regularizable.�8� In 1996 �Ref. 10� we studied SBC singularities in R3 and introduced what we call the

projective transformation near a SBC singularity and combined it with the product of severalMcGehee transformations to give a complete description of the flow near and on the collisionmanifold and also showed as an immediate corollary that a SBC singularity is Co blockregularizable. �For each binary we use one McGehee transformation centered at center ofmass of the binary. We use intrinsic energies to reduce the order of each binary singularity.The intrinsic angular momenta tend to zero on collision orbits. Therefore we define and useproperly rescaled angular momenta that do not tend to zero on collision orbits. The intrinsicenergies and rescaled angular momenta furnish a parametrization of different collision or-bits.�

�9� In 1999–2000 Martínez and Simó studied, in two nice pieces of work,25,26 the degree ofdifferentiability of block regularization of SBC singularities in some cases that can be re-duced to one-dimensional problems such as the collinear, bi-isosceles, trapezoidal, and thetetrahedral problems, and other symmetric n-body problems. However, their work is verysignificant because they show that SBC singularities in these problems are C�8/3�−� blockregularizable for any � 0, which implies that in the general case a SBC singularity cannotbe regularized any smoother.

D. Block regularization

In 1974 McGehee27 showed that triple-collision singularities in the collinear 3-body problemare not Co-block regularizable. In addition to the importance of this finding, McGehee introduceda new transformation that soon became known as the McGehee transformation and gave insightinto the geometric nature of collision singularities; for example, it brought to light the geometricnature of the facts that on a collision orbit of the N-body problem r� t2/3 and that the �-limit setof the configuration vector s=r−1x is a subset of the set of central configurations. In the McGeheecoordinates these two facts among a host of other facts are corollaries to the two facts that �a�collision orbits tend to the set of rest points on the collision manifold �no rest points exist outsidethe collision manifold� and �b� at srest points the configuration vector s is a central configuration.

The McGehee transformation opened the door to a host of investigations of collision singu-larities and the behavior of the flow near the collision manifold.

�1� In Refs. 3 and 4 Devaney studied the anisotropic Kepler problem and the planar isoscelesthree-body problem respectively.

�2� In Ref. 24 Martínez and Simó gave a detailed qualitative study of the planar isoscelesthree-body problem.

�3� In Refs. 29–31 Moeckel studied the flow near triple collisions of the three-body problem anddemonstrated the existence of chaotic behavior.

�4� In Ref. 7 we studied the isosceles three-body problem with small mass ratio � 0 andshowed that as �→0, the collision manifold undergoes a topological change from a spherewith four points deleted to a cylinder.



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�5� In Ref. 8 we generalized the McGehee transformation to the most general collision singu-larity of a simultaneous collapse of several clusters and proved, among other things, many ofthe results that were known for specific cases.

�6� In 1989 Xia50,51 resolved a long standing question by demonstrating the existence of non-collision singularities �a singularity is called noncollision if the distances between particlesbecome unbounded in finite time without having a collision� in the five-body problem. It isknown that there is no noncollision singularities in the two to four-body problem.

�7� In 1991 Gerver,13 using less geometric methods with very involved estimates, showed theexistence of noncollision singularities in the planar N-body problem with N large. Theweakness in this result is that the “N large” assumption does not answer the problem forsmall N.

Block regularization has an advantage over branch regularization because it takes into accountnear-collision and near-ejection orbits as well as collision and ejection orbits. This is an advantagebecause in practice one wants to know how well initial conditions predict the behavior of theparticles after their close encounter.

One complication that arises when working with block regularization is that traveling time�we call the time it takes a near-collision orbit to travel from the initial transversal cross section tothe final transversal cross section traveling time� does not necessarily depend continuously oninitial conditions. The reason is that usually one blows up the singularity to an invariant manifoldcalled the collision manifold. One place where this problem was overcome is our work on SBC inthe collinear n-body problem.9 In that work we removed the singularity without replacing it withan invariant manifold. Methods such as the one we used �which takes advantage of the fact that oncollision orbits r� t2/3� do not work in the planar case because the order of the singularity in theangular velocity is higher than the order of the radial singularity. In fact, in the McGehee trans-formation one rescales time by a factor of r3/2 because the singularity in the McGehee �rescaled�angular velocity is of order r−3/2. The singularity in the McGehee �rescaled� radial velocity is oforder r−1/2 only, and this is why we have to learn to live with an invariant collision manifold atr=0.

E. Summary of the main results

We show that 3D SBC singularities can be collectively branch regularized and that theircollision-ejection �CE� manifold in the GKS variables �parameter space� is also real analytic. Asan immediate corollary, we obtain the fact that the manifold of collision orbits in physical space isreal analytic, which we have shown previously in Ref. 10 where a complete description of thebehavior of collision and near-collision orbits is given.

A short description of the main theorems of the current work is as follows.

�1� In Theorem 1 we show that in the GKS variables, the totality SBC orbits and SBE orbits,together with the singularity manifold itself, form a real analytic submanifold. We call thismanifold the CE manifold, or CE manifold for short, and denote it by WK. The crucial pointis that we use a transformation that we call the KS-projective transformation �KSPT� toisolate collision and ejection orbits from near-collision and near-ejection orbits. Since all thetransformations we use are real analytic away from the singularity, it follows that the sub-manifold of collision orbits and the submanifold of ejection orbits in the physical space areboth real analytic submanifolds.

�2� In Theorem 2 we show that the singularity on the CE manifold WK is removed and we obtaina real analytic flow that pushes orbits through the singularity. The time rescaling we usereverses the regularized vector field on ejection orbits. Notice that Theorem 2 does not meanthat the singularity of the full flow can be removed. However, it is blown up and regularizedin a projective way.

�3� In Theorems 3 and 4 we prove some detailed results that will be given shortly when wedevelop the necessary notation.

�4� In Theorem 5 we prove the CAC of SBC singularities. We demonstrate this result geometri-



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cally and without writing down any power series expansions. We use the real analytic CEmanifold WK mentioned above.

�5� The affix KS in the “KSPT” is there to indicate that we do not use the standard quaternionprojective space HPM, but rather a different space that we denote by UPM, where the relativedirections of two vectors are defined via a nonassociative multiplication rule that we call KSmultiplication.

�6� For the sake of unnecessarily inviting the quaternions to the party, one can also interpret thismultiplication by viewing the vector space U=R4 as a left module over quaternions with themultiplication q⋄uªL�u�q ,q�H ,u�U, where the matrix L�u� is given by �1.4�.

�7� All this is well and good, but all what we need for the current work is to define the directionof v relative to u to be Sª �u�−2L�u�⊺v, which is equivalent to z2 /z1 in the complex plane.The nonassociativity of the KS multiplication should not deter us since multiplication ofoctonions is not associative either.

F. The LCT

The LCT for regularizing the perturbed planar Kepler problem is given by

x = y, y = −a

�x�3x + f , � = yTf , � =

�y�2

2−

a

�x��1.1�

In matrix notation the LCT is given by

�u,w,� � �x,y,t� ,

x = ��u� = L�u�u, y = �u�−2L�u�w ,

d

d= �u�2

d

dt, � =

1

2�y�2 −

a

�x�,

L�u� = �u1 − u2

u2 u1 = �u�1� u�2��, x = �x1,x2�T, u = �u1,u2�T, . . . , �1.2�

where � is the energy which is no longer an integral of motion. The LCT sends the perturbedKepler system �1.1� to the perturbed harmonic oscillator,

u� = 12w, w� = �u + �u�2L�u�Tf ,

�� = fTL�u�w, ��u�2 =1

2�w�2 − a . �1.3�

We introduce � as a variable in order to use it to reduce the order of the singularity in w before thetime rescaling from �u�−3 to �u�−1.

In complex number notation x=��u�=u2 and y=w / u. The LCM ��u� is locally real bianalytic,that is, locally it is a real analytic homeomorphism with a real analytic inverse.

G. The KSM „Ref. 43…

Let U=R4 ,X=R3� �0��R4 ,U*=U \ �0� and X*=X \ �0�. The KSM is given by

�:U* → X*, ��u� = L�u�u ,



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L�u� ª�u1 − u2 − u3 u4

u2 u1 − u4 − u3

u3 u4 u1 u2

u4 − u3 u2 − u1

�¬ �u�1� u�2� u�3� u�4�� , �1.4�

where u�j�ª Iju , j=1,2 ,3 ,4 and the Ij’s are given by �B5�. It is obvious that each Ij ,d=2,3 ,4,

defines an antisymmetric bilinear form on U*. In particular, I4 gives rise to antisymmetric bilinearform,

��u,v� ª �L�u�v�4 = u4v1 − u3v2 + u2v3 − u1v4 = vTI4u = �u�4�,v� .

Let x=L�u�u �B1�. Thus L�u�⊺L�u�= �u�2I, �x�= �u�2, det L�u�=−�u�4, and x4=��u ,u�=0.Moreover

��u,v� = 0, if and only if L�u�v = L�v�u ,

��u,v� = 0, if and only if �u�2L�v�v − 2�u,v�L�u�v + �v�2L�u�u = 0,

��u,v� = 0 if and only if L�u�TL�v�v = 2�u,v�v − �v�2u . �1.5�

The space U is called the parameter space.43

Lemma 1.8: �Reference 43� Let y4=0 and u�U. Then ��u ,L�u�⊺y�=0.Lemma 1.9: �The fibration of U �Ref. 43� �Ref. 12 for current notation��. The following are

true.

�1� Let x�y be two points in X*. Then �−1�x��−1�y�=�.�2� Let u��−1�x� be fixed but arbitrary. Then �−1�x� is given by

�−1�x� = ��u�2 = A�t�u�− �� t��

A�t� = etI4 = I1 cos t + I4 sin t = R14�− t� � R23�t� , �1.6�

where A�t� is given explicitly in �B4� and the meaning of R14�−t� and R23�t� should be clear.�3� �−1�x� is a circle of radius r=�x�= �u� lying in the plane Ruªspan�u�1� ,u�4��.�4� The tangent line to �−1�x� at u is given by

Tu��−1�x�� = span�u�, u = � d

d�A��u�

�=0= u�4� = I4u .

�5� It follows that

��u,v� = �u,v� = vTI4u, if and only if u,v � U .

Thus, v is normal to the fiber �−1�x� at u��−1�x� if and only if ��u ,v�=0 �.

Definition 1.10: Define a right action of (the compact Lie group) G= ��A�� 0,2�� onU* by

u · �ª u�ª A��u, u � U*, �� 0,2�� . �1.7�

Let �U*�=U* /G be the quotient space. Denote the G-orbit of u�U* by �u�. Sometimes we will talkabout S1 but we really mean G.

Remark: In Ref. 43 �p. 271, �9��, the fiber �−1�x� is given by ��u⊺A��−1�� 0,2��. Hence,the tangent vector to the fiber at u⊺ is −u⊺I4= �I4u�⊺. The difference here is that we are usingcolumn vectors rather than row vectors. For row vectors right actions are defined by multiplyingon the right by A��⊺=A��−1=A�−�� Ref. 42, p. 294�. Since u is a column vector, and since G is



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commutative, the action �1.7� can be viewed as a right action �A��u�⊺=u⊺A��⊺=u⊺A��−1.Corollary 1.12: The action �1.7� is free. That is, A��u=u⇐ ⇒A��= I. Moreover, �−1�x�

= �u� for any u��−1�x�. Since G is compact and acts freely, the quotient space �U*� is a realanalytic manifold �.

Proposition 1.13: �Reference 12� P= �U* ,� ,X* ,G� is a principal bundle. It follows that

�:�U*� → X*, ��u�� = ��u� �1.8�

is a real bianalytic map from �U*� on X* with inverse x��−1�x�. �

H. The KSM versus quaternionic multiplication

To show the difference between the KSM and quaternionic multiplication,12 we make thecorrespondence u�u1+ iu2+ ju3+ku4. We represent quaternionic multiplication by the matrixQ�u� given by �B3� and obtain

Q�u�v � uv, L�u�v � uv, L�u� = Q�u�N ,

v = v1 + iv2 + jv3 − kv4 � Nv . �1.9�

First, det L�u�=−�u�4 while det Q�u�= + �u�4. Let �e1 , . . . ,e4� be the standard basis of R4�X* and,to avoid confusion, let

k1 = �1 0 0 0�T, . . . , k4 = �0 0 0 1�T �1.10�

be the standard basis of U. Now we compare the solutions of the equations ��u�ªQ�u�u=�e1 tothose of the equations ��u�=�e1. Since �1��e1, these are the equations that correspond toz2=�1,z�C. From �1.6� we can see that the solution to ��u�=e1 is the circle,

�−1�e1� = �k1t � = �u�u1

2 + u42 = 1,u2 = u3 = 0� ,

and the solution to ��u�=−e1 is also a circle,

�−1�− e1� = �k2t � = �u�u2

2 + u32 = 1,u1 = u4 = 0� .

In fact, as we saw above, �−1�x� is always a circle. Moreover, P= �U* ,� ,X* ,S1� is a principalbundle.

By �B1� the map ��u� sends U* to R�X*. Moreover,

�−1�e1� = ��k1�, �−1�− e1� = �u�u22 + u3

2 + u42 = 1,u1 = 0� .

Hence, the dimension of �−1�u�=x is not constant and ��u� does not define a principal bundle.These observations make us believe that L�u�u is the appropriate generalization of the squar-

ing map z2 and that the KS matrices L�u� are fundamentally different from the quaternion matricesQ�u�. “And any attempt to substitute the theory of KS matrices by the more popular theory of thequaternion matrices leads, therefore, to failure or at least a very unwieldy formalism.” �Ref. 43, p.286�.

The KS multiplication L�u�v is not associative. But let us recall that multiplication of complexnumbers is both associative and commutative, multiplication of quaternions is only associative,and multiplication of Cayley numbers �octonians� is neither.

Remark 1.15: �Reference 12�

�1� We would like to point out that some authors make use of quaternions.45–48 In Ref. 48 theauthor uses “a new elegant way of handling the 3D case in complete analogy to the wellknown planar case by introducing an unconventional conjugation of quaternions �see defi-nition in Eq. (24) below�, first mentioned by Waldvogel (2006).” The “unconventional con-jugation” is v�v*=v1+ iv2+ jv3−kv4�Nv which is equivalent to v� v=−v1− iv2− jv3



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+kv4, which can be found in Ref. 43 �p. 286�. Then the author of Ref. 48 defines the KS mapas x=uu* which, except for the notation, is nothing more than L�u�u, also in Ref. 43. Thenhe reproduces a fragment of Chap. XI of Ref. 43 in the quaternionic notation. All this isnothing but the KS map in a cumbersome notation. It leads to the same Hopf fibration. Itdoes not lead to the squaring of quaternions u�u2�Q�u�u because squaring of quaternionsdoes not lead to a fibration at all as we saw above.

�2� The LCT as well as the KST for a single binary collision consist of a change in variables anda time rescaling. The order in which these two steps are performed does not matter. If weperform only the time rescaling we obtain the system

x� = �, �� =r�

rx� −

a

rx + r2f ,

which is still singular �Ref. 43, p. 20�. This makes the idea that only a time rescaling isneeded to regularize the Kepler problem a misconception.

In Ref. 12 we study the geometry of the KS map.

�1� We show that the KSM is the appropriate generalization of the squaring map z�z2 ,z�C.�2� We construct two square root branches from R3∖ �x1�− to the upper and lower halves of R3;

where �x1�− is the negative x1-axis in R3 and resembles the cut used to define the standardcomplex square root branches �z. We glue these two branches together.

�3� We introduce what we like to call KS cylindrical coordinates with a two dimensional axis ofrotation. We also introduce what we call KS toroidal and spherical coordinates.

�4� We use the KS cylindrical coordinates to define the full square root map on an S1-cover of R3

given by �R3�S1� /, where is an equivalence relation on �x1�−�S1�S1, and �x1�−

�S1 is the blown-up negative x1-axis in R3. The equivalence relation is �x1 ,� ,��x1� ,�� ,�� if and only if x1=x1��0 and �+�=��+��.

II. MAIN RESULTS

A. Notation and definitions

We study the gravitational Newtonian three-dimensional N-body problem in a neighborhoodof a SBC singularity of exactly M binaries involving 2M�N particles. The other KªN−2Mparticles stay apart and away from the colliding pairs.

In the collapse of one unperturbed cluster, one can take advantage of the conservation ofmomentum and fix an inertial frame of reference at the center of mass. One can also utilize therotational symmetry of the N-body problem and consider SO�3�-equivalent classes of configura-tions as one configuration. It is in that sense that one may ask whether the collapse of an unper-turbed cluster is an isolated singularity.

On the other hand, when two or more collision singularities take place simultaneously, eachcluster collapses at its center of mass. These centers of mass can be moved slightly, and we obtainanother singularity. Moreover, the relative orientations of the limiting configurations of differentclusters, when they exist, cannot be factored out because the system is not invariant if SO�3� actson each cluster by rotating it about its own center of mass. It is in that sense that simultaneouscollision singularities are not isolated.

We denote the M binaries by P1 , . . . , PM. Let zj �R3 be the center of mass of Pi and let mi and�i be the masses of its members. For 1� i� j�M, let zijªzj −zi. Let xi�R3 be the positionvector of �i relative to mi, rescaled so that the intrinsic kinetic energy of Pi is �1 /2��xi�2. Let xª �x1 , . . . ,xM�. Let the masses and positions of the remaining K particles be nk ,yk ,1�k�K. Letzª �z1 , . . . ,zM� and y= �y1 , . . . ,yK�. The SBC singularity set is defined as



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� = ��x,y,z��x = 0;zij � 0,i � j ;yk � yl,k � l;ym � zn, ∀ m,n� .

A singular point p=p�0,y ,z�� is not isolated since we can vary y and z and still obtain anotherp��. Moreover, subsingularities of the form xi1= ¯ =xiJ=0,J�M accumulate on p. However,as we showed in Refs. 8 and 10, upon blowing up the singularity, we find that p is isolated fromall singularities of any other type of clustering including the subsingularities we just mentioned.

We write the Hamiltonian of this system as Ho+H1, where Ho is the sum of the intrinsicHamiltonians of the M binaries and H1 is the Hamiltonian of the interaction between the differentbinaries; between the binaries and the remaining particles; and between the remaining particlesthemselves. Let x=X , y=Y, and z=Z. Thus, we have

H = Ho + H1, Ho = �i=1

M

�i, H1 = T + W, �i =1

2��Xi�2 −

ai2

�xi�� ,

T =1

2�i=1

M

Mi�Zi�2 +1

2�k=1

K

nj�Y j�2,

W = �1�i�j�M

Wij + �i=1

M

�k=1

K

Vik + �1�k�l�K

K

Rkl,

Wij = −aij

�zij + eixi − ejxj�−

bij

�zij + cixi − ejxj�−

cij

�zij + eixi + cjxj�−

dij

�zij − cixi + cjxj�,

1� i� j�M ,

Vij = −eik

�zi + cixi − yj�−

f ik

�zi + eixi − yj�, 1� i�M,1� k� K ,

Rkl = −njnl

�yl − yj�, 1� k� l� K . �2.1�

All the constants used here depend on the masses and are given explicitly in Appendix A.

B. Equations of motion in physical space

We call the space �x ,X ,y ,Y ,z ,Z� the physical space.The equations of motion are

xi = Xi, Xi = −ai

2

2�xi�3xi − �xi

W ,

y = Y, Y = − �yW ,

z = Z, Z = − �zW , �2.2�

where



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xi = �xi1,xi2,xi3,0�T, Xi = �Xi1,Xi2,Xi3,0�T, �xiW = � �W

�xi1,�W

�xi2,�W

�xi3,0 T

,

and �yjW and �zj

W are defined in a similar fashion.Lemma 2.3: �Appendix A� The following are true in a sufficiently small neighborhood of the

SBC singularity.

�1� For k=1, . . . ,M ,�xkW�x ,y ,z� is real analytic in all variables and has the following expan-

sion:

�xkW�x,y,z� = − �k�x�k�,y,z�xk − hk�x,y,z� = − fk�x,y,z� ,

x�k� = �x1, . . . ,xk−1,xk+1, . . . ,xM� ,

�k�x�k�,y,z� � 0, hk�x,y,z� = O��xk�2� . �2.3�

�2� The two vectors �ykW�x ,y ,z� and �zk

W�x ,y ,z� are real analytic in all variables.

The following lemma is a special case of a more general theorem in Ref. 8 on the simulta-neous collapse of several clusters. We will need it to define the generalized and projective KStransformations.

Lemma 2.4: �Asymptotic behavior in physical variables� �References 8 and 10� The followingquantities have finite limits on SBC collision and ejection orbits with k=1,2 , . . . ,M:

A ª ��,y,Y,z,Z� → A*, sk ªxk

�xk�→ s

k* � S2,

k ª�xk�

� j=1M �xj�2

→ k* � �0,1�, �k ª �xk�1/2xk → �

k* � 0. �2.4�

C. �-consistent vector fields and second order equations

In Sec. III we determine the class of vector fields on U* and the class vector fields on TU*

�and hence second order equations on U*� that are horizontal and equivariant �invariant under thefree action of the compact Lie group G�. We call them �-consistent. Each of these classes is inone-to-one correspondence with its counterpart on X*. A vector field is horizontal if it lies in thehorizontal bundle of the principal bundle. The horizontal bundle �also known as the principalconnection� is a sub-bundle of TU* such that each restriction

DuH�ª Du��HuU*:HuU* → T��u�X*�

is an isomorphism �Ref. 42, p. 289�. Equivariant horizontal vector fields are actually objectsdefined on the quotient space �U*�=U* /G which is a real analytic manifold, as we recalled above.

The bundle HPª �HU* ,TX* ,DH� ,G� is also a principal bundle. We find the appropriateconnection on HP needed to define horizontal second order equations.

We end Sec. III by defining the GKS transformation for one perturbed Kepler problem.

D. The GKS variables

In Sec. IV we define the KS transformation for M binaries. This is basically the product of Mone-binary KS transformations. For k=1, . . . ,M, let

xk = ��uk� = L�uk�uk, Xk =1

�uk�2L�uk�wk,



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�k =�wk�2 − ak

2

2�uk�2, �k =

wk

�wk�,

V = ��,�,y,Y,z,Z� . �2.5�

The variables �u ,V� are called the GKS variables.In the unperturbed Kepler problem the energy � is an integral of motion and is used to reduce

the order of the singularity. In our case, it is crucial that we replace each wk by ��k ,�k� in orderto reduce the order of the singularity from �uk�−4 to �uk�−2. Notice that the collection of equations,

wk2 = ak

2 + 2�k�uk�2, k = 1, . . . ,M ,

form a regular submanifold. Moreover, in Ref. 8 we show that in a general collision singularity inwhich M clusters collapse simultaneously, the intrinsic energy of each cluster has a finite definitelimit.

Now we can state some of our main results.Theorem 1: Let RC

K, REK, and ��

K,o be the collection of SBC orbits, the collection of SBEorbits, and the SBC singularity set, all in the GKS variables. In a sufficiently small neighborhoodof the singularity the following is true in the GKS variables.

�1� The three sets RCK ,RE

K, and ��K,o are all real analytic.

�2� The disjoint union WKªRC

K��K,o�REK �a square cup � indicates disjoint union� is the

graph of a real analytic function. Hence it is a real analytic manifold which we call the KSCE manifold.

�3� At each singular point in ��K,o a unique collision orbit and a unique ejection orbit meet. This

defines a one-to-one correspondence between collision and ejection orbits. If the singularpoint is added, we obtain a real analytic curve, which we call a CE orbit.

Theorem 2:

�1� The flow on RCK�RE

K can be extended to a real analytic flow on WK with no rest points. Thatis, the singularity on WK can be removed. Notice that this does not mean that the singularityof the full flow can be removed. It is just Co-block regularized.

�2� Let CK�RCK�EK�RE

K� be a submanifold of initial conditions that end (start) in SBC (SBE)singularity. The following hold:

�a� CK�EK� can be chosen to be a real analytic submanifold of codimension one in RCK�RE

K�.�b� The flow on WK of part (1) defines a real bianalytic (a real analytic homeomorphism with

a real analytic inverse) section map from CK to EK. This section map corresponds to theone-to-one correspondence between collision and ejection orbits given in Theorem 1.

Theorem 3: �Reference 10� In the GKS-space, sub-SBC orbits of J�M binaries do notaccumulate on any SBC orbit of M binaries.

Theorem 4: The following are true in the physical variables x ,X ,y ,Y ,z ,Z.

�1� Let RC�RE� be the set of collision (ejection) orbits near a SBC singularity in the physicalspace �x ,X ,y ,Y ,z ,Z�. Then, both RC and RE are real analytic.

�2� Let C�RC�E�RE� be a submanifold of initial conditions that end (start) in SBC (SBE)singularity.

�a� The set C�E� can be chosen to be a real analytic submanifold of codimension 1 inRC�RE�.

�b� The one-to-one correspondence between collision and ejection orbits of Theorem 1, whichis the same as the one of Theorem 2, defines a real bianalytic section map from C to E.

Theorem 5: The following hold in the physical space.

�1� Each CE orbit can be written as a convergent power series in t1/3 with coefficients that



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depend on initial conditions in C in a real analytic fashion. The power series takes the form

xk = t2/3qk�t1/3,V�, V � C, k = 1, . . . ,M , �2.6�

where qk�s ,V� is real analytic and qk�0,V��0,�2� For k=1,2 , . . . ,M , t1/3Xk can be written as a convergent power series in t1/3 with coefficients

that depend on initial conditions in C in a real analytic fashion. The power series takes theform

t1/3Xk = �k�t1/3,V�, V � C, k = 1, . . . ,M , �2.7�

where �k�s ,V� is real analytic and �k�0,V��0. Moreover, the classical resultslimt→0 t1/3Xk�t1/3 ,V��0 and limt→0�xk�1/2Xk�t1/3 ,V��0 follow.

The main step in proving Theorem 5 is to show that the CE manifold WK=RCK��K,o�RE

K isindeed a real analytic submanifold in the KS variables �Theorem 1, part 2� and that the singularityon WK can be removed and the flow on RC

K�REK can be extended to all of WK without creating

any rest points �Theorem 2, part 1�. In order to do that we need to isolate collision and ejectionorbits from nearby near-collision orbits and nearby near-ejection orbits. The first step is totranslate Lemma 2.4 on the asymptotic behavior of physical variables to the GKS variables�u ,� ,� ,y ,Y ,z ,Z� and obtain Lemma 5.2. This is done in Sec. V. We also show that the relativedirections of binaries have a finite limit in the sense that Sk

�j�ª �uj�−2L�uj�{uk , j�k, have finite

limits on SBC orbits. The vector Sk�j� corresponds to the quotient zk /zj of the planar problem. We

know from Lemma 2.4 that both �zk� / �zj� and �uk� / �uj� have nonzero finite limits on collision andejection orbits.

To appreciate this procedure let us recall how in Ref. 11 we manage to isolate collision andejection orbits from nearby near-collision orbits and nearby near-ejection orbits in the planarproblem. We identify the real plane R2 with the complex plane C and use Lemma 2.4 to focus onan open subset of the projective space CPM−1 by defining the projective transformation near aSBC singularity. If we identify z�C with �x ,y�{�R2, then

z1z2 L�z1�z2, z1z2 L�z1�Tz2,z2

z1

1

�z1�2L�z1�Tz2,

where L�z� is given by �1.2�. We can express a complex line along b�CM in the form

L�z�b = �L�z�b1, . . . ,L�z�bM�, z � C .

In the present investigation the variable u is in U*M, U*=R4 \ �0�.For clarity, we denote the standard basis of U by k1 , . . . ,k4 and the standard basis of R4 by

e1 , . . . ,e4 where

k1 = �1 0 0 0�T, . . . , k4 = �0 0 0 1�T,

When uj�0 we can write

S�j�ª �S1

�j�, . . . ,Sj−1�j� ,k1,Sj+1

�j� , . . . ,SM�j��, uj � 0,

Sk�j�ª

1

�uj�2L�uj�Tuk, k � j . �2.8�

Then we can write

u = L�uj�S�j�.

Define a relation � on U*M by



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u u� ⇔ �∃ j:1� j�M,uj � 0,uj� � 0

S�j� = S��j�.� �2.9�

Corollary 2.9:

�1� The relation � is an equivalence relation.�2� The quotient space U*M / is a topological manifold; that is, it is Hausdorff and second

countable.�3� Moreover,

u u� ⇔ �∃ j:1� j�M,uj � 0,uj� � 0

u� = �uj�−2L�uj��L�uj�Tu .� �2.10�

�4� Let u�u, uj�0�uj� �and hence S�j�=S��j��. If in addition ul�ul�, then S�l�=S��l�.�5� The transition functions are real analytic and given by

S�j� =1

�uj�2L�uj�TL�ul�S�l�, ul � 0 � uj . �2.11�

Proof: Only the fourth assertion needs demonstration: Notice that S�j�=S��j� implies that

1

�uj�2L�uj�TL�ul�S�l� = S�j� = S��j� =

1

�uj��2L�uj��

TL�ul��S��l�,

1

�uj�2L�uj�Tul =

1


Tul�.

The second equation and identity �B20� implies that

1

�uj�2L�uj�TL�ul� =

1

�uj�2NL�NL�uj�Tul� =

1

�uj��2NL�NL�uj��

Tul�� =1


TL�ul�� ,

and hence S�l�=S��l�. �

Definition 2.10: �The KS-projective space UPM−1� We define the KS-projective space to be thequotient space U*M / and denote it by UPM−1. The collection

S ª ��S�j�,Uj*M��j = 1, . . . ,M� ,

Uj*M

ª �u � U*M�uj � 0�, j = 1, . . . ,M , �2.12�

defines an atlas on UPM−1 whose transition functions are given by �2.11�. The transition functionsare real analytic. Hence UPM−1 is a real analytic manifold.

Definition 2.11: �KS lines� Fix 1� j�M and S�j�. The KS-line in Uj*M

through S�j� is definedas

u = L�w�S�j�, w � U . �2.13�

Lemma 2.12: Fix 1� j�M and 0�c�Uj*M

.

�1� Let

u = L�w�c, w � U . �2.14�

It follows that



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Sk�j� =

1

�cj�2L�Ncj�TNck, �2.15�

which is independent of w and hence (2.14) is a KS line.�2� The point c is not on the KS line (2.14).�3� The line through c is given by

u = L�w�Nc, w � U , �2.16�

Sk�j� =

1

�cj�2L�cj�Tck, k = 1,2, . . . ,M . �2.17�

Proof:

�1� Occasionally we write L�v� for L�v� for visual reasons. By �B13� with v=w and a=cj wehave

Sk�j� =

1

�uj�2L�L�w�cj�TL�w�ck =

1

�L�w�cj�2�L�w�NL�Ncj��TL�w�ck

=1

�w�2�cj�2L�Ncj�TNL�w�TL�w�ck =

1

�cj�2L�Ncj�TNck.

�2� Assume that c lay on the KS line �2.14�. Then c=L�w�c for some w. By �B10� w would haveto satisfy the identity

w = �ck�−2L�ck�TNck, k = 1, . . . ,M ,

which is not always possible.�3� c=L�k1�Nc since N=L�k1� and N2= I. Thus, the line through c is given by �2.16�.

If we compare �2.15� and �2.17� we see that it makes sense that the line through c is given by�2.16�. �

Definition 2.13: �The KSPT and variables� We define a transformation given by a collectionof charts �P j,r ,1� j�M ,1�r�4� near a SBC singularity. Each chart includes a time rescaling.We give the P1,1 chart explicitly. The others are defined similarly by relabeling the variables. TheP1,1 chart is defined near u1

1�0 and is given by

� = �1,1 = u11, � = �1,1 = �u1

2,u13,u1

4� ,

T = T1,1 =1

��1,1, E = E1,1 = �1,T1,1� = �1,T� ,

� = �1,1 = �11, � = �1,1 = ��1

2,�13,�1

4� ,

S = S�1� = �S2, . . . ,SM�, Sk = Sk�1� =

1

�u1�2L�u1�Tuk,

d

d=

d

d1,1= −

2�3�1 + �T�2��w1�

d

dt. �2.18�

The KS-projective variables near a SBC singularity are defined to be the collection of charts,

�P j,r� = �� j,r,Tj,r,S�j�,�,�,y,Y,z,Z��1� j�M,1� r� 4� . �2.19�



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Remarks:

�1� The time rescaling in �2.18� reverses the sense of time as �↘0, that is, on ejection orbits.�2� It is enough to work in P1,1 where �2.2� takes the real analytic nonsingular form �6.8� and

�6.9�.�3� The choice �=�1,1=u1

1 uniquely determines � and � since uk= ��wk� /2�uk�2��k, as seen by�5.8�.

�4� For each j=1, . . . ,M, the collection �Tj,r �1�r�4� is the standard open cover for RP3.

Theorem 6:On any collision (ejection) orbit, as t↗0�t↘0�, the projective variables�� ,T ,S ,V� have finite limits po= �0,To ,So ,Vo� which satisfies the following:

lim E = Eo = �1,To�, �1o = �o�1,To� = �oEo, �o = 1 + �To�2,

lim�wj� = aj, lim wj = wjo = aj� j

o, lim1

�uj�uj = � j

o, j = 1, . . . ,M .

• For k=2, . . . ,M,

lim�uk��u1�

= lim�Sk� = � ak

a1 1/3

,

lim1

�u1�2L�u1�Tuk = lim Sk = Sk

o = � ak

a1 1/3

L��1o�T�k

o = �o�ak/a1�1/3L�Eo�T�ko.

• It follows that for 1� j ,k�M, we have

lim�uk�3

� j=1M �uj�3

=ak

� j=1M aj

, lim1

�uj�2L�uj�Tuk = �ak

aj 1/3

L�� jo�T�k

o,

lim�uk��uj�

= �ak

aj 1/3

, limdir

d js=

1 + �Tiro �2

1 + �Tjso �2

,

� j = � jo + O�t4/3� . �2.20�

In Sec. VI we study the rest points of the vector field �6.8� and �6.9� in the KS-projectivevariables. In Sec. VI A we see that the SBC singularity is blown up via the KSPT to a real analyticmanifold ��

L. In article Sec. VI E we show that the singularity itself is lifted to a real analyticsubmanifold of rest points of the vector field �6.8� and �6.9� which we denote by ��

L,o��L. In

Sec. VI K we compute the linearization of the vector field �6.8� and �6.9� and show that ��L,o is a

real analytic normally hyperbolic submanifold of rest points and that the center manifold of eachrest point is ��

L,o itself. The local stable manifold of ��L,o, which we denote by WL=W��

L,o�,consists of three parts: The part with ��↗0�, which we denote by RC

L ; the part with �0↙��,which we denote by RE

L; and the lifted collision singularity ��L,o��=0�. Local stable manifolds are

graphs of functions of the same smoothness as the vector field. It follows that WL is the graph ofa real analytic map defined in an open neighborhood O of ��o ,Vo�= �0,Vo�, given by

�L:O → OLª �L�O� = WL,

�L��,V� ª ��,T,S,V� ,



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T = ��,V� � 0, S = ��,V� � 0,

where �, �= ��2 ,�3 , . . . ,�M�, are real analytic in �� ,V� and do not vanish. Thus, �L :O→OL isreal bianalytic and provides a coordinate chart for WL. In this coordinate chart, and after rescalingtime, our vector field �6.8� and �6.9� takes the real analytic form �6.21� for some real analyticGL�� ,V�,

�� = 1, V� = GL��,V� .

Each rest point in ��L,o corresponds to a lifted SBC singularity and has a one dimensional stable

manifold which we call a lifted CE orbit. One branch of it ��↗0� is a collision orbit, and theother �0↙�� is an ejection orbit. This yields the known result of the branch regularizability of asingle CE orbit in the KS-projective variables.

In Sec. VII we prove Theorems 1–4. First we show that WL projects down to a real analyticmanifold WK in the KS variables. In fact, we show that WK also is the graph of a real analyticfunction,

�K:O → OKª �K�O� = WK,

�K��,V� ª ��L��,V�� = �u,V� ,

��,T,S,V� = �u,V�, u1 = ��1,T�, uk = L�u1�Sk, k = 2, . . . ,M .

In this coordinate chart, and after rescaling time, �6.8� and �6.9� take the real analytic form �7.5�,

�� = 1, V� = GK��,V� ,

in fact, it is obvious that GK�� ,V�=GL�� ,V�.In Sec. VIII we prove Theorem 5 on CAC. Our proof is geometric and uses the fact that both

WL and WK are graphs of real analytic functions. We do not use power series nor any convergencetechniques.

In Sec. IX we prove Theorem 6 on the asymptotic behavior on collision orbits. The prooffollows immediately from the fact that collision and ejection orbits in the KS-projective variablesare the stable manifold of the collision singularity.

The reader who is familiar with lifting vector fields and second order equations in principalbundles and does not worry about the precise translations of small neighborhoods of the singu-larity from coordinate system to another, that reader, may go directly to the vector field �6.8� and�6.9�.

III. EQUIVARIANT VECTOR FIELDS ON U* AND TU*

In this section we describe the lifting of vector fields from X* and TX* to U* and TU*,respectively. Second order equations form a special class of vector fields on the tangent bundle.Therefore, we start by recalling how to lift vector fields from X* to U*. These are lifted to vectorfields on U* that are both horizontal and equivariant �invariant under the free action of thecompact Lie group G�. We call them �-consistent. A vector field is horizontal if it lies in thehorizontal bundle of the principal bundle. The horizontal bundle �also known as the principalconnection� is a sub-bundle of TU* invariant under the group action and is such that each restric-tion Du

H�ªDu��HuU*:HuU*→T��u�X* is an isomorphism. Equivariant horizontal vector fieldsare actually objects defined on the quotient space �U*�=U* /G which is a real analytic manifoldbecause the compact Lie group G acts freely on U*. For more details about general principalbundles, see Ref. 42 �p. 294�. See also Ref. 19 and 33.

In what follows, vector fields on X* and U* will be written in either of the two forms,



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x = y�x� or �x,y�x�� and u = v�x� or �u,v�u�� .

We call vector fields on the tangent bundles TX* and TU* systems of equations, or systems forshort, and write them as

x = ��x,y�, y = ��x,y�, or �x,y,��x,y�,��x,y�� ,

u = ��u,v�, v = ��u,v�, or �u,v,��u,v�,��u,v�� .

Sometimes we will write

��u,v�,��u,v�� and ��x,y�,��x,y�� .

Second order equations are those systems for which ��x ,y�=y and ��u ,v�=v. We will write themas

�x,y,��x,y�� and �u,v,��u,v��

A. The derivative of �

Let I40=diagonal�1,1 ,1 ,0�. For any u ,v�U*, we have �L�u�v�4=−�L�v�u�4 and �L�u�v� j

= �L�v�u� j , j=1,2 ,3. Thus, the derivative of � is given by

D�:TU* → TX*,

D��u,v� = ��u�,�uv� ,

�u ª Du�ª D��TuU*:TuU* → T��u�X*� ,

�uv ª �Du��v = L�u�v + L�v�u = 2I40L�u�v � T��u�X* � Y = R3� �0� � R4. �3.1�

Now, 2L�u�v�T��u�X* unless �L�u�v�4=��u ,v�=0. If so, 2I40L�u�v=2L�u�v. Thus, � has the niceproperty that

�L�u�u,2L�u�v� � T��u�X* ⇔ ��u,v� = 0.

B. The horizontal bundle HU*

Let V=R4 and

HU* = �u�U*

HuU*, HuU* = ��u,v� � TuU*�v � V,��u,v� = 0� .

Then the restriction of D� to HU* is given by

DH�ª D��HU*:HU* → TX*� ,

DH��u,v� ª ��u�,�uHv� ,

�uHª �u�HuU*:HuU* → T��u�X*� ,

�uHv ª 2L�u�v .

For �u ,v��HU* let



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u� = A��u, v� = A��v .

Since the right action u�u� is linear in u, R�*v=v�. Since A��=A�−��, it follows from �B29�that

v� · u��4� = vTA��TI4A��u = v · u�4�.

Thus

A��HuU* = Hu�U*,

�u�,v�� Hu�U* ⇔ �u,v� � HuU*.

It is obvious that HU* is invariant under G and that each �uH :HuU*→TuX* is an isomorphism.

Thus HU* is the horizontal bundle42 of the principal bundle P= �U* ,X* ,� ,G� and

DH��u,v� = DH��u, v� ⇔ ∃ �� 0,2��,�u, v� = �u�,v�� .

The following corollary is obvious.Corollary 3.4: HPª �HU* ,TX* ,DH� ,G� is a principal bundle.Definition 3.5: A vector field �u ,v�u�� on U* is said to be �-consistent if and only if it is

horizontal and invariant under the free action of the compact group G. That is, �u ,v�u�� satisfies

��u,v�u�� = 0, v�u�� = A��v�u�, u � U*,�� 0,2�� . �3.2�

Let VU*H,� be the collection of all C1 �-consistent vector fields. More precisely

VX* = �Y = �x,y�x�� C1��x,y�x�� TX*� ,

VU*H,� = �V = �u,v�u�� C1��u,v�u�� HU*,v�u�� = A��v�u�,�� 0,2�� .

We write x� to denote any point in �−1�x� when the expression does not depend on our choice.Define

L:VX* → VU*H,�, Y = �x,y�x��

LV = L�Y�, V = �u,

1

2�u�2L�u�Ty��u�� ,

L−1:VU*H,�→ VX*, V = �u,v�u��

L−1

Y = L−1�V�, Y = �x,2L�x��v�x�� . �3.3�

Proposition 3.6: L :VX*→VU*H,� is an isomorphism.

Proof: By �B32�, if Y �VX* then L�Y��VU*H,�. By �B32�, L�x��v�x�� is independent of x�

��−1�x�. Thus L−1�V��VX* for V�VU*H,�. By construction L and L−1 are inverse of each other.�

C. Vector fields on the quotient space †U*‡

A vector field V�VU*H,� gives rise to a unique well defined vector field �V� on the quotient

space �U*�. Conversely, each vector field defined on �U*� gives rise to a unique vector field inVU*

H,�. It follows that the following map is a homeomorphism:

�:VU*H,�→ V�U*� ª ��V��V � VU*

H,��, ��V� = �V� . �3.4�

Proposition 3.8: �Flows on X*, U*, and �U*�� Let xo=L�uo�uo. The following are equivalent.

�1� The map u�uo , t� is the flow of the vector field V�VU*H,� with u�uo ,0�=uo.



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�2� The map u��uo� , t�ª �u�uo , t�� is the flow of the vector field �V��V�U*� with u��uo� ,0�= �uo�.

�3� The map x�xo , t�ª��u�uo , t�� is the flow of the vector field F=L−1�V��VX* with c�0�=xo.

Proof: The proposition is true because HU* is the horizontal bundle of P and � �3.4� is ahomeomorphism. �

Lemma 3.9: Let x�t��X* ,a� t�b, be continuous on �a ,b� and C1 on �a ,b�. Let xo=x�0�and x*=x�b�. For every uo��−1�xo� there is a unique C1 curve u�t� ,a� t�b ,u�a�=uo, and u*ª limt↗b u�t� exists and u*��−1�x*�.

Proof: The existence of u�t��U* ,a� t�b, follows from proposition 3.8. Let us recall thatP= �U* ,X* ,� ,G� is a principal bundle and that G is compact and acts freely on U*. It follows thata curve x�t� is actually a curve x�t�=�−1�x�t��= �u�t�� U*� ,a� t�b with the same smoothnessand limt↗b �

−1�x�t��=�−1�x*��. It follows that d�u�t� ,�−1�x*��→0. Since ��u�t� , u�t��=0, it fol-lows that u�t� has a definite limit u*��−1�x*�. �

D. The second derivative of � :U*\X*

In order to be able to lift systems of the form x=��x ,y� , y=��x ,y� �and the special case when��,y�=y�, we need to define the horizontal bundle of the principal bundle HPª �HU* ,TX* ,DH� ,G�, which we denote by H2U*. To that end, we need to compute D�DH��.First we compute the second derivative of �. Let p�t� be C1 curve in TU* and set c�t�=D��p�t��. Then

p�t� = �u�t�v�t�

, c�t� = D��p�t�� = � L�u�t��u�t�2I40L�u�t��v�t�

,

c�t� = � 2I40L�u�t��u�t�2I40�L�u�t��v�t� + L�u�t��v�t��

.

If �u ,� ;� ,��= �u�to� ,v�to� ; u�to� , v�to��, then �� ,��T�u,v�TU*. Thus

D2��u,v;�,�� =�L�u�u

2I40L�u�v2I40L�u��

2I40�L��v + L�u��, ��,�� T�u,v�TU*. �3.5�

The linear map D2��u,v� :T�u,v�TU*→T�x,y�TX* is not an isomorphism, where x=L�u�u and y=2I40L�u�v. This is why we need to confine our curve p�t�= �u�t� ,v�t�� to HU*.

E. The derivative of DH� :HU*\TX*

Let �u ,v��HU* and �� ,��T�u,v�HU*. Then

�L�u�v�4 = ��u,v� = 0,

��,�� · ��u,v� = u�4� · � − v�4� · � = ��u,�� − ��v,�� = �L��v + L�u��4.

Thus,

T�u,v�HU* = ��u,v;�,��u,v� = 0,u�4� · � − v�4� · � = 0�

= ��u,v;�,��L�u�v��4 = 0,�L��v + L�u��4 = 0� .

It follows that we can drop I40 from the second and fourth vector components of �3.5� and obtain



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D�DH��u,v;�,�� =�L�u�u

2L�u�v2I40L�u��

2�L��v + L�u��, ��,�� T�u,v�HU*. �3.6�

Still we need to confine �u�t� , u�t�� to HU* in order to be able to handle �invert� 2I40L�u��=�.Before we proceed we note that symbolically we can write u�4� ·�−v�4� ·� as a determinant

u�4� · � − v�4� · � = �u�4� �

v�4� �� .

Definition 3.12: Let Y=F=X and

T2X* = X*� Y� F, H2U* = ��u,v��HU*

H�u,v�2 U*,

H�u,v�2 U* = ��u,v;�,�� T�u,v�HU*��u,�� = 0� = ��u,v;�,��u,v� = ��u,�� = 0,u�4� · � − v�4� · �

= 0� ,

DH2��u,v;�,�� ª � DH��u,v��u,v�

H2 ��,�� =�L�u�u

2L�u�v2L�u��

2�L��v + L�u�� .

Lemma 3.13: H2U* is the horizontal bundle (or principal connection) of the principal bundleHPª �HU* ,TX* ,DH� ,G�.

Proof: Once we specify u��−1�x�, we can invert 2L�u�v=y and 2L�u��=� to obtain �v ,��and then solve L��v+L�u��=� /2 for �. Thus, H2U* is the horizontal bundle of HP. �

Lemma 3.14: If �u ,v ,� ,��H2U*, then DH2��u ,v ,� ,��=D2��u ,v ,� ,��.Proof: If �� ,��H�u,v�

2 U*, then �L�u��4=0 and hence 2I40L�u��=2L�u��. Now �3.6� im-plies that DH2��u ,v ,� ,��=D2��u ,v ,� ,�� for �� ,��H�u,v�

2 U*. �

Definition 3.15: Let F= �x ,y ,��x ,y� ,��x ,y�� and �= �u ,v ,��u ,v� ,��u ,v��.

�1� We say that the system � is horizontal if and only if

�:HU* → H2U*.

�2� We say that the system � is G-invariant if and only if

��u�,v�� = A��u,v�, ��u�,v�� = A��u,v�, �� 0,2�� .

�3� We say that the system � is �-consistent if and only if it is horizontal and invariant under theaction of G.

�4� Define

VX*2 = �F:TX* → T2X*�F is C1� ,

VU*H2 = ��:HU* → H2U*�� is C1� ,

VU*H2,� = �� VU*

H2�� is �-consistent� .

�5� Let



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K:VX*2 → VU*

H2,�,

F = �x,y,��x,y�,��x,y�� = K�F� ,

� = �u,v,��u,v�,��u,v�� ,

��u,v� =1

2�u�2L�u�T��x,y� ,

��u,v� =1

2�u�2L�u�T��x,y� − 2L��v� ,

x = L�u�u, y = 2L�u�v .

The inverse K−1 is given by

K−1:VU*H2,�→ VX*

2 ,

� = �u,v,��u,v�,��u,v�� F = K−1�H� ,

F = �x,y,��x,y�,��x�� ,

��x,y� = 2L�x��x�,y�� ,

��x,y� = 2L�x��x�,y�� + 2L�z��y�,

y� =1

2�x�L�x��Ty, z� = ��x�,y��, x� � �−1�x� .

It is obvious that F�VX*2 , if and only if �=K�F��VU*

H2,�.Proposition 3.16: The map K :VX*

2 →VU*H2,�, is a homeomorphism. Moreover, DH2��

=K−1�� for ��VU*H2,�.

Proof: It follows from our construction that K and K−1 are well defined and are inverse ofeach other. �

F. Systems of equations on the quotient space †U*‡

A system ��VU*H2,� is horizontal and invariant under the free action of the compact Lie group

G, and hence gives rise to a system �� which is well defined on the quotient space �U*� and hasthe same smoothness as �. Conversely, each system on �U*� gives rise to a system on VU*

H2,�. Let

V�U*�2 = �� VU*

H2,�� .

Then the following map is a homeomorphism.

�:VU*H2,�→ V�U*�

2 , �� = �� 3.7�

Lemma 3.18: �Flows of systems of equations on X*, U* and, �U*�� Let xo

=L�uo�uo ,��uo ,vo�=0 and yo=2L�uo�vo. Let p= �x ,y� and w= �u ,v�. The following are equivalent.



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�1� The flow of ��VU*H2,� is given by ��wo , t�ª �u�wo , t� ,v�wo , t��, ��wo ,0�=wo.

�2� The flow of ��V�U*�2 is given by ��wo� , t�ª ��wo , t��= ��u�wo , t� ,v�wo , t��, ��wo� ,0�

= �wo�= ��uo ,vo��.�3� The flow of F=K−1��VX*

2 is given by

c�po,t� = �x�po,t�,y�po,t�� ª DH2��wo,t�� = �L�u�wo,t��u�wo,t�,2L�u�wo,t��v�wo,t��, c�0�

= �xo,yo� .

Proof: The Lemma follows from Lemma 3.13 and Lemma 3.14. �

G. Lifting second order equation from X* to U*

A second order equation on X* is a vector field on TX* of the form �x ,y ;y ,��x ,y��. A secondorder equation on U* is a system �u ,v ;v ,��u ,v��. Define the following:

E2X* = ��x,y ;��x,y ;y,�� T2X*� ,

EH2U* = ��u,v;��u,v� � HuU*,��u,�� = 0� = ��u,v;��u,v� = 0,��u,�� = 0� = ��u,v;��u�4� · v

= 0,u�4� · � = 0� .

We call EH2U* the second order horizontal bundle of U*. We say that the second order equation�= �u ,v ,��u ,v�� is �-consistent if and only if it is horizontal and invariant under the action of G.That is, if and only if

� = �u,v;��u,v��:HU* → EH2U*, ��u�,v�� = A��u,v�, �� 0,2�� . �3.8�

Let

EX*2 = �F = �x,y ;��x,y��x,y ;y,��x,y�� E2X*� ,

EU*H2,� = �� = �u,v;��u,v��u,v;v,��u,v�� EH2U*,�3.8� holds� . �3.9�

By abuse of notation, we will continue to call the restriction of K to EX*2 by K. Let K�F�=�. Then

��u,v� =1

2�u�2L�u�T��x,y� − 2L�v�v� ,

x = L�u�u, y = 2L�u�v ,

��x,y� = 2L�x��x�,y�� + 2L�y��y�,

y� =1

2�x�L�x��Ty, x� � �−1�x� . �3.10�

Proposition 3.20: The map K :EX*2 →EU*

H2,� is a homeomorphism. D2��=DH2��=K−1�� for ��EU*

H2,�. Moreover, ��u ,v� as given by (3.10) can be written as

��u,v� =1

2�u�2L�u�T��u�� − 2

�u,v��u�2

v +�v�2

�u�2u . �3.11�

Proof: It follows from our construction that K and K−1 are well defined and are inverse ofeach other. The form �3.11� follows from the identity �1.5�. �



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H. Second order equations on the quotient space †U*‡

A second order equation ��EU*H2,� is horizontal and invariant under the free action of the

compact Lie group G, and hence gives rise to a well defined second order equation on the quotientspace �U*� of the same smoothness which we denote by ��. Conversely, each second orderequation on �U*� gives rise to a second order equation in EU*

H2,�. Let

E�U*�2 = �� EU*

H2,�� .

Then, the following map is a homeomorphism:

�:EU*H2,�→ E�U*�

2 , �� = �� . �3.12�

Lemma 3.22: �Flows of second order equations on X*, U* and, �U*�� The following map is ahomeomorphism:

K:EX*2 → E�U*�

2 , K�F� ª �� K��F� = �K�F�� .

Let xo=L�uo�uo ,��uo ,vo�=0, and yo=2L�uo�vo. Let p= �x ,y� and w= �u ,v�. The following areequivalent.

�1� The map ��wo , t�ª �u�wo , t� ,v�wo , t�� is the flow of the second order equation ��EU*H2,� with

��wo ,0�=wo= �uo ,vo� and hence u�wo , t�=v�wo , t�.�2� The map

c�po,t� = �x�po,t�,y�po,t�� ª DH��wo,t�� = �L�u�wo,t��u�wo,t�,2L�u�wo,t��v�wo,t��

is the flow of the second order equation F=K−1��EX*2 with c�0�= �xo ,yo� and hence

x�po , t�=y�po , t�.�3� The curve ��wo� , t�ª ��wo , t��= ��u�wo , t� ,v�wo , t�� is the flow of the second order equa-

tion ��E�U*�2 with ��wo� ,0�= �wo�= ��uo ,vo��.

Proof: The map �3.12� is a homeomorphism. Thus the lemma is a special case of lemma3.18. �

I. The Kepler problem

In order to proceed with our regularization we need to perform a rescaling in HU* which isequivalent to the rescaling �x�r1/2x� of the McGehee variables. Let �u ,v��HU*. Define

w = 2�u�2v, �u,v� � HU*.

Thus

w = L�u�Ty, y = 2L�u�v .

Corollary 3.24: Let �= �u ,v ,��u ,v��=K�F��EU*H2,�, F= �x ,y ,��x ,y��. Then, straightforward

calculations lead to the following for the Kepler problem (1.1):

w = L�u�T��x,y� +�w�2

2�u�4u, �x,y� = �L�u�u,2L�u�v�, ��u,w� = 0.

The functions ��x ,y� and ��u ,v� are the forces acting on the particle in physical space andparameter space. In our problem the force in physical space is the gradient of a function. We needLemma 3.25.

Lemma 3.25: Let W be at least a C1 function of x�X and perhaps some other variables tobe specified when necessary. Thus ��xW�4= �DxW�4=0. Make the substitution x=��u�=L�u�u in



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W. We will abuse notation and continue to denote the new function by W. Then,

�uW = 2L�u�T�xW and ��u,�uW� = 0. �3.13�

Proof: Recall that the column vector �xW= �DxW�T. Thus �DxW�I40=DxW since �DxW�4=0,

DuW = �DxW��u = 2�DxW�I40L�u� = 2�DxW�L�u� .

The second assertion follows from the first assertion and Lemma 1.8. �

Proposition 3.26: �One binary in KS variables�. Consider a 3D perturbed Kepler problem ofone binary with the force,

��x� = −a2

2�x�3x + f�x�, f�x� = − �xW, � =

1

2��y�2 −

a2

�x� .

Then with x=��u� we have

��u��v = −a2

2�u�6L�u�u − �xW, L�u�T��u��v = −

a2

2�u�4u −

1

2�uW .

In the KS variables the 3D perturbed Kepler problem (where the energy � is not a constant) isgiven by

u =1

2�u�2w, w =

�

�u�2u −

1

2�uW, � = −

1

2�u�2�w,�uW�, � =

�w�2 − a2

2�u�2, ��u,w� = 0.

Proof: Lemma 3.25 implies that

L�u�T��x� = L�u�T��u�� = L�u�T�−a2

2�u�6L�u�u − �xW� = −

a2

2�u�4u −

1

2�uW .

It follows from Corollary 3.24 that

w =�w�2

2�u�4u + L�u�T��x� =

�w�2

2�u�4u −

a2

2�u�4u −

1

2�uW =

�

�u�2u −

1

2�uW .

As for the intrinsic energy �, we have

� = − yT�xW = −1

�u�2wTL�u�T�xW = −

1

2�u�2�w,�uW� . �3.14�

The identity �3.13� implies that ��u , w�=0. �

Remark: The 3D Kepler problem can be real analytically block regularized, and hence col-lectively analytically continued, by using a rescaled time

d

d= �u�2

d

dt.

IV. THE KS TRANSFORMATION FOR M BINARIES

The KS transformation for M binaries �Definition 4.15� is the product of M one-binarytransformations. The spaces we use are the product of M copies of the one-binary space. The KStransformation is defined in a block-diagonal fashion. Moreover all the definitions, statements, andproofs are obtained from those of Sec. III by adding a subscript j=1, . . . ,M.



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A. The spaces

We define the following product space:

U j = V j = M j = U = R4,

X j = Y j = F j = X = R3� �0� � R4, j = 1, . . . ,M ,

x = �x1,x2, . . . ,xL�T � X, u = �u1,u2, . . . ,uL�T, . . . ,

X* = �j=1

M

Xj*, Y = �

j=1

M

Y j, F = �j=1

M

F j ,

TX* = �j=1

M

TXj* = �

j=1

M

Xj*� Y j = X*�Y ,

T2X* = �j=1

M

T2Xj* = �

j=1

M

Xj*� Y j� F j = X*�Y� F ,

U = �j=1

M

U j, U* = �j=1

M

Uj*, V = �

j=1

M

V j, M = �j=1

M

M j ,

HU* = �j=1

M

HUj* = �

j=1

M

��uj,v j� � HUj*��uj,v j� = 0� ,

H2U* = �j=1

M

H2Uj*, H2U

j* = �

�uj,vj��HUj*H�uj,vj�

2 Uj*,

EH2U* = �j=1

M

EH2Uj*, EH2U

j* = ��uj,v j,� j��uj,v j� � HuU j

*,��uj,� j� = 0� .

B. The map

Define the following:

L�u� = blockdiag �L�u1�,L�u2�, . . . ,L�uL�� ,

�:U* → X*,

��u� = x = L�u�u = ��1�x1�,�2�x2�, . . . ,�M��xM� ,

� j:U j* → X

j*, � j�uj� = ��uj� = L�uj�uj, j = 1, . . . ,M . �4.1�



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C. The fibration of U*

Let u��−1�x�. For j=1, . . . ,M, let uj=uj

�4� be as in Lemma 1.9. Define the following:

tu = �u1,u2

, . . . ,uL�T,

l�u,v� = ��u1,v1�,��u1,v1�, . . . ,��u1,vL�� = �u1

T v1,u2

T v2, . . . ,uM

T vM� ,

TM = �j=1

M

Gj, Gj = G = S1, j = 1, . . . ,M .

Corollary 4.4: Let y�Y and u�U. Then, by Lemma 1.8 we have l�u ,L�u�Ty�=0.Proposition 4.5: The following are true.

�1� Let x�y be two points in X*. Then �−1�x��−1�y�=�.�2� Let u��−1�x� be fixed but arbitrary. Then

�−1�x� = �u = A��u�0� � j� 2�, j = 1, . . . ,M� = �j=1

M

� j−1�xj�

is an M-torus where = ��1 ,�2 , . . . ,�M�.�3� For any v�V or U we have

��uj,v j� = 0, j = 1, . . . ,M� ⇔ �l�u,v� = 0� ⇒ ��tu,v� = 0� �4.2�

and hence v is orthogonal to the tangent space Tu�−1�x�.�4� Notice that

�j=1

L

��uj,v j� = �tu,v� = 0 ⇏ l�u,v� = 0.

�5� The tangent space to �−1�x� at u is

Tu�−1�x� = �

j=1

M

TujU

j* = ��u,v��l�u,v� = 0� .

D. The principal bundle Qª„U* ,X* ,� ,TM…

It is obvious now that Q is the product principal bundle

Q = �j=1

M

P j, P j = �Uj*,X

j*,�,Sj

1� ,

Uj* = U*, X

j* = X*, Sj

1 = S1, j = 1, . . . ,M .

with horizontal bundle HU*. All the results for the principal bundle P= �U* ,X* ,� ,S1� can beextended to Q in the usual fashion.

E. Lifting vector fields to U*

Let u =A� �u, v =A� �v and let L be given by �3.3�. Define

VX* = �Y = �x,y�x�� C1��xj,yj�xj�� TXj*, j = 1, . . . ,M� ,



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VU*H,� = �V = �u,v�u�� C1��u,v�u�� HU*,v�u � = A� �v�u�� ,

L:VX* → VU*H,�, L = �L1, . . . ,LM� ,

L j = L:VXj*

H → VUj*

H,�, j = 1, . . . ,M .

Proposition 4.8: In view of Proposition �3.6�, the map L :VX*→VU*H,� is an isomorphism.

F. Second order equations to U*

Let F= �x ,y ;y ,��x�� and = �u ,v ;v ,��u ,v��. In view of �3.9� we define the following:

EX*2 = �F � C1��x,y;y,��x�� T2X*� ,

EU*H2,� = � � C1��u,v;v,��u,v�� EH2U*,��u ,v � = A� ��u,v�� .

We call a second order equation ��EU*H2,� �-consistent. Define

K:EX*2 → EU*

H2,�, K = �K1, . . . ,KM� ,

K j = K:EXj*

2 → EUj*

H2,�, j = 1, . . . ,M , �4.3�

where K and K−1 are given by �3.10�. It is obvious that K sends a second order equation on X*

to a �-consistent second order equation on U*.Proposition 4.10: In view of Proposition 3.16, the product map K :EX*

2 →EU*H2,� is an iso-

morphism.

G. The quotient spaces

Since we are working with product spaces and TM acts diagonally on both U* and HU*, thequotient spaces are also the product spaces,

�U*� = U*/TM = �j=1

M

�Uj*� = �

j=1

M

Uj*/G ,

�HU*� = HU*/TM = �j=1

M

�HUj*� = �

j=1

M

HUj*/G .

Notice that H2U* is the horizontal bundle of HU*, and we will call it the second horizontalbundle of U*.

H. Second order equations on the quotient space †U*‡

Let

E�U*� = �� EU*H2,�� = �

j=1

M

E�Uj*�.

Proposition 4.13: �Solutions of second order equations on X*, U*, and �U*�� Let K be themap given by (4.3). The following maps are homeomorphisms:



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�:EU*H2,�→ E�U*�, �� = �� ,

K:EX*2 → E�U*�, K�F� ª �� K��F� = �K�F�� .

Let xo=L�uo�uo, yo=2L�uo�vo, w= �u ,v� and p= �x ,y�. The following are equivalent.

�1� ��wo , t�ª �u�t� ,v�t��, ��wo ,0�=wo= �uo ,vo� is a solution of �EX*2 , and hence u�wo , t�

=v�wo , t�.�2� c�po , t�ª �x�po , t� ,y�po , t��= �L�u�wo , t��u�wo , t�, 2L�u�wo , t��v�wo , t��, c�0�= �xo ,yo� is a

solution of F=K−1��EU*H2,�, and hence x�po , t�=y�po , t�.

�3� ��t�ª ��u�wo , t� ,v�wo , t��, ��wo� ,0�= �wo� is a solution of ��E�U*�.

Proof: The map � :EU*H2,�→E�U*� is a homeomorphism since second order equations in

mcEU*H2,� are �-consistent. In view of Proposition 4.10, the map K=� �K :EX*

2 →E�U*� is anisomorphism. The rest of the proposition follows from Lemma 3.22. �

Proposition 4.14: Denote the second order equation (2.2) by FSBC. Then, FSBC�EX*2 and is

mapped by K to a unique second order equation in SBC�EU*H2,�. After performing the rescaling

wj = 2�uj�2v j, j = 1, . . . ,M , �4.4�

the vector field SBC takes the following form with k=1,2 , . . . ,M:

uk =1

2�uk�2wk, wk =

�k

�uk�2uk + L�uk�Tfk�L�uk�uk� =

�k

�uk�2uk −

1

2�uk

W ,

�k =�wk�2 − ak

2

2�uk�2,

y = Y, Y = − �yW, z = Z, Z = − �zW . �4.5�

Proof: The assertion follows from Proposition 3.26 since the map � is the product M KSMs�. �

Definition 4.15: �KS transformation and variables for M binaries�

• The KS transformation for M binaries consists of the map K :EX*2 →EU*

H2,� followed by therescaling (4.4).

• The variables �u ,w ,y ,Y ,z ,Z� are called the KS variables for M binaries.

V. THE GKS TRANSFORMATION

It follows from Propositions 3.26 and 4.10 that the second order equation �2.2� is lifted in ameaningful and unique way to the �-consistent second order equation �4.5� which belongs toEU*

H2,� and is the product of M second order equations of the type �3.14� in the variables,

u = �u1,u2, . . . ,uM�, w = �w1,w2, . . . ,wM�, l�u,w� = 0.

For the rest of this work we will always assume that l�u ,w�=0.Definition 5.1: Define the following for u with uk�0,k=1, . . . ,M:



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�k =uk

�uk�, �k = wk/�wk� = ak

2 + 2�k�uk�2, �k =�uk�

� j=1M �uj�2

, k = 1, . . . ,M . �5.1�

Lemma 2.4 tells us that on collision and ejection orbits each �k has a finite limit and hence �wk�→ak. In order to reduce the order of the singularity in each wk from �uk�−4 to �uk�−2, we replaceeach wk by �k and �k. In the following Lemma 5.2 we translate the asymptotic behavior describedin Lemma 2.4 to the KS variables.

Lemma 5.2: �Asymptotic behavior in KS variables� Recall quantities defined by (2.4), (2.5),(2.8), and (2.8). The following are true on collision and ejection orbits for k , j=1, . . . ,M.

�1� �k ,�k ,wk ,�k, have finite limits �k* ,�

k* ,w

k* ,�

k* satisfying

�k → �k* � �0,1�,

1

�uj�uk →

�k*

�j*�

k*,

wk → L��k*�T�

k* � 0, �wk� → ak,

�k = wk/�wk� →1

akL��

k*�T�

k*.

�2� Since each �k*� �0,1�, no binary collapses asymptotically faster than any of the others.

�3� The relative direction of each two binaries has a well defined limit in the sense that

Sk�j� =

1

�uj�2L�uj�Tuk →

�k*

�j*

L��j*�T�

k* � 0, k � j ,

1

�uj�2L�uj�uk →

�k*

�j*

L��j*��

k* � 0, k � j . �5.2�

Proof: Lemma 2.4 tells us that k*� �0,1�. Thus

�k = �xk�� j=1

M �xj�→ �

k* � 0, k = 1, . . . ,M ,

and since ��k2=1, each �

k*� �0,1�.

To prove the assertion about �k�t� we consider a collision orbit with t↗ t* ,0� t� t*. ByLemma 2.4, sk�t�→s

k* as t↗ t*. Hence, we have a continuous curve sk�t� ,0� t� t* which is also

C1 except perhaps at t= t*. Now the assertion follows from Lemma 3.9.As for wk, we have by Lemma 2.4

wk = L�uk�Txk = L��k�T�xk�1/2xk → L��k*�T�

k* � 0.

Since �k has a finite limit, �wk�=ak2+2�xk��k→ak, and hence �k=wk / �wk�→ak

−1L��k*�T�

k*. For 1

� j ,k�M, we have

1

�uj�uk =

�uk��uj��uk�

uk =�k

� j�k →

�k*

�j*�

k*.

The last two assertions follow immediately from the ones we proved. The assertions for ejectionorbits follow by reversing time. �



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A. The neighborhood

We would like to define the neighborhood of a SBC singularity which we use for the rest ofour work except for making it smaller. We would like also to take advantage of Lemma 5.2 tomake some choices that simplify our work but does not cost us any loss of generality. Fix asingularity point po= �uo ,wo ,Bo�, uo=0, B= �y ,Y ,z ,Z�. In view of Lemma 5.2 we may assumethat the following holds on the collision orbit that ends at po:

wjo � 0, j = 1, . . . ,M ,

�oª �1

1,o 0, �11,o � 0,

�14,o

�11,o 0,

Sk�1�,o � 0, k = 2, . . . ,M . �5.3�

Definition 5.4: �The GKS variables� Let �·� denote the standard Euclidean norm and �·� denotethe box norm. Recall definitions (2.12) and (2.18).

�1� Let B= �y ,Y ,z ,Z� and define

HU** = ��u,w� � HU*�wj � 0, j = 1, . . . ,M� = ��u,w��uj,wj� = 0,uj � 0,wj � 0, j

= 1, . . . ,M� .

�2� In HU**, �� ,T ,S ,� ,�� are well defined. Define the following for sufficiently small � 0:

N� = ��u,w,B� � HU**� R6K+6M��u�� ,�� − �o�� ,�B − Bo�

� ��T − To�� ,�S − So�� ,�� − �o�� , �5.5� holds� . �5.4�

�3� Take � 0 sufficiently small so that for some c 0 the following holds in N�:

�zi − zj� c, i � j, �yl − ym� c, l � m, �zi − ym� c ,

� = �11 c 0, � = u1

1 � 0, T3 =u1

4

u11 c 0,

�Sk� �Sk

o�2

, �wj� = aj2 + 2� j�uj�2 aj/2. �5.5�

�4� The GKS variables in a neighborhood of a SBC singularity are defined as

�u,�,�,B� = �u,�,�,y,Y,z,Z� .

�5� Let p= �u ,� ,� ,B� and define the following:

S2Uj* = ��uj,� j� � Huj

Uj*�� j� = 1�, j = 1, . . . ,M

S2U* = �j=1

M

S2Uj* = ��u,�� U*�T�l�u,�� = 0� ,

R = R6K� R6M, T = �k=1

M

Sk3, �5.6�



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N�K = �p � S2U*� RM�R��u�� ,�� − �o�� ,�B − Bo�� ,�T − To�� ,�S − So�

� �,�� − �o�� , �5.7�

where each Sk3=S3, the 3D unit sphere of R4.

�6� The GKS transformation is given by

K:N�K → N�, �u,�,�,B� � �u,w,B�, B = �y,Y,z,Z� .

The following corollary follows immediately from the definitions since wj�0, uj�0, j=1, . . . ,M in N�.

Corollary 5.5:

�1� The set N� is open in both HU* and HU**. Actually, HU** is open in HU*.�2� The set N�

K is open in S2U*�RM�R.�3� The GKS transformation K :N�

K→N� is a real bianalytic map.�4� It follows that (5.5) holds in N�

K.

B. Equations of motion in the GKS variables

Straightforward calculations show that in N�K the vector field �4.5� takes the form

uk =�wk�

2�uk�2�k, �k =

�k

�wk��uk�2gk +

1

�wk�Gk,

�k =1

�uk�2�wk,Fk�, �wk =

�k

2�uk�2uk + Fk ,

gk = uk − �uk,�k��k, Gk = Fk − �Fk,�k��k,

�wk�2 = ak2 + 2�uk�2�k,

Fk = − 12�uk

W = − L�uk�T��k�u�k�,y,z�L�uk�uk + hk�u,y,z�� ,

hk�u,y,z� = O��uk�4�, k = 1,2, . . . ,M . �5.8�

We do not need wk since we replace it by �k and �k. We include it in parentheses for completenessand to show that the order of the singularity has indeed been reduced from �uk�−4 to �uk�−2 even inwk.

The vector field �5.8� has no singularities in N�K. However, collision orbits leave N�

K at u=0.

Corollary 5.7: �A classical result� A single binary collision is real analytic block regulariz-able.

Proof: We work with the GKS variables and study the vector field �5.8� with M =1 near u1

=0. We introduce a new time � by d /d�= �u1�2�d /dt�. This yields a real analytic vector field. TheGKS transformation is real analytic away from the singularity. More over, the new vector fielddoes not vanish at u1=0. Thus, the section map from collision and near collision orbits to ejectionand near-ejection orbits is real bianalytic. Thus, the singularity is real analytic block regulariz-able. �

VI. THE PROJECTIVE KS TRANSFORMATION

In this section we define a transformation that we call the KSPT near a SBC singularity. Thepurpose of this transformation is to blow up the SBC singularity and separate collision and



131.181.251.130 On: Fri, 21 Nov 2014 21:36:41

ejection orbits from nearby near-collision and nearby near-ejection orbits. A near-collision orbit isan orbit that is close to a collision orbit of our M binaries and may be a collision orbit of asubsingularity of a smaller number of these M binaries. In physical space these orbits accumulateon collision orbits.

We used the asymptotic limits of Lemma 5.2 to focus on a neighborhood of the type �5.5� ofa fixed but arbitrary SBC point such as �5.3� and produce Corollary 5.5. These facts will allow usto blow up the SBC singularity and lift the vector field �5.8� from N�

K to a subset of

P ª R� RP3� UPM−1�T� RM�R .

An essential element in the definition of the KSPT is the existence of the KS projective spaceUPM−1 defined in Definition 2.10 by means of the matrix L�u�T. The very first component of thevector field �5.8� is

u11 =

�w1��11

2�u11�2�1 + �T�2�

, T =1

u11 �u1

2,u13,u1

4� .

Definition 6.1: Let �t ,u ,� ,� ,B��R�N�K. Recall that we take � 0 sufficiently small so that

(5.5) holds in N�. By Corollary 5.5, (5.5) holds also in N�K. Recall that in N�

K ,wj�0, j=1, . . . ,M.

�1� The KS projective variables are given by Definition 2.13.�2� Define the following using (2.18) and (5.2) with j=1:

pL � R� RP3� UPM−1�T� RM�R, B = �y,Y,z,Z� ,

pL = ��,T,S,�,�,B� = �uL,�,�,B�, uL = ��,T,S� � R� RP3� UPM−1,

�uL�# = ��1 + �T�2 max�1,�S��, �S� = max��S2�, . . . , �SM�� = ��E�max�1,�S�� .

To describe pL�N�K� explicitly we need to translate the conditions ��uj ,� j�=0 to the space P.

Corollary 6.2: Straightforward calculations yield the following in N�K for sufficiently small

� 0:

�u1� = ��E� = ��1 + �T�2, uk = L�u1�Sk = �L�E�Sk,

�uk� = ��E��Sk� = ��1 + �T�2�Sk�, �u� = �uL�# = ��E�max�1,�S�� ,

��u1,�1� = ��E,�1�, ��uk,�k� = ��L�E�Sk,�k�, k = 2,3, . . . ,M . �6.1�

For ��0, we have

��E,�1� = 0 ⇔ ��u1,�1� = 0, ��L�E�Sk,�k� = 0 ⇔ ��uk,�k� = 0, k = 2,3, . . . ,M .

A. A KS-projective chart

Define

S* = �S � �R4�M−1�Sk � 0,k = 2, . . . ,M� ,

HT,S* = ��T,S,�� R3�S*�T��E,�1� = 0,��L�E�Sk,�k� = 0,k = 2, . . . ,M� ,

M* = ��T,S��R3�S*

HT,S* , P* = R�M*� RM�R ,



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U�L = �pL � P*��uL�#� �,�� − �o�� ,�B − Bo�� ,�T − To�� ,�S − So�� ,�� − �o�� ,

��L = �pL � U�L�� = 0�, N�

L = U�L \��L. �6.2�

We call ��L the blown up singularity. In view of �5.5�, we have

��,T,S,�,�,B� � N�L ⇔ �u,�,�,B� � N�

K ⇔ �u,w,�,B� � N�.

Define a KS projective chart by

P:N�L → N�

K, �uL,�,�,B,� � �u,�,�,B,t� . �6.3�

The KS-projective chart blows up the SBC singularity near po to ��L�U�L, po is given by �5.3�.

Since �=u11 and each Sj = �u1�2L�u1�Tuj , j!2, we adopt the following notation temporarily:

u1,1 = uL, N�1,1 = N�, N�

K;1,1 = N�K,

U�L;1,1 = U�L, N�L;1,1 = N�

L, ��L;1,1 =��

L,

P1,1:N�L;1,1 → N�

K;1,1, P1,1�u1,1,�,B� = P�uL,�,B�, 1,1 = .

B. The KSPT

If instead of �2.18� we set

� = u12, � = �u1

1,u13,u1

4� ,

we can define in a similar fashion another chart P1,2 on an open set N�K,1,2→N�

L,1,2 for j=1 andr=2. Proceeding in this fashion, we obtain a collection of charts

C� = �P j,r:N�L;j,r → N�

K;j,r�1� j�M,1� r� 4� . �6.4�

The KSPT consists of the collection of real bianalytic charts and time rescalings,

P j,r:N�L;j,r → N�

K,j,r,

d

d j,r= −

2�ujr�3�1 + �Tj,r�2�

�wj�d

dt, j = 1,2, . . . ,M, r = 1,2,3,4. �6.5�

C. Transition functions for the KSPT

The transition functions for the spatial part of the transformation follow from �2.11� and forthe time rescaling from �6.5�,

S�j� =1

�uj�2L�uj�TL�ui�S�i� =

�i

� jL�� j�TL��i�S�i�, i � j , �6.6�

di,r

d j,s=

�wi��wj�

� �ujs�

�uir� 3� 1 + �Ti,r�2

1 + �Tj,s�2 , 1� i, j�M, i � j, 1� r,s� 4. �6.7�

Lemma 5.2 shows that the transition functions have finite limits on collision and ejection orbits.As for the transition function for the time rescaling �6.7� recall that each wk→ak. Moreover



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�ujs�

�uir�

= ��L�Ei,r�Sj�i��s�, Ei,r =

1

uirui � �1,Ti,r� .

Once we show that each Tk,r has a finite limit we will have proven that the transition functions forthe time rescalings have finite limits on collision and ejection orbits which is given in Theorem 6.

Remark: The analyses on different charts are similar. Therefore we work with �N�L;1,1 ,P1,1�

and go back to our simplified notation for P=P1,1.Corollary 6.7: The map P :N�

L→N�K is real bianalytic.

Proof: It follows from �6.3� that P is a homeomorphism. That P is real bianalytic followsfrom definitions and the fact that ��0, uj�0, wj�0, j=1, . . . ,M, for �u ,� ,� ,B��N�

K. �

D. The KS-projectivized vector field „6.8… and „6.9…

It consists of a collection of vector fields, one for each chart. It is enough to study one of themsince they are all similar. We denote differentiation with respect to by a prime “ �.” In KSPvariables the vector field �5.8� takes the following form:

�� = − ��, T� = �T − � ,

�� = −2�

�w1�2��1g1

1 + �2�1 + �T�2�G11�, �� = −

2�

�w1�2��1g1 + �2�1 + �T�2�G1� ,

�1� = − 2�3�w1��1,F1�, �1 = ��;�� ,

�w1� = a12 + 2�1�

2�1 + �T�2�, g1 = �g11; g1�, G1 = �G1

1;G1� . �6.8�

For k=2,3 , . . . ,M we have the following where g and G are given in �5.8�:

Sk� = −1

�Sk�2�1 + �T�2��wk��k − �w1��Sk�2L��1�Sk� ,

�k� = −2�

�w1��wk�2�Sk�2��kgk + �2�1 + �T�2��Sk�2Gk� , �6.9�

�k� = −2��wk�

�w1��Sk�2��k,Fk�, �wk� = ak

2 + 2�k�2�1 + �T�2��Sk�2. �6.10�

The time rescaling �2.18� resulted in reversing the direction of the vector field on ejection orbitswhere � 0. The choice �=�1,1=u1

1 in �2.18� uniquely determines � and � since uk

= ��wk� /2�uk�2��k, as seen by �5.8�. The following corollary is obvious.Corollary 6.9: Let � be small enough such that (5.5) holds.

�1� �6.8� and �6.9�-are real analytic in U�L (6.2) and leaves the blown up singularity ��L is

invariant.�2� Since a similar conclusion holds in every chart, the SBC singularity near po has been blown

up to a collision manifold given by the collection ��ª ��L;j,r�, where ��

L=��L;1,1.

�3� The KS-projectivized vector field (6.8) and (6.9) is real analytic near the collision manifold�� and leaves it invariant.

E. Rest points of the KS-projectivized vector field

At a rest point of �6.8� and �6.9�, �=�T. If �=0, then �=0, which would imply that 1= ��1�= �� ,��=0. Thus �=0 and ��0. �The requirement �5.5� includes the inequality � c



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0.� The other components except Sk-components have a factor of �. Now Sk�=0 tells us that�ak /a1��k= �Sk�2L��1�Sk. Since ��1�=1, L��1�TL��1�= I. Thus �Sk�= �ak /a1�1/3. It follows that restpoints are parametrized by �T ,S� as follows:

� = 0, � =1

1 + �T�2, 1 = ��1� = �1 + �T�2,

�1 = ��1,T� = �E = ��,�� ,

�k = �a1

ak 1/3

L��1�Sk = ��a1

ak 1/3

L�E�Sk, k = 2, . . . ,M ,

�Sk� = � ak

a1 1/3

, k = 2, . . . ,M, �wj� = aj, j = 1, . . . ,M . �6.11�

Since L�u1�Tu1= �u1�2k1, if we set S1ª �u1�−2L�u1�Tu1=k1 we have L�E�S1=E, and we can write�k=��a1 /ak�1/3L�E�Sk ,k=1, . . . ,M. However, we will continue to distinguish �1 and u1 from therest for obvious reasons.

F. Blowing up the singularity via the KS-projective variables pL= „ ,T ,S,� ,� ,B…

We saw in Sec. VI A that the KSPT blows up the SBC singularity to the set ��L�U�L. We will

see later that collision and ejection orbits constitute the stable manifold of the rest points. The restpoints of �6.8� and �6.9� form a proper subset of ��

L given by �where P* is given in �6.2��

��L,o = �pL ��

L��1 = �E,�k = ��a1/ak�1/3L�E�Sk,k = 2, . . . ,M,� = �1 + �T�2�−1/2� = �pL � P*��

= 0,�� − �o�� ,�B − Bo�� ,�� − �o�� ,�1 = �E,�k = ��a1/ak�1/3L�E�Sk,k = 2, . . . ,M,�

= �1 + �T�2�−1/2,�T − To�� ,�S − So�� .

Definition 6.12: We call ��L the blown up SBC singularity and ��

L,o the lifted SBC singularity.

G. Rest points as a graph

The set of rest points ��L,o is the graph of the real analytic map fL,

��ooª ��T,S,�,B� � R3�S*� RM�R�,�T − To�� ,�S − So�� ,�� − �o�

� �,�B − Bo�� ,

fL:��oo → �0��T � R�T, �T,S,�,B��

fL

��,�� = �0,�� ,

�1 = �E, �k = ��a1/ak�1/3L�E�Sk, k = 2, . . . ,M ,

� = �1 + �T�2�−1/2, E = �1,T� .

It follows that graph�fL�=��L,o. Moreover, we have a real bianalytic chart for ��

L,o given by

"L:��oo →��

L,o, �T,S,�,B��"L

�0,T,S,�,�,B�, �0,�� = fL�T,S,�,B� . �6.12�



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H. Extending the KSPT to all of UεL

Define the projection

�:U�L → S�K ª ��U�L� � U�T� RM�R ,

pL = ��,T,S,�,�,B� � ��pL� = �u,�,�,B� ,

u1 = �E = ��1,T�,uk = �L�E�Sk, k = 2, . . . ,M . �6.13�

We call � the extended KSPT.

I. Blowing up the singularity via the GKS variables „u,� ,� ,B…

We can blow up the SBC singularity from the physical variable to the KS-projective variables�� ,T ,S ,� ,� ,B� skipping the intermediary GKS variables. Then we project down ��

L and ��L,o to

obtain the blown up singularity in the GKS variables. Let

��Kª ��

L� � S�K = ��U�L�, ��K,o

ª ��L,o� ��

K.

The following corollary follows from the definitions by inspection.Corollary 6.16: The following are true for sufficiently small � 0.

�1� With M* as given in (6.2), we have, with p= �u ,� ,� ,B�,

��K = �p � U�T� RM�R��u� = 0,�� − �o�� ,�B − Bo�� ,�� − �o�

� �, ∃ �T,S�, with �T − To�� ,�S − So�� , such that �T,S,�� M*� ,

��K,o = �p � U�T� RM�R��u� = 0,�� − �o�� ,�B − Bo�� ,�� − �o�� ,�1

= �E,�k = ��a1/ak�1/3L�E�Sk,k = 2, . . . ,M,� = �1 + �T�2�−1/2,E

= �1,T�,�T − To�� ,�S − So�� . �6.14�

�2� With N�K as given in (5.7), we have

S�K = N�K ��

K, N�K ��

K = � . �6.15�

�3� The subset ��K,o=graph�fK�, where fK is a real analytic map defined as follows:

fK:��oo → U�T, �T,S,�,B��

fK

�u,�� ,

u = 0, �1 = �E, � = �1 + �T�2�−1/2, E = �1,T� ,

�k = ��a1/ak�1/3L�E�Sk, k = 2, . . . ,M .

�4� We also have a real bianalytic chart for ��K,o given by

"K:��oo →��

K,o, �T,S,�,B��"K

�u,�,�,B� = �0,�,�,B�, �0,�� = fK�T,S,�,B� .

�6.16�

�5� Let P be the KSPT given by (6.3). The restriction ��N�L is real bianalytic and

��N�L:N�

L� → N�K, � � P−1 = idN

�K. �6.17�

�6� The restriction



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��L,o� = "K � "L−1:��

L,o →��K,o �6.18�

is a real bianalytic map. In fact, if we use the two charts "L and "K, the map ��L,o :��L,o

→��K,o is represented by the identity. That is

"K−1 � � � "L�T,S,�,B� = �T,S,�,B�, �T,S,�,B� ��

oo.

�7� The set ��K is not necessarily a graph of a function since there can be two points �T ,S ,��

and �T ,S , �� in M*.�8� ��

L is invariant under the vector field (6.8) and (6.9).

We can think of the set ��K as a fan, so to speak, attached to N�

K at u=0, and made up of allpossible values of � that can be seen at a rest point, equivalently, � can be attained on the limitof a collision orbit. Since �k=��a1 /ak�1/3L�E�Sk ,k=2, . . . ,M, we can see that ��

K is made up of allpossible limiting directions of collision orbits.

J. Notation

We will consider the restriction of � to the subsets of N�L, ��

L, and ��L,o. To keep notation

under control, we continue to denote these restrictions by � as long as there is no ambiguity.

K. Linearization of the vector field „6.8… and „6.9… at rest points

The rest points of the vector field �6.8� and �6.9� constitute the real analytic submanifold ��L,o.

To compute the linearization at rest points, we group the variables as follows:

pL = ��,T,S,V�, S = �S2,S3, . . . ,SM�, V = ��,�,y,Y,z,Z� = ��,�,B� .

Recall �5.5� and that �=�11 0. Straightforward computations show that the linearization is given

by

A =�− � 0 � �

0 �I3 � �

0 0 B �

0 0 0 0�, B =�

B2 0 0 0

0 B3 0 0

0 0 � 0

0 0 0 BM

� ,

Bk = a1��I4 + Nk�, Nk = � ak

a1 −2/3

SkSkT = L��1�T�k�k

TL��1� .

The first eigenvalue is �=−��0 with eigenspace in the �-axis. Then we have an eigenvalue �=� 0 with multiplicity 3 with eigenspace in the T-directions. It remains to compute the eigen-values of each Ck= I4+Nk. But # is an eigenvalue of Nk if and only if �=1+# is an eigenvalue ofCk. Moreover, we have

Nk = bkbkT, bk = L��1�T�k, Nk

ij = bki bk

j ,

where superscripts refer to components. It follows that

bkTbk = �bk�2 = �k

TL��1�L��1�T�k = 1, bkbkTbk = bk.

Thus, Nk has an eigenvalue #1=1 with eigenvector Ek,1=bk. Moreover, any vector v with bk{v

=0, that is, any v in the three-dimensional hyperplane normal to bk, satisfies



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bkbkTv = 0.

Hence, Nk has an eigenvalue #2=0 with multiplicity 3 and eigenspace the hyperplane �v �bk{v

=0�. It follows that the eigenvalues of Ck are �=1+#=2,1 ,1 ,1. Since a1 ,� 0, the eigenvaluesof Bk are

�2a1�,a1�,a1�,a1��

and they are all positive. We have proven the following proposition.Proposition 6.20: The following are true for sufficiently small � 0.

�1� The vector field (6.8) and (6.9) is real analytic in U�L which is an open neighborhood of theblown up SBC singularity ��

L.�2� The lifted SBC singularity ��

L,o is a normally hyperbolic submanifold consisting of all therest points.

�3� Each point in ��L,o is a partially hyperbolic rest point. Its zero-center manifold is ��

L,o itself.All the other eigenvalues are the same for all the rest points and are as follows.

�a� �−1� with multiplicity 1 and eigenspace in the �-direction.�b� �=�1

1 0 with multiplicity 3 and eigenspace in the T-direction.�c� For each k=2,3 , . . . ,M, there are four positive eigenvalues

�2a1�,a1�,a1�,a1��, � = �11 0,

with eigenspace in the Sk direction. The eigenspace of 2a1� is in the direction of

ck = �0, . . . ,0,bk,0, . . . ,0�, bk = L��1�T�k.

The three-dimensional eigenspace of a1� is the hyperplane determined by ck{vk=0, where

vk = �0, . . . ,0,vk,0, . . . ,0�, bkTvk = 0.

This is a three-dimensional subspace of the four-dimensional Sk-direction.�d� These positive eigenvalues add up to 4�M −1� positive eigenvalues.�e� The remaining eigenvalues are zeros with eigenspace in the direction of V

= �� ,� ,y ,Y ,z ,Z�. There are M +6K+6M of them. Moreover, the corresponding zero-center manifold is ��

L,o itself.

We describe the stable manifold of ��L,o in Proposition 6.23 which follows from Proposition 6.20.

Proposition 6.21: By Proposition 6.20, the following are true for sufficiently small � 0.

�1� Each point in the lifted SBC singularity ��L,o has a one-dimensional real analytic local stable

manifold in the direction of the variable �.�2� ��

L,o is normally hyperbolic.�3� The local stable manifold of ��

L,o, which we denote by WL=W��L,o�, is real analytic.

Definition 6.22: Write the local stable manifold WL as the disjoint union,

WL = RCL ��L,o � RE

L , �6.19�

where a square cup � denotes a disjoint union, RCL is the part on which �↗0, RE

L is the part onwhich �↘0, and ��

L,o is the lifted collision singularity given by �=0.Proposition 6.23: The local stable manifold WL is the graph of a real analytic map from an

open neighborhood O of ��o ,Vo�= �0,Vo� given by

T = ��,V� � 0, Sk =�k��,V� � 0, k = 2,3, . . . ,M ,

where �, �= ��2 ,�3 , . . . ,�M�, are real analytic in �� ,V� and do not vanish. Thus, we have a realbianalytic chart for WL given by



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�L:O → OLª �L�O� = WL,

�L��,V� ª ��,T,S,V� = ��,��,V�,��,V�,V� . �6.20�

Proof: The proposition follows immediately from Proposition 6.20. �

Now the following Proposition 6.24 is obvious.Proposition 6.24: In the real bianalytic coordinate chart �L :O→WL, after rescaling time so

that ��=1, the restriction of the vector field �6.8� and �6.9� to WL takes the form

�� = 1, V� = GL��,V� , �6.21�

where GL�� ,V� is real analytic. Choose � small and set

CL = ��,T,S,V� � RCL �� = − ��, EL = ��,T,S,V� � RE

L�� = �� .

It is obvious that CL and EL are real analytic because RCL and RE

L are. In view of �6.21�, CL�RCL

has codimension 1 in RCL . Similarly, EL�RE

L has codimension 1 in REL.

The real analytic flow of the vector field �6.21� defines a real analytic section map from thesection CL to the section EL.

Recall that the time rescaling in �2.18� resulted in reversing the direction of the vector field onRE

L which is made up of ejection orbits. Using the vector field �6.21�, we can bring back the senseof time on RE

L.Corollary 6.26: In view of the vector field �6.21�, we have removed the collision singularity

on (and only on) the lifted CE manifold WL.

VII. THE CE MANIFOLD

A. Proof of Theorem 1, part „1…

We know that the local stable manifold WL is the graph of a real analytic function and has areal bianalytic chart �L :O→OL

ª�L�O�=WL given by �6.20�. The real analytic projection� :U�L→S�Kª��U�L�, �6.13�, is a homeomorphism from WL onto

WKª ��WL� � U�T� RM�R . �7.1�

We call WK the KS CE manifold. Since WL=REL ��L,o�RC

L , �6.19�, WK=��WL� is the disjointunion,

WK = REK ��K,o � RC

K, RCK = ��RC

L� ,

REK = ��RE

L�, ��K,o = ��

L,o� .

Corollary 6.16 says that the two restrictions ��N�L :N�

L→N�K �6.17� and ��L,o :��

L,o→��K,o �6.18�

are real bianalytic. Moreover, RCK and RC

L are real analytic submanifolds of N�L. Thus, RC

K, REK,

and ��K,o are real analytic submanifolds. This proves part �1� of Theorem 1. �

B. Proof of Theorem 1, part „2…

To show that WK is real analytic it suffices to show that it is the graph of a real analytic map.This proves the assertion and provides us with a real bianalytic chart which will prove useful later.In what follows, �0,V� will stand for �� ,V� ,�=0 while �0� ,V� will stand for �u ,V� ,u=0� . Thecomposition of the real analytic map �, �6.13�, and the real bianalytic map �L, �6.20�, provide uswith the real analytic map

� � �L:O → WK � U�T� RM�R, ��,V� � ��L��,V�� = �u,V� ,

u1 = �u11,�� = ��1,T��,V�� = �E��,V�, E��,V� = �1,��,V�� ,



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uk = �L�E��,V��Sk��,V�, k = 2, . . . ,M ,

=�L�E��,V��k��,V�, V = ��,�,y,Y,z,Z� . �7.2�

In view of Proposition 6.23, we describe WK using �� ,V�, �� ,V�, and E�� ,V�= �1,�� ,V��,

��u =$��,V� =�

��,V��L�E��,V��2��,V�

]

�L�E��,V��k��,V�]

�L�E��,V��M��,V�� , u1 = ��,��, u = �u2, . . . ,uk�T.

The derivative of $�� ,V� at a point �0,V� is given by

D$�0,V� =��0,V� 0

L�E�0,V��2�0,V� 0

] ]

L�E�0,V��k�0,V� 0

] ]

L�E�0,V��M�0,V� 0

� ,

where each 0 is a zero row with the same number of components as V, the first column givesdifferentiation with respect to �, and the second column �with the 0’s� gives differentiation withrespect to V. Recall that we are working in a neighborhood of a singularity,

� = 0, V = Vo, T = ��0,Vo� = To,

�k�0,Vo� = Sko = � ak

a1 1/3

L��1o��k

o, k = 2,3, . . . ,M .

Thus D$�0,Vo� is finite and hence for sufficiently small � 0, there is an open neighborhood ofthe singularity in which D$�� ,V� is finite. Thus WK is the graph of a real analytic function. �

Lemmas 7.4 and 7.4 are immediate.Lemma 7.3: �A real analytic coordinate chart near �0� ,V�� It follows that the map

�K� � �L:O → �K�O� = WK¬ OK �7.3�

given by �7.2� is a real bianalytic chart near �0� ,V�.Lemma 7.4: It follows that the map

�ª ��WL = �K � ��L�−1�:OL = WL → WK = OK �7.4�

is real bianalytic where � is given by �6.13�. Moreover, the restriction �−1��WK \��K,o� coincides

with the restriction of the hyperprojective transformation P��WK \��K,o�.

C. Proof of Theorem 1, part „3…

There is a one-to-one correspondence between points p= �0� ,V��K,o rest points pL

= �0,T ,S ,V��L,o. By Proposition 6.23 pL has a one-dimensional stable manifold W�pL�. In

view of part �1� of this theorem and Lemma 7.4, the branch �↑0− of W�pL� is mapped down viathe real bianalytic map � to a unique collision orbit Cp. The branch �↓0+ of W�pL� is mappeddown to a unique ejection orbit Ep. The two orbits meet at p=��pL�. Since � is real bianalytic, theCE curve,



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Cp � �p� � Ep

is real analytic. This finishes the proof of Theorem 1. �

D. Proof of Theorem 2

In the real bianalytic chart �K :O→OK�O�=WK, after rescaling time so that ��=1, the re-striction of the vector field �5.8� to WK \��

K,o=RCK�RE

K can be extended to WK and takes the form

�� = 1, V� = GK��,V� , �7.5�

where GK�� ,V� is real analytic. In fact, GK�� ,V�=GL�� ,V�.Choose � small and set

CK = ��u,V� � RCK�� = − ��, EK = ��u,V� � RE

K�� = �� .

It follows that

CK = ��CL� = ��CL�, EK = ��EL� = ��EL� ,

where CL and EL are the real analytic submanifolds defined in Proposition 6.24. It is obvious thatCK and EK are real analytic because ��N�

L is real bianalytic �6.17�.In view of �7.5�, CK�RC

K �EK�REK� has codimension 1 in RC

K �REK�. The real analytic flow of

�7.5� defines a real analytic section map from CK to EK. This finishes the proof of Theorem 2. �

E. Proof of Theorem 3

In Proposition 6.23 we saw that ��L,o is normally hyperbolic with stable manifold WL in the

�-direction. The unstable directions are �T ,S�. It follows that nearby near-collision orbits arerepelled in forward time and nearby near-ejection orbits are repelled in backward time. Thisshows that subsingularity of J�M binaries do not accumulate on the SBC singularity of Mbinaries. �

F. Proof of Theorem 4

We need to project RCK and RE

K from the parameter space with GKS variables�u ,� ,� ,y ,Y ,z ,Z� to the physical space with variables �x ,X ,y ,Y ,z ,Z�. Recall that for suffi-ciently small � 0, and away from the singularity, both the map �u ,w�� u ,� ,�� and the changein variables �u ,v�� u ,w�, given by �4.4�, are real bianalytic. It follows that a real analyticinvariant manifold for the vector field �5.8� gives rise to a real analytic invariant manifold for thevector field �4.5�. Therefore, it is enough to consider the relationship between solutions of secondorder equations in �u ,v� and those of second order equations in �x ,X�.

Without loss of generality we will not distinguish between the invariant manifolds in thevariables �u ,� ,� ,y ,Y ,z ,Z�, �u ,w ,y ,Y ,z ,Z�, and �u ,v ,y ,Y ,z ,Z�. Let

WK*ª WK \��

K,o = RCK � RE

K.

Recall the following.

• The principal bundle Q �Sec. IV D� is locally trivial.• CK and EK are real analytic submanifolds of the real analytic manifold WK*.• WK* does not intersect the singularity ��

K,o and hence no uj�0 at any point in WK* for allj.

• Our second order equation �5.8� is defined on Q �Sec. IV D� and is horizontal andTM-equivariant.

Therefore we have the following.

• WK*, RCK, RE

K, CK, and EK are invariant under the action of TM.



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• The quotients �WK*�, �RCK�, �RE

K�, �CK�, and �EK� are also real analytic and give rise to realanalytic submanifolds in the physical space �x ,X ,y ,Y ,z ,Z� which we denote by W*, RC,RE, C, and E, respectively.

Now, Theorem 4 follows from Theorems 1 and 2 and Proposition 4.13. In short, horizontalequivariant second order equations on U* are really second order equations on the quotient space�U*�. But �U*� is in a real bianalytic correspondence with X* �.

VIII. CAC

In this section we prove Theorem 5.

A. The existence of power series expansion for the KS variables

We start by showing that there is a power series expansion for u, then we show the existenceof an expansion for x.

In view of �7.5�, each CE orbit on WK can be written as a power series of the form

2u = u��,V*�, V = V��,V*� ,

V�− �,V*� = V*, u�− �,V*� = u*, u�0,V*� = 0

that depends real analytically on initial conditions c*= �u* ,V*��CK, and hence u* is completelydetermined by the initial condition �−� ,V*� as follows:

u1* = �− ��1,��− �,V*��, u

k* = L�u1

*��k�− �,V*�, k = 2, . . . ,M ,

p = �u,V� = �u,�,�,B� = �u,�,�,y,Y,z,Z� .

Thus, after fixing � 0 sufficiently small, u* is completely determined by V*. To simplify nota-tion we denote the initial conditions by V* and write V=V�� ,V*� instead of V=V�� ,u* ,V*� , . . .. Near the singularity we choose � 0 small enough so that �5.5� holds, �1

1

=� 0 and �wk� ak /2. It follows from �2.18�, �6.20�, and �7.3� that we can express these orbits inthe form

2T = ��,V��,V*��, � = �11 = �1 + �T�2�−1/2,

u1 = ��1,T� = ��1,��,V��,V*��, uk = L�u1��k��,V��,V*�� ,

�w1� = + a1 + 2�1�2�1 + �T�2�, �wk� = + ak

2 + 2�k�2�1 + �T�2��k��,V*��2,

dt

d�=

2�2�1 + �T�2��w1��,V*�

= ��2f��,V*��, k = 2, . . . ,M, c* = c*�− �,V*� � CK,

where V�� ,V*� is as above, f�� ,V*� is real analytic, and f�0,V��0. It follows that

t = �3�h�V*� + P��,V*��, h�V*� � 0, P�0,V*� = 0,

and both h�V*� and P�� ,V*� are real analytic. Let t=s3. Then

s = ��h�V*� + P��,V*��1/3.

Since h�V*��0, an elementary application of the inverse function theorem allows us to write thefollowing with both g�V*� and Q�s ,V*� are real analytic,



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� = t1/3�g�V*� + Q�t1/3,V*��, g�V*� � 0, Q�0,V*� = 0.

Now, for sufficiently small � 0, the collection of CE orbits on the real analytic CE manifold WK

can be written as convergent power series in t1/3 with coefficients that depends on initial condi-tions �u* ,V*��CK real analytically,

uk = t1/3pk�t1/3,V*�, k = 1, . . . ,M ,

where for each k=1, . . . ,M , pk�s ,V*� is real analytic and pk�0,V*��0.

B. Proof of Theorem 5, part „1…

We obtain the power series expansion of x using the KS map,

xk = t2/3L�pk�t1/3,V*��pk�t1/3,V*� = t2/3qk�t1/3,V*� , �8.1�

where for each k=1, . . . ,M ,qk�s ,V*� is real analytic and qk�0,V*��0.The power series �8.1� depends on initial conditions in the real analytic submanifold CK�WK.

As above, let

WK*ª WK \��

K,o = REK � RC

K.

Recall the last paragraph in Sec. VII F. We know that for any �x ,X�, if we specify u��−1�x�, thenv= �1 /2�u�2�L�u�⊺X is uniquely determined. Hence we can view the parameters V* as points inC. �

C. Proof of Theorem 5, part „2…

Recall that Xk�t1/3 ,V*�= xk�t1/3 ,V*�. Thus,

t1/3Xk�t1/3,V*� =2

3qk�t1/3,V*� +

1

3

�qk

�s�t1/3,V*� ,

which is real analytic in t1/3 and V*. Moreover,

limt→0

t1/3Xk�t1/3,V*� = 23qk�0,V*� � 0.

This finishes the proof of Theorem 5. �

IX. ASYMPTOTIC BEHAVIOR OF COLLISION ORBITS

Proof of Theorem 6: Recall that collision and ejection orbits approach rest points of thevector field �6.8� and �6.9� st t→0%. All the limits given in theorem follow from the descriptionof the rest points which can be found in �6.11�. It remains to prove �2.20�. It follows from �3.13�and �5.8� that for k=1, . . . ,M,

�k = −1

�uk�2�wk,L�uk�T�xk

W� = −1

�uk�2�L�uk�wk,�xk

W� =1

�uk�2�L�uk�wk,�k�x�k�,y,z�xk + hk�x,y,z��

= t1/3�ck + Ak�t1/3��, ck � 0,

where Ak�s� is real analytic and Ak�0�=0. Integrating, we get

�k�t� = �ko + O�t4/3� .

This finishes the proof of our last Theorem 6.

APPENDIX A: PROOF OF LEMMA 2.3

Let j=1, . . . ,M. Let



131.181.251.130 On: Fri, 21 Nov 2014 21:36:41

Mj =mj� j

mj + � j, cj =

mj

�mj + � j�Mj

, ej =� j

�mj + � j�Mj

.

Let � j and � j be the position vectors of mj and � j, respectively. Let

xj = Mjxj, xj = � j − � j .

Thus

� j = zj − ejxj, � j = zj + cjxj .

For 1� i , j�M , i� j, let

2aj2 = 2mj� j

Mj ,

aij = mimj, bij = �imj ,

cij = mi� j, bij = �i� j .

For j=1, . . . ,M and k=1,2 , . . . ,K, let

eik = �ink, f ik = mink.

Proof of Lemma 2.3: The assertion that all these vector functions are real analytic near a SBCsingularity follows from the fact that at such a singularity zij�0, i� j, yl�yk , l�k, and yk�zi forall i and k.

The assertions about the expansion of W follow by the standard calculations used to computea power series. The following relations are useful:

eiaij = cibij, eicij = cidij, eiaki = cicki,

eibki = cidki, cieik = eif ik, . . . .

APPENDIX B: BASIC IDENTITIES FOR KS MULTIPLICATION

Let u�j�= Iju and x�j�=L�u�u�j� , j=1, . . . ,4. Then

L�u�u =�u1

2 − u22 − u3

2 + u42

2�u1u2 − u3u4�2�u1u3 + u2u4�

0�, x�2� = L�u�u�2� =�

− 2�u1u2 + 2u3u4�u1

2 − u22 + u3

2 − u42

2�u1u4 − u2u3�0

� , �B1�

x�3� = L�u�u�3� =�2�− u1u3 + u2u4�− 2�u1u4 + u2u3�u1

2 + u22 − u3

2 − u42

0�, x�4� = L�u�u�4� =�

0

0

0

�u�2� ,

�x�j�� = �u�k��2, j,k = 1, . . . ,4 L�u�b =�u1b1 − u2b2 − u3b3 + u4b4

u2b1 + u1b2 − u4b3 − u3b4

u3b1 + u4b2 + u1b3 + u2b4

u4b1 − u3b2 + u2b3 − u1b4

� , �B2�



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Q�u� = L�u�N =�u1 − u2 − u3 − u4

u2 u1 − u4 u3

u3 u4 u1 − u2

u4 − u3 u2 u1

�, N =�1 0 0 0

0 1 0 0

0 0 1 0

0 0 0 − 1� ,

Q�u�u =�u1

2 − u22 − u3

2 − u42

2u1u2

2u1u3

2u1u4

�, Q�u�b =�u1b1 − u2b2 − u3b3 + u4b4

u2b1 + u1b2 − u4b3 + u3b4

u3b1 + u4b2 + u1b3 − u2b4

u4b1 − u3b2 + u2b3 + u1b4

� �B3�

A�� =�cos � 0 0 sin �

0 cos � − sin � 0

0 sin � cos � 0

− sin � 0 0 cos ��, M =�

1 0 0 0

0 − 1 0 0

0 0 − 1 0

0 0 0 − 1� ,

�B4�

Let

I1 =�1 0 0 0

0 1 0 0

0 0 1 0

0 0 0 1� = � I 0

0 I , I2 =�

0 − 1 0 0

1 0 0 0

0 0 0 1

0 0 − 1 0� = �J 0

0 − J ,

I3 =�0 0 − 1 0

0 0 0 − 1

1 0 0 0

0 1 0 0� = �0 − I

I 0 , I4 =�

0 0 0 1

0 0 − 1 0

0 1 0 0

− 1 0 0 0� = � 0 − J

− J 0 ,

�B5�

L�u� = �u�1� u�2� u�3� u�4��, u�j� = Iju, j = 1,2,3,4. �B6�

Straightforward calculations yield the following:

I2I3 = I4, I3I4 = I2, I4I2 = I3,

IjIk = − IkIj, k, j = 2,3,4, j , �B7�

u = − u�4�, �B8�

L�u�1�� = �u�1� u�2� u�3� u�4�� = L�u� ,

L�u�2�� = �u�2� − u�1� − u�4� u�3�� = L�u�I2,

L�u�3�� = �u�3� u�4� − u�1� − u�2�� = L�u�I3,

L�u�4�� = �u�4� − u�3� u�2� − u�1�� = − L�u�I4. �B9�



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It follows from �B9� that

L�a�b = NL�b�a , �B10�

L�a�Tb = ML�b�Ta , �B11�

L�a�L�b� = NL�b�L�a�N , �B12�

L�L�v�a� = L�v�NL�Na� , �B13�

L�L�v�a�b = NL�b�L�v�a = L�v�NL�Na�b , �B14�

1

�a�2L�L�v�a�Ma =

1

�a�2NL�Ma�L�v�a = v , �B15�

L�Q�v�a� = L�v�NL�a� , �B16�

L�a�TNL�v� = L�v�L�Ma�N , �B17�

L�Mv� = ML�v�M , �B18�

L�v�T = NL�Mv�N , �B19�

L�v�TL�b� = NL�NL�v�Tb� , �B20�

L�v�L�b�T = L�L�v�NMb�N , �B21�

Q�v�T = Q�Mv� ,

1

�b�2NL�b�Tb =

1

�b�2L�b�Tb = k1, �B22�

1

�b�2Q�b�Tb =

1

�b�2Q�Mb�b =

1

�b�2NQ�Mb�b = k1, �B23�

1

�b�2L�b�TNL�u�b = u , �B24�

1

�b�2L�b�ML�u�Tb = u . �B25�

Let R14�� be rotation with angle � in the �1,4� direction. Similarly, let R23�� be rotation withangle � in the �2,3� direction.

A�� = e�I4 = I cos � + I4 sin � = R14�− �� R23�� , �B26�

A��T = A�− �� = A��−1, �B27�



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I1A�� = A��I1,

I2A�� = I2 cos � − I3 sin � = A�− ��I2,

I3A�� = I3 cos � + I2 sin � = A�− ��I3,

I4A�� = A��I4. �B28�

Let

u��j� = Iju� = IjA��u, j = 1,2,3,4.

It follows from �B28� that

u��1� = u�1� cos � + u�4� sin � = �u�1��,

u��2� = u�2� cos � − u�3� sin � = �u�2��−�,

u��3� = u�2� sin � + u�3� cos � = �u�3��−�,

u��4� = − u�1� sin � + u�4� cos � = �u�4��. �B29�

Hence,

A��L�u� = �u��1� u−��2� u−��3� u��4�� = L�u�NA��N = L�u��R14�� R23�� ,

R14�� R23�� =�cos � 0 0 − sin �

0 cos � − sin � 0

0 sin � cos � 0

sin � 0 0 cos �� , �B30�

L�u�� = �u��1� u��2� u��3� u��4�� = ��u�1�� u�2��−� �u�3��−� �u�4�� = L�u��I1 cos �

− I4 sin �� = L�u�A�− �� = L�u�A��−1 = L�u�A��T, �B31�

A��L�u�T = L�u��T, �B32�

u� = u��4� = u�4�� = u�. �B33�

For any u�U we have

�u� = �− u� = �u�4��, �u�2�� = �u�3�� Qu ª span�u�2�,u�3�� .

In fact, u�3�=A�� /2�u�2�, u�4�=A�� /2�u�1�, and −u�1�=A��u�1� Thus �B32� shows that

L�u�3��u�3� = L�u�2��u�2� = − L�u�u ,

L�u�4��u�4� = L�u�u = L�− u��− u� . �B34�

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