On the C 8/3 -Regularisation of Simultaneous Binary Collisions in the Collinear 4-Body Problem
OON THE C / -REGULARISATION OF SIMULTANEOUS BINARYCOLLISIONS IN THE COLLINEAR 4-BODY PROBLEM NATHAN DUIGNAN AND HOLGER R. DULLIN
Abstract.
The singularity at a simultaneous binary collision is explored in the collinear 4-body problem. It is known that any attempt to remove the singularity via block regularisationwill result in a regularised flow that is no more than C / differentiable with respect to initialconditions. Through a blow-up of the singularity, this loss of differentiability is investigatedand a new proof of the C / regularity is provided. In the process, it is revealed that thecollision manifold consists of two manifolds of normally hyperbolic saddle singularities whichare connected by a manifold of heteroclinics. By utilising recent work on transitions near suchobjects and their normal forms, an asymptotic series of the transition past the singularity isexplicitly computed. It becomes remarkably apparent that the finite differentiability at 8 / Introduction
Of central importance in the n -body problem is the fact that isolated binary collisions can beregularised; a singular change of space and time variables allows trajectories to pass analyticallythrough binary collisions unscathed. This so called Levi-Civita regularisation provides a flowsmooth with respect to initial conditions. Curiously, when two binary collisions occur simulta-neously, we are not so fortunate. In [18], Martinez and Sim´o gave strong evidence to conjecturethe regularised flow, in a neighbourhood of the simultaneous binary collision, is at best C / .Remarkably, the conjecture was confirmed for some sub-problems of the 4-body problem [19],including the collinear and trapezoidal problems. Despite this, the conjecture remains open forthe collinear or planar n -body with n >
4, and the planar 4-body problem [27].Let q i ( t ) ∈ R be the position of the i th body on the line for i = 1 , . . . ,
4. A simultaneousbinary collision occurs at some time t c when two pairs of binaries, say ( q , q ) and ( q , q ),satisfy q ( t c ) = q ( t c ) , q ( t c ) = q ( t c ) but q ( t c ) (cid:54) = q ( t c ). Throughout the paper only a spatialneighbourhood of the simultaneous binary collisions between these two distressed binaries isconsidered. Denote the set of all such simultaneous binary collisions by C .In essence, regularisation concerns the continuation of solutions to differential systems pastsingular points. Solutions that approach the singularity in forward (resp. backward) time arecalled ingoing (resp. outgoing) asymptotic orbits. If they can be extended through the singularpoint in some meaningful manner, then the singularity is deemed regularisable . There are twoprimary notions of what is meant by ‘meaningful’. The first asks, when considered as a powerseries about t c in t , whether each asymptotic orbit has an analytic continuation. This is referredto as branch regularisation or regularisation with respect to time . It has its foundation in celestialmechanics in the work of Sundman [26] and has been considered for simultaneous binary collisionsin [1, 14, 22, 23, 25].However, we are concerned with the alternate approach to regularisation whereby a singularityis regularisable if there exists an extension of the asymptotic orbits that is at least continuouswith respect to initial conditions. Conley and Easton [4, 8] provide a precise definition of this a r X i v : . [ m a t h . D S ] A ug NATHAN DUIGNAN AND HOLGER R. DULLIN notion, referred to as block regularisation . They link the regularisability of a set of singularitiesto the behaviour of the flow in an isolating block N around the singularities. Essentially, oneconstructs a homeomorphism π by flowing ingoing points on the boundary of N to outgoingpoints. Note that π is only defined for points which are not in an asymptotic orbit. If π admits aunique C k extension π that maps ingoing asymptotic orbits to outgoing asymptotic orbits thenthe set of singularities is said to be C k -regularisable and π is denoted the block map .Many examples connecting regularity and isolating blocks are given in [6]. In the context ofthe n -body problem, the ingoing and outgoing asymptotic orbits are called collision and ejection orbits respectively. They are denoted by E + and E − respectively and their union is denoted by E . If the block map π is already known to be C , then an isolating block can be constructed fromany two transverse sections Σ , Σ of E + , E − respectively [4]. It will be reproved in Theorem 3.3that the set of simultaneous binary collisions C is at least C regularisable. Hence, the followingis a simpler working definition of regularisation for the simultaneous binary collisions. Definition 1.1.
The set of simultaneous binary collisions C is C k -regularisable if there existstwo transverse sections Σ , Σ to the collision and ejection orbits respectively and the block-map π : Σ → Σ is C k .With the given definition of regularisation, Easton proved that isolated binary collisions areanalytically regularisable [8, 9]. Yet, through the use of blow-up, it was shown by McGehee thatthe triple collision is not even C regularisable [20]. Despite being a limiting behaviour of twoisolated binary collisions, the following curious result has been conjectured. Conjecture 1.2 (Martinez and Sim´o [18] (1999)) . The set of simultaneous binary collisions C is exactly C / -regularisable in the planar 4-body problem. Remarkably, this odd behaviour of orbits near collision has been confirmed by Martinez andSim´o [19] for some sub-problems.
Theorem 1.3 (Martinez and Sim´o [19] (2000)) . The set of simultaneous binary collisions C is exactly C / -regularisable for the collinear, trapezoidal, bi-isosceles and tetrahedron 4-bodyproblems. The conjecture remains open for the collinear or planar n -body with n >
4, and the planar4-body problem.There have been several authors researching work towards this conjecture. Elbialy [11] used theblow-up method, first introduced to celestial mechanics by McGehee in [20], to take a very generalapproach to the problem. Multiple collision singularities were investigated and some asymptoticbehaviour of collision and ejection orbits in the n -body problem was given. Elbialy’s research wasfollowed by the work of Sim´o and Lacomba [28] which proved the simultaneous binary collisionis C -regularisable in the n -body problem through the use of perturbative techniques. Two keypapers by Elbialy, one on the collinear problem [13] and the other on the planar problem [12],produced a set of coordinates, the generalised Levi-Civita coordinates , which showed clearly theresult is C in the planar problem, and further, at least C in the collinear problem. It wasin 1999 that Martinez and Sim´o reproved the C result in the plane and provided numericalevidence to support Conjecture 1.2 for the trapezoidal 4-body problem [18]. Then, a year later,they proved Theorem 1.3 on the C / regularisation in some sub-problems of the planar 4-bodyproblem. Their proof involved a type of Picard iteration to explicitly compute, order by order,some trajectories nearby collision as a function of time. After a few iterations a power of 8 / EGULARISATION FOR SIMULTANEOUS BINARY COLLISIONS 3 collisions. In doing so, they required the time difference t d between the two binary collisions,with simultaneous binary collision occuring when t d = 0. However, in the planar problem, anorbit can be near collision without undergoing an isolated binary collision. As a result, theredoes not seem a simple extension of their method to the planar case.In this paper Theorem 1.3 is reproved for the collinear 4-body problem. However, in an at-tempt to construct a method of proof that may extend to the planar problem, the more geometricpath paved by Elbialy is followed. A geometrical explanation of the generalised Levi-Civita co-ordinates, first used by Elbialy in [11], is given in Section 2. It is shown that these coordinatesregularise independent binary collisions but produce a codimension 2 set of degenerate equilibriacorresponding to simultaneous binary collisions.In the proceeding Section 3, a blow-up and desingularisation produces the collision manifold.Proposition 3.1 is proved, revealing the collision manifold as two, 3:1 and 1:3 resonant, normallyhyperbolic manifolds of singularities that are connected by a manifold of heteroclinics. A similarresult was first observed in [15]. The proposition gives the topological structure of the flowin a neighbourhood of the set of singularities. Ultimately, this fact leads to a proof of the C -regularisation in Theorem 3.3.Section 4 constitutes the bulk of the paper. It provides the necessary theory required to provethe main theorem, Theorem 4.13, on the C / -regularisation of the simultaneous binary collisions.We begin the section by contemplating the existence of a foliation into normal, invariant 2-planesin a tubular neighbourhood of C . Through a study of the homological operator associated tothe normal form of the set of collision singularities, the existence of the foliation is linked tothe existence of a set of formal, local integrals. With this normal form procedure, a notion ofhow well a normal space admits a smooth, invariant foliation is defined. In particular, for thesimultaneous binary collisions, a computation of the normal form in Proposition 4.2 concludesthat the foliation fails to exist at order 8. Remarkably, the term preventing the foliation is thefirst term in the potential coupling the two distressed binaries. This proves a heuristic observationgiven by Martinez and Sim´o [18] on the crucial role the coupling term plays.We continue Section 4 by noticing that the structure of the collision manifold admits a proce-dure for explicitly computing the asymptotic series of the block map π . The relevant theory tocompute the asymptotic orbit is detailed in [5]. This theory is summarised in several propositions.It is used to prove Theorem 4.11 which asserts that the block map is generically quasi-regular.In fact, it is seen that the block map for the simultaneous binary collisions is asymptotic to apower series in terms of θ / , where θ will be defined as some measure of the distance away froma collision orbit. The 1 / / C .2. Coordinates Near Simultaneous Binary Collision
Difference Vectors
Suppose there are 4 collinear bodies consisting of two binaries undergoing collision in differentregions of configuration space at precisely the same time t c . Further suppose that the bodieswith mass m and m undergo one of the binary collisions and bodies with masses m and m undergo the other. Let the signed distance between the bodies in each binary be given by Q , Q respectively and let x be the signed distance between the two centre of masses of the binaries. NATHAN DUIGNAN AND HOLGER R. DULLIN Q Q x Figure 2.1.
The configuration variables near simultaneous binary collisionThe coordinates are depicted Figure 2.1. If P , P , y are the conjugate momenta of Q , Q , x ,the dynamics of the collinear 4-body problem is given by the Hamiltonian,(2.1) H ( Q, x, P, y ) = (cid:88) i =1 (cid:18) M i P i − k i | Q i | − (cid:19) + 12 µy − ˆ K ( Q , Q , x ) , with the standard symplectic form ω = dQ ∧ dP + dQ ∧ dP + dx ∧ dy . The smooth functionˆ K contains the potential terms coupling the two binaries and M i , k i , µ, d i , c i > K = d | x + c Q − c Q | + d | x + c Q + c Q | + d | x − c Q − c Q | + d | x − c Q + c Q | M = m m m + m , M = m m m + m , k = m m , k = m m ,µ = m + m + m + m ( m + m )( m + m ) ,d = m m , d = m m , d = m m , d = m m ,c = m − M , c = m − M , c = m − M , c = m − M . This choice of coordinates is a reduction of the system by translational symmetry via a coordinatetransform that preserves the diagonal structure of the mass metric.It is more convenient to work with rescaled variables ˜ Q i , ˜ P i defined via the symplectic trans-formation(2.3) ˜ Q i = 4 k i M i Q i , ˜ P i = 14 M i k i P i . Under this scaling the Hamiltonian is(2.4) H ( ˜ Q, x, ˜ P , y ) = (cid:88) i =1 a i (cid:18) ˜ P i − | ˜ Q i | − (cid:19) + 12 µ | y | − ˜ K (cid:16) ˜ Q, x (cid:17) , where a i = 16 M i k i and ˜ K (cid:16) ˜ Q i , x (cid:17) = ˆ K (cid:16) k i M i ˜ Q i , x (cid:17) .2.2. Levi-Civita Regularisation of Binaries
In an attempt to regularise the simultaneous binary collisions, it is natural to first regularisethe binary collisions of each distressed binary. This is done by passing to the Levi-Civita variables(˜ z i , u i ) through the symplectic map˜ Q i = 12 ˜ z i , ˜ P i = ˜ z − i u i . EGULARISATION FOR SIMULTANEOUS BINARY COLLISIONS 5
The result of this transformation is the partially regularised, translational reduced Hamiltonian,(2.5) H (˜ z, x, u, y ) = (cid:88) i =1 a i ˜ z − i (cid:0) u i − (cid:1) + 12 µy − ¯ K (˜ z , ˜ z , x ) , with ¯ K (˜ z , ˜ z , x ) := ˜ K (cid:0) ˜ z , ˜ z , x (cid:1) .The Hamiltonian is said to be partially regularised for the following reason. Time can berescaled to dt = z z dτ by using the Poincar´e trick of moving to extended phase space andrestricting to a constant energy surface. That is, by introducing the Hamiltonian(2.6) H (˜ z, x, u, y ) = ˜ z ˜ z ( H (˜ z, x, u, y ) − h )= 12 a ˜ z (cid:0) u − (cid:1) + 12 a ˜ z (cid:0) u − (cid:1) + ˜ z ˜ z (cid:18) µy − ¯ K (˜ z, x ) − h (cid:19) , and restricting to a constant energy surface in the original Hamiltonian, H = h , yielding H = 0.The flow on H = 0 is equivalent to the flow on H = h up to time rescaling. As desired, theHamiltonian H is regular at ˜ z = 0 or ˜ z = 0 and so the binary collision singularities have beenregularised. The set of simultaneous binary collisions ˜ z = ˜ z = 0, denoted by C , is a criticalpoint of H and the associated Hamiltonian differential equation,(2.7) ˙˜ z = a ˜ z u ˙˜ z = a ˜ z u ˙ x = µ ˜ z ˜ z y ˙ u = ˜ z (cid:18) z (cid:18) h + ¯ K (˜ z, x ) − µy (cid:19) − a (cid:0) u − (cid:1) + ˜ z ˜ z ∂ ¯ K∂ ˜ z (cid:19) ˙ u = ˜ z (cid:18) z (cid:18) h + ¯ K (˜ z, x ) − µy (cid:19) − a (cid:0) u − (cid:1) + ˜ z ˜ z ∂ ¯ K∂ ˜ z (cid:19) ˙ y = ˜ z ˜ z ∂ ¯ K∂x has a manifold of singularities given by C . Essentially, when rescaling time to regularise thebinary collisions, time was over-scaled at the set of simultaneous binary collisions, slowing downorbits as they approach the singularity and creating an equilibrium. Instead of a simultaneousbinary collision occurring at some finite time t c , it now occurs as τ → ±∞ for collision andejection orbits respectively.The following proposition gives crucial properties of the collision and ejection orbits. It hasbeen proved in, for example, [11, 18, 26]. We state it here in the Levi-Civita coordinates. Proposition 2.1.
Suppose that (˜ z , ˜ z , x, u , u , y ) is a collision (resp. ejection) orbit. Then, u i → ε i , ˜ z ˜ z → δ (cid:18) a a (cid:19) , x → x ∗ , y → y ∗ , as τ → ∞ (resp. τ → −∞ ). Here | x ∗ | , | y ∗ | < ∞ , ε i = ± and δ = ε ε . Moreover, for eachchoice of ε i , the set E of collision and ejection orbits is a 5 dimensional manifold. A geometrical proof can be constructed using the methods of blow-up and desingularisation.The curious reader is referred to [11] for details of the proof. The different values of ε i resultfrom the Levi-Civita transformation being a double cover of the original phase space.2.3. Generalised Levi-Civita Coordinates
Blow-up and desingularisation methods in the Levi-Civita coordinates (˜ z i , u i ) have been im-plemented in [18, 11] to produce useful asymptotic results. However, it can be argued they arenot ideal coordinates to see that the set of simultaneous binary collisions is block regularisable.A set of singularities is only C -regularisable if each ingoing asymptotic orbit can be mapped toa unique outgoing asymptotic orbit. For the set of simultaneous binary collisions C , this requiresthat each collision orbit map to a unique ejection orbit under π . However, from Proposition 2.1, NATHAN DUIGNAN AND HOLGER R. DULLIN E + and E − are both 5 dimensional. Consequently, if there is no obvious constraint on how orbitson E + must map to orbits on E − then any block map π can not possibly be extended uniquely to a map π on the whole block.A natural constraint on how E + maps to E − can be made by demanding that collision orbitsmap to ejection orbits with the same asymptotic properties. In fact, this technique is prominentin blow-up methods for algebraic geometry problems; see for instance [10]. Proposition 2.1shows that all collision orbits approach C with the same value of u ∗ i and with the same tangent˜ z ∗ / ˜ z ∗ . Therefore, if it is desired to distinguish between distinct collision orbits, we must use theasymptotic value of the second derivative of the collision orbits as they approach C . That is, wemust use the asymptotic of the “curvature” of each collision orbit in the ( u i , ˜ z i ) plane. This canbe achieve by introducing a new coordinate κ i through u i = a i + κ i ˜ z i .However, the intrinsic energy of each distressed binary is given by(2.8) ˜ h i = 12 a i ˜ z − i ( u i − . Re-arranging this for u i and expanding at the collision point u i = 1 , ˜ z i = 0 in z i gives u i = 1 + 1 a i ˜ h i ˜ z i + . . . . Consequently, the more physical intrinsic energies ˜ h i can be used instead of the curvature κ i to distinguish between distinct collision orbits. But introducing the intrinsic energies as newcoordinates is precisely what is done by Elbialy in [11]! We take a slight vairation to Elbialy byusing a rescaling of(2.9) z i = a − / i ˜ z i , h = 2 a − / i ˜ h i to produce a version of the generalised Levi-Civita coordinates ( z i , h i , x, y ).The Hamiltonian in the generalised Levi-Civita coordinates and the symplectic form are,(2.10) H = 12 a / h + 12 a / h + 12 µy − K ( z , z , x ) ,ω = 12 a / z u dz ∧ dh + 12 a / z u dz ∧ dh + dx ∧ dy. where K ( z , z , x ) := ¯ K ( a / z , a / z , x ). Of course, (2.8) is only invertible when h i z i + 1 > u i is made. Each of the choices will cover at least one of the simultaneousbinary collision equilibria and a sufficiently small neighbourhood of z = z = 0 can be chosen.So, without loss of generality, make the choice u i = + (cid:112) h i z i .Using Hamilton’s equations and rescaling by dt = z z dτ as before, the collinear 4-bodyproblem is given in the generalised Levi-Civita coordinates by the system(2.11) z (cid:48) = z (cid:113) z h z (cid:48) = z (cid:113) z h x (cid:48) = µz z y h (cid:48) = 2 a − / z (cid:113) z h ∂K∂z h (cid:48) = 2 a − / z (cid:113) z h ∂K∂z y (cid:48) = z z ∂K∂x . Denote by X the vector field associated to System (2.11).In the generalised Levi-Civita coordinates, some properties of the flow near simultaneousbinary collision become clear. Firstly, each of the binary collisions (e.g. z = 0 , z (cid:54) = 0) areregular points of the flow, hence regularisable. There is a co-dimension 2 manifold of equilibria(0 , , h ∗ , h ∗ , x ∗ , y ∗ ) ∼ = R corresponding to the simultaneous binary collisions C in the chosenchart. Moreover, the equilibria in this manifold are degenerate in that they have vanishing EGULARISATION FOR SIMULTANEOUS BINARY COLLISIONS 7
Jacobian. Each fixed point in C corresponds to the asymptotic values of a collision (resp.ejection) orbit as time approaches ∞ , (resp. −∞ ). That is, a fixed point in C gives the value ofthe intrinsic energies of each distressed binary ( h ∗ , h ∗ ), the distance between the two collisions x ∗ , and the momentum at which the two binaries are moving from each other y ∗ at collision.The fact that these equilibria are degenerate obfuscates even the topological properties of theflow in a neighbourhood of the collision set. Though, once the blow-up and desingularisation pro-cess is done in these coordinates, determining if the simultaneous binary collision is regularisable,and quantifying the degree to which it is, is a less formidable task.3. C -regularity of block map The primary aim of this section is to reprove Theorem 3.3 on the C -regularisation of simul-taneous binary collisions. A didactic example, referred to as the uncoupled problem, is used tomotivate the techniques and computations in this section and in Section 4. A blow-up and desin-gularisation of the set of collisions C in the generalised Levi-Civita coordinates and a study of theflow on the resultant collision manifold ultimately leads to the desired proof of C -regularisationin Theorem 3.3.3.1. The Uncoupled Problem
If one uncouples the interaction between the two distressed binaries the result is the directproduct of two Kepler systems. This so called uncoupled problem is integrable. Consequently,many of the properties of the flow, such as C -regularisation, will follow with minimal effort.Using some of the integrals of the uncoupled problem, a lower dimensional problem can beproduced and visualised. The terms in the generalised Levi-Civita system (2.11) influenced bythe coupling terms K are of high order in z , z . So, in the study of a tubular neighbourhood of C , where z = z = 0, removing the coupling terms should still capture the essential dynamicsof the full problem.Explicitly, the uncoupled Kepler problem is given by the system,(3.1) z (cid:48) = z (cid:113) z h z (cid:48) = z (cid:113) z h x (cid:48) = µz z y. with the other variables integrals, h (cid:48) = h (cid:48) = y (cid:48) = 0. By making a choice of h , h , y the systemcan be considered as a vector field on R . Similar to the coupled problem, simultaneous binarycollision at ( z , z ) = (0 ,
0) corresponds to a co-dimension 2 set of degenerate fixed points. Eachfixed point is parameterised by x ∗ . A qualitative plot of the dynamics is given in Figure 3.1.From Figure 3.1, the C -regularity of the block map π is clear. There is a manifold of collision E + (resp. ejection E − ) orbits asymptotic to C in forward (resp. backward) time. Orbits oneither side E + pass around the set of singularities C and meet one another on the other side near E − . In the next section this qualitative picture is validated for both the uncoupled and coupledproblems.3.2. Study of the Collision Manifold
In order to get the asymptotic and topological structure of the flow in a neighbourhood of C , blow-up and desingularisation can be performed. The use of blow-up in celestial mechanicswas introduced by McGehee [20] in his study of the triple collision. Later it was implementedin investigations of the simultaneous binary collision by Elbialy [11], and Mart´ınez and Sim´o[18]. This section follows similarly to the work of Elbialy [12]. We are less ambitious in our NATHAN DUIGNAN AND HOLGER R. DULLIN C E + E − x ∗ Figure 3.1.
A qualitative plot of the uncoupled problem. Using blow-up meth-ods in Section 3.2 this qualitative picture is validated. The C -regularity of awell chosen block map π is clearly apparent.treatment of the problem in comparison to the general framework presented by Elbialy where l pairs of binaries are undergoing collision simultaneously inside the n -body problem. By onlytreating two binaries in the 4-body problem, some simplifications and more concise statementsof the flow near C can be made.In the generalised Levi-Civita coordinates, the blow-up is easily achieved by introducing polarcoordinates in the position variables(3.2) z = r cos θ, z = r sin θ and the desingularisation by rescaling time d ¯ τ = rdτ . The result is the blown-up system,(3.3) r (cid:48) = r sin θ cos θ (cid:18) cos θ (cid:113) h r sin θ + sin θ (cid:112) h r cos θ (cid:19) θ (cid:48) = (cid:18) cos θ (cid:113) h r sin θ − sin θ (cid:112) h r cos θ (cid:19) x (cid:48) = µr y sin θ cos θh (cid:48) = 2 a − / r cos θ (cid:112) h r cos θ ∂K∂z ( r cos θ, r sin θ, x ) h (cid:48) = 2 a − / r sin θ (cid:113) h r sin θ ∂K∂z ( r cos θ, r sin θ, x ) y (cid:48) = r sin θ cos θ ∂K∂x ( r cos θ, r sin θ, x ) . Denote the vector field associated to the system by X θ .In these coordinates, the set of simultaneous binary collisions corresponds to r = 0. Theintroduction of polar coordinates and consequent time rescaling by r has replaced the set ofequilibria occurring at C = (0 , × R with the cylinder C = 0 × S × R . The cylinder C isreferred to as the collision manifold . EGULARISATION FOR SIMULTANEOUS BINARY COLLISIONS 9
By studying the fictitious dynamics on the collision manifold, qualitative information onnearby orbits can be gathered. The flow on the C is given by setting r = 0 in (3.3),(3.4) r (cid:48) = 0 θ (cid:48) = cos θ − sin θx (cid:48) = h (cid:48) = h (cid:48) = y (cid:48) = 0 . Remarkably, as noted in the work of Elbialy [11], not only is the collision manifold invariantunder the flow, but x, y, h , h remain constant. In other words, the collision manifold C isfoliated by invariant S . The flow on these invariant circles is independent of the choice ofconstant ( x ∗ , y ∗ , h ∗ , h ∗ ). Furthermore, each S has equilibria when tan θ = 1. This agrees withthe results given in Proposition 2.1. Combining these facts with a study the blown-up systemallows us to prove the following proposition. Proposition 3.1.
The collision manifold C is a heteroclinic connection between two normallyhyperbolic invariant manifolds of fixed points. Moreover, the following properties hold:(i) The normally hyperbolic manifolds are given by the ( r, θ ) = (0 , π/ and ( r, θ ) = (0 , − π/ .Denote them by N − , N + respectively.(ii) The normal bundle of each manifold is 2-dimensional in the ( r, θ ) directions.(iii) The heteroclinic connection is foliated by invariant S .(iv) Restricted to the normal bundle, the N + , N − are resonant hyperbolic saddles with the ratioof stable to unstable eigenvalue given by and respectively.Proof. The Jacobian of X θ (system (3.3)) is given by,(3.5) DX θ = sin θ cos θ (cos θ + sin θ ) 00 − θ cos θ (cos θ + sin θ )
00 0 + O ( r ) . Evaluating on the manifolds of fixed points ( r, θ ) = (0 , π/
4) and ( r, θ ) = (0 , − π/
4) yields,(3.6) DX θ | N − = − / − · − /
00 0 , DX θ | N + = − − /
00 3 · − /
00 0 . Hence, both N + , N − are normally hyperbolic with central directions ( x, h , h , y ) and each is ahyperbolic saddle with 1:3 and 3:1 resonances respectively. The unstable manifold of each fixedpoint in N + begins in the θ -direction. Due to the invariant foliation of the collision manifoldinto S , the unstable manifold must coincide with the stable manifold of a fixed point in N − with the same values of ( x ∗ , h ∗ , h ∗ , y ∗ ). (cid:3) Each invariant S is blown-down to a single point on the manifold of simultaneous binarycollisions C . The stable manifold of N + leaves the collision manifold in the r direction. Thus,the portion of the stable manifold with r > E + . Similarly, the portion of theunstable manifold with r > N − is the set of ejection orbits E − . Because of the heteroclinicconnection between the two normally hyperbolic manifolds, when the system is blown-down, E + , E − are glued together with each collision orbit connected to the unique ejection orbit withthe same asymptotic values of ( x ∗ , h ∗ , h ∗ , y ∗ ). The following nice corollary can be concluded. Corollary 3.2.
Each collision orbit is connected to a unique ejection orbit.
Both the proposition and corollary are visually represented by a diagram of the uncoupledproblem in Figure 3.2. In particular, the normal hyperbolicity of the two manifolds at θ = π/ , − π/
4, the foliation of C into invariant S , and that each collision orbit is uniquely connected to an ejection orbit via a heteroclinic. Note that the flow on C coincides for the uncoupled andcoupled problem. N + N − Figure 3.2.
Blow-up of the uncoupled problem. The collision manifold C isrepresented by the cylinder and some trajectories on and nearby are given.Finally, we are in a position to prove the first key theorem, already known in [28, 13, 18]. Theorem 3.3.
The set of simultaneous binary collisions is at least C -regularisable in thecollinear 4-body problem.Proof. The blow-up is a diffeomorphism on R \ C and so the inverse, the blow-down, exists. Theblow-down preserves the topological structure of the flow on ( R + × S × R ) \C . The C -regularityshould now be clear. A section Σ transverse to the manifold of collision orbits E + is split intothree sets depending on whether a point is in E + ∩ Σ or which side of E + it is on. Points onthe two halves of Σ can be flowed around the collision manifold C where they eventually meetagain at a transverse section of the ejection orbits E − , say Σ . One can then glue each collisionorbit in E + ∩ Σ to its unique ejection orbit in E − ∩ Σ to produce a C block map π . (cid:3) C / -regularity of the Block Map In this section Theorem 4.13 on the C / -regularisation of simultaneous binary collisions isproved. Firstly, a heuristic argument motivates the normal form computation to degree 9 of X in a neighbourhood of an arbitrary simultaneous binary collision on C . Approximate integralsare computed through a normal form procedure. An obstacle to increasing their degree is foundto occur at order 8 in ( z , z ) in the intrinsic energy components h i of X . The non-smoothness ofthe block map is then established in Theorem 4.11, where a deeper investigation of the flow nearthe collision manifold reveals the quasi-regularity of the block map. We give a geometric sketch EGULARISATION FOR SIMULTANEOUS BINARY COLLISIONS 11 of how one computes the exact regularity of the block map π . Finally, the sketch is implementedto achieve a direct asymptotic expansion of the block map and confirm the C / -regularity. Theloss of differentiability at 8 / Nonlinear Normal Form Theory
The journey to proving the finite differentiability begins with a heuristic. Suppose that C was indeed smoothly regularisable. If this is to mean that the singularities share properties akinto regular points, then the existence of a “generalised flow-box” theorem may be expected. Atheorem of this nature would imply the existence of a transformation that flattens the vectorfield in a neighbourhood of any point in C . In particular, a transformation could be found thatflattens the vector field in the ( x, h , h , y ) variables, in turn reducing the computation to a 2-dimensional problem in the ( z , z ) variables. If this foliation exists, not only would the numberof terms in the series expansions required to compute the block map be dramatically reduced,but the theory developed in previous work on regularisation for planar vector fields [6] could beutilised. In other words, we want to know if there exists a foliation of a tubular neighbourhoodof C into invariant 2-planes normal to C . Of course, there could be some obstruction to thisfoliation and this could have an implication for the regularity of the block map.The heuristic is supported by a study of the uncoupled problem (3.1). As the Kepler problem isanalytically regularisable and the uncoupled problem is the direct product of two Kepler systems,the uncoupled problem must too be analytically regularisable. Moreover, the uncoupled problemadmits a set of smooth integrals. These integrals give a foliation of a tubular neighbourhood of C into analytic, invariant 2-planes. Hence, for the uncoupled problem, we have establishing alink between the existence of the foliation and the regularity of C .We turn to normal form theory to investigate the existence of a foliation in the collinear 4-body problem. Normal form theory has a long history resulting in several different normal form‘styles’. For an overview see [21]. The vector field X , given in (2.11), has vanishing Jacobian on C , rendering the common semi-simple style useless. For this reason, the style of both Belitskiiand Elphick et al [2, 3, 16], referred to as the inner product normal form , will be used. All stylesbegin the same; assume a vector field X on R n has a fixed point at 0 and decompose X accordingto some filtration. Usually this is done by taking the Taylor series of X at 0 and decomposing itinto homogeneous components, X = X + X + . . . , with X the leading order homogeneous component of degree s and X d ∈ H d + s − the space ofdegree s + d homogeneous vector fields.If one applies a near identity, formal transformation of the formˆ φ − = I + U d + . . . , where U d is homogeneous of degree d + 1, then the transformed vector field ˆ φ ∗ X at order d isgiven by the equation(4.1) ( ˆ φ ∗ X ) d = X d + [ X , U d ] , whilst the terms of lower order remain unchanged. Here [ · , · ] denotes the usual Lie bracketbetween vector fields. Because the lower terms remain unchanged, an iterative procedure on theorder d can be constructed to produce a transformation putting X into the ‘simplest’ form.Of course a choice must be made about what is meant by ‘simplest’ form for φ ∗ X . Letting L X := [ X , · ], it can be seen that L X is a linear operator acting on H d . Hence, (4.1) is alinear equation denoted the cohomological equation and L X the cohomological operator . Denoteby L d := L X : H d → H s + d − when it is clear what X is and there is a need to differentiate between which order L X is acting. Note that in the typical case the Jacobian at the equilibriumis non-vanishing and hence the degree of of X is s = 1. In such a case L d maps H d to itself.The leading order terms in the vector field 2.11 near C are quadratic, thus we are interested inthe case s = 2.If the simplest form for ˆ φ ∗ X is deemed the one with fewest possible terms, the question thenbecomes, what terms of X d can be removed by L X ? Looking at the cohomological equation,terms in Im L X ⊂ H d + s − can be removed by a choice of U d . However, any terms in thecomplement of Im L X can not be killed by the formal transformation ˆ φ . Such terms are called resonant .There are many choices for a space complement to Im L X , however, Belitskii highlights thenatural choice given by H d + s = Im L X ⊕ ker L ∗ X , where the adjoint is defined with respect tosome choice of inner product on H d + s . We follow Belitskii [3] by taking the Fischer inner producton H d . The following theorem combines the works of Belitskii [3], and Stolovitch and Lombardi[17]. Theorem 4.1 ([3, 17]) . There exists a formal transformation ˆ φ − = I + (cid:80) U d with U d ∈ Im L ∗ d that formally conjugates X = X + (cid:80) X d to the normal form, (4.2) ˆ φ ∗ X = X + (cid:88) d ≥ N d , with N d ∈ ker L ∗ d Theorem 4.1 completely characterises the formal normal form for arbitrary vector fields. More-over, it explicitly gives a way of computing both the formal transformation and the formal normalform of a given vector field X .In practice, as the computation of the normalising transformation φ is done order by order, oneonly knows the normal form up to some truncated degree. Throughout the remaining sectionsit will become apparent that, for the determination of the C / regularity, the vector field canbe truncated at degree 9.Let us now refocus on the problem at hand. The leading order term at any point in thesingular manifold of simultaneous binary collisions is given by X = ( z , z , , , , w = ( w , . . . , w ) ∈ H d +1 and denoting by ˜ X = z ∂ z + z ∂ z the leading order vector field asa derivation, the adjoint of the cohomological operator is given by,(4.3) L ∗ d w = (cid:16) ˜ X ∗ w − ∂ z w , ˜ X ∗ w − ∂ z w , ˜ X ∗ w , ˜ X ∗ w , ˜ X ∗ w , ˜ X ∗ w (cid:17) , where ˜ X ∗ = z ∂ z + z ∂ z is the adjoint of ˜ X . Note that L ∗ d : H d → H d − . What is immediatefrom the form of L ∗ d is the decoupling of the z , z components and each of the x, h , h , y components from one another. This is a consequence of the fact that X decouples into the z , z system and a trivial vector field in the other variables. The decoupling inevitably leads toa proof of Lemma 4.3 in the next section and an answer to the question on the existence of aninvariant foliation.4.2. Computation of the Formal Normal Form
The normal form near an arbitrary simultaneous binary collision to degree 9 will now becomputed. For this calculation, the Taylor series of the vector field X , which is given by (2.11),around the fixed point (0 , , x ∗ , h ∗ , h ∗ , y ∗ ) to degree 9 is required. Therefore, the coupled termsfrom the potential K ( z , z , x ) must be expanded to degree 8. K is given by the sum of terms of the form, d i | x + C l z + C m z | = d i | x | | C l z /x + C m z /x | . EGULARISATION FOR SIMULTANEOUS BINARY COLLISIONS 13
Taking the sum of the four expressions of this form in K and expanding in ( z , z ) to degree 8produces a series of the form b / | x | (1 + P ( z /x, z /x )) with P some degree 4 polynomial and b a function of the masses. Remarkably, the coefficients of monomials z i z j with degree less than 8vanish, provided both i > , j > z x , z x vanish. Thus, the desired expansion takes the form,(4.4) K ( z , z , x ) = 1 | x | (cid:18) b + K (cid:18) z x (cid:19) + K (cid:18) z x (cid:19) + b c z z x (cid:19) + . . . , K i ( Q ) = (cid:88) j =2 b ij Q j , where b ij , b c , b are functions of the masses given by,(4.5) b = d + d + d + d ,b = C ( d + d ) + C ( d + d ) , b = C ( d + d ) + C ( d + d ) ,b = C ( d + d ) − C ( d + d ) , b = C ( d + d ) − C ( d + d ) ,b = C ( d + d ) + C ( d + d ) , b = C ( d + d ) + C ( d + d ) ,b c = 6( C ( C d + C d ) + C ( C d + C d )) ,C = a / k M c , C = a / k M c , C = a / k M c , C = a / k M c . The first coupled monomial b c z z will be seen to play a crucial role in the arrival of non-vanishing resonant terms and ultimately the finite differentiability of the block map. This provesa heuristic observation given by Martinez and Sim´o [18] on the crucial role of the coupling term.The following result on the normal form near an arbitrary simultaneous binary collision cannow be given. Due to the scaling symmetry of the Hamiltonian H , it can be assumed that x hasthe asymptotic value x ∗ = 1. Proposition 4.2.
The normal form X in a neighbourhood of the simultaneous binary collisionwith asymptotic values ( x, h , h , y ) = (1 , h ∗ , h ∗ , y ∗ ) is given to degree 9 by (4.6) z (cid:48) = z + ( h + h ∗ ) R , ( z , z ) + ( h + h ∗ ) R , ( z , z ) z (cid:48) = z + ( h + h ∗ ) R , ( z , z ) + ( h + h ∗ ) R , ( z , z ) x (cid:48) = 0 h (cid:48) = b c a − / R h ( z , z ) h (cid:48) = b c a − / R h ( z , z ) y (cid:48) = 0 where R ,i , R h are homogeneous polynomials of degree 6 and 9 respectively. Each is given by (4.7) R , ( z , z ) = 87195 (cid:0) − z z (cid:0) z − z (cid:1)(cid:1) ,R , ( z , z ) = 87195 (cid:0) z + 10 z z − z (cid:1) ,R h ( z , z ) = 419 ( z − z ) (cid:0) z + z z + z (cid:1) (cid:0) z − z z + z (cid:1) , Proof.
In order to check that this is indeed the normal form, the existence of a transformationtaking system (2.11) to system (4.6) must be computed. Then, one can check that the higherorder terms in the normal form system (4.6) are elements of ker L ∗ X by simply applying L ∗ X given in (4.3) to the higher order terms. In fact, ˜ X ∗ R h = 0 and this is the unique degree9 polynomial (up to scaling) in the kernel. Due to the large number of terms in this normalform transformation it is far too unwieldy to include in this paper. The transformation can beprovided upon request. (cid:3) There is a lot of information to unpack from Proposition 4.2. Firstly, the normal form proce-dure concludes with the appearance of resonant terms at degree 9 in z i for the h i components of X . This fact gives an answer to our question on existence of an invariant foliation of the normalspace to C . Lemma 4.3.
It is not possible to construct analytic invariants diffeomorphic to h i .Proof. Let φ be the transformation bringing X into the normal form X . Note that φ is adegree 8 polynomial in z , z . Then from the fact x (cid:48) = y (cid:48) = 0 in the normal form (4.6) it isclear the x, y components of φ are invariant to order 8 in z , z . Moreover, due to the resonantterms R h of degree 9, the h i components of φ fail to be invariants at order 8 in z i . To provethe lemma it is sufficient to show that there does not exist a transformation of the form,˜ h i = h i + F ( z , z , x, h , h , y ) , with F some analytic function such that ˜ h (cid:48) i = 0. We will prove this for h as the case h follows analogously. To show this, decompose F into, F = (cid:80) d ≥ F d ( z , z ), where each F d is ahomogeneous polynomial in ( z , z ) of degree d + 1 and with coefficients analytic functions in x, h , h , y . Recall that ˜ X is the derivation associated to a vector field X and let X be thedegree 6 components of the normal form (4.6). Then assuming there exists ˜ h with ˜ h (cid:48) = 0 wehave, ˜ h (cid:48) = ˜ X (˜ h )0 = ˜ X ( h ) + ˜ X (cid:88) d ≥ F d ( z , z ) h (cid:48) + ( ˜ X + ˜ X + . . . )( F + F + . . . )0 = b c a − / R h ( z , z ) + ˜ X ( F ) + ˜ X ( F ) + · · · + ˜ X ( F ) + . . . . (4.8)Now ˜ X : H d → H d +1 , ˜ X : H d → H d +5 and R h ( z , z ) ∈ H . Taking all elements on the rhs in H we obtain the equation,0 = b c a − / R h ( z , z ) + ˜ X ( F ) + ˜ X ( F ) . So, we require F or F to be found which cancels with the R h term if the approximate integral˜ h i exists. But, by the normal form procedure R h ∈ ker L ∗ d which, from (4.3) implies R h ∈ ker ˜ X ∗ .As Im ˜ X is the orthogonal complement to ker ˜ X ∗ we are guaranteed that R h / ∈ Im ˜ X , thus nosuch F can be found. Moreover, by collecting the terms in H terms in the expansion (4.8), weobtain ˜ X ( F ) = 0. That is, F ∈ ker ˜ X . Dynamically this says that F is invariant under theflow of the leading order terms X . But it is easily verified that the only invariants of X arepolynomials in the leading order invariantˆ κ = z − z . As any homogeneous polynomial in ˆ κ must be degree 3 j and hence in H j − , it follows that F = 0. Thus, no such F or F exists and the lemma follows in consequence. (cid:3) Remark 4.4.
Note that, after blow-up, degree 9 terms become degree 8 terms due to therescaling by dτ = rdt . Therefore, an obstacle to the foliation is occurring at degree 8 in theblow-up space. Remark 4.5.
The resonant term R h appearing in the intrinsic energy components have b c asa factor, implying these terms come from the first coupled monomial z z from the expandedpotential K in (4.4). Moreover, the absence of b ij , b in the normal form show K , K makeno contribution to the resonant terms in the normal form. Lastly, the only terms in X from(2.11) which are independent of the mass constants are the terms in the z i components. As R ,j EGULARISATION FOR SIMULTANEOUS BINARY COLLISIONS 15 are independent of the masses as well, they must then come from the kinetic energy terms. AHamiltonian with only kinetic terms has no singularities and is thus analytically regularisable.Under this reasoning, it must follow that the R ,j terms make no contribution to any finitedifferentiability of the block map π . This will be explicitly confirmed in the computation of theasymptotic series of π in Section 4.5. Remark 4.6.
There are three invariants up to order 8 given by(4.9) H = a − / h + a − / h and the transformed x and y variables. There is in fact another. The degree 3 polynomialˆ κ = z − z , an integral of X , was essential in the proof of Lemma 4.3. This integral can beextended to a degree 7 integral of the normal form X ,(4.10) κ ( z , z , h , h ) = 16 ( z − z ) + ( h + h ∗ ) κ ( z , z ) − ( h + h ∗ ) κ ( z , z ) ,κ ( z , z ) = 150365 z (cid:0) z − z z + 308 z (cid:1) . In a sense, the existence of this integral reflects Remark 4.5 and the heuristic argument thatsmooth block-regularisation may imply a type of flow box theorem. The integral κ will play acentral role in showing the R terms do not affect the 8 / Geometric Sketch of Proof
A procedure for determining the finite differentiability of the block map π : n + → n − is nowsketched. Recall from Proposition 3.1 the topological structure of the flow near C . If T is atubular neighbourhood of C , then Proposition 3.1 reveals a natural decomposition of T into fouroverlapping neighbourhoods T = U + ∪ U − ∪ V + ∪ V − , where U + , U − are tubular neighbourhoods of the normally hyperbolic invariant manifolds N + , N − respectively, and V + , V − are a tubular neighbourhoods of one of the two manifolds of hetero-clinic orbits. The decomposition splits T into regions where the flow is topologically equivalentto a neighbourhood of a normally hyperbolic saddle ( U + , U − ) and regions where the flow istopologically equivalent to a regular flow ( V + , V − ).Now, the collision-ejection manifold E := E + ∪ E − splits T into two disjoint segments, onecontaining V + and the other V − . It follows that π can be split into its restriction to these twosegments, say π + and π − . The key to the calculation is to introduce two intermediate sections,Σ +1 ⊂ U + ∩ V + and Σ +2 ⊂ V + ∩ U − , that are both transversal to the flow and intersect theheteroclinic connection (see Figure 4.3). In doing so, the block map π + can be decomposed into π + = D +2 ◦ T + ◦ D +1 where D +1 : n + → Σ +1 , T + : Σ +1 → Σ +2 and D +2 : Σ +2 → n − . Analogously, take π − = D − ◦ T − ◦ D − where D − : n + → Σ − , T − : Σ − → Σ − and D − : Σ − → n − . D +1 , D +2 are transitions near a normally hyperbolic manifold of hyperbolic saddles and T + isa regular transition map. It is now obvious what needs to be done; first compute the hyperbolicpassages D ± , D ± , glue them together with the relevant regular transition map T ± to get π ± ,and compare the one-sided asymptotics towards E ± of π + and π − . In fact, with some knowl-edge of hyperbolic transition maps, it is already possible from this sketch to see how the finitedifferentiability of the block map will creep in.4.4. The Block-map is Quasi-Regular
Due to their relevance to Hilbert’s sixteenth problem, the asymptotic properties of hyperbolictransition maps have been well studied for planar vector fields, see for instance [24, 7]. In this N + N − Σ Σ Σ +1 Σ +2 Figure 4.1.
A geometric sketch of the required computation. The dynamicslocal to the manifold has been split into hyperbolic regions (Σ → Σ +1 andΣ +2 → Σ ) and a smooth transition region (Σ +1 → Σ +2 ).context they are called Dulac maps. The Dulac maps have been shown to be quasi-regular (alsoreferred to as almost regular ). We extend the definition of quasi-regular as given by Roussarie[24] to higher dimensions. Definition 4.7.
Let ( x, u ) ∈ R + × R k . A germ of a map f : R + × R k → R + × R k at 0 is called quasi-regular in x if there exists a, b > f has a representative on [0 , a ) × ( − b, b ) k that is C ∞ on (0 , a ) × ( − b, b ) k .ii. lim x → f ( x, u ) = (0 , Au ), for some linear map A : R k → R k .iii. The components of f − (0 , Au ) = ( f , f , . . . , f k ) are asymptotic to the Dulac series,ˆ f k ( x, u ) = ∞ (cid:88) j =1 x ρ j P kj ( u, ln x ) , with 0 (cid:54) = ρ j ∈ R + , P kj is a sequence of polynomials in x with coefficients smooth in u , andthe sum is taken with ordering 0 < ρ j ≤ ρ j +1 .Define f as a quasi-regular homeomorphism in x if f is quasi-regular and P ( x, u ) = p ( u )with p ( u ) positive on ( − b, b ) k and A is invertible. Remark 4.8.
The set of all quasi-regular homeomorphisms in x , denoted by D , is a group undercomposition. Further, the group of diffeomorphisms with f (0 , u ) = (0 , Au ), for some invertible A , and ∂f ∂x (0 , >
0, is a subgroup of D .In the current work we wish to obtain asymptotic properties of transition maps near mani-folds of normally hyperbolic saddle singularities. Specifically, let N be a manifold of normally EGULARISATION FOR SIMULTANEOUS BINARY COLLISIONS 17 hyperbolic saddle singularities of co-dimension 2 inside some vector field X . Take u ∈ N anddenote the non-zero eigenvalues of X at u by λ ( u ) < < λ ( u ). Without loss of generalitytake coordinates ( x, y, u ) ∈ R × R × R k local to u such that u is at the origin and N is given by x = y = 0. Moreover, assume that centre-stable W s ( N ) and centre-unstable W u ( N ) manifoldsof u are aligned with the x, y axis.From the work of [5], many details about normal forms and transition maps near manifoldsof normally hyperbolic singularities are known. Let u ∈ N . A crucial object in the study of thetransition map is ρ ( u ) := − λ ( u ) /λ ( u ), the so called ratio of hyperbolicity . From Proposition3.1, the ratio of hyperbolicity of N + , N − takes the constant value 1 / u ∈ N + or N − . The following proposition from [5] gives the normal form near a point u ∈ N assumed to be 0. Proposition 4.9 ([5]) . If, for all u ∈ N , ρ = pq ∈ Q , with p, q co-prime, then there exists a C ∞ , near identity transformation Φ and a smooth time rescaling bringing X into the normalform, (4.11) ˙ x = x ˙ y = − ρy + 1 q y (cid:88) j ≥ α j ( u )( x p y q ) j ˙ u i = (cid:88) j ≥ δ ij ( u )( x p y q ) j , i = 1 , . . . , k with α j ( u ) , δ j ( u ) smooth functions in u . If ρ / ∈ Q then α j = δ j = 0 . Now, denote the normal form of X by X N and consider the transverse sections σ = { y = 1 } and σ = { x = 1 } in X N . Let ( x , u ) , ( y , u ) be coordinates on σ , σ respectively. Define thetransition map D : σ → σ which is given in components by y = D y ( x , y , u ) , u = D u ( x , y , u ) . As in the literature on planar vector fields, call this specific transition map the
Dulac map . Adiagram of the Dulac map is given in Figure 4.2 for the case N is dimension 1 inside R .Σ N W s ( N ) W u ( N ) Σ D Figure 4.2.
Diagram of the case N is co-dimension 2 in R The following proposition from [5] gives the asymptotic structure of the Dulac map D . Proposition 4.10 ([5]) . If, for all u ∈ N , ρ ( u ) = pq ∈ Q with p, q co-prime, then the Dulacmap D = ( D x , D u ) has the asymptotic series (4.12) D y ( x , u ) ∼ x ρ (cid:88) j ≥ P jy ( u ; ln x ) x jp D u ( x , u ) ∼ u + (cid:88) j ≥ P ju ( u ; ln x ) x jp , where P jy , P ju i are polynomial in ln x with coefficients smooth in u and P jy (0 , u ) = P ju i (0 , u ) =0 . Moreover, P jy and P ju i are polynomial in α l ( u ) , δ l ( u ) for l ≤ j with vanishing constant term.If ρ ( u ) / ∈ Q then P jy = P ju = 0 for all j ≥ . In either case D is quasi-regular. In contrast to the quasi-regularity of the hyperbolic transition maps, the regular transitionmaps T ± are smooth. This can be deduced from the fact that V ± has no singularities and aconsequent use of the flow-box theorem. Combining this fact with Remark 4.8, the followingtheorem is concluded. Theorem 4.11.
The block maps π ± are quasi-regular. More precisely, if Z = ( θ, x, h , h , y ) then, ¯ π + ( θ, Z ) ∼ γ Z + i + jρ 3, the asymptotic series of the block map is of the form θ i + j/ and possiblysome θ m ln θ terms. Moreover, Theorem 4.11 shows that the finite differentiability is generic.For the map to be smooth, and hence smoothly regularisable, it is required that all the γ i,j and γ m vanish. That is, finite differentiability should be expected and any smoothness of the blockmap is remarkable. Remark 4.12. It is important to study what mechanisms give rise to the coefficients γ i,j , γ m .The θ m ln θ terms come from the first resonance term α m ( u )( x p y q ) m − or δ km ( u )( x p y q ) m in thenormal form (4.11). If α m (resp. δ km ) does not vanish then a term of type γ mp +1 ( u ) θ mp +1 ln θ (resp. γ mp ( u ) θ mp ln θ ) arises in the Dulac map. This can be seen from Proposition 4.10. Thevalue of m is called the order of resonance . It will be shown that this mechanism does not causethe finite differentiability in the collinear 4-body problem. EGULARISATION FOR SIMULTANEOUS BINARY COLLISIONS 19 On the other hand, the γ i,j ( u ) coefficients arise from the interaction of the two Dulac maps D , D and the smooth transition T . Ignoring any higher order resonance terms, then fromProposition 4.10, the Dulac maps near N behave approximately like D ( θ, x, y, h , h ) = ( θ / , x, y, h , h ) , D ( r, x, y, h , h ) = ( r , x, y, h , h ) . Now, if the smooth transition has the form, say, T ( r, x, y, h , h ) = (cid:0) r, x, y, h + ar j , h + br j (cid:1) ,for some a, b ∈ R , then the composition of the maps π = D ◦ T ◦ D ( Z ) = (cid:16) θ, x, y, h + aθ j/ , h + bθ j/ (cid:17) , showing the arrival of a θ j/ term. It is this mechanism that will ultimately lead to the finitedifferentiability of π .4.5. Asymptotic Expansion of the Block Map With the block map realised as quasi-regular and the mechanisms leading to finite differentia-bility discussed, we are now in a position to determine the precise regularity of the simultaneousbinary collision singularities C for the collinear 4-body problem. We make a particular choiceof sections Σ , Σ transverse to E + , E − respectively and the intermediate sections Σ + i to explic-itly compute the hyperbolic transitions and smooth transition map. Recall the procedure fromSection 4.3:(1) Take the truncated normal form X and blow-up and desingularise the set of simultane-ous binary collisions to produce X θ . From Proposition 3.1, a collision manifold that istwo normally hyperbolic invariant manifolds connected by heteroclinics is obtained.(2) In the blow-up system X θ , compute the hyperbolic transition D +1 and D +2 near anarbitrary point of each normally hyperbolic invariant manifold N + , N − . This is doneby first computing the normal form X N to a desired order near each N + , N − and usingProposition 4.10.(3) In the blow-up system X θ , solve the variational equations to a desired order along( r, x, h , h , y ) = 0 to get the smooth transition map from Σ +1 to Σ +2 .(4) Compose the maps, π + = D +2 ◦ T + ◦ D +1 to obtain the asymptotic series of the blockmap π + .Whilst theoretically this procedure is sound, there are three obstacles faced in carrying out anexplicit calculation. The most easily resolved obstacle is with the blow-up method described inSection 3.2. Here, polar coordinates in ( z , z ) were introduced to blow-up the set of simultaneousbinary collisions. Whilst this certainly achieved the blow-up, it introduced trig functions intothe equations. This is at odds with the normal form methods needed to compute the hyperbolictransitions D + i . The normal form procedure is iterative on the homogeneous components ofthe vector field X θ . In order to get the homogeneous components, one needs to Taylor expandthe vector field X θ ; a computation that involves the Taylor expansion of many trig functions.Further, the expansion needs to be done twice at both N + and N − .Fortunately, one can circumvent both the introduction of trig functions and the need to do thenormal form procedure twice by performing a directional blow-up instead of the polar blow-up performed in Section 3.2. Firstly, it is more convenient to work with X rotated clockwise by π/ z , z ) plane through the transformation (˜ z , ˜ z ) = ( z + z , z − z ). This aligns N + , N − with θ = π, z (cid:48) = ˜ z + ˜ z + ˜ R , (˜ z , ˜ z )˜ z (cid:48) = − z ˜ z + ˜ R , (˜ z , ˜ z ) x (cid:48) = 0 h (cid:48) = b c a − / ˜ R h (˜ z , ˜ z ) h (cid:48) = − b c a − / ˜ R h (˜ z , ˜ z ) y (cid:48) = 0 where(4.14) ˜ R , (˜ z , ˜ z ) = 12 ( R , (˜ z + ˜ z , ˜ z − ˜ z ) + R , (˜ z + ˜ z , ˜ z − ˜ z ))˜ R , (˜ z , ˜ z ) = 12 ( R , (˜ z + ˜ z , ˜ z − ˜ z ) − R , (˜ z + ˜ z , ˜ z − ˜ z ))˜ R h (˜ z , ˜ z ) = R h (˜ z + ˜ z , ˜ z − ˜ z )The directional blow-up is no more difficult, it merely involves taking charts of S by per-forming the transformations,(4.15) ˜ z -direction, ( z , z ) = (ˆ u, ˆ u ˆ v )˜ z -direction, ( z , z ) = (¯ u ¯ v, ¯ v ) , followed by rescaling d ˆ τ = ˆ udτ, d ¯ τ = ¯ vdτ to produce the desingularised z and z directionalblow-ups ˆ X, ¯ X .The ˜ z and ˜ z -directional blow-up, and the order to which we know them after truncating thenormal form X at degree 9, are,(4.16)ˆ u (cid:48) = ˆ u (cid:0) v (cid:1) + ˆ u ˜ R , (1 , ˆ v ) + O (ˆ u )ˆ v (cid:48) = − v (cid:0) / v (cid:1) + ˆ u (cid:16) ˜ R , (1 , ˆ v ) − ˆ v ˜ R , (1 , ˆ v ) (cid:17) + O (ˆ u ) x (cid:48) = 0 + O (ˆ u ) h (cid:48) = b c a − / ˆ u ˜ R h (1 , ˆ v ) + O (ˆ u ) h (cid:48) = − b c a − / ˆ u ˜ R h (1 , ˆ v ) + O (ˆ u ) y (cid:48) = 0 + O (ˆ u )and(4.17)¯ u (cid:48) = 1 + 3¯ u + ¯ v (cid:16) ˜ R , (¯ u, − ¯ u ˜ R , (¯ u, (cid:17) + O (¯ v )¯ v (cid:48) = − u ¯ v + ¯ v ˜ R , (¯ u, 1) + O (¯ v ) x (cid:48) = 0 + O (¯ v ) h (cid:48) = b c a − / ¯ v ˜ R h (¯ u, 1) + O (¯ v ) h (cid:48) = − b c a − / ¯ v ˜ R h (¯ u, 1) + O (¯ v ) y (cid:48) = 0 + O (¯ v )respectively.As desired, system (4.16) and (4.17) are free of trig functions and polynomial in the variables.The collision manifold r = 0 from the polar blow-up has become the manifolds ˆ u = 0 and ¯ v = 0in the two charts. Further, the two normally hyperbolic invariant manifolds in the polar blow-up X θ have been reduced to a single normally hyperbolic manifold in the ˜ z -chart at (ˆ u, ˆ v ) = (0 , N . In the ˜ z -chart ¯ X , the collision manifold ¯ v = 0 is free of singularitiesand the flow is given trivially on it. A projection of the charts into the (˜ z , ˜ z ), (ˆ u, ˆ v ) and (¯ u, ¯ v )planes is provided in Figure 4.3.The second obstacle again involves the normal form procedure near the normally hyperbolicmanifold. In the computation, it is desired to choose the nicest possible sections Σ , Σ transverseto E + , E − respectively. Due to the complicated form of the asymptotic series given in Theorem4.11, it is wise to choose sections that minimise the complexity of D +1 and D +2 . To do this, letΦ be the normalising transform near the normally hyperbolic invariant manifold N and P ˜ z , P ˜ z the maps from X to ˆ X, ¯ X respectively. Then choose the sections(4.18) Σ = P − z ◦ Φ( σ ) , σ := {− } × ( − δ, δ ) × B δ (0) Σ = P − z ◦ Φ( σ ) , σ := { } × ( − δ, δ ) × B δ (0) Σ +1 = P − z ◦ Φ( σ ) , σ := ( − δ, × {− } × B δ (0) Σ +2 = P − z ◦ Φ( σ ) , σ := [0 , δ ) × { } × B δ (0) Σ Σ Σ +2 Σ +1 z z f Σ +2 Σ +1 uvD +2 D +1 (cid:98) Σ (cid:98) Σ (cid:98) Σ +2 (cid:98) Σ +1 ˆ u ˆ vP z P z Figure 4.3. Intermediate sections and their desingularisations for the upperblock map π + .with some choice of 0 < δ ≤ B δ (0) the open ball of radius δ in R . Further, denoterespectively by ˆΣ i , ¯Σ i the images of σ i in the ˜ z and ˜ z directional charts. See Figure 4.3 for adepiction of the sections. With these sections, coordinates on all images of each σ i can be givenby the normal form coordinates, between which the hyperbolic transition maps are simply theDulac map.The third obstacle still arises when trying to compute the transition T + between the normallyhyperbolic sectors. The transition is between σ and σ in the normal form X N . In order tocalculate the transition T , coordinates in X N need to be transformed to coordinates in ¯ X . Herelies the problem; at any iteration in its computation, Φ is only known up to some truncatedorder. Consequently, the image of σ i in ¯ X will only be known to some truncated order. Thekey to clearing this obstacle is to observe that the block map π + is independent of the choiceof intermediate sections Σ +1 and Σ +2 . Hence, there is freedom in the choice of these sections. Inparticular, sections in X N can be chosen by(4.19) σ ν := ( − δ, × {− ν } × B δ (0) σ ν := [0 , δ ) × { ν } × B δ (0) for 0 < ν (cid:28) νi , ˆΣ νi , Σ νi the image of σ i in ¯ X, ˆ X, X respectively. As the final series doesnot depend on this choice, then it must be that the series does not depend on ν , and inevitably, the limit ν → ν is a book keeping measure to ensure the images of σ i are known to sufficiently high order.With the obstacles adequately navigated, we are ready to proceed with the calculation of π + .4.5.1. Computing the hyperbolic transitions Take (0 , , x ∗ , h ∗ , h ∗ , y ∗ ) ∈ N in the ˜ z -directional blow-up ˆ X . We wish to compute thehyperbolic transition maps D ν , D ν in a neighbourhood of this point. From Proposition 4.10 andProposition 4.9 this can be done by first computing the normal form X N near N .It is possible to iteratively compute the normal form of ˆ X through the cohomological equationsorder by order. However, this can be avoided by using the approximate integral κ computed in(4.10). First, writing κ in the rotated ˜ z , ˜ z coordinates,(4.20) ˜ κ (˜ z , ˜ z ) = 13 ˜ z (3˜ z + ˜ z ) + ( h + h ∗ ) ˜ κ (˜ z , ˜ z ) − ( h + h ∗ ) ˜ κ (˜ z , − ˜ z ) , ˜ κ (˜ z , ˜ z ) = κ (˜ z + ˜ z , ˜ z − ˜ z ) . Projecting ˜ κ into the ˜ z -directional chart induces an integral to order O (ˆ u ) in ˆ X near thecollision manifold ˆ u = 0,(4.21) ˜ κ = ˆ u (cid:18) ˆ v + 13 ˆ v (cid:19) + ˆ u (cid:0) ( h + h ∗ ) ˜ κ (1 , ˆ v ) − ( h + h ∗ ) ˜ κ (1 , − ˆ v ) (cid:1) + O (ˆ u ) . As ˆ u = 0 is invariant it is possible to write ˆ u (cid:48) = ˆ uG (ˆ u, ˆ v ) for some smooth function G . Lookingat the form of ˆ u (cid:48) in (4.16) we have that, G (ˆ u, ˆ v ) = (cid:0) v (cid:1) + ˆ u ˜ R , (1 , ˆ v ) + O (ˆ u ) . The normal form to sufficiently higher order can then be computed by making the smooth timerescaling dτ = G (ˆ u, ˆ v ) − d ˜ τ , and introducing u, v, h i as the normal form coordinates through the near identity transformationΦ,(4.22) Φ : u = ˆ u, v = ˆ u − ˜ κ (ˆ u, ˆ v, h , h ) , h = h − u b c v a / , h = h + u b c v a / . Noting that dd ˜ τ ˜ κ = 0 + O ( u ), the normal form X N near N is given by,(4.23) u (cid:48) = u + O ( u ) v (cid:48) = − v + O ( u ) x (cid:48) = 0 + O ( u ) h (cid:48) = h + O ( u , v ) h (cid:48) = h + O ( u , v ) y (cid:48) = 0 + O ( u ) , with (cid:48) denoting derivative with respect to ˜ τ .The truncated normal form X N , is remarkably simple; it is merely the leading order terms ofˆ X . This truncation admits x, y, h , h and ˜ κ = u v as integrals. From these integrals the hyper-bolic transitions D ν : ( − , − v, x, h , h , y ) (cid:55)→ ( − u, − ν, x, h , h , y ) and D ν : ( u, ν, x, h , h , y ) (cid:55)→ (1 , v, x, h , h , y ) are easily computed. What needs to be determined is the order to which D and D is known if the normal form is truncated at order 9. From Remark 4.12 any resonantmonomial with non-vanishing coefficient appearing in the normal form will produce terms oftype u mp ln( u ), where p = 1 for D ν and p = 3 for D ν (because the ratio of hyperbolicity is 1 / X N and hencewe can conclude there are no terms in D ν of the from v ln v, v ln v , and no terms of the form u ln u, u ln u in D ν . EGULARISATION FOR SIMULTANEOUS BINARY COLLISIONS 23 It follows that the hyperbolic transitions are simply(4.24) D ν ( v, x, h , h , y ) = ( ν − / v / , x, h , h , y ) + O ( v ln v ) ,D ν ( u, x, h , h , y ) = ( νu , x, h , h , y ) + O ( u ln u ) . Smooth Transition Map We will use the ˜ z -directional blow-up, system (4.17), to compute the smooth transition T + : σ ν → σ ν . Recall that, σ ν := { v = − ν } , σ ν := { v = ν } , where v is the normal form coordinate of X N . The idea is to compute T + by considering T + : ¯Σ ν → ¯Σ ν with ¯Σ νi the images of P ˜ z ◦ P − z ◦ Φ − ( σ i ) parameterised by the normal formcoordinates. The transition T + will be computed using the variational equations. One cancompute and solve the variational equations using the coordinates ¯ u, ¯ v . However, by making useof the approximate integral ˜ κ , the task becomes much more manageable.Writing ˜ κ in the ˜ z -directional chart yields,(4.25) ˜ κ = ( 13 + ¯ u )¯ v + ¯ v (cid:0) ( h + h ∗ ) ˜ κ (¯ u, − ( h + h ∗ ) ˜ κ (¯ u, − (cid:1) + O (¯ v ) , with ddτ ˜ κ = 0+ O (¯ v ). Hence, by replacing ¯ v by the new coordinate w through the diffeomorphism w = ˜ κ (¯ u, ¯ v, h , h ) , and noting that ¯ u (cid:48) > v sufficiently small, system (4.17) is transformed to the non-autonomous system,(4.26) dwd ¯ u = 0 + O ( w ) dxd ¯ u = 0 + O ( w ) dh d ¯ u = 3 / b c a − / (cid:0) u (cid:1) − / ˜ R h (¯ u, w + O ( w ) dh d ¯ u = − / b c a − / (cid:0) u (cid:1) − / ˜ R h (¯ u, w + O ( w ) dyd ¯ u = 0 + O ( w )System (4.26) has an explicit solutions on the collision manifold given by( w, x, h , h , y ) = (0 , ¯ x , ¯ h , ¯ h , ¯ y ) , for each choice of xxx = (¯ x , ¯ h , ¯ h , ¯ y ) ∈ R . We seek a variation of this solutions in the w direction, that is, we want to compute the variation,(4.27) w ( w , xxx , ¯ u ) = w (1) ( xxx ; ¯ u ) w + (cid:88) w ( j ) ( xxx ; ¯ u ) w j x ( w , xxx , ¯ u ) = ¯ x + (cid:88) x ( j ) ( xxx ; ¯ u ) w j h ( w , xxx , ¯ u ) = ¯ h + (cid:88) h ( j )1 ( xxx ; ¯ u ) w j h ( w , xxx , ¯ u ) = ¯ h + (cid:88) h ( j )2 ( xxx ; ¯ u ) w j y ( w , xxx , ¯ u ) = ¯ y + (cid:88) W ( j ) ( xxx ; ¯ u ) w j with w (1) ( xxx , 0) = 1 and otherwise η ( j ) ( xxx , 0) = 0 , η = w, x, h , h , y , so that at ¯ u = 0 the vari-ation has the initial conditions ( w, x, h , h , y ) = ( w , ¯ x , ¯ h , ¯ h , ¯ y ). The coefficient functions η ( j ) can be computed using the variational equations. These equations are derived by differenti-ating both sides of (4.27) by d/d ¯ u , replacing the lhs by the non autonomous system (4.26) andsubstituting the variables ( w, x, h , h , y ) with their variations. The coefficients of w j are thenequated to get a linear, non-autonomous system in η ( j ) called the j th order variational equations. Due to the absence of lower order w terms in (4.26), it is immediate that, w (1) ( xxx ; ¯ u ) = 1 , w ( j ) ( xxx ; ¯ u ) = 0 , j = 2 , . . . , ,x ( j ) ( xxx ; ¯ u ) = y ( j ) ( xxx ; ¯ u ) = 0 , j = 1 , . . . , h ( j )1 ( xxx ; ¯ u ) = h ( j )2 ( xxx ; ¯ u ) = 0 , j = 1 , . . . , th variation of h i is given by h (8)1 = b c a − / ¯ H (8) (¯ u ) , h (8)2 = − b c a − / ¯ H (8) (¯ u ) , where, ¯ H (8) (¯ u ) = 3 / (cid:90) ¯ u (cid:0) u (cid:1) − / ˜ R h ( u, du = − / ¯ u (cid:32) (cid:0) ¯ u + 2¯ u − (cid:1) (3¯ u + 1) / − F (cid:18) , 23 ; 32 ; − u (cid:19)(cid:33) , and F is the hypergeometric function.In summary, the variation is computed as,(4.28) w = w + O ( w ) x = ¯ x + O ( w ) h = ¯ h + b c a − / ¯ H (8) (¯ u ) w + O ( w ) h = ¯ h − b c a − / ¯ H (8) (¯ u ) w + O ( w ) y = ¯ y + O ( w ) . One can think of the variation (4.28) as the flow ψ ¯ u ( w , xxx ) of the non-autonomous system(4.26) up to some order in w . With this view, a method for computing the smooth transition T + : ¯Σ ν → ¯Σ ν becomes apparent. Let i = ( w i , ¯ x i , ¯ h i , ¯ h i , ¯ y i ) be the coordinates on ¯Σ νi . Thenthere exists ¯ u i = ¯ u i ( i ) such that, = ψ ¯ u ( w , xxx ) = ψ ¯ u ( w , xxx ) . The transition T + inthese coordinates is hence computed as, = ψ ¯ u ◦ ψ − ¯ u ( ) . This computation yields,(4.29) w = w + O ( w )¯ x = ¯ x + O ( w )¯ h = ¯ h + b c a − / (cid:16) ¯ H (8) (¯ u ) − ¯ H (8) (¯ u ) (cid:17) w + O ( w )¯ h = ¯ h − b c a − / (cid:16) ¯ H (8) (¯ u ) − ¯ H (8) (¯ u ) (cid:17) w + O ( w )¯ y = ¯ y + O ( w ) . The transition T + : σ ν → σ ν will follow after replacing in (4.29) by their respective pa-rameterisation through P ˜ z ◦ P − z ◦ Φ − ( σ i ). Let ( − u , − ν, x , h , h , y ) , ( u , ν, x , h , h , y )be the normal form coordinates on σ , σ respectively. Using the fact that w = ˜ κ / = uv / andsubstituting into (4.29) it follows that u = u + O ( u ) . It is also evident x = x + O ( u ) , y = y + O ( u ). To get the h transitions, we need to explicitlycompute the parameterisation. The first step is to find the inverse of Φ from its definition in EGULARISATION FOR SIMULTANEOUS BINARY COLLISIONS 25 (4.22). We have, − ν = 13 ˆ v (3 + ˆ v ) + O (ˆ u ) = ⇒ ˆ v ∼ − ν + O ( ν , u ) . Then, using that (¯ u, ¯ v ) = P ˜ z ◦ P − z (ˆ u, ˆ v ) = (ˆ v − , ˆ u ˆ v ) we obtain¯ u ( ν ) = ( − ν ) − + O ( ν , u ) , ¯ u ( ν ) = ( ν ) − + O ( ν , u ) . Again using the normal form coordinates (4.22), the h parameterisations are computed as¯ h = h + u b c ν a / + O ( ν , u ) , ¯ h = h − u b c ν a / + O ( ν , u ) . A similar expression is obtained for ¯ h , ¯ h . Finally, substituting each parameterisation into(4.29) the transition map T + : σ ν → σ ν is explicitly computed as(4.30) u = u + O ( u ) x = x + O ( u ) h = h + b c a − / H ( ν ) u + O ( u ) h = h − b c a − / H ( ν ) u + O ( u ) y = y + O ( u )where H ( ν ) = 43295 ν + (cid:16) ¯ H (8) ( − ν − ) − ¯ H (8) ( ν − ) (cid:17) + O ( ν ) = − · / √ π Γ ( − / / ν / + O ( ν ) . Gluing Together At last we are in a position to give the asymptotic expansion of the block map π + . Composingthe maps π + ν = D ν ◦ T + ν D ν and taking the limit ν → π + ν = D ν ◦ T + ν ◦ D ν ( v, x, h , h , y )= D ν ◦ T + ν (cid:16)(cid:16) ν − / v / , x, h , h , y (cid:17) + O ( v ln v ) (cid:17) = D ν (cid:16) ν − / v / , x, h + ˜ b c a − / v / , h + ˜ b c a − / v / , y ) + O (cid:16) ν / , v ln v (cid:17)(cid:17) = (cid:18) ν (cid:16) ν − / v / (cid:17) , x, h + ˜ b c a − / v / , h + ˜ b c a − / v / , y ) + O (cid:16) ν / , v ln v (cid:17)(cid:19) lim ν → π + ν = (cid:16) v, x, h + ˜ b c a − / v / , h + ˜ b c a − / v / , y (cid:17) + O ( v ln v ) , where ˜ b c = − b c · / √ π Γ( − / / . With this calculation, and noting that ˜ b c is a strictly positivefunction of the masses, we have shown the main theorem. Theorem 4.13. For any choice of masses, the simultaneous binary collision is precisely C / -regularisable in the collinear 4-body problem. Concluding Remarks A new proof of the C / -regularity of simultaneous binary collisions in the collinear 4-bodyproblem has been given. In the process, new results about the problem have been shown. The C / differentiability is now known to hold for any choice of masses and for almost all directionalderivatives of π . The exception is when v = 0, that is, a derivative is taken in the direction alongthe set of collision orbits E + . This last result was known to Elbialy [12]. Moreover, a heuristic of Martinez and Sim´o on the crucial role of the first coupling term b c z z was proved by explicitlycomputing the asymptotics of the block map.We sought a more geometric proof to that in [18]. It is now clear that is not possible toconstruct a set of integrals local to the set of singularities. Another geometric notion essentialto the proof was an investigation of hyperbolic transitions near normally hyperbolic manifoldsof fixed points. From this work, the following can now be concluded about the mysterious 8 / • The 1 / N + , N − . • The 8 results from the inability to construct an invariant foliation normal to the set ofsimultaneous binary collisions C at order 8 in the intrinsic energies h i .A physical interpretation of the non-smoothness is as follows. Near the simultaneous binarycollision the energy of the individual binaries before and after collision is a non-smooth functionof a measure of their difference in coordinates. Expressing the final variable v on the section u = ± Q i gives to leading order v = t / − t / , t i = Q i M i k i . Here t i is just an abbreviation, but it has units of time and can be considered as the leadingorder term in the solution Q i ( t i ) near collision Q i = 0.Whilst the explanation of the finite differentiability given is certainly succinct, the computationrequired to prove the theorem is overly cumbersome. 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