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NONCOLLISION SINGULARITIES IN THE NEWTONIAN

N-BODY PROBLEM: RESULTS AND PERSPECTIVES

JINXIN XUE

1. Introduction

In this paper, we survey the state of the art of the Painleve conjecture on theexistence of noncollision singularities in Newtonian N-body problem. Consider Npoint masses mi, i = 1, 2, . . . , N , with position vector Qi ∈ R3. The equation ofmotion is

mid2Qidt2

=

N∑j=1,j 6=i

Gmimj(Qj −Qi)‖Qj −Qi‖3

.

It is a fundamental problem in celestial mechanics to describe the set of initial condi-tions of the Newtonian N-body problem leading to global solutions. The complementto this set splits into the initial conditions leading to the collision and non-collisionsingularities.

It is shown in [Sa1] that the set of initial conditions leading to collisions has zeromeasure. Much less is known about the non-collision singularities. The main moti-vation for our work is provided by following basic problems.

Conjecture 1. The set of non-collision singularities has zero measure for all N > 3.

Conjecture 2. The set of non-collision singularities is non-empty for all N > 3.

Conjecture 1 is mentioned by several authors, see e.g. [Sim, Sa3, K]. It is knownin [Sa1] that Conjecture 1 is true for N = 4.

Conjecture 2 was explicitly mentioned in Painleve’s lectures [Pa] where the authorproved that for N = 3 there are no non-collision singularities. Soon after Painleve,von Zeipel showed that if the system of N bodies has a non-collision singularity thensome particle should fly off to infinity in finite time (see for instance [G3] for a proofof the two results).

The first breakthrough towards proving Conjecture 2 was made by Mather andMcGehee [MM], where the authors constructed a collinear four-body problem inwhich initial conditions can be found such that the four bodies escape to infinity

Date: October 27, 2015.

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in finite time. However, infinitely many collisions are not avoidable. Later, Xiain [Xia] constructed a spatial five-body problem and independently Gerver in [G1]constructed a planar 3N -body problem, in which non collision singularities areshown to exist. We will explain these works in more details in Section 5. TheN = 4 case of Conjecture 2 remains open. In a series of papers [DX] and [Xue],based on an ideal model of Gerver in [G2], we give an affirmative answer to thePainleve conjecture in the N = 4 case. Since the papers are long, we explain thekey ideas using some non rigorous but plausible arguments in this paper.

1.1. Our work. Based on [G2], we show the existence of non collision singularitiesin a planar two-center-two-body problem (2C2BP) in [DX] and a planar four-bodyproblem (4BP) in [Xue]. In both models we assume m1 = m2 = 1 and m3 = m4 =µ � 1. One of the light bodies (Q3 or Q4) is captured by Q2 and the other istraveling back and forth between Q1 and Q2. In the 2C2BP model, we fix Q1 at(−χ, 0) where χ � 1 and Q2 at (0, 0), while in the 4BP model, all the four bodiesmove under their mutual gravitational forces.

Figure 1. The configuration of the models

1.1.1. Main theorem. Now let us state our main theorem. We will exhibit a richvariety of singular solutions. Fix a small ε0. Let ω = {ωj}∞j=1 be a sequence of 3sand 4s.

Definition 1 (Singular solutions). we say that (Qi(t), Qi(t)), i = 1, 2, 3, 4, is asingular solution with symbolic sequence ω if there exists a positive increasingsequence {tj}∞j=0 such that

(1) in the 4BP case,(i) t∗ = limj→∞ tj <∞.(ii) |Q3 −Q2|(tj) ≤ ε0, |Q4 −Q2|(tj) ≤ ε0.

(iii) For t ∈ [tj−1, tj ], |Q7−ωj − Q2|(t) ≤ ε0 and {Qωj (t)}t∈[tj−1,tj ] windsaround Q1 exactly once.

(iv) supt |Qi(t)|, |Qi(t)| → ∞ and as t→ t∗, i = 1, 2, 3, 4.(2) In the 2B2CP case, we replace (iv) above by

(iv′) supt |Qi(t)| → ∞ and as t→ t∗, i = 3, 4.

NONCOLLISION SINGULARITIES 3

During the time interval [tj−1, tj ] we refer to Qωj as the traveling particle and toQ7−ωj as the captured particle. Thus ωj prescribes which particle is the travelerduring the j trip.

We denote by Σω the set of initial conditions of singular orbits with symbolic se-quence ω. Our main theorem is stated as follows.

Theorem 1 ( [DX,Xue]). For both the 2C2BP and 4BP models, there exists µ∗ � 1such that for µ < µ∗ the set Σω 6= ∅.Moreover there is an open set U in the phase space and a foliation of U by two-dimensional surfaces such that for any leaf S of our foliation Σω ∩ S is a Cantorset.

In the rest of the paper, we fix a sequence of symbols and always call Q4 the travelerand Q3 the captured particle.

The paper is organized as follows. We describe Gerver’s ideal model and give heuris-tic arguments revealing the hyperbolicity in Section 2. In Section 3, we explain ourframework of partially hyperbolic dynamics and the proof of the main theorem. InSection 4, we explain the idea of proving the two main technical lemmas of theprevious section. We use two mechanisms of producing hyperbolicity, the scatteringand kicked shears. In Section 5, we revisit the previous works of Mather-McGehee,Gerver and Xia. In Section 6, we explain various remaining detailed problems inthe proof of the main theorem. In Section 7, we first formulate two questions onnon collision singularities, then we further explain our two mechanisms of producinghyperbolicity using billiard dynamics.

2. Gerver’s model and its hyperbolicity

In this section, we describe Gerver’s ideal model in [G2] and explain heuristicallyits hyperbolicity.

2.1. Gerver’s model.

In [G2], Gerver proposed the following model (see Figure 2). Let us send Q1 toinfinity so that we focus on the interaction of Q2, Q3, Q4. We assume m3 = m4 =µ = 0 so that Q3 and Q4 do not interact with each other unless they come to exactcollision. When they collide, we treat their collision as an elastic collision meaningthat during the collision process, energy and momentum conserve. Otherwise, theirmotions are standard Kepler elliptic and hyperbolic motions respectively.

We assume that the asymptotes of the hyperbolic motions of Q4 before and afterthe elastic collision with Q3 are horizontal. Moreover, we require that the majoraxis of the Q3 ellipse stays always vertical.

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(a) Angular momentum transfer (b) Energy transfer

Figure 2. Gerver’s construction

Gerver’s construction consists of two steps. In the first step, (see Figure 2 (A)),Q3 moves on the ellipse with smaller eccentricity (rounder) and Q4 comes to collidewith it at a particular point that we call Gerver’s collision point. After collision,Q4 flies off to negative infinity with horizontal asymptote and Q3 moves on anotherellipse with smaller semi minor and the same semi major. In the second step, (seeFigure 2 (B)), Q3 moves on the ellipse obtained from the first step and Q4 comesto collide with it at another Gerver’s collision point. After collision, again Q4 fliesoff to negative infinity with horizontal asymptote and Q3 moves on a final ellipsewith the same eccentricity as the initial one but smaller semi major. If we zoom inthe final ellipse and reflect it along the x-axis, we get back the initial ellipse, so weexpect to repeat this procedure for infinitely many steps.

For Kepler elliptic motion, energy is related to the semimajor through the relationE3 = − 1

2a . Suppose we can iterate the above procedure for infinitely many steps,we get that the energy of Q3 grows to −∞ like E3 ∼ −λn, where λ > 1 is the ratioof the semi majors of the initial and final ellipses. Energy conservation implies thatthe energy of Q4 grows to ∞ like E4 ∼ λn. For most of the time Q4 is far from thetwo large bodies so that most of its energy is kinetic energy and the speed of Q4

grows exponentially like |Q4| ∼ λn/2.In our 2C2BP, the distance between Q1 and Q2 is fixed. So the time it takes Q4 tocomplete one return is ∼ χ/λn/2, which decays exponentially so that infinitely many

returns can be completed in finite time and |Q4|, |Q3| grow to infinity in finite time.

In our 4BP, the velocity of Q4 changes direction by almost π when Q4 turningaround Q1, momentum conservation implies that a fixed proportion of energy istransferred to Q1 and the speed of Q1 grows like |Q1| ∼ µλn/2, which is still much

slower than |Q4| ∼ λn/2. So Q4 can still complete a return within time χ(λ−cµ)−n/2


for some constant c. Infinitely many returns can be completed in finite time and anon collision singularity can be constructed.

2.2. Heuristic arguments. To see how we can carry out Gerver’s strategy forinfinitely many steps in the µ > 0, χ < ∞ case, we notice there are two sources ofstrong expansion in the problem. The methods of hyperbolic dynamics is often usedin celestial mechanics (see for instance [Mo1]).

2.2.1. The first expanding direction, prescribing angular momentum. The first ex-panding direction comes from the hyperbolicity of the hyperbolic Kepler motion.Namely, if we shoot a bunch of orbits of Q4 towards Q1 with the same initial veloc-ity and slightly different y component of the initial position, then the bunch of orbitswill diverge after turning around Q1 (see Figure 3 (A)). This actually enables us tochange the vertical component of Q4 arbitrarily when Q4 comes to a neighborhoodof the pair Q2, Q3.

2.2.2. The second expanding direction, phase stretching and synchronization. Thesecond expanding direction is a bit subtler. Let us use ψ ∈ [0, 2π) to denote thephase Q3 on its ellipse.

Let us look at Figure 3 (B). Imagine that Q4 and Q3 collide at a point slightlydifferent from Gerver’s collision point and we require again the outgoing asymptoteof Q4 to be horizontal. We expect that the semi major a+ of the Q3 ellipse aftercollision is slightly different from that of Gerver’s case, i.e.

(1)∂a+

∂ψ6= 0.

Kepler’s law a3

T 2 = 1(2π)2

implies that the two ellipses (red and black in Figure 3 (A))

have different periods, hence during one period of Gerver’s standard elliptic orbit,Q3 on the other ellipse has a small phase difference. It takes long time (of order χ)for Q4 to complete a return and come to the next collision with Q3. During this longtime, two different elliptic motions have accumulated a huge phase difference. Thisis the second expanding direction along which a small phase difference is stretchedto a huge phase difference. The second expanding direction allows us to synchronizeQ3 and Q4. Namely, by adjusting the phase of collision in the present step slightly,we can arrange that Q3 and Q4 come to the same point at the same time in the nextstep. We also note that if we adjust the phase in the present step slightly further,Q3 and Q4 may miss each other in the next step, but if we continue to adjust thephase in the present step furthermore, it is possible to control Q3 and Q4 to cometo close encounter again in the next step, but differ from the previous encounter by2π in phase. All these adjustments are done in a very small interval of the phasevariable since the phase stretching rate is huge, as we will see in the next section.This is the reason why we get a Cantor set as initial conditions.

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(a) First expanding direction (b) The second expanding direction

Figure 3. Expansions

3. Partial hyperbolicity

In this section, we sketch the proof of the main theorem based on two technicallemmas. Our idea is to compute the derivative of the Poincare map and showthat there is a two dimensional invariant subspace that is strongly expanding underthe map. These two strongly expanding directions are described in the previoussection. The phase space is obtained by reducing the translation invariance, fixingthe zeroth energy level and picking the Poincare section {x = −2}. We will introducecoordinates later to parametrize the phase space of the 4BP as T ∗(T3×R2) and thatof the 2C2BP as T ∗T3. We will not consider all the points in the section {x = −2}as initial condition. Instead, we pick a small number δ and define U1(δ), U2(δ) as δneighborhoods of Gerver’s first and second collision points respectively, traced backto the section {x = −2} along the flow. We define two maps G and L called theglobal map and local map respectively. See figure 4. The Poincare section {x = −2}cuts the orbit into two different pieces. The right piece is defined as the local mapL and the left piece is defined as the global map G. The Poincare return map isdefined as the composition P = G ◦L whose domain is contained in Uj(δ), j = 1, 2.We also define a renormalization map R to zoom in the position space by λ andslow down the velocity by dividing

√λ, where λ is the energy of the Q3 after two

steps of interactions in Gerver’s construction.

3.1. C 1 control: the derivatives of local and global maps.


Figure 4. Poincare sections

Lemma 3.1 (Lemma 3.1 of [DX, Xue]). If the incoming asymptote θ− and theoutgoing asymptotes θ+ satisfy |θ−| ≤ Cµ and |θ+ − π| ≤ Cµ for some constant C,then for x ∈ Uj , j = 1, 2 being a initial point on the section {x = −2} there existlinear functionals l(x), a vector fields u(x) and matrices B(x) which are uniformlybounded such that

(2) dL(x) =1

µu(x)⊗ l(x) +B(x) + o(1), 1/χ� µ→ 0.

Lemma 3.2 (Lemma 3.2 of [DX,Xue]). If the y coordinates of the initial and finalpositions of Q4 are bounded when apply G, Then there exist linear functionals l(x)

and ¯l(x) and vectorfields u(y) and ¯u(y) such that

(3) dG(x) = χ2u(y)⊗ l(x) + χ¯u(y)⊗ ¯l(x) +O(µχ), 1/χ� µ→ 0,

where we denote x the initial point and y = G(x) the final point.

Let us forget about the o(1), O(µχ) perturbations in dL, dG respectively, whichcan be handled by introducing invariant cones. After application of dG we geta plane span{u, ¯u}. We next apply dL to get a plane span{u, BY } where Y ∈(Kerl)∩span{u, ¯u}. To apply dG again, we want to guarantee that the planespan{u, BY } is not collapsed into a line or a point so that we need the follow-ing transversality condition

(4) (Kerl ∩Kerl) is transversal to span{u, BY }.

This condition is equivalent to det

(l(u) lBY¯l(u) ¯lBY

)6= 0, which can be verified by

working out the vectors and matrices explicitly (see Section 3.3 of [DX] and Section3.4 of [Xue]).

3.2. The Cantor set construction. With the above lemmas on dG, dL, and thetransversality condition, we establish the strong expansion of the Poincare map. Thenext lemma is stated for the 4BP, whose 2C2BP version is obtained by changingthe phase space from T ∗(T3 × R2) to T ∗T3.

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Lemma 3.3 (Lemma 2.5 of [DX,Xue]). There are cone families K1 on Tx(T ∗(T3×R2)), x ∈ U1(δ) and K2 on Tx(T ∗(T3 × R2)), x ∈ U2(δ), each of which contains atwo dimensional plane, such that

• Invariance: dP(K1) ⊂ K2, d(R ◦ P)(K2) ⊂ K1.• Expansion: If v ∈ K1, then ‖dP(v)‖ ≥ cχ‖v‖. If v ∈ K2, then ‖d(R ◦P)(v)‖ ≥ cχ‖v‖.

The cone can be defined as the set of vectors forming some small angle η with theplane span{u, ¯u}.

Definition 2. We call a C1 surface S1 ⊂ U1(δ) (respectively S2 ⊂ U2(δ) admissibleif TS1 ⊂ K1 (respectively TS2 ⊂ K2).

With these preparations, we can now construct the Cantor set in the main theorem.We can control the orbit of Q3 such that it always stays close to that in Gerver’smodel, hence we have exponential energy growth described in Section 2.1. Take apiece of admissible surface S2 and look at the pre image (RP2)−1S2, which consistsof many copies of tiny (due to expansion in Lemma 3.3) pieces of admissible (dueto invariance in Lemma 3.3) surfaces. The reason why we have many copies isbecause one of the expanding direction, the phase variable ψ, is defined up to 2π(see Section 2.2.2). Finally, our Cantor set is constructed as limj(RP2)−jS2j , whereS2j is a piece of admissible surface at the 2j-th step.

3.3. The coordinates. The proof of the technical Lemma 3.1 and 3.2 involvesestimating the solution to the variational equations of the Hamiltonian equations incertain suitable coordinates. We explain in this section our choice of coordinates.

3.3.1. The Jacobi coordinates. The basic strategy is to treat the motion of Q4 as aperturbed Kepler motion focused at the mass center of Q2 and Q3 when it is closerto Q2 than to Q1, and treat it as a perturbed Kepler motion focused at Q1 whenit is closer to Q1 than to Q2. This is easy for the 2C2BP. In the 4BP, we needto introduce a new set of coordinates to reduce the translation invariance of thesystem. When Q4 is closer to Q2 we use the classical Jacobi coordinates (see theupper figure in Figure 5). Namely, we denote x3 = Q3 − Q2, define x4 to be thedistance from Q4 to the mass center of Q3 and Q2 and define x1 to be the distancefrom Q1 to the mass center of Q4, Q3, Q2. There is a corresponding linear changeof coordinates on the momentum side v1, v3, v4 to make the new set of coordinatessymplectic in the reduced phase space. Similarly, we construct coordinates when Q4

is closer to Q1 (see the lower figure of Figure 5 and see Section 4 of [Xue] for moredetails).

The advantage of the coordinates is that they reduce the Hamiltonian system intothree Kepler motions with “controllable” perturbations. To see the meaning of


Figure 5. Jacobi coordinates eliminating the translation invariance

“controllable”, we show one example. There is a term in the potential µ|Q4−Q3| .

When we integrate over time t assuming Q4 moves away from Q3 linearly in t,the integral blows up like ln t as t → ∞. However, in the new coordinates, the

interaction between x4, x3 is given by a term of the form µ 〈x3,x4〉|x4|3 whose t integral

is now convergent as t → ∞ knowing that x3 is bounded and x4 is linear in t (seeSection 4 and 6.2 of [Xue]).

3.3.2. The Delaunay coordinates. Now we have perturbed Kepler motions for (x, v)3,1,4.We next introduce the classical Delaunay coordinates for the hyperbolic motion(x4, v4) and for the elliptic motion (x3, v3). Delaunay coordinates (L, `,G, g) arethe action-angle coordinates for Kepler motion, hence are symplectic. Delaunaycoordinates have clear physical and geometrical meanings. Namely, L2 is the semimajor a; LG is the semi minor b; G is the angular momentum; g is the argumentof periapsis and ` is the mean anomaly, which is proportional to the area swiped by

the moving particle within time t. In particular, the Hamiltonian H = |P |22 −

1|Q| of

two-body problem has a simple form in terms of Delaunay coordinates, which canbe written as H = − 1

2L2 for elliptic motion, and H = 12L2 for hyperbolic motion.

We convert (x, v)3,4 into Delaunay coordinates (L, `,G, g)3,4. By fixing an energylevel, for instance 0, and picking Poincare sections, we can eliminate L4, `4 from ourlist of variables by solving L4 as a function of other variables and treating `4 as thenew time. So we get six variables for the 2C2BP. And for the 4BP, we stick to x1, v1

without reducing the rotational invariance, so we get totally ten variables.

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4. Two mechanisms of producing hyperbolicity: scattering and kickedshears

In this section, we explain the Lemma 3.1 and 3.2 on the derivatives of local andglobal map under various simplifying assumptions. These simple derivations lie inthe heart of our lengthy calculations in [DX,Xue].

4.1. The derivative of the local map, scattering.

Figure 6. shadowing the elastic collision

4.1.1. Elastic collision. Elastic collision is a collisional process where energy conser-vation and momentum conservation are obeyed. We consider elastic collision in R2

only. Consider two equal mass particles Q3, Q4 undergoing an elastic collision. Wedenote by v−3 , v

−4 ∈ R2 the velocities before collision and v+

3 , v+4 the velocities after

collision. Energy and momentum conservation requires |v+3 |2 + |v+

4 |2 = |v−3 |2 + |v−4 |2,and v+

3 +v+4 = v−3 +v−4 . Suppose we know v−3 , v

−4 and we want to know v+

3 , v+4 . We

have four variables and three equations so that we have one free variable denotedby α. Solving the equations, we get

(5)

{v+

3 = 12R(α)(v−3 − v

−4 ) + 1

2(v−3 + v−4 ),

v+4 = −1

2R(α)(v−3 − v−4 ) + 1

2(v−3 + v−4 ),where R(α) =

[cosα − sinαsinα cosα

]


is a rotation by angle α, where α can be either of the two angles smaller than π inthe first figure in Figure 6.

4.1.2. The Rutherford Scattering. When we turn on the Newtonian interaction be-tween the two particles, we get hyperbolic Kepler motions to shadow the elasticcollision picture (see the lower two figures in Figure 6). The momentum and energyare still conserved. The rotation angle α, now measured as the angle formed by twoasymptotes of the hyperbolas, is no longer a free variable but determined by theinitial conditions. To determine the rotation angle α, we introduce the Rutherford’sscattering formulas. We consider two particles of masses m3,m4 interacting via theNewtonian potential − k

|Q−| , where Q− = Q3 − Q4 is the relative position. The

rotation angle is given by (see [LL])

(6) α = 2 arctan

(1

|x0|

), where x0 = b

m∗v2−

k,

m∗ = m3m4m3+m4

is the reduced mass, v− = v−3 − v−4 is the relative velocity of the two

bodies and the quantity b = v−|v−| ×Q− is the most important quantity in scattering

theory called impact parameter, which is a measurement of the closest distancebetween the two particles during the scattering process.

For the Q3 and Q4 interaction, we have

k = µ2, m∗ = µ/2, v2− = O(1), as µ→ 0.

We expect the deflection angle α to be bounded away from 0, π, so we must haveb = O(µ). When we compute how the outgoing velocities v+

3 , v+4 depend on incoming

velocities v−3 , v−4 , we notice that v+

3 , v+4 has implicit dependence on v−3 , v

−4 through

b and also explicit dependence. So we compute the derivative as follows

∂α

∂b=−2(m∗(v−)2/k)

1 + x20

= O

(1

µ

),

∂α

∂(v2−)

=2bm∗/k

1 + x20

= O(1), as µ→ 0.

∂(v3, v4)+

∂(v3, v4)−= O

(1

µ

)∂(v3, v4)+

∂α⊗ ∂b

∂(v3, v4)−+ (derivative not through b).

This calculation shows that the most effective way to change v+3,4 significantly is to

change the impact parameter b.

This simple derivation explains the structure of the derivative of the local map inLemma 3.1. Moreover, both the tensor part and the B part in dL can be computedexplicitly. However, when we take into account of the perturbation coming from Q1

andQ2, we must have a oµ→0(1) perturbation to equation (5). It is nontrivial to showthat this perturbation is also C 1 small so that it enters dL as a o(1) perturbation.It turns out that in the limit 1/χ � µ → 0, we have u is the partial derivativeof all the outgoing variables with respect to α and l is the partial derivative of b

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with respect to all the outgoing variables, where α and b are defined for the relativemotion of Q3 and Q4. Details of the proof are presented in Section 10 of [DX,Xue].

4.2. The O(χ) term in dG, shears. The O(χ) term in dG involves mainly themotion of Q3. We forget about the perturbation coming from Q4 for simplicity to seethe ideas. Actually, many pages are devoted to controlling the perturbations in [DX]and [Xue]. The Hamiltonian for elliptic Kepler motion in Delaunay coordinatesH3 = − 1

2L23. We integrate the Hamiltonian equations from time 0 to time T to get

L3 = 0,˙3 = 1

L33,

G3 = 0,

g3 = 0.

=⇒

L3(T ) = L3(0),

`3(T ) = `3(0) + TL3

3(0),

G3(T ) = G3(0),

g3(T ) = g3(0).

The derivative matrix has the following decomposition

(7)∂(L, `,G, g)3(T )

∂(L, `,G, g)3(0)=

1 0 0 0

− 3TL4

3(0)1 0 0

0 0 1 00 0 0 1

= − 3T

L43(0)

0100

⊗ [1, 0, 0, 0] +O(1).

We have estimate − 3TL43(0)

= O(χ) if we choose T = Oχ→∞(χ) as the time for Q4 to

complete a return. This gives the χ¯u ⊗ ¯l part of the dG. We see that the vectors¯u and ¯l can be obtained explicitly. It turns out that in the limit 1/χ� µ→ 0, we

have ¯u→ ∂∂`3

and ¯l→ dL3. See Section 6, 7 of [DX] and Section 7, 8 of [Xue].

4.2.1. Shear-induced chaos. Notice the above matrix (7) is only a shear, which hasno hyperbolicity. There is a mechanism called shear-induced chaos (see [LY] for theterm), namely, hyperbolicity can be created by a shear combined with a kick. Tosee why, we multiply another matrix (kick) to a shear matrix to get

(8)

[1 0χ 1

] [1 10 1

]=

[1 1χ χ+ 1

]The resulting matrix has two eigenvalues, O(χ) and O(1/χ). Actually, to get aneigenvalue of order χ, we only need to require that in the above second matrixthe bold 1 entry is nonzero and independent of χ, while the other entries can bearbitrary.

In our case, dG gives us a shear matrix and dL gives us a kick so that the com-position of local and global map give rise to an expansion with rate χ. We have a

transversality condition (4), which essentially requires ∂a+

∂ψ 6= 0 in (1) (see the proof

of the transversality condition in Section 3.3 of [DX] and Section 3.4 of [Xue]). Thisrequirement amounts to requiring the bold 1 entry is nonzero in (8).


4.3. The O(χ2) term in dG, scattering. Here we explain the O(χ2) term in dG.Details can be found in Section 8 of [DX] and Section 9 of [Xue].

We pick two more Poincare sections {x = −χ/2} to cut the orbit of the global mapinto three pieces denoted by (I), (III), (V ) (see Figure 4). The (I), (V ) pieces oforbits are considered as a perturbed hyperbolic Kepler motion focused at the masscenter of Q2, Q3, while the (III) piece of orbit is treated as a perturbed hyperbolicKepler motion focused at Q1. We need two coordinates changes when the orbitcrosses the sections denoted by (II), (IV ).

The χ2 term in dG comes mainly from the motion of Q4 as described in Section 2.2.1.Since L, ` are reduced by fixing an energy level and picking a Poincare section, wehave only Delaunay variables G, g to characterize the motion of Q4, whose meaningsare respectively the angular momentum and the argument of periapsis (direction)of the hyperbolic motion. We define the angle of asymptotes of the hyperbola as

(9) θ = g ± arctanG

L

since g is the direction of the symmetric axis of the hyperbola and 2 arctan ba =

2 arctan GL is the angle formed by the two asymptotes. We make the following

simplifying assumptions.

• Assume L is a constant, say, 1.• Assume v4 = (±1, θ).• Assume the two hyperbolas to the left and right of the section {x = −χ/2}

share the same asymptote angle θ.

In [DX, Xue], removing these assumptions produces only small errors in the esti-mates.

Let us now look at Figure 4. When we change coordinates from (I) to (III), we arechanging the focus of the hyperbolic motion of Q4. During this coordinate change,the two different hyperbolic motions share nearly the same asymptotes, which weidentify. So we first convert (G, g) in the right to variables (G, θ) in the right, thenwe convert (G, θ) in the right to (G, θ) in the left, and the last step is to convert(G, θ) back to (G, g) in the left. We summarize the steps as follows. The coordinateschanges (II) from the right (subscript R) to the left (subscript L) is the compositionof (iii)(ii)(i),

(G, g)R(i)−→ (G, θ)R

(ii)−→ (G, θ)L(iii)−→ (G, g)L,

where the maps (i), (ii), (iii) are given explicitly as follows:

(i) :

{GR = GR,

θR = gR − arctanGR,(ii) :

{GL = GR + χθ,

θR = θL,(iii) :

{GL = GL,

gL = θL − arctanGL.

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The g and θ relations are obtained from (9), while GL = GR + χθ comes from thedefinition of angular momentum since we move the origin from (0, 0) toQ1 = (−χ, 0),

GL = vL × xL = vR × (xR − (χ, 0)) = GR − vR × (χ, 0) = GR + χθ.

The derivatives for (II) = (iii)(ii)(i) is

D[(iii)(ii)(i)] =

[1 0] 1

] [1 χ0 1

] [1 0] 1

],

where we use ] to denote some constants that can be computed explicitly. Similarly,for matrix (IV ) going from the left to the right, we get

D[(iii′)(ii′)(i′)] =

[1 0] 1

] [1 −χ0 1

] [1 0] 1

].

Next we look at the derivative of the composition (IV )(III)(II) assuming D(III) =id, since G, g are constants of motion for the Kepler problem (Unfortunately, thisassumption is wrong. It turns out that the perturbation from the pair Q2, Q3 has anontrivial contribution to D(III). We can get an explicit expression for G, g partof D(III), which is not id but does not cause any trouble).

D[(IV )(III)(II)] =

[1 0] 1

] [1 −χ0 1

] [1 0] 1

]·[

1 0] 1

] [1 χ0 1

] [1 0] 1

]=

[1 0] 1

] [1 + ]χ −]χ2

] −]χ+ 1

] [1 0] 1

]= ]

[1 0] 1

](χ2

[1

1/χ

]⊗ [1/χ, 1] +O(1)

)[1 0] 1

].

This calculation shows the χ2 part of dG. We see that the u, l can again be calculatedexplicitly. It turns out that in the limit 1/χ� µ→ 0, we have l→ dθ+

4 , where θ+4

is the outgoing asymptote of the Q4 hyperbola after close encounter with Q3, and

u→(

01×8; 1,− L4

L24+G2

4

)T, which shows that after the application of the global map

there is a linear relation between G4 and g4 forced by the fact that the asymptoteof Q4 must be close to horizontal when Q4 comes close to Q2.

5. Previous results revisited

In this section, we briefly review the previous results [MM,Xia,G1,G4]. We mainlyfocus on revealing the strong hyperbolicity. Please refer to the original papers andthe surveys [SX,C,G3] etc for more details.

5.1. Mather-McGehee’s collinear four-body problem. We describe the modelof [MM] as follows. In a line, there are four particles labeled by Q1, Q4, Q3, Q2 fromthe left to the right. The pair Q2, Q3 is doing Kepler collision-ejection periodicmotion if there are no external force. Next, Q4 travels back and forth between Q3


and Q1 to exchange energy. We may think the Mather-McGehee model as a limitingcase of our four-body problem where the eccentricities of all the Kepler motions goto one.

The total phase space has eight dimensions since each body has one-dimensionalposition and one-dimensional velocity. We get rid of two dimensions by imposingthe momentum conservations. Two more dimensions can be removed by fixing anenergy level and pick a Poincare section. Hence we get four dimensions phase spacefor the Poincare map and the Poincare map can be defined by taking a snapshot atthe time when Q4 is crossing the mass center (set as the origin) for the left to theright or vice versa.

To see the idea clearer, we make the following simplifications: we assume Q3 and Q2

interact via Newtonian potential but the Q4, Q1 interaction, and Q4, Q3 interactionare elastic collisions. We next think the Q3, Q4 motion as a Kepler elliptic motion.We only need two variables L, ` to characterize their relative motion, where L2 isthe semi major and `4 characterizes the relative position of the two bodies. TheDelaunay variables are singular at the points `4 = 0, π, but still well behaved awayfrom the two points. The two other variables (G, g) degenerate to zero in thecollinear case. With the two variables L, `, we still have the Hamiltonian equationsL = 0, ˙ = 1

L3 so that the derivative of the Poincare map is a shear matrix and theQ3, Q4 interaction will give rise to a kick, hence we again get hyperbolicity from (8).In [MM], the authors constructed Cantor set along a curve.

5.2. Gerver’s five-body problem and 3N-body problem. In [G4], Gerver pro-posed a planar five-body model in hopes of finding non collision singularities. Themasses satisfy m1 � m2 � m3 < m4 = m5. The three large bodes form a triangle.One light body m2 is captured by m3. The orbit of m2 is nearly circular. Anothereven lighter body m1 is traveling along the edge of the triangle. We may arrangethe phase of Q2 when close encountering Q1 such that Q1 gains energy from thepair Q2 − Q3 so that the semi-major of the nearly elliptic orbit of Q2 − Q3 getssmaller. After reducing the all the invariance (space and time translation, rotation)and picking a Poincare section, we get a Poincare map in a phase space of twelvedimensions. In the twelve dimensions, we can still get two strongly expanding onesin the same way as we did in the 4BP. One source of strong expansion is given by thehyperbolic motion when Q1 turns around Q4 and Q5. The other is again inducedby the shears. Namely, a small phase difference of Q2 at the time of close encounterwith Q1 will result in a huge phase difference of Q2 at the next encounter. Weexpect a complete proof of the existence of a Cantor set of non collision singularitiescan be given using our method.

Later, Gerver proved the existence of non collision singularity in a planar 3N -bodyproblem in [G1] by introducing symmetry to the above five body model. In the3N-body problem, N(� 1) pairs of elliptic motion Q2i − Q3i, i = 1, 2, . . . , N are

16 JINXIN XUE

(a) The spacial 5-body problem ofXia (b) The planar 3N-body of Gerver

Figure 7. Xia’s and Gerver’s models

located around the vertices of a regular N -polygon, and there are N smaller bodiesQ1i, i = 1, 2, . . . , N , each travels from one pair to the next (see Figure 7 (B)).The symmetry reduces the dimension of the phase space where the Poincare map isdefined from twelve to seven. The two sources of strong expansion are the same asthe five-body case (see [G1,G3]).

5.3. Xia’s spatial five-body problem. Xia constructed in [Xia] a model of spatialfive-body problem and proved the existence of non collision singularities therein.

The model is as follows. The equal mass pair Q1, Q2 and the equal mass pair Q4, Q5

are always parallel to the x − y plane, and a smaller mass Q3 is restricted to thez-axis (see Figure 7 (A)). The underlying mechanism for non collision singularitiesis called the Sitnikov-Alekseev mechanism by [G3]. The motions of each of thepairs are highly eccentric. The idea is that Q1, Q2 achieve their closest distanceimmediately after Q3 crosses the plane containing Q1, Q2, so that Q1 and Q2 exerta strong force to pull Q3 back. The same thing is expected to occur when Q3 comesclose to the pair Q4, Q5. To guarantee this scenario to occur for infinitely manytimes, an accurate timing is needed. This is again given by adjusting the relativephase of each pair when Q3 is getting close, hence two strongly expanding directionscan be obtained by the shear-induced chaos mechanism.


6. Miscellaneous issues

Besides the main hyperbolicity problem discussed in the previous sections, thereare still several remaining issues to handle. In this section, we briefly discuss thefollowing problems: (1) resolving the singular behavior near double collision, (2)excluding collisions and show the existence of returning orbit, (3) controlling thedynamics of x1, v1, (4) controlling the shape of Q3 ellipse to maintain the almostself-similarity, (5) switching the roles of Q3 and Q4 to construct symbolic sequence,(6) measuring the Cantor set.

6.1. Singularity resolution. Both Mather-McGehee’s and Xia’s results involvetriple collision blow-up technique, which is not needed in our work. However, weneed to analyze the two-body close encounter of Q3 and Q4. In two-body problem,when two bodies come to collision, Q→ 0 and |P | → ∞. So when µ > 0 but small,the Kepler problem is a singular perturbation of the elastic collision. Traditionally,people use Levi-Civita or Moser ( [Mo2]) regularization to resolve the singularity.However, the these methods do not fit into our framework very well. Instead, weregularize double-collision using hyperbolic Delaunay coordinates. This methodworks well when the eccentricity e is bounded away from 1, which is the case ofthe Q3, Q4 interaction. However, for the Q1, Q4 interaction, since the eccentricitye can be close to 1, the Delaunay coordinates are singular close to double collision.It can be shown that the variables G4, g4 are not singular and since we eliminateL4, g4 from our list of variables, we never take derivatives with respect to `4 in theHamiltonian equations and the variational equations.

6.2. Collision exclusion and the existence of returning orbit. We need toexclude the possibility of the collisions between the pair Q3, Q4 and the pair Q1, Q4.The pair Q3, Q4 is easily done using the formula (6) since the rotation angle α is notclose to π in Gerver’s construction so that b/µ is bounded away from zero using (6).Next, we explain how to exclude the Q1, Q4 collision. Recall that in the two-bodyproblem, if two bodies collide they will bounce back. Suppose we have a collisionof Q1 and Q4, then we reverse the time for the piece of orbit coming to collisionand compare it with the bouncing back orbit. We work in Delaunay coordinatesso that the collisional singularity is resolved. We can measure the difference of thetwo orbits by integrating the variational equations (derivative of the Hamiltonianequations). It turns out that the deviation of the two orbits is at most O(µ) when wetrace the orbit to the section {x = −2}. However, the returning orbit that we wantshould stay close to Gerver’s model and the y coordinates of Q4 for two consecutivevisits to the section {x = −2} differ by a O(1) number. So we conclude that thereis no collision between Q1 and Q4. See Section 11.1 of [DX] and Section 6.6 of [Xue]for more details.

18 JINXIN XUE

6.3. The angular momentum conservation and the dynamics of x1, v1.When we analyze the total angular momentum G =

∑i=3,1,4 vi × xi where xi, vi

are the Jacobi coordinates in the right case, we find that (x1, v1) will eventuallyget all the angular momentum when G 6= 0. Indeed, G3, G4 decay exponentially tozero in the way λ−n/2 as L3 does. In the renormalized system, G3, G4 are rescaledto the unit size hence G1 behaves like λn/2. On the other hand, the renormalizedv1 has unit length and the renormalized x1 grows like (λ + cµ)nχ0 for some con-stant c > 0, where χ0 is the initial distance, λ appears due to the rescaling and cµappears due to the motion of Q1. This shows that the angle ∠(x1, v1) decays like

(λ+ cµ)−n/2/χ0, hence the motion of x1, v1 approaches a line in the limit. The case

G = 0 is different where G1 also decays to zero like λ−n/2 hence the angle ∠(x1, v1)decays like (λ+ cµ)−n/χ0.

6.4. How to control the shape of Q3 ellipse. In order to control the phasespace dynamics such that the image of the admissible surface always visits the fixedneighborhood U(δ), we look at our list of variables (L3, `3, G3, g3;x1, v1;G4, g4).First L3 is always rescaled to the unit size by the renormalization and x1, v1 can becontrol by the angular momentum in the previous Subsection. Next `3, G4, whichcan be used to parametrized the admissible surface, can be chosen arbitrarily due tothe strong expansion, and g4 is also determiend since the asymptote of Q4 is almosthorizontal. The only remaining variables are G3, g3, which we want to control suchthat they stay close to Gerver’s values.

We notice that the Q3 ellipse may deviate from Gerver’s standard case and lose self-similarity. The problem was notice by Gerver and solved in [G2]. Consider againGerver’s ideal model. We have four variables L, `,G, g to characterize the ellipticmotion. The variable ` is almost the same as the phase ψ which can be controlledby the strong hyperbolicity and L is related to the semi major which can alwaysbe rescaled to one applying the renormalization R. It remains to control G andg such that they do not deviate too far. The observation of Gerver is that duringeach collision, there is a phase of Q3 that can be adjusted. So after the two steps inGerver’s model, we get two phases ψ1, ψ2 on which the final orbit parameters ¯G, ¯g

depend smoothly. One only need to verify that the Jacobian ∂( ¯G,¯g)∂(ψ1,ψ2) 6= 0 in order

to control ¯G, ¯g through adjusting the phases ψ1, ψ2.

6.5. How to switch the roles of Q3 and Q4. We see from Figure (6) that thereare two hyperbolic motions shadowing the same elastic collision picture. This impliesthat by choosing the rotation angle α correctly, one can switch the roles of themessenger and the captured particle after each Q3 and Q4 close interaction. Forthis reason, for any given symbolic sequence ω, a Cantor set Σω of non collisionsingularities can be constructed.


6.6. The measure and Hausdorff dimension of the Cantor set. We noticethat each time when we apply the renormalizationR by zooming in the configurationspace by λ, the distance between Q1 and Q2 get multiplied by λ, hence χ in Lemma3.1 and 3.2 grows exponentially to infinity. Since χ is the expansion rate in Lemma3.3. In each step of the Cantor set construction, we preserves only 1/χ2 of thetotal measure on the initial admissible surface. Since 1/χ2 decays exponentiallyto zero, we conclude that the Hausdorff dimension of the Cantor set restricted toeach two dimensional admissible surface is 0. Hence our Cantor set of non collisionsingularities is a zero measure set of codimension 2 in the phase space.

7. Outlooks

In this section, we first ask two questions on the existence of non collision singularity.After that we relate our two mechanisms of producing hyperbolicity to that inbilliard dynamics.

7.1. Nonperturbative noncollision singularities. Since our technique is per-turbative and it is necessary that µ� 1, we ask the following questions.

Question 1: Are there noncollision singularities for a four-body problem in whichall the four bodies have comparable masses?

In fact it is possible that the following stronger result holds.

Question 2: Is it true that for any choice of positive masses (m1,m2,m3,m4) ∈RP 3 the corresponding four-body problem has noncollision singularities?

We need to develop some nonperturbative techniques for the first question and weneed to exploit the obstructions for the existence of noncollision singularities for thesecond.

7.2. Hyperbolic billiards. We have explained how the two mechanisms, scatter-ing and kicked shears, enable us to construct non collision singularities in celestialmechanics. In fact, they are the fundamental mechanisms in various hyperbolicdynamical systems. For instance, the latter mechanism was used in [D] to studythe random behavior of adiabatic invariants in the presence of resonances in slow-fast systems, where the hyperbolicity is created due to the combination of strongshearing away from resonances with the kicks near the resonances.

In hyperbolic (planar) billiard theory, the above two mechanisms are the only knownmechanisms for hyperbolicity. A billiard ball in a domain moves freely until it hitsthe boundary. Then the path is continued according to the reflection law of light.When the particle is moving freely the Jacobi field satisfies{

J(t) = J(0) + J ′(0)t

J ′(t) = J ′(0).

20 JINXIN XUE

(a) Sinai’s billiard (b) Bunimovich’s billiard

Figure 8. Billiard analogue of the mechanisms for hyperbolicity

When the particle collides the boundary, the Jacobi field behaves as follows (see forinstance Lemma 3.2 of [Do]) {

J+ = −J−J ′+ = 2k

sin θJ− − J′−,

where J− (J+ respectively) is the Jacobi field immediately before (after respec-tively) colliding the boundary, θ is the reflection angle and k is the curvature of theboundary. We compose the derivative of a collision followed by a free motion to get

(10)

[1 t0 1

] [−1 02k

sin θ −1

]=

[−1 + t 2k

sin θ −t2k

sin θ −1

].

Please compare (10) with (8) to see the similarity. The product in (10) has deter-minant 1 and trace −2 + 2kt

sin θ . When k < 0 and t > 0, we know that there aretwo roots whose modulus are not 1. This is the hyperbolicity mechanism for Sinai’sdispersing billiard (see [S] and Figure 8 (A), and compare with Figure 3a). Whenk > 0, there may be eigenvalues of modulus 1, for instance, we choose t = 2 sin θ/ksuch that the trace is 2. However, if we have that the free motion time t is greaterthan 2 sin θ/k, we again have eigenvalues that are not modulus 1. This is the hy-perbolicity mechanism for Bunimovich’s defocusing billiard (see [B, W] and Figure8 (B)).


Acknowledgment

I would like express my deepest thanks to my former thesis advisor Prof. DmitryDolgopyat for his constantly intensive guidance and support. My thanks also goto Professor A. Chenciner, A. Albouy and J. Gerver who read the manuscriptcarefully and gave insightful remarks. This work is supported by the NSF grantDMS-1500897.

References

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[C] Chenciner, Alain, A l’infini en temps fini. Seminaire Bourbaki 39 (1997): 323-353.[D] Dolgopyat, Dmitry. Repulsion from resonances. Societe mathematique de France, 2012.[Do] Victor J. Donnay, Using integrability to produce chaos: billiards with positive entropy, Com-

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the University of Chicago, Chicago, IL, 60637

E-mail address: [email protected]

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