The existence of oscillatory and superhyperbolic motion in Newtonian systems

JOURNAL

OF DIFFERENTIAL

EQUATIONS

82,

342-355

(1989)

The Existence of Oscillatory and Superhyperbolic Motion in Newtonian Systems DONALD G. SAARI AND ZHMONG XIA* Department

of

Received

December

Mathematics, Northwestern Evanston, Illinois 60208 5, 1985; revised

March

University, 21, 1988

We establish the existence of solutions for the collinear Newtonian four body problem where the maximum spacing between particles exceeds any given function of time as t--t co. As an example, there exist solutions where the maximum spacing separates faster than any constant multiple of time. Also, it is shown that the three and four body problems admit oscillatory solutions. 0 1989 Academic Press, Inc.

1.

IN-~R~DuCTI~N

Can solutions of the Newtonian n-body problem permit the distances between particles to exceed any constant multiple of time as t + co ? This is impossible for n < 3, but, as we prove here, there are solutions of the four body problem that violate this property of relativity. More generally, by using the fact that Newtonian systems allow velocities of any magnitude, we construct the appropriate symbolic or “chaotic” dynamics to answer three longstanding questions about the Newtonian n-body problem. The first is the issue just raised. Namely, if rjk(t) is the distance between particles j and k, and if R(t) = max(rjk(t)}, we show there are solutions where R(t)/t + co as t + co. If scalar multiples of t do not bound R(t), then what does? So the next obvious question, probably first raised by Harry Pollard [S], is whether R(t) admits any functional upper bound. For instance, will a scalar multiple of exp(exp(exp(t))), or 2exp(exp(exp(‘)))eventually bound R(t) as t -+ co ? We show that the answer is no; for the four body problem R(t) has no universal functional upper bound. We also prove that a special kind of motion, oscillatory motion, exists for the three and four body problem. Moreover, because all of our conclusions are based on a symbolic dynamic argument, our proofs establish the existence of several other kinds of motion, Finally, because it takes only minor modiftcations to extend our construction to all values of n 2 4, our basic conclusions hold for all n > 4. The notation is standard. Let m,, ri, vi be, respectively, the mass, posi* Current 02138.

address:

OO22-0396/89

$3.00

Department

of Mathematics,

342 Copynght 0 1989 by Academic Press, Inc. All rights 01 reproduction in any form reserved.

Harvard

University,

Cambridge,

MA

OSCILLATORY

AND

SUPERHYPERBOLIC

MOTION

343

tion vector, and velocity vector of the ith particle where it is assumed that the center of mas is fixed at the origin. The equations of motion are m,r,!’ = Vi U,

where U=xjck mjmk/rjk energy integral is

and Vi is the gradient T=$~m,vj?=U+h,

(1.1) with respect to ri. The

(1.2)

where h is a constant of integration. A prime motivation for our results is that they resolve several remaining questions about the evolution of n-body systems as t -+ co. To see what they are, recall from the asymptotic classification of the n-body system given by Saari [lo] and Marchal and Saari [3] that there are two general settings. The first one is superhyperbolic motion where both R(t)/t -+ cc and r(t) = min,,, rjk(t) -+ 0 as t + 00. (Because r(t) becomes arbitrarily small, such motion requires point masses.) Superhyperbolic motion emerges as a logical possibility for n > 4, but its existence is not established in these papers. (Indeed, the possible existence of orbits where lim sup R(t)/t = co has been raised at least since Painleve’s lectures [7] in the 1890’s.) Our first assertion proves that superhyperbolic motion exists. In fact, our second assertion proves that superhyperbolic motion admits no functional upper bound. Using this and an inequality due to Pollard [S], it follows that there is no restriction on how fast r approaches zero as t --+ co. Returning to the asymptotic classification of Newtonian orbits, it states that whenever the solution is not superhyperbolic, then each particle has a representation r,(t) = Ait + O(t2j3) where Ai is a vector constant, i = 1, .... n. This forces the particles to cluster according to the value of Ai. If Ai # A,, then the two particles mi and mj separate hyperbolically. If the motion is not completely hyperbolic (i.e., if Ai = Aj for some indices), then there is a further subdivision between parabolic and oscillatory motion. Parabolic motion is where the particles separate from each other like t2j3. (See Saari [ 1 l] for more details.) Oscillatory motion is where, for some indices, lim sup rij(t) = 00, while lim inf r,(t) < cc. In either setting, the distance between particles is bounded above by a scalar multiple of t2j3. Both motions occur in the three body problem; indeed, oscillatory motion first arose as a logical possibility in Chazy’s [ 1] development of the asymptotic behavior of the three body problem. It is easy to prove the existence of hyperbolic and parabolic motion in many settings, but the existence of oscillatory motion has been established only for the spatial three body problem (Sitnikov [12]). Sitnikov uses, quite strongly, a symmetrical model where one particle is restricted to the

344

SAARI

AND

XIA

z axis while the motion of the other two particles is such that the three particles always form an isoscles triangle. Again by use of symmetry of isosceles three body motion, R. Moeckel [6] uses different methods to establish such motion in another setting. For no other situations has oscillatory motion been shown to exist; however, Easton [2] and Robinson [9] have contributed to our understanding of this issue for the coplanar problem. We show here that oscillatory motion exists for the collinear three and four body problems, and, by extension, to all n-body problems for n > 3. All of our results concern the collinear problem, so we’ll assume that ri= (xi, 0,O) and vi = (ui, 0,O) where x1
OSCILLATORY

AND

SUPERHYPERBOLIC

MOTION

345

By using the fact that arbitrarily high velocities can be attained by the escaping particle, Mather and McGehee [4] proved the existence of motion for the collinear four body problem where R(t) --) co in finite time. In their construction, one particle, m3, commutes between a binary consisting of m, and m2 and an escaping particle m4. Particle m3 comes close to a triple collision, on the correct side of the collision manifold, so that m3 is ejected with a very high velocity. The particles are restricted to the x axis, so if the ejection velocity of m3 is sufficiently large, then m3 must eventually catch and collide with m4. After this binary collision, m3 rebounds. With simple computations involving the “center of mass” of the binary (m3, m4), it follows that if the mass of m3 is sufficiently small with respect to m4, and if m3 entered this binary collision with sufficiently large velocity, then m3 rebounds from its binary collision with m4 with a sufficiently large velocity to return to collide again with the binary (m,, m2). It is possible to force the time when m3 next collides with m2 so that the system has another near-triple collision on the correct side of this triple collision manifold. Thus, with an iteration argument, the motion is established. Note that their construction requires keeping the orbit arbitrarily close and on the correct side of the triple collision manifolds; in this manner they constructed a Cantor set of initial conditions leading to the desired orbit. Our examples use a similar construction, but we need a tighter control on the ejection velocity of m3. We show that orbits exist where not only when m3 returns to a near-triple collision is the trajectory on the appropriate side of the triple collision manifold, but also its distance from the manifold is within a tight bound-not too close, but close enough. We need control on the distance from the manifold to control the ejection velocity. For instance, to show that oscillatory motion exists for 12= 3, we need to ensure that m3 is ejected fast enough so that r,,(t) eventually exceeds any fixed constant. On the other hand, oscillatory motion requires m3 to return for another close encounter with a triple collision. This means we need to control the ejection velocity so that m3 will not escape. (The desired behavior is obtained by iterating this construction.) To prove the existence of superhyperbolic motion where R(t) exceeds any specified function g(t), we need to slow down the ejection velocity of m3 so that all the action isn’t concluded in a finite time-we need the motion to last infinitely long. On the other hand, this ejection velocity needs to be large enough so that R(t) grows faster than the specified g(t). To gain the required control over the ejection velocities, we use both sides of the triple collision manifold. To suggest how this is done, let n = 3 and x3 be the position of m3. Wherever xi = 0, particle 3 returns to the binary. While m3 is returning, the dynamics of the binary (m,, mJ is one of continual collisions and rebounds. The number of collisions that occur until m3 finally hits m2 depends on the value of x3; the larger the value, the

346

SAARI

AND

XIA

more time provided for m, and m2 to collide. Conversely, we can use the count of the number of collisions between (m,, mz) as a crude measure of the initial value of x3. In this manner, we can divide the values for x3 into intervals [a,, CI,, i]; if xX E [an, a n + i ), then (m 1, m2) has precisely n collisions until m3 hits m2. The end points of [a,,, a,, + i] correspond to initial conditions for xg where a triple collision occurs; one endpoint is where the triple collision occurs just as (m,, m2) started its nth collision, and the other is where the binary is entering its (n + 1)th collision. This means that if x3 is sufficiently close to the left endpoint of [a,,, a,,+ ,I, particle 3 arrives just after a binary collision, while if x3 is sufficiently close to the right hand endpoint, particle 3 arrives just before a binary collision. Consequently the two ends of the interval (a,, a,, + , ) h ave orbits on different sides of a triple collision manifold, and for x3 E (a,, a,+ I ) the orbit does not have a triple collision (at least during this pass of the binary). This means that the evolution of the set of initial conditions (a,, a,, ,) emerges from the close triple collision in a set that covers the total spectrum of possibilities ranging from where m, is ejected with arbitrarily high velocity to where m3 is ejected. Using continuity considerations, it follows that the original curve of initial conditions defines a new curve of initial conditions. A portion of this curve has a continuum where xi = 0. For this new curve, we can define Cd, ab+ 1] to be the distance of x3 where (m,, m2) will have k collisions from when xi = 0 to when m2 and m3 have their next collision. The new curve includes all intervals [a;, a; + , ] for k 2 1. Thus, for any value of n and k, there are orbits where r3 is first in [a,, a,+,] and then in Ca~~a~+Il. BY choosing k sufficiently large, R(t) becomes large. By choosing k small, the motion is slowed down. In this manner, the iterative symbolic dynamics is established to create almost any kind of behavior and to prove our assertions. While the above description emphasized the value of x3 after a triple collision, it could just as well have emphasized the other side of the x axis where m, approaches the binary (m,, m3). In this manner all sorts of other motion can be established. For instance, we can establish an effect where m3 collides with the binary, and m, is ejected. Then m, returns ot collide with the binary (m,, m3) and m3 is ejected. This can be made into a periodic motion, into oscillatory motion, etc. By using the full time axis, we can establish the existence of one kind of motion as t + -co, but another kind as t + co. Indeed, the kinds of motion that are substituted are nearly arbitrary; e.g., the motion could be parabolic (or superhyperbolic) as t + co, but, say, approaching a periodic orbit as t + + co. In this manner, the symbolic dynamics characterizes all three-body collinear motion.

OSCILLATORY AND SUPERHYPERBOLIC MOTION

347

2. THE BASIC CONSTRUCTION APPLIED TO OSCILLATORYMOTION In this section, we derive the properties of the driving mechanism that lead to the various conclusions, and we use them to establish the existence of oscillatory motion. In particular, we prove that THEOREM 1. For the collinear three body problem, there are positive masses and initial conditions so that lim sup R(t) = GOwhile lim inf R(t) < co as t+cO.

We know from Chazy [l] that the motion in Theorem 1 occurs only if h < 0; this is our assumption on the energy level. For fixed h ~0, let Q denote the energy manifold in phase space. Because T 3 0, it follows from Eq. (1.2) that on Q (2.1)

As already indicated, our basic approach is to develop the appropriate symbolic dynamics to establish our claims. This is based on when a specified particle reaches a maximum distance. To do this, choose any continuous curve r, in Q parameterized by u E (-co, 00) with the following properties. For some value of DQc,/lhl, if cr> D, then x3=c(, and x;=O; if a<-D, then X,=X, x2-x12x3-xX2, x,-x,2x,--x,, and xi = 0; and if Ial < D, the curve is selected in such a fashion to satisfy the continuity restriction. (Thus the particle whose position is given by CI is at critical point, and the distance between the other two particles defines the minimum spacing between particles.) The existence of such curves follows from elementary properties of the energy integral and 52. For instance, if tl> D, the choice of r, determines continuous functions & gj, j= 1, 2, that satisfy the following: x3 = a, xi = 0, xj =fi(a), xi = g,(a), ~~~raz;l;>r?(~)>r;O, m,fi(a)+m2f2(u)+m3a=0, m,g,(a)+m,g,(a) 3

i(m, g: + m2g:) = Wi(cO, .fAu), Co+ h. A point on r, is denoted by the parameter value LX An initial condition tl> D has x; < 0, so in the subsequent motion m3 collides with m2 within finite time. A similar description requires a collision between m, and m2 for a < -D. Call the particle designated by a the commuting particle and the other two particles the binary. We define N,(M) to designate the number of collisions between the binary until the commuting particle collides with m2. Namely, if tl > D, let N,(a) be the number of collisions between m, and m2 that occur up to and including the time when m,

348

SAARI

AND

XIA

collides the first time with m3. If c1< -D, let -N,(a) be the number of collisions between m2 and m3 that occur up to and including the time when m2 collides the first time with m,. Thus the sign of N,(a) designates which particles form the binary. In the definition of N,,, the last collision in the count between the binaries could be a triple collision that involves the commuting particle. These initial conditions are designated by +$ = { GIE r,, 1in the definition of N,(a), the collision is a triple collision}. LEMMA 1. (i)

(ii) (iii)

lNO(a)l + co as Ial + co. No(a) changes value onZy at a~[~.

‘?&,is not bounded above nor below.

Proof. Assertion (ii) follows from the continuity of solutions with respect to initial conditions. (Our system is regularized, so the solution is regular at binary collisions.) Assertion (iii) follows from (i) and (ii). Thus, we need only prove (i), and, by symmetry considerations, only for a 3 D. The proof is simple. If tl is sufficiently large, then until m3 is sufficiently close to m,, the minimum spacing, r(t), between the particles is determined by the binary, (m,, mz). Thus, the equations of motion for m3 are those for a perturbed central force problem. (The perturbation term is of the order r(t)/x:.) Because x;(O) = 0, the length of time until x3 = 42 is essentially that given by the central force problem. In particular, for any value T, a can be selected sufficiently large so that it takes longer than T units of time before x3 = a/2. On the other hand, as long as the binary is sufficiently far from x3, the equation for the binary is also a perturbed form of the central force equation. Because O 2c, Ihl, there is a maximum time, 6, between collisions. By choosing a sufficiently large, N,,(a) 3 T/o. Because T is arbitrary, the conclusion (i) follows immediately. Very little is specified about curve r,,, so it may be that there are no values of a so that N,(a) = 0. For instance such a situation occurs if for small values of Ial, the curve has m, at the midpoint between the other two particles and the velocities of m,, m2 are very large and pointed toward each other. On the other hand, with a compactness argument, it ;follows immediately that there is an integer q so that for all curves r, there are both positive and negative values for a so that [No(a)1 , D. Choose the value of a so that m2 is midway between m, and m,. Because the curve must be on the energy surface, the maximum value for a = c2 occurs if x; = xi = 0. If x; > 0, then the particles start at a distance closer than c2 and m, must first approach m3 before it returns to collide with m, ; indeed, for this curve and a, it may

OSCILLATORY

AND

SUPERHYPERBOLIC

MOTION

349

be that N,(a) = 0. If xi < 0, the particles start closer than c2 and m, approach each other. The collision between m, and m2 is an elastic collision, so the particles rebound with a large velocity that forces m2 toward m3. To try to allow N,(a) = 2, we need to reduce the rebounding velocity toward m3 and increase the starting distance between them; this requires x;(O) =x;(O) =O. In this situation, if m, =m3, then it follows from standard arguments that the first collision is a triple collision. (The solution has x,(t) = 0 and xl(t) = -xX(t).) If m3 > m,, then N,(a) = 0. So the worst case situation is if m3 4 m 1 and xi = x’, = 0. The proof can then be completed by going to the extreme situation of the restricted problem. (In what follows, we use the value of q = 1, but this is not necessary.) Although No(a) is an unbounded, step function, it need not be monotonic. (In part this is because we imposed very few conditions on r,.) Nevertheless, there are intervals where N,, does have a monotonic flavor. LEMMA 2. (a) the range of N,.

The value of D can be selected so that 1 and - 1 are in

(b) For any integer n, InI 22, there are ai, ai~Gf$,, a;< al, so that N,,(a;) = n, N,(ai) =n + 1, and if a E (ah, a:), then a# VO, and N,,(a) = n. Moreover, for E > 0, there is an n so that aA > F and a’.., c - E. The value of D specified in part (a) of this lemma is the choice made for the curve r,. Proof: (a) This is discussed above. Actually, all we use is that there is a finite value of q so that q and -q are in this range. (b) We prove this part of the lemma for n >O; a similar proof holds for negative values of n. Let ai = inf{a E G$,/N,,(a) =n + l}, and al:=sup{aE%$/a
To gain control over the independent variable, let T,,(a) be the time when the commuting particle collides with m2. The following lemma is immediate. Indeed, it can be shown that T,,(a) is continuous for all a, even those values that lead to a triple collision. However, Lemma 3 suffices for our purposes. LEMMA 3. T,(a) is continuous for a 4 %$,, and T,(a) + co as Ial + 00. rf K is a bounded set on the real axis, then {T,(a) 1a E K} is bounded.

Next we follow the evolution of the set of initial conditions (a;, a:). For each a from this interval, select a particular point from the trajectory after

350

SAARI

AND

XIA

time To(a) that is a candidate for defining a new curve, rl, with properties similar to that of r,. This means we want to select a point at some time after the near-triple collison where the expelled particle has velocity xi(t) = 0. However, it isn’t sufficient to define the point by this property alone. This is because the dynamics of the near-triple collision involves N repeated binary collisions. To see the difficulty, suppose m3 eventually will be expelled. After the first collision, xi > 0. However, when m2 emerges from a collision with m, to collide again with m3, the acceleration on m3 becomes infinitely large and xi + - co. Thus at some time after T,(a), but before m3 is expelled, x;(t) = 0. We do know that for initial conditions c1 sufficiently close to an endpoint, there will be precisely N binary collisions (where, again, N is determined by the masses of the particles) before a particle is expelled. On the other hand, this need not be true for values of a with orbits not sufficiently close to the triple collision manifold. The next terms are introduced to handle both possibilities. Define E(a) = inf{ t 2 T,,(a) I the system has precisely N binary collisions for t> T,,(a)}. Let D(a) = {t> To(a)! two particles collide and the next binary collision is between the same two particles.} For a E (a,,, a,,+ 1), let C,(a) = inf(t > Min(D(a), E(a)) Ix;(t) = 0} and C,(a) = inf(t > Min(D(a), E(a)); x;(t) =O}. Let r(a) = min(C,(a), C,(a)). The term C,(a), j= 1, 3, refers to particle mj; it is the first time after T,,(a) and any near triple collision dynamics when xi = 0 (so mj starts its return toward mz). Thus, T(a) defines a new situation similar to that described by points on the curve Z-,. According to the dynamics of near triple collisions described in the introductory section, it is possible for r(a) = co. This situation occurs when the trajectory is arbitrarily close to the triple collision manifold, so one particle is ejected with arbitrarily high velocity. Because this ejected particle escapes, C,(a) = C,(a) = co. Indeed, it is easy to show that t(a)= 00 iff R(r) + cc for t > r,. In this section, we consider only those points where T(a) < co. So, for j= 1,3, let Gj= {the point on the trajectory defined by the initial condition a at t = z(a) 1 InI > 1, a E (a, .a,,+ 1), z(a) = Cj(a) < a}. LEMMA 4. There exist values of the masses, m, , m2, m3, positive constants D, and D,, and a continuous curve rl such that rl is in the orbit of the interval (a,,, a,,+ 1). rl can be parametrized with PE (-CO, co) in the following manner. If /I< -D,, then xl=/? and thepoint is in G,. If p2D3, then x3 = /I and the point is in G3.

ProoJ Binary collisions solution exists at least until some time after T,(a). Thus, are continuous with respect

are regularized so for a E (a,, a,, + , ), a smooth the trajectory has a triple collision which is at on any compact interval of time, the solutions to a.

OSCILLATORY

AND

SUPERHYPERBOLIC

MOTION

351

By using McGehee’s results (described in the introductory section), there are choices of masses so that the following occurs. For those initial conditions arbitrarily close to one end of the interval (LX,, ~1,+ , ), m, is expelled with arbitrarily high velocity, while for the initial conditions arbitrarily close to the other endpoint, m3 is expelled with arbitrarily high velocity. The next terms are introduced to consider escaping particles on a compact interval of time in order to invoke continuity of solutions with respect to a. So, for any integer M> c,/lhl, let T,(U) =min(z(cl), inf{t > To(cl)i either x2(t) - x,(t) = M, or x3(t) - x2(t) = M}. For each CIE (a,, a,,+ 1), TM(a) < co. There are orbits where x2 -xi = A4 and orbits where x3 -x2 = A4. It follows from the continuity of solutions with respect to initial conditions and the intermediate value theorem that there is a smooth curve in the orbit of (a,, a,+ 1) with the desired properties of ri, except that the desired parameter values range only over -Md /I < M. The conclusion follows by letting the value of M approach infinity. We now complete the proof of Theorem 1. Each interval (a,, a,,+ 1) on the curve r,, gives rise to a new curve r,. The same construction is repeated. The subscripts for the intervals [a,, a,, 1] are augmented with an index i to indicate that it is on curve ri where the original intervals (akcoJyakco)+ 11= (ak(oxoy akco)+ l,. ). Thus, given any sequence of integers (k(O), k(l), ...}. for each i30 there is a curve ri+, with the properties of r. that is contained in the image of (akcij,i, akciJ+ 1,i). Conversely, the set of points in (ak+ l),i- 1, ak(;- I)+ l,i- 1) that is mapped to CQ;),i, ak(i)+ l,iI is a closed, bounded subset. By continuing to take inverse images, this defines the closed, bounded subset of (a,(,,, a,+,)+ ,) where at the jth stage, the image in [a of the is,,,,,,yy$+(k;;.j1 f;r ‘_= ;’ ...’i. Thus by using the first i terms , i - , .... we obtain a nested sequence of closed, bounded subseis of ‘(&o) , a,@)+ i). By a standard theorem about nested, decreasing sequences of compact sets, there exists an initial condition a so that the behavior described by the sequence is realized. Theorem 1 follows from Lemma 1 by selecting any sequence {k(O), k(l), ...} such that lim sup /k(i)1 = co as i+ co. Several other results follow immediately from the symbolic dynamics. To include the possibility of a particle escaping, we introduce the symbols - cc and cc to correspond, respectively, to where m, escapes to infinity and to where m3 escapes to infinity. Thus, for example, the sequence { -5, 6, - 7, 8, -2, cc } corresponds to the initial conditions where m, and m3 alternate in the roles as commuting particle until m3 is ejected to infinity. The integer corresponds to the number of collisions of the binary from the time the commuting particle is at a critical distance until it collides with m,. The n-body system is time reversible, so a similar construction holds as

352

SAARI AND XIA

t -+ - co. Thus, a sequence { .... k( -2), k( - l), k(O), k(l), ...} describes the evolution of a trajectory for all time. For instance, the sequence { - co, -3, 5, 1, 8, ...> corresponds to where m, is captured (equivalently, m, escapes as t -+ - co) and the integers describe the subsequent behavior. The following is immediate from the above construction. THEOREM 2. For the collinear three body problem, there exist choices of masses so that there are initial conditions realizing any sequence consisting of the nonzero integers and f co.

Other results, such as the structure of the set of initial conditions leading to a triple collision and the usual Cantor set assertions about the initial conditions in Theorem 2 (when f co are not admitted), are immediate by applying standard techniques from symbolic dynamics. 3. MOTION FASTER THAN TIME-SUPERHYPERBOLIC ORBITS The dynamical behavior described in Section 2 was made possible by the oscillating behavior created by a commuting particle reaching a maximum distance and then returning to collide with m2. This is the only way a particle can be forced to return for n = 3. On the other hand, if the particle escapes, then the attractive force law forces the velocity of the ejected particle to decrease. Thus the escape velocity determines an upper bound for v so that eventually R(t) < Ct. The four body problem admits more possibilities. The necessary recurrance to establish a symbolic dynamic description of possible motion can be created with another mechanism-collisions. For instance, if the commuting particle, m3, collides with m4, the rebound could force m3 to return to collide again with m2. With this kind of oscillatory behavior, we can harness the large velocities resulting from a close triple collision to determine new kinds of motions for the four body problem. In doing so, the symbol cc from Section 2 is further relined into a set of symbols describing different ejection velocities. It is in this manner that we establish our remaining claims. THEOREM 3. For the collinear 4 body problem, there exist choices of masses and initial conditions so that R(t)/t + COas t -+ CO.

To explain the restrictions on the masses, note that if there is a binary collision between m3 and mq and if x2 is sufficiently far away (so the center of mass, C3,4, of the binary (m3, m4) has the acceleration, C;,,, of the order (x3 -x2)-*), then C;,, both just before and after the binary collision is essentially the same. The collision is essentially an elastic one. This means

OSCILLATORY

AND

SUPERHYPERBOLIC

MOTION

353

that relative to C;,, the velocity of each particle is essentially the same as the velocity relative to C;,, entering the collision. A simple computation shows that m3 must be lighter than m4 to allow such motion to exist. A similar argument shows there must be restrictions on m3 relative to the masses of m, and m2. The same requirement occurs in the Mather-McGehee paper. Thus it follows from their computations with these center of mass considerations that a sufficient condition on the masses are those that satisfy Cm, - m3Mm4 + 4

> hAmI

+ 4.

(3.1)

If the masses satisfy Eq. (3.1), then it is easy to show that Theorem 3 is true. A simple way is to force particle m3 to escape with arbitrary high velocity and then to collide with m4. This can be done periodically using an argument similar to that in Section 2. To ensure that all motion is not completed in finite time, we can waste as much time as desired by interspersing these high velocity escapes with oscillatory motion for m3 of the kind described in Section 2. However, we take a different approach to prove the more general Theorem 4. THEOREM 4. Let g(t) be a monotonically increasing function on (0, co). the masses satisfy Eq. (3.1), there exist sets oj initial conditions so that

lim(R(t)/g(t))

-+ co

as

[f

2 -+ co

To prove this theorem, replace the curve r,, from Section 2 with two curves r,, , and r,,, . The first curve is similar to that of r, in Section 2 except that an upper limit is imposed on the value of a. Formally, let rO,, be a curve in the energy surface for the four particles parameterized by u where the properties for the first three particles are the same as for r, with the following modifications: At all points on the curve, xi > 0. When c( specifies the location of either m, or m3, it is relative to the center of mass of the first three particles. The upper limit on czis where x; = 0. r,,, is a curve in the energy surface parameterized by b E [ 1, co) in the following manner. C;,, --co, C;,,>O, xZ-x16xq-x3=1
354

SAARI

AND

XIA

for a, and hence for the value of n > 0 so that (c$, olz+ i) exists. The greater the distance between x1 and x4, the larger the permissible value of ~1and the larger the defined values of n. Because the value of /I represents the return velocity for m3, it is clear that &, > flz + , . Also, because this return velocity can be arbitrarily large (/I E (0, co)), with the constraint of Eq. (3.1), a choice of p can be made so that m3 collides in as short of a time interval as desired. This means that (/I; + i, /Ii) intervals exist for all choices of n 2 1. (Actually, it exists for all values n 2 0.) With an argument similar to that used in Section 2, let each interval (4, al+l) and Pk+19 PiJ evolve beyond the first near triple collision. The image of each interval contains a set of curves Zi,, and Z,,* with properties of the curves Z,,j, j= 1,2. (The proof that Z1,2 exists is immediate with a standard center of mass argument using the mass inequalities given in Eq. (3.1). This argument is essentially the same as that given in Mather and McCehee to show that m3 can rebound with arbitrarily high velocity, so we do not repeat it here.) Thus, the symbolic dynamics can be established. To distinguish between an n corresponding to an interval from fi, i or Zi,2, we use superscripts. Thus, the sequence (5l, 65’, ...) corresponds to m3 starting from rest and colliding with m2 after 5 collisions, being expelled, and rebounding from a collision with m4. After it is unit distance from m4, the binary (m,, mz) collides 65 times before m3 collides with m,, etc. It remains to show that there exists a sequence from the symbolic dynamics so that the solution does as asserted. The first step is to note that for any sequence (n:, ni, ... ), when the force on m3 effects the velocity, x;’ is of the order (x4 - x3)-2. This means that in any short period of time, the change in the velocity is minimal, so xi is of the order of 0. Namely, xi is bounded below by a constant multiple of /I between the consecutive close triple collisions. Second, by using the distance x4-x2 and varying the value of 0, the length of time between consecutive triple collisions is controlled. All that remains is to establish that choices of /I and the length of time between triple collisions can be made to force R(t)/g(t) --f 00. This is accomplished by using a multiple of the polar moment of inertia defined in Eq. (3.2). Because the center of mass is fixed at 0, if M = xi mj, then (3.2) i

k
It is well known that (e.g., see Pollard [S]) there exists positive constants A and B, depending on the masses, so that AR* Q Id BR2. Therefore, the assertion of Theorem 4 is equivalent to showing that z/g* + co

as t-+co.

(3.3)

355

OSCILLATORY AND SUPERHYPERBOLIC MOTION

The growth of Z (see, for example, [S, lo]) is given by Y=T+h.

(3.3)

Thus it follows from the definition of T that 2Z” > WI~{X;(~)}~ + 2h. Let h(t) be a function defined on (0, co) so that h(t) > t{ g(t)}’ and h” > 0. By selecting values of /Ij to control the time between consecutive near triple collisions, a step function exists so that on the jth time interval m3{/?,}’ > /r”(t). To complete the proof, we prove there exists an orbit where xi is bounded below by pj on the appropriate time interval. To do this, note that the sequence {/I?,} defines a sequence {n:, ni, . ..}. where if pj is in an interval (pz+, , /II,), then this is the value of nj. If pi is not in such an interval, select the first one with pi+ r > pi. According to the above symbolic dynamic argument, there is an initial condition with the desired properties. ACKNOWLEDGMENT The research for both authors was partially supported by an NSF grant IST-8415348.

REFERENCES 1. J. CHAZY, Sur l’allure du mouvement dans le probltme des trois corps quand le temps coroit indetinement, Ann. Sci. Ecole Norm. Sup. 39 (1922), 29-130. 2. R. EASTON, Parabolic orbits for the planar three body problem, J. Differential Equations 52 (1984), 116134. 3. C. MARCHAL AND D. G. SAARI, On the final evolution of the n-body problem, J. Differenrid Equations 20 (1976), 150-186. 4. J. MATHER AND R. MCGEHEE, Solutions of the collinear four-body problem which become unbounded in finite time, in “Dynamical Systems Theory and Applications,” p. 573-587 (J. Moser, Ed.), Lecture Notes in Physics, Springer-Verlag, New York, 1974. 5. R. MCGEHEE, Triple collisions in the collinear three-body problem, Invent. Math. (1974) 191-227. 6. R. MOEKEL, Heteroclinic phenomena in the isosceles three body problem, SIAM J. Math. Anal. 15 (1984), 8.57-876. 7. P. PAINLEVE, LeGons sur la thtorie analytique des equations differentielles, Herman, Paris, 1987. 8. H. POLLARD, Gravitational systems, J. Math. Mech. 17 (1967) 60-617. 9. C. ROBINSON, Homoclinic orbits and oscillation for the planar three body problem, J. Differenlial Equations 52 (1984), 356377. 10. D. G. SAARI, Expanding gravitational systems, Trans. Amer. Math. SOC. 156 (1971) 219-242. 11. D. G. SAARI, The manifold structure for collision and for hyperbolic-parabolic orbits in the n-body problem, J. Differential Equations 55 (1984), 300-329. 12. K. SITNIKOV, The existence of oscillatory motion in the three-body problem, Soaier Phys. Dokl. 5 (1964), 647-656.