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Central configurations with many small masses
JOURNAL
OF DIFFERENTIAL
Central
91, 168-179 (1991)
EQUATIONS
Configurations
with
Many Small Masses
ZHIHONG XIA * Department of Mathematics, Harvard University, Cambridge, Massachusetts 02138 Communicated by Jack K. Hale Received October 30. 1989
By using the method of analytical continuation, we find the exact numbers of central configurations for some open sets of n positive masses for any choice of n. It turns out that the numbers increase dramatically as n increases; e.g., for some open set of 18 positive masses, some 2.08766 x 10” classes of distinctive central configurations are found. In the mean time, we obtained some results about the Hausdorff measure for the set of n positive masses where degenerate central configuration arises. ‘0 1991 Academic Press, Inc.
1. INTRODUCTION Consider N point massesm,, m,, .... mN > 0 with positions ql, q2, .... qN, qi E R2 in the Euclidean plane. These points form a central configuration, if, for some positive constant 1, the following system of algebraic equations is satisfied mimj 3 (qj-qj)=$
Rm,qi= C j#i
I
llqi-qjll
(1)
for i= 1, 2, .... N, and where
u,q
m,m, i
1/4i-4jll
The central configurations are important in n-body problems because they are bifurcation points for the topological classification of the co-planar n-body problem (see Easton [6] and Smale [12]); they are the limiting configurations for colliding particles and for completely parabolic orbits (see D. Saari [9, 111); they are the only configurations which can be maintained all the time in the n-body system; and often they are the starting * Research supported in part by an NSF grant.
168 0022-0396/91 $3.00 Copyright 0 1991 by Academic Press, Inc. All rights of reproduction in any form reserved.
CENTRAL CONFIGURATIONS WITH SMALL MASSES
169
points for finding some new classes of periodic solutions. Questions about these conligurations were raised by Wintner [lo], Smale [12], Palmore [S], and many others. Recently there has been an active interest in finding and classifying the central configurations, e.g., the recent work of Meyer and Schmidt [ 11, Hall [2], Moeckel [3], and others. For n = 3, this problem is completely solved and it is a classical result: There are exactly 3 collinear solutions and 2 equilateral triangular solutions for any 3 positive masses. In case of n = 4, even for 4 equal masses, the exact number of central configurations is unknown. On the other hand, Arenstorf [S] obtained the number of central configurations with one of four masses being very small. He started with one zero mass and then analytically continue it into positive masses. A very interesting phenomenon arises in this case, i.e., there are two sets of masses such that the numbers of central configurations corresponding to these two sets are different. In this paper, we are going to find the exact number of central configuration for some open sets of N masses for any N. It turns out that for any N 2 4, the same phenomenon arises; there are always two open sets of masses such that the numbers of central configurations corresponding to these masses are different. From this we conclude that for any Nb4, the set of masses for which the degenerate central configuration exists has a positive (N- 1 )-dimensional measure (Haussdorff). This was proved first by Palmore [S], but it seems difficult to complete all the steps in his proof. The method we are going to use is analytical continuation. We start with two zero masses and find the corresponding central configurations, especially the ones in which two zero masses are at the same point. Under certain conditions, these central conligurations can be analytically continued into a full n-body problem with all masses positive. We will see that the central conliguration with two zero masses at the same point will bifurcate into several distinctive central configurations of a full n-body problem. By repeating this processes, for any choice of II, we can find the exact number of central configurations for at least some open subset of N positive masses. The central configurations are also called relative equilibria. However, throughout this paper we only use the name central configuration and save the relative equilibria for those special central configurations with one or more zero masses.
2. RESTRICTED PROBLEM OF (n+2)-BODY Let q* = (q:, qf, .... 9.:) be a solution of Eq. (1), i.e., qr, q:, .... 4.: form a central configuration; then it is easy to see that kq* = (kq:, kqr, .... kq,;) and Aq* = (Aq:, Aq:, .... Aq,i$) is also a solution of (l), where k is a constant and A E SO( 2, R j is a rotation matrix. We can define an equivalence
170
ZHIHONG
XIA
relation by q-kq and q- Aq. By a central configuration, we will actually mean an equivalence class of central configurations. An equivalent delinition of central configuration is the critical points of the potential function U restricted to the sphere I= 1, where I= (l/2) x mj(lqi((* is the moment of inertia. Then (1) is exactly the equation given by the Lagrange multiplier method for finding the critical point and d is the Lagrange multiplier. Note that there is a degenerate direction for U restricted on the sphere I= 1 due to SO(2, R) action. To remove this degeneracy, let S/h be the quotient space of the sphere I= 1 with relation q- Aq, A E SO(2, R); then the equivalence classes of central configurations are the critical points of VCql) on s/-, where [q] is the equivalent class of q. A central configuration is called nondegenerate if its equivalence class is a nondegenerate critical point of U( [q]; i.e., the Hessian is nonsingular at the critical point. By the implicit function theorem, any nondegenerate central conliguration can be analytically continued to nearby masses. We remark that, in (l), the equation for central configuration, one may fix i. = 1. This only fixes the scale of the central configuration. For any equivalence class of central configurations, the one with i = 1 can be found. From now on, we fix J = 1. Let q* = (q:, qf, .... q;) be a nondegenerate central configuration for the N-body problem; now we consider an n-body system with n = IV+ 2 in which we add two small massesto the N-body system. We want to find the central configurations to this system in which q = (q,, q2, .... qN) is close to q* = (q;p, q;, .... 4%). The corresponding equations for the central configurations are
mim.w+2
- 4;) + IlqN+Z-qrl13 (qN+Z for i = 1, 2, .... N, and
N c !=, Ilq,+,-qil13(q~-q4.y+1) n1im.w
mNf
Iq.%‘+
I =
+ I
mlv, ,nlJv,2 --qN+l) + IIqNfI -9.v+zI13 (qY+z mzv+2qN+2=
(2)
;$, llq;yy&
(3)
(4i - qN+2) (4)
CENTRAL
CONFIGURATIONS
WITH
SMALL
MASSES
171
Note that in Eqs. (3) and (4) there are factors nz,,,+, and mN+?, respectively. When dealing with small massesit is convenient to factor out nlN+ It nl,v+2. Assume this is done. Our next step is to consider the extreme For m,V+,=m,V+2=0, (2) reduces to (l), case where ~~~+,=rn,+~=O. and (3 ) and (4) become .\ m, q.v+1= c t-l lIq,~+,-q:l13~q‘y+L-qr*) m,
q.v+2= i !=I
IlqY+2-qi
* 3 (q,v+z-q?). II
Note that the equations for q ,,,+ , , qNfz are identical; we may write them in a different way as D Wq .v+ I I= 0,
DV,(q ,v+2)=07
(7)
where
v,\r(.Y) =;=,i Ilq,*-*ul[ mi +P12. l Let I* be a solution of DV,,(x) =O; then it is called a relative equilibrium of the restricted (n + l)-body problem, and it is nondegenerate if the Jacobian of DVJx) has a rank 2; i.e., if ID’V,(s)l # 0 at x = .Y*. Since D’V,(x) is symmetric, it has two real valued eigenvalues, say a, and a2, and by selecting appropriate coordinates, we may write
The objective of this section is to find some central configurations of the (n + 2)-body problem with two small massesfrom a known relative equilibrium of the restricted (n + l)-body problem. The following proposition shows how to find the central configurations where two small massesare far apart from each other. PROPOSITION 1. Let q:, q?, .... q,;! he a nondegenerate central configuration for m L, m3, ..,, m,, > 0. Let q$ + , , 4%+ z be nondegenerate relative equilibria of the restricted (n + l)-body problem, and q$+ , # q$+ 2; then the central configuration q?, qz, .... qg, qz+ ,, qz.+: for the masses m,, 1112, . . . . nz,\r, 0, 0, can be analyticallJ1 continued to a central configuration for an)’ masses in an open neighborhood qf m,, m,, .... mN, 0, 0 in iW,yf2.
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In this proposition, qE+ , # q$+ 2 is required so that the influences of the two small masses on each other are of high order, therefore the implicit function theorem applies. In fact, the Jacobian at q:, qf, .... q:, q$+ ,, qEr+* for (2), (3) and (4) breaks into three parts, one for (2) at q;, q:, ..., q.:, and the other two for (3) and (4) at qX+, and q$+?, respectively. All of them are nondegenerate, so the implicit function theorem gives the proposition. Observe that, if we have k nondegenerate relative equilibria for the restricted (n + 1)-body problem, this proposition gives k. (k - 1) distinctive central configurations for the full (n + 2)-body problem with two small masses and it is easy to see that, for rn.&,+ ,, HZ,,,+?sufficiently small, this gives all the central configurations with m,, rn2, .... rnl:v near q:, qf, .... qX and mNtl, rnN+2 away from one another. It is still possible that there are some central configurations for which the two small massesnzN+,, mN+Z are close together; i.e., as the limits of mN+ ,, m,,,+l tend to zero, t~z:v+,, mN+2 occupy the same point. We consider this case next. In case of qz,+, = qjcy+2, the influences between two small massesbecome very large as their massesbecome positive, since the distance between them is very small. Therefore we can not apply the implicit function theorem q$+, =qz+? is a singular point for directly. In fact, m,y+l =O=mJv+2, Eqs. (2), (3), and (4), we will see that the bifurcation of the central configuration occurs here. When mN+I >O, m,v+2 > 0 are very small, and qN + , - q.v+ ? is very small, it is convenient to introduce some new coordinates for mN+ , and nzNt2; let
and
Then, for
qN+,-&+I,
qN+r-q~~+z
very small, Eqs. (2), (3), and (4)
become
i=
1, 2, .... N
(8)
(9)
CENTRAL
CONFIGURATIONS
WITH
SMALL
MASSES
173
We still have a singularity for (10) at Mu+, =O, rr~~+~=O, and r,=O; a scaling of variables will remove this singularity. Let rz= (mv+, +mN+2),‘3r;; (10) becomes
rz = 0 is no longer a singular point for (8) (9) Now, m lV+,=nl:~+~=O, and (11) and therefore we may use this set of equations for finding the central configurations with two small masses close together. Let m ,,,+,=O, mNtl=O and let r:=qE,+,=qz,+, be a nondegenerate relative equilibrium of the restricted (N+ 1)-body problem; (11) gives
1
0
aI - W~l13 0 [
ri = 0,
a, - l/llrSII’
(12)
where a, and a, are the eigenvalues of DV,,,(r~) and a, # 0, a2 #O. There are three cases for the solutions of this equation depending on different values of a I and a, : (i) a, < 0, a, < 0, i.e., r: is a local maximum of V(X); then there is no solution at all. (ii) a, < 0, a, > 0 (or similarly, a, > 0, a, < 0), r: is a saddle point of V(X); there are two solutions, ri = &(O, CI;‘,~), and the Jacobian at these solutions is a,--a2 0
0
=3(a,-a,)a~#o;
30,
so these two solutions can be continued into configurations for a full (n + 2)-body problem. (iii)
two classes of central
a, > 0, a, > 0; there are four solutions in this case, rk= _
+(0 u-“~) --r2
,
r2 = +(a;’
3, 0),
and the Jacobians at these solutions are al--a2 0
0 I
I 3a,
0
3a, ’
0
a,--a,
.
Both of them do not vanish provided that a, #a?, and hence if a, #a, these four solutions can be continued into four different classes of central configurations for the full (N+ 2)-body problem with two small masses.
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ZHIHONG
XIA
The following graph illustrates the relative positions of two small masses in a (4 +2)-body problem.
* mI
* “22 1124 *
* *+* * *-* 7
7
minimum
saddle
m, * Now we have a way for finding the central configurations of the (N+ 2)body problem with two small masses even when these two small masses have the same limit position as their masses tend to zero. We point out that the central configurations thus found exhaust all the possible ones with two sufficient small masses and with the large masses m,, m,, .... mN close to q1*, q2*, .... 4:. In case (iii), when a, = a, < 0, more complicated bifurcation occurs, and it depends on high order terms of V,v(x) at s*, it is possible that we have more than four solutions that can be continued into a full (N+ 2)-body problem instead of four solutions in the case of a, # a,, a, < 0, and a, < 0. For example, let us put N equal masses at the vertices of a regular N-polygon; then the origin is a relative equilibrium and is a local minimum of V,,v(x), by symmetry, a, = a, < 0. One can show that at least 2N classes of central configurations of the (N + 2)-problem, with two small masses at the origin, can be obtained, again by symmetry. As an application of above results, let us consider a 4-body problem with two small masses. There are live relative equilibria for a restricted 3-body problem; two of them are equilateral triangle solutions which are minima of Vz(.u), and two eigenvalues of D’V(x) are different at these two points, and the other three of them are collinear solutions which are saddle points of VJx). All of the live relative equilibria are nondegenerate. Thus, Proposition 1 gives 5 x 4= 20 classes of central conligurations for the 4-body problem with two small masses which are away from each other and there are 2 x 4 + 3 x 2 = 14 classes of central conligurations with two small masses close to each other; therefore, there are totally 34 classes of central configurations for the 4-body problem with two small masses. This same result was obtained by Arenstorf [8] by a different method.
CENTRAL
CONFIGURATIONS
WITH
SMALL
175
MASSES
3. THE NUMBERSOF CENTRAL CONFIGURATIONS
In last section, we developed a method to find the number of relative equilibria of the (N+ 2)-body problem from that of the restricted (N+ 1jbody problem, and we continued these into the central configurations of the full (N+ 1)-body problem. If we keep M,~+ z = 0 and only continue rnN+ i into positive values, then what we obtain are some relative equilibria of the restricted ((N + 1) + 1)-body problem, so we can use the same technique again to obtain some central configurations of the (N+ 3)-body problem, and so on. Before we do that, we must first find the eigenvalues of the restricted of D2N,v+ i(s) at these new relative equilibria ((N + 1) + 1 )-body problem. For the relative equilibria given by Proposition 1, i.e., when two small masses are away from each other, for rnh’+, small enough, D2V,,,+ ,(I) has the same property as that of D2Vy(x), and therefore the method of the last section applies again. In the case of two small masses close to each other, if m,, , is small enough, then I’,,.+ ,(x) can be approximated by
with D’V,v(.x)=
[0 a2 1 a’
’
Corresponding to that three cases of the last section, local properties of I’,,,.+ i(x) can be shown to be the following. (i) a, < 0, a2 < 0; no relative equilibrium problem; (ii)
a,
a?>0
(and similarly for a,>O,
again, the relative equilibria (iii)
a,>O,
a2>0,
a?
r>=
&(0,a;‘,3),
are saddle points;
a,#az;
and for the solution t-i = +(a;
for the (N + 1) + 1)-body
for the solution r;=
+(0,a;‘:3),
‘;3, 0),
1). 1+O(?v+
176
ZHIHONG
XIA
So two of the relative equilibria are saddle points, while the other two are minimal of V,,,+ ,(x). We conclude that, for any saddle point of V,,(x), then for each central configuration continued from this relative equilibrium, with m,, , sufficiently small, there are always two saddle points of V,+,(x) nearby mN+ , , and correspondingly, for any minimum of V,%(X), there will always be two minima and two saddle points nearby mN+ 1 for VI”+ ,(x). Now, we come back to the 4-body problem we discussed at the end of the last section. First consider the case with m, and m, large and m3 sufficiently small. From the above result, there are together 2 x 4 + 3 x 2 + 5 x 4 = 34 relative equilibria. Among these 34 relative equilibria of the restricted (3 + I)-body problem, 16 belong to the equilateral triangle formed by m,, m2, and m, with 2 x 3 = 6 local minima and 2 x 5 = 10 local saddle points. The other 18 belong the collinear central configurations of m, , m,, and m3 with 3 x 2 local minima and 3 x 4 local saddle points. By a very careful counting, we conclude that there are ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 3 x 4 x 2 + 3 x 4 x 5 = 294 relative equilibria for the restricted (4 + 1)-body problem. We now turn our attention to the central configurations with three large masses. First for any three positive masses, there are five classes of central configurations; three of them are collinear and two of them are equilateral. Arenstorf [S] and Palmore [4] showed that for any collinear central configuration there are six relative equilibria for the restricted 4-body problem; two of them are local minima and the other four of them are saddles. For the equilateral central configuration, an interesting phenomenon happens. Arenstorf [S] showed that there are 8 or 10 relative equilibria of the restricted 4-body problem depending on the three positive masses, and for most of the masses they are all nondegenerate. A simple topological argument shows that in the nondegenerate case, among these 8 (or 10) relative equilibria, 3 (or 4) of them must be local minima and the other 5 (or 6) must be saddle points. Since thee eigenvalues continuously depend on the masses, it can be shown easily that for most of the masses, at the minima relative equilibria, D2 V3(x) has distinct eigenvalues. Therefore for the restricted 4-body problem, for some open set of three large masses, there are totally 34 relative equilibria. Among them, 22 are saddle points and the other 12 are local minima, and for some other open set of three large masses, the total number of relative equilibria is 36. Among them 24 are saddle points and 14 are local minima. From these results about the restricted (3 + 1 =4)-body problem and the technique we developed above, it is easy to see (again, by carefully counting the numbers), for a 5-body problem, for some open set of masses (with two large masses, one small mass, and two even smaller masses), there are 294
CENTRAL
CONFIGURATIONS
WITH
SMALL
177
MASSES
classes of central configurations, while for some other open set of masses (with three large masses and two small masses), there are 374 classes of central configurations, and all of them are nondegenerate. Repeating the above procedure, for any N there are two open sets of massesfor which we can find the exact number of central configurations corresponding to these two sets of masses.For n from 4 through 10, the numbers thus found are listed in Table I. It would be nice if we could develop a general formula depending on the number of bodies for the number of central configurations thus found. It turns out that it is even hard to write a recursive formula for these numbers. In fact, the numbers in Table I are produced by a small Fortran program. It is interesting to see how fast the number of the central configurations increasesas n increases.Let t, be the total number of central configurations we find for some n positive masses. Since local minima produce more central configurations when continued into positive masses, t,, must be greater than the number obtained by considering all local minima as saddle points. In this way, we have a simple estimate: for n>4.
t,, > (n + 2)!/24
By a topological argument, Palmore [7] showed that there is a lower bound, (3n - 4)(n - 1)!/2, on the numbers of central configurations for any choice of n positive masses.It is interesting to note that the numbers that are actually found here far exceedsthe lower bound as n becomes large. Observe that, for any n 2 4, there are two open sets of n positive masses such that all the corresponding central configurations are nondegenerate and the numbers of central configurations corresponding to these two sets are different. This shows the complexity of the problem of central configurations and also give us some information about degeneracy of central configurations.
TABLE Numbers
n 2 Large masses 3 Large Lower
masses bound
of Central
4
5
6
1
34
294
3096
38,250
38
374
4512
24
132
840
I Configurations 8
9
10
5,536,800
9,036,729
165,191,OOO
64,248
1.051,440
19,399,OOO
397,737,OOO
6,120
50,400
463.680
4,717.440
178
ZHIHONG XIA 4. MEASUREOF DEGENERATECENTRAL CONFIGURATION
A long-standing problem in central configurations of the n-body problem is whether there is only a finite number of central configurations for any choice of n positive masses. This problem concerns the nature of degenerate central configurations. The first example of degenerate central configurations was given by Palmore [S] in a 4-body problem. By using some results of the last section, we can prove the following theorem concerning of the masses for which the degenerate central configuration exists. THEOREM. Let .Z, be the set of masses (m,, m2, .... mN) E G2.yfor which the degenerate central configuration exists; then .Z,v has a positive (N - l)dimensional Hausdorff measure. This is a theorem of Palmore [S], but there is a step in his proof that seems difficult to complete. Here we give a different proof. Proof: We have shown that for any n 24, there exist m E 52’: and m*E[Wy, such that m$Z‘, and rn* $Z,, and the numbers of central configurations corresponding to m and rn* are different. Let r be any arc in R’y joining m and m *, then l-r\ Z,\, # 4; otherwise, by analytical continuation of the central configuration along the arc, we will get the same number for both m and m* but that is impossible. This proves the theorem.
5. SOME REMARKS While it seems impossible to find all central configurations and classify them for any choice of n positive masses, the method we introduced here can solve this problem for at least some open sets of n positive masses for any choice of n. It is amazing to see how fast the number of central confiigurations grows as ii increases. Note that our open sets of masses consist of two or three large masses and a small mass, an even smaller mass, .... and so on. An immediate way to extend our results is to consider three, four, or even more zero masses as a starting point, i.e., consider first the restricted (N + n)-body problem and find corresponding central configurations and then continue them into the full (N+ n)-body problem. It turns out that, after some changing and resealing of variables, the resulting equations are not much easier to solve than solving the original equations with N + n positive masses. However, it might be interesting to solve the problem for n = 3. Finally, we remark that the above method can also be used to find the central configurations in R3 and in this way, the author suspects that the
CENTRAL
CONFIGURATIONS
WITH SMALL MASSES
179
nonplanar central configurations contribute most to the total number of central configurations as the number of bodies becomes large.
ACKNOWLEDGMENTS Part of the writing of this paper was done at Northwestern University and the University of Cincinnati. The author thanks D. Saari, K. Meyer, D. Schmidt, and G. Hall for reading the original maniscript and making many useful suggestions. The author also thanks R. Moeckel for pointing out an error in the original manuscript.
REFERENCES I. K. MEYER AND D. SCHMIDT, Bifurcations of relative equilibria in the n-body and Kirchhoff problems, preprint. 2. G. R. HALL, Central configurations in the planar I tn body problem, preprint. 3. R. MOECKEL, Relative equilibria of the 4-body problem, Ergodic Theory Dynamical Systems 5, 3 (1985). 4. J. PALMORE.Collinear relative equilibria of the planar n-body problem. Celestial Mech. 28. 9 (1982), 17-23. 5. J. PALMORE, Measure of degenerate relative equilibria, I. Ann. qf Math. 104 (1976). 421429. 6. R. EASTON, Some topology of n-body problems, .I. D#erenfial Equations 19 (1975), 258-269. 7. J. PALMORE,Classifying relative equilibria, I, Bull. Amer. Mafh. Sot. 79, 5 (1973). 904-907. 8. R. ARENSTORF,Central configurations of 4-body with one inferior mass. Celestial Mech. 28, 9. 9. D. SAARI, The manifold structure for collision and for hyperbolic-parabolic orbits in the n-body problem, J. Differential Equations 55, 3 (1984). 300-329. 10. A. WINTNER, “The Analytical Foundations of Celestial Mechanics,” Princeton Univ. Press, Princeton, NJ, 1941. 1 I. D. SAARI. On the role and properties of n-body central configurations, Celestial Mech. 21 (1980), 9-20. I?. S. SMALE, Topology and mechanics, I, Incenr. Math. 10 (1970), 305-331.
Primed in Belgium
OF DIFFERENTIAL
Central
91, 168-179 (1991)
EQUATIONS
Configurations
with
Many Small Masses
ZHIHONG XIA * Department of Mathematics, Harvard University, Cambridge, Massachusetts 02138 Communicated by Jack K. Hale Received October 30. 1989
By using the method of analytical continuation, we find the exact numbers of central configurations for some open sets of n positive masses for any choice of n. It turns out that the numbers increase dramatically as n increases; e.g., for some open set of 18 positive masses, some 2.08766 x 10” classes of distinctive central configurations are found. In the mean time, we obtained some results about the Hausdorff measure for the set of n positive masses where degenerate central configuration arises. ‘0 1991 Academic Press, Inc.
1. INTRODUCTION Consider N point massesm,, m,, .... mN > 0 with positions ql, q2, .... qN, qi E R2 in the Euclidean plane. These points form a central configuration, if, for some positive constant 1, the following system of algebraic equations is satisfied mimj 3 (qj-qj)=$
Rm,qi= C j#i
I
llqi-qjll
(1)
for i= 1, 2, .... N, and where
u,q
m,m, i
1/4i-4jll
The central configurations are important in n-body problems because they are bifurcation points for the topological classification of the co-planar n-body problem (see Easton [6] and Smale [12]); they are the limiting configurations for colliding particles and for completely parabolic orbits (see D. Saari [9, 111); they are the only configurations which can be maintained all the time in the n-body system; and often they are the starting * Research supported in part by an NSF grant.
168 0022-0396/91 $3.00 Copyright 0 1991 by Academic Press, Inc. All rights of reproduction in any form reserved.
CENTRAL CONFIGURATIONS WITH SMALL MASSES
169
points for finding some new classes of periodic solutions. Questions about these conligurations were raised by Wintner [lo], Smale [12], Palmore [S], and many others. Recently there has been an active interest in finding and classifying the central configurations, e.g., the recent work of Meyer and Schmidt [ 11, Hall [2], Moeckel [3], and others. For n = 3, this problem is completely solved and it is a classical result: There are exactly 3 collinear solutions and 2 equilateral triangular solutions for any 3 positive masses. In case of n = 4, even for 4 equal masses, the exact number of central configurations is unknown. On the other hand, Arenstorf [S] obtained the number of central configurations with one of four masses being very small. He started with one zero mass and then analytically continue it into positive masses. A very interesting phenomenon arises in this case, i.e., there are two sets of masses such that the numbers of central configurations corresponding to these two sets are different. In this paper, we are going to find the exact number of central configuration for some open sets of N masses for any N. It turns out that for any N 2 4, the same phenomenon arises; there are always two open sets of masses such that the numbers of central configurations corresponding to these masses are different. From this we conclude that for any Nb4, the set of masses for which the degenerate central configuration exists has a positive (N- 1 )-dimensional measure (Haussdorff). This was proved first by Palmore [S], but it seems difficult to complete all the steps in his proof. The method we are going to use is analytical continuation. We start with two zero masses and find the corresponding central configurations, especially the ones in which two zero masses are at the same point. Under certain conditions, these central conligurations can be analytically continued into a full n-body problem with all masses positive. We will see that the central conliguration with two zero masses at the same point will bifurcate into several distinctive central configurations of a full n-body problem. By repeating this processes, for any choice of II, we can find the exact number of central configurations for at least some open subset of N positive masses. The central configurations are also called relative equilibria. However, throughout this paper we only use the name central configuration and save the relative equilibria for those special central configurations with one or more zero masses.
2. RESTRICTED PROBLEM OF (n+2)-BODY Let q* = (q:, qf, .... 9.:) be a solution of Eq. (1), i.e., qr, q:, .... 4.: form a central configuration; then it is easy to see that kq* = (kq:, kqr, .... kq,;) and Aq* = (Aq:, Aq:, .... Aq,i$) is also a solution of (l), where k is a constant and A E SO( 2, R j is a rotation matrix. We can define an equivalence
170
ZHIHONG
XIA
relation by q-kq and q- Aq. By a central configuration, we will actually mean an equivalence class of central configurations. An equivalent delinition of central configuration is the critical points of the potential function U restricted to the sphere I= 1, where I= (l/2) x mj(lqi((* is the moment of inertia. Then (1) is exactly the equation given by the Lagrange multiplier method for finding the critical point and d is the Lagrange multiplier. Note that there is a degenerate direction for U restricted on the sphere I= 1 due to SO(2, R) action. To remove this degeneracy, let S/h be the quotient space of the sphere I= 1 with relation q- Aq, A E SO(2, R); then the equivalence classes of central configurations are the critical points of VCql) on s/-, where [q] is the equivalent class of q. A central configuration is called nondegenerate if its equivalence class is a nondegenerate critical point of U( [q]; i.e., the Hessian is nonsingular at the critical point. By the implicit function theorem, any nondegenerate central conliguration can be analytically continued to nearby masses. We remark that, in (l), the equation for central configuration, one may fix i. = 1. This only fixes the scale of the central configuration. For any equivalence class of central configurations, the one with i = 1 can be found. From now on, we fix J = 1. Let q* = (q:, qf, .... q;) be a nondegenerate central configuration for the N-body problem; now we consider an n-body system with n = IV+ 2 in which we add two small massesto the N-body system. We want to find the central configurations to this system in which q = (q,, q2, .... qN) is close to q* = (q;p, q;, .... 4%). The corresponding equations for the central configurations are
mim.w+2
- 4;) + IlqN+Z-qrl13 (qN+Z for i = 1, 2, .... N, and
N c !=, Ilq,+,-qil13(q~-q4.y+1) n1im.w
mNf
Iq.%‘+
I =
+ I
mlv, ,nlJv,2 --qN+l) + IIqNfI -9.v+zI13 (qY+z mzv+2qN+2=
(2)
;$, llq;yy&
(3)
(4i - qN+2) (4)
CENTRAL
CONFIGURATIONS
WITH
SMALL
MASSES
171
Note that in Eqs. (3) and (4) there are factors nz,,,+, and mN+?, respectively. When dealing with small massesit is convenient to factor out nlN+ It nl,v+2. Assume this is done. Our next step is to consider the extreme For m,V+,=m,V+2=0, (2) reduces to (l), case where ~~~+,=rn,+~=O. and (3 ) and (4) become .\ m, q.v+1= c t-l lIq,~+,-q:l13~q‘y+L-qr*) m,
q.v+2= i !=I
IlqY+2-qi
* 3 (q,v+z-q?). II
Note that the equations for q ,,,+ , , qNfz are identical; we may write them in a different way as D Wq .v+ I I= 0,
DV,(q ,v+2)=07
(7)
where
v,\r(.Y) =;=,i Ilq,*-*ul[ mi +P12. l Let I* be a solution of DV,,(x) =O; then it is called a relative equilibrium of the restricted (n + l)-body problem, and it is nondegenerate if the Jacobian of DVJx) has a rank 2; i.e., if ID’V,(s)l # 0 at x = .Y*. Since D’V,(x) is symmetric, it has two real valued eigenvalues, say a, and a2, and by selecting appropriate coordinates, we may write
The objective of this section is to find some central configurations of the (n + 2)-body problem with two small massesfrom a known relative equilibrium of the restricted (n + l)-body problem. The following proposition shows how to find the central configurations where two small massesare far apart from each other. PROPOSITION 1. Let q:, q?, .... q,;! he a nondegenerate central configuration for m L, m3, ..,, m,, > 0. Let q$ + , , 4%+ z be nondegenerate relative equilibria of the restricted (n + l)-body problem, and q$+ , # q$+ 2; then the central configuration q?, qz, .... qg, qz+ ,, qz.+: for the masses m,, 1112, . . . . nz,\r, 0, 0, can be analyticallJ1 continued to a central configuration for an)’ masses in an open neighborhood qf m,, m,, .... mN, 0, 0 in iW,yf2.
172
ZHIHONG
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In this proposition, qE+ , # q$+ 2 is required so that the influences of the two small masses on each other are of high order, therefore the implicit function theorem applies. In fact, the Jacobian at q:, qf, .... q:, q$+ ,, qEr+* for (2), (3) and (4) breaks into three parts, one for (2) at q;, q:, ..., q.:, and the other two for (3) and (4) at qX+, and q$+?, respectively. All of them are nondegenerate, so the implicit function theorem gives the proposition. Observe that, if we have k nondegenerate relative equilibria for the restricted (n + 1)-body problem, this proposition gives k. (k - 1) distinctive central configurations for the full (n + 2)-body problem with two small masses and it is easy to see that, for rn.&,+ ,, HZ,,,+?sufficiently small, this gives all the central configurations with m,, rn2, .... rnl:v near q:, qf, .... qX and mNtl, rnN+2 away from one another. It is still possible that there are some central configurations for which the two small massesnzN+,, mN+Z are close together; i.e., as the limits of mN+ ,, m,,,+l tend to zero, t~z:v+,, mN+2 occupy the same point. We consider this case next. In case of qz,+, = qjcy+2, the influences between two small massesbecome very large as their massesbecome positive, since the distance between them is very small. Therefore we can not apply the implicit function theorem q$+, =qz+? is a singular point for directly. In fact, m,y+l =O=mJv+2, Eqs. (2), (3), and (4), we will see that the bifurcation of the central configuration occurs here. When mN+I >O, m,v+2 > 0 are very small, and qN + , - q.v+ ? is very small, it is convenient to introduce some new coordinates for mN+ , and nzNt2; let
and
Then, for
qN+,-&+I,
qN+r-q~~+z
very small, Eqs. (2), (3), and (4)
become
i=
1, 2, .... N
(8)
(9)
CENTRAL
CONFIGURATIONS
WITH
SMALL
MASSES
173
We still have a singularity for (10) at Mu+, =O, rr~~+~=O, and r,=O; a scaling of variables will remove this singularity. Let rz= (mv+, +mN+2),‘3r;; (10) becomes
rz = 0 is no longer a singular point for (8) (9) Now, m lV+,=nl:~+~=O, and (11) and therefore we may use this set of equations for finding the central configurations with two small masses close together. Let m ,,,+,=O, mNtl=O and let r:=qE,+,=qz,+, be a nondegenerate relative equilibrium of the restricted (N+ 1)-body problem; (11) gives
1
0
aI - W~l13 0 [
ri = 0,
a, - l/llrSII’
(12)
where a, and a, are the eigenvalues of DV,,,(r~) and a, # 0, a2 #O. There are three cases for the solutions of this equation depending on different values of a I and a, : (i) a, < 0, a, < 0, i.e., r: is a local maximum of V(X); then there is no solution at all. (ii) a, < 0, a, > 0 (or similarly, a, > 0, a, < 0), r: is a saddle point of V(X); there are two solutions, ri = &(O, CI;‘,~), and the Jacobian at these solutions is a,--a2 0
0
=3(a,-a,)a~#o;
30,
so these two solutions can be continued into configurations for a full (n + 2)-body problem. (iii)
two classes of central
a, > 0, a, > 0; there are four solutions in this case, rk= _
+(0 u-“~) --r2
,
r2 = +(a;’
3, 0),
and the Jacobians at these solutions are al--a2 0
0 I
I 3a,
0
3a, ’
0
a,--a,
.
Both of them do not vanish provided that a, #a?, and hence if a, #a, these four solutions can be continued into four different classes of central configurations for the full (N+ 2)-body problem with two small masses.
174
ZHIHONG
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The following graph illustrates the relative positions of two small masses in a (4 +2)-body problem.
* mI
* “22 1124 *
* *+* * *-* 7
7
minimum
saddle
m, * Now we have a way for finding the central configurations of the (N+ 2)body problem with two small masses even when these two small masses have the same limit position as their masses tend to zero. We point out that the central configurations thus found exhaust all the possible ones with two sufficient small masses and with the large masses m,, m,, .... mN close to q1*, q2*, .... 4:. In case (iii), when a, = a, < 0, more complicated bifurcation occurs, and it depends on high order terms of V,v(x) at s*, it is possible that we have more than four solutions that can be continued into a full (N+ 2)-body problem instead of four solutions in the case of a, # a,, a, < 0, and a, < 0. For example, let us put N equal masses at the vertices of a regular N-polygon; then the origin is a relative equilibrium and is a local minimum of V,,v(x), by symmetry, a, = a, < 0. One can show that at least 2N classes of central configurations of the (N + 2)-problem, with two small masses at the origin, can be obtained, again by symmetry. As an application of above results, let us consider a 4-body problem with two small masses. There are live relative equilibria for a restricted 3-body problem; two of them are equilateral triangle solutions which are minima of Vz(.u), and two eigenvalues of D’V(x) are different at these two points, and the other three of them are collinear solutions which are saddle points of VJx). All of the live relative equilibria are nondegenerate. Thus, Proposition 1 gives 5 x 4= 20 classes of central conligurations for the 4-body problem with two small masses which are away from each other and there are 2 x 4 + 3 x 2 = 14 classes of central conligurations with two small masses close to each other; therefore, there are totally 34 classes of central configurations for the 4-body problem with two small masses. This same result was obtained by Arenstorf [8] by a different method.
CENTRAL
CONFIGURATIONS
WITH
SMALL
175
MASSES
3. THE NUMBERSOF CENTRAL CONFIGURATIONS
In last section, we developed a method to find the number of relative equilibria of the (N+ 2)-body problem from that of the restricted (N+ 1jbody problem, and we continued these into the central configurations of the full (N+ 1)-body problem. If we keep M,~+ z = 0 and only continue rnN+ i into positive values, then what we obtain are some relative equilibria of the restricted ((N + 1) + 1)-body problem, so we can use the same technique again to obtain some central configurations of the (N+ 3)-body problem, and so on. Before we do that, we must first find the eigenvalues of the restricted of D2N,v+ i(s) at these new relative equilibria ((N + 1) + 1 )-body problem. For the relative equilibria given by Proposition 1, i.e., when two small masses are away from each other, for rnh’+, small enough, D2V,,,+ ,(I) has the same property as that of D2Vy(x), and therefore the method of the last section applies again. In the case of two small masses close to each other, if m,, , is small enough, then I’,,.+ ,(x) can be approximated by
with D’V,v(.x)=
[0 a2 1 a’
’
Corresponding to that three cases of the last section, local properties of I’,,,.+ i(x) can be shown to be the following. (i) a, < 0, a2 < 0; no relative equilibrium problem; (ii)
a,
a?>0
(and similarly for a,>O,
again, the relative equilibria (iii)
a,>O,
a2>0,
a?
r>=
&(0,a;‘,3),
are saddle points;
a,#az;
and for the solution t-i = +(a;
for the (N + 1) + 1)-body
for the solution r;=
+(0,a;‘:3),
‘;3, 0),
1). 1+O(?v+
176
ZHIHONG
XIA
So two of the relative equilibria are saddle points, while the other two are minimal of V,,,+ ,(x). We conclude that, for any saddle point of V,,(x), then for each central configuration continued from this relative equilibrium, with m,, , sufficiently small, there are always two saddle points of V,+,(x) nearby mN+ , , and correspondingly, for any minimum of V,%(X), there will always be two minima and two saddle points nearby mN+ 1 for VI”+ ,(x). Now, we come back to the 4-body problem we discussed at the end of the last section. First consider the case with m, and m, large and m3 sufficiently small. From the above result, there are together 2 x 4 + 3 x 2 + 5 x 4 = 34 relative equilibria. Among these 34 relative equilibria of the restricted (3 + I)-body problem, 16 belong to the equilateral triangle formed by m,, m2, and m, with 2 x 3 = 6 local minima and 2 x 5 = 10 local saddle points. The other 18 belong the collinear central configurations of m, , m,, and m3 with 3 x 2 local minima and 3 x 4 local saddle points. By a very careful counting, we conclude that there are ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 3 x 4 x 2 + 3 x 4 x 5 = 294 relative equilibria for the restricted (4 + 1)-body problem. We now turn our attention to the central configurations with three large masses. First for any three positive masses, there are five classes of central configurations; three of them are collinear and two of them are equilateral. Arenstorf [S] and Palmore [4] showed that for any collinear central configuration there are six relative equilibria for the restricted 4-body problem; two of them are local minima and the other four of them are saddles. For the equilateral central configuration, an interesting phenomenon happens. Arenstorf [S] showed that there are 8 or 10 relative equilibria of the restricted 4-body problem depending on the three positive masses, and for most of the masses they are all nondegenerate. A simple topological argument shows that in the nondegenerate case, among these 8 (or 10) relative equilibria, 3 (or 4) of them must be local minima and the other 5 (or 6) must be saddle points. Since thee eigenvalues continuously depend on the masses, it can be shown easily that for most of the masses, at the minima relative equilibria, D2 V3(x) has distinct eigenvalues. Therefore for the restricted 4-body problem, for some open set of three large masses, there are totally 34 relative equilibria. Among them, 22 are saddle points and the other 12 are local minima, and for some other open set of three large masses, the total number of relative equilibria is 36. Among them 24 are saddle points and 14 are local minima. From these results about the restricted (3 + 1 =4)-body problem and the technique we developed above, it is easy to see (again, by carefully counting the numbers), for a 5-body problem, for some open set of masses (with two large masses, one small mass, and two even smaller masses), there are 294
CENTRAL
CONFIGURATIONS
WITH
SMALL
177
MASSES
classes of central configurations, while for some other open set of masses (with three large masses and two small masses), there are 374 classes of central configurations, and all of them are nondegenerate. Repeating the above procedure, for any N there are two open sets of massesfor which we can find the exact number of central configurations corresponding to these two sets of masses.For n from 4 through 10, the numbers thus found are listed in Table I. It would be nice if we could develop a general formula depending on the number of bodies for the number of central configurations thus found. It turns out that it is even hard to write a recursive formula for these numbers. In fact, the numbers in Table I are produced by a small Fortran program. It is interesting to see how fast the number of the central configurations increasesas n increases.Let t, be the total number of central configurations we find for some n positive masses. Since local minima produce more central configurations when continued into positive masses, t,, must be greater than the number obtained by considering all local minima as saddle points. In this way, we have a simple estimate: for n>4.
t,, > (n + 2)!/24
By a topological argument, Palmore [7] showed that there is a lower bound, (3n - 4)(n - 1)!/2, on the numbers of central configurations for any choice of n positive masses.It is interesting to note that the numbers that are actually found here far exceedsthe lower bound as n becomes large. Observe that, for any n 2 4, there are two open sets of n positive masses such that all the corresponding central configurations are nondegenerate and the numbers of central configurations corresponding to these two sets are different. This shows the complexity of the problem of central configurations and also give us some information about degeneracy of central configurations.
TABLE Numbers
n 2 Large masses 3 Large Lower
masses bound
of Central
4
5
6
1
34
294
3096
38,250
38
374
4512
24
132
840
I Configurations 8
9
10
5,536,800
9,036,729
165,191,OOO
64,248
1.051,440
19,399,OOO
397,737,OOO
6,120
50,400
463.680
4,717.440
178
ZHIHONG XIA 4. MEASUREOF DEGENERATECENTRAL CONFIGURATION
A long-standing problem in central configurations of the n-body problem is whether there is only a finite number of central configurations for any choice of n positive masses. This problem concerns the nature of degenerate central configurations. The first example of degenerate central configurations was given by Palmore [S] in a 4-body problem. By using some results of the last section, we can prove the following theorem concerning of the masses for which the degenerate central configuration exists. THEOREM. Let .Z, be the set of masses (m,, m2, .... mN) E G2.yfor which the degenerate central configuration exists; then .Z,v has a positive (N - l)dimensional Hausdorff measure. This is a theorem of Palmore [S], but there is a step in his proof that seems difficult to complete. Here we give a different proof. Proof: We have shown that for any n 24, there exist m E 52’: and m*E[Wy, such that m$Z‘, and rn* $Z,, and the numbers of central configurations corresponding to m and rn* are different. Let r be any arc in R’y joining m and m *, then l-r\ Z,\, # 4; otherwise, by analytical continuation of the central configuration along the arc, we will get the same number for both m and m* but that is impossible. This proves the theorem.
5. SOME REMARKS While it seems impossible to find all central configurations and classify them for any choice of n positive masses, the method we introduced here can solve this problem for at least some open sets of n positive masses for any choice of n. It is amazing to see how fast the number of central confiigurations grows as ii increases. Note that our open sets of masses consist of two or three large masses and a small mass, an even smaller mass, .... and so on. An immediate way to extend our results is to consider three, four, or even more zero masses as a starting point, i.e., consider first the restricted (N + n)-body problem and find corresponding central configurations and then continue them into the full (N+ n)-body problem. It turns out that, after some changing and resealing of variables, the resulting equations are not much easier to solve than solving the original equations with N + n positive masses. However, it might be interesting to solve the problem for n = 3. Finally, we remark that the above method can also be used to find the central configurations in R3 and in this way, the author suspects that the
CENTRAL
CONFIGURATIONS
WITH SMALL MASSES
179
nonplanar central configurations contribute most to the total number of central configurations as the number of bodies becomes large.
ACKNOWLEDGMENTS Part of the writing of this paper was done at Northwestern University and the University of Cincinnati. The author thanks D. Saari, K. Meyer, D. Schmidt, and G. Hall for reading the original maniscript and making many useful suggestions. The author also thanks R. Moeckel for pointing out an error in the original manuscript.
REFERENCES I. K. MEYER AND D. SCHMIDT, Bifurcations of relative equilibria in the n-body and Kirchhoff problems, preprint. 2. G. R. HALL, Central configurations in the planar I tn body problem, preprint. 3. R. MOECKEL, Relative equilibria of the 4-body problem, Ergodic Theory Dynamical Systems 5, 3 (1985). 4. J. PALMORE.Collinear relative equilibria of the planar n-body problem. Celestial Mech. 28. 9 (1982), 17-23. 5. J. PALMORE, Measure of degenerate relative equilibria, I. Ann. qf Math. 104 (1976). 421429. 6. R. EASTON, Some topology of n-body problems, .I. D#erenfial Equations 19 (1975), 258-269. 7. J. PALMORE,Classifying relative equilibria, I, Bull. Amer. Mafh. Sot. 79, 5 (1973). 904-907. 8. R. ARENSTORF,Central configurations of 4-body with one inferior mass. Celestial Mech. 28, 9. 9. D. SAARI, The manifold structure for collision and for hyperbolic-parabolic orbits in the n-body problem, J. Differential Equations 55, 3 (1984). 300-329. 10. A. WINTNER, “The Analytical Foundations of Celestial Mechanics,” Princeton Univ. Press, Princeton, NJ, 1941. 1 I. D. SAARI. On the role and properties of n-body central configurations, Celestial Mech. 21 (1980), 9-20. I?. S. SMALE, Topology and mechanics, I, Incenr. Math. 10 (1970), 305-331.
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