Sep 26, 2009 - csc Ï n k is the potential of a regular n-gon of unit size and unit masses. The ...... âMeasure of Degenerate Equilibria Iâ, Ann. ...

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arXiv:0909.4890v1 [math-ph] 26 Sep 2009

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Department of Mathematics, University of California, Irvine and Zhou Pei-Yuan Center for Applied Mathematics, Tsinghua University, Beijing, 100084 China, email: [email protected] (2) Department of Mathematics, University of California, Irvine, Irvine CA, 92697 USA, email: [email protected] Abstract. In this paper we find a class of new degenerate central configurations and bifurcations in the Newtonian n-body problem. In particular we analyze the Rosette central configurations, namely a coplanar configuration where n particles of mass m1 lie at the vertices of a regular n-gon, n particles of mass m2 lie at the vertices of another n-gon concentric with the first, but rotated of an angle π/n, and an additional particle of mass m0 lies at the center of mass of the system. This system admits two mass parameters µ = m0 /m1 and ǫ = m2 /m1 . We show that, as µ varies, if n > 3, there is a degenerate central configuration and a bifurcation for every ǫ > 0, while if n = 3 there is a bifurcations only for some values of ǫ. Keywords: N-body problem, central configurations, bifurcations, degenerate central configurations

1. Introduction In the planar Newtonian n-body problem the simplest possible motions are such that the whole system of particles rotates as a rigid body about its center of mass. In this case the configuration of the bodies does not change with time. Only some special configurations of point particles are allowed such motions. These configurations are called central configurations. Many questions were raised about the set of central configurations. The main general open problem is the Chazy-Wintner-Smale conjecture: given n positive masses m1 , . . . , mn interacting by means of the Newtonian potential, the set of equivalence classes of central configurations is finite. Such conjecture was proved for n = 4, in the case of equal masses, by Albouy (1995) and (1996) and in the general case by Hampton and Moeckel (2006). Chazy believed in a stronger statement: namely that any equivalence class of central configuration is non-degenerate. This statement is known to be false: Palmore (Palmore, 1975; Palmore, 1976) showed Supported by NNSFC (National Natural Science Foundation of China) grant No. 10301006. ∗

c 2009 Kluwer Academic Publishers. Printed in the Netherlands.

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J. Lei and M. Santoprete

the existence of degenerate central configurations in the planar n-body problem with n ≥ 4. His example consists of n − 1 particles lying at the vertices of a regular polygon and one particle at the centroid. Unfortunately only few examples of degenerate central configurations are known. In this paper we find a new family of degenerate central configurations that arise from some highly symmetrical configurations. Another interesting problem, that is strictly related to the study of degenerate central configurations, is the study of bifurcations in the n-body problem. The interest in this problem arises because, at a bifurcation, the structure of the phase space changes. Several authors studied bifurcations in the n-body problem (see Sekiguchi (2004) for a list of references), in particular M. Sekiguchi analyzed a highly symmetrical configuration of 2n + 1-bodies. He considered a rosette configuration, i.e. a planar configuration where 2n particles of mass m lie at the vertices of two concentric regular n-gons, one rotated an angle of π/n from the other and another particle of mass m0 lies at the center of the two n-gons. He showed that there is a bifurcation in the number of classes of central configurations for any n ≥ 3. In this paper we generalize Sekiguchi example and we allow the masses on the two concentric n-gons to be different. This considerably complicates the analysis. Indeed, if one considers two concentric n-gons one with particles of mass m1 and the other (rotated of an angle π/n from the first) with particles of masses m2 and a mass m0 in the center, one has to deal with two mass parameters µ = m0 /m1 and ǫ = m2 /m1 . In this case we prove that, as µ varies, if n > 3, there is a degenerate central configuration and a bifurcation for every ǫ > 0. On the other hand the case n = 3 is special and, in this case, as µ is varied, there is a bifurcation for some values of ǫ but not for others. This paper is organized as follows. In the next section we introduce the equation of the n-body problem. In Section 3 we discuss central configurations. In the following section we introduce the highly symmetrical configurations that are the object of the paper. In Section 5 we present and prove the main results of the paper: the existence, for any n > 3, of a bifurcation in the number of classes of central configurations and of a new family of degenerate central configurations. In the last section we analyze the special case where n = 3.

2. Equations of Motion The planar n-body problem concerns the motion of n particles with masses mi ∈ R+ and positions qi ∈ R2 , where i = 1, . . . , n. The motion

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Rosette Central Configurations

3

is governed by Newton’s law of motion mi q¨i =

∂U . ∂qi

(1)

Where U (q) is the Newtonian potential U (q) =

X mi mj i

|qi − qj |

.

(2)

Let q = (q1 , . . . , qn ) ∈ R2n and M = diag[m1 , m2 , . . . , mn ]. Then the equations of motion can be written as q¨ = M −1

∂U . ∂q

(3)

In studying this problem it is natural to assume that the center of mass of the system is at the origin, i.e. m1 q1 + . . . + mn qn = 0, and that the configuration avoids the set ∆ = {q : qi = qj for some qi 6= qj }. 3. Central Configurations Definition 1. A configuration q ∈ R2 \ ∆ is called a central configuration if there is some constant λ such that M −1

∂U = λq. ∂q

Central configurations, as it was shown by Smale (see (Abraham and Marsden, 1978; Smale, 1970)), can be viewed as rest points of a certain gradient flow. Introduce a metric in R2n such that hq, qi = q T M q and let S = {q : hq, qi = 1, m1 q1 + . . . + mn qn = 0}

denote the unit sphere S 2n−3 with respect to this metric in the subspace where the center of mass is at the origin. The scalar product I = hq, qi is called moment of inertia. Let S ∗ = S \ ∆. The vector field X = M −1 ∂U ∂q + λq where λ = U (q) is the gradient of US , the restriction of U to the unit sphere S with respect to the metric h·, ·i. This is because X is tangent to S , it has rest points at exactly the central configurations with hq, qi = 1 and hX(q), vi = DU (q)v for every q ∈ S and v ∈ Tq S. Furthermore the rest points of X are exactly the central configurations in S. Note that, since the Newtonian potential is an homogeneous function, any central configuration is homothetic to one in S. Therefore the problem of finding central configurations is essentially

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that of finding rest points of the gradient flow of US or, equivalently, finding the critical points of US . The gradient flow preserves some sets of configurations with symmetry. In this paper we study one of such sets of configurations with symmetry. We denote by Cn the set of central configuration of the n-body problem. We say that two relative equilibria in S ∗ are equivalent (and belong to the same equivalence class) if one is obtained from the other by a rotation and an homothety. The set C˜n is the set of equivalence classes of central configurations. Clearly I and ∆ are invariant under the action of S 1 . Thus, we can conclude that S ∗ is diffeomorphic to the (2n − 3)-dimensional sphere S 2n−3 (it is actually an ellipsoid E 2n−3 ) with all the points ∆ removed, that is S ∗ = E 2n−3 \ (E 2n−3 ∩ ∆) ≈ S 2n−3 \ (S 2n−3 ∩ ∆). ˜S : Since US is invariant under the action of S 1 it defines a map U ∗ 1 ∗ ∗ 1 S /S → R. If we let π : S → S /S denote the canonical projection, ˜ = π(E 2n−3 ∩∆), and recalling that E 2n−3 /S 1 ≈ S 2n−3 /S 1 ≈ CP n−2 , ∆ complex projective space, we are led to the investigation of the critical ˜S : CP n−2 \ ∆ ˜ → R. points of U Consequently one can show that the set of equivalence classes of central configurations is given by the set of critcal points of the map ˜S : CP n−2 \ ∆ ˜ → R. More precisely we have the following result of U Smale (see (Abraham and Marsden, 1978; Smale, 1970; Smale, 1971)) Proposition 1. For any n ≥ 2 and any choices of the masses in the planar n-body the set of equivalence classes of central configurations is ˜S : CP n−2 \ ∆ ˜ → diffeomorphic to the set of critical points of the map U R. ˜S . A critical point of U ˜S is degenerate Let q be a critical point of U 2 ˜ provided that the hessian D US (q) has a nontrivial nullspace. We have the following definition Definition 2. An equivalence class of central configurations is degenerate (nondegenerate) provided that the corresponding critical point q ˜S is degenerate (nondegenerate). of U

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Rosette Central Configurations

N2

N1

m2

r1 m1

m0 r2

Figure 1. Rosette configuration for n = 6

4. Symmetrical Configurations Consider the set Σ of all the configuration in R2 consisting of two concentric regular n-gons, one rotated of an angle π/n from the other, with a mass in their common center of symmetry (see Figure 1). Let m0 , m1 and m2 be the masses in the center of mass, on the n-gon N1 and on the n-gon N2 respectively. Then it follows from the ˜S is tangent to Σ ˜ symmetry of the configuration that the gradient of U 2n−3 ˜ (where Σ = π(E ∩ Σ)). Thus to find equivalence classes of central ˜ it is sufficient to study the critical points of U ˜S | ˜ . configurations in Σ Σ ˜ is one dimensional, only one parameter is needed to describe Since Σ such symmetric configuration. This is a great simplification. Figure 1 shows two parameters (r1 , r2 ) which can be used to describe such a configuration. The potential in these coordinates is U (q) = (nm1 )2 U (r1 , r2 ) where µ U (r1 , r2 ) = n

ǫ 1 + +kn r1 r2

n ǫ 1 ǫ2 1X q + + r1 r2 n k=1 r 2 + r 2 − 2r r cos φ 1 2 k 1 2

!

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with, µ = m0 /m1 , ǫ = m2 /m1 , φk = (2k − 1)π/n and kn =

X 1 n−1 π csc k 4n k=1 n

is the potential of a regular n-gon of unit size and unit masses. The last term in U is the moment of inertia I(q) = hq, qi = I(r1 , r2 ) = m1 n(r12 + ǫr22 ). ˜S | ˜ = 0 The central configurations are the solutions of the equation ∇U Σ λ or ∇U = 2 ∇I (with λ = U ), that in this case can be written as ∂U =m1 nλr1 ∂r1 ∂U =m1 nλǫr2 . ∂r2

(4)

Solving the equations above for λ one gets 1 F (x) = 0 r23

(5)

where n (1 − ǫ) − x1 (1 − ǫx2 ) cos φk x3 X µ (1 − x3 ) + kn (ǫ − x3 ) + n n k=1 (1 + x2 − 2x cos φk )3/2 (6) and x = r2 /r1 . The equation for the central configuration above depends only on one parameter and is invariant under the transformation (x, ǫ, µ) → ( x1 , 1ǫ , µǫ). Thus it suffices to study the central configurations with 0 < ǫ ≤ 1. The case ǫ = 1 was studied in detail by Sekiguchi (2004) that proved the following

F (x, ǫ, µ) =

Theorem 1. If n = 2 the number of central configurations is one for any value of µ. If n ≥ 3 the number of central configurations is three for µ < µc (n) and one for µ ≥ µc (n), where n cos φk 1 X − nkn µc (n) = 12 k=1 sin3 (φk /2)

It is therefore sufficient to analyze the problem with 0 < ǫ < 1. Proposition 2. For every µ > 0 and ǫ > 0 there is at least one rosette central configuration.

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Rosette Central Configurations

Proof. Since limx→0 F (x) = nµ + kn > 0 and limx→∞ F (x) = −∞ < 0, by the intermediate value theorem, the equation F (x) = 0 has at least one solution.

When n = 2 it can be shown that, for every value of ǫ, there is only one class of central configurations and no bifurcation occur, or more precisely we have the following Proposition 3. If n = 2 for every µ > 0 and ǫ > 0 there is only one rosette central configurations. Proof. In this case F (x, ǫ, µ) =

µ 1 x3 (1 − ǫ) (1 − x3 ) + (ǫ − x3 ) + 2 8 (1 + x2 )3/2

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limx→0 F (x) = µ2 + 8ǫ and limx→∞ F (x) = −∞ so F (x) = 0 has at least one solution. We need only to prove the statement for 0 < ǫ ≤ 1. If ǫ = 1 F (x) is a monotonically decreasing and the statement follows. If 0 < ǫ < 1 consider ′

F (x) = 3x

2

µ 1 + − 2 8

(1 − ǫ) + (1 + x2 )5/2

.

(8)

Clearly one solution of F ′ (x) = 0 is x = 0. The other solutions can be found studying the equation η(x) = µ2 + 18 , where η(x) =

(1 − ǫ) (1 + x2 )5/2

is a monotonically decreasing function and η(0) = (1−ǫ). The equation η(x) = µ2 + 18 has no solutions if µ ≥ 47 or µ < 74 and ǫ ∈ ( 78 − µ2 , 1). It has one solution x∗ if µ < 47 and ǫ ∈ (0, 78 − µ2 ]. Consequently if µ ≥ 47 or µ < 47 and ǫ ∈ ( 78 − µ2 , 1) F ′ (x) is always negative, F (x) monotonically decreasing and F (x) = 0 has only one solution. On the other hand, if µ < 74 and ǫ ∈ (0, 78 − µ2 ], F ′ (x) is positive for x ∈ (0, x∗ ) and negative for x ∈ (x∗ , ∞). Thus F (x) is increasing for x ∈ (0, x∗ ), decreasing for x ∈ (x∗ , ∞) and F (x) = 0 has one solution since F (0) > 0. 5. Bifurcations and degenerate central configurations for n>3 In this section we consider the rosette central configurations for n > 3. The main result is the existence of a bifurcations for every value of ǫ as the parameter µ increases. The case n = 3 is studied in the next section. More precisely we prove the following

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Theorem 2. For any n > 3 and ǫ > 0 there is at least one value µ0 corresponding to a bifurcation in the number of equivalence classes of rosette central configurations as the parameter µ > 0 increases. An important consequence of the existence of a bifurcation is the existence of a degenerate equivalence class of rosette central configurations Corollary 1. For any n > 3 and ǫ > 0 there is at least one value µ0 of µ for which there is a degenerate equivalence class of rosette central configuration. ˜S (q; µ) Proof. The proof is by contradiction. Consider the potential U for the configuration under discussion in this paper, where we put into evidence the dependence on the mass µ. Let q10 , . . . , ql0 be the ˜S for µ = µ0 , where µ0 is the bifurcation value. critical points of U Assume that the class of central configurations is nondegenerate for ˜S (q 0 ; µ) has bounded inverse. But then every ql0 . This means that D 2 U l by the implicit function theorem, there exist a neighborhood B of µ0 and unique functions {ql (µ)}nl=1 defined in B, such that ql (µ0 ) = ql0 ˜S (ql (µ); µ) = 0. This contradicts the assumption that µ0 is a and D U bifurcation value. The proof of Theorem 2 requires several preparations. The reminder of this section is devoted to such preparations and to the proof of Theorem 2 First of all observe that the central configurations, when ǫ 6= 1 can also be viewed as the solutions of h(x, ǫ) = µ where n X ǫ − x3 x (1 − ǫ) − (1 − ǫ x2 ) cos φk x2 h(x, ǫ) = −n kn − 1 − x3 (1 − x3 ) k=1 (1 + x2 − 2 x cos φk )3/2

Hereinafter, we say x to be a rosette central configuration if x is solution of the equation µ = h(x, ǫ). Let uk = cos φk then h(x, ǫ) = h0 (x) + (1 − ǫ)h1 (x) where n uk x2 (1 − x2 ) X h0 (x) = −nkn + (1 − x3 ) k=1 (1 + x2 − 2xuk )3/2

and h1 (x) =

n x3 X (1 − xuk ) nkn − . 3 3 (1 − x ) 1 − x k=1 (1 + x2 − 2xuk )3/2

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5.1. The case x > 1, ǫ ∈ (0, 1) We now study the number of central configurations for x > 1 and ǫ ∈ (0, 1). It is easy to show that, for any µ > 0, there is at least one rosette central configuration with x > 1. This follows from the limits lim h(x, ǫ) = ∞,

lim h(x, ǫ) = −n kn < 0

x→∞

x→1+

and an application of the Intermediate Value Theorem. The first limit is lim+ h(x, ǫ) = ∞ × sgn ((ǫ − 1)An ) x→1

where An = nkn −

n φk 1X csc 4 k=1 2

(9)

and sgn ((ǫ − 1)An ) = 1 since ǫ − 1 < 0 and An < 0 by the following Lemma Lemma 1. For all n ≥ 2,

An < 0.

Proof. Clearly n−1 X

n kπ kπ X π csc − csc − n n 2n k=1 k=1

1 An = 4

!

(10)

therefore when n is even one has n−1 X

csc

k=1

=

n kπ π kπ X − csc − n n 2 n k=1

n/2 X

k=1

+

kπ kπ π csc − csc − n n 2n

n−1 X

+1 k= n 2

π π + − csc 2 2n

kπ π kπ − csc + csc n n 2n

(11)

<0

while when n is odd n−1 X

n kπ X kπ π csc − csc − n n 2n k=1 k=1

=

(n−1)/2

X

csc

k=1

+

n−1 X

k=(n+1)/2

kπ kπ π − csc − n n 2n

csc

− csc

kπ π kπ − csc + n n 2n

π 2

(12)

< 0.

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We now want to show that when µ is large enough, for every ǫ ∈ (0, 1), there is exactly one rosette central configuration with x > 1, i.e., we prove the following Proposition 4. For every ǫ ∈ (0, 1) there exists a µ ˆ such that for every µ>µ ˆ there is one and only one rosette central configuration Proof. Observe that one can write h(x, ǫ) = −

(1 − ǫ)An + O((x − 1)0 ). 3(x − 1)

(13)

Therefore there exist µ ˆ0 > 0 and δ > 0 such that for any µ > µ ˆ0 the equation h(x, ǫ) = µ has a unique solution in (1, 1 + δ). Moreover the function h(x, ǫ) has a maximum value µ ˆ1 in [1 + δ, ∞), since limx→∞ h(x, ǫ) = −nkn . Let µ ˆ = max(ˆ µ0 , µ ˆ1 ) then, if µ > µ ˆ, the equation h(x, ǫ) = µ has a unique solution. 5.2. The case 0 < x < 1, ǫ ∈ (0, 1) We now study the number of central configurations for x < 1 and ǫ ∈ (0, 1). In particular we show that Proposition 5. For any n > 3 and ǫ ∈ (0, 1), 1. there is a µ∗n > 0 such that for every 0 < µ < µ∗n there are at least two rosette central configurations, with x ∈ (0, 1) 2. there is a µ ˇ ≥ µ∗n such that for every µ > µ ˇ there are no rosette central configurations with x ∈ (0, 1) . Proof. (a) Observe that, if ǫ ∈ (0, 1), h(0, ǫ) = −n kn ǫ < 0,

lim h(x, ǫ) = −∞ × sgn ((ǫ − 1)An )) = −∞,

x→1−

where the limit follows from Lemma 1. If there exists x∗n ∈ (0, 1) such that h1 (x∗n ) = 0 and h0 (x∗n ) > 0, then by the Intermediate Value Theorem, µ = h(x, ǫ) has at least two solutions for every 0 < µ < µ∗n = h0 (x∗n ). To complete the proof it is necessary to show the existence of x∗n . The existence of x∗n will be proved in Lemma 3. (b) Since h(0, ǫ) < 0 and limx→1− h(x, ǫ) = −∞ < 0 the function h(x, ǫ) has a maximum value in [0, 1]. Let µ ˇ be such maximum. Then the equation µ = h(x, ǫ) has no solutions for x ∈ (0, 1) if µ > µ ˇ.

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To complete the proof of the proposition above, and prove Lemma 3 we need the following technical result Lemma 2. Let kn−

1 ln = 4π

then for all n ≥ 3

1 + cos nπ 1 − cos πn

!

+

1 4n sin nπ

(14)

kn > kn− .

Moreover kn− is monotonically increasing with n. Proof. The sum: n−1 X

csc

k=1

πk n

can be estimated using the trapezoidal rule. Since g(u) = csc πu n is convex on [1, n − 1] the trapezoidal rule gives an upper bound for the integral over [1, n − 1]: Z

1

n−1

1 1 g(x) dx < g(1) + g(2) + . . . + g(n − 2) + g(n − 1). 2 2

This gives the formula for kn− . Moreover kn− is monotonically increasing since the derivative of the function, obtained replacing πn in kn− with the continuous variable u, is negative. We can finally prove the following 54 , 1) such that Lemma 3. For any n > 3, there exists a x∗n ∈ ( 100 h1 (x∗n ) = 0 and h0 (x∗n ) > 0.

Proof. Verifying these conclusions numerically for small n is trivial by common mathematical software (for example, Mathematica1 or Matlab2 ). Numerical results for 4 ≤ n ≤ 106 are given at Figure 2 (The solutions x∗n are found numerically through the function FindRoot provided by Mathematica). The proof for n ≥ 107 is given below. The proof will be completed by showing firstly h1 (x) = 0 has solu54 54 , 1) and secondly h0 (x) > 0 for any x ∈ ( 100 , 1) such that tion x∗n ∈ ( 100 h1 (x) = 0. 1. We first show that for any n ≥ 107, the equation h1 (x) = 0 has 54 , 1). To this end, it is sufficient to show at least one solution x∗n ∈ ( 100 54 that h1 ( 100 ) > 0 and h1 (1) < 0. Equivalentlly, let ˜ 1 (x) = (1 − x3 ) h1 (x) = nkn − x3 h 1 2

n X

1 − xuk (1 + x2 − 2xuk )3/2 k=1

http://www.wolfram.com/ http://www.mathworks.com/

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J. Lei and M. Santoprete 0.76 0.76

160

140

0.74 0.74

120 0.72 0.72 0.72

**

h0x(x nn )

**

h0x(x nn )

100 0.7 0.70.7

80

0.68 0.68 0.68 60 0.66 0.66 0.66

40

0.64 0.64 0.64

20

0.62 0.62 0.62 0 0 0

0 2020 20

4040 40

60 60 60 nn

80 80 80

100 100 100

120 120

0

20

40

60 n

80

100

120

Figure 2. The root x∗n such that h1 (x∗n ) = 0 and corresponding h0 (x∗n ) for 4 ≤ n ≤ 106.

˜ 1 ( 54 ) > 0 and h ˜ 1 (1) < 0. we will show that h 100 When u ∈ [−1, 1] and x ∈ [0, 1], we have 1 − xu 1 < . 2 3/2 (1 − x)2 (1 + x − 2xu) and therewith ˜ 1 (x) > n kn − h

x3 (1 − x)2

!

Thus, when n ≥ 107

3 54 100 54 2 − 100 )

˜ 1 ( 54 ) > n h kn − 100 (1

> n (kn− −

75 75 − ) ≥ n (k107 − )>0 100 100

− where k107 = 0.7514096544 was computed using Lemma 5.2 and kn− > − k107 since kn− is monotonically increasing with n. A simple computation shows that

˜ 1 (1) = An = 1 h 4

n−1 X

n kπ X kπ π csc − csc( − ) , n n 2n k=1 k=1

!

(15)

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Rosette Central Configurations

13

˜ 1 (1) < 0 for any n. Hence, we conclude that and thus, by Lemma 1, h 54 for any n ≥ 3, there exist x∗n ∈ ( 100 , 1), such that h1 (x∗n ) = 0. 54 , 1) such that 2. We now show that for any n ≥ 107 and x∗n ∈ ( 100 ∗ ∗ h1 (xn ) = 0, h0 (xn ) > 0. Let 1 − x3 (h0 (x) + (1 − x3 ) h1 (x)) h2 (x) = 2 x (1 − x5 ) then

h0 (x∗n ) = where

x∗n 3 h2 (x∗n ) R1 (x∗n )

x (1 + x + x2 ) 1 + x + x2 + x3 + x4

R1 (x) =

54 , 1). To Thus, it is sufficient to prove that h2 (x) > 0 for any x ∈ ( 100 this end, introduce the notations

R2 (x) = 0.15 R1 (x) + 0.85 u − R1 (x) g(x, u) = (1 + x2 − 2 x u)3/2 then h2 (x) =

n X

g(x, uk ).

k=1

It is easy to have 0 < R1 (x) < R2 (x) < 1, ∀x ∈ (0, 1) Thus, grouping the subscripts k in the summation as following J1 = {k |1 ≤ k ≤ n, uk < R1 (x)} J2 = {k | 1 ≤ k ≤ n, uk ≥ R2 (x)} we have h2 (x) ≥

X

k∈J1

g(x, uk ) +

X

g(x, uk ).

k∈J2

54 Now, the function g(x, u) of u ∈ [0, 1] (with given x ∈ ( 100 , 1)) has minimum at 3 x R1 (x) − 1 − x2 u = u− (x) = x and is increasing when R2 (x) < u < 1. Thus, we have when k ∈ J1 ,

0 > g(x, uk ) > g(x, u− (x))

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J. Lei and M. Santoprete 80

70

60

h3(x,107)

50

40

30

20

10

0 0.5

0.55

0.6

0.65

0.7

0.75 x

0.8

0.85

0.9

0.95

1

Figure 3. The function h3 (x; n) with n = 107 and x ∈ (0.54, 1)

and when k ∈ J2 ,

g(x, uk ) ≥ g(x, R2 (x)) > 0

The number of elements in J1 and J2 are respectively n arccos R1 (x) ) < (π − arccos R1 (x)) N (J1 ) = n (1 − π π 1 n n arccos R2 (x) + ≥ arccos R2 (x) − 1 N (J2 ) = 2 2π 2 π

Therefore, we have h2 (x) >

X

k∈J1

g(x, u− (x)) +

X

g(x, R2 (x))

k∈J2

= N (J1 ) g(x, u− (x)) + N (J2 ) g(x, R2 (x)) n > ((π − arccos R1 (x)) g(x, u− (x)) + arccos R2 (x) g(x, R2 (x))) − g(x, R2 (x)) π := h3 (x; n) Now, we only need to verify h3 (x; n) > 0 for any n ≥ 107 and 54 , 1). It is evident that h3 (x, n) is increasing with respect to x ∈ ( 100 n, and thus h3 (x; 107) > 0, which is shown at Figure 3, is enough to complete the proof. The Lemma has been proved. 5.3. Proof of Theorem 2 With all the preparations above we are now well on our way to proving Theorem 2.

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Rosette Central Configurations

15

On one hand, using Proposition 4 and 5, we have, for every ǫ ∈ (0, 1), that if µ > max(ˆ µ, µ ˇ) the equation µ = h(x, ǫ) has a unique solution. On the other hand, by Proposition 5, we have, for every ǫ ∈ (0, 1), that if µ < µ∗n the equation µ = h(x, ǫ) has at least two solutions for x ∈ (0, 1) and at least one solution for x > 1. Moreover if ǫ 6= 1, x = 1 is not a solution of µ = h(x, ǫ). Thus the number of rosette central configurations changes as the parameter µ increases. The fact that this result holds for every ǫ > 0 follows from Theorem 1 and the invariance under the transformation (x, ǫ, µ) → ( x1 , 1ǫ , µǫ). This concludes the proof of Theorem 2. 6. The case n = 3 The case n = 3 is special, indeed for n = 3 the proof of Lemma 3 fails. 54 but h(x∗3 ) = −0.188154 < 0. This is because x∗3 = 0.617364 > 100 In this case we study numerically the maximum hmax (ǫ) of the function h(x, ǫ) (as a function of x) on the interval (0, 1). Figure 4(a) depicts hmax (ǫ) for ǫ ∈ (0, 1). Figure 4(b) shows a magnification of Figure 4(a) near ǫ = 0 making apparent that, near ǫ = 0, hmax (ǫ) > 0. From Figure 4(a)-(b) it is apparent that hmax (ǫ) is always negative except when ǫ is close to 0 or to 1. More precisely we find that hmax (ǫ) > 0 for ǫ ∈ (0, ǫ1 ) and ǫ ∈ (ǫ2 , 1) while hmax (ǫ) < 0 for ǫ ∈ (ǫ1 , ǫ2 ), where ǫ1 = 0.00076760883 and ǫ2 = 0.97198893434. On the other hand it can be proved that h(x, ǫ) is a monotone decreasing function with respect to x for x ∈ (1, ∞) and ǫ ∈ (0, 1). In fact, when n = 3, we have h(x, ǫ) = h0 (x) + (1 − ǫ)h1 (x) where

√ x2 (1 + x) 1 3 1 + − h0 (x) = − 3 1 + x + x2 (1 − x + x2 )3/2 (1 + x)3 √ 1 x3 2−x 3 − + h1 (x) = 3(1 − x3 ) 1 − x3 (1 + x)2 (1 − x + x2 )3/2

When x > 1, we have h′0 (x) < 0 and h′0 (x) + h′1 (x) < 0. From which it is easy to conclude that h′x (x, ǫ) < 0 for any x > 1 and ǫ ∈ (0, 1). Detailed computations will be omitted. Consequently for every ǫ ∈ (ǫ1 , ǫ2 ) there is one and only one rosette central configuration for every value of µ > 0. On the other hand, our numerical study shows that, if ǫ ∈ (0, ǫ1 ) or ǫ ∈ (ǫ2 , 1) there is a µ∗ such that if 0 < µ < µ∗ there are three rosette central configurations and if µ > µ∗ there is only one. In conclusion, we have the following.

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16

J. Lei and M. Santoprete

0.05

0.006 0.004

0 0.002 0

−0.05

−0.002 −0.1

−0.004 −0.006

−0.15 −0.008 −0.2

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

−0.01

0

0.0005

(a)

0.001

0.0015

0.002

0.0025

0.003

0.0035

(b)

Figure 4. (a) The maximum of the function h(x, ǫ) as a function of x for 0 < ǫ < 1. (b) Magnification of (a) near ǫ = 0.

Proposition 6. For n = 3 and ǫ ∈ (ǫ1 , ǫ2 ), there is exactly one rosette configuration for any µ > 0. For n = 3 and ǫ ∈ (0, ǫ1 ) or ǫ ∈ (ǫ2 , 1), there exists a value µ0 (ǫ) > 0, such that when µ > µ0 (ǫ), µ = µ0 (ǫ), and µ < µ0 (ǫ), there are exactly one, two and three rosette configurations, respectively.

Acknowledgements MS wish to thank Giampaolo Cicogna and Donald Saari for their comments and suggestions regarding this work. References Abraham, R. and Marsden, J.E. (1978). “Foundation of Mechanics”, (2nd ed.), Benjamin, New York . Albouy, A. (1995). “Sym´etrie des configurations centrales de quatre corps” , C.R. Acad. Sci. Paris, 320 , 217-220. Albouy, A. (1996). “The symmetric central configurations of four equal masses”, Contemporary Mathematics, 198, 131-135. Hampton, M. and Moeckel, R. (2006). “Finiteness of Relative Equilibria of the Four-Body Problem”, Inv. Math., to appear. Palmore, J.I. (1975). “Classifying relative equilibria II”, Bull. Amer. Math Soc., 81 , 489-491. Palmore, J.I. (1976). “Measure of Degenerate Equilibria I”, Ann. Math., 104 , 421429. Sekiguchi, M. (2004). “Bifurcations of central configurations in the 2N + 1 body problem”, Cel. Mech. & Dyn. Astr., 90 , 355 - 360. Smale, S. (1970). “Topology and Mechanics II”, Inv. Math, 11 , 45-64. Smale, S. (1971). “Problems on the nature of relative equilibria in celestial mechanics”, Lecture Notes in Math., 197 , 194–198.

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