Convex central configurations of the n-body problem
which are not strictly convex
Kuo-Chang Chen, Jun-Shian Hsiao
Department of Mathematics
National Tsing Hua University
Hsinchu 30013, Taiwan
Abstract. It is well-known that if a planar central configuration for the Newtonian 4body problem is convex, then it must be strictly convex. In some literature, same conclusion were believed to hold for the case of five or even more bodies but rigorous treatments
are absent. In this paper we provide concrete examples of central configurations which are
convex but not strictly convex. Our examples include planar central configurations with
five bodies and spatial central configurations with seven bodies.
1. Introduction
The classical n-body problem concerns the motion of n mass points moving in space in
accordance with Newton’s law of gravitation:
(1)
q̈k =
X mi (qi − qk )
i6=k
|qi − qk |3
,
k = 1, 2, · · · , n.
Here qk ∈ Rd (1 ≤ d ≤ 3) is the position of mass mk > 0. Alternatively the system (1) can
be written
(2)
mk q̈k =
∂
U (q),
∂qk
k = 1, 2, · · · , n
where
U (q) :=
X
1≤i<j≤n
mi mj
|qi − qj |
is the potential of the system. The position vector q = (q1 , · · · , qn ) ∈ (Rd )n is often referred
to as the configuration of the system, and vectors {qk } are vertices of the configuration q.
Let M = m1 + · · · + mn be the total mass and
q̂ =
1
(m1 q1 + · · · + mn qn )
M
Key words and phrases. central configuration, n-body problem.
1
2
KUO-CHANG CHEN AND JUN-SHIAN HSIAO
be the mass center of the configuration q. The set ∆ of collision configurations is the
algebraic variety defined by
∆ = {q ∈ (Rd )n : qi = qj for some i 6= j}.
A configuration q ∈ (Rd )n \ ∆ is called a central configuration if there exists some positive
constant λ, called the multiplier, such that
X mi (qi − qk )
−λ(qk − q̂) =
(3)
,
|qi − qk |3
k = 1, 2, · · · , n.
i6=k
The assumption that λ being positive is redundant as it can be easily verified that there
is no negative λ that can possibly satisfy (3). The definition of central configuration can
be extended to cases with some zero masses but we shall only consider positive masses
throughout this paper.
The set of central configurations are invariant under three classes of transformations on
(Rd )n which act equitably on each copy of Rd : translations (qk 7→ qk + a, a ∈ Rd ), scalings
(qk 7→ µqk , µ 6= 0), and orthogonal transformations (qk 7→ Aqk , A ∈ O(d)). Scaling a central
configuration with multiplier λ by multiplying µ results in a central configuration with
multiplier λ/|µ|3 . Conventionally two central configurations are considered equivalent if one
can be obtained from the other via translations, scalings, and SO(d)-actions. This clearly
defines an equivalence relation on central configurations, and two central configurations
are different in that aspect if they differ from each other by a reflection with respect to a
hyperplane in Rd . The term “central configurations” is often referred to equivalence classes
of central configurations.
There are several reasons why central configurations are of special importance in the
study of the n-body problem, see [5] for details.
Without loss of generality we consider only central configurations q with mass centers q̂
at the origin. In this case, using the Leibniz formula for the moment of inertia
I(q) :=
n
X
mk |qk |2 =
1 X
mi mj |qi − qj |2
M
i<j
k=1
the system (3) can be written in a compact form
λ
− ∇I(q) = ∇U (q),
2
(4)
or equivalently,
X
i6=k
mi
λ
1
−
M
|qi − qk |3
(qi − qk ) = 0,
k = 1, 2, · · · , n.
CONVEX CENTRAL CONFIGURATIONS WHICH ARE NOT STRICTLY CONVEX
3
By suitable scaling, we may assume without loss of generality that the multiplier λ is equal
to the total mass M . Then equations for central configurations become
(5)
n
X
mi (1 − sik ) (qi − qk ) = 0,
k = 1, 2, · · · , n,
i=1
where
sij =
1
|qi − qj |3
for i 6= j;
sii = 1
for each i.
This is the system of equations we will be working with.
The convex hull Conv(q) of the configuration q ∈ (Rd )n is the convex hull in Rd for its
vertices {qk }. We say the configuration q is convex if each qk is on the boundary of Conv(q)
(as a subset of Rd ); it is strictly convex if no qk is on the convex hull for other vertices.
Concave configurations are configurations which are not strictly convex. Strictly concave
configurations are configurations which are not convex. The dimension of a configuration q
is defined as the dimension dim Conv(q) of Conv(q).
Strictly convex and strictly concave configurations are clearly two open sets in (Rd )n and
their boundaries intersect at configurations that are both convex and concave (convex but
not strictly convex, in other words). The intersection also includes some collision configurations. According to the above definition, a collinear configuration q (i.e. dim Conv(q) ≤ 1)
in (Rd )n \ ∆ with n ≥ 3 is concave if d = 1 but is both convex and concave if d ≥ 2.
Likewise, a coplanar configuration q (i.e. dim Conv(q) ≤ 2) with n ≥ 4 is convex if d ≥ 3,
or d = 2 and no qk is in the interior of the convex hull for other vertices, or d = 2 and q
is collinear. In literature the terms convex and concave configurations are often referred to
noncollinear configurations.
Other than the 12 collinear central configurations [6], the planar four-body problem
(d = 2) has at least 14 concave central configurations [2, 3], and at least 6 non-collinear
convex central configurations, one for each of the 6 cyclic orderings of the four bodies [4].
It follows easily from the perpendicular bisector theorem [5] that all of these convex central
configurations are strictly convex. By calculating the derivative of the normalized potential
on the boundary of the set of convex configurations, Xia shows that no local minimum can
possibly fall on this boundary, following from which he provides a simple alternative proof
for the existence of a convex central configuration for each of the 6 cyclic orderings. There
it was proposed that such a method may also work for the planar five-body problem, or
even planar and spatial n-body problems in general [8, §2]. In [7, §2 and §8-Theorem 8.1]
Williams also claimed that convex five-body central configurations cannot have three masses
4
KUO-CHANG CHEN AND JUN-SHIAN HSIAO
along the same line. Our main result shows the existence of central configurations which
disprove above stated assertions.
Theorem 1. For some positive masses there are 2-dimensional central configurations for
the 5-body problem which are convex but not strictly convex. For some positive masses there
are 3-dimensional central configurations for the 7-body problem which are convex but not
strictly convex.
2. Planar central configurations with five bodies
In this section we construct a 2-dimensional central configuration with five bodies which
is convex but not strictly convex.
Consider a planar convex configuration q ∈ (R2 )5 whose convex hull Conv(q) is an isosceles trapezoid. Vertices are
q1 = (−α, −γ1 ),
q2 = (α, −γ1 ),
q3 = (−β, γ2 ),
q4 = (β, γ2 ),
where α, β, γ1 , γ2 are positive numbers. The fifth vertex q5 is located (0, −γ1 ), making the
configuration q convex but not strictly convex.
Let m1 = m2 = m5 = 1, m3 = m4 = µ. The height γ = γ1 + γ2 of the isosceles trapezoid
and γ1 , γ2 are related by
2µγ
3γ
, γ2 =
3 + 2µ
3 + 2µ
so that the mass center q̂ of the configuration q is at the origin.
γ1 =
Apparently we have
(6)
s13 = s24 = (α − β)2 + γ 2
− 3
s14 = s23 = (α + β)2 + γ 2
− 3
s35 = s45 = β 2 + γ 2 2 ,
− 32
2
,
,
s15 = s25 = 8s12 = α−3 ,
s34 = (2β)−3 ,
sij = sji for any i, j.
From these symmetries, in (5) equations for k = 1, 2 are identical, equations for k = 3, 4
are also identical. Equations (5) for central configurations become
(1 − s12 )(q2 − q1 ) + µ(1 − s13 )(q3 − q1 ) + µ(1 − s14 )(q4 − q1 ) + (1 − s15 )(q5 − q1 ) = 0
(1 − s13 )(q1 − q3 ) + (1 − s23 )(q2 − q3 ) + µ(1 − s34 )(q4 − q3 ) + (1 − s35 )(q5 − q3 ) = 0
(1 − s15 )(q1 − q5 ) + (1 − s25 )(q2 − q5 ) + µ(1 − s35 )(q3 − q5 ) + µ(1 − s45 )(q4 − q5 ) = 0.
CONVEX CENTRAL CONFIGURATIONS WHICH ARE NOT STRICTLY CONVEX
5
This is actually a system of six equations with positive unknowns α, β, γ, µ, and sij :
2(1 − s12 )α + µ(1 − s13 )(α − β) + µ(1 − s14 )(α + β) + (1 − s15 )α = 0
(1 − s13 )(−α + β) + (1 − s23 )(α + β) + 2µ(1 − s34 )β + (1 − s35 )β = 0
−(1 − s15 )α + (1 − s25 )α − µ(1 − s35 )β + µ(1 − s45 )β = 0
(7)
µ(1 − s13 )γ + µ(1 − s14 )γ = 0
(1 − s13 )γ + (1 − s23 )γ + (1 − s35 )γ = 0
µ(1 − s35 )γ + µ(1 − s45 )γ = 0
The third identity is obvious as it follows immediately from (6). The fourth and the sixth
identities are equivalent to
(8)
s13 + s14 = 2,
s35 = 1.
These two identities together with s14 = s23 imply the fifth identity in (7). The second
identity in (7) can therefore be simplified to
µ(1 − s34 )β = (1 − s13 )α.
(9)
By (6), (8), (9), the first identity in (7) can be written
(3 − 10s12 )(s34 − 1) + 2(s13 − 1)2 = 0.
(10)
The system (7) is now reduced to (8), (9), (10).
We will soon see that every variable can be expressed in terms of s13 = θ. Observe that
2
−2
2
−2
θ− 3 = s133 = (α − β)2 + γ 2 .
(2 − θ)− 3 = s143 = (α + β)2 + γ 2 ,
Thus
2
α=
2
θ− 3 + (2 − θ)− 3 − 2
2
!1
2
2
,
2
(2 − θ)− 3 − θ− 3
β=
,
4α
both of which are positive since
2 − θ = s14 < s13 = θ ∈ (1, 2).
6
KUO-CHANG CHEN AND JUN-SHIAN HSIAO
Every sij can now be expressed in terms of θ via (6) and (8), and so is γ =
p
1 − β 2 . In
terms of θ, (10) becomes
(11)


2 3
2
−
−
3
3
3
(2 − θ) − θ
√ 2
2
2


3 2 θ− 3 + (2 − θ)− 3 − 2 − 5 1 − √
3 
− 23
− 32
2 2 θ + (2 − θ) − 2 2
2
2 3
+
(2 − θ)− 3 − θ− 3 (θ − 1)2 = 0.
Let R(θ) be the function in (11). Then it increases on [1.4, 1.6] from R(1.4) ≈ −0.647 to
R(1.6) ≈ 0.866. The unique root θ∗ in there is approximately 1.506654. One can easily
check that the corresponding γ, µ are both strictly positive. This proves the existence of a
2-dimensional central configuration for the five-body problem with suitable masses.
Numerical data accurate to the 24th decimal places are given in table 1. Figure 1 shows
a homographic solution for this particular central configuration with period 2π and eccentricity 0.6.
From (4), central configurations are exactly the set of critical points of the normalized
√
potential Ũ = IU . The central configuration we found is of local minimum type as one
can easily check that eigenvalues of D2 Ũ are all positive, except the two zero eigenvalues
due to the invariance under rotations and scalings. Therefore, the idea of perturbing the
intermediate mass q5 downward so as to decrease the normalized potential (see [8]) is not
applicable to this case.
2
0
-2
-4
-4
-2
0
2
4
Figure 1. A convex but not strictly convex central configuration with five bodies.
CONVEX CENTRAL CONFIGURATIONS WHICH ARE NOT STRICTLY CONVEX
θ
= 1.506654068878065383203023,
µ
7
= 11.23156072828415553841745,
α = 0.425756342462430700206462,
γ1 = 0.767184777048876570481630,
β = 0.493679334448944202819131,
γ2 = 0.102459239050841954854995.
Table 1. Numerical data for the example of planar five-body central configuration.
Same calculations can be carried out for the more general case with m5 = ν as a new
parameter. The system is reduced to (8), (9), and an identity similar to (10):
(2 + ν − (2 + 8ν)s12 )(s34 − 1) + 2(s13 − 1)2 = 0.
(12)
Following the procedure demonstrated above we may reduce (12) to an equation in θ = s13
with an additional parameter ν. By solving the equation numerically, as ν increases the
mass µ increases and the shape of the isosceles trapezoid deforms closer and closer to a
rectangle. Figure 2 shows one central configuration with ν = 10−4 and the other one with
ν = 104 . The corresponding µ are approximately 2.758 and 81952.332, respectively. The
one with ν = 104 is of local minimum type but the one with ν = 10−4 is not. A degenerate
central configuration is found when ν reaches a threshold value around 0.518085751, for
which case µ is approximately 7.215.
0.2
0.0
0.0
-0.2
-0.2
-0.4
-0.4
-0.6
-0.6
-0.8
-0.4
-0.2
0.0
0.2
0.4
-0.4
-0.2
0.0
0.2
0.4
Figure 2. Central configurations with ν = 10−4 (left) and ν = 104 (right).
3. Spatial central configurations with seven bodies
In this section we construct a 3-dimensional central configuration with seven bodies
which is convex but not strictly convex. Its convex hull is an octahedron with two planes
of symmetry. The idea is to add two mass points along the vertical line through the
circumcenter of a horizontal five-body isosceles trapezoidal configuration similar to the one
in the previous section.
8
KUO-CHANG CHEN AND JUN-SHIAN HSIAO
Consider a 3-dimensional configuration q ∈ (R3 )7 with vertices
q1 = (−α, −ζ, 0),
q2 = (α, −ζ, 0),
q3 = (−β, γ − ζ, 0),
q4 = (β, γ − ζ, 0),
q5 = (0, −ζ, 0),
q6 = (0, δ − ζ, η),
q7 = (0, δ − ζ, −η).
Here α, β, γ, η are all assumed to be positive. The term δ is given by
δ =
−α2 + β 2 + γ 2
2γ
so that q6 and q7 are equally distant from the vertices q1 , q2 , q3 , q4 of a trapezoid. The
configuration is clearly convex but not strictly convex as q5 is the midpoint of the edge q1 q2 .
Let m1 = m2 = m5 = 1, m3 = m4 = µ, m6 = m7 = ν. By setting
ζ =
2(µγ + νδ)
3 + 2µ + 2ν
the mass center q̂ of the configuration q is exactly the origin. We will show that q is indeed
a central configuration for some positive α, β, γ, η, µ, and ν.
The main idea is the same as the planar case. First we write down equations of central
configurations in terms of α, β, γ, η, µ, ν, and sij ’s, and then look for solutions with all
of these variables positive. Each sij can be easily expressed in terms of α, β, γ, η and
are subject to some symmetry constraints. By straightforward reductions, variables µ and
ν can be also easily expressed in terms of these four variables, and we are left with four
equations with four variables.
We begin with some observations on sij ’s:
3
s13 = s24 = ((α − β)2 + γ 2 )− 2 ,
3
s14 = s23 = ((α + β)2 + γ 2 )− 2 ,
3
s35 = s45 = (β 2 + γ 2 )− 2 ,
s15 = s25 = 8s12 = α−3 ,
(13)
s34 = (2β)−3 ,
3
s17 = s27 = s37 = s47 = s16 = s26 = s36 = s46 = (α2 + δ 2 + η 2 )− 2 ,
3
s56 = s57 = (δ 2 + η 2 )− 2 ,
s67 = (2η)−3 ,
sij = sij
for any i, j.
Using merely symmetries among sij ’s, equations (5) are reduced to cases k = 1, 3, 5, 7 and
they are expressed in terms of α, β, γ, η, µ, ν, and s12 , s13 , s14 , s34 , s35 , s17 , s57 , s67 . The
CONVEX CENTRAL CONFIGURATIONS WHICH ARE NOT STRICTLY CONVEX
9
third component of (5) is clearly zero when k = 1, 3, 5, and the first component of (5) is
also zero when k = 5, 7. There are seven equations left:
α(3 − 10s12 ) + µα(2 − s13 − s14 ) + 2να(1 − s17 ) + µβ(s13 − s14 ) = 0
µγ(2 − s13 − s14 ) + 2νδ(1 − s17 ) = 0
α(s13 − s14 ) + β(3 − s13 − s14 − s35 ) + 2µβ(1 − s34 ) + 2νβ(1 − s17 ) = 0
γ(3 − s13 − s14 − s35 ) + 2ν(γ − δ)(1 − s17 ) = 0
(14)
µγ(1 − s35 ) + νδ(1 − s57 ) = 0
2(µ(γ − δ) − δ)(1 − s17 ) − δ(1 − s57 ) = 0
2(1 + µ)(1 − s17 ) + (1 − s57 ) + 2ν(1 − s67 ) = 0.
For brevity we keep the notation δ in these equations. One can easily check linear dependence of the second, fourth, fifth, and the sixth equations. By the substitutions
νδ
(1 − s57 ) + 1
µγ
2(µγ − µδ − δ)
= 1−
(1 − s17 )
δ
µγ
=
(2 − s13 − s14 ) + 1
2νδ
s35 =
(15)
s57
s17
the system is reduced to three equations, two of which can be used to determine masses µ
and ν:
(16)
µ =
ν =
α(s14 − s13 )
β(s13 + s14 − 2s34 )
s
µγ 2 − s13 − s14
.
δ
2(1 − s67 )
In particular we have s13 + s14 − 2s34 < 0, since s14 < s13 . The remaining equation can be
written
(17)
(3 − 10s12 )(s13 + s14 − 2s34 )
α(δ − γ)
= (s13 − s14 )2 +
(s13 − s14 )(2 − s13 − s14 ) .
βδ
With the help of (13) and (16), equations (15) and (17) are now four equations in four
unknowns α, β, γ, η. The Newtonian method converges rapidly near (0.4, 0.5, 0.87, 0.77),
and resulting masses µ and ν are both positive. This shows the existence of a central
configuration with seven bodies which is convex but not strictly convex, and finishes the
proof of Theorem 1.
10
KUO-CHANG CHEN AND JUN-SHIAN HSIAO
α = 0.396619093609962801426587,
γ = 0.871188462806049795533515,
β = 0.495555327813506194775772,
η = 0.772004237900502489476959,
δ
ζ
= 0.486253980284487121746795,
µ = 14.09290585257097996322574,
= 0.777294735425231051490145,
ν = 0.540451308652400689509046.
Table 2. Numerical data for the example of spatial seven-body central configuration
Numerical data accurate to the 24th decimal places are given in table 2. Figure 3 shows
the corresponding central configuration. This central configuration is not of local minimum type since the second derivative of the normalized potential D2 Ũ has two negative
eigenvalues that are approximately −134.616 and −66.881.
As in the previous section, we may consider the more general case with m5 = σ as
a parameter, then obtain a family of convex spatial central configurations which are not
strictly convex. The spectrum of D2 Ũ evaluated at the central configuration varies with σ
and, among those we computed, the number of negative eigenvalues are two or three. A
degenerate central configuration can be obtained when σ ≈ 0.504336299.
0.0
-0.2
-0.6
-0.4
0.5
0.0
-0.5
0.5
0.0
-0.5
Figure 3. A convex but not strictly convex central configuration with seven bodies.
Convex central configurations which are not strictly convex should be rare; we have
searched miscellaneous seemly possible candidates and the one in this section is the only
one we found for the spatial seven-body central configuration.
Now we finish this paper with some remarks on spatial central configurations with less
bodies. There is no 3-dimensional convex central configuration with five bodies which is
CONVEX CENTRAL CONFIGURATIONS WHICH ARE NOT STRICTLY CONVEX
11
not strictly convex. If there were, then there would be a plane π which contains exactly
four masses. One way of excluding this possibility is by using a spatial version of the
perpendicular bisector theorem [5, pp.511]. Another simple approach is by using a result in
[1, Proposition 5] which implies in this case that the four masses on π must be cocircular,
thus forming a strictly convex polygon. Consequently, since the fifth body is not on π, the
configuration is strictly convex.
It would be interesting to know if there exists any 3-dimensional convex central configuration with six bodies which is not strictly convex. We don’t know the answer. A case
we find difficult to exclude is a hexahedron with an isosceles triangular base and with three
collinear bodies on its top which fall on the perpendicular bisecting plane of the base.
Acknowledgement.
We are grateful to Alain Albouy and Rick Moeckel for helpful conversations during preparation of this work. Our research is partly supported by the National Science Council and
the National Center for Theoretical Sciences in Taiwan.
References
[1] Albouy, A., On a paper of Moeckel on central configurations. Regul. Chaotic Dyn. 8 (2003), 133–142.
[2] Hampton, M., Convex central configurations in the four body problem. Thesis, University of Washington (2002).
[3] Hampton, M., Moeckel, R., Finiteness of relative equilibria of the four-body problem, Invent. Math.
163 (2006), 289–312.
[4] MacMillan, W. D.; Bartky, W., Permanent configurations in the problem of four bodies. Trans.
Amer. Math. Soc. 34 (1932), 838–875.
[5] Moeckel, R., On central configurations. Math Z. 205 (1990), 499–517.
[6] Moulton, F. R., The straight line solutions of the problem of N bodies. Ann. of Math. (2) 12 (1910),
1–17.
[7] Williams, W. L., Permanent configurations in the problem of five bodies. Trans. Amer. Math. Soc.
44 (1938), 563–579.
[8] Xia, Z., Convex central configurations for the n-body problem. J. Differential Equations 200 (2004),
185–190.
Kuo-Chang Chen, Department of Mathematics, National Tsing Hua University, Taiwan
E-mail address: [email protected]
Jun-Shian Hsiao, Department of Mathematics, National Tsing Hua University, Taiwan
E-mail address: [email protected]