This document discusses central configurations (CC) of systems with multiple twisted rings of bodies. It begins by introducing the problem and providing relevant equations. For two twisted rings, the mass ratio and sizes are related by an equation involving trigonometric functions. Results are presented for specific numbers of bodies, showing the possible CC depend on the mass ratio and ring sizes. For three twisted rings, the CC equations involve three unknown masses and sizes. The document outlines findings for multiple rings and numbers of bodies in each ring.

1. Introduction CC of two twisted rings CC of three twisted rings
On Central Configurations of Twisted Rings
E. Barrab´es1
J.M. Cors 2
G. Roberts3
1
Universitat de Girona
2
Universitat Polit`ecnica de Catalunya
3
College of the Holy Cross
HamSys – Bellaterra, 2014/06/02
Barrab´es, Cors and Roberts (HamSys 2014) Twisted CC 1 / 27

2. Introduction CC of two twisted rings CC of three twisted rings
Outline
Introduction
CC of two twisted rings
CC of three twisted rings
Barrab´es, Cors and Roberts (HamSys 2014) Twisted CC 2 / 27

3. Introduction CC of two twisted rings CC of three twisted rings
Planar Central Configuration of κn Bodies
We consider
κ groups of n bodies: qji ∈ R2
, i = 1, . . . , n, j = 1, . . . , κ
All bodies in the same group have equal mass mj, j = 1, . . . , κ
κ
j=1
mj(qj1 + · · · + qjn) = 0
A central configuration is q = (q11, q12, . . . , qκn) ∈ R2κn
such that, for some
value of w, satisfies the algebraic equation
U(q) + w2
Mq = 0
U(q) =
κ
j=1
n
i=1


n
l=i+1
m2
j
||qji − qjl||
+
n
l=j+1
n
m=1
mjml
||qji − qlm||


Barrab´es, Cors and Roberts (HamSys 2014) Twisted CC 3 / 27

4. Introduction CC of two twisted rings CC of three twisted rings
Planar Central Configuration of κn Bodies
The particles within the same group are located at the vertices of an n-gon.
Nested rings Twisted rings
Vertices align Rotation of π/n
Barrab´es, Cors and Roberts (HamSys 2014) Twisted CC 4 / 27

5. Introduction CC of two twisted rings CC of three twisted rings
Previous Results
Moeckel and Sim´o (1995)
Two nested rings for any n: for every ratio m1/m2, there are exactly two
planar central configurations, one with the ratio of the sizes of the two
polygons less than 1, and the other one greater than 1.
Zhang and Zhou (2002)
Two rings: they take all the masses different, imposing CC equations, the
masses within the same group must be equal.
Corollary (MacMillan-Bartky): in the case of two twisted rings of two
bodies, the ratio of the sizes of the two polygons a ∈ (1/
√
3,
√
3).
Yu and Zhang (2012)
CC of two rings: if two rings, one rotated with respect the other an angle
θ, are a CC then θ = 0 (nested) or θ = π/n (twisted)
Barrab´es, Cors and Roberts (HamSys 2014) Twisted CC 5 / 27

6. Introduction CC of two twisted rings CC of three twisted rings
Previous Results
Corbera, Delgado, Llibre (2009)
Nested κ n-gons: there exist planar central configurations of κ nested
rings of n-gons
Llibre, Melo (2009)
Particular cases of twisted rings: existence of central configurations of
twisted rings for κ = 3 and n = 2, 3 and κ = 4 and n = 2.
Roberts (1999 PhD), Sekiguchi (2004), Lei and Santoprete (2006)
CC of two rings and a central mass: number of central configurations and
bifurcations varying the value of the central mass
Barrab´es, Cors and Roberts (HamSys 2014) Twisted CC 6 / 27

7. Introduction CC of two twisted rings CC of three twisted rings
Equations (1)
Size of the n-gon: aj, j = 1, . . . , κ
Argument of the first particle in a group: j, j = 1, . . . , κ
First group: m1 = 1, a1 = 1 and 1 = 0
qj1 = ajei j
, qjk = qj1ei2π(k−1)/n
, k = 2, . . . , n, j = 1, . . . , κ.
Barrab´es, Cors and Roberts (HamSys 2014) Twisted CC 7 / 27

8. Introduction CC of two twisted rings CC of three twisted rings
Equations (2)
Using the symmetry of the problem, the CC equation reduces to
(Eqj) :
∂U
∂qj1
+ w2
mjqj1 = 0 −→
(Eqj) = 0
(Eqj) = 0
for all j = 1, . . . , κ.
(Eqj) = 0 if and only if:
κ
l=1
l=j
mlal
n
k=1
sin( l − j + 2πk/n)
(a2
l + a2
j − 2alaj cos( l − j + 2πk/n))3/2
= 0
Barrab´es, Cors and Roberts (HamSys 2014) Twisted CC 8 / 27

9. Introduction CC of two twisted rings CC of three twisted rings
Equations (2)
Using the symmetry of the problem, the CC equation reduces to
(Eqj) :
∂U
∂qj1
+ w2
mjqj1 = 0 −→
(Eqj) = 0
(Eqj) = 0
for all j = 1, . . . , κ.
(Eqj) = 0 if and only if:
κ
l=1
l=j
mlal
n
k=1
sin( l − j + 2πk/n)
(a2
l + a2
j − 2alaj cos( l − j + 2πk/n))3/2
0
if k=0,π/n, ∀k
= 0
Barrab´es, Cors and Roberts (HamSys 2014) Twisted CC 8 / 27

10. Introduction CC of two twisted rings CC of three twisted rings
Equations (3)
Fixed { j}j=1,...,κ with j ∈ {0, π/n},
(Cj1 − Snaj)m1 +
κ
l=2
l=j
(Cjl − ajC1l)ml +
Sn
a2
j
− ajC1j mj = 0,
j = 2, . . . , κ
κ − 1 equations with 2κ − 2 unknowns: m2, . . . , mκ, a2, . . . , aκ
Sn =
1
4
n−1
k=1
1
sin(kπ/n)
,
Cjl = Cjl(aj, al) =
n
k=1
aj − al cos( j − l + 2kπ/n)
(a2
j + a2
l − 2ajal cos( j − l + 2kπ/n))3/2
.
Barrab´es, Cors and Roberts (HamSys 2014) Twisted CC 9 / 27

11. Introduction CC of two twisted rings CC of three twisted rings
Outline
Introduction
CC of two twisted rings
CC of three twisted rings
Barrab´es, Cors and Roberts (HamSys 2014) Twisted CC 10 / 27

12. Introduction CC of two twisted rings CC of three twisted rings
Two twisted rings
The unknowns are m = m2 and a = a2
(C21(a) − Sna) +
Sn
a2
− aC12(a) m = 0
where
C21(a) = −φ (a), C12(a) = φ(a) + aφ (a)
and
φ(a) =
κ
k=1
1
(1 + a2 − 2a cos (2k−1)π
n )1/2
Barrab´es, Cors and Roberts (HamSys 2014) Twisted CC 11 / 27

13. Introduction CC of two twisted rings CC of three twisted rings
Two twisted rings
Inverse problem
m = H(a) =
a2
(Sna − C21(a))
Sn − a3C12(a)
, a > 0.
1 H(1) = 1. If the two n-gons have the same size, all the masses are equal.
2 H(1/a) = 1/H(a).
If (m, a) corresponds to a CC −→ (1/m, 1/a) is also a CC
3 The CC is convex if
cos
π
n
≤ a ≤
1
cos
π
n
Barrab´es, Cors and Roberts (HamSys 2014) Twisted CC 12 / 27

14. Introduction CC of two twisted rings CC of three twisted rings
Results for n = 2
For each value of the mass ratio m, there exist only one central configuration of
two twisted 2-gons of the 4-body problem (rhomboidal central configuration)
Admissible sizes a ∈ (1/
√
3,
√
3)
a = 1/
√
3 → m = ∞
a =
√
3 → m = 0
All the CC are convex
Barrab´es, Cors and Roberts (HamSys 2014) Twisted CC 13 / 27

15. Introduction CC of two twisted rings CC of three twisted rings
Results for n = 3
Theorem
For n = 3, the set of admissible values of a such that there exist a CC is
(0, z1) ∪ (1/z2, z2) ∪ (1/z1, ∞),
where z1 < 1 < z2 are the only two positive zeros of m = H(a) = 0, and
z1z2 < 1
convex
Barrab´es, Cors and Roberts (HamSys 2014) Twisted CC 14 / 27

16. Introduction CC of two twisted rings CC of three twisted rings
Results for n = 3
Theorem
For n = 3, the set of admissible values of a such that there exist a CC is
(0, z1) ∪ (1/z2, z2) ∪ (1/z1, ∞),
where z1 < 1 < z2 are the only two positive zeros of m = H(a) = 0, and
z1z2 < 1
convex
Barrab´es, Cors and Roberts (HamSys 2014) Twisted CC 14 / 27

17. Introduction CC of two twisted rings CC of three twisted rings
Results for n = 3
Theorem
For n = 3, the set of admissible values of a such that there exist a CC is
(0, z1) ∪ (1/z2, z2) ∪ (1/z1, ∞),
where z1 < 1 < z2 are the only two positive zeros of m = H(a) = 0, and
z1z2 < 1
convex
Barrab´es, Cors and Roberts (HamSys 2014) Twisted CC 14 / 27

18. Introduction CC of two twisted rings CC of three twisted rings
Results for n = 3
Theorem
Given a value of the mass ratio m ≥ 1, the number of different CC are
three for any m ∈ (1, m∗
) ∪ (m∗∗
, ∞)
one for any m ∈ (m∗
, m∗∗
)
two for m = 1, m∗
, m∗∗
.
m∗
1.000768, m∗∗
35.700177
Barrab´es, Cors and Roberts (HamSys 2014) Twisted CC 15 / 27

19. Introduction CC of two twisted rings CC of three twisted rings
Convex CC for the 6-body problem
Moeckel: numerical exploration of CC for the N-body problem with equal
masses (N = 4, . . . , 8). In particular, for N = 6 there exist two convex
CC:
a = 1 a 1.077171777
For any m ∈ [1, 1.000768) there exist two convex CC of the 6-body
problem
Barrab´es, Cors and Roberts (HamSys 2014) Twisted CC 16 / 27

20. Introduction CC of two twisted rings CC of three twisted rings
Results for n ≥ 4
Proposition
There exist at least two positive solutions, z1 and z2, of m = H(a) = 0 with
z1 < 1 < z2 and and z1z2 > 1.
Conjecture
For n ≥ 4 the equation H(a) = 0 has only two solutions z1, z2 in (0, ∞).
Barrab´es, Cors and Roberts (HamSys 2014) Twisted CC 17 / 27

21. Introduction CC of two twisted rings CC of three twisted rings
Results for n ≥ 4
Theorem
Suppose that the Conjecture is true. Then, for n ≥ 4, the set of admissible
values of a such that there exist a CC of two twisted n-gons is
(0, 1/z2) ∪ (z1, 1/z1) ∪ (z2, ∞)
where z1 < 1 < z2 are the only two positive zeros of m = H(a) = 0, and
z1z2 > 1
Notice that m ∈ (0, ∞) for a in any of the three admissible intervals
Barrab´es, Cors and Roberts (HamSys 2014) Twisted CC 18 / 27

22. Introduction CC of two twisted rings CC of three twisted rings
Results for n ≥ 4
Theorem
Suppose that the Conjecture is true. Then, for n ≥ 4, the set of admissible
values of a such that there exist a CC of two twisted n-gons is
(0, 1/z2) ∪ (z1, 1/z1) ∪ (z2, ∞)
where z1 < 1 < z2 are the only two positive zeros of m = H(a) = 0, and
z1z2 > 1
Notice that m ∈ (0, ∞) for a in any of the three admissible intervals
Theorem
For n > 4 any CC of two twisted n-gons with a ∈ (z1, 1/z1) is convex
Barrab´es, Cors and Roberts (HamSys 2014) Twisted CC 18 / 27

23. Introduction CC of two twisted rings CC of three twisted rings
Convex CC for n = 4
(0.69738051, 0.707106781) (0.707106781, 1.414213563) (1.414213563, 1.433937407)
convex
not all the configurations a ∈ (z1, 1/z1) are convex.
convex configurations correspond to m ∈ (0.062278912, 16.05679941)
there are not convex CC of twisted rings for certain values of m
Barrab´es, Cors and Roberts (HamSys 2014) Twisted CC 19 / 27

24. Introduction CC of two twisted rings CC of three twisted rings
Results for n ≥ 4
Theorem
Suppose that the Conjecture is true. Then, for two nested rings with n ≥ 4
bodies in each ring, and for any positive value of the mass ratio m there exists
at least three different central configurations.
Barrab´es, Cors and Roberts (HamSys 2014) Twisted CC 20 / 27

25. Introduction CC of two twisted rings CC of three twisted rings
Outline
Introduction
CC of two twisted rings
CC of three twisted rings
Barrab´es, Cors and Roberts (HamSys 2014) Twisted CC 21 / 27

26. Introduction CC of two twisted rings CC of three twisted rings
Three twisted rings
Fixed values for 2 and 3, the unknowns are m2, m3 and a2, a3.
(Sn − a3
2C12(a2)) m2 + a2
2(C23(a2, a3) − a2C13(a3)) m3 = a2
2(Sna2 − C21(a2)),
a2
3(C32(a2, a3) − a3C12(a2)) m2 + (Sn − a3
3C13(a3)) m3 = a2
3(Sna3 − C31(a3)).
Nested: 2 = 3 = 0
Twisted: one of the rings is rotated with respect the other two. It is
enough to restrict the study to
2 = π/n, 3 = 0, and 0 < a2, 0 < a3 < 1
Barrab´es, Cors and Roberts (HamSys 2014) Twisted CC 22 / 27

27. Introduction CC of two twisted rings CC of three twisted rings
Results for n = 2
Theorem
Consider three rings of two bodies in each ring, all of them with equal masses.
Then, if they are in a twisted central configuration, then a3 < a2 < 1, that is,
the twisted ring is in the middle of the other two.
Barrab´es, Cors and Roberts (HamSys 2014) Twisted CC 23 / 27

28. Introduction CC of two twisted rings CC of three twisted rings
Numerical results
Region of admissible values of (a2, a3) for n = 2
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0.4 0.6 0.8 1 1.2 1.4 1.6 1.8
a3
a2
m2<m3
m2>m3
Barrab´es, Cors and Roberts (HamSys 2014) Twisted CC 24 / 27

29. Introduction CC of two twisted rings CC of three twisted rings
Numerical results
Region of admissible values of (a2, a3) for n = 3
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8
a3
a2
m2<m3
m2>m3
Barrab´es, Cors and Roberts (HamSys 2014) Twisted CC 25 / 27

30. Introduction CC of two twisted rings CC of three twisted rings
Numerical results
Region of admissible values of (a2, a3) for n = 4
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
a3
a2
m2<m3
m2>m3
Barrab´es, Cors and Roberts (HamSys 2014) Twisted CC 25 / 27

31. Introduction CC of two twisted rings CC of three twisted rings
Numerical results for a2 = a3
Conjecture If a2 = a3 then m2 > m3
n = 2
0
2
4
6
8
10
0.54 0.55 0.56 0.57 0.58 0.59 0.6 0.61 0.62
a2=a3
m2
m3
n = 3
0
0.001
0.002
0.003
0.004
0.005
0.006
0.007
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14
a2=a3
m2
m3
0
2
4
6
8
10
0.58 0.59 0.6 0.61 0.62 0.63 0.64
a2=a3
m2
m3
Barrab´es, Cors and Roberts (HamSys 2014) Twisted CC 26 / 27

32. Introduction CC of two twisted rings CC of three twisted rings
Numerical results for a2 = a3
Conjecture If a2 = a3 then m2 > m3
n = 4
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35
a2=a3
m2
m3
0
2
4
6
8
10
0.61 0.6105 0.611 0.6115 0.612
a2=a3
m2
m3
Barrab´es, Cors and Roberts (HamSys 2014) Twisted CC 26 / 27

33. Introduction CC of two twisted rings CC of three twisted rings
Thank you for your attention
Barrab´es, Cors and Roberts (HamSys 2014) Twisted CC 27 / 27