Symmetry-breaking for a restricted n-body problem in the Maxwell-ring configuration

31.8 MB·

by  Renato Calleja

EPJ manuscript No.
(will be inserted by the editor)
Symmetry-breaking for a restricted nnn-body
problem in the Maxwell-ring configuration
Renato Calleja1,a, Eusebius Doedel2,b, and Carlos Garc´ıa-Azpeitia3,c
1 Matema´ticas y Meca´nica, IIMAS, Universidad Nacional Auto´noma de M´exico, Admon.
No. 20, Delegacio´n Alvaro Obrego´n, 01000 M´exico D.F.
6 2 Department of Computer Science, Concordia University, 1455 boulevard de Maisonneuve
1 O., Montr´eal, Qu´ebec H3G 1M8, Canada
0 3 DepartamentodeMatema´ticas,FacultaddeCiencias,UniversidadNacionalAuto´nomade
2
M´exico, 04510 M´exico DF, Mexico

n
a
J
Abstract. We investigate the motion of a massless body interacting
2
with the Maxwell relative equilibrium, which consists of n bodies of
2
equal mass at the vertices of a regular polygon that rotates around

a central mass. The massless body has three equilibrium Z -orbits
] n
S fromwhichfamiliesofLyapunovorbitsemerge.Numericalcontinuation
D of these families using a boundary value formulation is used to con-
struct the bifurcation diagram for the case n=7, also including some
.
h secondary and tertiary bifurcating families. We observe symmetry-
at breakingbifurcationsinthissystem,aswellascertainperiod-doubling
m bifurcations.
[

Introduction
1
v
0
Inhis1859essay[9]MaxwellproposedamodeltostudytheringsofSaturn.Hismodel
6
consists of n bodies of equal mass at the vertices of a regular polygon that rotates
1
around a massive body at the center. Maxwell used Fourier analysis and dispersion
6
relations in the determination of the stability of the ring. The Maxwell equilibrium
0
. has been studied in several papers since then. In particular, Moeckel proved in [10]
1
that the equilibrium is stable if n 7 and the body at the center massive enough.
0 ≥
See also [4,11,13] and references therein.
6
1 Inthispaperweconsiderthemotionofasatelliteunderthegravitationaleffectof
: theMaxwellequilibrium.Severalpapershavebeendevotedtostudythestabilityand
v
bifurcation of periodic solutions for the restricted N-body problem in the Maxwell
i
X configuration.Forexample,astudyoftheexistenceandlinearstabilityofequilibrium
positions can be found in [1], an analysis of the bifurcation of planar and vertical
r
a families of periodic solutions in [3], and a numerical exploration in [6].
Given that a change of stability occurs at µ 584 when n = 7, we present
1
≈
a numerical exploration of the motion of the satellite for this stable system, taking
a e-mail: [email protected]
b e-mail: [email protected]
c e-mail: [email protected]
2 Will be inserted by the editor
µ=103.WefollowtheplanarandverticalLyapunovfamiliesthatwereprovedtoex-
ist in [3], and we compute new, secondary families that bifurcate from the Lyapunov
families. These numerical results allow us to construct a bifurcation diagram in Fig-
ure 2 that shows primary, secondary, and tertiary bifurcating families. The results
also allow us to present the isotropy lattice in Figure 3 which shows the symmetries
of these families.
The numerical computations in this paper are done using continuation methods
andboundaryvaluetechniquesfordeterminingtheperiodicorbitsthatemanatefrom
the equilibrium orbits. Python scripts that make the AUTO software perform the
calculations reported in this paper will be made freely available. Similar techniques
have been applied to the restricted 3-body problem; see for example [14], where a
detailed bifurcation diagram with various families of periodic orbits can be found.
This paper is organized as follows. In Section 1 we recall some key results from
the literature concerning the equilibria and the Lyapunov orbits of the problem. In
Section 2 we present a bifurcation diagram and an isotropy lattice for the restricted
N-body problem in the Maxwell configuration, with n=7 and µ=103. In Section 3
we describe the isotropy groups of the Lyapunov families. In Section 4 we address
secondary bifurcations, and in Section 5 describe some tertiary families. We also
present evidence of the existence of invariant tori foliated by periodic orbits. Finally,
inSection6,weconsiderthebreakingofsymmetriesofplanarinterplanetaryperiodic
orbits.
1 The restricted N-body problem
The Maxwell relative equilibrium consists of a body of mass µ at aaa = 0 R3, and
0
n bodies of mass 1 located at aaa = (eijζ,0) R3, j = 1, ,n, where ζ∈= 2π/n.
j
∈ ···
ThesepositionsandmassescorrespondtoarelativeequilibriumsolutionofNewton’s
equations, qqq (t) = eJtaaa , when the masses are renormalized by µ+s [1,3], where
j j
=diag(J,0),
J
(cid:18)0 1(cid:19) 1n(cid:88)−1 1
J = − , s= .
1 0 4 sin(jπ/n)
j=1
The equation of a satellite in rotating coordinates,qqq(t)=eJtuuu, is
uuu¨+2 uuu˙ = V, (1)
J ∇
whereuuu=(x,y,z) R3 and
∈
1 µ 1 (cid:88)n 1 1
V(uuu)= (x,y) 2+ + . (2)
2(cid:107) (cid:107) s+µ uuu s+µ uuu (eijζ,0)
(cid:107) (cid:107) j=1 (cid:107) − (cid:107)
The first term of the potential V(uuu) in Equation (2) corresponds to the centrifugal
force, the second term is the interaction with the mass µ, and the third term models
the interaction with the n primaries of mass 1; see [1] and [3]. The equilibrium posi-
tionsofthesatellitearecriticalpointsofV,definedinEquation(2).Moreover,dueto
the particular form of the Maxwell configuration, the potential V is Z -invariant. In
n
this respect the existence of three Z equilibrium orbits for any value of µ is proved
n
in [1,3]. For small µ, two additional equilibrium orbits appear close to the origin [3];
see Figure 1.
In [3], taking advantage of the symmetries of the equations, the authors analyze
the bifurcation of periodic solutions from the Z -orbits of equilibria. Here we state
n
these results for the equilibria , and of the three Z -orbits:
1 2 3 n
L L L
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Fig. 1. TheMaxwellconfigurationwiththemassescoloredblueandtheequilibriumorbits
colored red. Left: the 35 equilibria when µ=3. Right: the 21 equilibria when µ=1000.
Theorem 1 [Ize & Garc´ıa-Azpeitia [3].] The libration point , has one global bifur-
1
L
cation family of planar periodic solutions that will be denoted by L , and one global
1
family of vertical solutions, denoted by V . Similarly, the libration point has one
1 2
L
global family of planar periodic solutions, denoted L , and one family of vertical so-
2
lutions, V . For µ > µ , the equilibrium has two global bifurcations of planar
2 1 3
L
solutions, one of which has longer period, denoted L , and another planar family of
3l
shorter period, L . There is also a bifurcating family of vertical periodic solutions
3s
that will be denoted by V . Moreover, as a consequence of the symmetries, the shape
3
of all vertical solutions close to the equilibrium resembles a spatial figure eight.
The global property guarantees that the family is a continuum that either goes to
infinity in Sobolev norm or period, ends in a collision, or ends at a bifurcation point.
Indeed, each one of these possibilities appears in the numerical continuation of the
families, as illustrated in the bifurcation diagram in Figure 2.
2 Breaking of symmetries
In this section we discuss the breaking of symmetries of Equation (1) for the case
n= 7 and µ=103, as observed in the numerically computed Lyapunov families that
emergefromthelibrationpointsandthesecondaryfamiliesthatbifurcatefromthem.
The 2π/ν-periodic solutions of Equation (1) are zeros of the map
fff(uuu;ν)=ν2uuu¨+ νuuu˙ V(uuu),
J −∇
defined in a set of 2π-periodic collisionless functionsuuu; see [3]. Since the potential is
Z -invariant and the equations are autonomous, the map fff is equivariant under the
7
action of (ζ,ϕ) Z S1 given by
7
∈ ×
ρ(ζ,ϕ)uuu(t)=eJζuuu(t+ϕ),
whereζ =2π/7.Inadditiontheequationsaresymmetricwithrespecttoreflectionof
y about the xz plane, while reversing time, and with respect to reflection of z about
the xy plane. In this regard we define the reflections κ and κ by
y z
ρ(κ )uuu(t)=R uuu( t) and ρ(κ )uuu(t)=R uuu(t),
y y z z
−
whereR =diag(1, 1,1)andR =diag(1,1, 1).Thereforethemapfff isequivariant
y z
− −
under the full symmetry group
G=(cid:0)Z S1 κ (Z S1)(cid:1) Z (κ ). (3)
7 y 7 2 z
× ∪ × ×
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We will use the property that the group orbit of a functionuuu,
G(uuu)= ρ(γ)uuu:γ G ,
{ ∈ }
is isomorphic to G/G , where G is the isotropy group defined as
uuu uuu
G = uuu:ρ(γ)uuu=uuu, γ G .
uuu
{ ∀ ∈ }
In this sense the libration equilibria , for j =1,2,3, have group orbits G( ) Z
j j 7
L L (cid:39)
and isotropy group
G =(S1 κ S1) Z (κ ).
Lj ∪ y × 2 z
We therefore present the breaking of symmetries only for the libration points ,
j
L
j =1,2,3.
V
3
B
3
A
C 3.2
3
L3S P3
L3L A3
L3
V
2
B
A 2
3.1
A
2
V
L 1
H2L2 2 A1 B1
C
2
L
L1H1 1
C
1
Fig. 2. Bifurcationdiagramforµ=103:Thegreencubesdenotetheequilibriumpositions
L , L , and L , the white spheres represent bifurcation points, the blue cubes represent
1 2 3
families that apparently end in collisions, and tetrahedra indicate that the family goes to
infinity in period or in Sobolev norm.
The bifurcation diagram for the case µ = 103 is given in Figure 2. The blue
lines represent the vertical Lyapunov families V and the planar Lyapunov families
j
L , j = 1,2,3. Planar families are positioned in the plane of the bodies, as are the
j
black lines that represent planar interplanetary orbits. The red lines are the result
of a secondary symmetry-breaking, with two solutions per equilibrium, as for the
Halo orbits H and the Axial orbits A , j = 1,2,3. The green lines correspond to a
j j
tertiary symmetry-breaking, and as such they have a trivial isotropy group and four
symmetry-related branches per libration point.
Remark 1 :Someofthefamiliesinthebifurcationdiagramthatendatatetrahedron,
infactterminateasaheteroclinicorbit.Fortherestrictedthree-bodyproblemsimilar
heteroclinic orbits are given in [2,8], and the existence of heteroclinic connections is
provedin[7].Infutureworkwewillpresentmanyotherheteroclinicconnectionsthat
we have located by continuation of orbits in stable/unstable manifolds. Furthermore,
in future work we will present evidence of families of planar orbits that interconnect
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the planar families L and L , as in [5]. All results are accompanied by scripts that
3s 3l
allow their reproduction.
The breaking of symmetries in the bifurcation diagram gives rise to the isotropy
lattice in Figure 3.
Libration equilibria
O(2)×Z (κ )
2 z
↓
Planar asymmetric Interplanetary
Z (κ ) Z (π)×Z (κ )×Z (κ )
2 z 2 2 y 2 z
↑ (cid:46) ↓ (cid:38)
Planar Lyapunov Vertical Lyapunov Interplanetary
Z (κ )×Z (κ ) Z (κ )×Z (π,κ ) Z (πκ )×Z (κ )
2 y 2 z 2 y 2 z 2 y 2 z
↓ (cid:46) ↓ (cid:46)
Halo Axial
Z (κ ) Z (πκ ,κ )
2 y 2 y z
(cid:38) ↓
{e}
Fig. 3. Isotropy lattice
3 Lyapunov families
The first bifurcation occurs when the S1-symmetry is broken, giving rise to the Lya-
punov families (the blue lines in Figure 2) from the equilibria , j = 1,2,3. The
j
L
numberofsymmetry-relatedLyapunovbranchesisequaltotheorderofG/G ,which
uuu
is 7, since each isotropy group is isomorphic to Z Z .
2 2
×
The isotropy group of the planar Lyapunov orbits that emerge from the libration
equilibria is
G =Z (κ ) Z (κ )<G . (4)
Lj 2 y × 2 z Lj
Theseperiodicsolutionshavethepropertythatx(t)iseven,y(t)isodd,andz(t)=0.
Inparticular,weobservethattheplanarorbitsareinvariantunderthetransformation
that takes y to y.
−
For the vertical Lyapunov orbits the isotropy group is
G =Z (κ ) Z (π,κ )<G . (5)
Vj 2 y × 2 z Lj
Here a solution is fixed by G if it satisfies
Vj
uuu(t)=ρ(κ,π)uuu(t)=R uuu(t+π),
z
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Fig. 4. Top Left: The planar Lyapunov family L , which ends in a collision orbit. Top
1
Right:TheverticalfamilyV untilitssecondbifurcationorbit.CenterLeft:The“Southern”
1
branch of the Halo family H , which ends in a collision orbit. Center Right: One branch of
1
the Axial family A , which forms a “bridge” between L and V . Bottom Left: One branch
1 1 1
of the third family that bifurcates from L , which ends in a collision orbit. Bottom Right:
1
One branch of the fourth family that bifurcates from L , which approaches an orbit that is
1
homoclinic to a periodic orbit.
which is equivalent to assuming that x(t) is a π-periodic even function, y(t) is a π-
periodic odd function, and z(t) = z(t+π). Therefore these solutions follow the
−
planar π-periodic curve (x,y) twice; one time with the spatial coordinate z and a
second time with z. This fact was proved in [3].
−
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Fig. 5. Top Left: Part of the torus generated by the family A ; the torus closes up after
3.1
making two full rounds. Top Right: Bifurcation orbits along the torus generated by A .
3.1
Bottom Left:Partof the torusgenerated by the familyA ; Thistorus also closesup after
3.2
makingtwofullrounds.BottomRight:BifurcationorbitsalongthetorusgeneratedbyA .
3.2
4 Secondary families
The bifurcations from the Lyapunov families coincide with the second breaking of
symmetries. The families that emanate from such bifurcation points (the red lines in
Figure2)havethreekindsofisotropygroups,eachoneisomorphictoZ .Thenumber
2
ofsymmetry-relatedbranchesisequalto2 7.Therefore,therearetwosuchbranches
×
per equilibrium.
There is a symmetry-breaking from the planar families to solutions with isotropy
group
Z (κ ). (6)
2 z
These solutions have vertical component z = 0. In this case the κ -symmetry is
y
broken, so these planar orbits are asymmetric with respect to the transformation
that takes y to y. Such solutions are observed for the third bifurcating family along
−
L and the fourth family that bifurcates from L ; see Figure 4.
2 1
The Halo families H and H bifurcate from the Lyapunov families L and L ,
1 2 1 2
respectively. Each of these has isotropy group
Z (κ ), (7)
2 y
i.e., their solutions have the property that x(t) is even and y(t) is odd. Thus these
spatial orbits are invariant under the transformation that takes y to y.
−
TheAxialfamiliesA ,andsimilarlyfamiliesB ,bifurcatefromV ,forj =1,2,3.
j j j
Here the symmetry-breaking is from the group G to the group
Vj
Z (πκ ,κ ). (8)
2 y z
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This means that the Axial solutions satisfy uuu(t) = R R ( t+π) or, setting uuu˜(t) =
z y
−
uuu(t+π/2), that
uuu˜(t)=R R uuu˜( t).
z y
−
Then x(t) is even, and y(t) and z(t) are odd. Therefore these spatial orbits are in-
variant under the transformation that takes (y,z) to ( y, z).
− −
5 Tertiary families
Tertiarysymmetry-brokenfamilies(thegreenlinesinFigure2)correspondtofamilies
thatbifurcatefromsolutionswithisotropygroups(6),(7),and(8).Sincethesegroups
areisomorphictoZ ,thetertiarysymmetry-brokensolutionshavethetrivialisotropy
2
group. Therefore, the number of symmetry-related branches of the tertiary families
is 4 7, i.e. four per equilibrium. In particular, the symmetry-breaking bifurcations
×
along the families A and B give rise to families that generate invariant surfaces.
3 3
Fig. 6. Top Left: The Vertical family V , from the libration point L until its second
3 3
bifurcation orbit, with the bifurcation orbits colored blue. Top Right: The Axial family A ,
3
from the orbit it shares with V and until the orbit it shares with the planar family P .
3 3
Bottom Left: The family B , from the orbit it shares with V and until the orbit it shares
3 3
with the planar family P . Bottom Right: One of several families that bifurcate from B .
3 3
Theorbitsofthisfamilygenerateatorusthatclosesupaftermakingtwofullroundsaround
the z-axis.
Indeed, the surface generated by the A family in Figure 5 reconnects to A
3.1 3
after a complete loop around the central body, but to an orbit that is symmetric
to the original one. Following this surface, i.e., its orbits, for a second loop around
the central body, we obtain a double surface that interconnects the 7 pairs of Axial
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bifurcation orbits that emanate from each of the 7 libration points symmetry-related
to . The surface generated by the orbits of the family A contains an additional
3 3.1
L
2 7 bifurcation orbits that connect to Halo families H . Consequently there is a
2
×
continuous path in the bifurcation diagram between any of the 7 symmetry-related
libration points and .
2 3
L L
The surface generated by the orbits of the family A in Figure 5 is similar to
3.2
that of A in that it interconnects 2 7 bifurcation orbits along the Axial families
3.1
×
A . These bifurcation orbits are distinct from the bifurcation orbits along A that
3 3
are interconnected via the family A . The family A has 2 7 extra bifurcation
3.1 3.2
×
orbits that connect to Halo-like families that we do not describe here.
Several families bifurcate from the family B with trivial isotropy group, one of
3
which is illustrated in the bottom right panel of Figure 6. This family is similar to
A and A in that it connects one Axial family, with isotropy group Z (πκ ,κ ),
3.1 3.2 2 y z
to one Halo-like family, with isotropy group Z (κ ). In future work we will report
2 z
other families bifurcating from B that connect two Axial-like families with isotropy
3
group Z (πκ ,κ ).
2 y z
Fig.7. Left:Threeorbitsfromtheplanar,interplanetaryfamilyC fromwhichthevertical
2
family V bifurcates via a period-doubling. Right: Three orbits from the planar, interplane-
2
tary family P that bifurcates from the Axial family A .
3 3
6 Interplanetary orbits
Several families connect to planar families that encompass more than one body and
that do not correspond to a planar Lyapunov family. Such families are referred to
as interplanetary in [6], and indicated by black lines in the bifurcation diagram of
Figure 2.
The family V has the symmetry group (5) and connects to a planar family C
2 2
in Figure 7 via a reverse period-doubling bifurcation, i.e., the vertical family V
2
arisesfromtheplanarfamilyviaaperiod-doublingbifurcation.Indeed,thefamilyV
2
bifurcates from the interplanetary family C with isotropy group
2
Z (π) Z (κ ) Z (κ ). (9)
2 2 y 2 z
× ×
These planar solutions are π-periodic and their orbits are invariant under the trans-
formation that takes y to y. This is consistent with the symmetry-breaking phe-
−
nomenon, since theisotropygroup (5) is contained in the isotropy group ofthe inter-
planetary orbits (9).
10 Will be inserted by the editor
The Lyapunov families V and V also connect to planar interplanetary families,
1 3
C andC ,respectively,viaaperiod-doublingbifurcation.Actually,C andC corre-
1 3 1 3
spondtothesamefamily,i.e.,C =C ,andmoreover,bothV andV bifurcatefrom
1 3 1 3
exactly the same orbit along C = C , via a degenerate period-doubling bifurcation
1 3
with two Floquet multipliers at 1.
−
AnotherfamilythatendsinaninterplanetaryfamilyisA ;seeFigure7.Herethe
3
isotropy group of the planar family P is
3
Z (πκ ) Z (κ ).
2 y 2 z
×
These solutions have the property that x(t) is even, y(t) is odd, and z = 0, so that
the orbits are planar and symmetric with respect to the transformation that takes
y to y. Therefore the Axial family A , with group Z (πκ ,κ ), corresponds to a
3 2 y z
−
symmetry-breaking from the interplanetary family P .
3
Acknowledgements. We thank Ramiro Chavez Tovar for his assistance in pro-
ducing the bifurcation diagram.
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