Families o symme ic pe iodic o bi s in he h ee body p oblem
and he igu e eigh
F. J. Mu˜noz-Alma az†, J. Gal´an‡and E. F ei e‡
†Depa amen o de Ma em´a ica Aplicada y Compu aci´on.
Facul ad de Ciencias. Uni e sidad de Valladolid.
47005 Valladolid, Spain.
‡Depa amen o de Ma em´a ica Aplicada II.
Escuela Supe io de Ingenie os. Uni e sidad de Se illa.
41092 Se illa, Spain.
Monog a ´ıas de la Real Academia de Ciencias de Za agoza. 25: 229–240, (2004).
Abs ac
In his pape we show a echnique o he con inua ion o symme ic pe iodic o -
bi s in sys ems wi h ime- e e sal symme ies. The geome ic idea o his echnique
allows us o gene alize he “cylinde ” heo em o his kind o sys ems. We s a e
he main heo e ical esul wi hou p oo ( o be published elsewhe e). We ocus on
he applica ion o his scheme o he h ee body p oblem (TBP), aking as s a ing
poin he igu e eigh o bi [3] o ind amilies o symme ic pe iodic o bi s.
Key wo ds and exp essions: Hamil onian and conse a i e sys ems, pe iodic
solu ions, nume ical con inua ion, bounda y alue p oblems, h ee-body p oblem,
igu e-8 o bi .
MSC: 34C25, 34C30, 34C14, 37J15, 65L10.
1 In oduc ion
Ou in e es is abou inding symme ic pe iodic solu ions in he h ee body p oblem. In
a p e ious pape [7] we s udied he con inua ion o pe iodic o bi s in he TBP ollowing
he scheme gi en in [13] o con inua ion in conse a i e sys ems. Unlike o he me hods,
his scheme does no make use o symplec ic educ ion be o e nume ic calcula ion. Se e al
amilies o pe iodic o bi s we e shown in [7] and he nume ical s udy o hei linea s abili y
(cha ac e is ic mul iplie s). Ano he applica ions o his echnique appea ed in [6] whe e
229
a lo o amilies o pe iodic o bi s in he TBP and he es ic ed h ee body p oblem
we e ound. Some o hese amilies a e o med by symme ic pe iodic o bi s wi h espec
o ime- e e sal symme ies. Howe e , while he scheme gi en in [13] does no ake
ad an age o hese symme ies, classical me hods do [1, 4]. Ou scope in his pape is o
adap he scheme in [13], so ha amilies o symme ic pe iodic o bi s can be compu ed
in a mo e p ope way han in [7, 6] whe e ime- e e sal symme ies we e no used o he
compu a ions.
Chencine and Mon gome y [3] p o ed ecen ly he exis ence o a spec acula solu ion
in he TBP called he igu e eigh , since he h ee bodies ollow he same cu e (cho eog-
aphy) wi h his shape. The ini ial condi ions o he igu e eigh , which we e compu ed
nume ically by Sim´o, a e shown in [3] oo. This kind o solu ion had been p edic ed by
Moo e [11] in 1993. A lo o wo k has been de eloped a ound his new solu ion o he
las wo yea s. Sim´o also s udied he egion o s abili y nea he igu e eigh and ound
coun less examples o cho eog aphies [14, 15]. Fo o he esul s on con inua ion, see [9, 2].
The igu e eigh is symme ic wi h espec o se e al ime- e e sal symme ies; he e-
o e, we hink ha i is a good s a ing poin o look o symme ic pe iodic solu ions.
In his in oduc ion we gi e a sho e ision abou he main concep s ela ed o sym-
me ies in he TBP and we desc ibe wi h espec o which ime- e e sal symme ies he
igu e eigh is symme ic ( o mo e de ails, see [3, 9]).
In he second sec ion we show geome ically when a i s in eg al plays a ole in he
con inua ion o symme ic pe iodic o bi s. In he hi d sec ion we a e going o s a e he
main heo e ical esul in his pape abou pe sis ence (o con inua ion) o symme ic
pe iodic o bi s. As we ha e said in he abs ac , his esul is a gene aliza ion o he
“cylinde ” heo em. We adap he ideas om e e ence [13] whe e con inua ion o pe iodic
o bi s in conse a i e sys ems is based on a p ope un olding o hese sys ems. Fo
symme ic pe iodic o bi s con inua ion i is necessa y o un old sys ems wi h i s in eg als
which e i y a ce ain p ope y. In he las sec ion we apply hese ideas o he igu e eigh
in he TBP.
1.1 On symme ies in he TBP
Le nbe he dimension o he con igu a ion space, so i n= 2, he plana TBP is se and
i n= 3, i is he spa ial TBP. Fo each smoo h unc ion F:R6n7→ R he ec o ield
XF(x) = J∇F(x) is de ined, whe e Jis he ma ix o size 6n×6n
J=

03n−I3n
I3n03n

.(1)
230
The TBP ( o bodies o masses mjwi h j= 1,2,3) is a sys em ˙
x=XH(x) whose
Hamil onian His gi en by
H(x) =
3
X
j=1
1
2mj
kpjk2−X
1≤i<j≤3
mimj
kqi−qjk,(2)
wi h x= (q1,q2,q3,p1,p2,p3) whe e qjand pja e he posi ion ec o and momen um
ec o o he j- h body (wi h j= 1,2,3).
Besides he Hamil onian, he TBP has o he i s in eg als, as he componen s o he
o al linea momen um and he o al angula momen um. Mo eo e , e e y quad a ic i s
in eg al can be w i en as
Fa,A(x) = a∗

3
X
j=1
pj
+
3
X
j=1
q∗
jApj,(3)
whe e a∈Rnand Ais a skew symme ic ma ix. Mo eo e , in his pape , he anspose
o a ec o x∈Rnis deno ed by x∗. The ec o s o he canonical basis in R2a e deno ed
as e1and e2. In a simila way, o R3a e deno ed as e1,e2and e3.
The Noe he ’s symme y heo em [10] s a es ha he lux o ˙
x=XFa,A (x) is a sym-
plec ic con inuous g oup o symme ies, i.e. he sys em ˙
x=XH(x) emains in a ian
unde he changes o a iables induced om he lux o ˙
x=XFa,A (x).
We can easily check ha ΨQ,ba e symme ies o he TBP
ΨQ,b:R6n→Rn×Rn×Rn×Rn×Rn×Rn
x7→ (Qq1+b, Qq2+b, Qq3+b, Qp1, Qp2, Qp3)(4)
whe e Qis an o hogonal ma ix o size n×nand bis any ec o in Rn.
I wo bodies ha e equal masses, hen he e is ano he symme y which exchanges
posi ions and momen a o he bodies. Fo ins ance, wi h m2=m3, he TBP has he
symme y
C: (q1,q2,q3,p1,p2,p3)7→ (q1,q3,q2,p1,p3,p2).(5)
We ecall ha he composi ion o symme ies is a symme y.
A ma ix Ro size 6n×6nsuch ha R2=I6nis said o be a ime- e e sal symme y o
he TBP i XH(Rx) = −R XH(x). The ixed poin subspace o a ime- e e sal symme y
is Fix(R) = {x∈R6n:Rx=x}. In he TBP a ime- e e sal symme y is
RN=

I3n03n
03n−I3n

.(6)
Any composi ion o a symme y wi h RNis ano he ime- e e sal symme y. The e o e,
he TBP is ull o ime- e e sal symme ies. I is well known (see [8]) ha an o bi wi h
wo poin s in Fix(R) is a symme ic pe iodic o bi , i.e. he o bi is in a ian wi h espec
o R.
231
In he TBP he e exis s a escale Φλ(wi h λ∈R {0}) o he posi ions and he
conjuga e momen a which maps a solu ion in o ano he solu ion wi h a new ime scale.
This unc ion is
Φλ:R6n→R6n
(q1,q2,q3,p1,p2,p3)7→ (λ−2q1, λ−2q2, λ−2q3, λp1, λp2, λp3),(7)
and i s ime– escale is λ−3.
1.2 The igu e eigh as a symme ic pe iodic o bi
Chencine and Mon gome y [3] looked o a plana cu e which is a minimum o he ac ion
o he TBP wi h he bodies o uni mass,
ZT
0(1
2
3
X
j=1
kpj( )k2+X
1≤i<j≤3
1
kqi( )−qj( )k)d , (8)
o e he subspace o cu es which e i y ha :
•Ini ial condi ion: The i s body is in he middle o he o he wo bodies.
•Final condi ion: The second body (which was in an ex emum in he ini ial condi-
ion) is now equidis an o he i s and he hi d bodies.
They p o ed ha he minimum is eached in a cu e which poin s gene a e a solu ion
o he TBP whe e he h ee bodies ollow he same cu e (cho eog aphy) wi h a shape o
an eigh , so his o bi was called he igu e eigh by hem. In Figu e 1, we see he cu e
ollowed by he bodies and he ini ial and inal condi ions o he minimiza ion p oblem.
−1.5 −1 −0.5 0 0.5 1 1.5
−0.4
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
0.4
m1
m2
m3
−1.5 −1 −0.5 0 0.5 1 1.5
−0.4
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
0.4
m1
m2
m3
Ini ial condi ion Final condi ion
Figu e 1: Figu e eigh wi h he ini ial and inal condi ions o he minimiza ion p oblem.
The igu e eigh is symme ic wi h espec o six ime- e e sal symme ies in he plana
case and nine ime- e e sal symme ies in he spa ial case. These ime- e e sal symme ies
a e gi en by he composi ion o RNwi h an exchange o he bodies ( o example, Cis one
232
o he h ee possible exchanges) and wi h a symme y ΨS,0. The ma ix Scan be chosen
om he ollowing lis o he plana case:
SM1=

1 0
0−1
, SE1=

−1 0
0−1
(9)
and o he spa ial case:
SM13D=




1 0 0
0−1 0
0 0 −1





, SΠ13D=




1 0 0
0 1 0
0 0 −1





and SE13D=




−1 0 0
0−1 0
0 0 −1





.(10)
In o de o simpli y (and ollowing [3]), he ime- e e sal symme y RN◦C◦ΨSM1,0
is called M1and RN◦C◦ΨSE1,0is called E1 h oughou his pape . The ixed poin s o
M1ha e he i s body in he axis e1and his body o ms an isosceles iangle wi h he
o he wo bodies whose symme y axis is e1; he conjuga e momen um o he posi ion o
he i s body, p1, is in he di ec ion e2and he conjuga e momen a o he h ee bodies
(p1,p2and p3) o m an isosceles iangle wi h symme y axis in e2. See Figu e 2 o a
g aphical in e p e a ion.
e2
e1
p2
p2
p3
p3p1
p1
Figu e 2: Fixed poin s o M1.
The ixed poin s o E1a e gi en by he i s body in he o igin and in he middle
poin o he o he wo bodies; he conjuga e momen a o his wo bodies a e iden ical,
i.e., q1=0,q2=−q3and p2=p3.
2 Geome ic In e p e a ion o he Con inua ion wi h Time-Re e sal Sym-
me ies
The e exis s a “cylinde ” heo em o sys ems wi h a ime- e e sal symme y (see [8]). A
simple geome ic easoning s a es ha pe iodic o bi s gene ically a ise as a one-pa ame e
amily (we a e assuming ha he dimension o Fix(R) is hal he whole dimension).
233

Howe e , he TBP is a non-gene ic sys em, since mos o he o bi s do no belong o one-
pa ame e amilies. We ha e s udied why some i s in eg als a oid a gene ic beha io o
he TBP.
We ocus on he ime- e e sal symme y M1 o ou explana ion. An analogous si u-
a ion appea s o E1. A ansla ion o a ixed poin o M1in he di ec ion e1p oduces
ano he ixed poin gene a ing a symme ic pe iodic o bi and he unc ion Φλmaps ixed
poin s o M1in o ixed poin s o M1 oo. This ema k sugges s ha , unde gene ic con-
di ions, ixed poin s o M1gene a ing symme ic pe iodic o bi s o m a wo-dimensional
submani old. By heo e ical esul s and nume ical compu a ion, we a e able o s a e ha
he igu e eigh belongs o a 2-pa ame e amily o symme ic pe iodic o bi s wi h espec
o M1.
3 Resul on he Con inua ion o Symme ic O bi s
The p e ious geome ic idea s ill holds in a mo e gene al si ua ion: a sys em wi h a
ime- e e sal symme y Rand a i s in eg al Fsuch ha he lux o he ec o ield XF
maps ixed poin s o Rin o ixed poin s o Rhas he same non-gene ic beha io . I R
is an i-symplec ic (i.e RTJR =−J), his condi ion is equi alen o he de i a i e o he
unc ion F es ic ed o Fix(R) is ze o. The e o e, a i s in eg al cons an on Fix(R)
implies ha he symme ic pe iodic o bi does no belong o a one–pa ame e amily any
mo e.
The symplec ic amewo k is no necessa y and i is possible o o mula e a gene al
esul o any kind o sys em wi h a ime- e e sal symme y. So, we assume in his sec ion
ha X:R2m→R2mis a ec o ield wi h a ime- e e sal symme y Rand he dimension
o Fix(R) is m. These assump ions a e no manda o y, bu hey simpli y he heo em
and hey hold in mos physical applica ions. We de ine Rx0as he se o de i a i es o
i s in eg als which a e cons an on Fix(R), i.e.
Rx0=


DF (x0) : Fis a i s in eg al o ˙
x=X(x) and
Fis cons an on Fix(R)


(11)
and le ϕ (x) be he lux o he sys em ˙
x=X(x). We deno e he subspace spanned by
he ec o x∈R2mwi h Rx. We s a e he ollowing heo em o he con inua ion o
symme ic pe iodic o bi s
Theo em 3.1 Le be x0∈Fix(R)and T > 0. I ϕT(x0)∈Fix(R)and
Im((I−R)DϕT(x0)(I+R)) + RX(ϕT(x0)) = R⊥
ϕT(x0)∩Im(I−R),(12)
hen he symme ic pe iodic o bi gene a ed by x0belongs o a (dim(Rx0) + 1)-pa ame e
amily o symme ic pe iodic o bi s wi h espec o R.
234
A p oo o his esul can be ound in [12] and will be published elsewhe e. The
idea o his p oo ollows he one gi en in [13] whe e “a i icial” pa ame e s (o un olding
pa ame e s) a e added in he ec o ield. I Fj(wi h 1 ≤j≤k) a e i s in eg als such
ha {DFj(x0)}1≤j≤kis a basis o Rx0, hen we conside he lux ˜ϕ o he sys em
˙
x=X(x) +
k
X
j=1
βj∇Fj(x) (13)
whe e βja e he “a i icial” pa ame e s and we apply he implici unc ion heo em o
he unc ion
(x, , β1, . . . , βk)∈Fix(R)×R×Rk7→ (I−R) ˜ϕ (x, β1,...,βk),(14)
ge ing a submani old o ze os o his unc ion and, inally, checking ha he a i icial
pa ame e s a e ze o along ha mani old.
4 Nume ical Resul s abou he Figu e Eigh
The scheme o con inua ion has been applied o he igu e eigh as he mass o one o
he bodies is allowed o a y. When he masses a e iden ical (m1=m2=m3= 1),
a consequence o he o me heo em is ha he igu e eigh o bi belongs o a wo-
pa ame e amily gi en by he ansla ion in he di ec ion e1and by mapping Φλ. So,
o hese alues o he masses he e a e no mo e symme ic pe iodic o bi s wi h espec
o M1“nea ” (wi h he pe iod close o he o iginal one) he igu e eigh . I he mass m1
is a ied, he sys em s ill has he M1 ime- e e sal symme y; he e o e we will y o
con inue aking m1as a con inua ion pa ame e , looking o symme ic pe iodic o bi s.
The only quad a ic i s in eg al cons an on Fix(R) is he i s componen o he o al
linea momen um F1(x) = e∗
1³P3
j=1 pj´whe e x= (q1,q2,q3,p1,p2,p3). The e o e, we
se he ollowing bounda y alue p oblem





























Find x∈ C1([0,1]; R12) such ha
˙
x=T(J∇H(x) + β1∇F1(x)),
e∗
2q1(0) = 0 e∗
2q1(1) = 0,
q3(0) = SM1q2(0) q3(1) = SM1q2(1),
e∗
1p1(0) = 0 e∗
1p1(1) = 0,
p3(0) = −SM1p2(0) p3(1) = −SM1p2(1),
(15)
whe e he hal -pe iod Tis ixed, SM1is he ma ix de ined in (9) and β1is he “a i icial”
pa ame e needed o he con inua ion.
The ini ial condi ions o his p oblem o m a wo-dimensional submani old whe e
e e y ansla ion in he di ec ion e1o i s poin s is also included in he submani old. This
235
si ua ion is a oided wi h an addi ional in eg al condi ion
ZT
0
DF1(x0(s))J(x(s)−x0(s)) ds = 0,(16)
whe e x0( ) is he p e ious compu ed solu ion. The sys em (15) oge he wi h his in eg al
condi ion can be nume ically con inued wi h a package as AUTO [5]. We ge a cu e o
ini ial condi ions on Fix(M1) which gene a es symme ic pe iodic o bi s. Downwa ds
solu ions in he amily a e shown in Figu e 3 and upwa ds in Figu e 4, o his las case
he e exis s a old in he amily o m1= 1.00004.
−1.5 −1 −0.5 0 0.5 1 1.5
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
−1 −0.5 0 0.5 1
−1.5
−1
−0.5
0
0.5
1
1.5
−1 −0.5 0 0.5 1
−1.5
−1
−0.5
0
0.5
1
1.5
m1= 0.9m1= 0.7m1= 0.5
Figu e 3: We show he amily downwa ds o which he igu e eigh belongs whe e he
ini ial condi ions in he ixed poin subspace o M1a e ma ked.
−1.5 −1 −0.5 0 0.5 1 1.5
−0.4
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
0.4
−1.5 −1 −0.5 0 0.5 1 1.5
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
−1.5 −1 −0.5 0 0.5 1 1.5
−1.5
−1
−0.5
0
0.5
1
1.5
m1= 1.00 m1= 0.97 m1= 0.92
Figu e 4: Family upwa ds o which he igu e eigh belongs.
We do no de ec any poin whe e he condi ion (12) is no ul illed along his amily,
bu i Tis aken as i e imes he hal pe iod and as s a ing solu ion i e hal windings
o he igu e eigh , hen subha monic bi u ca ions a e de ec ed when he condi ion (12)
is no ul illed. Using AUTO we ge a b anch o pe iodic o bi s which s a om a 5-
subha monic bi u ca ion loca ed a m1= 0.9618044. In Figu e 5 some o hese solu ions
a e shown.
Using he ime– e e sal symme y E1 he i s amily can be also con inued as sym-
me ic pe iodic o bi s wi h espec o E1, bu in his case he i s in eg al cons an on
236
−1.5 −1 −0.5 0 0.5 1 1.5
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
−1.5 −1 −0.5 0 0.5 1 1.5
−1
−0.5
0
0.5
1
−1.5 −1 −0.5 0 0.5 1 1.5
−1
−0.5
0
0.5
1
m1= 0.96 m1= 0.95 m1= 0.94
Figu e 5: We plo some o bi s go om a amily which s a s om a 5-subha monic
bi u ca ion.
Fix(E1) is he angula momen um
G1(x) =
3
X
i=1
q∗
i

0 1
−1 0 
pi.
The o me heo y d i es o o mula e he ollowing bounda y alue p oblem





















Find x∈ C1([0,1]; R12) such ha
˙
x=T(J∇H(x) + β1∇G1(x)),
q1(0) = 0 q1(1) = 0,
q2(0) = −q3(0) q2(1) = −q3(1),
p2(0) = p3(0) p2(1) = p3(1),
(17)
and o add he in eg al condi ion RT
0DG1(x0(s))J(x(s)−x0(s)) ds = 0. He e β1is, again,
an “a i icial” pa ame e o allow he con inua ion. Along he amily which was plo ed
in Figu e 3 and 4 an o bi whe e condi ion (12) is no ul illed is de ec ed o he alue
m1= 0.699779 and so a new amily o symme ic pe iodic o bi s appea s om he solu ion
wi h his alue o m1. Some o bi s in his amily a e shown in Figu e 6.
−1.5 −1 −0.5 0 0.5 1 1.5
−1
−0.5
0
0.5
1
−1.5 −1 −0.5 0 0.5 1 1.5
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
−1.5 −1 −0.5 0 0.5 1 1.5
−1
−0.5
0
0.5
1
m1= 0.65 m1= 0.55 m1= 0.45
Figu e 6: We show symme ic pe iodic o bi s wi h espec o E1(ini ial condi ions in
he ixed subspace a e ma ked) which belong o a amily appea ing om a poin whe e
condi ion (12) is no ul illed o E1.
Finally, we wan o ema k ha he same scheme is also use ul in he spa ial case. Fo
he ime- e e sal symme y gi en by he composi ion RN◦C◦SM13D he e a e wo i s
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