a Xi :ma h-ph/0602038 1 14 Feb 2006
k-cosymplec ic o malism in classical ield heo y: he Skinne –Rusk app oach
Angel M. Rey
Depa amen o de Xeome ´ıa e Topolox´ıa, Facul ade de Ma em´a icas,
Uni e sidade de San iago de Compos ela, 15706-San iago de Compos ela, Spain
e-mail: [email p o ec ed]n a.es
Na ciso Rom´an-Roy
Depa amen o de Ma em´a ica Aplicada IV, Edi icio C-3, Campus No e UPC,
C/ Jo di Gi ona 1, E-08034 Ba celona, Spain
e-mail: n @ma4.upc.edu
Modes o Salgado
Depa amen o de Xeome ´ıa e Topolox´ıa, Facul ade de Ma em´a icas,
Uni e sidade de San iago de Compos ela, 15706-San iago de Compos ela, Spain
e-mail: modes o@zma .usc.es
Abs ac
The k-cosymplec ic Lag angian and Hamil onian o malisms o i s -o de ield heo ies a e
e iewed and comple ed. In pa icula hey a e s a ed o singula almos - egula sys ems. A e
ha , bo h o malisms a e uni ied by gi ing an ex ension o he Skinne -Rusk o mula ion on
classical mechanics o i s -o de ield heo ies.
M.S. Classi ica ion (2000): 70S05, 53D05, 53Z05
Key wo ds:k-cosymplec ic o ms, classical ield heo y, Lag angian o malism, Hamil onian o -
malism.
1
1 In oduc ion
The k-symplec ic o malism [9, 18] is he gene aliza ion o ield heo ies o he s anda d symplec ic
o malism in mechanics, which is he geome ic amewo k o desc ibing au onomous dynamical
sys ems. In his sense, he k-symplec ic o malism is used o gi e a geome ic desc ip ion o ce ain
kindS o ield heo ies: in a local desc ip ion, hose whose Lag angian does no depend on he
coo dina es in he basis (in many o hem, he space- ime coo dina es); ha is, i is only alid o
Lag angianS L(qi, i
A) and Hamil onianS H(qi, pA
i) ha depend on he ield coo dina es qiand on
he pa ial de i a i es o he ield i
A. Le us poin ou ha k-symplec ic o malism has as i s base
he k-symplec ic mani olds in oduced by Awane [1, 2, 3].
The k-cosymplec ic o malism is he gene aliza ion o ield heo ies o he s anda d cosymplec ic
o malism in mechanics, which is he geome ic amewo k o desc ibing non au onomous dynam-
ical sys ems [14, 15]. This o malism desc ibes ield heo ies in ol ing he coo dina es in he basis
( 1,..., k) on he Lag angian L( A, qi, i
A) and on he Hamil onian H( A, qi, pA
i).
The k-cosymplec ic o malism has as i s base he k-cosymplec ic mani olds in oduced in [14,
15]. One o he ad an ages o his o malism, and o he G¨un he o malism (k-symplec ic o
polysymplec ic o malism), is ha only he angen and co angen bundle o a mani old a e equi ed
o de elop i . In addi ion, he e a e also o he polysymplec ic o malisms o desc ibing ield heo ies
such as hose de eloped by G. Sa danash ily e al [7, 8, 24], and by I. Kana chiko [10], as well as
he n-symplec ic o malism o L. K. No is [16, 19, 20, 21, 22].
The Skinne -Rusk o malism [25] was de eloped in o de o gi e a geome ical uni ied o malism
o desc ibing mechanical sys ems. I inco po a es all he cha ac e is ics o Lag angian and Hamil-
onian desc ip ions o hese sys ems (including dynamical equa ions and solu ions, cons ain s,
Legend e map, e olu ion ope a o s, equi alence, e c.). This o malism has been gene alized o
ime-dependen mechanical sys ems [4], o he mul isymplec ic desc ip ion o i s -o de ield he-
o ies [6, 11], and also o he k-symplec ic o mula ion o ield heo ies [23].
The main aim o his pape is o ex end his uni ied amewo k o he k-cosymplec ic desc ip ion
o i s -o de classical ield heo ies [14, 15], and o show how his desc ip ion comp ises he main
ea u es o he Lag angian and Hamil onian o malisms, bo h o he egula and singula cases.
P e iously, he k-cosymplec ic o malism o singula ield heo ies is s a ed, imp o ing p e ious
de elopmen s on his opic [15].
The o ganiza ion o he pape is as ollows: Sec ion 2 is de o ed o e iewing he main ea u es
o he k-cosymplec ic o malism [14, 15] o Lag angian and Hamil onian ield heo ies, and o s a ing
hese o malisms o singula sys ems. Fi s , he ield heo e ic phase o he Hamil onian app oach
space is Rk×(T1
k)∗Q, whe e (T1
k)∗Q=T∗Q⊕k
... ⊕T∗Qis he Whi ney sum o k-copies o he
co angen bundle T∗Qo a mani old Q. This space is he canonical example o a k-cosymplec ic
mani old. Wi h he in oduc ion o hese mani olds and using he Da boux heo em, we desc ibe
he Hamil onian o malism.
The ield phase o he Lag angian desc ip ion is Rk×T1
kQ, whe e T1
kQ=TQ⊕k
...⊕TQ is he
he Whi ney sum o k-copies o he angen bundle T Q o a mani old Q. This mani old T1
kQhas
he canonical k- angen s uc u e, gi en by kcanonical enso ields o ype (1,1) sa is ying ce ain
algeb aic p ope ies. This s uc u e on T1
kQcan be li ed o Rk×T1
kQ. Using he ex ended enso
ields o he Legend e map, we can cons uc a cosymplec ic s uc u e on Rk×T1
kQwhich enables
us o de elop he Lag angian o malism.
2
In Sec ion 3 we de elop he uni ied o malism o ield heo ies, which is based on he use o
he Whi ney sum M=Rk×T1
kQ⊕Rk×QRk×(T1
k)∗Q. The e a e canonical ”p ecosymplec ic”
o ms on i ( he pull-back o he canonical cosymplec ic o ms on each R×T∗Q) and a na u al
coupling unc ion, which is de ined by he con ac ion be ween ec o s and co ec o s. Then, gi en
a Lag angian L∈C∞(Rk×T1
kQ), we can s a e a ield equa ion on M. This equa ion has solu ion
only on a submani old ML, which is he g aph o he Legend e map. Then we p o e ha i
Z= (Z1,...,Zk) is an in eg able k- ec o ield, which is a solu ion o his equa ion and angen
o ML, hen he p ojec ion on o he i s ac o T1
kQo he in eg al sec ions o Za e solu ions
o he Eule -Lag ange ield equa ions. I Lis egula , he con e se also holds. Fu he mo e, we
es ablish he ela ionship be ween Zand he Hamil onian and he Lag angian k- ec o ields o he
k-cosymplec ic o malism, XHand XL.
This pape concludes wi h he s udy o he Skinne -Rusk o malism o ield heo ies using he
k-symplec ic [9, 18] and he k-cosymplec ic o malisms [14, 15].
Mani olds a e eal, pa acompac , connec ed and C∞. Maps a e C∞. Sum o e c ossed epea ed
indices is unde s ood.
2 The k-cosymplec ic o malism in ield heo y
2.1 The Hamil onian app oach [14]
2.1.1 The geome ic elemen s
Le Qbe a di e en iable mani old, dim Q=n, and τ∗:T∗Q→Qi s co angen bundle.
Deno e by (T1
k)∗Q=T∗Q⊕k
... ⊕T∗Q, he Whi ney sum o kcopies o T∗Q. The mani old
(T1
k)∗Qcan be iden i ied wi h he mani old J1(Q, Rk)0o 1-je s o mappings om Q o Rkwi h
a ge a 0 ∈Rk, ha is
J1(Q, Rk)0≡T∗Q⊕k
...⊕T∗Q
j1
q,0σ≡(dσ1(q),...,dσk(q))
whe e σA=πA◦σ:Q−→ Ris he A h componen o σ, and πA:Rk→Ris he canonical
p ojec ion on o he A h componen , o 1 ≤A≤k. (T1
k)∗Qis called he bundle o k1-co eloci ies
o he mani old Q,.
The mani old J1πQo 1-je s o sec ions o he i ial bundle πQ:Rk×Q→Qis di eomo phic
o Rk×(T1
k)∗Q, ia he di eomo phism gi en by
J1πQ→Rk×(T1
k)∗Q
j1
qφ=j1
q(φQ, IdQ)7→ (φQ(q), α1
q,...,αk
q)
whe e φQ:Qφ
→Rk×QπRk
→Rk,αA
q=d(φQ)A(q),1≤A≤kand (φQ)A:QφQ
→RkπA
→Ris he
A h componen o φQ.
Th oughou he pape we use he ollowing no a ion o he canonical p ojec ions
Rk×(T1
k)∗Q(πQ)1,0
−→ Rk×QπQ
−→ Q
3
and (πQ)1=πQ◦(πQ)1,0, whe e
πQ( , q) = q, (πQ)1,0( , α1
q,...,αk
q) = ( , q),(πQ)1( , α1
q,...,αk
q) = q ,
wi h ∈Rk,q∈Qand (α1
q,...,αk
q)∈(T1
k)∗Q.
I (qi) a e local coo dina es on U⊆Q, hen he induced local coo dina es (qi, pi), 1 ≤i≤n,
on (τ∗)−1(U) = T∗U⊂T∗Q, a e gi en by
qi(αq) = qi(q), pi(αq) = αq∂
∂qiq,
and, in he same way, he induced local coo dina es ( A, qi, pA
i) on [(πQ)1]−1(U) = Rk×(T1
k)∗Ua e
gi en by
A(j1
qφ) = (φQ(q))A, qi(j1
qφ) = qi(q), pA
i(j1
qφ) = d(φQ)A(q)∂
∂qiq,
o equi alen ly, o 1 ≤i≤nand 1 ≤A≤k,
A( , α1
q,...,αk
q) = A, qi( , α1
q,...,αk
q) = qi(q), pi
A( , α1
q, . . . , αk
q) = αA
q∂
∂qiq.
On Rk×(T1
k)∗Q, we de ine he di e en ial o ms
ηA
0= (πA
1)∗d A, θA
0= (πA
2)∗θ0, ωA
0= (πA
2)∗ω0,1≤A≤k
whe e πA
1:Rk×(T1
k)∗Q→Rand πA
2:Rk×(T1
k)∗Q→T∗Qa e he p ojec ions de ined by
πA
1( , (α1
q,...,αk
q)) = A, πA
2( , (α1
q,...,αk
q)) = αA
q,1≤A≤k,
ω0=−dθ0=dqi∧dpiis he canonical symplec ic o m on T∗Qand θ0=pidqiis he Liou ille
1- o m on T∗Q. Ob iously ωA
0=−dθA
0.
In local coo dina es we ha e
ηA
0=d A, θA
0=pA
idqi, ωA
0=dqi∧dpA
i,1≤A≤k(1)
Mo eo e , le V0= ( (πQ)1,0)∗. Then
V0=∂
∂p1
i
,..., ∂
∂pk
ii=1,...,n
A simple inspec ion o he exp essions in local coo dina es (1) shows ha he o ms ηA
0and ωA
0
a e closed, and he ollowing ela ions hold
1. η1
0∧ · · · ∧ ηk
06= 0, (ηA
0)|V0= 0,(ωA
0)|V0×V0= 0,
2. (∩k
A=1 ke ηA
0)∩(∩k
A=1 ke ωA
0) = {0},dim(∩k
A=1 ke ωA
0) = k,
Then, om he abo e geome ical model, he ollowing de ini ion is in oduced in [14]:
4
De ini ion 2.1 Le Mbe a di e en iable mani old o dimension k(n+1)+n. A amily (ηA, ωA, V ; 1 ≤
A≤k), whe e each ηAis a 1- o m, each ωAis a 2- o m and Vis an nk-dimensional dis ibu ion
on M, such ha
1. η1∧ · · · ∧ ηk6= 0,ηA|V= 0, ωA|V×V= 0,
2. (∩k
A=1 ke ηA)∩(∩k
A=1 ke ωA) = {0},dim(∩k
A=1 ke ωA) = k,
is called an almos k–cosymplec ic s uc u e, and Mis said o be an almos k–cosymplec ic mani-
old.
The ollowing heo em has been p o ed in [14].
Theo em 2.1 (Da boux Theo em) I he o ms ηAand ωAa e closed and Vis in eg able, hen
a ound each poin o M he e exis local coo dina es ( A, qi, pA
i; 1 ≤A≤k, 1≤i≤n)such ha
ηA=d A, ωA=dqi∧dpA
i, V =∂
∂p1
i
,..., ∂
∂pk
ii=1,...,n
.
In his case Mis called a k–cosymplec ic mani old.
The canonical model o hese geome ical s uc u es is (Rk×(T1
k)∗Q, ηA
0, ωA
0, V0).
Fo e e y k-cosymplec ic s uc u e (ηA, ωA, V ) on M, he e exis s a amily o k ec o ields
{RA,1≤A≤k}cha ac e ized by he ollowing condi ions
ıRAηB=δB
A, ıRAωB= 0,1≤A, B ≤k
They a e called he Reeb ec o ields associa ed o he k–cosymplec ic s uc u e. In he canonical
model RA=∂/∂ A,1≤A≤k. Obse e ha he ec o ields {∂/∂ A,1≤A≤k}a e de ined
in insically in Rk×(T1
k)∗Q, and span locally he e ical dis ibu ion wi h espec o he canonical
p ojec ion Rk×(T1
k)∗Q→(T1
k)∗Q.
2.1.2 k- ec o ields and in eg al sec ions
Le Mbe an a bi a y mani old, T1
kM he Whi ney sum TM⊕k
... ⊕TM o kcopies o TM, and
τM:T1
kM−→ Mi s canonical p ojec ion. τM:T1
kM−→ Mis usually called he angen bundle
o k1- eloci ies o M, he eason o his name will be explained la e in Sec ion 2.2.1
De ini ion 2.2 A sec ion X:M−→ T1
kMo he p ojec ion τMis called a k- ec o ield on M.
5

Since T1
kMis he Whi ney sum TM⊕k
... ⊕TM o kcopies o T M, we deduce ha o gi e a
k- ec o ield Xis equi alen o gi ing a amily o k ec o ields X1, . . . , Xkon Mby p ojec ing
Xon o e e y ac o . Fo his eason we will deno e a k- ec o ield by (X1,...,Xk).
De ini ion 2.3 An in eg al sec ion o he k- ec o ield (X1,...,Xk)passing h ough a poin
x∈Mis a map φ:U0⊂Rk→M, de ined on some neighbo hood U0o 0∈Rk, such ha
φ(0) = x, φ∗( )∂
∂ A =XA(φ( )) o all ∈U0,1≤A≤k .
We say ha a k- ec o ield (X1,...,Xk)on Mis in eg able i he e is an in eg al sec ion passing
h ough each poin o M.
Obse e ha , i k= 1, his de ini ion coincides wi h he de ini ion o in eg al cu e o a ec o
ield. In he k-cosymplec ic o malism, he solu ions o he ield equa ions a e desc ibed as he
in eg al sec ions o some k- ec o ields.
2.1.3 Hamil onian o malism
Le (M, ηA, ωA, V ) be a k-cosymplec ic mani old, and H:M→Ra Hamil onian unc ion. Le
X= (X1,...,Xk) be a k- ec o ield on Mwhich is a solu ion o he ollowing equa ions
ηA(XB) = δA
B,1≤A, B ≤k
k
X
i=1
ıXAωA=dH −
k
X
A=1
RA(H)ηA,
using Da boux coo dina es we know ha RA=∂/∂ Aand ηA=d A, hen we can w i e locally he
abo e equa ions as ollows
d A(XB) = δA
B,1≤A, B ≤k
k
X
i=1
ıXAωA=dH −
k
X
A=1
∂H
∂ Ad A.(2)
Using Da boux coo dina es, i X= (X1,...,Xk) is an in eg able k- ec o ield, locally gi en by
XA= (XA)B∂
∂ B+ (XA)i∂
∂qi+ (XA)B
i
∂
∂pB
i
hen
(XA)B=δB
A,∂H
∂pA
i
= (XA)i,∂H
∂qi=−
k
X
A=1
(XA)A
i,(3)
and i φ:Rk→M, locally gi en by φ( ) = (φA( ), φi( ), φA
i( )), is an in eg al sec ion o X, hen
∂φA
∂ B=δAB,∂φi
∂ B= (XB)i,∂φA
i
∂ B= (XB)A
i.
6
The e o e, om (3) we ob ain ha φ( ) is a solu ion o he Hamil onian ield equa ions
∂H
∂qi=−
k
X
A=1
∂φA
i
∂ A,∂H
∂pA
i
=∂φi
∂ A,(1 ≤A≤k, 1≤i≤n) (4)
So, equa ions (2) can be conside ed as a geome ic e sion o he Hamil onian ield equa ions.
Rema k 2.1 I (M, ηA, ωA, V )is a k-cosymplec ic mani old we can de ine he ec o bundle mo -
phism
Ω♯:T1
kM−→ T∗M
(X1,...,Xk)→Ω♯(X1,...,Xk) =
k
X
A=1
ıXAωA+ηA(XA)ηA
and deno ing by Mk(C∞(M)) he space o ma ices o o de kwhose en ies a e unc ions on M
we can also de ine he ec o bundle mo phism
η♯:T1
kM−→ Mk(C∞(M))
(X1,...,Xk)→η♯(X1,...,Xk) = (ηA(XB)) .
Then, he solu ions o (2) a e gi en by (X1,...,Xk) + (ke Ω♯∩ke η♯), whe e (X1,...,Xk)is a
pa icula solu ion.
2.2 The Lag angian app oach [15]
2.2.1 The geome ic elemen s
The mani old Rk×T1
kQ
Le τ:TQ →Qbe he angen bundle o Q. Le us deno e by T1
kQ he Whi ney sum
TQ⊕k
... ⊕TQ o kcopies o T Q. Nex we see ha he mani old Rk×T1
kQis a cosymplec ic
mani old when a egula Lag angian L:Rk×T1
kQ→Ris gi en.
T1
kQcan be iden i ied wi h he mani old J1
0(Rk, Q) o he k1- eloci ies o he mani old Q, ha
is, he mani old o 1-je s o maps σ:Rk→Qwi h sou ce a 0 ∈Rk, say
J1
0(Rk, Q)≡TQ⊕k
...⊕TQ
j1
0,qσ≡( 1q,..., kq)
whe e q=σ(0), and Aq=σ∗(0)[(∂/∂ A)(0)],1≤A≤k. Fo his eason T1
kQis called he angen
bundle o k1- eloci ies o Q,(see [17]).
The mani old J1πRko 1-je s o sec ions o he i ial bundle πRk:Rk×Q→Rkis di eomo phic
o Rk×T1
kQ, ia he di eomo phism gi en by
J1πRk→Rk×T1
kQ
j1
φ=j1
(IdRk, φQ)→( , 1,..., k)
7
whe e φQ:Rkφ
→Rk×QπQ
→Q, and
A= (φQ)∗( )∂
∂ A ,1≤A≤k .
Deno e by ρ:Rk×T1
kQ→Q he canonical p ojec ion, ha is ρ( , 1q,..., kq) = q. I
(qi) a e local coo dina es on U⊆Q, hen he induced local coo dina es (qi, i), 1 ≤i≤n, on
τ−1(U) = TU ⊂TQ, a e gi en by
qi( q) = qi(q), i( q) = q(qi),
and hen he induced local coo dina es ( A, qi, i
A) on ρ−1(U) = Rk×T1
kUa e gi en by
A(j1
φ) = A, qi(j1
φ) = qi(φQ( )) , i
A(j1
φ) = ∂(qi◦φQ)
∂ A( )
o equi alen ly
A( , 1q,..., kq) = A;qi( , 1q,..., kq) = qi(q); i
A( , 1q, . . . , kq) = Aq(qi),
whe e 1 ≤i≤n, 1≤A≤k.
Th oughou he pape we use he ollowing no a ion o he canonical p ojec ions
Rk×(T1
k)Q(πRk)1,0
−→ Rk×QπRk
−→ Rk
and (πRk)1=πRk◦(πRk)1,0, whe e
πRk( , q) = , (πRk)1,0( , 1q,..., kq) = ( , q),(πRk)1( , 1q,..., kq) = ,
wi h ∈Rk,q∈Qand ( 1q,..., kq)∈T1
kQ.
Canonical ec o ields and enso ields on Rk×T1
kQ
Deno e by C he canonical ec o ield (Liou ille ec o ield) o he ec o bundle (πRk)1,0:
Rk×T1
kQ→Rk×Q. This ec o ield Cis he in ini esimal gene a o o he ollowing low
R×(Rk×T1
kQ)−→ Rk×T1
kQ
(s, ( , 1q,..., kq)) −→ ( , es 1q,...,es kq),
and in local coo dina es i has he o m
C=X
i,A
i
A
∂
∂ i
A
,
Ccan be w i en as he sum C=
k
X
A=1
CA, whe e each ec o ield CAis he gene a o in ini esimal
o he ollowing low
R×(Rk×T1
kQ)−→ Rk×T1
kQ
(s, ( , 1q,..., kq)) −→ ( , 1q,..., A−1q, es Aq, A+1q,..., kq).
8
De ini ion 2.4 Fo a ec o Xqa Q, and o A= 1,...,k, we de ine i s e ical A-li (Xq)A
as he local ec o ield on τQ−1(q)⊂T1
kQgi en by
(Xq)A(wq) = d
dss=0 wq+ (0,...,0, s A
Xq,0,...,0)
o e e y poin wq= ( 1q,..., kq)∈T1
kQ.
In local coo dina es, o a ec o Xq=ai∂
∂qiwe ha e
(Xq)A=ai∂
∂ i
A
.(5)
The canonical k- angen s uc u e on T1
kQis he se (S1,...,Sk) o enso ields o ype (1,1)
de ined by
SA(wq)(Zwq) = (τ∗(wq)(Zwq))A, o all Zwq∈Twq(T1
kQ), wq= ( 1q,..., kq),
F om (5), in local coo dina es we ha e
SA=∂
∂ i
A
⊗dqi(6)
The enso s SAcan be ega ded as he (0,...,0,A
1,0,...,0)-li o he iden i y enso on Q o
T1
kQde ined in [17].
In an ob ious way we conside he ex ension o SA o Rk×T1
kQ, which we also deno e by SA,
and hey ha e he same local exp essions (6).
The k- angen mani olds we e in oduced as a gene aliza ion o he angen mani olds in [12, 13].
The canonical model o hese mani olds is T1
kQwi h he s uc u e gi en by (S1,...,Sk).
As in he case o mechanical sys ems, hese enso ields SAallow us o in oduce he o ms θA
L
and ωA
Lon Rk×T1
kQas ollows
θA
L=dL ◦SA, ωA
L=−dθA
L,1≤A≤k ,
wi h local exp essions
θA
L=∂L
∂ i
A
dqiωA
L=dqi∧d∂L
∂ i
A,1≤A≤k . (7)
These o ms play an impo an ole in he Lag angian o mula ion.
Finally, on Rk×T1
kQwe can conside he enso ields o ype (1,1) de ined by
ˆ
SA=SA−CA⊗d A,1≤A≤k .
These enso ields will be used o cha ac e izing he second o de pa ial di e en ial equa ions.
9
hey a e no sopde, in gene al. Thus, in o de o eco e he Eule -Lag ange equa ions (13), he
ollowing condi ion mus be added o he equa ions (14) (see p oposi ion 2.1):
ˆ
SA(XB) = 0
I he Lag angian is almos - egula , hen he e exis s H0∈C∞(P) such ha (FL0)∗H0=EL,
whe e FL0:Rk×T1
kQ→ P is de ined by 0◦FL0=FL. The Hamil onian ield equa ion analogous
o (2) should be
∗
0(ηA
0)((X0)B) = δA
B,
k
X
i=1
ı(X0)A(∗
0(ωA
0)) = dH0−
k
X
A=1
∂H0
∂ Aj∗
0(ηA
0),1≤A, B ≤k .
whe e X0= ((X0)1,...,(X0)k) (i i exis s) is a k- ec o ield on P. The exis ence o a k- ec o
ield X0in Psolu ion o he abo e equa ions is no assu ed excep , pe haps, in a submani old o
P.
3 Skinne -Rusk o mula ion
3.1 Geome ic elemen s
Le us conside he Whi ney sum M=Rk×T1
kQ⊕Rk×QRk×(T1
k)∗Q, wi h na u al coo dina es
( A, qi, i
A, pA
i). I has na u al bundle s uc u es o e Rk×T1
kQand Rk×(T1
k)∗Q. Le us deno e
by p 1:M → Rk×T1
kQ he p ojec ion in o he i s ac o , p 1( A, qi, i
A, pA
i) = ( A, qi, i
A) and by
p 2:M → Rk×(T1
k)∗Q he p ojec ion in o he second ac o , p 2( A, qi, i
A, pA
i) = ( A, qi, pA
i).
Le (η1
0,...,ηk
0, ω1
0,...,ωk
0) be he canonical o ms o he canonical k-cosymplec ic s uc u e on
Rk×(T1
k)∗Q. We deno e
ϑA= (p 2)∗ηA
0=d A,ΩA= (p 2)∗ωA
0,1≤A≤k ,
and so we ha e he amily (ϑ1,...,ϑk,Ω1,...,Ωk) in M.
Now, aking he k- ec o ield ∂
∂ 1,..., ∂
∂ kin Rk×(T1
k)∗Q, we can de ine a amily o k- ec o
ields (ξ1,...,ξk) in Msuch ha
(p 2)∗ξA=∂
∂ A,1≤A≤k .
These k- ec o ields (ξ1,...,ξk) sa is y ha , o 1 ≤A, B ≤k,
ıξAϑB=ıξA(p ∗
2ηB
0) = p ∗
2(ı∂
∂ Ad B) = δB
A
ıξAΩB=ıξA(p ∗
2ωB
0) = p ∗
2(ı∂
∂ AωB
0) = 0
and hey a e locally gi en by
ξA=∂
∂ A+ (ξA)i
B
∂
∂ i
B
1≤A≤k . (19)
16

whe e (ξA)i
Ba e a bi a y local unc ions in M. Hence, his k- ec o ield is no unique.
Finally, he coupling unc ion in M, deno ed by C, is de ined as ollows:
C:M=Rk×T1
kQ⊕Rk×QRk×(T1
k)∗Q−→ R
( , 1q,..., kq, α1
q,...,αk
q)7→
k
X
A=1
αA
q( Aq)
3.2 The Skinne -Rusk o malism o k-cosymplec ic ield heo ies
Gi en a Lag angian L∈C∞Rk×T1
kQ, we can de ine he Hamil onian unc ion H ∈ C∞(M) as
H=C − p ∗
1L(20)
which, in coo dina es, is gi en by
H=pA
i i
A−L( A, qi, i
A).(21)
Then, in his o malism, we ha e he ollowing p oblem:
S a emen 3.1 Le us suppose ha he e exis s an in eg able k- ec o ield Z= (Z1,...,Zk)on
M, such ha
ϑA(ZB) = δA
B,
k
X
A=1
ıZAΩA=dH −
k
X
A=1
ξA(H)ϑA,(22)
now he p oblem is o ind he in eg al sec ions ψ:Rk→ M o Z= (Z1,...,Zk).
Equa ions (22) gi e di e en kinds o in o ma ion. In ac , w i ing locally each ZAas
ZA= (ZA)B∂
∂ B+ (ZA)i∂
∂qi+ (ZA)i
B
∂
∂ i
B
+ (ZA)B
i
∂
∂pB
i
,
om (1), (21) and (22) we ob ain
(ZA)B=δB
A(23)
pA
i=∂L
∂ i
A
◦p 1(24)
(ZA)i= i
A(25)
k
X
A=1
(ZA)A
i=∂L
∂qi◦p 1(26)
whe e 1 ≤A≤k , 1≤i≤n. Then he ec o ields ZAa e locally gi en by
ZA=∂
∂ A+ i
A
∂
∂qi+ (ZA)i
B
∂
∂ i
B
+ (ZA)B
i
∂
∂pB
i
.(27)
17
whe e he coe icien s (ZA)B
ia e ela ed by he equa ions (26). Obse e ha hese equa ions do
no depend on he a bi a y unc ions (ξA)i
B, ha is, on he amily o ec o ields {ξA} ha we
ha e chosen o ex end he ec o ields ∂
∂ A.
So, in pa icula , we ha e ob ained in o ma ion o ou di e en classes:
1. The cons ain equa ions (24), which a e algeb aic (no di e en ial) equa ions de ining a
submani old MLo Mwhe e he equa ion (22) has solu ion. Obse e ha his submani old
is jus he g aph o he Legend e map FL de ined by he Lag angian L.
2. Le us obse e ha , as a consequence o (24), he k- ec o ield Z= (Z1,...,Zk), ZA∈X(M),
sa is ies equa ion (22) only on ML.
3. Equa ions (25), called he sopde condi ion, will be used in he ollowing subsec ion (see
Theo em 3.1), o show ha he in eg al sec ions o Z= (Z1, . . . , Zk) can be ob ained om
i s p olonga ions φ[1] o maps φ:Rk→Q.
4. Equa ions (26) which, aking in o accoun (23), (24) and (25), will gi e he classical Eule -
Lag ange equa ions o he in eg al sec ions o Z(see Theo em 3.1).
5. F om (23), (24), (25) and (26) we deduce ha he solu ions o equa ions (22) do no depend
on he k- ec o ield (ξ1,...,ξk) chosen.
We deno e by :ML→ M he na u al imbedding, and by
p 0
1:ML→Rk×T1
kQ , p 0
2:ML→Rk×(T1
k)∗Q
he es ic ed p ojec ions o p 1and p 2.
Rema k 3.1 Obse e ha , as MLis he g aph o FL, i is di eomo phic o Rk×T1
kQ, and his
means ha p 0
1is eally a di eomophism.
I Z= (Z1,...,Zk) is a solu ion o (22), hen each ZAis angen o he submani old MLi ,
and only i , he unc ions ZA pB
j−∂L
∂ j
B
◦p 1! anish a he poin s o ML, o e e y 1 ≤A, B ≤
k , 1≤j≤n. Then om (27) we deduce ha his is equi alen o he ollowing equa ions
(ZA)B
j=∂2L
∂ A∂ j
B
+ i
A
∂2L
∂qi∂ j
B
+ (ZA)i
C
∂2L
∂ i
C∂ j
B
.(28)
which a e condi ions o he coe icien s (ZA)i
C.
Taking in o accoun ha he k- ec o ields Zmus be angen o he submani old ML, he
abo e p oblem can be s a ed in ML, ins ead o in M. Fi s obse e ha he amily made o he k
ec o ields (ξ1,...,ξk) on Ma e angen o MLi and only i
∂2L
∂ A∂ i
B
◦p 1+ (ξA)j
C
∂2L
∂ j
C∂ i
B
◦p 1= 0 ,1≤i≤n , 1≤A, B ≤k ,
since he cons ain unc ion de ining MLis pA
i−∂L
∂ i
A
◦p 1. Thus aking in o accoun 3, we can
s a e
18
S a emen 3.2 To ind he in eg al sec ions ψ:Rk→ML⊂ M o in eg able k- ec o ields ZL=
((ZL)1,...,(ZL)k)on MLsolu ion o he ollowing equa ions
(∗ϑA)((ZL)B) = δA
B,
k
X
A=1
ı(ZL)A(∗ΩA) = d(∗H)−∗"k
X
A=1
ξA(H)#(∗ϑA),(29)
O cou se, ∗(ZL)A=ZA|ML, whe e Z= (Z1,...,Zk)is he k- ec o ield on Msolu ion o (22).
I is in e es ing o ema k ha :
1. In gene al, equa ions (22) (o , wha is equi alen , equa ions (29)) do no ha e a unique
solu ion. Solu ions o (22) a e gi en by (Z1,...,Zk) + ke Ω♯∩ke ϑ♯, whe e (Z1,...,Zk)
is a pa icula solu ion, Ω♯is he mo phism de ined by
Ω♯:T1
kM −→ T∗M
(Y1,...,Yk)→Ω♯(Y1,...,Yk) =
k
X
A=1
ıYAΩA+ϑA(YA)ϑA,
and, deno ing by Mk(C∞(M)) he space o ma ices o o de kwhose en ies a e unc ions
on M, he ec o bundle mo phism ϑ♯is de ined by
ϑ♯:T1
kM −→ Mk(C∞(M))
(Y1,...,Yk)→ϑ♯(Y1,...,Yk) = (ϑA(YB)) .
2. I Lis egula , hen aking in o accoun (23), (25) and (26) we can de ine a local k- ec o
ield (Z1,...,Zk) on a neighbo hood o each poin in MLwhich is a solu ion o (22). Each
ZAis locally gi en by
(ZA)B=δB
A,(ZA)i= i
A,(ZA)B
i=1
k
∂L
∂qiδB
A,
wi h (ZA)i
Bsa is ying (28). Now, by using a pa i ion o he uni y, one can cons uc a global
k- ec o ield which is a solu ion o (22).
When he Lag angian unc ion Lis singula we canno ensu e he exis ence o solu ions o he
equa ions (22) o (29). Then we mus de elop a cons ain algo i hm o ob aining a cons ain
submani old (i i exis s) whe e hese solu ions exis . Nex , we ou line his p ocedu e (see also [11],
whe e a simila algo i hm is ske ched in he mul isymplec ic o mula ion).
Assuming ha he Lag angian is almos - egula , we s a wi h P0=ML. Then, le P1be he
subse o P0composed o hose poin s whe e a solu ion o (29) exis s, ha is,
P1={z∈P0| ∃((ZL)1,...,(ZL)k)∈(T1
k)zP0solu ion o (29)}
I P1is a submani old o P0, hen he e exis s a sec ion o he canonical p ojec ion τP0:T1
kP0→P0
de ined on P1which is a solu ion o (29), bu which does no de ine a k- ec o ield on P1, in
gene al. In o de o ind solu ions aking alues in o T1
kP1, we de ine a new subse P2o P1as
ollows
P2={z∈P1| ∃((ZL)1,...,(ZL)k)∈(T1
k)zP1solu ion o (29)}
19
I P2is a submani old o P1, hen he e exis s a sec ion o he canonical p ojec ion τP1:T1
kP1→P1
de ined on P2which is a solu ion o (29), bu which does no de ine, in gene al, a k- ec o ield on
P2. P occeding u he , we ge a amily o cons ain mani olds
. . . ֒→P2֒→P1֒→P0=ML֒→ M
I he e exis s a na u al numbe such ha P +1 =P and dim P > k, hen we call P he
inal cons ain submani old o e which we can ind solu ions o equa ion (29). Obse e ha he
solu ions a e no unique (e en in he egula case) and, in gene al, hey a e no in eg able. In o de
o ind in eg able solu ions o equa ion (29), a cons ain algo i hm based on he same idea mus
be de eloped.
3.3 The ield equa ions o sec ions
Le Z= (Z1,...,Zk) be an in eg able k- ec o ield solu ion o (22). E e y in eg al sec ion ψ: ∈
Rk→(ψA( ), ψi( ), ψi
A( ), ψA
i( )) ∈ M o Zis o he o m ψ= (ψL, ψH), wi h ψL=p 1◦ψ:Rk→
Rk×T1
kQ, and i ψ akes alues in ML hen ψH=FL ◦ψL. In ac , om (24) we ob ain
ψH( ) = (p 2◦ψ)( ) = (ψA( ), ψi( ), ψA
i( )) = ψA( ), ψi( ),∂L
∂ i
A
(ψL( ))= (FL ◦ψL)( ).
In his way, e e y cons ain , di e en ial equa ion, e c. in he uni ied o malism can be ans-
la ed o he non au onomous Lag angian o Hamil onian o malism by es ic ion o he i s o
second ac o s o he p oduc bundle. In pa icula , condi ions (24) gene a e, by p 2-p ojec ion,
he p ima y cons ain s o he Hamil onian o malism o singula Lag angians (i.e., he image o
he Legend e ans o ma ion, FL(Rk×T1
kQ)⊂Rk×(T1
k)∗Q), and hey can be called he p ima y
Hamil onian cons ain s.
Hence he main esul in his subsec ion is he ollowing:
Theo em 3.1 Le Z= (Z1,...,Zk)be an in eg able k- ec o ield in Msolu ion o (22), and le
ψ:Rk→ML⊂ M be an in eg al sec ion o Z= (Z1,...,Zk), wi h ψ= (ψL, ψH) = (ψL, FL ◦ψL).
Then ψLis he canonical li φ[1] o he p ojec ed sec ion φ=ρ◦p 0
1◦ψ:Rkψ
→ML
p 0
1
≈Rk×T1
kQρ
→
Q, and φis a solu ion o he Eule -Lag ange ield equa ions (13).
20
M
p 1p 2






+
QQQQQQ
Qs

6
Rk×T1
kQ
p 0
1
ML
p 0
2-Rk×(T1
k)∗Q
-
FL
ρ(πQ)1
ψL=φ[1] ψH=FL ◦φ[1]
ψ
φ
Q
Rk
?
HHHHHHj







S
S
S
S
S
S
S
S
So

7
6
6
6
P oo : I
ψ( ) = ψA( ), ψi( ), ψi
A( ), ψA
i( ) = ∂L
∂ i
A
(ψL( ))
is an in eg al sec ion o Z= (Z1,...,Zk), hen
ZA(ψ( )) = ∂ψB
∂ A( )∂
∂ Bψ( )+∂ψi
∂ A( )∂
∂qiψ( )+∂ψB
i
∂ A( )∂
∂pB
iψ( )+∂ψi
B
∂ A( )∂
∂ i
Bψ( )(30)
F om (23), (24), (25) and (30) we ob ain
∂ψB
∂ A( ) = (ZA)B(ψ( )) = δB
A(31)
ψA
i( ) = pA
i(ψ( )) = ∂L
∂ i
A
◦p 1(ψ( )) = ∂L
∂ i
A
(ψL( )) (32)
ψi
A( ) = i
A(ψ( )) = (ZA)i(ψ( )) = ∂ψi
∂ A( ) (33)
∂ψB
i
∂ A( ) = (ZA)B
i(ψ( )) (34)
The e o e om (26), (32) and (34) we ob ain
∂L
∂qi(ψL( )) =
k
X
A=1
(ZA)A
i(ψ( )) =
k
X
A=1
∂ψA
i
∂ A( ) =
k
X
A=1
∂
∂ A∂L
∂ i
A
(ψL( ))
and om (31) we ob ain ψA( ) = A+cA. Taking cA= 0, om (33) we ha e
ψL( ) =  , ψi( ),∂ψi
∂ A( ),
and om he las wo equa ions we deduce ha ψL=φ[1] and φ=ρ◦p 0
1◦ψ:Rkψ
→ML
p 0
1
≈
Rk×T1
kQρ
→Q, is a solu ion o he Eule -Lag ange ield equa ions (13), whe e φ( ) = (ψi( )).
21

Fu he mo e, o he egula case we can p o e:
P oposi ion 3.1 Acco ding o he hypo hesis o Theo em 3.1, i Lis egula hen ψH=FL◦ψLis
a solu ion o he Hamil on ield equa ions (4), whe e he Hamil onian His gi en by H◦FL =EL.
P oo : Since Lis egula , FL is a local di eomo phism, and hus we can choose o each poin in
Rk×T1
kQan open neighbo hood U⊂Rk×T1
kQsuch ha FL|U:U→FL(U) is a di eomo phism.
So we can de ine HU:FL(U)→Ras HU= (EL)|U◦(FL|U)−1.
Deno ing by H≡HU,EL≡(EL)|Uand FL ≡FL|U, we ha e EL=H◦FL, which p o ides
he iden i ies ∂H
∂pA
i
◦FL = i
A,∂H
∂qi◦FL =−∂L
∂qi.(35)
Now conside ing he open subse V=ψ−1
L(U)⊂Rkwe ha e ψ|V:V⊂Rk→U⊕FL(U)⊂ML,
whe e (ψL)|V:V⊂Rk→U⊂Rk×T1
kQand (ψH)|V=FL ◦(ψL)|V:V⊂Rk→FL(U)⊂
Rk×(T1
k)∗Q.
The e o e om (26), (33), (34) and (35), o e e y ∈V⊂Rkwe ob ain
∂H
∂pA
i
(ψH( )) = ∂H
∂pA
i
◦FL(ψL( )) = i
A(ψL( )) = ∂ψi
∂ A( )
and ∂H
∂qi(ψH( )) = ∂L
∂qi◦FL(ψL( )) = −∂L
∂qi(ψL( )) = −(ZA)A
i(ψ( )) = −∂ψA
i
∂ A( )
om which we deduce ha (ψH)|Vis a solu ion o he Hamil on ield equa ions (4).
Con e sely, we can s a e:
P oposi ion 3.2 I Lis egula and X= (X1,...,Xk)is a solu ion o (14) hen:
1. The k- ec o ield Z= (Z1,...,Zk)gi en by ZA= (IdRk×T1
kQ⊕FL)∗(XA),1≤A≤kis a
solu ion o (22).
2. I ψL:Rk→Rk×T1
kQis an in eg al sec ion o X= (X1,...,Xk)(and hus, om Rema k
2.2 and om Theo em 2.2, φ=ρ◦ψL:RkψL
→Rk×T1
kQρ
→Qis a solu ion o he Eule -
Lag ange ield equa ions) hen ψ= (ψL, FL ◦ψL) : Rk→ML⊂ M is an in eg al sec ion o
Z= (Z1,...,Zk).
22
P oo :
1. I Lis egula and X= (X1,...,Xk) is a solu ion o (14), hen om Theo em 2.2 we know
ha XAis a sopde and hus XAis locally gi en by
XA=∂
∂ A+ i
A
∂
∂qi+ (XA)i
B
∂
∂ i
B
(36)
whe e (XA)i
Bsa is y
∂2L
∂ A∂ i
A
+ j
A
∂2L
∂qj∂ i
A
+ (XA)j
B
∂2L
∂ j
B∂ i
A
=∂L
∂qi(37)
Since he map IdRk×T1
kQ⊕FL :Rk×T1
kQ→ML⊂ M, is locally gi en by
( A, qi, i
A)7→  A, qi, i
A,∂L
∂ i
A,
om (36) and (1) we ob ain
ZA= (IdRk×T1
kQ⊕FL)∗(XA) = ∂
∂ A+ i
A
∂
∂qi+ (XA)i
B
∂
∂ i
B
+ ∂2L
∂ A∂ j
C
+ i
A
∂2L
∂qi∂ j
C
+ (XA)i
B
∂2L
∂ i
B∂ j
C!∂
∂pC
j
(38)
Then om (3.2), (37) and (38) we ha e ha
(ZA)B=δB
A,(ZA)i= i
A
k
X
A=1
(ZA)A
j=∂2L
∂ A∂ j
A
+ i
A
∂2L
∂qi∂ j
A
+ (XA)i
B
∂2L
∂ i
B∂ j
A
=∂L
∂qj
ZApB
k−∂L
∂ k
B= 0 ,
ha is, he k- ec o ield Z= (Z1,...,Zk) is a solu ion o (22) and each ZAis angen o ML
o A= 1,...,k.
2. Since ψLis in eg al sec ion o X= (X1,...,Xk) we ha e
XA(ψL( )) = (ψL)∗( )∂
∂ A 
and hen
ZA(ψ( )) = (IdRk×T1
kQ⊕FL)∗(ψL( )) (XA(ψL( )))
= ((IdRk×T1
kQ⊕FL)◦ψL)∗( )∂
∂ A =ψ∗( )∂
∂ A .
23
Rema k 3.2 The las esul eally holds o egula and almos - egula Lag angians. In he almos -
egula case, assuming as addi ional hypo hesis ha XLis a sopde, he p oo is he same, bu he
sec ions ψ,ψLand ψH ake alues no on ML,Rk×T1
kQand Rk×(T1
k)∗Q, bu in he inal cons ain
submani old P and on he p ojec ion submani olds p 1(P )⊂Rk×T1
kQand p 2(P )⊂Rk×(T1
k)∗Q,
espec i ely.
3.4 The ield equa ions o k- ec o ields
The aim o his subsec ion is o es ablish he ela ionship be ween k- ec o ields ha a e solu ions
o (14) and k- ec o ields ha a e solu ions o (22) o , wha is equi alen , solu ions o (29).
Fi s , obse e ha :
Lemma 3.1 Fo e e y 1≤A≤kwe ha e ha
∗ϑA= (p 0
1)∗d A, ∗ΩA= (p 0
1)∗ωA
L,(39)
P oo : In ac , aking in o accoun ha FL ◦p 0
1=p 2◦j, we ob ain
∗ϑA=∗(p 2)∗ηA
0= (FL ◦p 0
1)∗ηA
0= (p 0
1)∗FL∗ηA
0= (p 0
1)∗d A,
∗ΩA=∗(p 2)∗ωA
0= (FL ◦p 0
1)∗ωA
0= (p 0
1)∗FL∗ωA
0= (p 0
1)∗ωA
L.
Then, he main esul is he ollowing:
Theo em 3.2 a) Le L:Rk×T1
kQ→Rbe a Lag angian and le ZL= ((ZL)1,...,(ZL)k)be a k-
ec o ield on MLsolu ion o (29). Then he k- ec o ield XL= ((XL)1,...,(XL)k)on Rk×T1
kQ
de ined by
XL◦p 0
1=T1
k(p 0
1)◦ZL(40)
is a k- ec o ield solu ion o (14), whe e T1
k(p 0
1): T1
k(ML)→T1
k(Rk×T1
kQ)is he na u al ex ension
o p 0
1, in oduced in (8).
Con e sely, e e y k- ec o ield XLsolu ion o (14) can be eco e ed in his way om a k- ec o
ield ZLin MLsolu ion o (29).
b) The k- ec o ield ZLis in eg able i , and only i , he k- ec o ield XLis an in eg able
sopde.
24
P oo : a) Since p 0
1:ML→Rk×T1
kQis a di eomo phism, hen he k- ec o ield XLon Rk×T1
kQ
de ined by (40) is gi en by
(XL)A=(p 0
1)−1∗(ZL)A,1≤A≤k . (41)
Fu he mo e, we ob ain ha
∗H=∗(C − (p 1)∗L) = ∗C − ∗(p 1)∗L= (p 0
1)∗(C(L)) −(p 0
1)∗L= (p 0
1)∗EL.(42)
F om (39) and (41) we deduce ha
∗ϑA((ZL)B) = (p 0
1)∗d A(p 0
1)∗(XL)B= (p 0
1)∗d A((XL)B)(43)
and om (19), (20), (21) and (24)
∗[ξA(H)] = ∗ ∂
∂ A+ (ξA)i
B
∂
∂ i
B(pC
j j
C−(p ∗
1L))
=∗(ξA)i
BpB
i−∂L
∂ i
B
◦p 1−p ∗
1∂L
∂ A=−(p 0
1)∗∂L
∂ A(44)
The e o e om (29), (39), (41), (42) and (44) we ob ain
k
X
A=1
ı(ZL)A∗ΩA−d(∗H) + ∗"k
X
A=1
ξA(H)#(∗ϑA)
=
k
X
A=1
ı(p 0
1)∗(XL)A(p 0
1)∗ωA
L−d((p 0
1)∗EL)−
k
X
A=1
(p 0
1)∗∂L
∂ A(p 0
1)∗d A
= (p 0
1)∗ k
X
A=1
ı(XL)AωA
L−dEL−
k
X
A=1
∂L
∂ Ad A!.
(45)
Since p 0
1is a di eomo phism, om (43) and (45) we deduce ha he k- ec o ield ZLis a solu ion
o (29) i , and only i , he k- ec o ield XLis a solu ion o (14). This inishes a).
b) Suppose now ha he k- ec o ield ZLis in eg able. Le ϕ:Rk→Rk×T1
kQbe an in eg al
sec ion o XL, ha is, (XL)A(ϕ( )) = ϕ∗( )∂
∂ A . Thus
(ZL)A((p 0
1)−1◦ϕ( )) = ((p 0
1)−1)∗(XL)A((p 0
1)−1◦ϕ( )) = ((p 0
1)−1)∗(ϕ( ))((XL)A(ϕ( )))
= ((p 0
1)−1)∗(ϕ( )) ϕ∗( )∂
∂ A = ((p 0
1)−1◦ϕ( ))∗∂
∂ A ,
which means ψ= (p 0
1)−1◦ϕ:Rk→MLis an in eg al sec ion o ZL.
Since ψ:Rk→ML hen we know ha he in eg al sec ion j◦ψ:Rk→ M is gi en by
((j◦ψ)L, FL ◦(j◦ψ)L), and om Theo em 3.1, we know ha (j◦ψ)L=φ[1], whe e φ=ρ◦ψ:
Rkψ
→ML≈Rk×T1
kQρ
→Q. Then we ha e
φ[1] = (j◦ψ)L=p 1◦j◦ψ=p 0
1◦ψ=ϕ
Since e e y in eg al sec ion ϕo XLis a i s p olonga ion φ[1] o a map φ:Rk→Qspace we
deduce om Lema 2.2 ha XLis a sopde.
25