k-cosymplectic formalism in classical field theory: the Skinner–Rusk approach

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