Lagrangian–Hamiltonian Unified Formalism for Field Theory

JOURNAL OF MATHEMATICAL PHYSICS VOLUME 45, NUMBER 1 JANUARY 2004

Lagrangian–Hamiltonian unified formalism for field theory Arturo Echeverrıá-Enrı´quez Departamento de Matema´tica Aplicada IV, Edificio C-3, Campus Norte UPC, C/Jordi Girona 1, E-08034 Barcelona, Spain Carlos Lo´peza) Departamento de Matema´ticas, Campus Universitario. Fac. Ciencias, 28871 Alcala´ de Henares, Spain Jesu´s Marıń-Solanob) Departamento de Matema´tica Econo´mica, Financiera y Actuarial, UB Avenida Diagonal 690, E-08034 Barcelona, Spain Miguel C. Munõz-Lecandac) and Narciso Romań-Royd) Departamento de Matema´tica Aplicada IV, Edificio C-3, Campus Norte UPC, C/Jordi Girona 1, E-08034 Barcelona, Spain ͑Received 27 November 2002; accepted 15 September 2003͒ The Rusk–Skinner formalism was developed in order to give a geometrical unified formalism for describing mechanical systems. It incorporates all the characteristics of Lagrangian and Hamiltonian descriptions of these systems ͑including dynamical equations and solutions, constraints, Legendre map, evolution operators, equivalence, etc.͒. In this work we extend this unified framework to first-order classical field theories, and show how this description comprises the main features of the Lagrangian and Hamiltonian formalisms, both for the regular and singular cases. This formulation is a first step toward further applications in optimal control theory for partial differential equations. © 2004 American Institute of Physics. ͓DOI: 10.1063/1.1628384͔

I. INTRODUCTION

In ordinary autonomous classical theories in mechanics there is a unified formulation of Lagrangian and Hamiltonian formalisms,1 which is based on the use of the Whitney sum of the ϭ ϵ ϫ ͑ tangent and cotangent bundles W TQ T*Q TQ QT*Q the velocity and momentum phase spaces of the system͒. In this space, velocities and momenta are independent coordinates. There is a canonical presymplectic form ⍀͑the pull-back of the canonical form in T*Q), and a natural i coupling function, locally expressed as piv , is defined by contraction between vectors and cov- ෈ ϱ ϭ i ectors. Given a Lagrangian L C (TQ), a Hamiltonian function, locally given by H piv ϪL(q,v), is determined, and, using the usual constraint algorithm for the geometric equation i(X)⍀ϭdH associated to the Hamiltonian system (W,⍀,H), we obtain that iϭ ץ ץ ͒ ͑ 1 The first constraint submanifold W1 is isomorphic to TQ, and the momenta L/ v pi are determined as constraints. ͑2͒ The geometric equation contains the second order condition viϭdqi/dt. ͑ ͒ ϵ 3 The identification W1 TQ allows us to recover the Lagrangian formalism. ͑4͒ The projection to the cotangent bundle generates the Hamiltonian formalism, including constraints. The Legendre map and the time evolution operator are straightforwardly obtained by the previous identification and projection.2

a͒Electronic mail: [email protected] b͒Electronic mail: [email protected] c͒Electronic mail: [email protected] d͒Electronic mail: [email protected]

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It is also worth noticing that this space is also appropriate for the formulation of different kinds of problems in optimal control.3–7 Furthermore, in Refs. 8 and 9 this unified formalism has been extended for nonautonomous mechanical systems. Our aim in this paper is to reproduce the same construction for first-order field theories, generating a unified description of Lagrangian and Hamiltonian formalisms and its correspondence, starting from the multisymplectic description of such theories. ͑See, for instance, Refs. 10–18, for some general references on this formalism. See, also, Refs. 19–25, for other geometric formulations of field theories.͒ As is shown throughout the paper, characteristics analogous to those pointed out for mechanical systems can be stated in this context. In Ref. 9, a first approach to this subject has been made, focusing mainly on the constraint algorithm for the singular case. The organization of the paper is as follows: Sec. II is devoted to reviewing the main features of the multisymplectic description of Lagrangian and Hamiltonian field theories. In Sec. III we develop the unified formalism for field theories: starting from the extended jet-multimomentum bundle ͑analogous to the Whitney sum in mechanics͒, we introduce the so-called extended Hamil- tonian system and state the field equations for sections, m-vector fields, connections, and jet fields in this framework. It is also shown how the standard Lagrangian and Hamiltonian descriptions are recovered from this unified picture. As a typical example, the minimal surface problem is described in this formalism in Sec. IV. Finally, we include an Appendix where basic features about connections, jet fields, and m-vector fields are displayed. Throughout this paper ␲:E→M will be a fiber bundle (dim Mϭm, dim EϭNϩm), where M is an oriented manifold with volume form ␻෈⍀m(M). ␲1:J1E→E is the jet bundle of local 1 1 1 ␣ A A sections of ␲, and ␲¯ ϭ␲ؠ␲ :J E→M gives another fiber bundle structure. (x ,y ,v␣) will denote natural local systems of coordinates in J1E, adapted to the bundle E→M (␣ϭ1,...,m; A ϭ ␻ϭ 1∧ ∧ mϵ m 1,...,N), and such that dx ••• dx d x. Manifolds are real, paracompact, connected, and Cϱ. Maps are Cϱ. Sum over crossed repeated indices is understood.

II. GEOMETRIC FRAMEWORK FOR CLASSICAL FIELD THEORIES

A. Lagrangian formalism

͑For details concerning the contents of this and the next section, see, for instance, Refs. 10–13, 17, 18, and 26–31. See, also, the Appendix͒. A classical field theory is described by giving a configuration fiber bundle ␲:E→M and a Lagrangian density, which is a ␲¯ 1-semibasic m-form on J1E usually written as LϭL␲¯ 1*␻, where L෈Cϱ(J1E) is the Lagrangian function determined by L and ␻. The Poincare´–Cartan m and (mϩ1)-forms associated with the Lagrangian density L are defined using the vertical endo- morphism V of the bundle J1E ͑see Ref. 30͒

⌰ ͑V͒LϩL෈⍀m͑ 1 ͒ ⍀ Ϫ ⌰ ෈⍀mϩ1͑ 1 ͒ Lªi J E ; Lª d L J E . 1 A Lagrangian system is a couple (J E,⍀L). It is regular if ⍀L is a multisymplectic (mϩ1)-form ͑a closed m-form, mϾ1, is called multisymplectic if it is one-nondegenerate; elsewhere it is pre-multisymplectic͒. In natural charts in J1E we have

ץ ץ Vϭ͑ AϪ A ␣͒ dy v␣dx A ␯ , xץ v␯ץ

and

Lץ Lץ A mϪ1 A m ⌰Lϭ dy ∧d x␮Ϫͩ v ϪL ͪ d x, A A ␮ v␮ץ v␮ץ

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2Lץ 2Lץ 2Lץ B A mϪ1 B A mϪ1 A B m ⍀LϭϪ dv␯ ∧dy ∧d x␣Ϫ dy ∧dy ∧d x␣ϩ v␣dv␯ ∧d x B A B A B A ␣vץ v␯ץ ␣vץ yץ ␣vץ v␯ץ

2Lץ Lץ 2Lץ ϩͩ vAϪ ϩ ͪ dy B∧dmx ␣ B A B ␣ B ␣vץ xץ yץ ␣vץ yץ

mϪ1 ␣ m 2 A B v␯(¯y))Þ0, forץ␣vץ/L ץ)x )d x); the regularity condition is equivalent to detץ/ץ)where d x␣ϵi͑ every ¯y෈J1E. 1 The Lagrangian problem associated with a Lagrangian system (J E,⍀L) consists in ﬁnding sections ␾෈⌫(M,E), the set of sections of ␲, which are characterized by the condition

1␾͒ ͒⍀ ϭ ෈ 1 ͒ ͑ j *i͑X L 0, for every X X͑J E .

In natural coordinates, if ␾(x)ϭ(x␣,␾A(x)), this condition is equivalent to demanding that ␾ satisfy the Euler–Lagrange equations

Lץ ץ Lץ ͯ Ϫ ͩ ͪͯ ϭ0 ͑for Aϭ1,...,N͒. ͑1͒ A ␣ A ␣vץ xץ yץ j1␾ j1␾

The problem of ﬁnding these sections can be formulated equivalently as follows: ﬁnding a distribution D of T(J1E) such that it is integrable ͑that is, involutive͒, m-dimensional, ␲¯ 1-transverse, and the integral manifolds of D are the image of sections solution of the above equations ͑therefore, lifting of ␲-sections͒. This is equivalent to stating that the sections solution to the Lagrangian problem are the integral sections of one of the following equivalent elements:

m 1 • A class of holonomic m-vector fields ͕XL͖ʚX (J E), such that i(XL)⍀Lϭ0, for every XL෈͕XL͖. 1 1 . A holonomic connection ٌL in ␲¯ :J E→M such that i(ٌL)⍀Lϭ(mϪ1)⍀L • 1 1 1 • A holonomic jet field ⌿L :J E→J J E, such that i(⌿L)⍀Lϭ0 ͑the contraction of jet fields with differential forms is defined in Ref. 11͒.

Semi-holonomic locally decomposable m-vector fields, jet fields, and connections which are solution to these equations are called Euler–Lagrange m-vector fields, jet fields, and connections for 1 1 (J E,⍀L). In a natural chart in J E, the local expressions of these elements are

ץ ץ ץ m A A XLϭ f ∧ ͩ ϩF ϩG ͪ , ␣ ␣␯ ␣ A A v␯ץ yץ xץ ϭ1␣

ץ ץ ץ ␣ A A , ͪ Lϭdx ͩ ϩF ϩGٌ ␣ ␣␯ ␣ A A v␯ץ yץ xץ

␣ A A A A ⌿Lϭ͑x ,y ,v␣ ,F␣ ,G␣␩͒,

A A with F␣ϭv␣ ͑which is the local expression of the semi-holonomy condition͒, and where the A coefﬁcients G␣␯ are related by the system of linear equations

2ץ 2ץ ץ 2ץ L A L L L A G␣␯ϭ Ϫ Ϫ v␯ ͑A,Bϭ1,...,N͒. ͑2͒ A B B ␯ B A B v␯ץ yץ v␯ץ xץ yץ v␯ץ␣vץ

ϱ 1 f ෈C (J E) is an arbitrary nonvanishing function. A representative of the class ͕XL͖ can be 1 selected by the condition i(XL)(␲¯ *␻)ϭ1, which leads to f ϭ1 in the above local expression. 1 ␮ A A ␯ A A ␣ ,x , and henceץ/ ␾ץx ) is an integral section of XL , then v␣ϭץ/ ␾ץ, Therefore, if j ␾ϭ(x ,␾ B the coefﬁcients G␣␯ must satisfy the equations

2␾Aץ ␾Aץ A ͩ ␣ ␾A ͪ ϭ ͑ ϭ ␩ ␯ϭ ͒ G␯␩ x , , A 1,...,N; , 1,...,m . x␯ץx␩ץ ␣xץ

As a consequence, the system ͑2͒ is equivalent to the Euler–Lagrange Eq. ͑1͒ for ␾. 1 If (J E,⍀L) is a regular Lagrangian system, the existence of classes of Euler–Lagrange m-vector fields for L ͑or what is equivalent, Euler–Lagrange jet fields or connections͒ is assured. For singular Lagrangian systems, the existence of this kind of solutions is not assured except 1 ͑ perhaps on some submanifold SJ E. Furthermore, solutions of the field equations can exist in general, on some submanifold of J1E), but none of them are semi-holonomic ͑at any point of this submanifold͒. In both cases, the integrability of these solutions is not assured, except perhaps on a smaller submanifold I such that the integral sections are contained in I.

B. Hamiltonian formalism For the Hamiltonian formalism of field theories, we have the extended multimomentum bundle M␲, which is the bundle of m-forms on E vanishing by contraction with two ␲-vertical vector fields ͓or equivalently, the set of affine maps from J1E to ␲*⌳mT*M ͑Refs. 10 and 32͔͒, and the restricted multimomentum bundle J1*EϵM␲/␲*⌳mT*M. We have the natural projections

.␶1:J1*E→E, ¯␶1ϭ␲ؠ␶1:J1*E→M, ␮:M␲→J1*E, ␮ˆ ϭ¯␶1ؠ␮:M␲→M

Given a system of coordinates adapted to the bundle ␲:E→M, we can construct natural coordi- ␣ A ␣ ␣ϭ ϭ M␲ ϭ m nates (x ,y ,pA ,p)( 1,...,m; A 1,...,N)in , corresponding to the m-covector p pd x ϩ ␣ A∧ mϪ1 ෈M␲ ␣ A ␣ 1 ͓ ͔ϭ ␣ A∧ mϪ1 ϩ͗ m ͘ pAdy d x␣ , and (x ,y ,pA)inJ *E, for the class p pAdy d x␣ d x ෈J1*E. 1 Now,if(J E,⍀L) is a Lagrangian system, the extended Legendre map associated with L, gFL:J1E→M␲, is deﬁned as

͓gFL͑¯ ͔͒͑ ͒ ͑⌰ ͒ ͑¯ ¯ ͒ ͑ ͒ y Z1 ,...,Zm ª L ¯y Z1 ,...,Zm , 3 ෈ ¯ ¯ ෈ 1 ␲1¯ ϭ where Z1 ,...,Zm T␲1(¯y)E, and Z1 ,...,Zm T¯yJ E are such that T¯y Z␣ Z␣ . Then the re- L FL ␮ؠgFL stricted Legendre map associated with is ª . Their local expressions are Lץ Lץ gFL* ␣ϭ ␣ gFL* Aϭ A gFL* ␣ϭ gFL* ϭ Ϫ A x x , y y , pA A , p L v␣ A , ␣vץ ␣vץ

Lץ FL* ␣ϭ ␣ FL* Aϭ A FL* ␣ϭ x x , y y , pA A . ␣vץ

1 Therefore, (J E,⍀L)isaregular Lagrangian system if FL is a local diffeomorphism ͑this deﬁ- 1 nition is equivalent to that given above͒. Elsewhere (J E,⍀L)isasingular Lagrangian system. As 1 a particular case, (J E,⍀L)isahyper-regular Lagrangian system if FL is a global diffeomor- 1 ⍀ P FL 1 phism. A singular Lagrangian system (J E, L)isalmost-regular if: ª (J E) is a closed 1 ͑ P 1 FL submanifold of J *E we will denote the natural imbedding by  : J *E), is a submersion onto its image, and for every ¯y෈J1E, the ﬁbres FLϪ1(FL(¯y)) are connected submanifolds of J1E. 1 In order to construct a Hamiltonian system associated with (J E,⍀L), recall that the multi- cotangent bundle ⌳mT*E is endowed with a natural canonical form ⌰෈⍀m(⌳mT*E), which is

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␶ → the tautological form deﬁned as follows: let E :T*E E be the natural projection, and ⌳m␶ ⌳m → ෈⌳m ͑ ϭ ␤ E : T*E E its natural extension; then, for every ¯p T*E where ¯p (y, ), with y ෈ ␤෈⌳m ෈ ⌳m E and T*y E), and for every X1 ,...,Xm X( T*E) we have

m m m ͓⌰͑X ,...,X ͔͒¯ ͓͑⌳ ␶ ͒*␤͔͑X ,...,X ͒ϭ␤͑T¯⌳ ␶ ͑X ͒,...,T¯⌳ ␶ ͑X ͒͒. 1 m pª E 1¯p m¯p p E 1¯p p E m¯p ⍀ Ϫ ⌰෈⍀mϩ1 ⌳m M␲ϵ⌳m Thus we also have the multisymplectic form ª d ( T*E). But 1 T*E is ⌳m ␭ ⌳m ⌳m ⌰ ␭*⌰ a subbundle of T*E. Then, if : 1 T*E T*E is the natural imbedding, ª and ⍀ Ϫ ⌰ϭ␭*⍀ M␲ ª d are canonical forms in , which are called the multimomentum Liouville m and ϩ ⌰ ϭ ␶ ؠ␮ ෈M␲ (m 1) forms. In particular, we have that (p) ( 1 )*p, for every p . Their local expressions are

⌰ϭ ␣ A∧ mϪ1 ϩ m ⍀ϭϪ ␣∧ A∧ mϪ1 Ϫ ∧ m ͑ ͒ pAdy d x␣ pd x, dpA dy d x␣ dp d x. 4 g g Observe that FL*⌰ϭ⌰L , and FL*⍀ϭ⍀L . 1 ⍀ P˜ gFL 1 Now,if(J E, L) is a hyper-regular Lagrangian system, then ª (J E) is a one- codimensional and ␮-transverse imbedded submanifold of M␲ ͑we will denote the natural im- ˜ P˜ M␲ 1 ␮Ϫ1 ␮ bedding by  : ), which is diffeomorphic to J *E. This diffeomorphism is , when is P˜ gFLؠFLϪ1 restricted to , and also coincides with the map hª , when it is restricted onto its image ͑which is just P˜ ͒. This map h is called a Hamiltonian section, and can be used to construct the Hamilton-Cartan m and (mϩ1) forms of J1*E by making

⌰ ϭ ⌰෈⍀m 1 ͒ ⍀ ϭ ⍀෈⍀mϩ1 1 ͒ h h* ͑J *E , h h* ͑J *E . 1 ⍀ The couple (J *E, h) is said to be the Hamiltonian system associated with the hyper-regular 1 Lagrangian system (J E,⍀L). Locally, the Hamiltonian section h is speciﬁed by the local Hamil- ϭ ␣ LϪ1 AϪ LϪ1 ␣ A ␣ ϭ ␣ A ␣ Ϫ tonian function H pA(F )*v␣ (F )*L, that is, h(x ,y ,pA) (x ,y ,pA , H). Then we have the local expressions

⌰ ϭ ␣ A∧ mϪ1 Ϫ m ⍀ ϭϪ ␣∧ A∧ mϪ1 ϩ ∧ m h pAdy d x␣ Hd x, h dpA dy d x␣ dH d x. FL ⌰ ϭ⌰ FL ⍀ ϭ⍀ Of course * h L and * h L . 1 ⍀ The Hamiltonian problem associated with the Hamiltonian system (J *E, h) consists in ﬁnding sections ␺෈⌫(M,J1*E), which are characterized by the condition

␺ ͒⍀ ϭ ෈ 1 ͒ *i͑X h 0, for every X X͑J *E . ␺ ϭ ␣ A ␣ In natural coordinates, if (x) (x ,y (x),pA(x)), this condition leads to the so-called Hamilton–De Donder–Weyl equations ͑for the section ␺͒. The problem of ﬁnding these sections can be formulated equivalently as follows: ﬁnding a distribution D of T(J1*E) such that D is integrable ͑that is, involutive͒, m-dimensional, ¯␶1-transverse, and its integral manifolds are the sections solution to the above equations. This is equivalent to stating that the sections solution to the Hamiltonian problem are the integral sections of one of the following equivalent elements:

1 m 1 • A class of integrable and ¯␶ -transverse m-vector ﬁelds ͕XH͖ʚX (J *E) satisfying that ⍀ ϭ ෈ i(XH) h 0, for every XH ͕XH͖. ␶1 1 → ٌ ⍀ ϭ Ϫ ⍀ ٌ • An integrable connection H in ¯ :J *E M such that i( H) h (m 1) h . ⌿ 1 → 1 1 ⌿ ⍀ ϭ • An integrable jet ﬁeld H :J *E J J *E, such that i( H) h 0.

¯␶1-transverse and locally decomposable m-vector fields, orientable jet fields, and orientable connections, which are solutions of these equations, are called Hamilton–De Donder–Weyl (HDW) 1 ⍀ m-vector fields, jet fields, and connections for (J *E, h). Their local expressions in natural coordinates are

ץ ץ ץ m A ␩ XHϭ f ∧ ͩ ϩF ϩG ͪ , ␣ ␣ ␩ ץ A A ץ ␣ ץ ␣ϭ1 x y pA

⌿ ϭ͑ ␣ A ␣ A ␩ ͒ H x ,y ,pA ;F␣ ,GA␣ ,

ץ ץ ץ ␣ A ␯ , ͪ Hϭdx ͩ ϩF ϩGٌ ␣ ␣ ␯ ץ A A ץ ␣ ץ x y pA ෈ ϱ 1 A ␩ where f C (J *E) is a nonvanishing function, and the coefﬁcients F␣ , GA␣ are related by the system of linear equations

ץ ץ A H ␯ H F␣ϭ , G ␯ϭϪ . A ץ A ␣ ץ pA y ␺ ϭ ␣ A ϭ␺A ␣ ϭ␺␣ Now, if (x) (x ,y (x) (x),pA(x) A(x)) is an integral section of XH then

␣ ␺ץ Hץ ␺Aץ Hץ ϭ Aؠ␺ϭ Ϫ ͯ ϭ ␣ ؠ␺ϭ A ͯ ␣ F␣ ; GA␣ ␣ , ץ A ץ ␣xץ ץ pA ␺ y ␺ x

which are the Hamilton–De Donder–Weyl equations for ␺. As above, a representative of the class 1 ͕XH͖ can be selected by the condition i(XH)(¯␶ *␻)ϭ1, which leads to f ϭ1 in the above local expression. The existence of classes of HDW m-vector fields, jet fields, and connections is assured. 1 In an analogous way, if (J E,⍀L) is an almost-regular Lagrangian system, the submanifold P 1 ␮ P˜ M␲ : J *E, is a fiber bundle over E and M. In this case the -transverse submanifold is diffeomorphic to P. This diffeomorphism is denoted by ␮˜ :P˜ →P, and it is just the restriction of ␮ P˜ ˜ ˜ؠ␮˜ Ϫ1 the projection to . Then, taking the Hamiltonian section hª , we define the Hamilton– Cartan forms

⌰0ϭ˜ ⌰ ⍀0ϭ˜ ⍀ h h* ; h h* , FL ⌰0ϭ⌰ FL ⌰0ϭ⍀ ͑ FL FL which verify that 0* h L and 0* h L where 0 is the restriction map of onto P͒ P ⍀0 . Then ( , h) is the Hamiltonian system associated with the almost-regular Lagrangian system 1 (J E,⍀L), and we have Diagram 1.

͑5͒ P ⍀0 Then, the Hamiltonian problem associated with the Hamiltonian system ( , h), and the equations for the sections of ⌫(M,P) solution to the Hamiltonian problem are stated as in the regular case. Now, the existence of the corresponding Hamilton–De Donder–Weyl m-vector ﬁelds, jet P ⍀0 P ﬁelds, and connections for ( , h) is not assured, except perhaps on some submanifold P of , where the solution is not unique. From now on we will consider only regular or almost-regular systems.

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III. UNIFIED FORMALISM

A. Extended Hamiltonian system

Given a ﬁber bundle ␲:E→M over an oriented manifold (M,␻), we deﬁne the extended W W jet-multimomentum bundle and the restricted jet-multimomentum bundle r as W 1 ϫ M␲ W 1 ϫ 1 ªJ E E , rªJ E EJ *E, ␣ A A ␣ ␣ A A ␣ whose natural coordinates are (x ,y ,v␣ ,pA ,p) and (x ,y ,v␣ ,pA), respectively. We have the natural projections ͑submersions͒

␳ W→ 1 ␳ W→M␲ ␳ W→ ␳ W→ 1 : J E, 2 : , E : E, M : M, ͑6͒ ␳r W → 1 ␳r W → 1 ␳r W → ␳r W → 1 : r J E, 2 : r J *E, E : r E, M : r M. ␲1ؠ␳ ϭ␶1ؠ␮ؠ␳ ϭ␳ Note that 1 2 E . In addition, there is also the natural projection ␮ W→W W : r ,

.y,͓p͔͒¯͑ۋy,p͒¯͑

The bundle W is endowed with the following canonical structures: Definition 1: ͑ ͒ W C ␳ 1 The coupling m-form in , denoted by , is an m-form along M which is defined as follows: ෈ 1 ␲1 ϭ␲ ϭ ෈ ෈M ␲ ϵ ෈W for every¯ y JyE, with ¯ (¯y) (y) x E, and p y , let w (¯y,p) y , then C͑ ͒ ͑ ␾͒ w ª Tx *p, where ␾:M→E satisfies that j1␾(x)ϭ¯y. Cˆ෈⍀m W ␳ C Then, we denote by ( ) the M-semibasic form associated with . ͑ ͒ ⌰ ෈⍀m W ⌰ ␳*⌰ 2 The canonical m-form W ( ) is defined by Wª 2 , and it is therefore ␳ E-semibasic. ϩ ⍀ Ϫ ⌰ ϭ␳*⍀ The canonical (m 1)-form is the pre-multisymplectic form Wª d W 1 ෈⍀mϩ1(W).

Cˆ ␳ ˆ ෈ ϱ W Cˆϭ ˆ ␳ ␻ ⍀ Being a M-semibasic form, there is C C ( ) such that C( *M ). Note also that W ␳ W ⍀ is not one-nondegenerate, its kernel being the 2-vertical vectors; then, we call ( , W) a pre- multisymplectic structure. This deﬁnition of the coupling form is in fact an alternative ͑obviously equivalent͒ presentation of the extended multimomentum bundle as the set of afﬁne maps from the jet bundle J1E to ␲-basic m-forms. The local expressions for ⌰W and ⍀W are the same as ͑4͒, and for Cˆ we have

Cˆ͑ ͒ϭ͑ ϩ ␣ A͒ m w p pAv␣ d x. L෈⍀m 1 Lˆ ␳*L෈⍀m W Given a Lagrangian density (J E), we denote ª 1 ( ), and we can write Lˆ ϭ ˆ ␳ ␻ ˆ ϭ␳ ෈ ϱ W L( *M ), with L 1*L C ( ). We deﬁne a Hamiltonian submanifold W ͕ ෈W͉Lˆ ͑ ͒ϭCˆ͑ ͖͒ 0ª w w w . W W ˆ Ϫ ˆ ϭ So, 0 is the submanifold of deﬁned by the constraint function C L 0. In local coordinates this constraint function is

ϩ ␣ AϪ ˆ ͑ ␯ B B͒ϭ p pAv␣ L x ,y ,v␯ 0.

W W ͑ ͒ We have the natural imbedding 0 : 0 , as well as the projections submersions ␳0 W → 1 ␳0 W →M␲ ␳0 W → ␳0 W → 1: 0 J E, 2: 0 , E: 0 E, M: 0 M,

W ͑ ͒ ␳0ϭ␮ؠ␳0 W → 1 which are the restrictions to 0 of the projections 6 , and ˆ 2 2: 0 J *E. So we have Diagram 2.

W ␣ A A ␣ Local coordinates in 0 are (x ,y ,v␣ ,pA), and we have that

␳0͑ ␣ A A ␣͒ϭ͑ ␣ A A͒ 1 x ,y ,v␣ ,pA x ,y ,v␣ ,

͑ ␣ A A ␣͒ϭ͑ ␣ A A ␣ Ϫ A ␣͒ 0 x ,y ,v␣ ,pA x ,y ,v␣ ,pA ,L v␣pA ,

␳0͑ ␣ A A ␣͒ϭ͑ ␣ A ␣ Ϫ A ␣͒ 2 x ,y ,v␣ ,pA x ,y ,pA ,L v␣pA ,

␳0͑ ␣ A A ␣͒ϭ͑ ␣ A ␣͒ ˆ 2 x ,y ,v␣ ,pA x ,y ,pA . W ␮ W Proposition 1: 0 is a one-codimensional W-transversal submanifold of , diffeomorphic W to r .

͒ ͒෈W ͒ϵ ˆ ͒ϭ ˆ ͒ ͑Proof For every ͑¯y,p 0 , we have L͑¯y L͑¯y,p C͑¯y,p ,

and

͒ ␮ ؠ ͒ ͒ϭ␮ ͒ϭ ␮ ͒͒ϭ ͑ W 0 ͑¯y,p W͑¯y,p ͑¯y, ͑p ͑¯y,͓p͔ . ␮ ؠ ෈W First, W 0 is injective: let (¯y 1 ,p1), (¯y 2 ,p2) 0 , then we have

͒ ␮ ؠ ͒ ͒ϭ ␮ ؠ ͒ ͒⇒ ␮ ͒͒ϭ ␮ ͒͒⇒ ϭ ␮ ͒ϭ␮ ͑ W 0 ͑¯y 1 ,p1 ͑ W 0 ͑¯y 2 ,p2 ͑¯y 1 , ͑p1 ͑¯y 2 , ͑p2 ¯y 1 ¯y 2 , ͑p1 ͑p2 ,

hence,

͒ϭ ͒ϭ ˆ ͒ϭ ˆ ͒ L͑¯y 1 L͑¯y 2 C͑¯y 1 ,p1 C͑¯y 2 ,p2 .

In a local chart, third equality gives

͑ ͒ϩ ␣͑ ͒ A͑ ͒ϭ ͑ ͒ϩ ␣͑ ͒ A͑ ͒ p p1 pA p1 v␣ ¯y 1 p p2 pA p2 v␣ ¯y 2 , ␮ ϭ␮ but (p1) (p2) implies that

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␣͑ ͒ϭ ␣͓͑ ͔͒ϭ ␣͓͑ ͔͒ϭ ␣͑ ͒ pA p1 pA p1 pA p2 pA p2 , ϭ ϭ therefore, p(p1) p(p2), and hence, p1 p2 . ͔ ␮ ؠ ෈W ෈ W ͓ ͔ϭ͓ Second, W 0 is onto: Let (¯y,p) r , then there exists (¯y,q) 0( 0) such that q p . ͓ ͔ 1 ϫ M␲ϭW In fact, it suffices to take q in such a way that, in a local chart of J E E ␣͑ ͒ϭ ␣͓͑ ͔͒ ͑ ͒ϭ ␣͓͑ ͔͒ A͑ ͒Ϫ ͑ ͒ pA q pA p , p q pA p v␣ ¯y L ¯y . W ˆ Ϫ ˆ ␮ Finally, observe that 0 is defined by the constraint function L C and, as ker W * Ϫ ˆ ϭ W W ˆ ץ ץ ץ ץ ϭ ͕ / p͖ and / p(L C) 1, then 0 is a 1-codimensional submanifold of and ␮W-transversal. ᭿ W ˆ W →W As a consequence of this property, the submanifold 0 induces a section h: r of the projection ␮W . Locally, hˆ is specified by giving the local Hamiltonian function Hˆ ϭϪLˆ ϩ ␣ A ˆ ␣ A A ␣ ϭ ␣ A A ␣ Ϫ ˆ ˆ pAv␣ ; that is, h(x ,y ,v␣ ,pA) (x ,y ,v␣ ,pA , H). In this sense, h is said to be a Hamil- tonian section of ␮W . ␮ ˆ W →W Remark:It is important to point out that, from every Hamiltonian W-section h: r in the extended unified formalism, we can recover a Hamiltonian ␮-section ˜h:P→M␲ in the standard Hamiltonian formalism. In fact, given ͓p͔෈J1*E, the section hˆ maps every point (¯y,͓p͔) ෈ ␳r Ϫ1 ͓ ͔ ␳Ϫ1͓␳ ˆ ͓ ͔ ͔ ( 2) ( p ) into 2 2(h(¯y, p )) . So, the crucial point is the projectability of the local A ␳ ˆ ␳ ץ ץ ␳ ˆ function H by 2 . But, being / v␣ a local basis for ker 2 , H is 2-projectable if, and only if, ෈P*ϭ FLʚ 1͔ ͓ ␣ ץ ץϭ␣ pA L/ vA , and this condition is fulfilled when p Im J *E, which implies that ␳ ͓ ˆ ␳r Ϫ1 ͓ ͔ ෈P˜ ϭ FLʚM␲ ˜ 2 h( 2) )( p ))] Im . Hence, the Hamiltonian section h is defined as follows:

ϭ͑␳ ؠ ˆ ͓͒͑␳r ͒Ϫ1͑ ͓͑ ͔͔͒͒ ͓ ͔෈P͔͒ ͓͑¯ h p 2 h 2  p , for every p .

So we have Diagram 3 ͑see also Diagram 1͒.

͑For ͑hyper͒ regular systems this diagram is the same with Im FLϭJ1*E.) Finally, we can deﬁne the forms

⌰ *⌰ ϭ␳0*⌰෈⍀m͑W ͒ ⍀ *⍀ ϭ␳0*⍀෈⍀mϩ1͑W ͒ 0ª j0 W 2 0 , 0ª j0 W 2 0 , with local expressions

⌰ ϭ͑ Ϫ ␣ A͒ m ϩ ␣ A∧ mϪ1 0 L pAv␣ d x pAdy d x␣ , ͑7͒ ⍀ ϭ ͑ ␣ AϪ ͒∧ m Ϫ ␣∧ A∧ mϪ1 0 d pAv␣ L d x dpA dy d x␣ , ͑ ͒ W ⍀ and we have obtained a pre-multisymplectic Hamiltonian system ( 0 , 0), or equivalently W ˆ ⍀ ( r ,h* W).

B. Field equations for sections W ⍀ The Lagrange-Hamiltonian problem associated with the system ( 0 , 0) consists in ﬁnding ␺ ෈⌫ W sections 0 (M, 0) which are characterized by the condition ␺ ͑ ͒⍀ ϭ ෈ ͑W ͒ ͑ ͒ 0*i Y 0 0 0, for every Y 0 X 0 . 8

This equation gives different kinds of information, depending on the type of the vector ﬁelds Y 0 ␳0 involved. In particular, using vector ﬁelds Y 0 which are ˆ 2-vertical, we have:

␳ˆ 0 ෈ V( 2) W ␳0 ⍀ ␳0 Lemma 1: If Y 0 X ( 0) (i.e., Y 0 is ˆ 2-vertical), then i(Y 0) 0 is M-semibasic. A v␣͖ as aץ/ץ͕ Proof͒ A simple calculation in coordinates leads to this result. In fact, taking͑ ␳0 ͑ ͒ local basis for the ˆ 2-vertical vector ﬁelds, and bearing in mind 7 we obtain

ץ ץ ␣ L iͩ ͪ ⍀ ϭͩ p Ϫ ͪ dmx, A 0 A A ␣vץ ␣vץ

␳0 ᭿ which are obviously M-semibasic forms. ෈ V(␳ˆ 0) W ͑ ͒ As an immediate consequence, when Y 0 X 2 ( 0), condition 8 does not depend on the ␺ ͑ ͒ derivatives of 0 : is a pointwise algebraic condition. We can deﬁne the submanifold W ϭ͕͑ ͒෈W ͉ ͑ ͒͑⍀ ͒ ϭ ෈ ͑␳0͖͒ 1 ¯y,p 0 i V0 0 ͑¯y,p͒ 0, for every V0 V ˆ 2 ,

which is called the first constraint submanifold of the Hamiltonian pre-multisymplectic system W ⍀ ␺ ͑ ͒ W W W ( 0 , 0), as every section 0 solution to 8 must take values in 1 . We denote by 1 : 1 0 the natural embedding. A ץ ץW W ␣ϭ Locally, 1 is defined in 0 by the constraints pA L/ v␣ . Moreover: W gFL W ϭ gFL ෈W͉ ෈ 1 Proposition 2: 1 is the graph of ; that is, 1 ͕(¯y, (¯y)) ¯y J E͖. ͑Proof͒ Consider ¯y෈J1E, let ␾:M→E be a representative of ¯y, and pϭgFL(¯y). For every ෈ ϭ ␾ ¯ ϭ 1␾ U T␲¯ 1(¯y)M, consider V T␲¯ 1(¯y) (U) and its canonical lifting V T␲¯ 1(¯y) j (U). From the defi- ͑ ͒ ␲ gFL ϭ ⌰ nition of the extended Legendre map 3 we have that (T¯y )*( (¯y)) ( L)¯y , then

¯ ͒ ␲1͒ gFL ͒͒ ϭ ¯ ͒ ⌰ ͒ i͑V ͓͑T¯y *͑ ͑¯y ͔ i͑V ͑ L ¯y .

Furthermore, as pϭgFL(¯y), we also have that

¯ ͒ ␲1͒ gFL ͒͒ ϭ 1␾ ͒͒ ␲1͒ ͒ i͑V ͓͑T¯y *͑ ͑¯y ͔ i͑T␲¯ 1͑¯y͒ j ͑U ͓͑T¯y *p ϭ 1␾ ͒ ͒ ϭ ␾ ͒͒ ϭ ͒ i͑T␲1͑¯y͓͒͑T␲¯ 1͑¯y͒ j ͑U ͔ p i͑T␲¯ 1͑¯y͒ ͑U p i͑V p.

Therefore, we obtain

͒ ␾ ͒ϭ ͒ 1␾͒ ⌰ ͒ i͑U ͑ *p i͑U ͓͑j *͑ L ¯y͔,

and bearing in mind the deﬁnition of the coupling form C, this condition becomes

͒ C ͒͒ϭ ͒ 1␾͒ ⌰ ͒ i͑U ͑ ͑¯y,p i͑U ͓͑j * L ¯y]. ෈ C ϭ 1␾ ⌰ Since it holds for every U T␲¯ 1(¯y)M, we conclude that (¯y,p) ͓( j )* L͔¯y , or equivalently, Cˆ(¯y,p)ϭLˆ (¯y,p), where we have made use of the fact that ⌰L is the sum of the Lagrangian density L and a contact form i(V͒L ͑vanishing by pull-back of lifted sections͒. This is the con- W gFL ෈W ෈ 1 dition defining 0 , and thus we have proved that (¯y, (¯y)) 0 , for every ¯y J E; that is, FLgʚW FL W W graph 0 . Furthermore, graph and 1 are defined as subsets of 0 by the same local Aϭ gFLϭW ᭿ ץ ץϪ␣ conditions: pA L/ v␣ 0. So we conclude that graph 1 . W FL 1 ␺ →W Being 1 the graph of , it is diffeomorphic to J E. Every section 0 :M 0 is of the ␺ ϭ ␺ ␺ ␺ ϭ␳0ؠ␺ → 1 ␺ W ␺ ϭgFL form 0 ( L , H), with L 1 0 :M J E, and if 0 takes values in 1 then H ؠ␺L . In this way, every constraint, differential equation, etc., in the unified formalism can be translated to the Lagrangian or the Hamiltonian formalisms by restriction to the first or the second factors of the product bundle.

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͑ ͒ W However, as was pointed out before, the geometric condition 8 in 0 , which can be solved ␺ →W ʚW ␺ ⍀ ϭ ͓ only for sections 0 :M 1 0 , is stronger than the Lagrangian condition *L i(Z) L 0, for ෈ 1 1 W every Z X(J E)] in J E, which can be translated to 1 by the natural diffeomorphism between 0 them. The reason is that TW W ϭTW VW (␳ ), so the additional information comes therefore 1 0 1 1 1 ␳0 from the 1-vertical vectors, and it is just the holonomic condition. In fact: ␺ →W ␺ ϭ ␺ ␺ ϭ ␺ gFLؠ␺ Theorem 1: Let 0 :M 0 be a section fulﬁlling Eq. (8), 0 ( L , H) ( L , L), ␺ ϭ␳0ؠ␺ where L 1 0 . Then: ␺ ␾ϭ␳0 ؠ␺ → ␺ (1) L is the canonical lift of the projected section E 0 :M E (that is, L is a holonomic section). 1 The section ␺Lϭ j ␾ is a solution to the Lagrangian problem, and the section ␮ؠ␺H (2) g 1 .ϭ␮ؠFLؠ␺LϭFLؠ j ␾ is a solution to the Hamiltonian problem

Conversely, for every section ␾:M→E such that j1␾ is solutions to the Lagrangian problem FLؠ 1␾ ␺ (and hence j is solution to the Hamiltonian problem) we have that the section 0 .ϭ( j1␾,gFLؠ j1␾), is a solution to (8) ͑see Diagram 4͒

͑Proof ͒ ␳0 ͖␣ ץ ץ͕ ͒ ͑ 1 Taking / pA as a local basis for the 1-vertical vector ﬁelds:

ץ A m A mϪ1 iͩ ͪ ⍀ ϭv d xϪdy ∧d x␣ , ␣ 0 ␣ ץ pA

␺ so that for a section 0 , we have y Aץ ץ 0ϭ␺*ͫ iͩ ͪ ⍀ ͬϭͩ vA͑x͒Ϫ ͪ dmx, ␣ ץ ␣ 0 ␣ ץ 0 pA x

and thus the holonomy condition appears naturally within the uniﬁed formalism, and it is not A ␺ ץ ץ ␣ ץ A ץ ␺ ␺ ϭ ␣ A necessary to impose it by hand to 0 . Thus, we have that 0 (x ,y , y / x , L/ v␣), since 0 W ␺ ϭ 1␾ gFLؠ 1␾ ␾ϭ ␣ A ϭ␳0 ؠ␺ takes values in 1 , and hence, it is of the form 0 ( j , j ), for (x ,y ) E 0 .

͑ ͒ ␺ →W ͑ ͒ W 2 Since sections 0 :M 0 solution to 8 take values in 1 , we can identify them with ␺ →W ␺ ␺ ⍀ ϭ sections 1 :M 1 . These sections 1 verify, in particular, that 1*i(Y 1) 1 0 holds for every ෈ W ␺ ϭ ؠ␺ W gFL ␳1ϭ␳0 Y 1 X( 1). Obviously, 0 1 1 . Moreover, as 1 is the graph of , denoting by 1 1 ؠ W → 1 W 1 ⍀ ϭ ⍀ 1 : 1 J E the diffeomorphism which identiﬁes 1 with J E, if we deﬁne 1 1* 0 ,we ⍀ ϭ␳1 ⍀ ␳1 Ϫ1 ϭ gFL ෈ 1 ␳2ؠ have that 1 1* L . In fact; as ( 1) (¯y) (¯y, (¯y)), for every ¯y J E, then ( 0 1 ؠ ␳1 Ϫ1 ϭgFL ෈M␲ ( 1) )(¯y) (¯y) , and hence,

⍀ ϭ͑␳2ؠ ؠ͑␳1͒Ϫ1͒ ⍀ϭ͓͑͑␳1͒Ϫ1͒ ؠ ؠ␳2 ͔⍀ϭ͓͑͑␳1͒Ϫ1͒ ؠ ͔⍀ ϭ͑͑␳1͒Ϫ1͒ ⍀ L 0 1 1 * 1 * 1* 0* 1 * 1* 0 1 * 1 .

Now, let X෈X(J1E). We have

1␾͒ ͑ ͒⍀ ϭ͑␳0ؠ␺ ͒ ͑ ͒⍀ ϭ͑␳0ؠ ؠ␺ ͒ ͑ ͒⍀ ͑ j *i X L 1 0 *i X L 1 1 1 *i X L

1 1 Ϫ1 1 ϭ͑␳ ؠ␺ ͒*i͑X͒⍀Lϭ␺*i͑͑␳ ͒ X͒͑␳ *⍀L͒ϭ␺*i͑Y ͒⍀ ͑9͒ 1 1 1 1 * 1 1 1 1

ϭ␺ ͑ ͒͑ ⍀ ͒ϭ͑␺ ؠ ͒ ͑ ͒⍀ ϭ␺ ͑ ͒⍀ 1*i Y 1 1* 0 1* 1* i Y 0 0 0*i Y 0 0 ,

෈ W ϭ ␺ ⍀ ϭ ෈ W where Y 0 X( 0) is such that Y 0 1 Y 1 . But as 0*i(Y 0) 0 0, for every Y 0 X( 0), then 1 * 1 we conclude that ( j ␾)*i(X)⍀Lϭ0, for every X෈X(J E). 1 1 1 1 Conversely, let j ␾:M→J E such that ( j ␾)*i(X)⍀Lϭ0, for every X෈X(J E), and deﬁne ␺ →W ␺ 1␾ gFLؠ 1␾ ͑ ␺ W 0 :M 0 as 0ª( j , j ) observe that 0 takes its values in 1). Taking into account W ෈ W ϭ 1ϩ 2 1෈ W that, on the points of 1 , every Y 0 X( 0) splits into Y 0 Y 0 Y 0, with Y 0 X( 0) tangent to ␳0 W 2෈ V( 1) W 1 , and Y 0 X ( 0), we have that

␺ ͑ ͒⍀ ϭ␺ ͑ 1͒⍀ ϩ␺ ͑ 2͒⍀ ϭ 0*i Y 0 0 0*i Y 0 0 0*i Y 0 0 0,

1 ͑ ͒ because for Y 0, the same reasoning as in 9 leads to

␺ ͑ 1͒⍀ ϭ͑ 1␾͒ ͑ 1͒⍀ ϭ 0*i Y 0 0 j *i X0 L 0

Ϫ ͓where X1ϭ(␳1) 1Y 1] and for Y 2, following also the same reasoning as in ͑9͒, a local calculus 0 1 * 0 0 gives y Aץ ␺* ͑ 2͒⍀ ϭ͑ 1␾͒*ͫͩ ␣͑ ͒ͩ AϪ ͪͪ m ͬϭ 0 i Y 0 0 j f A x v␣ d x 0, ␣xץ

since j1␾ is a holonomic section. -The result for the sections FLؠ j1␾ is a direct consequence of the equivalence theorem be tween the Lagrangian and Hamiltonian formalisms ͑see, for instance, Refs. 12 and 31͒. ᭿ Remark: The results in this section can also be recovered in coordinates taking an arbitrary ෈ W ␣ ץ ץ ␣ A ϩ ץ ץ A ϩ A ץ ץ ϭ A local vector ﬁeld Y 0 f ( / y ) g␣( / v␣) hA( / pA) X( 0), then

A͒͒ m ץ ץA͒ m ϩ A ␣∧ mϪ1 ϩ A͑ ␣Ϫ͑ ץ ץ⍀ ϭϪ A͑͒ ͑ i Y 0 0 f L/ y d x f dpA d x␣ g␣ pA L/ v␣ d x ϩ ␣ A m Ϫ ␣ A∧ mϪ1 hAv␣d x hAdy d x␣

␺ ͑ ͒ and, for a section 0 fulﬁlling 8 ,

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A ץ ץ ץ ␣ ץ pA L ␣ L ␣ y 0ϭ␺*i͑Y ͒⍀ ϭͫ f Aͩ Ϫ ͪ ϩgAͩ p Ϫ ͪ ϩh ͩ vAϪ ͪͬdmx ␣ ␣ 0 0 0 ␣ A A A A ␣ xץ ␣vץ yץ xץ

reproduces the Euler–Lagrange equations, the restricted Legendre map ͑that is, the definition of the momenta͒, and the holonomy condition. Summarizing, Eq. ͑8͒ gives different kinds of information, depending on the type of vertical- lity of the vector fields Y 0 involved. In particular, we have obtained equations of three different classes: ͑ ͒ ͑ ͒ W W 1 Algebraic not differential equations, determining a subset 1 of 0 , where the sections solution must take their values. These can be called primary Hamiltonian constraints, and in ␳0 fact they generate, by ˆ 2 projection, the primary constraints of the Hamiltonian formalism for singular Lagrangians, i.e., the image of the Legendre transformation, FL(J1E)ʚJ1*E. ͑ ͒ ␺ 2 The holonomic differential equations, forcing the sections solution 0 to be lifting of ␲-sections. This property is similar to the one in the unified formalism of Classical Mechanics, and it reflects the fact that the geometric condition in the unified formalism is stronger than the usual one in the Lagrangian formalism. ͑3͒ The classical Euler–Lagrange equations.

C. Field equations for m-vector fields, connections, and jet fields The problem of finding sections solution to ͑8͒ can be formulated equivalently as follows: W ͑ ͒ finding a distribution D0 of T( 0) such that it is integrable that is, involutive , m-dimensional, ␳0 M-transverse, and the integral manifolds of D0 are the sections solution to the above equations. ͑Note that we do not ask them to be lifting of ␲-sections; that is, the holonomic condition.͒ This is equivalent to stating that the sections solution to this problem are the integral sections of one of the following equivalent elements: ␳0 ͕ ͖ʚ m W • A class of integrable and M-transverse m-vector fields X0 X ( 0) satisfying that ͒⍀ ϭ ෈ ͑ ͒ i͑X0 0 0, for every X0 ͕X0͖. 10 → ␳0 W ٌ • An integrable connection 0 in M: 0 M such that ͒ ͑ ⍀ ϭ Ϫ ͒⍀͒ ٌ i͑ 0 0 ͑m 1 0 . 11 ⌿ W → 1W • An integrable jet field 0 : 0 J 0 , such that ⌿ ͒⍀ ϭ ͑ ͒ i͑ 0 0 0. 12

␳0 Locally decomposable and M-transverse m-vector fields, orientable jet fields, and orientable connections, which are solutions of these equations will be called Lagrange–Hamiltonian W ⍀ m-vector fields, jet fields, and connections for ( 0 , 0). W Recall that, in a natural chart in 0 , the local expressions of a connection form, its associated jet field, and the m-multivector fields of the corresponding associated class are

ץ ץ ץ ץ ␯ , ͪ ϭdx␣ ͩ ϩFA ϩGA ϩH ٌ ␣ ␣␯ ␣ ␯ ץA A ץ A ץ ␣ ץ 0 x y v␯ pA

⌿ ϭ͑ ␣ A A A A ␯ ͒ ͑ ͒ 0 x ,y ,v␣ ,F␣ ,G␣␩ ,H␣A , 13

ץ ץ ץ ץ m ␯ X ϭ f ∧ ͩ ϩFA ϩGA ϩH ͪ , ␣ ␣␯ ␣ ␯ ץA A ץ A ץ ␣ ץ 0 ␣ϭ1 x y v␯ pA

where f ෈Cϱ(J1E) is an arbitrary nonvanishing function. A representative of the class ͕X͖ can be ␳0 ␻ ϭ ϭ selected by the condition i(X)(¯ M* ) 1, which leads to f 1 in the above local expression. Now, the equivalence of the unified formalism with the Lagrangian and Hamiltonian formalisms can be recovered as follows: W Theorem 2: Let ͕X0͖ be a class of integrable Lagrange–Hamiltonian m-vector fields in 0 , m 0 whose elements X :W →⌳ TW are solutions to (10), and let ٌ :W →␳ *T*M W TW be 0 0 0 0 0 M 0 0 its associated Lagrange–Hamiltonian connection form [which is a solution to (11)], and ⌿ W → 1W 0 : 0 J 1 its associated Lagrange–Hamiltonian jet field [which is a solution to (12)]. ෈ 1 →⌳m 1 (1) For every X0 ͕X0͖, the m-vector field XL :J E TJ E defined by

ؠ␳0ϭ⌳m ␳0ؠ XL 1 T 1 X0 ,

1 is a holonomic Euler–Lagrange m-vector field for the Lagrangian system (J E,⍀L) (where ⌳m ␳0 ⌳m W →⌳m 1 ␳0 T 1: T 0 TJ E is the natural extension of T 1). Conversely, every holonomic Euler–Lagrange m-vector field for the Lagrangian system 1 (J E,⍀L) can be recovered in this way from an integrable Lagrange–Hamiltonian m-vector field m X ෈XW (W ). 0 1 0

␲1 1→ 1 ٌ (2) The Ehresmann connection form L :J E ¯ *T*M J1ETJ E deﬁned by

0 , Lؠ␳ ϭ␬W ؠٌٌ 1 0 0

1 is a holonomic Euler–Lagrange connection form for the Lagrangian system (J E,⍀L) (where ␬W is deﬁned as the map making the following diagram commutative) ͑see Diagram 5͒. 0

Conversely, every holonomic Euler–Lagrange connection form for the Lagrangian system 1 (J E,⍀L) can be recovered in this way from an integrable Lagrange–Hamiltonian connection ٌ form 0 . 1 1 1 (3) The jet ﬁeld ⌿L :J E→J J E deﬁned by

⌿ ؠ␳0ϭ 1␳0ؠ⌿ L 1 j 1 0 ,

1 is a holonomic Euler–Lagrange jet field for the Lagrangian system (J E,⍀L). Conversely, every 1 holonomic Euler–Lagrange jet field for the Lagrangian system (J E,⍀L) can be recovered in this ⌿ way from an integrable Lagrange–Hamiltonian jet field 0 . ͑ ͒ ␳0 W ͑ ͒ Proof Let X0 be a M-transversal m-vector field on 0 solution to 10 . As sections ␺ →W ͑ ͒ W 0 :M 0 solution to the geometric equation 8 must take value in 1 , then X0 can be m m identified with a m-vector field X :W →⌳ TW ͑i.e., ⌳ T ؠX ϭX ͉W ), and hence, there 1 0 1 1 1 0 1 m 1 ϭ⌳m ␳1 Ϫ1ؠ ෈ m W⌳→ 1 exists XL :J E TJ E such that X1 T( 1) XL X ( 1). Therefore as a consequence ͑ ͒ ␺ ͑ ͒ 0 ෈ m 1␾ of item 1 in theorem 1, for every section 0 solution to 8 , there exists XL X ( j (M)) such m ؠ 0 ϭ ͉ 1␾→ ␲1⌳ that T␾ XL XL j1␾(M) , where ␾ : j E is the natural imbedding. So, XL is ¯ -transversal ⍀ ϭ␳1 ⍀ and holonomic. Then, bearing in mind that 1* 0 1* L , we have

͑ ͒⍀ ϭ ͑ ͒͑ ⍀ ͒ϭ ͑ ͒͑␳1 ⍀ ͒ϭ␳1 ͑ ͒⍀ 1*i X0 0 i X1 1* 0 i X1 1* L 1*i XL L ,

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⍀ ϭ ⇒ ⍀ ϭ then i(X0) 0 0 i(XL) L 0. Conversely, given an holonomic Euler–Lagrange m-vector field XL , from i(XL)⍀Lϭ0, and ⍀ ෈ W 0 ͓ taking into account the above chain of equalities, we obtain that i(X0) 0 ͓X( 1)͔ the anni- W hilator of X( 1)]. Moreover, being XL holonomic, X0 is holonomic, and then the extra condition V(␳0) i(Y )i(X )⍀ ϭ0 is also fulfilled for every Y ෈X 1 (W ). Thus, remembering that TW W 0 0 0 0 0 1 0 0 ϭTW VW (␳ ), we conclude that i(X )⍀ ϭ0. 1 1 1 0 0 The proof for Ehresmann connections and jet fields is straightforward, taking into account that they are equivalent alternative descriptions in the Lagrangian formalism. ᭿ This statement also holds for nonintegrable classes of m-vector fields, connections, and jet W fields in 0 , but now the corresponding classes of Euler–Lagrange m-vector fields, connections and jet fields in J1E will not be holonomic ͑but only semi-holonomic͒. To prove this assertion it suffices to compute Eq. ͑10͒ in coordinates, using the local expressions ͑7͒ and ͑13͒, concluding A A then that, in the expressions ͑13͒, F␣ϭv␣ , which is the local expression of the semi-holonomy condition ͑see, also, Ref. 9͒. Finally, the Hamiltonian formalism is recovered in the usual way, by using the following: 1 ⍀ Theorem 3: Let (J *E, h) be the Hamiltonian system associated with a (hyper) regular 1 Lagrangian system (J E,⍀L).

m 1 m 1 (1) (Equivalence theorem for m-vector ﬁelds͒ Let XL෈X (J E) and XH෈X (J *E) be the m-vector ﬁelds solution to the Lagrangian and the Hamiltonian problems respectively. Then

m ,TFLؠXLϭ fXHؠFL ⌳

ϱ 1 for some f ෈C (J *E) (we say that the classes ͕XL͖ and ͕XH͖ are FL-related).

(2) (Equivalence theorem for jet fields and connections) Let YL and YH be the jet fields solution of the Lagrangian and the Hamiltonian problems respectively. Then 1 j FLؠYLϭYHؠFL (we say that the jet fields YL and YH are FL-related). As a consequence, their associated .connection forms, ٌL and ٌH respectively, are FL-related, too (For almost-regular systems the statement is the same, but changing J1*E for P͒. ͑Proof͒ See Ref. 31. ͑The proof for the almost-regular case follows in a straight-forward way.͒ ᭿ As a consequence of these latter theorems, similar comments to those made at the end of Secs. II A and II B about the existence, integrability, and nonuniqueness of Euler–Lagrange and Hamilton–de Donder–Weyl m-vector fields, connections, and jet fields, can be applied to their associated elements in the unified formalism. In particular, for singular systems, the existence of S W these solutions is not assured, except perhaps on some submanifold 1 , and the number of arbitrary functions which appear depends on the dimension of S and the rank of the Hessian matrix of L ͑an algorithm for finding this submanifold is outlined in Ref. 9͒. The integrability of these solutions is not assured ͑even in the regular case͒, except perhaps on a smaller submanifold I S I such that the integral sections are contained in .

3 IV. EXAMPLE: MINIMAL SURFACES „in R … ͓In Ref. 9 we ﬁnd another interesting example, the bosonic string ͑which is a singular model͒, described in this uniﬁed formalism.͔

A. Statement of the problem: Geometric elements

The problem consists in looking for mappings ␸:UʚR2→ such that their graphs have minimal area as sets of R3, and satisfy certain boundary conditions.

For this model, we have that MϭR2, EϭR2ϫR, and

J1Eϭ␲*T*R2 Rϭ␲*T*Mϭ␲*T*R2,

M␲ϭ␲ ϫ ͒ 1 ␲ ⌳2 ͒ *͑TM ME ͑afﬁne maps from J E to * T*M ,

J1*Eϭ␲*TMϭ␲*TR2 ͑classes of afﬁne maps from J1E to ␲*⌳2T*M ͒. 1 1 M␲ 1 2 1 2 1 2 The coordinates in J E, J *E and are denoted (x ,x ,y,v1 ,v2), (x ,x ,y,p ,p ), and (x1,x2,y,p1,p2,p), respectively. If ␻ϭdx1∧dx2, the Lagrangian density is

Lϭ ϩ ͒2ϩ ͒2 1/2 1∧ 2ϵ 1∧ 2 ͓1 ͑v1 ͑v2 ͔ dx dx Ldx dx ,

and the Poincare´–Cartan forms are

v v v 2 v 2 ⌰ ϭ 1 ∧ 2Ϫ 2 ∧ 1ϩ ͩ Ϫͩ 1 ͪ Ϫͩ 2 ͪ ͪ 1∧ 2 L dy dx dy dx L 1 dx dx , L L L L

v v v 2 v 2 ⍀ ϭϪ ͩ 1 ͪ ∧ ∧ 2ϩ ͩ 2 ͪ ∧ ∧ 1Ϫ ͫ ͩ Ϫͩ 1 ͪ Ϫͩ 2 ͪ ͪͬ∧ 1∧ 2 L d dy dx d dy dx d L 1 dx dx . L L L L

The Legendre maps are

v1 v2 FL͑x1,x2,y,v ,v ͒ϭͩ x1,x2,y, , ͪ , 1 2 L L

͒2 ͒2 v1 v2 ͑v1 ͑v2 gFL͑x1,x2,y,v ,v ͒ϭͩ x1,x2,y, , ,LϪ Ϫ ͪ , 1 2 L L L L

and then L is hyperregular. The Hamiltonian function is

HϭϪ͓1Ϫ͑p1͒2Ϫ͑p2͒2͔1/2. ͑14͒

So the Hamilton–Cartan forms are

⌰ ϭ 1 ∧ 2Ϫ 2 ∧ 1Ϫ 1∧ 2 h p dy dx p dy dx Hdx dx ,

⍀ ϭϪ 1∧ ∧ 2ϩ 2∧ ∧ 1ϩ ∧ 1∧ 2 h dp dy dx dp dy dx dH dx dx .

B. Uniﬁed formalism

For the uniﬁed formalism we have

Wϭ␲ ϫ ␲ ϫ ͒ W ϭ␲ ϫ ␲ ϭ␲ ϫ ͒ *T*M E *͑TM ME , r *T*M E *TM *͑T*M MTM .

ϭ 1 2 1 2 ͒෈W w ͑x ,x ,y,v1 ,v2 ,p ,p ,p ,

the coupling form is

Cˆϭ 1 ϩ 2 ϩ ͒ 1∧ 2 ͑p v1 p v2 p dx dx ,

therefore,

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W ϭ 1 2 1 2 ͒෈W͉ ϩ ͒2ϩ ͒2 1/2Ϫ 1 Ϫ 2 Ϫ ϭ 0 ͕͑x ,x ,y,v1 ,v2 ,p ,p ,p ͓1 ͑v1 ͑v2 ͔ p v1 p v2 p 0͖,

and we have the forms

⌰ ϭ ϩ ͒2ϩ ͒2 1/2Ϫ 1 Ϫ 2 ͒ 1∧ 2Ϫ 2 ∧ ϩ 1 ∧ 0 ͓͑1 ͑v1 ͑v2 ͔ p v1 p v2 dx dx p dy dx1 p dy dx2 ,

⍀ ϭϪ ϩ ͒2ϩ ͒2 1/2Ϫ 1 Ϫ 2 ͒∧ 1∧ 2ϩ 2∧ ∧ Ϫ 1∧ ∧ 0 d͓͑1 ͑v1 ͑v2 ͔ p v1 p v2 dx dx dp dy dx1 dp dy dx2 . ץ ץ ␳0 Taking ﬁrst ˆ 2-vertical vector ﬁelds / v␣ we obtain

␣v ץ ϭ ͩ ͪ ⍀ ϭͩ ␣Ϫ ͪ 1∧ 2 0 i 0 p dx dx , v␣ Lץ

W ϭ gFL ͑ 1 which determines the submanifold 1 graph diffeomorphic to J E), and reproduces the ␣ ץ ץ ␳0 expression of the Legendre map. Now, taking 1-vertical vector ﬁelds / p , the contraction ⍀ ␣ϭ ␣ ץ ץ i( / p ) 0 gives, for 1,2, respectively, 1∧ 2Ϫ ∧ 2 1∧ 2ϩ ∧ 1 v1dx dx dy dx , v2dx dx dy dx ,

so that, for a section

␺ ϭ 1 2 1 2͒ 1 2͒ 1 2͒ 1 1 2͒ 2 1 2͒͒ 0 ͑x ,x ,y͑x ,x ,v1͑x ,x ,v2͑x ,x ,p ͑x ,x ,p ͑x ,x , W taking values in 1 , we have that the condition

ץ ␺*ͫ ͩ ͪ ⍀ ͬϭ 0 i 0 0 ␣pץ

leads to yץ yץ ͩ Ϫ ͪ 1∧ 2ϭ ͩ Ϫ ͪ 1∧ 2ϭ v1 dx dx 0, v2 dx dx 0, x2ץ x1ץ y we haveץ/ץ which is the holonomy condition. Finally, taking the vector ﬁeld ץ iͩ ͪ ⍀ ϭϪdp2∧dx1ϩdp1∧dx2, y 0ץ ␺ and, for a section 0 fulﬁlling the former conditions, the equation ץ 0ϭ␺*ͫiͩ ͪ ⍀ ͬ, y 0ץ 0

leads to p1ץ p2ץ 0ϭͩ ϩ ͪ dx1∧dx2 x1ץ x2ץ ץ ץ v1 v2 ϭͫ ͩ ͪ ϩ ͩ ͪͬdx1∧dx2 x2 Lץ x1 Lץ

2 2 2yץ yץ yץ 2yץ yץ 2yץ yץ 1 ϭ ͫͩ 1ϩͩ ͪ ͪ ϩͩ 1ϩͩ ͪ ͪ Ϫ2 ͬdx1∧dx2, x2ץx1ץ x2ץ x1ץ x1ץx1ץ x2ץ x2ץx2ץ x1ץ L3

which gives the Euler–Lagrange equation of the problem. Now, bearing in mind ͑14͒, and the expression of the Legendre map, from the Euler–Lagrange equations we get p2ץ p1ץ y p2ץ y p1ץ ϭϪ , ϭϪ ; ϭϪ , x2ץ x1ץ x2 Hץ x1 Hץ

which are the Hamilton–De Donder–Weyl equations of the problem. The m-vector fields, connections and jet fields which are the solutions to the problem in the unified formalism are

ץ 2 ץ ץ 1 ץ ץ ץ ץ ץ ץ ץ v1 v2 p p X ϭ f ͩ ϩv ϩ ϩ ϩ ϩ ͪ 2 1 1 1 ץ 1 ץ 1 ץ1 1 0 pץ xץ pץ xץ x v2ץ x v1ץ x yץ

ץ 2 ץ ץ 1 ץ ץ ץ ץ ץ ץ ץ v1 v2 p p ∧ͩ ϩv ϩ ϩ ϩ ϩ ͪ , 2 2 1 2 ץ 2 ץ 2 ץ2 2 pץ xץ pץ xץ x v2ץ x v1ץ x yץ p2ץ p2ץ p1ץ p1ץ vץ vץ vץ vץ ⌿ ϭͩ 1 2 1 2 1 1 2 2 ͪ 0 x ,x ,y,p ,p ;v1 ,v2 , , , , , , , , , x2ץ x1ץ x2ץ x1ץ x2ץ x1ץ x2ץ x1ץ

ץ 2 ץ ץ 1 ץ ץ ץ ץ ץ ץ ץ v1 v2 p p ͪ ϭdx1 ͩ ϩv ϩ ϩ ϩ ϩ ٌ 2 1 1 1 ץ 1 ץ 1 ץ1 1 0 pץ xץ pץ xץ x v2ץ x v1ץ x yץ

ץ 2 ץ ץ 1 ץ ץ ץ ץ ץ ץ ץ v1 v2 p p ϩdx2 ͩ ϩv ϩ ϩ ϩ ϩ ͪ 2 2 1 2 ץ 2 ץ 2 ץ2 2 pץ xץ pץ xץ x v2ץ x v1ץ x yץ

␯ 2 ␯ ␣ x are related by theץ xץ/y ץx ϭץ/ ␣vץ f being a nonvanishing function͒, where the coefficients͑ –x␯ are related by the Hamilton–De Donderץ/␣pץ Euler–Lagrange equations, and the coefficients Weyl equations ͑the third one͒. Hence, the associated Euler–Lagrange m-vector fields, connections and jet fields which are the solutions to the Lagrangian problem are

ץ ץ ץ ץ ץ ץ ץ ץ ץ ץ ץ ץ v1 v2 v1 v2 XLϭ f ͩ ϩv ϩ ϩ ͪ ∧ͩ ϩv ϩ ϩ ͪ , ץ 2 ץ 2 ץ2 2 ץ 1 ץ 1 ץ1 1 x v2ץ x v1ץ x yץ x v2ץ x v1ץ x yץ vץ vץ vץ vץ 1 2 1 2 1 1 2 2 ⌿Lϭͩ ͪ x ,x ,y,p ,p ;v1 ,v2 , , , , , x2ץ x1ץ x2ץ x1ץ

ץ vץ ץ vץ ץ ץ ץ vץ ץ vץ ץ ץ 1 1 2 2 1 2 , ͪ Lϭdx ͩ ϩv ϩ ϩ ͪ ϩdx ͩ ϩv ϩ ϩٌ ץ 2 ץ 2 ץ2 2 ץ 1 ץ 1 ץ1 1 x v2ץ x v1ץ x yץ x v2ץ x v1ץ x yץ

and the corresponding Hamilton–De Donder–Weyl m-vector ﬁelds, connections, and jet ﬁelds which are the solutions to the Hamiltonian problem are

ץ p2ץ ץ p1ץ ץ p2 ץ ץ p2ץ ץ p1ץ ץ p1 ץ XHϭ f ͩ Ϫ ϩ ϩ ͪ ∧ͩ Ϫ ϩ ϩ ͪ , p2ץ x2ץ p1ץ x2ץ yץ x2 Hץ p2ץ x1ץ p1ץ x1ץ yץ x1 Hץ p2ץ p2ץ p1ץ p1ץ p1 p2 1 2 1 2 ⌿Hϭͩ x ,x ,y,p ,p ;Ϫ ,Ϫ , , , , ͪ , x2ץ x1ץ x2ץ x1ץ H H

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ץ p2ץ ץ p1ץ ץ p2 ץ ץ p2ץ ץ p1ץ ץ p1 ץ 1 2 . ͪ Hϭdx ͩ Ϫ ϩ ϩ ͪ ϩdx ͩ Ϫ ϩ ϩٌ p2ץ x2ץ p1ץ x2ץ yץ x2 Hץ p2ץ x1ץ p1ץ x1ץ yץ x1 Hץ

V. CONCLUSIONS AND OUTLOOK We have generalized the Rusk–Skinner unified formalism to first-order classical field theories. ϫ 1 Corresponding to the Whitney sum TQ QT*Q in autonomous mechanics, here we take J E ϫ M␲ W ʚ 1 ϫ M␲ E as standpoint, but the field equations are stated in a submanifold 0 J E E .Asa particular case of this situation, the unified formalism for nonautonomous mechanics is recovered, 1 ϫ ␲ → 8,9 the Whitney sum being now J E ET*E, where :E R is the configuration bundle. Once the suitable ͑pre͒ multisymplectic structures are introduced, the field equations can be written in ͑ ͒ W ͑ ͒ several equivalent ways: using sections and vector fields 8 in 0 , m-vector fields 10 , connections ͑11͒, or jet fields ͑12͒. ͑ ͒ W Starting from Eq. 8 , we have seen how, when different kinds of vertical vector fields in 0 ␳0 are considered, this equation gives a different type of information. In particular, using ˆ 2-vertical W W vector fields, we can define a submanifold 1 0 , which turns out to be the graph of the ͑extended͒ Legendre transformation ͑and hence diffeomorphic to J1E). Furthermore, the field W equations are only compatible in 1 . As sections solution to the field equations take values in W ␺ ϭ ␺ ␺ ͑ ␺ → 1 1 , they split in a natural way into two components, 0 ( L , H), with L :M J E, and ͒ ͑ ␺ ϭgFLؠ␺ ␳0 H L). Then, taking 1-vertical vector fields in 8 , we have proved that the sections solution to the field equations in the unified formalism are automatically holonomic, even in the ␺ singular case. They are so in the following sense: for every section 0 solution in the unified formalism, the corresponding section ␺L is holonomic. ͑As a special case, nonintegrable m-vector fields, connections and jet fields which are solutions to the field equations are semi-holonomic.͒ W ͑ ͒ These solutions only exist in general in a submanifold of 1 . Finally, considering 8 for a generic vector field, the Euler–Lagrange equations for ␺L , and the Hamilton–De Donder–Weyl equations for ␮ؠ␺HϭFLؠ␺L arise in a natural way. Conversely, starting from sections ␺L ϭ 1␾ FLؠ␺ ␺ j and L solutions to the corresponding field equations, we can recover sections 0 solution to ͑8͒. Thus, we have shown the equivalence between the standard Lagrangian and Hamiltonian formalisms and the unified one. This equivalence has been also proved for m-vector fields, connections and jet fields. Although the subject is not considered in this work, K operators ͑i.e., the analogous operators in field theories to the so-called evolution operator in mechanics͒, in their different alternative definitions,33 can easily be recovered from the unified formalism, similarly to the case of classical mechanics. In a forthcoming paper, this formalism will be applied to give a geometric framework for optimal control with partial differential equations. Although this subject has been dealt with in the context of functional analysis, to our knowledge there has been no geometric treatment of it to date.

ACKNOWLEDGMENTS The authors acknowledge the ﬁnancial support of Ministerio de Ciencia y Tecnologı´a, BFM2002-03493 and BFM2000-1066-C03-01. The authors thank Jeff Palmer for his assistance in preparing the English version of the manuscript.

APPENDIX: m-VECTOR FIELDS, JET FIELDS, AND CONNECTIONS IN JET BUNDLES ͑See Refs. 17 and 27 for the proofs and other details of the following assertions.͒ Let E be a n-dimensional differentiable manifold. For mрn, sections of ⌳m(TE) are called m-vector fields in E ͑they are contravariant skew-symmetric tensors of order m in E͒. We denote by Xm(E) the set of m-vector fields in E. Y෈Xm(E) is said to be locally decomposable if, for ෈ ʚ ෈ every p E, there exists an open neighborhood U p E and Y 1 ,...,Y m X(U p) such that Ӎ ∧ ∧ Y Y 1 ••• Y m . Contraction of m-vector fields and tensor fields in E is the usual one. Up

We can define the following equivalence relation: if Y,YЈ෈Xm(E) are nonvanishing m-vector fields, then YϳYЈ if there exists a nonvanishing function f ෈Cϱ(E) such that YЈϭ fY ͑perhaps only in a connected open set UʕE). Equivalence classes will be denoted by ͕Y͖. There is a one-to-one correspondence between the set of m-dimensional orientable distributions D in TE and the set of the equivalence classes ͕Y͖ of nonvanishing, locally decomposable m-vector fields in E. Then, there is a bijective correspondence between the set of classes of locally decomposable and ␲-transverse m-vector fields ͕Y͖ʚXm(E), and the set of orientable jet fields ⌿:E→J1E; that is, -the set of orientable Ehresmann connection forms ٌ in ␲:E→M. This correspondence is charac .(terized by the fact that the horizontal subbundle associated with ⌿͑and ٌ͒ coincides with D(Y If Y෈Xm(E) is nonvanishing and locally decomposable, the distribution associated with the class ͕Y͖ is denoted D(Y). A nonvanishing, locally decomposable m-vector field Y෈Xm(E)is ͑ ͒ D said to be integrable respectively, involutive if its associated distribution U(Y) is integrable ͑respectively, involutive͒. Of course, if Y෈Xm(E) is integrable ͑respectively, involutive͒, then so is every m-vector field in its equivalence class ͕Y͖, and all of them have the same integral manifolds. Moreover, Frobenius’ theorem allows us to say that a nonvanishing and locally decomposable m-vector field is integrable if, and only if, it is involutive. Of course, the orientable jet field ⌿, and the connection form ٌ associated with ͕Y͖ are integrable if, and only if, so is Y, for every Y෈͕Y͖. Let us consider the following situation: if ␲:E→M is a fiber bundle, we are concerned with the case where the integral manifolds of integrable m-vector fields in E are sections of ␲. Thus, ෈ m ␲ ෈ ␲ ␤ Þ ␤ Y X (E) is said to be -transverse if, at every point y E,(i(Y)( * ))y 0, for every ෈⍀m(M) such that ␤(␲(y))Þ0. Then, if Y෈Xm(E) is integrable, it is ␲-transverse if its integral manifolds are local sections of ␲. In this case, if ␾:UʚM→E is a local section with ␾(x)ϭy and ␾ ␾ D ␾ (U) is the integral manifold of Y through y, then Ty(Im )is y(Y). Integral sections of the → m ␾ϭ ؠ␾ؠ␴ ␴ ⌳m⌳ ͖ ͕ class Y can be characterized by the condition T fY M , where M : TM M is the natural projection, and f ෈Cϱ(E) is a nonvanishing function. As a particular case, let ͕X͖:J1E→DmTJ1Eʚ͕⌳mTJ1E͖ be a class of nonvanishing, locally decomposable and ␲¯ 1-transverse m-vector fields in J1E, ⌿:J1E→J1J1E its associated jet field, ␲1 1→ 1 ٌ and :J E ¯ *TM J1ETJ E its associated connection form. Then, these elements are said to be holonomic if they are integrable and their integral sections ␸:M→J1E are holonomic. Further- more, consider the (1,m)-tensor field in J1E defined by J i(V)(␲¯ 1*␻), whose local expression A A ␣ mϪ1 A ª 1 1 v␯ . A connection form ٌ in ␲¯ :J E→M ͑and its associatedץ/ץ is Jϭ(dy Ϫv␣dx )∧d x␯ jet field ⌿:J1E→J1J1E) are said to be semi-holonomic ͑or a second order partial differential equation͒,if

ٌ where h denotes the horizontal projector associated with ٌ.If͕X͖ʚXm(J1E) is the associated class of ␲¯ 1-transverse multivector ﬁelds, then this condition is equivalent to J(X)ϭ0, for every ,X෈͕X͖. Then the class ͕X͖, and its associated jet ﬁeld ⌿ and connection form ٌ are holonomic if and only if, they are integrable and semi-holonomic.

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