extended hamiltonian systems in multisymplectic field theories

Extended Hamiltonian systems in multisymplectic field theoriesArturo Echeverría-Enríquez, Manuel de León, Miguel C. Muñoz-Lecanda, and Narciso Román-Roy Citation: J. Math. Phys. 48, 112901 (2007); doi: 10.1063/1.2801875 View online: http://dx.doi.org/10.1063/1.2801875 View Table of Contents: http://jmp.aip.org/resource/1/JMAPAQ/v48/i11 Published by the American Institute of Physics. Related ArticlesD-module representations of N = 2,4,8 superconformal algebras and their superconformal mechanics J. Math. Phys. 53, 043513 (2012) On a kind of Noether symmetries and conservation laws in k-cosymplectic field theory J. Math. Phys. 52, 022901 (2011) Substituting fields within the action: Consistency issues and some applications J. Math. Phys. 51, 122903 (2010) Loop vertex expansion for Φ2k theory in zero dimension J. Math. Phys. 51, 092304 (2010) Rigid symmetries and conservation laws in non-Lagrangian field theory J. Math. Phys. 51, 082902 (2010) Additional information on J. Math. Phys.Journal Homepage: http://jmp.aip.org/ Journal Information: http://jmp.aip.org/about/about_the_journal Top downloads: http://jmp.aip.org/features/most_downloaded Information for Authors: http://jmp.aip.org/authors

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Extended Hamiltonian systems in multisymplectic fieldtheories

Arturo Echeverría-EnríquezDepartamento de Matemática Aplicada IV, Campus Norte UPC, Edificio C-3, C/JordiGirona 1, E-08034 Barcelona, Spain

Manuel de Leóna�

Instituto de Matemáticas y Física Fundamental, CSIC, C/Serrano 123, E-28006 Madrid,Spain

Miguel C. Muñoz-Lecandab� and Narciso Román-Royc�

Departamento de Matemática Aplicada IV, Campus Norte UPC, Edificio C-3, C/JordiGirona 1, E-08034 Barcelona, Spain

�Received 21 September 2006; accepted 2 October 2007;published online 1 November 2007�

We consider Hamiltonian systems in first-order multisymplectic field theories. Wereview the properties of Hamiltonian systems in the so-called restricted multimo-mentum bundle, including the variational principle which leads to the Hamiltonianfield equations. In an analogous way to how these systems are defined in theso-called extended �symplectic� formulation of nonautonomous mechanics, we in-troduce Hamiltonian systems in the extended multimomentum bundle. The geomet-ric properties of these systems are studied, the Hamiltonian equations are analyzedusing integrable multivector fields, the corresponding variational principle is alsostated, and the relation between the extended and the restricted Hamiltonian sys-tems is established. All these properties are also adapted to certain kinds of sub-manifolds of the multimomentum bundles in order to cover the case of almost-regular field theories. © 2007 American Institute of Physics.�DOI: 10.1063/1.2801875�

I. INTRODUCTION

The Hamiltonian formalism of dynamical systems, and the study of the properties of Hamil-tonian dynamical systems in general, is a fruitful subject in both applied mathematics and theo-retical physics. From a generic point of view, the characteristics of these kinds of systems makethem specially suitable for analyzing many of their properties: for instance, symmetries and relatedtopics such as the existence of conservation laws and reduction, the integrability �including nu-merical methods�, and the possible quantization of the system, which is based on the use of thePoisson bracket structure of this formalism. Moreover, it is also important to point out the exis-tence of dynamical Hamiltonian systems which have no Lagrangian counterpart �see an examplein Ref. 45�.

From the geometrical viewpoint, many of the characteristics of the autonomous Hamiltoniansystems arise from the existence of a “natural” geometric structure with which the phase space ofthe system is endowed: the symplectic form �a closed, nondegenerated two form�, which allows theconstruction of Poisson brackets. In this model, the dynamic information is carried out by theHamiltonian function, which is not coupled to the geometry. This is not the case for nonautono-

a�Electronic mail: [email protected]�Electronic mail: [email protected]�Electronic mail: [email protected]

JOURNAL OF MATHEMATICAL PHYSICS 48, 112901 �2007�

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http://dx.doi.org/10.1063/1.2801875

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mous Hamiltonian systems, which have different geometric descriptions. One of the most fre-quently used formulations for these systems is in the framework of contact geometry, which takesplace in the restricted phase space T*Q�R, where Q is the configuration manifold �see Ref. 11and references therein�. Here, the physical information is given by the Hamiltonian function,which allows us to construct the contact form in T*Q�R. However, a more appropriate descrip-tion is the symplectic or extended formulation of nonautonomous mechanics,17,29,36,42,46 which isdeveloped in the extended phase space T*�Q�R�. Now, the natural symplectic structure ofT*�Q�R� and the physical information, given by the extended Hamiltonian function, are decou-pled, and this provides us with a Hamiltonian description similar to the autonomous case.

When first-order field theories are considered, the usual way to work is with the Lagrangianformalism1,3,9,18,15,16,35,39,44 because their Hamiltonian description presents different kinds of prob-lems. First, several Hamiltonian models can be stated, and the equivalence among them is notalways clear �see, for instance, Refs. 2, 10, 19, 21–23, 40, and 43�. Furthermore, there are equiva-lent Lagrangian models with nonequivalent Hamiltonian descriptions.26–28 Among the differentgeometrical descriptions to be considered for describing field theories, we focus our attention onthe multisymplectic models,7,20,24,25,38 where the geometric background is in the realm of multi-symplectic manifolds, which are manifolds endowed with a closed and 1-nondegenerate k-form,with k�2. In these models, this form plays a similar role to the symplectic form in mechanics.

The main aim of this paper is to generalize the Hamiltonian symplectic formulation of non-autonomous mechanics to first-order multisymplectic field theories. The motivation and basicfeatures of this formulation are the following. As is well known, there is no canonical model forthe Hamiltonian first-order field theory. Hence, the first problem to be considered is the choice ofa suitable multimomentum bundle to develop the formalism. The most frequently used choice is totake the so-called restricted multimomentum bundle, denoted by J1�*, that is, analogous toT*Q�R in the mechanical case. The Hamiltonian formalism in J1�* has been extensivelystudied.6,12,31,37 Nevertheless, this bundle does not have a canonical multisymplectic form and thephysical information, given by a Hamiltonian section, is used to obtain the geometric structure.This is a problem when other aspects of Hamiltonian field theories are considered, such as thedefinition of Poisson brackets, the notion of integrable system, the problem of reduction bysymmetries, and the quantization procedure. An attempt to overcome these difficulties is to workin a greater dimensional manifold, the so-called extended multimomentum bundle, denoted byM�, which is analogous to the extended phase space T*�Q�R� of a nonautonomous mechanicalsystem. M� has a canonical multisymplectic form since it is a vector sub-bundle of a multico-tangent bundle. In this manifold M�, the physical information is given by a closed one form, theHamiltonian form. Then, Hamiltonian systems can be introduced as in autonomous mechanics, byusing certain kinds of Hamiltonian multivector fields. The resultant extended Hamiltonian formal-ism is the generalization to field theories of the extended formalism for nonautonomous mechani-cal systems29,11 and, to our knowledge, it was introduced for the first time in field theories in Ref.41

The goal of our work is to carry out a deeper geometric study of these kinds of systems. Themain results are the following. First, to every Hamiltonian system in the extended multimomentumbundle, we can associate in a natural way a class of equivalent Hamiltonian systems in therestricted multimomentum bundle �Theorem 5� and conversely �Proposition 7�. The solutions tothe field equations in both models are also canonically related. In addition, the field equations forthese kinds of systems can be derived from an appropriate variational principle �Theorem 6�,which constitutes a first attempt to tackle variational principles for field theories with nonholo-nomic constraints �see Ref. 47 for a geometrical setting of these theories�. Furthermore, theintegral submanifolds of the Hamiltonian 1-form can be embedded into the extended multimo-mentum phase space similar to the way in which the constant energy surfaces are coisotropicallyembedded in T*�Q�R� in nonautonomous mechanics �Proposition 6�. Finally, the case of non-regular Hamiltonian systems is considered and, after a careful definition of what an almost-regularHamiltonian system is, the above results are adapted to this situation in a natural way. We hopethat all these results could be a standpoint from which to study Poisson brackets, the quantization

112901-2 Echeverría-Enríquez et al. J. Math. Phys. 48, 112901 �2007�


problem, and also the reduction by symmetries of field theories in further research works.The paper is organized as follows. In Sec. II, we review basic concepts and results, such as

multivector fields and connections, multisymplectic manifolds and Hamiltonian multivector fields,and the restricted and extended multimomentum bundles with their geometric structures. SectionIII is devoted to reviewing the definition and characteristics of Hamiltonian systems in the re-stricted multimomentum bundles, in particular, the definitions of Hamiltonian sections and densi-ties, the variational principle which leads to the Hamilton–De Donder–Weyl equations, and the useof multivector fields for writing these equations in a more suitable geometric way. Sections IV andV contain the most relevant material of the work. Thus, Hamiltonian systems in the extendedmultimomentum bundle are introduced in Sec. IV; in particular, their geometric properties, theirrelation with those introduced in Sec. III, and the corresponding variational principle are studied.In Sec. V, we adapt the above definitions and results in order to consider Hamiltonian systemswhich are not defined everywhere in the multimomentum bundles, but in certain submanifolds ofthem: here, these are the so-called almost-regular Hamiltonian systems. Finally, as typical ex-amples, in Sec. VI, we review the standard Hamiltonian formalism associated with a Lagrangianfield theory, both in the regular and singular �almost-regular� cases, and the Hamiltonian formal-isms of time-dependent dynamical systems in the extended and restricted phase space, which arerecovered as a particular case of this theory.

All manifolds are real, paracompact, connected, and C�. All maps are C�. Sum over crossedrepeated indices is understood. Throughout this paper, � :E→M will be a fiber bundle �dim M=m, dim E=n+m�, where M is an oriented manifold with volume form ��m�M�, and �x� ,yA��with �=1, . . . ,m; A=1, . . . ,n� will be natural local systems of coordinates in E adapted to thebundle, such that �=dx1∧ ¯ ∧dxm�dmx.

II. PREVIOUS DEFINITIONS AND RESULTS

A. Multivector fields and connections

See Ref. 13 for details. Let M be a n-dimensional differentiable manifold. Sections of�m�TM� are called m-multivector fields in M �they are the contravariant skew-symmetric tensorsof order m in M�. We will denote by Xm�M� the set of m-multivector fields in M.

If Y�Xm�M�, for every p�M, there exists an open neighborhood Up�M and Y1 , . . . ,Yr

�X�Up� such that Y=Up�1i1¯imrf

i1¯imYi1∧ ¯ ∧Yim

, with f i1¯im �C��Up� and mrdim M. Then, Y�Xm�M� is said to be locally decomposable if, for every p�M, there existsan open neighborhood Up�M and Y1,. . .,Ym�X�Up� such that Y=Up

Y1∧ ¯ ∧Ym.A nonvanishing m-multivector field Y�Xm�M� and an m-dimensional distribution D�TM

are locally associated if there exists a connected open set U�M such that �Y�U is a section of�m�D�U. If Y ,Y��Xm�M� are nonvanishing multivector fields locally associated with the samedistribution D, on the same connected open set U, then there exists a nonvanishing function f�C��U� such that Y�=UfY. This fact defines an equivalence relation in the set of nonvanishingm-multivector fields in M, whose equivalence classes will be denoted by �YU. Then, there is aone-to-one correspondence between the set of m-dimensional orientable distributions D in TMand the set of the equivalence classes �YM of nonvanishing, locally decomposable m-multivectorfields in M.

If Y�Xm�M� is nonvanishing and locally decomposable, and U�M is a connected openset, the distribution associated with the class �YU is denoted by DU�Y�. If U=M, we write D�Y�.

A nonvanishing, locally decomposable multivector field Y�Xm�M� is said to be integrable�involutive� if its associated distribution DU�Y� is integrable �involutive�. Of course, if Y�Xm�M� is integrable �involutive�, then so is every other in its equivalence class �Y, and all ofthem have the same integral manifolds. Moreover, Frobenius theorem allows us to say that anonvanishing and locally decomposable multivector field is integrable if and only if it is involu-tive. Nevertheless, in many applications, we have locally decomposable multivector fields Y�Xm�M� which are not integrable in M but integrable in a submanifold of M. A �local� algo-rithm for finding this submanifold has been developed.13

112901-3 Extended Hamiltonian systems in field theories J. Math. Phys. 48, 112901 �2007�


The particular situation in which we are interested is the study of multivector fields in fiberbundles. If � :M→M is a fiber bundle, we will be interested in the case where the integralmanifolds of integrable multivector fields in M are sections of �. Thus, Y�Xm�M� is said to be� transverse if, at every point y�M, �i�Y��*��y �0, for every ��m�M� with ��y��0.Then, if Y�Xm�M� is integrable, it is � transverse if and only if its integral manifolds are localsections of � :M→M. In this case, if � :U�M→M is a local section with ��x�=y and ��U� isthe integral manifold of Y through y, then Ty�Im ��=Dy�Y�.

Finally, it is clear that classes of locally decomposable and �-transverse multivector fields�Y�Xm�M� are in one-to-one correspondence with orientable Ehresmann connection forms � in� :M→M. This correspondence is characterized by the fact that the horizontal sub-bundle asso-ciated with � is D�Y�. In this correspondence, classes of integrable locally decomposable and�-transverse m-multivector fields correspond to flat orientable Ehresmann connections.

B. Hamiltonian multivector fields in multisymplectic manifolds

See Refs. 4 and 33 for details. Let M be a n-dimensional differentiable manifold and ��m+1�M�. The couple �M ,�� is said to be a multisymplectic manifold if � is closed and1-nondegenerate; that is, for every p�M, and Xp�TpM, we have that i�Xp��p=0 if and only ifXp=0.

If �M ,�� is a multisymplectic manifold, X�Xk�M� is said to be a Hamiltoniank-multivector field if i�X�� is an exact �m+1−k�-form; that is, there exists ��m−k�M� such that

i�X�� = d , �1�

where is defined as modulo closed �m−k�-forms. The class � ��m−k�M� /Zm−k�M� defined by

is called the Hamiltonian for X, and every element in this class � � is said to be a Hamil-tonian form for X. Furthermore, X is said to be a locally Hamiltonian k-multivector field if i�X��is a closed �m+1−k�-form. In this case, for every point x�M, there is an open neighborhoodW�M and ��m−k�W� such that

i�X�� = d �on W� .

As above, changing M by W, we obtain the Hamiltonian for X, � ��k−m−1�W� /Zk−m−1�W�, andthe local Hamiltonian forms for X.

Conversely, ��k�M� � ��k�W�� is said to be a Hamiltonian k-form �a local Hamiltoniank-form� if there exists a multivector field X�Xm−k�M� �X�Xm−k�W�� such that �1� holds �on W�.In particular, when k=0, that is, if �C��M�, then the existence of Hamiltonian m-multivectorfields for is assured �see Ref. 4�.

C. Multimomentum bundles

See, for instance, Ref. 10. Let � :E→M be the configuration bundle of a field theory �withdim M =m, dim E=n+m�. There are several multimomentum bundle structures associated with it.

First, we have �2mT*E, which is the bundle of m-forms on E vanishing by the action of two

�-vertical vector fields. Furthermore, if J1�→E→M denotes the first-order jet bundle over E, theset made of the affine maps from J1� to �mT*M, denoted as Aff �J1� ,�mT*M�, is another bundleover E which is canonically diffeomorphic to �2

mT*E.6,10 We will denote

M� � �2mT*E Aff�J1�,�mT*M� .

It is called the extended multimomentum bundle, and its canonical submersions are denoted as

�: M� → E, � = � � �: M� → M .

M� is a sub-bundle of �mT*E, the multicotangent bundle of E of order m �the bundle of m-formsin E�. Then, M� is endowed with canonical forms. First, we have the “tautological form” �



��m�M�� which is defined as follows. Let �x ,��2mT*E, with x�E and ��2

mTx*E; then, for

every X1 , . . . ,Xm�T�x,��M��,

��x,��;X1, . . . ,Xm� ª ��x;T�x,��X1�, . . . ,T�q,��Xm�� .

Thus, we define the multisymplectic form

� ª − d� � �m+1�M�� .

They are known as the multimomentum Liouville m- and �m+1�-forms.We can introduce natural coordinates in M� adapted to the bundle � :E→M, which are

denoted by �x� ,yA , pA� , p�, and such that �=dmx. Then, the local expressions of these forms are

� = pA�dyA ∧ dm−1x� + pdmx, � = − dpA

� ∧ dyA ∧ dm−1x� − dp ∧ dmx �2�

�where dm−1x�ª i�� /�x��dmx�.Consider �1

mT*E��*�mT*M, which is another bundle over E, whose sections are the�-semibasic m-forms on E and denote by J1�* the quotient �2

mT*E /�1mT*E�M� /�1

mT*E. Wehave the natural submersions

� : J1�* → E, � = � � � : J1�* → M .

Furthermore, the natural submersion � :M�→J1�* endows M� with the structure of an affinebundle over J1�*, with �*�1

mT*E as the associated vector bundle. J1�* is usually called therestricted multimomentum bundle associated with the bundle � :E→M. Natural coordinates inJ1�* �adapted to the bundle � :E→M� are denoted by �x� ,yA , pA

��. We have the diagram

Hamiltonian systems can be defined in M� or in J1�*. The construction of the Hamiltonianformalism in J1�* was pionered in Ref. 6 �see also Refs. 12 and 10�, while a formulation in M�has been stated recently.41 In the following sections, we review the main concepts of the formalismin J1�* and we make an extensive development of the formalism in M�.

III. HAMILTONIAN SYSTEMS IN J1�*

First, we consider the standard definition of Hamiltonian systems in field theory, which isstated using the restricted multimomentum bundle J1�*.

A. Restricted Hamiltonian systems

Definition 1: Consider the bundle � :J1�*→M.

�1� A section h :J1�*→M� of the projection � is called a Hamiltonian section of �.�2� The differentiable forms

�h ª h*�, �h ª − d�h = h*�



are called the Hamilton-Cartan m- and �m+1�-forms of J1�* associated with the Hamil-tonian section h.

�3� The couple �J1�* ,h� is said to be a restricted Hamiltonian system �or just a Hamiltoniansystem�.

In a local chart of natural coordinates, a Hamiltonian section is specified by a local Hamil-tonian function h�C��U�, U�J1�*, such that h�x� ,yA , pA

��x� ,yA , pA� , p=−h�x� ,yB , pB

��. Thelocal expressions of the Hamilton-Cartan forms associated with h are

�h = pA�dyA ∧ dm−1x� − hdmx, �h = − dpA

� ∧ dyA ∧ dm−1x� + dh ∧ dmx . �3�

Remark 1: Notice that �h is 1-nondegenerate, that is, a multisymplectic form �as a simplecalculation in coordinates shows�.

Hamiltonian sections can be obtained from connections. In fact, if we have a connection � in� :E→M, it induces a linear section h� :J1�*→M� of �.6 Then, if � is the canonical m-form in�m�M��, the forms

�h� ª h�*� � �m�J1�*�, �h� ª − d�h� � �m+1�J1�*� �4�

are the Hamilton-Cartan m- and �m+1�-forms of J1�* associated with the connection �. In asystem of natural coordinates in J1�*, if �=dx� � �� /�x��+��

A�� /�yA�� is the local expression ofthe connection �, the local expressions of these Hamilton-Cartan forms associated with � are

�h� = pA��dyA − ��

Adx�� ∧ dm−1x� = pA�dyA ∧ dm−1x� − pA

��Admx ,

�h� = − dpA� ∧ dyA ∧ dm−1x� + ��

AdpA� ∧ dmx + pA

�d��A ∧ dmx .

Observe that a local Hamiltonian function associated with h� is h�= pA��

A.

B. Variational principle and field equations

Now, we establish the field equations for restricted Hamiltonian systems. They can be derivedfrom a variational principle. In fact, first, we state the following.

Definition 2: Let �J1�* ,h� be a restricted Hamiltonian system. Let ��M ,J1�*� be the set ofsections of �. Consider the map

H: ��M,J1�*� → R ,

� � �M

�*�h

(where the convergence of the integral is assumed). The variational problem for this restrictedHamiltonian system is the search for the critical (or stationary) sections of the functional H, withrespect to the variations of � given by �t=�t ��, where ��t is the local one-parameter group ofany compact-supported Z�XV��J1�*� [where XV��J1�*� denotes the module of �-vertical vec-tor fields in J1�*], that is,

� d

dt�

t=0�

M

�t*�h = 0.

This is the so-called Hamilton-Jacobi principle of the Hamiltonian formalism.Then, the following fundamental theorem is proven �see also Ref. 10�.Theorem 1: Let �J1�* ,h� be a restricted Hamiltonian system. The following assertions on a

section ��M ,J1�*� are equivalent.

(1) � is a critical section for the variational problem posed by the Hamilton-Jacobi principle.



(2) �*i�Z��h=0, for every Z�XV��J1�*�.(3) �*i�X��h=0, for every X�X�J1�*�.(4) If �U ;x� ,yA , pA

�� is a natural system of coordinates in J1�*, then � satisfies the followingsystem of equations in U:

��yA � ��x� =

�h

�pA� � � � � �h

�pA� �

�

,��pA

� � ��x� = −

�h

�yA � � � − � �h

�yA��

, �5�

where h is a local Hamiltonian function associated with h. They are known as theHamilton–De Donder–Weyl equations of the restricted Hamiltonian system.

Proof: �1⇔2� Let Z�XV��J1�*� be a compact-supported vector field and V�M an open setsuch that �V is an �m−1�-dimensional manifold and that ��supp�Z��V. Then,

� d

dt�

t=0�

M

�t*�h = � d

dt�

t=0�

V

�t*�h = � d

dt�

t=0�

V

�*��t*�h� = �

V

�* limt→0

�t*�h − �h

t�

= �V

�*�L�Z��h� = �V

�*�i�Z�d�h + di�Z��h� = − �V

�*�i�Z��h − di�Z��h�

= − �V

�*�i�Z��h� + �V

d��*�i�Z��h�� = − �V

�*�i�Z��h� + ��V

�*�i�Z��h�

= − �V

�*�i�Z��h�

�as a consequence of Stoke’s theorem and the hypothesis made on the supports of the verticalfields�. Thus, by the fundamental theorem of the variational calculus, we conclude that�d /dt�t=0�V�t

*�h=0 if and only if �*�i�Z��h�=0, for every compact-supported Z�XV��J1�*�.However, as compact-supported vector fields generate locally the C��J1�*� module of vectorfields in J1�*, it follows that the last equality holds for every Z�XV��J1�*�.

�2⇔3� If p� Im �, then TpJ1�*=Vp�� Tp�Im ��. So, if X�X�J1�*�, then

Xp = �Xp − Tp�� Xp�� + Tp�� Xp� � XpV + Xp

�

and therefore

�*�i�X��h� = �*�i�XV��h� + �*�i�X��h� = �*�i�X��h� = 0

since �*�i�XV��h�=0 by the above item. Furthermore, Xp��Tp�Im �� and dim�Im ��=m, with

�h��m+1�J1�*�. Hence, we conclude that �*�i�X��h�=0, for every X�X�J1�*�. The converse isobvious.

�3⇔4� If X=�� /�x��+�A�� /�yA�+�A�� /�pA

��X�J1�*�, taking into account the local ex-pression �3� of �h, we have

i�X��h = �− 1�� dpA� ∧ dyA ∧ dm−2x�� −

�h

�pA� dpA

� ∧ dm−1x�� + �A dpA� ∧ dm−1x� +

�h

�yAdmx�+ �A

� − dyA ∧ dm−1x� +�h

�pA� ∧ dmx� .

However, if �= �x� ,yA�x�� , pA��x��, then



�*i�X��h = �− 1��+�� yA � ��x� − � �h

�pA� �

�� pA

� � ��x� dmx + �A ��pA

� � ��x� + � �h

�yA��dmx

+ �A� −

��yA � ��x� + � �h

�pA� �

��dmx ,

and, as this holds for every X�X�J1�*�, we conclude that �*i�X��h=0 if and only if theHamilton–De Donder–Weyl equations hold for �. �

Remark 2: It is important to point out that Eqs. �5� are not covariant since the Hamiltonianfunction h is defined only locally, and hence it is not intrinsically defined. In order to write a setof covariant Hamiltonian equations, we must use a global Hamiltonian function, that is, a Hamil-tonian density �see Refs. 6 and 10 for comments on this subject�.

Observe also that the solution to these equations is not unique.

C. Hamiltonian equations for multivector fields

See Refs. 13 and 8 for more details. Let �J1�* ,h� be a restricted Hamiltonian system. Theproblem of finding critical section solutions of the Hamilton-Jacobi principle can be formulatedequivalently as follows: to find a distribution D of T�J1�*� satisfying that

• D is m dimensional;• D is � transverse;• D is integrable �that is, involutive�;• the integral manifolds of D are the critical sections of the Hamilton-Jacobi principle.

However, as explained in Sec. II A, these kinds of distributions are associated with classes ofintegrable �i.e., nonvanishing, locally decomposable, and involutive� �-transverse multivectorfields in J1�*. The local expression in natural coordinates of an element of one of these classes is

X = ∧�=1

m

f �

�x� + F�A �

�yA + GA��

�pA� � , �6�

where f �C��J1�*� is a nonvanishing function.Therefore, the problem posed by the Hamilton-Jacobi principle can be stated in the following

way:12,33

Theorem 2: Let �J1�* ,h� be a restricted Hamiltonian system and �X�Xm�J1�*� a class ofintegrable and �-transverse multivector fields. Then, the integral manifolds of �X are the criticalsection for the variational problem posed by the Hamilton-Jacobi principle if and only if

i�Xh��h = 0 for every Xh � �Xh . �7�

Remark 3: The �-transversality condition for the multivector field solution to �7� can bestated by demanding that i�Xh��*��0. In particular, if we take i�Xh��*��=1, we are choosinga representative of the class of �-transverse multivector field solution to �7�. �This is equivalent toputting f =1 in the local expression �6�.�

Thus, the problem posed in Definition 2 is equivalent to looking for a multivector field Xh

�Xm�J1�*� such that

�1� i�Xh��h=0;�2� i�Xh��*��=1;�3� Xh is integrable.

From conditions �1� and �2�, using the local expressions �3� of �h and �6� for Xh, we obtainthat f =1 and



F�A =

�h

�pA� , GA�

� = −�h

�yA .

If ��x�= �x� ,yA�x�� , pA��x�� must be an integral section of Xh, then

��yA � ��x� = F�

A � �,��pA

� � ��x� = GA�

� � � .

Thus, the Hamilton–De Donder–Weyl equations �5� for � are recovered from �7�.Remark 4: Classes of locally decomposable and �-transverse multivector fields are in one-

to-one correspondence with connections in the bundle � :J1�*→M �see Sec. II A�. Then, it can beproven12 that the condition stated in Theorem 2 is equivalent to finding an integrable connection�h in J1�*→M satisfying the equation

i��h��h = �m − 1��h,

whose integral sections are the critical sections of the Hamilton-Jacobi problem. Of course, �h isthe connection associated with the class �Xh solution to �7� and Xh is integrable if and only if thecurvature of �h vanishes everywhere.

The expression of �h in coordinates is

�h = dx��

�x� + F�A �

�yA + GA��

�pA� � .

Definition 3: Xh�Xm�J1�*� will be called a Hamilton–De Donder–Weyl �HDW�-multivectorfield for the system �J1�* ,h� if it is � transverse, locally decomposable, and verifies the equationi�Xh��h=0. Then, the associated connection �h is called a HDW connection for �J1�* ,h�.

For restricted Hamiltonian systems, the existence of Hamilton–De Donder–Weyl multivectorfields or connections is guaranteed, although they are not necessarily integrable.13,18

Theorem 3: �Existence and local multiplicity of HDW-multivector fields� Let �J1�* ,h� be arestricted Hamiltonian system. Then, there exist classes of HDW-multivector fields �Xh. In a localsystem, the above solutions depend on n�m2−1� arbitrary functions.

Remark 5: In order to find a class of integrable HDW-multivector fields �if it exists�, we mustimpose that Xh verify the integrability condition: the curvature of �h vanishes everywhere. Hence,the number of arbitrary functions will in general be less than n�m2−1�. If this integrable multi-vector field does not exist, we can eventually select some particular HDW-multivector field solu-tion and apply an integrability algorithm in order to find a submanifold I�J1�* �if it exists�,where this multivector field is integrable �and tangent to I�.

IV. HAMILTONIAN SYSTEMS IN M�

Now, we introduce Hamiltonian systems in the extended multimomentum bundle M� and westudy their relation with those defined in the above section.

A. Extended Hamiltonian systems

Now, we have the multisymplectic manifold �M� ,�� and we are interested in definingHamiltonian systems on this manifold which are suitable for describing Hamiltonian field theories.Thus, we must consider Hamiltonian or locally Hamiltonian m-multivector fields and forms of aparticular kind. In particular, bearing in mind the requirements in Remark 3, we can state thefollowing.

Definition 4: The triple �M� ,� ,�� is said to be an extended Hamiltonian system if

(1) ��Z1�M�� (it is a closed 1-form);(2) there exists a locally decomposable multivector field X��Xm�M�� satisfying that



i�X�� = �− 1�m+1�, i�X��*�� = 1 �� transversality� . �8�

If � is an exact form, then �M� ,� ,�� is an extended global Hamiltonian system. In thiscase, there exist H�C��M�� such that �=dH, which are called Hamiltonian functions ofthe system. (For an extended Hamiltonian system, these functions exist only locally, and theyare called local Hamiltonian functions).

The condition that � is closed plays a crucial role �see Proposition 2 and Sec. IV B�. Thefactor �−1�m+1 in the definition will be justified later �see Proposition 1 and Remark 7�.

Observe that if �M� ,� ,�� is an extended global Hamiltonian system, giving a Hamiltonian

function H is equivalent to giving a Hamiltonian density H�H��*��m�M��.In natural coordinates of M�, the most general expression for a locally decomposable mul-

tivector field X��Xm�M�� is

X� = ∧�=1

m

f �

�x� + F�A �

�yA + GA��

�pA� + g�

�

�p� , �9�

where f �C��M�� is a nonvanishing function which is equal to 1 if the equation i�X��*��=1 holds.

Remark 6: In addition, bearing in mind Remark 5, the integrability of X� must be imposed.Then, all the multivector fields in the integrable class �X� have the same integral sections.

A first important observation is that not every closed form ��m�M�� defines an extendedHamiltonian system.

Proposition 1: If �M� ,� ,�� is an extended Hamiltonian system, then i�Y��0, for every�-vertical vector field Y �XV��M��, Y �0. In particular, for every system of natural coordi-nates �x� ,yA , pA

� , p� in M� adapted to the bundle � :E→M (with �=dmx),

i �

�p�� = 1. �10�

Proof: In order to prove this, we use natural coordinates of M�. The local expression of � isgiven in �2�, and a �-vertical vector field is locally given by Y = f�� /�p�. Then, if X�

�Xm�M�� is a multivector field solution to Eq. �8�, we have

i�Y�� = �− 1�m+1i�Y�i�X�� = �− 1�m+1�− 1�mi�X��i�Y�� = − i�X��i f�

�p��− dp ∧ dmx

− dpA� ∧ dyA ∧ dm−1x�� = − i�X��i f

�

�p��− dp ∧ dmx� = fi�X��dmx = f ,

and, as Y �0⇔ f �0, the first result holds. In particular, taking f =1, the expression �10� isreached. �

As a consequence of this result we have the following.Proposition 2: If �M� ,� ,�� is an extended Hamiltonian system, locally �=dp+�, where �

is a closed and �-basic local 1-form in M�.Proof: As a consequence of �10�, �=dp+� locally, where � is a �-semibasic local 1-form.

However, as � is closed, so is �. Hence, for every Y �XV��M��, we have that L�Y��= i�Y�d�+di�Y��=0, and � is � basic. �

Therefore, by Poincaré’s lemma, on an open set U�M�, � has necessarily the followingcoordinate expression:

� = dp + dh�x�,yA,pA�� , �11�

where h=�*h, for some h�C��U��. Then, if H is a �local� Hamiltonian function for �, that is,such that �=dH �at least locally�, we have that �see also Ref. 41�



H = p + h�x�,yA,pA�� , �12�

where h�x� ,yA , pA�� is determined up to a constant.

Conversely, every closed form ��1�M�� satisfying the above condition defines an ex-tended Hamiltonian system since, in an analogous way to Theorem 3, we can prove the following.

Theorem 4: Let ��Z1�M�� satisfy the condition stated in Propositions 1 and 2. Then, thereexist locally decomposable multivector fields X��Xm�M�� (not necessarily integrable) satisfy-ing Eq. (8) [and hence �M� ,� ,�� is an extended Hamiltonian system]. In a local system, theabove solutions depend on n�m2−1� arbitrary functions.

Proof: We use the local expressions �2�, �11�, and �9� for �, �, and X�, respectively. Then,

i�X��*��=1 leads to f =1. Furthermore, from i�X��= �−1�m+1�, we obtain that the equality forthe coefficients on dpA

� leads to

F�A =

�H

�pA� =

�h

�pA� �for every A,�� . �13�

For the coefficients on dyA, we have

GA�� = −

�H

�yA = −�h

�yA �A = 1, . . . ,n� , �14�

and for the coefficients on dx�, using these results, we obtain

g� = −�H

�x� + F�AGA�

� − F�AGA�

� = −�H

�x� +�H

�pA� GA�

� −�H

�pA�GA�

� = −�h

�x� +�h

�pA� GA�

� −�h

�pA�GA�

� �A

= 1, . . . ,n;� � �� , �15�

where the coefficients GA�� are related by Eq. �14�. Finally, the coefficient on dp are identical,

taking into account the above results.Thus, Eq. �13� makes a system of nm linear equations which determines univocally the

functions F�A, Eq. �14� is a compatible system of n linear equations on the nm2 functions GA�

� , andEq. �15� makes a system of m linear equations which determines univocally the functions g�. Inthis way, solutions to Eq. �8� are determined locally from the relations �13� and �15� and throughthe n independent linear equations �14�. Therefore, there are n�m2−1� arbitrary functions. Theseresults assure the local existence of X�. The global solutions are obtained using a partition of unitysubordinated to a covering of M� made of natural charts. �

A further local analysis of these multivector field solutions and other additional details can befound in Refs. 12 and 41.

Remark 7: With regard to this result, it is important to point out that if X��Xm�M�� is asolution �not necessarily integrable� to Eq. �8�, then every multivector field X�� X�, that is, such

that X�� = fX� �where f �C��M�� is nonvanishing� is a solution to the equations

i�X�� = f�− 1�m+1�, i�X��*�� = f �� transversality� .

In particular, if we have a 1-form �=dH �locally�, with 0��H/�p�1, then the �-transversalitycondition must be stated as i�X��*��=−�H/�p, and the solutions X� to the equation i�X��= �−1�m+1� have the local expression �9� with f =−�H/�p, and the other coefficients being solu-tions to the system of equations

f F�A =

�H

�pA� , f GA�

� = −�H

�yA , f g� = −�H

�x� +�H

�pA� GA�

� −�H

�pA�GA�

� �� .

Thus, in an analogous way to restricted Hamiltonian systems in J1�*, we define the following.



Definition 5: X��Xm�M�� will be called an extended Hamilton–De Donder–Weyl multivec-tor field for the system �M� ,� ,�� if it is � transverse, locally decomposable, and verifies theequation i�X��= �−1�m+1�. Then, the associated connection �� in the bundle � :M�→M iscalled an extended Hamilton–De Donder–Weyl connection for �M� ,� ,��.

Now, if �X� is integrable and ��x�= �x� ,yA�x�� , pA��x�� , p�x�� must be an integral section of

X�, then

��yA � ��x� = F�

A � �,��pA

� � ��x� = GA�

� � �,��p � ��

�x� = g� � � , �16�

so Eqs. �13�–�15� give partial differential equations �PDE’s� for �. In particular, the Hamilton–DeDonder–Weyl equations �5� are recovered from �13� and �14�.

As for restricted Hamiltonian systems, in order to find a class of integrable extended HDW-multivector fields �if it exists�, we must impose that X� verify the integrability condition, that is,that the curvature of �� vanishes everywhere, and thus the number of arbitrary functions will ingeneral be less than n�m2−1�. Just as in that situation, we cannot assure the existence of anintegrable solution. If it does not exist, we can eventually select some particular extended HDW-multivector field solution and apply an integrability algorithm in order to find a submanifold ofM� �if it exists�, where this multivector field is integrable �and tangent to it�.

B. Geometric properties of extended Hamiltonian systems

Most of the properties of the extended Hamiltonian systems are based in the following generalresults.

Lemma 1: Let � :M→F be a surjective submersion, with dim M=dim F+r. Consider�1 , . . . ,�r��1�M� such that ��1∧ . . . ∧�r is a closed r-form, and ��p��0, for every p�M. Finally, let ��0

ª �Z�X�M� � i�Z��=0 be the annihilator of �. Therefore,

(1) ��0ª �Z�X�M� � i�Z��i=0, ∀ i=1. . .r;

(2) ��0 generates an involutive distribution in M of corank equal to r, which is called thecharacteristic distribution of � and is denoted D�.

If, in addition, the condition i�Y��0 holds, for every Y �XV��M�, then(3) D� is a �-transverse distribution;(4) the integral submanifolds of D� are r-codimensional and �-transverse local submanifolds of

M;(5) TpM=Vp�� p

0, for every p�M;(6) if S is an integral submanifold of D�, then ��S :S→F is a local diffeomorphism;(7) for every integral submanifold S of D�, and p�S, there exists W�M, with p�W, such that

h= ��W�S�−1 is a local section of � defined on ��W�S� (which is an open set of F).

Proof: First, observe that for every p�M, ��p��0 implies that �i�p�, for every i=1, . . . ,r,are linearly independent, then

0 = i�Z�� = �i=1

r

�− 1�i−1i�Z��i��1 ∧ ¯ ∧ �i−1 ∧ �i+1 ∧ ¯ ∧ �r� ⇔ i�Z��i = 0.

Hence, the statement in item �1� holds and, as a consequence, we conclude that ��0 generates adistribution in M of rank equal to dim F.

Furthermore, if � is closed, for every Z1, Z2� ��0, we obtain that �Z1 ,Z2�� 0 because

i��Z1,Z2�� = L�Z1�i�Z2�� − i�Z2�L�Z1�� = − i�Z2��i�Z1�d� − di�Z1�� = 0.

Then, D� is involutive.



The other properties follow straightforwardly from these results and the condition i�Y��0,for every Y �XV��M�.

Now, from this lemma we have that the following.Proposition 3: If �M� ,� ,�� is an extended Hamiltonian system, then

(1) D� is a �-transverse involutive distribution of corank equal to 1;(2) the integral submanifolds S of D� are 1-codimensional and �-transverse local submanifolds

of M� (we denote by jS :S→M� the natural embedding);(3) for every p�M�, we have that TpM�=Vp�� D��p and thus, in this way, � defines an

integrable connection in the affine bundle � :M�→J1�*;(4) if S is an integral submanifold of D�, then ��S :S→J1�* is a local diffeomorphism;(5) For every integral submanifold S of D�, and p�S, there exists W�M�, with p�W, such

that h= ��W�S�−1 is a local Hamiltonian section of � defined on ��W�S�.

Remark 8: Observe that if �M� ,� ,�� is an extended Hamiltonian system, as �=dH �lo-cally�, every local Hamiltonian function H is a constraint defining locally the integral submani-folds of D�. Thus, bearing in mind the coordinate expression �12�, the local Hamiltonian sectionsassociated with these submanifolds are locally expressed as

h�x�,yA,pA�� = �x�,yA,pA

� ,p = − h�x�,yB,pB�� ,

where �*h=h.A relevant question is under what condition the existence of global Hamiltonian sections is

assured. The answer is given by the following.Proposition 4: Let �M� ,� ,�� be an extended global Hamiltonian system. If there is a

Hamiltonian function H�C��M��, and k�R, such that ��H−1�k��=J1�*, then there exists aglobal Hamiltonian section h��J1�* ,M��.

Proof: In order to construct h, we prove that for every q�J1�*, we have that �−1�q��Sk

contains only one point. Let �U ;x� ,yA , pA� , p� be a local chart in M�, with q��U�. By Propo-

sition 2, we have that H�U= p+�*h, for h�C��U��. If p��−1�q��Sk, we have that

k = H�p� = p�p� + ��*h��p = p�p� + h��p�� ,

then p�p� is determined by q, and p is unique. This allows to define a global section h :J1�*

→M� by h�q�ª�−1�q��Sk, for every q�J1�*, which obviously does not depend on the localcharts considered. �

Observe that if the first de Rahm cohomology group H1�M��=0, then every extended Hamil-tonian system is a global one, but this does not assure the existence of global Hamiltoniansections, as we have shown.

In addition, we have the following.Proposition 5: Given an extended Hamiltonian system �M� ,� ,��, every extended HDW-

multivector field X��Xm�M�� for the system �M� ,� ,�� is tangent to every integral submani-fold of D�. As a consequence, if X� is integrable, then its integral sections are contained in theintegral submanifolds of D�.

Proof: By definition, an extended HDW-multivector field is locally decomposable, so locallyX�=X1∧ ¯ ∧Xm, with X1 , . . . ,Xm�X�M��. Then, X� is tangent to every integral submanifold Sof D� if and only if X� are tangent to S, for every �=1, . . . ,m. However, as D� is the characteristicdistribution of �, this is equivalent to jS

*i�X��=0, and this is true because

i�X�� = i�X��i�X�� = i�X��i�X1 ∧ ¯ ∧ Xm�� = 0.

The last consequence is immediate. �

Remark 9: Observe that if X�=X1∧ ¯ ∧Xm locally, using the local expressions �9� and �11�and Eqs. �13� and �14�, the conditions i�X��=0 lead to



0 =�h

�x� + F�A �h

�yA + GA�� h

�pA� + g� =

�h

�x� −�h

�pA� GA�

� + GA�� h

�pA� + g� =

�h

�x� −�h

�pA� �GA�

� + GA��

+ GA�� h

�pA� + GA�

� �h

�pA� + g� =

�h

�x� −�h

�pA� GA�

� +�h

�pA�GA�

� + g� �� = 1, . . . ,m;� fixed,� � �� ,

which are Eq. �15�. So these equations are consistency conditions �see also the comment inRemark 10�.

Finally, we have the following result.Proposition 6: The integral submanifolds of D� are m-coisotropic submanifolds of �M� ,��.Proof: Let S be an integral submanifold of D�. First, remember that for every p�S, the

m-orthogonal multisymplectic complement of S at p is the vector space

TpS�,mª �Xp � TpM��i�Xp ∧ Xp��p = 0 for every Xp � ∧mTpS � ∧m�D��p ,

and S is said to be an m-coisotropic submanifold of �M� ,�� if TpS�,m�TpS.5,33 Then, for everyXp�TpS�,m, if X� is a HDW-multivector field for the extended Hamiltonian system �M� ,� ,��,as �X��p��mTpS, by Proposition 5, we have

0 = i�Xp�i��X��p��p = i�Xp��p.

Therefore, Xp� �D��p=TpS. �

This statement generalizes a well-known result in time-dependent mechanics �see the examplein Sec. VI B�: considering the line bundle � :T*Q�T*R→T*Q�R, the zero section gives acanonical coisotropic embedding of the submanifold T*Q�R into the symplectic manifoldT*�Q�R�T*Q�T*R. Furthermore, in field theories, every maximal integral submanifold S ofD� gives a local m-coisotropic embedding of U��S��J1�* into M�, given by ��S�−1, whichis obviously not canonical.

C. Relation between extended and restricted Hamiltonian systems

Now, we can establish the relation between extended and restricted Hamiltonian systems inJ1�*. Taking into account the considerations made in the above section, we can state the follow-ing.

Theorem 5: Let �M� ,� ,�� be an extended global Hamiltonian system and �J1�* ,h� arestricted Hamiltonian system such that Im h=S is an integral submanifold of D�. For everyX��Xm�M�� solution to Eq. (8),

i�X�� = �− 1�m+1�, i�X��*�� = 1,

there exists Xh�Xm�J1�*� which is h related with X� and is a solution to the equations

i�Xh��h = 0, i�Xh��*�� = 1

[i.e., satisfying the conditions (1) and (2) in Remark 3].Furthermore, if X� is integrable, then Xh is integrable too, and the integral sections of Xh are

recovered from those of X� as follows: if � :M→M� is an integral section of X�, then �=�

� � :M→J1�* is an integral section of Xh.Proof: Given S=Im h, let jS :S→M� be the natural embedding and hS :J1�*→S the diffeo-

morphism between J1�* and Im h, then h=jS �hS.If X��Xm�M�� is a solution to Eq. �8�, by Proposition 5, it is tangent to S, then there exists

XS�Xm�S� such that �mjS*XS=X��S. Let Xh�Xm�J1�*� be defined by Xh=�mhS*−1XS. Therefore,

from the equation i�X��= �−1�m+1� and the condition jS*�=0 �which holds because S is an

integral submanifold of D��, we obtain



0 = h*�i�X�� − �− 1�m+1�� = �jS � hS�*�i�X�� − �− 1�m+1�� = �hS* � jS

*��i�X�� − �− 1�m+1��

= hS*�i�XS�jS

*� − �− 1�m+1jS*�� = i�Xh��hS

* � jS*�� = i�Xh�h*� = i�Xh��h.

Furthermore, bearing in mind that � �h=IdJ1�*, we have that

i�Xh��*�� = i�Xh�� h�*�� = h*�i�X��*�� ,

and, if i�X��*��=1, this equality holds, in particular, at the points of the image of h; therefore,i�Xh��*��=1. Then, Xh is the desired multivector field since Xh�S=XS=�mhS�Xh.

Finally, if X� is integrable, as it is tangent to S, the integral sections of X� passing through anypoint of S remain in S, and hence they are the integral sections of XS, so X� is integrable and, asa consequence, Xh is integrable too. �

All of these properties lead to establish the following.Definition 6: Given an extended global Hamiltonian system �M� ,� ,�� and considering all

the Hamiltonian sections h :J1�*→M� such that Im h are integral submanifolds of D�, we havea family ��J1�* ,h��, which will be called the class of restricted Hamiltonian systems associatedwith �M� ,� ,��.

As it is obvious, in general, the above result holds only locally.The following result show how to obtain extended Hamiltonian systems from restricted

Hamiltonian ones, at least locally.Proposition 7: Given a restricted Hamiltonian system �J1�* ,h�, let jS :S=Im h→M� be the

natural embedding. Then, there exists a unique local form ��1�M�� such that

(1) ��Z1�M�� (it is a closed form);(2) jS

*�=0;(3) i�Y��0, for every nonvanishing Y �XV��M�� and, in particular, such that i�� /�p��

=1, for every system of natural coordinates �x� ,yA , pA� , p� in M�, adapted to the bundle

� :E→M (with �=dmx).

Proof: Suppose that there exist �, �� satisfying the above conditions. Taking into account thecomments after Proposition 1, we have that locally in U�M�, �=dp+� and ��=dp+��, where

�=�*�, ��=�*�, with �, ��B1��U�� they are exact 1-forms�. From condition �2� in thestatement, we have that jS

*�=jS*��; hence,

0 = jS*�� − �� ⇔ 0 = jS

*�� − �� = �� jS�*�� − �� ⇒ 0 = �� jS � hS�*�� − �� ⇒ � − �� = 0 ⇔ �

= �� ⇔ � = ��

since � �jS �hS=� �h=Id��U�. This proves the uniqueness.The existence is trivial since, locally, every section h of � is given by a function h

�C��U�� such that p=−h�x� ,yA , pA��. Hence, ��U�=dp+d��*h��dp+dh. �

Definition 7: Given a restricted Hamiltonian system �J1�* ,h�, let ��1�M�� be the localform satisfying the conditions in the above proposition. The couple �M� ,�� will be called the�local� extended Hamiltonian system associated with �J1�* ,h�.

As a consequence of the last proposition, if �=dp+�*�, there exists a class �h�C��U�� /R, such that �=dh, where h is a representative of this class.

Corollary 1: Let � be the unique local 1-form verifying the conditions of Proposition 7,associated with a section h. Consider its characteristic distribution D� and let �h� the family oflocal sections of � such that Im h�S are local integral submanifolds of D�. Then, for every h�

��h�, we have that Im h� is locally a level set of the function H= p+�*h� p+h.Proof: If S=Im h� then, for every p�S, we have TpS= �D��p, which is equivalent to d�p

+�*h��TpS=0, and this holds if and only if H�S��p+�*h��S=ctn.Bearing in mind these considerations, we can finally prove the following.Proposition 8: Let �M� ,� ,�� be an extended Hamiltonian system and ��J1�* ,h�� the class



of restricted Hamiltonian systems associated with �M� ,� ,��. Consider the submanifolds �Sh

=Im h, for every Hamiltonian section h in this class, and let jSh:Sh→M� be the natural em-

beddings. Then, the submanifolds �Sh ,jSh

* �� are multisymplectomorphic.Proof: Let h1, h2� �h and S1=Im h1, S2=Im h2. We have the diagram

Denote �1=jS1

* �, �2=jS2

* �. As a consequence of the above corollary, if �h1=h1

*�, �h2

=h2*�, we have that �h1

=�h2. However,

�h1= �h2

⇔ �jS1� hS1

�*� = �jS2� hS2

�*� ⇔ hS1

* �1 = hS2

* �2.

Then, the map �¬hS2��1 :S1→S2 is a multisymplectomorphism. In fact, it is obviously a dif-

feomorphism and

�*�2 = �hS2� �1�*�2 = �1

*hS2

* �2 = �1*hS1

* �1 = �hS1� �1�*�1 = �1 . �

As an immediate consequence of this, if X��Xm�M�� is a solution to Eq. �8�, the multivec-tor fields XSh

�Xm�Sh� such that �m�jSh�*XSh

=X��S, for every submanifold Sh of this family �seethe proof of Theorem 5�, are related by these multisymplectomorphisms.

D. Variational principle and field equations

As in the case of restricted Hamiltonian systems, the field equations for extended Hamiltoniansystems can be derived from a suitable variational principle.

First, denote by X��M�� the set of vector fields Z�X�M�� which are sections of thesub-bundle D� of TM�, that is, satisfying that i�Z��=0 �and hence they are tangent to all theintegral submanifolds of D��. Let X�

V��M��X��M�� be those which are also � vertical.

Furthermore, as we have seen in previous sections, the image of the sections � :M→M�,which are solutions to the extended field equations, must be in the integral submanifolds of thecharacteristic distribution D�, that is, they are also integral submanifolds and hence j

�

*�=0 �where

j� : Im �→M� is the natural embedding�. We will denote by ��M ,M�� the set of sections of �

satisfying that j�

*�=0.

Taking all of this into account, we can state the following.Definition 8: Let �M� ,� ,�� be an extended Hamiltonian system. Consider the map

H�:��M,M�� → R ,

� � �M

�*�

�where the convergence of the integral is assumed�. The variational problem for this extended

Hamiltonian system is the search for the critical �or stationary� sections of the functional H�, with

respect to the variations of ��M ,M�� given by �t=�t � �, where ��t is the local one-parameter group of any compact-supported vector field Z�X�

V��M��, that is,



� d

dt�

t=0�

M

�t*� = 0.

This is the extended Hamilton-Jacobi principle.Observe that, as � is closed, the variation of the set ��M ,M�� is stable under the action of

X�V��M��. In fact, being � closed, for every Z�X�

V��M��, we have that L�Z��= i�Z�d�

+di�Z��=0, that is, �t*�=�. Hence, if ��M ,M��, we obtain

�t*� = ��t � ��*� = �*�t

*� = �*� = 0.

Then, we have the following fundamental theorems.Theorem 6: Let �M� ,� ,�� be an extended Hamiltonian system. The following assertions on

a section ��M ,M�� are equivalent:

(1) � is a critical section for the variational problem posed by the extended Hamilton-Jacobiprinciple;

(2) �*i�Z��=0, for every Z�X�V��M��;

(3) �*i�X��=0, for every X�X��M��;(4) if �U ;x� ,yA , pA

� , p� is a natural system of coordinates in M�, then � satisfies the followingsystem of equations in U:

��yA � ��x� =

�h

�pA� � �,

��pA� � ��x� = −

�h

�yA � �,��p � ��

�x� = −��h � ��

�x� , �17�

where h=�*h, for some h�C��U��, is any function such that ��U=dp+dh�x� ,yA , pA��.

These are the extended Hamilton–De Donder–Weyl equations of the extended Hamiltoniansystem.

Proof: �1⇔2� Let Z�X�V��M�� be a compact-supported vector field and V�M an open set

such that �V is an �m−1�-dimensional manifold and that ��supp�Z��V. Then,

� d

dt�

t=0�

M

�t*� = � d

dt�

t=0�

V

�t*� = � d

dt�

t=0�

V

�*��t*�� = �

V

�* limt→0

�t*� − �

t� = �

V

�*�L�Z��

= �V

�*�i�Z�d� + di�Z�� = − �V

�*�i�Z�� − di�Z�� = − �V

�*�i�Z��

+ �V

d��*�i�Z�� = − �V

�*�i�Z�� + ��V

�*�i�Z�� = − �V

�*�i�Z��

�as a consequence of Stoke’s theorem and the hypothesis made on the supports of the verticalfields�. Thus, by the fundamental theorem of the variational calculus, we conclude that

d /dt�t=0�V�t*�=0 if and only if �*�i�Z��=0, for every compact-supported Z�X�

V��M��. How-ever, as compact-supported vector fields generate locally the C��M�� module of vector fields inM�, it follows that the last equality holds for every Z�X�

V��M��.�2⇔3� If p� Im � and S is the integral submanifold of D� passing through p, then

�D��p = �Vp�� D��p� � Tp�Im �� .

So, for every X�X��M��,

Xp = �Xp − Tp�� Xp�� + Tp�� Xp� � XpV + Xp

�

and therefore



�*�i�X�� = �*�i�XV�� + �*�i�X�� = �*�i�X�� = 0

since �*�i�XV��=0 by the above item. Furthermore, Xp��Tp�Im �� and dim�Im ��=m, with �

��m+1�M��. Hence, we conclude that �*�i�X��=0, for every X�X��M��. The converse isobvious.

�3⇔4� The local expression of any X�X��M�� is

X = ��

�x� + �A �

�yA + �A� �

�pA� − �� h

�x� + �B �h

�yB + �B� �h

�pB��

�p.

Then, taking into account the local expression �2� of �, if �= �x� ,yA�x�� , pA��x�� , p�x��, we obtain

�*i�X�� = �� p � ��x� + � �h

�x��

�

− ��

� �h

�pA� �

�

��pA� � ��x� − � �h

�pA��

�

��pA� � ��x� ��dmx

+ �A ��pA� � ��x� + � �h

�yA��dmx + �A

� −��yA � ��

�x� + � �h

�pA� �

��dmx

and, as this holds for every X�X��M�� i.e., for every �� ,�A ,�A��, we conclude that �*i�X��h

=0 if and only if

��yA � ��x� = � �h

�pA� �

�

,��pA

� � ��x� = − � �h

�yA��

, �18�

��p � ��x� = − � �h

�x��

�

+ ��

� �h

�pA� �

�

��pA� � ��x� − � �h

�pA��

�

��pA� � ��x� � . �19�

Using Eqs. �18� in �19�, we obtain

��p � ��x� = − � �h

�x��

�

+ � �h

�pA� �

�

��pA� � ��x� −

��pA� � ��x� � − � �h

�pA� �

�

��pA� � ��x�

− � �h

�pA� �

�

��pA� � ��x� � = − � �h

�x��

�

− � �h

�yA��

��yA � ��x� − � �h

�pA� �

�

��pA� � ��x�

= −��h � ��

�x� �� = 1, . . . ,m;� fixed� . �

Remark 10: It is important to point out that the last group of Eqs. �17� are consistency

conditions with respect to the hypothesis made on the sections �. In fact, this group of equations

leads to p � �=−h � �+ctn or, which means the same thing, ��M ,M�� see also the commentin Remark 9�. The rest of Eqs. �17� are just the Hamilton–De Donder–Weyl equations �5� of the

restricted case since the local expressions of the functions h and h are the same.Theorem 7: Let �M� ,� ,�� be an extended Hamiltonian system and X�Xm�M�� an inte-

grable multivector field verifying the condition i�X��*��=1. Then, the integral manifolds of Xare the critical section for the variational problem posed by the extended Hamilton-Jacobi prin-ciple if and only if X satisfies the condition i�X��= �−1�m+1�.

Proof: �⇐� Let S be an integral submanifold of X. By the �-transversality conditioni�X��*��=1, S is locally a section of �. Then, for every p�S, there are an open set U�M, with

��p��U, and a local section � :U�M→M� of �, such that Im �= �S��−1�U�. Now, let q�U andu1 , . . . ,um�TqM, with i�u1∧ ¯ ∧um��p��=1. Then, there exists ��R such that



�*�u1 ∧ ¯ ∧ um� = �X��q��.

Therefore,

i��*�u1 ∧ ¯ ∧ um��q�� = ��− 1�m+1��q�� .

Thus, for every X�X��M��, we obtain that

i�X��q��i��*�u1 ∧ ¯ ∧ um��q�� = 0.

Hence, �*i�X��=0, for every X�X��M��, and � is a critical section by the third item of the lasttheorem.

�⇒� Let p�M�; by the hypothesis, there exists a section � :M→M� such that

�1� ��p��=p;

�2� � is a critical section for the extended Hamilton-Jacobi variational problem, that is,

�*i�X��=0, for every X�X��M��;�3� Im � is an integral submanifold of X.

Now, let u1 , . . . ,um�T��p�M, with i�u1∧ ¯ ∧um��p��=1. Then, there exists ��R suchthat

�*�u1 ∧ ¯ ∧ um� = �Xp,

but the condition imposed to u1 , . . . ,um leads to �=1. Therefore,

i��*�u1 ∧ ¯ ∧ um��p�� = i�Xp��p�� .

Thus, for every X�X��M��, as � is a critical section, we obtain that

i�Xp�i�Xp��p�� = 0

and hence i�X�i�X��=0, for every X�X��M��. This implies that i�X��= f�, for some nonva-nishing f �C��M��. Nevertheless, as �M� ,� ,�� is an extended Hamiltonian system, in any

local chart, we have that �=dp+dh�x� ,yA , pA��, which by the condition i�X��*��=1, and bearing

in mind the local expression of �, leads to f = �−1�m+1. So the result holds. �

Observe that the extended Hamilton–De Donder–Weyl equations �17� can also be obtained asa consequence of this last theorem, taking into account Eqs. �13�–�16�.

V. ALMOST-REGULAR HAMILTONIAN SYSTEMS

There are many interesting cases in Hamiltonian field theories where the Hamiltonian fieldequations are established not in J1�* but rather in a submanifold of J1�* �for instance, whenconsidering the Hamiltonian formalism associated with a singular Lagrangian�. Next, we considerthis kind of systems in J1�*, as well as in M�.

A. Restricted almost-regular Hamiltonian systems

Definition 9: A restricted almost-regular Hamiltonian system is a triple �J1�* ,P ,hP�, where

(1) P is a submanifold of J1�* with dim P�n+m and such that if jP :P�J1�* denotes thenatural embedding, the map �P=� �jP :P→E is a surjective submersion (and hence so is themap �P= � �jP=� ��P :P→M);

(2) hP :P→M� satisfies that � �hP=jP, and it is called a Hamiltonian section of � on P.



Then, the differentiable forms

�hPª hP

* �, �hPª − d�hP

= hP* �

are the Hamilton-Cartan m- and �m+1�-forms on P associated with the Hamiltonian section hP.We have the diagram

Remark 11: Notice that �hPis, in general, a 1-degenerate form and hence it is premultisym-

plectic. This is the main difference with the regular case.Furthermore, if we make the additional assumption that P→E is a fiber bundle, the Hamilton-

Jacobi variational principle of Definition 2 can be stated in the same way, now using sections of�P :P→M and the form �hP

. So we look for sections �P��M ,P� which are stationary withrespect to the variations given by �t=�t ��P, where ��t is a local one-parameter group of anycompact-supported �P-vertical vector field ZP�X�P�, i.e., such that

� d

dt�

t=0�

M

�t*�hP

= 0.

Then, these critical sections will be characterized by the condition �analogous to Theorem 1�

�P* i�XP��hP

= 0 for every XP � X�P� .

As in the case of restricted Hamiltonian systems �Theorem 2�, we have the following.Theorem 8: The critical sections of the Hamilton-Jacobi principle are the integral sections

�P��M ,P� of a class of integrable and �P-transverse multivector fields �XhP�Xm�P� satisfy-

ing that

i�XhP��hP

= 0 for every XhP� �XhP

or, equivalently, the integral sections of an integrable multivector field XhP�Xm�P� such that

(1) i�XhP��hP

=0;(2) i�XhP

��P* ��=1.



A multivector field XhP�Xm�P� will be called a Hamilton–De Donder–Weyl multivector field

for the system �J1�* ,P ,hP� if it is �P transverse, locally decomposable, and verifies the equationi�XhP

��hP=0. Then, the associated connection �hP

, which is a connection along the submanifoldP �see Refs. 30, 31, and 34�, is called a Hamilton–De Donder–Weyl connection for �J1�* ,P ,hP�and satisfies the equation

i��hP��hP

= �m − 1��hP.

Remark 12: It should be noted that, as �hPcan be 1-degenerate, the existence of the corre-

sponding Hamilton–De Donder–Weyl multivector fields for �J1�* ,P ,hP� is in general not assuredexcept perhaps on some submanifold S of P, where the solution is not unique. A geometricalgorithm for determining this submanifold S has been developed.32

B. Extended almost-regular Hamiltonian systems

Definition 10: An extended almost-regular Hamiltonian system is a triple �M� , P ,�P� suchthat

(1) P is a submanifold of M� and if jP : P�M� denotes the natural embedding, then

(a) �P=� � jP : P→E is a surjective submersion (and hence so is the map �P= � � jP=�

��P : P→M),

(b) �� jP��P��P is a submanifold of J1�*, and

(c) P=M��P�, that is, for every p� P we have that �−1��p��=p+�1mT��p�

* E� P;

(2) �P�Z1�P� (it is a closed 1-form in P);

(3) there exists a locally decomposable multivector field X�P�Xm�P� satisfying that

i�X�P��P = �− 1�m+1�P, i�X�P

��P*

�� = 1 ��P transversality� , �20�

where �P= jP*

�.

If �P is an exact form, then �M� , P ,�P� is an extended almost-regular global Hamiltonian

system. In this case, there exist functions HP�C��P�, which are called Hamiltonian functions ofthe system, such that �P=dHP. (For an extended Hamiltonian system, these functions exist onlylocally and they are called local Hamiltonian functions).

Remark 13: As straightforward consequences of this definition, we have the following.

• condition �1�-�c� of Definition 20 imply, in particular, that dim P�dim E+1. Furthermore, it

means that P is the union of fibers of �.• � � jP is a surjective submersion if and only if � �� jP. This means that P� Im�� jP� is a

submanifold verifying the conditions stated in the first item of Definition 9 and such that

dim P=dim P−1, as a consequence of the properties given in item �1� of Definition 10. This

submanifold is diffeomorphic to P /�1mT*E.

Denoting �P=� � jP : P→J1�* and �P : P→P its restriction to the image �that is, such that�P= jP � �P�, we have the diagram



Remark 14: In addition, as for extended Hamiltonian systems �see Remarks 5 and 6�, theintegrability of X�P

is not assured, so it must be imposed. Then, all the multivector fields in the

integrable class �X�P have the same integral sections.

As in Propositions 1 and 2, we have the following.

Proposition 9: If �M� , P ,�P� is an extended almost-regular Hamiltonian system, then

i�YP��P�0, for every nonvanishing �P-vertical vector field YP�XV��P��P�. In particular, for

every system of natural coordinates in P adapted to the bundle� :E→M (with �=dmx),

i �

�p��P = 1.

Proof: As a consequence of condition �1�-�c� of Definition 10, we have the local expression

�P = jP*

� = jP* �− dpA

� ∧ dyA ∧ dm−1x� − dp ∧ dmx� = jP* �− dpA

�� ∧ dyA ∧ dm−1x� − dp ∧ dmx

and YP= f�� /�p�, for every �-vertical vector field in P. Therefore, the proof follows the samepattern as in the proof of Proposition 1.

The last part of the proof is a consequence of condition �1�-�c� given in Definition 10, from

which we have that every system of natural coordinates in P adapted to the bundle � :E→M

contains the coordinate p of the fibers of � and the coordinates �x�� in E. This happens because Preduces only the degrees of freedom in the coordinates pA

� of M�. �

Proposition 10: If �M� , P ,�P� is an extended Hamiltonian system, locally �P=dp+�P,

where �P is a closed and �P-basic local 1-form in P.

A multivector field X�P�Xm�P� will be called an extended Hamilton–De Donder–Weyl mul-

tivector field for the system �M� , P ,�P� if it is �P transverse, locally decomposable, and verifiesthe equation i�X�P

��P= �−1�m+1�P. Then, the associated connection ��Pis called a Hamilton–De

Donder–Weyl connection for �M� , P ,�P�,Remark 15: Notice that �P is usually a 1-degenerate form and hence premultisymplectic.As a consequence, the existence of extended Hamilton–De Donder–Weyl multivector fields

for �M� , P ,�P� is not assured, except perhaps on some submanifold S of P, where the solutionis not unique.



C. Geometric properties of extended almost-regular Hamiltonian systems: Variationalprinciple

Let �M� , P ,�P� be an extended almost-regular Hamiltonian system and the submanifold

�P�P��P. As for the general case, we can define the characteristic distribution D�Pof �P. Then,

following the same pattern as in the proofs of the propositions and theorems given in Sec. IV B,we can prove the following.

Proposition 11:

(1) D�Pis an involutive and �P-transverse distribution of corank equal to 1.

(2) The integral submanifolds of D�Pare �P-transverse submanifolds of M�, with dimension

equal to dim P−1. (We denote by jS :S� P the natural embedding.)

(3) For every p� P, we have that TpP=Vp��P� � �D�P�p and thus, in this way, �P defines a

connection in the bundle �P : P→P.(4) If S is an integral submanifold of D�P

, then ��P�S :S→P is a local diffeomorphism.

(5) For every integral submanifold S of D�Pand p�S, there exists W� P, with p�W, such that

h= ��W�S�−1 is a local Hamiltonian section of �P defined on �P�W�S�.

If �M� , P ,�P� is an extended almost-regular Hamiltonian system, as �P=dHP �locally�,every local Hamiltonian function HP is a constraint defining the local integral submanifolds of DP.

If �M� , P ,�P� is an extended almost-regular global Hamiltonian system, the Hamiltonian func-tions HP are globally defined, and we have the following.

Proposition 12: Let �M� , P ,�P� be an extended almost-regular global Hamiltonian system.

If there is a global Hamiltonian function HP�C��P� and k�R, such that ��HP−1�k��=P, then

there exists a global Hamiltonian section hP��P , P�.Proposition 13: Given an extended almost-regular Hamiltonian system �M� , P ,�P�, every

extended HDW-multivector field X�P�Xm�P� for the system �M� , P ,�P� is tangent to every

integral submanifold of D�P.

At this point, the extended Hamilton-Jacobi variational principle of Definition 8 can be stated

in the same way, now using sections �P of �P : P→M, satisfying that j�P* �P=0 �where

j�P: Im �P� P denotes the natural embedding�. Thus, using the notation introduced in Sec. IV D,

we look for sections �P��P�M , P� which are stationary with respect to the variations given by

�t=�t ��P, where ��t is a local one-parameter group of every compact-supported Z�X�P

V��P��P�,

that is,

� d

dt�

t=0�

U

�t*�P = 0.

Then, the statements analogous to Theorems 6 and 7 can be established and proven in the presentcase.

D. Relation between extended and restricted almost-regular Hamiltonian systems

Finally, we study the relation between extended and restricted almost-regular Hamiltoniansystems. �The proofs of the following propositions and theorems are analogous to those in Sec.IV C.�

First, bearing in mind Remark 13, we have the following.

Theorem 9: Let �M� , P ,�P� be an extended global Hamiltonian system and �J1�* ,P ,hP� a



restricted Hamiltonian system such that dim P=dim P+1, and Im hP=S is an integral submani-

fold of D�P. Then, for every X�P

�Xm�P� solution to the equations

i�X�P��P = �− 1�m+1�P, i�X�P

��P*

�� = 1

[i.e., an extended HDW-multivector field for �M� , P ,�P�], there exists XhP�Xm�P� which is hP

related with X� and is a solution to the equations

i�XhP��hP

= 0, i�XhP��P

* �� = 1

[i.e., a HDW multivector field for �J1�* ,P ,hP�]. Furthermore, if X�Pis integrable, then XhP

is

integrable too.As a consequence, the following definition can be established.

Definition 11: Given an extended almost-regular global Hamiltonian system �M� , P ,�P� andconsidering all the Hamiltonian sections hP :P→M� such that Im hP are integral submanifoldsof D�P

, we have a family ��J1�* ,P ,hP��P, which will be called the class of restricted almost-

regular Hamiltonian systems associated with �M� , P ,�P�.Remark 16: Observe that, for every Hamiltonian section hP in this class, Im hP is a submani-

fold of P. Therefore, we have induced a Hamiltonian section hP :P→P of �P such that hP=jP�hP.

Proposition 14: Let �J1�* ,P ,hP� be a restricted almost-regular Hamiltonian system.

(1) There exists a unique submanifold P of M� satisfying the conditions of Definition 10 and

such that �P�P�=P.

(2) Consider the submanifold S=Im hP and the natural embedding jS :S=Im hP� P. Then, there

exists a unique local form �P��1�P� such that:

(a) �P�Z1�P� (it is a closed form),

(b) jS*�P=0, and

(c) i�YP��P�0, for every nonvanishing YP�XV��P��P�, and, in particular, such that

i�� /�p��P=1, for every system of natural coordinates in P, adapted to the bundle � :E→M (with �=dmx).

Proof: The existence and uniqueness of the submanifold P is assured since it is made of all thefibers of � at every point of P. The rest of the proof is like in Proposition 7. �

So we have the diagram

Bearing in mind Remark 16, we can also state the following.

Corollary 2: Let �P ,�P� be the couple associated with a given restricted almost-regularHamiltonian system �J1�* ,P ,hP� by Proposition 14. Consider the characteristic distribution D�P



and let �h�Pbe the family of local sections of �P such that Im h�S are local integral submani-

folds of D�P. Then, for every hP� � �h�P

, we have that Im hP� is locally a level set of a function HPsuch that ��P�S= ��dHP��S, locally.

Definition 12: Given a restricted almost-regular Hamiltonian system �J1�* ,P ,hP�, let �P ,�P�be the couple associated with �J1�* ,P ,hP� by Proposition 14. The triple �M� , P ,�P� will becalled the (local) extended almost-regular Hamiltonian system associated with �J1�* ,h�.

Proposition 15: Let ��J1�* ,P ,hP� be the class of restricted almost-regular Hamiltonian

systems associated with an extended almost-regular Hamiltonian system �M� , P ,�P�. Consider

the submanifolds �ShP=Im hP, for every Hamiltonian section h in this class, and let jShP

:ShP� P

be the natural embeddings. Then, the submanifolds �S=Im hP, for every Hamiltonian section hPin this class, are premultisymplectomorphic.

As a consequence of this, if X�P�Xm�P� is a solution to Eq. �20�, the multivector fields

XShP�Xm�ShP

� such that �m�jShP�*XShP

= �X�P�ShP

, for every submanifold ShPof this family, are

related by these presymplectomorphisms.

VI. EXAMPLES

A. Restricted Hamiltonian system associated with a Lagrangian system

A particular but relevant case concerns �first-order� Lagrangian field theories and their Hamil-tonian counterparts.

In field theory, a Lagrangian system is a couple �J1� ,�L�, where J1� is the first-order jetbundle of � :E→M and �L��m+1�J1�� is the Poincaré-Cartan �m+1�-form associated with theLagrangian density L describing the system �L is a �1-semibasic m-form on J1�, which is writtenas L= £ �1*�� £�, where £�C��J1�� is the Lagrangian function associated with L and ��. TheLagrangian system is regular if �L is 1-nondegenerate; elsewhere, it is singular.

The extended Legendre map associated with L, FL :J1�→M�, is defined by

�FL˜�y��Z1, . . . ,Zm� ª ��L�y�Z1, . . . ,Zm� ,

where Z1 , . . . ,Zm�T�1�y�E and Z1 , . . . , Zm�TyJ1� are such that Ty�

1Z�P=Z�P

. �FL can also be

defined as the “first order vertical Taylor approximation to £.”6� We have that FL*�=�L. If�x� ,yA ,v�

A� is a natural chart of coordinates in J1� �adapted to the bundle structure and such that�=dx1∧ ¯ ∧dxm�dxm�, the local expressions of these maps are

FL˜*x� = x�, FL˜*yA = yA, FL˜*pA� =

�£

�v�A , FL˜*p = £ − v�

A �£

�v�A ,

FL*x� = x�, FL*yA = yA, FL*pA� =

�£

�v�A .

Using the natural projection � :M�→J1�*, we define the restricted Legendre map associated

with L as FLª� �FL.Then, �J1� ,�L� is a regular Lagrangian system if FL is a local diffeomorphism �this defi-

nition is equivalent to that given above�. Elsewhere, �J1� ,�L� is a singular Lagrangian system. Asa particular case, �J1� ,�L� is a hyper-regular Lagrangian system if FL is a global diffeomor-phism. Finally, a singular Lagrangian system �J1� ,�L� is almost regular if PªFL�J1�� is aclosed submanifold of J1�*, FL is a submersion onto its image, and for every y�J1�, the fibersFL−1�FL�y�� are connected submanifolds of J1E.

If �J1� ,�L� is a hyper-regular Lagrangian system, then FL�J1�� is a 1-codimensional em-bedded submanifold of M�, which is transverse to the projection �, and is diffeomorphic to J1�*.



This diffeomorphism is �−1, when � is restricted to FL�J1��, and also coincides with the map

hªFL �FL−1, when it is restricted onto its image �which is just FL�J1��. This map h is theHamiltonian section needed to construct the restricted Hamiltonian system associated with�J1� ,�L�. In other words, the Hamiltonian section h is given by the image of the extendedLegendre map.

Using charts of natural coordinates in J1�* and M�, we obtain that the local Hamiltonianfunction h representing this Hamiltonian section is

h�x�,yA,pA�� = �FL−1�* v�

A �£

�v�A − £ � = pA

��FL−1�*v�A − FL−1*

£ .

Of course, if �M� ,� ,�� is any extended Hamiltonian system associated with �J1�* ,h�, then

FL�J1�� is an integral submanifold of the characteristic distribution of �.In an analogous way, if �J1� ,�L� is an almost-regular Lagrangian system and the submani-

fold jP :P�J1�* is a fiber bundle over E and M, the �-transverse submanifold

j :FL�J1��M� is diffeomorphic to P. This diffeomorphism � :FL�J1��→P is just the re-

striction of the projection � to FL�J1��. Then, taking the Hamiltonian section hPªj � �−1, wedefine the Hamilton-Cartan forms

�hP= hP

* �, �hP= hP

* � ,

which verify that FLP* �h

0=�L and FLP* �h

0=�L �where FLP is the restriction map of FL ontoP�. Once again, this Hamiltonian section hP is given by the image of the extended Legendre map.Then, �J1�* ,P ,hP� is the Hamiltonian system associated with the almost-regular Lagrangiansystem �J1� ,�L�, and we have the following diagram:

The construction of the �local� extended almost-regular Hamiltonian system associated with�J1�* ,P ,hP� can be made by following the procedure described in Sec. V D. Of course, if

�M� , P ,�P� is the extended Hamiltonian system associated with �J1�* ,P ,hP�, then FL˜�J1�� isan integral submanifold of the characteristic distribution of �P.

B. Nonautonomous dynamical systems

Another example consists in showing how the so-called extended formalism of time-dependentmechanics �see Refs. 17, 29, 36, 42, and 46� can be recovered from this more general framework.

The starting point consists in giving the configuration bundle, which for a large class ofnonautonomous dynamical systems can be taken to be � :E�Q�R→R, where Q is an-dimensional differentiable manifold endowed with local coordinates �qi� and R has as a globalcoordinate t. The extended and restricted momentum phase spaces are

M� T*E � T*�Q � R� T*Q � R � R*, J1�* T*Q � R .

Then, the following projections can be defined:



pr1:M� → T*Q, �:M� → T*Q � R ,

pr2:M� → R � R*, p:M� → R*.

If �Q�Z2�T*Q� and �R�Z2�R�R*� denote the natural symplectic forms of T*Q and R�R*,then the natural symplectic structure of M� is just

� = pr1*�Q + pr2

*�R.

Then, we define the so-called extended time-dependent Hamiltonian function

H ª �*h + p � C��T*�Q � R�� ,

where the dynamical information is given by the “time-dependent Hamiltonian function”h�C��T*Q�R�.

Now, we have that �T*�Q�R� ,� ,��, with �=dH, is an extended global Hamiltonian system,and then the equations of motion are

i�XH�� = dH, i�XH�dt = 1 with XH � X�T*�Q � R�� . �21�

In order to analyze the information given by this equation, we take a local chart of coordinates�qi , pi , t , p� in T*�Q�R�, and one can check that the unique solution to these equations is

XH =�H

�pi

�

�qi −�H

�qi

�

�pi+

�

�t−

�H

�t

�

�p=

��*h��pi

�

�qi −��*h�

�qi

�

�pi+

�

�t−

��*h��t

�

�p. �22�

If ��t�= �qi�t� , pi�t� , t , p�t�� denote the integral curves of this vector field, the last expression leadsto the following system of extended Hamiltonian equations:

d�qi � ��dt

=��*h�

�pi� �,

d�pi � ��dt

= −��*h�

�qi � �,d�p � ��

dt= −

��*h��t

� � . �23�

Observe that the last equation corresponds to the last group of Eq. �17� in the general case of fieldtheories. In fact, using the other Hamilton equations, we get

d�p � ��dt

= −��*h � ��

dt= − � ��*h�

�t�

�

− � ��*h��qi �

�

��qi � ��t

− � ��*h��pi

��

��pi � ��t

= − � ��*h��t

��

− � ��*h��qi �

�� *h�

�pi�

�

+ � ��*h��pi

�� *h�

�qi ��

= − � ��*h��qi �

�

.

However, as the physical states are the points of T*Q�R and not those of T*�Q�R�, thevector field which gives the real dynamical evolution is not XH, but the other one in T*Q�Rwhich, as XH is � projectable, is just �*XH=Xh�X�T*Q�R�, that is, in local coordinates �qi , pi , t�of T*Q�R,

Xh =�h

�pi

�

�qi −�h

�qi

�

�pi+

�

�t. �24�

Thus, the integral curves ��t�= �qi�t� , pi�t� , t� of Xh are the � projection of those of XH and they aresolutions to the system of Hamilton equations

d�qi � ��dt

=�h

�pi� �,

d�pi � ��dt

= −�h

�qi � � .



This result can also be obtained by considering the class of restricted Hamiltonian systemsassociated with �T*�Q�R� ,� ,dH�. In fact, T*�Q�R� is foliated by the family of hypersurfaces ofT*�Q�R� where the extended Hamiltonian function is constant, that is,

S ª �p � T*�Q � R��H�p� = r�ctn� ,

which are the integral submanifolds of the characteristic distribution of �=dH. Thus, every S isdefined in T*�Q�R� by the constraint ªH−r, and the vector field given in �22�, which is thesolution to �21�, is tangent to all of these submanifolds. Then, taking the global Hamiltoniansections,

h:�qi,pi,t� � �qi,pi,t,p = r − �*h� ,

we can construct the restricted Hamiltonian systems �T*Q�R ,h� associated with �T*�Q�R� ,� ,dH�. Therefore, �24� is the solution to the equations

i�Xh��h = 0, i�Xh�dt = 1 with XH � X�T*�Q � R�� ,

where

�h = h*� = �Q − dh ∧ dt � �2�T*Q � R� .

The dynamics on each one of these restricted Hamiltonian systems is associated to a givenconstant value of the extended Hamiltonian. Observe also that, on every submanifol S, the globalcoordinate p is identified with the physical energy by means of the time-dependent Hamiltonianfunction �*h, and hence the last equation �23� shows the known fact that the energy is notconserved on the dynamical trajectories of time-dependent systems.

In this way, we have also recovered one of the standard Hamiltonian formalisms for time-dependent systems �see Ref. 11�.

VII. CONCLUSIONS AND OUTLOOK

The usual way of defining Hamiltonian systems in first-order field theory consists in workingin the restricted multimomentum bundle J1�*, which is the natural multimomentum phase spacefor field theories, but J1�* has no a natural multisymplectic structure. Thus, in order to definerestricted Hamiltonian systems, we use Hamiltonian sections h :J1�*→M�, which carry the“physical information” and allow us to pullback the natural multisymplectic structure of M� toJ1�*. In this way, we obtain the Hamilton-Cartan form �h��m+1�J1�*�, and then the Hamil-tonian field equations can be derived from the Hamilton-Jacobi variational principle. As a conse-quence, both the geometry and the physical information are coupled in the noncanonical multi-symplectic form �h.

The alternative way that we have introduced consists in working directly in the extendedmultimomentum bundle M�, which is endowed with a canonical multisymplectic structure ��m+1�M��. Then, we define extended Hamiltonian systems as a triple �M� ,� ,��, where ��Z1�M�� is a �-transverse closed form, and the Hamiltonian equation is i�X��= �−1�m+1�, withX�Xm�M��. Thus, in these models, the geometry � and the physical information � are notcoupled, and geometric field equations can be expressed in an analogous way to those of mechani-cal autonomous Hamiltonian systems.

The characteristic distribution D� associated with �, being involutive, has 1-codimensionaland �-transverse integrable submanifolds of M�, where the section solutions to the field equa-tions are contained. These integrable submanifolds can be locally identified with local sections ofthe affine bundle � :M�→J1�*. Each one of them allows us to define locally a restricted Hamil-tonian system, although all those associated with the same form � are, in fact, multisymplecto-morphic. The conditions for the existence of global Hamiltonian sections have also been analyzed.Conversely, every restricted Hamiltonian system is associated with an extended Hamiltonian sys-tem �at least locally�.



In addition, the extended Hamiltonian field equations can be obtained from an extendedHamilton-Jacobi variational principle, stated on the set of sections of the bundle � :M�→M,which are integral sections of the characteristic distribution of �, taking the variations given by theset of the �-vertical vector fields incident to �. In fact, a part of the local system of differentialequations for the critical sections of an extended Hamiltonian system is the same as for theassociated restricted Hamiltonian system. Nevertheless, there is another part of the whole systemof differential equations which leads to the condition that the critical sections must also be integralsubmanifolds of the characteristic distribution D�.

Restricted and extended Hamiltonian systems for submanifolds of J1�* and M� �satisfyingsuitable conditions� have been defined in order to include the almost-regular field theories in thispicture. Their properties are analogous to the former case.

The extended Hamiltonian formalism has already been used for defining Poisson brackets infield theories.14 It could provide new insights into some classical problems, such as the reductionof multisymplectic Hamiltonian systems with symmetry, integrability, and quantization of multi-symplectic Hamiltonian field theories.

ACKNOWLEDGMENTS

We acknowledge the financial support of Ministerio de Educación y Ciencia, Project Nos.BFM2002-03493, MTM2004-7832, and MTM2005-04947. We wish to thank Mr. Jeff Palmer forhis assistance in preparing the English version of the manuscript. We thank the referee for hissuggestions and comments.

1 Aldaya, V., and de Azcárraga, J. A., “Geometric formulation of classical mechanics and field theory,” Riv. NuovoCimento 3, 1–66 �1980�.

2 Awane, A., “k-symplectic structures,” J. Math. Phys. 32, 4046–4052 �1992�.3 Binz, E., Sniatycki, J., and Fisher, H., The Geometry of Classical Fields �North-Holland, Amsterdam, 1988�.4 Cantrijn, F., Ibort, A., and de León, M., “Hamiltonian Structures on Multisymplectic manifolds,” Rend. Semin. Mat.Torino 54, 225–236 �1996�.

5 Cantrijn, F., Ibort, L. A., and de León, M., “On the Geometry of Multisymplectic Manifolds,” J. Austral. Math. Soc. Ser.66, 303–330 �1999�.

6 Cariñena, J. F., Crampin, M., and Ibort, L. A., “On the multisymplectic formalism for first order Field Theories,” Diff.Geom. Applic. 1, 345–374 �1991�.

7 Castrillón-López, M., and Marsden, J. E., “Some remarks on Lagrangian and Poisson reduction for field theories,” J.Geom. Phys. 48, 52–83 �2003�.

8 Echeverría-Enríquez, A., Muñoz-Lecanda, M. C., and Román-Roy, N., “A Geometrical Analysis of the Field equationsin Field Theories,” Int. J. Math. Math. Sci. 29, 687–699 �2002�.

9 Echeverría-Enríquez, A., Muñoz-Lecanda, M. C., and Román-Roy, N., “Geometry of Lagrangian first-order classicalfield theories,” Fortschr. Phys. 44, 235–280 �1996�.

10 Echeverría-Enríquez, A., Muñoz-Lecanda, M. C., and Román-Roy, N., “Geometry of Multisymplectic HamiltonianFirst-order Field Theories,” J. Math. Phys. 41, 7402–7444 �2000�.

11 Echeverría-Enríquez, A., Muñoz-Lecanda, M. C., and Román-Roy, N., “Geometrical setting of time-dependent regularsystems. Alternative models,” Rev. Math. Phys. 3, 301–330 �1991�.

12 Echeverría-Enríquez, A., Muñoz-Lecanda, M. C., and Román-Roy, N., “Multivector Field Formulation of HamiltonianField Theories: Equations and Symmetries,” J. Phys. A 32, 8461–8484 �1999�.

13 Echeverría-Enríquez, A., Muñoz-Lecanda, M. C., and Román-Roy, N., “Multivector Fields and Connections. SettingLagrangian Equations in Field Theories,” J. Math. Phys. 39, 4578–4603 �1998�.

14 Forger, M., Paufler, C., and Römer, H., “A general construction of Poisson brackets on exact multisymplectic manifolds,”Rep. Math. Phys. 51, 187–195 �2003�.

15 García, P. L., “The Poincaré-Cartan invariant in the calculus of variations,” Symp. Math. 14, 219–246 �1974�.16 Giachetta, G., Mangiarotti, L., and Sardanashvily, G., New Lagrangian and Hamiltonian Methods in Field Theory

�World Scientific, Singapore, 1997�.17 Giachetta, G., Mangiarotti, L., and Sardanashvily, G., “Differential geometry of time-dependent mechanics,” e-print

arXiv:dg-ga/9702020v1.18 Goldschmidt, H., and Sternberg, S., “The Hamilton-Cartan formalism in the calculus of variations,” Ann. Inst. Fourier

23, 203–267 �1973�.19 Gotay, M. J., in Mechanics, Analysis and Geometry: 200 Years after Lagrange, edited by M. Francaviglia �Elsevier, New

York, 1991�, pp. 203–235.20 Gotay, M. J., Isenberg, J., Marsden, J. E., and Montgomery, R., “Momentum maps and classical relativistic fields:

Covariant theory,” e-print arXiv:physics/9801019v2.21 Günther, C., “The polysymplectic Hamiltonian formalism in the Field Theory and the calculus of variations I: the local

case,” J. Diff. Geom. 25, 23–53 �1987�.



22 Hélein, F., and Kouneiher, J., “Finite dimensional Hamiltonian formalism for gauge and quantum field theories,” J. Math.Phys. 43, 2306–2347 �2002�.

23 Kanatchikov, I. V., “Canonical structure of Classical Field Theory in the polymomentum phase space,” Rep. Math. Phys.41, 49–90 �1998�.

24 Kijowski, J., and Szczyrba, W., “Multisymplectic Manifolds and the Geometrical Construction of the Poisson Bracketsin the Classical Field Theory,” Géométrie Symplectique et Physique Mathématique �Colloq. Int. C. N. R. S., Aix-en-Provence, 1974�, 347–378. Editions Centre Nat. Recherche Sci., Paris, 1975.

25 Kijowski, J., and Tulczyjew, W. M., A Symplectic Framework for Field Theories, Lecture Notes in Physics Vol. 170�Springer-Verlag, Berlin, 1979�.

26 Krupka, D., “Regular Lagrangians and Lepagean forms,” Differential Geometry and its Applications �Brno, 1986�,111–148, Math. Appl. �East European Ser.�, 27, Reidel, Dordrecht, 1987.

27 Krupkova, O., and Smetanova, D., “Legendre transformation for regularizable Lagrangians in field theory,” Lett. Math.Phys. 58, 189–204 �2002�.

28 Krupkova, O., and Smetanova, D., “On regularization of variational problems in first-order field theory,” Rend. Circ.Mat. Palermo 66, 133–140 �2001�.

29 Kuwabara, R., “Time-dependent mechanical symmetries and extended Hamiltonian systems,” Rep. Math. Phys. 19,27–38 �1984�.

30 de León, M., Marín-Solano, J., and Marrero, J. C., “Ehresmann Connections in Classical Field Theories,” An.de Fís.Monogr. 2, 73–89 �1995�.

31 de León, M., Marín-Solano, J., and Marrero, J. C., in Proceedings on New Developments in Differential Geometry, editedby L. Tamassi and J. Szenthe �Kluwer, Dordrecht, 1996�, pp. 291–312.

32 de León, M., Marín-Solano, J., Marrero, J. C., Muñoz-Lecanda, M. C., and Román-Roy, N., “Multisymplectic constraintalgorithm for field theories,” Int. J. Geom. Methods Mod. Phys. 2, 839–871 �2005�.

33 de León, M., Martín de Diego, D., and Santamaría-Merino, A., in Applied Differential Geometry and Mechanics, editedby W. Sarlet and F. Cantrijn �Academia, Gent, 2003�, pp. 21–47.

34 de León, M., Martín de Diego, D., and Santamaría-Merino, A., “Symmetries in classical field theories,” Int. J. Geom.Methods Mod. Phys. 1, 651–710 �2004�.

35 de León, M., Merino, E., and Salgado, M., “k-cosymplectic manifolds and Lagrangian field theories,” J. Math. Phys. 42,2092–2104 �2001�.

36 Mangiarotti, L., and Sardanashvily, G., Gauge Mechanics �World Scientific, Singapore, 1998�.37 Marsden, J. E. and Shkoller, S., “Multisymplectic Geometry, Covariant Hamiltonians and Water Waves,” Math. Proc.

Cambridge Philos. Soc. 125, 553–575 �1999�.38 Martínez, E., “Classical field theory on Lie algebroids: multisymplectic formalism,” e-print arXiv:math.DG/0411352.39 Martínez, E., “Classical field theory on Lie algebroids: variational aspects,” J. Phys. A 38, 7145–7160 �2005�.40 Munteanu, F., Rey, A. M., and Salgado, M., “The Günther’s formalism in classical field theory: momentum map and

reduction,” J. Math. Phys. 45, 1730–1751 �2004�.41 Paufler, C., and Römer, H., “Geometry of Hamiltonian n-vector fields in multisymplectic field theory,” J. Geom. Phys.

44, 52–69 �2002�.42 Rañada, M. F., “Extended Legendre transformation approach to the time-dependent Hamiltonian formalism,” J. Phys. A

25, 4025–4035 �1992�.43 Sardanashvily, G., Generalized Hamiltonian Formalism for Field Theory. Constraint Systems �World Scientific, Sin-

gapore, 1995�.44 Saunders, D. J., The Geometry of Jet Bundles, London Mathematical Society Lecture Notes Series Vol. 142 �Cambridge

University Press, Cambridge, 1989�.45 Souriau, J. M., Structure des Systémes Dynamiques �Dunod, Paris, 1969�.46 Struckmeier, J., “Hamiltonian dynamics on the symplectic extended phase space for autonomous and non-autonomous

systems,” J. Phys. A 38, 1275–1278 �2005�.47 Vankerschaver, J., Cantrijn, F., de León, M., and Martín de Diego, D., “Geometric aspects of nonholonomic field

theories,” Rep. Math. Phys. 56, 387–411 �2005�.



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