JOURNAL OF GEOMETRIC doi:10.3934/jgm.2012.4.1 American Institute of Mathematical Sciences 4, 1, March 2012 pp. 1–26

CLASSICAL THEORIES OF FIRST ORDER AND LAGRANGIAN SUBMANIFOLDS OF PREMULTISYMPLECTIC

Cedric´ M. Campos Dept. Matem´aticaFundamental Universidad de La Laguna, ULL Avda. Astrof´ısicoFco. S´anchez 38206 La Laguna, Tenerife, Spain Elisa Guzman´ and Juan Carlos Marrero ULL-CSIC Geometr´ıaDiferencial y Mec´anicaGeom´etrica Dept. Matem´aticaFundamental Universidad de La Laguna, ULL Avda. Astrof´ısicoFco. S´anchez 38206 La Laguna, Tenerife, Spain

(Communicated by Manuel de Le´on)

Abstract. A description of classical field theories of first order in terms of Lagrangian submanifolds of premultisymplectic manifolds is presented. For this purpose, a Tulczyjew’s triple associated with a fibration is discussed. The triple is adapted to the extended Hamiltonian formalism. Using this triple, we prove that Euler-Lagrange and Hamilton-De Donder-Weyl are the local equations defining Lagrangian submanifolds of a premultisymplectic .

1. Introduction. It is well-known that the Lagrangian formulation of -inde- pendent Mechanics may be developed using the of the TQ of the configuration Q and the Hamiltonian formulation may be given using the canonical symplectic Ω of the T ∗Q (see, for instance, [1]). But Lagrangian and may also be formulated in terms of Lagrangian submanifolds of symplectic manifolds [34, 35] (see also [28]). In fact, the Euler-Lagrange equations for a Lagrangian on TQ and the Hamilton equations for a Hamiltonian function on T ∗Q are just the local equations defining Lagrangian submanifolds of T (T ∗Q). Here, the symplectic structure on T (T ∗Q) is just the complete lift Ωc of Ω. Moreover, in this construction (the so- called Tulczyjew’s triple for ), an important role is played by the canonical involution of the double tangent bundle T (TQ) and by the isomorphism

2000 Subject Classification. Primary: 70S05; Secondary: 70H03, 70H05, 53D12. Key words and phrases. Field theory, multisymplectic structure, Lagrangian submanifold, Tul- czyjew’s triple, Euler-Lagrange , Hamilton-De Donder-Weyl equation. The authors have been partially supported by MEC (Spain) grants MTM2009-13383 and MTM2009-08166-E and by the grants of the Canary government SOLSUBC200801000238 and ProID20100210. CC wishes to thank JCM and E. Padr´onfor a short postdoc at the ULL. EG wishes to thank the CSIC for a JAE-predoc grant. We also would like to thank D. Mart´ınde Diego for their helpful comments.

1 2 CEDRIC´ M. CAMPOS, ELISA GUZMAN´ AND JUAN CARLOS MARRERO between T (T ∗Q) and T ∗(T ∗Q) induced by the canonical symplectic structure of T ∗Q. For time-dependent Mechanics, the situation is more complicated. Anyway, the restricted (respectively, extended) formalism of Lagrangian and Hamiltonian time- dependent Mechanics may be formulated in terms of Lagrangian submanifolds of Poisson (respectively, presymplectic) manifolds. In this case, we have a fibration π : E −→ R, where the total space E is the configuration manifold, which for instance may be E = R × Q. As in the time-independent situation, an important role is played by the canonical involution of T (TE) and the canonical isomorphism between T (T ∗E) and T ∗(T ∗E). In fact, using these tools, in a recent paper [18] the autors prove that, in the restricted formalism, the Euler-Lagrange and Hamilton equations are the local equations defining Lagrangian submanifolds of the Poisson 1 ∗ ∗ ∗ manifold J (π1 ). Here, π1 : Vert (π) −→ R is the canonical fibration over R of the dual bundle of the vertical bundle Vert(π) of π. They also prove that, in the extended formalism, the dynamical equations are just the local equations defining 1 ∗ Lagrangian submanifolds of the presymplectic manifold J π˜E, whereπ ˜E : T E −→ R is the projection of T ∗E onto the real line. Other different descriptions of time- dependent Mechanics in the Lagrangian submanifold setting have been proposed by several authors (see [17, 20, 23, 27]). may be of a generalization of Classical Mechanics so as the under study does not only depend on a one dimensional parameter, the time line, but on multi-dimensional one, space-time for instance. In this setting, a first order theory is described by a Lagrangian function, that is, a real function L on the 1- bundle J 1π, where π : E −→ M is a fibration (Classical Mechanics corresponds to the particular case in which M is the real line). A solution of the Euler-Lagrange equations for L is a local section φ of π : E −→ M such that its first jet prolongation j1φ satisfies   ∂L d ∂L 2 α − i α ◦ (j φ) = 0 . (1.1) ∂u dx ∂ui Here, (xi, uα) are local coordinates on E which are adapted to the fibration π and i α α 1 (x , u , ui ) are the corresponding local coordinates on J π. J 1π is an affine bundle modeled on the vector bundle V (J 1π) = π∗(T ∗M) ⊗ Vert(π). The dual bundle J 1π◦ of V (J 1π) is the so-called restricted multimomentum bundle and a section of the canonical projection µ: J 1π† −→ J 1π◦ is said to be a Hamiltonian section, where J 1π† is the extended multimomentum bundle, that is, 1 ◦ 1 ◦ the affine dual to J π. A section τ of the canonical projection π1 : J π −→ M is a solution of the Hamilton-De Donder-Weyl equations for h if α i ∂u ∂H ∂pα ∂H i = i ◦ τ , i = − α ◦ τ, (1.2) ∂x ∂pα ∂x ∂u i α i 1 ◦ where (x , u , pα) are local coordinates on J π and i α i i α i α i i h(x , u , pα) = (x , u , −H(x , u , pα), pα) . Classical field theories are closely related with multisymplectic geometry (see [21, 36]; see also [11, 12, 13, 14]). In fact, a geometric formulation of Eqs. (1.2) may be given using the canonical multisymplectic structure Ω of the extended multimo- mentum bundle J 1π† and the homogeneous real function H¯ on J 1π† induced by the Hamiltonian section h. It is the extended Hamiltonian formalism for classical field theories of first order (see [2,6]). Hamilton-De Donder-Weyl equations may also be 1ST ORDER CFT AND LAGRANGIAN SUBMANIFOLDS 3 obtained in an intrinsic form using the η on M and the multisymplectic structure on J 1π◦ induced by Ω and h. It is the restricted Hamiltonian formalism (see, for instance, [2,5,7,8,9, 10, 25, 30, 32]). An extension of the previous Lagrangian and Hamiltonian formalism for classical field theories may be developed for the more general case when the base manifold is not, in general, orientable (see, for instance, [2, 10]). Other different geometric approaches to classical field theories have been pro- posed by several authors using polysymplectic, k–symplectic or k–cosymplectic (see [32] and references therein). However, these formulations only cover some special types of classical field theories. Now, a natural question arises: Is it possible to develop a geometrical descrip- tion of Lagrangian and Hamiltonian field theories in the Lagrangian submanifold setting?. A positive answer to this question was given in [24] for the particular case of k–symplectic (and k–cosymplectic) field theory (see also [15, 31]). On the other hand, a Tulczyjew’s triple for multisymplectic classical field theories was proposed in [26]. The triple is adapted to the restricted Hamiltonian formalism. The Hamiltonian section is involved in the definition of the Hamiltonian side of the triple, which differs form the original Tulczyjew’s triple, and the constructions in the Lagrangian side are local. Even though, some ideas in [26] are of great interest and it can be considered as the first clear attempt to construct a multisymplectic Tulczyjew triple for classical field theories of first order. Very recently, a Tulczyjew’s triple for classical field theories of first order was proposed in [16]. This triple is adapted to the restricted Hamiltonian formalism and the basic concepts of the variational calculus are used in order to construct it. Multisymplectic geometry is not used in this approach. In this paper, we discuss a description of Lagrangian and Hamiltonian classical field theories in terms of Lagrangian submanifolds of premultisymplectic manifolds. For this purpose, we will construct a Tulczyjew’s triple associated with a fibration π : E −→ M. For the sake of simplicity, we consider the case where M is an oriented manifold with a fixed volume form. However, in the Appendix we extend the con- struction for the more general case where M is not necessarily an oriented manifold. Our triple is adapted to the extended Hamiltonian formalism. The multisymplectic structure of the extended multimomentum bundle and the involution ex ∇ of the iterated associated with π and a linear connection ∇ on M (see [22, 29]) play an important role in this construction. We remark that ex ∇ is used in order to construct the Lagrangian side of the triple and that ex ∇ depends on the linear connection ∇. However, the structural maps of the triple do not depend on the linear connection. In this sense, the triple is canonical. The paper is structured as follows. In Section 2, we present some basic notions on (pre)multisymplectic vector and manifolds. The multisymplectic formu- lation of classical field theory is reviewed in Section 3. In Section 4, we discuss the Lagrangian and Hamiltonian side of our Tulczyjew’s triple for classical field theories of first order. In Section 5, we present our conclusions and a description of future research directions. The paper ends with an Appendix. In this Appendix, we propose an extension of the Tulczyjew’s triple for a fibration π : E −→ M such that M needs not to be orientable and therefore a volume form on M is no longer required. 4 CEDRIC´ M. CAMPOS, ELISA GUZMAN´ AND JUAN CARLOS MARRERO

2. Multisymplectic structures. In the same way in which Classical Mechanics is modeled on , one of the geometric approaches of Classical Field Theories is multisymplectic geometry (for the definition and properties of a multisymplectic structure, see [4]). In this section, we give some basic notions on multisymplectic vector spaces and manifolds.

2.1. Multisymplectic vector spaces. Along this section, V will denote a real of finite and W ⊂ V a vector subspace of the former. Definition 2.1. A(k + 1)–form Ω on V is said to be multisymplectic if it is non- degenerate, that is, if the k ∗ [Ω : V −→ Λ V

v 7−→ [Ω(v) := ivΩ is injective. In such a case, the pair (V, Ω) is said to be a multisymplectic vector space of order k + 1. Definition 2.2. Let (V, Ω) be a multisymplectic vector space of order k + 1. Given a vector subspace W ⊂ V of V , we define the l–orthogonal complement of W , with 1 ≤ l ≤ k, as the subspace of V ⊥,l W := {v ∈ V : iv∧w1∧...∧wl Ω = 0 , ∀w1, . . . , wl ∈ W } . Moreover, we say that W is • l–isotropic if W ⊂ W ⊥,l; • l–coisotropic if W ⊥,l ⊂ W ; • l–Lagrangian if W = W ⊥,l; • multisymplectic if W ∩ W ⊥,k = {0}. It easily seen that W ⊥,1 ⊆ ... ⊆ W ⊥,k. Besides, if W is a multisymplectic subspace of V then, the pullback i∗Ω of the multisymplectic form Ω by the canonical inclusion i: W,→ V is a multisymplectic form on W of degree k + 1. A natural problem that arises in some particular Classical Field Theories is that the dynamical form under study fails to be non-degenerate, which is a case of further study. Definition 2.3. A premultisymplectic structure of order k+1 on a real vector space V of finite dimension is a (k + 1)–form Ω on V . The pair (V, Ω) is said to be a premultisymplectic vector space of order k + 1. Let (V, Ω) be a premultisymplectic vector space of order k+1. Then, the quotient ˜ vector space V := V/ ker [Ω admits a natural multisymplectic structure Ωe of order k + 1 which is characterized by the following condition

µ∗Ωe = Ω , where µ: V −→ V˜ = V/ ker [Ω is the canonical projection. Definition 2.4. Let (V, Ω) be a premultisymplectic vector space of order k + 1 and W be a subspace of V . Given 1 ≤ l ≤ k, then W is said to be l–isotropic (resp., l–coisotropic, l–Lagrangian, multisymplectic) if µ(W ) is an l–isotropic (resp., l– coisotropic, l–Lagrangian, multisymplectic) subspace of the multisymplectic vector space V˜ = V/ ker [Ω. 1ST ORDER CFT AND LAGRANGIAN SUBMANIFOLDS 5

Note that W is l–isotropic (resp., l–coisotropic, l–Lagrangian, multisymplectic) for (V, Ω) if, and only if, the quotient space W/(W ∩ker [Ω) is an l–isotropic (resp., l– coisotropic, l–Lagrangian, multisymplectic) subspace of the multisymplectic vector space (V,˜ Ω).˜

2.2. Multisymplectic manifolds. Along this section, both P and E represent real smooth manifolds of finite dimension. Definition 2.5. A premultisymplectic structure of order k + 1 on a manifold P is a closed (k + 1)–form Ω on P . If, furthermore, (TxP, Ω(x)) is multisymplectic for each x ∈ P , then Ω is said to be a multisymplectic structure of order k + 1 on P . The pair (P, Ω) is called a (pre)multisymplectic manifold of order k + 1. The canonical example of a multisymplectic manifold is the bundle of forms over a manifold E, that is, the manifold P = ΛkE. Example 2.6 (The bundle of forms). Let E be a smooth manifold of dimension n,ΛkE be the bundle of k–forms on E and ν :ΛkE −→ E be the canonical projec- tion. The Liouville or tautological form of order k is the k–form Θ over ΛkE given by

Θ(ω)(X1,...,Xk) := ω((Tων)(X1),..., (Tων)(Xk)), k k for any ω ∈ Λ E and any X1,...,Xk ∈ Tω(Λ E). Then, the canonical multisym- plectic (k + 1)–form is Ω := −dΘ . i i If (y ) are local coordinates on E and (y , pi1...ik ), with 1 ≤ i1 < . . . < ik ≤ n, are the corresponding induced coordinates on ΛkE, then

X i1 ik Θ = pi1...ik dy ∧ ... ∧ dy , (2.1)

i1<...

X i1 ik Ω = −dpi1...ik ∧ dy ∧ ... ∧ dy . (2.2)

i1<...

i α m−1 m 1 m m−1 m pαdu ∧ d xi, where d x = dx ∧ ... ∧ dx and d xi = i ∂ d x. Therefore, ∂xi we have that Θ2 and Ω2 are locally given by the expressions m i α m−1 Θ2 = pd x + pαdu ∧ d xi , (2.3) and

m i α m−1 Ω2 = −dp ∧ d x − dpα ∧ du ∧ d xi . (2.4) C Before ending this section, we recall the definition of l–Lagrangian submanifolds of a multisymplectic manifold and the natural extension of this notion to premulti- symplectic manifolds. Definition 2.8. Let (P, Ω) be a multisymplectic manifold of order k + 1. A sub- manifold S of P is l–Lagrangian, with 1 ≤ l ≤ k, if TxS is an l–Lagrangian subspace of the multisymplectic vector space (TxP, Ω(x)), for all x ∈ S. In other words, S is an l–Lagrangian submanifold of P if ⊥,l (TxS) = TxS, ∀x ∈ S. (2.5) Definition 2.9. Let (P, Ω) be a premultisymplectic manifold of order k + 1 and k ∗ [Ω : TP −→ Λ T P be the corresponding vector bundle morphism. A submanifold S of P is l–Lagrangian, with 1 ≤ l ≤ k, if TxS/(TxS ∩ ker [Ω(x)) is an l–Lagrangian subspace of the multisymplectic vector space TxP/ ker [Ω(x), for all x ∈ S. 3. Multisymplectic formulation of classical field theory. In this section, we recall the basics of the geometric formulation of Classical Field Theory within a multisymplectic framework of jet bundles. The theory is set in a configuration fiber bundle π : E −→ M, whose sections represent the fields of the system. Then, one may choose to develop a Lagrangian formalism, by considering a Lagrangian density L: J 1π −→ ΛmM (or a Lagrangian function L: J 1π −→ R, if a volume form on M has been fixed), and derive the Euler-Lagrange equations. Or one may choose to develop a Hamiltonian formalism, by considering a Hamiltonian density H: J 1π† −→ ΛmM (or a Hamiltonian section h: J 1π◦ −→ J 1π†), and derive the Hamilton’s equations. The literature on this subject is vast. For a more concise study, the interested reader is referred, for instance, to [2,3,5,6,8, 25, 30, 33]. From here on, π : E −→ M will always denote a fiber bundle of n over an m– dimensional manifold, i.e. dim M = m and dim E = m + n. Fibered coordinates on E will be denoted by (xi, uα), 1 ≤ i ≤ m, 1 ≤ α ≤ n; where (xi) are local coordinates on M. The shorthand dmx = dx1 ∧ ... ∧ dxm will represent the local volume form i m−1 1 m that (x ) defines, but we also use the notation d xi = i∂/∂xi dx ∧ ... ∧ dx for the contraction with the field. Many bundles will be considered over M and E, but all of them vectorial or affine. For these bundles, we will only consider natural coordinates. In general, indexes denoted with lower case Latin letters (resp. Greek letters) will range between 1 and m (resp. 1 and n). The Einstein sum convention on repeated crossed indexes is always understood. Furthermore, we assume M to be orientable with fixed orientation, together with a determined volume form η. Its pullback to any bundle over M will still be denoted η, as for instance π∗η. In , local coordinates on M will be chosen compatible with η, which means that dmx = η. The most part of the objects that will be defined on the subsequent sections will depend in some way or another on the volume form η. An alternative definition 1ST ORDER CFT AND LAGRANGIAN SUBMANIFOLDS 7 independent of the volume form may be given for each one of them (as an overall volume independent theory). In particular, the aim of the Appendix is to provide a construction of the Tulczyjew’s triple for field theory independent of the volume form.

3.1. Lagrangian formalism. As already mentioned, the Lagrangian formulation of Classical Field Theory is stated on the first jet manifold J 1π of the configuration bundle π : E −→ M. This manifold is defined as the collection of tangent maps of local sections of π. More precisely,

1 J π := {Txφ : φ ∈ Secx(π), x ∈ M} .

1 1 The elements of J π are denoted jxφ and called the 1st-jet of φ at x. Adapted co- i α i α α 1 α 1 ordinates (x , u ) on E induce coordinates (x , u , ui ) on J π such that ui (jxφ) = α i 1 ∂φ /∂x |x. It is clear that J π fibers over E and M through the canonical projec- 1 1 tions π1,0 : J π −→ E and π1 : J π −→ M, respectively. In local coordinates, these i α α i α i α α i projections are given by π1,0(x , u , ui ) = (x , u ) and π1(x , u , ui ) = (x ). Fiberwise, J 1π may be seen as a set of linear maps, namely for each u ∈ E,

1 Juπ = {z ∈ Lin(Tπ(u)M,TuE): Tuπ ◦ z = idTπ(u)M } , which is an affine space modeled on the vector space

1 ∗ Vu(J π) = Tπ(u)M ⊗ Vertu(π) = {z¯ ∈ Lin(Tπ(u)M,TuE): Tuπ ◦ z¯ = 0} . Thus, the first jet manifold J 1π is an affine bundle over E modeled on the vector 1 ∗ ∗ bundle V (J π) = π (T M) ⊗E Vert(π). The of a Lagrangian field system are governed by a Lagrangian density, a fibered map L: J 1π −→ ΛmM over M. The real valued function L: J 1π −→ R that satisfies L = Lη is called the Lagrangian function. Both Lagrangians permit to define the so-called Poincar´e-Cartanforms:

m 1 m+1 1 ΘL = Lη + hSη, dLi ∈ Ω (J π) and ΩL = −dΘL ∈ Ω (J π) , (3.1)

1 where Sη is a special and canonical structure of J π called vertical endomorphism and whose local expression is

α α j m−1 ∂ Sη = (du − uj dx ) ∧ d xi ⊗ α . (3.2) ∂ui In local coordinates, the Poincar´e-Cartanforms read   α ∂L m ∂L α m−1 ΘL = L − ui α ∧ d x + α du ∧ d xi , (3.3) ∂ui ∂ui     α α j ∂L m ∂L m−1 ΩL = − (du − uj dx ) ∧ α d x − d α ∧ d xi . (3.4) ∂u ∂ui It is important to note that, even though the Poincar´e-Cartan m–form (and there- fore, the Poincar´e-Cartan(m+1)–form) depends on the Lagrangian function L and the volume form η, actually it only depends on the Lagrangian density L = Lη (see Appendix). A critical point of L is a (local) section φ of π such that

1 ∗ (j φ) (iX ΩL) = 0, 8 CEDRIC´ M. CAMPOS, ELISA GUZMAN´ AND JUAN CARLOS MARRERO for any vector field X on J 1π. A straightforward would show that this implies that   2 ∗ ∂L d ∂L (j φ) α − i α = 0 , 1 ≤ α ≤ n . (3.5) ∂u dx ∂ui The above equations are called Euler-Lagrange equations.

3.2. Hamiltonian formalism. The dual formulation of the Lagrangian formalism is the Hamiltonian one, which is set in the affine dual bundles of J 1π. The (extended) affine dual bundle J 1π† is the collection of real-valued affine maps defined on the 1 fibers of π1,0 : J π −→ E, namely 1 † 1  1 (J π) := Aff(J π, R) = A ∈ Aff(Juπ, R): u ∈ E . The (reduced) affine dual bundle J 1π◦ is the quotient of J 1π† by constant affine maps, namely 1 ◦ 1 (J π) := Aff(J π, R)/{f : E −→ R} . It is again clear that J 1π† and J 1π◦ are fiber bundles over E but, in contrast to J 1π, they are vector bundles. Moreover, J 1π† is a principal R–bundle over J 1π◦. † 1 † ◦ 1 ◦ The different canonical projections are denoted π1,0 : J π −→ E, π1,0 : J π −→ E and µ: J 1π† −→ J 1π◦. The natural pairing between the elements of J 1π† and those of J 1π will be denoted by the usual angular bracket, 1 † 1 h · , · i : J π ×E J π −→ R . 1 ◦ 1 ∗ ∗ We note here that J π is isomorphic to the dual bundle of V (J π) = π (T M)⊗E Vert(π). Besides of defining the affine duals of J 1π, we also consider the extended and reduced multimomentum spaces m ◦ m m Mπ := Λ2 E and M π := Λ2 E/Λ1 E. By definition, these spaces are vector bundles over E and we denote their canonical projections ν : Mπ −→ E, ν◦ : M◦π −→ E and µ: Mπ −→ M◦π (some abuse of notation here). Again, µ: Mπ −→ M◦π is a principal R–bundle. We recall from Example 2.7 that Mπ has a canonical multisymplectic structure which we denote Ω. On the contrary, M◦π has no canonical multisymplectic structure, but Ω can still be pulled back by any section of µ: Mπ −→ M◦π to give rise to a multisymplectic structure on M◦π. An interesting and important fact is how the four bundles we have defined so far are related. We have that J 1π† =∼ Mπ and J 1π◦ =∼ M◦π , (3.6) although these isomorphisms depend on the base volume form η. In fact, the bundle isomorphism Ψ : Mπ −→ J 1π† is characterized by the equation 1 ∗ 1 1 Ψ(ω), jxφ η = φx(ω) , ∀jxφ ∈ Jν(ω)π , ∀ω ∈ Mπ . We therefore identify Mπ with J 1π† (and M◦π with J 1π◦) and use this isomor- phism to pullback the nature of J 1π† to Mπ. ◦ i α i Adapted coordinates in Mπ (resp. M π) will be of the form (x , u , p, pα) (resp. i α i (x , u , pα)), such that m i α m−1 m m m pd x + pαdu ∧ d xi ∈ Λ2 E (pd x ∈ Λ1 E) . 1ST ORDER CFT AND LAGRANGIAN SUBMANIFOLDS 9

Under these coordinates, the canonical projections have the expression i α i i α ◦ i α i i α ν(x , u , p, pα) = (x , u ) , ν (x , u , pα) = (x , u ) and i α i i α i µ(x , u , p, pα) = (x , u , pα); and the paring takes the form i α i i α α i α (x , u , p, pα), (x , u , ui ) = p + pαui . We also recall the local description of the canonical multisymplectic form Ω of Mπ, m i α m−1 Ω = −dp ∧ d x − dpα ∧ du ∧ d xi . Now, we focus on the principal R–bundle structure of µ: Mπ −→ M◦π. This structure arises from the R– R × Mπ −→ Mπ

(t, ω) 7−→ t · ην(ω) + ω . In coordinates,

i α i i α i (t, (x , u , p, pα)) 7−→ (x , u , t + p, pα) .

We will denote by Vµ ∈ X(Mπ) the infinitesimal generator of the action of R on ∂ Mπ, which in coordinates is nothing else but Vµ = ∂p . Since the of this action are the fiber of µ, Vµ is also a generator of the vertical bundle Vert(µ). The dynamics of a Hamiltonian field system are governed by a Hamiltonian sec- tion, a section h: M◦π −→ Mπ of µ: Mπ −→ M◦π. In presence of the base volume form η, the set of Hamiltonian sections Sec(µ) is in one-to-one correspondence with ∞ the set of functions {H¯ ∈ C (Mπ): Vµ(H¯ ) = 1} and with the set of Hamiltonian m densities, fibered maps H: Mπ −→ Λ M over M such that iVµ dH = η. Given a Hamiltonian section h: M◦π −→ Mπ, the corresponding Hamiltonian density is H(ω) = ω − h(µ(ω)) , ∀ω ∈ Mπ . Conversely, given a Hamiltonian density H: Mπ −→ ΛmM, the corresponding Hamiltonian section is characterized by the condition imh = H−1(0) . Obviously, H = Hη¯ . In adapted coordinates, i α i i α i α i i h(x , u , pα) = (x , u , p = −H(x , u , pα), pα) , (3.7) i α i i α i  m H(x , u , p, pα) = p + H(x , u , pα) d x . (3.8) The locally defined function H is called the Hamiltonian function and it must not be confused with the globally defined function H¯ such that H = Hη¯ . A critical point of H is a (local) section τ of π ◦ ν : Mπ −→ M that satisfies the (extended) Hamilton-De Donder-Weyl equation ∗ τ iX (Ω + dH) = 0 , (3.9) for any vector field X on Mπ.A critical point of h is a (local) section τ of π ◦ ν◦ : M◦π −→ M that satisfies the (reduced) Hamilton-De Donder-Weyl equation ∗ (h ◦ τ) iX (Ω + dH) = 0 , (3.10) 10 CEDRIC´ M. CAMPOS, ELISA GUZMAN´ AND JUAN CARLOS MARRERO for any vector field X on Mπ. A straightforward but cumbersome computation shows that both are equivalent to the following set of local equations known as Hamilton’s equations:

α i ∂(u ◦ τ) ∂H ∂(pα ◦ τ) ∂H i = i ◦ τ , i = − α ◦ τ . (3.11) ∂x ∂pα ∂x ∂u

3.3. Equivalence between both formalisms. In this section, we present the equivalence between the Lagrangian and Hamiltonian formalisms of classical field theories for the case when the Lagrangian function is (hyper)regular (see below). Let L be a Lagrangian density. The (extended) Legendre transform is the bundle 1 morphism LegL : J π −→ Mπ over E defined as follows: 1 Leg (j φ)(X ,...,X ) := (Θ ) 1 (X ,..., X ), (3.12) L x 1 m L jxφ e1 em

1 1 1 for all j φ ∈ J π and X ∈ T E, where X ∈ T 1 J π are such that T π (X ) = x i φ(x) ei jxφ 1,0 ei Xi. The (reduced) Legendre transform is the composition of LegL with µ, that is, the bundle morphism

1 ◦ legL := µ ◦ LegL : J π −→ M π . (3.13) In local coordinates,   i α α i α ∂L α ∂L LegL(x , u , ui ) = x , u ,L − α ui , α , (3.14) ∂ui ∂ui   i α α i α ∂L legL(x , u , ui ) = x , u , α , (3.15) ∂ui where L is the Lagrangian function associated to L, i.e. L = Lη. ∗ ∗ From the definitions, we deduce that (LegL) (Θ) = ΘL, (LegL) (Ω) = ΩL, where Θ is the Liouville m–form on Mπ and Ω is the canonical multisymplectic (m + 1)– 1 ◦ form. In addition, we have that the legL : J π −→ M π is a local diffeomorphism, if and only if, the Lagrangian function L is regular, that is,   ∂2L 1 ◦ the Hessian α β is a regular . When legL : J π −→ M π is a global ∂ui ∂uj diffeomorphism, we say that the Lagrangian L is hyper-regular. In this case, we may define the Hamiltonian section h: M◦π −→ Mπ

−1 h = LegL ◦ legL , (3.16) whose associated Hamiltonian density is −1 −1 H(ω) = ω, legL (µ(ω)) η − (L ◦ legL )(µ(ω)) , ∀ω ∈ Mπ . (3.17) In coordinates,

i α i i α i α α i α i h(x , u , pα) = (x , u ,L(x , u , ui ) − pαui , pα) , (3.18) α α −1 i α i where ui = ui (legL (x , u , pα)). Accordingly, i α i i α  m H(x , u , p, pα) = p + pαui − L d x (3.19) and the Hamiltonian function is

i α i i α H(x , u , pα) = pαui − L. (3.20) 1ST ORDER CFT AND LAGRANGIAN SUBMANIFOLDS 11

Theorem 3.1 (ref. [8, 25, 30]). Assume L is a hyper-regular Lagrangian density. 1 If φ is a solution of the Euler-Lagrange equations for L, then ω = legL ◦j φ is a solution of the Hamilton’s equations for h. Conversely, if ω is a solution of the −1 1 Hamilton’s equations for h, then legL ◦ ω is of the form j φ, where φ is a solution of the Euler-Lagrange equations for L.

4. Tulczyjew’s triple. In 1976, W. Tulczyjew presented in a of papers [34, 35] a complete geometric construction relating the triple T (T ∗Q), T ∗(TQ) and T ∗(T ∗Q). The canonical symplectic structures of these spaces are carried between them maintaining their symplectic character. However, the relevance of his is not only this but the fact that, when one introduces dynamics into the picture, a Lagrangian and a , then the dynamics are lifted to this triple giving the natural correspondence between these two formalisms and showing again the very geometric nature of Lagrangian and Hamiltonian mechanics. In this section, we will obtain the main results of the paper. In fact, we will introduce a Tulczyjew’s triple for Classical Field Theory associated with a fibration π : E −→ M, where M is an oriented manifold with a fixed volume form η. First, we will present the Lagrangian side of the triple and then the Hamiltonian side. Finally, we will introduce the equivalence between the two formalisms in this setting. 4.1. Lagrangian formalism. Recall that if Q is a smooth manifold, there is a ∗ ∗ canonical vector bundle isomorphism AQ : T (T Q) −→ T (TQ), named after Tul- czyjew’s work [35], whose local representation is i i i i AQ(q , , q˙ , p˙i) = (q , q˙ , p˙i, pi) , i i i i where (q ) are local coordinates on Q and (q , pi) (respectively, (q , pi, q˙ , p˙i)) are the corresponding adapted local coordinates on T ∗Q (respectively, T (T ∗Q)). The Tulczyjew’s isomorphism AQ is built by means of the canonical involution of TTQ, ∗ κQ : TTQ −→ TTQ, and the tangent lift of the natural pairing h · , · i : T Q πQ×τQ ∗ TQ −→ R, where τQ : TQ −→ Q and πQ : T Q −→ Q are the canonical tangent and cotangent projections. Note that, while the second argument of the tangent pairing (the tangent lift of h · , · i) ∗ T h · , · i : TT Q T πQ ×T τQ TTQ −→ R fibers over TQ through T τQ, thanks to the involution κQ, the second argument of the nondegenerate pairing ∗ T h · , · i ◦ (id, κQ): TT Q T πQ×τTQ TTQ −→ R

fibers over TQ through τTQ. In fact, AQ is the vector bundle isomorphism that this later map induces from T (T ∗Q) to T ∗(TQ). In what follows, we will discuss the construction of the Tulczyjew’s morphism for jet bundles. We shall define an affine morphism (between the proper spaces) by mimicking Tulczyjew’s construction. Therefore, we will need an involution of the 1 iterated jet bundle J π1 and a “tangent” pairing. Like the iterated tangent bundle TTQ of a manifold Q, the iterated jet bundle 1 1 J π1 of a fiber bundle π : E −→ M has two different affine structures over J π. The 1 first is the one in which we think of J π1 as the 1st-jet bundle of the fiber bundle 1 1 1 π1 : J π −→ M, that is the affine bundle (π1)1,0 : J π1 −→ J π modeled over the 1 ∗ ∗ vector bundle V (J π1) = π1 (T M) ⊗J 1π Vert(π1). The second is the one in which 1 1 we think of J π1 as the 1st-jet prolongation of the morphism π1,0 : J π −→ E, 12 CEDRIC´ M. CAMPOS, ELISA GUZMAN´ AND JUAN CARLOS MARRERO

1 1 1 that is the affine bundle j (π1,0): J π1 −→ J π modeled over the vector bundle J 1(V (J 1π)).

j1(π ) ex 1 1,0 1 1 ∇ 1 J π1 / J π J π1 o / J π1

(π1)1,0 π1,0 1 (π1)1,0 j (π1,0)

 π1,0   Ô J 1π / E J 1π

Figure 1. The iterated Figure 2. The ex- jet bundle change map

But unlike the iterated tangent bundle TTQ, there isn’t a canonical involution of 1 the iterated jet bundle J π1 of a fiber bundle π : E −→ M (besides of the identity map). However, Kol´aˇrand Modugno showed in [22] that the natural involutions of 1 the iterated jet bundle J π1 depend on the symmetric linear connections of the base manifold M. In fact, given a symmetric linear connection ∇ on M, they introduced 1 1 an affine bundle isomorphism ex ∇ over the identity of J π from (π1)1,0 : J π1 −→ 1 1 1 1 J π to j (π1,0): J π1 −→ J π. In local coordinates, the exchange map ex ∇ : 1 1 J π1 −→ J π1 has the expression i α α α α i α α α α α α k ex ∇(x , u , ui , u¯j , uij) = (x , u , u¯i , uj , uji + (¯uk − uk )Γji) , (4.1) k where Γij are the Christoffel’s symbols of the symmetric linear connection ∇, that is ∂ k ∂ ∇ ∂ j = Γij k , for all i, j . (4.2) ∂xi ∂x ∂x Remark 1. Using (4.2) as the definition of the Christoffel symbols of ∇ with respect to the local coordinates (xi), we have that (4.1) is the right definition of the map ex ∇. On the other hand, if we adopt other conventions for the definition of the Christoffel symbols of ∇ then the local expression of the exchange map will be different ([22, 29]). But, obviously, the exchange map is the same. C

Note that ex ∇ ◦ ex ∇ = idJ 1π. The definition of the “tangent” pairing needs of an extra element that also de- pends on the linear connection ∇ and on the volume form η too. We define the R–linear map d∇,η : C∞(M) −→ Ω1(M) characterized by the condition d∇,ηf ⊗ η = ∇(f · η) , ∀f ∈ C∞(M) . (4.3) For a (xi) compatible with the volume form, i.e. η = dmx, we have that  ∂f  d∇,ηf = − f · Γj dxi . ∂xi ij Now, we consider the natural pairing between Mπ and J 1π 1 h · , · i : Mπ ν×π1,0 J π −→ R 1 1 ∗ 1 (ω, jxφ) 7−→ ω, jxφ such that φx(ω) = ω, jxφ η(x) 1ST ORDER CFT AND LAGRANGIAN SUBMANIFOLDS 13 and we lift it to the map ∇,η 1 1 ∗ 1 1 d h · , · i : J (π ◦ ν) j ν×j (π1,0) J π1 −→ T M 1 1 ∇,η (jxω, jxσ) 7−→ d hω, σi (x) . In local coordinates, i α i i α α i α (x , u , p, pα), (x , u , ui ) = p + pαui . Therefore, ∇,η i α i α i i α α α α d (x , u , p, pα, u¯j , pj, pαj), (x , u , ui , u¯j , uij) =  i α i α i α j  k = pk + pαkui + pαuik − (p + pαui )Γkj dx . At this point, we remark that the second argument of the prolonged pairing is 1 1 1 the iterated jet bundle J π1 fibering over J π by j (π1,0). Thus, by composition with the exchange map ex ∇, we obtain a morphism ∇,η 1 1 ∗ 1 d h · , · i ◦ (id × ex ∇): J (π ◦ ν) j ν×(π1)1,0 J π1 −→ T M 1 1 where now the second argument is the iterated jet bundle J π1 fibering over J π by (π1)1,0. In local coordinates, ∇,η i α i α i i α α α α d (x , u , p, pα, uj , pj, pαj), ex ∇(x , u , ui , u¯j , uij) =  i α i α α α l i α j  k = pk + pαku¯i + pα(uki + (¯ul − ul )Γki) − (p + pαu¯i )Γkj dx . ∇,η Besides, observe that d h · , · i◦(id × ex ∇) is a vector valued map that takes values 1 1 from a vector bundle, J (π ◦ ν), and an affine bundle, J π1. Hence it induces a vector bundle morphism over the identity of J 1π ]∇,η 1 1 ∗ Aπ : J (π ◦ ν) −→ Affπ1 (J π1,T M) , where 1 ∗ 1 ∗ Aff (J π ,T M) = ∪ 1 Aff(J π ,T M) . π1 1 z∈J π z 1 π1(z) i α α j ij Considering the adapted coordinates (x , u , ui , p¯k, p¯αk, p¯αk) on the vector bundle 1 ∗ Affπ1 (J π1,T M),

]∇,η i α i α i Aπ (x , u , p, pα, uj , pj, pαj) = i α α i α l j i i l i i j ij j i = (x , u , ui , p¯k = pk −pαul Γki −pΓkj, p¯αk = pαk +pαΓkl −pαΓkj, p¯αk = pαδk) .

Lemma 4.1. Given an arbitrary fiber bundle πE,M : E −→ M, let πF,E : F −→ E and πV,M : V −→ M be an affine and a vector bundle, respectively. We have that, the vector bundles

AffπE,M (F,V ) := ∪y∈E Aff(Fy,VπE,M (y)) , and † ∗ ∗ F ⊗E πE,M (V ) = Aff(F, R) ⊗E πE,M (V ) are isomorphic. Proof. The lemma’s assertion is a simple algebraic fact. Given a point x ∈ M, let † us fix a point y ∈ Ex in its fiber, πE,M (y) = x. Then, we define a map from Fy ×Vx † to Aff(Fy,Vx) as follows: Given ω ∈ Fy and v ∈ Vx, we consider the affine map (ω : v): Fy −→ Vx given by (ω : v)(z) = ω(z)v, for z ∈ Fy. The map † (ω, v) ∈ Fy × Vx 7−→ (ω : v) ∈ Aff(Fy,Vx) 14 CEDRIC´ M. CAMPOS, ELISA GUZMAN´ AND JUAN CARLOS MARRERO is obviously bilinear and, therefore, it induces a linear map from the † Fy ⊗ Vx to Aff(Fy,Vx). It only rests to prove that, in fact, it is an isomorphism. i † Consider a (vj) of Vx and a basis {1, ω } of Fy dual to a reference system (o, fi) of Fy, i.e. i j i j i 1(o + r fi) = 1 and ω (o + r fi) = r , ∀r ∈ R . i † Then {1 ⊗ vj, ω ⊗ vj} is a basis of Fy ⊗ Vx and, as it is easy to check, its image i {(1 : vj), (ω : vj)} is a basis of Aff(Fy,Vx).

∇,η Using the previous lemma, we transform the range space of A]π , obtaining a new morphism

]∇,η 1 1 † ∗ ∗ Aπ : J (π ◦ ν) −→ (J π1) ⊗J 1π (π1) (T M) ,

]∇,η 1 † which we continue to denote by Aπ . Note that (J π1) may be identified with the m 1 bundle of m–forms Mπ1 = Λ2 J π. Under this identification, the local expression ∇,η of A]π is

]∇,η i α i α i Aπ (x , u , p, pα, uj , pj, pαj) = m i α m−1 ij α m−1 k = (¯pkd x +p ¯αkdu ∧ d xi +p ¯αkdui ∧ d xj) ⊗ dx , i α l j i i l i i j ij j i wherep ¯k = pk − pαul Γki − pΓkj,p ¯αk = pαk + pαΓkl − pαΓkj,p ¯αk = pαδk. 1 † ∗ ∗ m+1 1 Considering the natural morphism (J π1) ⊗J 1π (π1) (T M) −→ Λ2 J π given by the wedge product, we finally obtain the vector bundle morphism η 1 m+1 1 Aπ : J (π ◦ ν) −→ Λ2 J π i α i α i i α i α m (4.4) (x , u , p, pα, uj , pj, pαj) 7−→ (pαidu + pαdui ) ∧ d x , which we call the Tulczyjew’s morphism. It is important to note that this morphism does no longer depend on the linear connection ∇, while it still depends on the volume form η even though it is not explicitly noted.

η 1 m+1 1 Theorem 4.2. The Tulczyjew’s morphism Aπ : J (π ◦ ν) −→ Λ2 J π, locally given by Equation (4.4), is a vector bundle epimorphism over the identity of J 1π. Moreover, it is canonical in the sense that it only depends on the original fiber bundle π : E −→ M and, a pripori, on the base volume form η.

m+1 1 According to Example 2.7,Λ2 J π has a canonical multisymplectic structure given by the form α m i α m Ω m+1 1 = −dp¯α ∧ du ∧ d x − dp¯ ∧ du ∧ d x , (4.5) Λ2 J π α i i α α i m+1 1 where (x , u , ui , p¯α, p¯α) are natural coordinates on Λ2 J π. This structure is 1 η pulled back to J (π ◦ ν) by Aπ defining a premultisymplectic structure given by the (m + 2)–form

η ∗ i α m i α m Ωe := (A ) (Ω m+1 1 ) = −dp ∧ du ∧ d x − dp ∧ du ∧ d x . (4.6) π Λ2 J π αi α i The local basis of the kernel of Ωe is generated by *( )+ ∂ ∂ ∂ i ∂ ker Ωe = , , i − δj k , (4.7) ∂p ∂pj ∂pαj ∂pαk 1ST ORDER CFT AND LAGRANGIAN SUBMANIFOLDS 15 where 1 ≤ k ≤ m is a fixed index (no Einstein convention here, take for instance k = 1). Next, let L: J 1π −→ ΛmM be an arbitrary Lagrangian density such that L = Lη. Then, we obtain the following result.

η −1 1 Proposition 1. SL = (Aπ) (dL(J π)) is an (m + 1)–Lagrangian submanifold of the premultisymplectic manifold (J 1(π ◦ ν), Ω)e .

Proof. From (4.4), we obtain that the submanifold SL is locally given by   i α i α i i ∂L i ∂L SL = (x , u , p, pα, uj , pj, pαj): pα = α , pαi = α , (4.8) ∂ui ∂u thus *( )+ i ∂ ∂ ∂ i ∂ TSL = Xi,Uα,Uα, , , i − δj 1 , (4.9) ∂p ∂pj ∂pαj ∂pα1 where ∂ ∂2L ∂ ∂2L ∂ Xi = + + , (4.10) ∂xi i β j ∂xi∂uβ ∂p1 ∂x ∂uj ∂pβ β1 ∂ ∂2L ∂ ∂2L ∂ Uα = + + , (4.11) ∂uα α β j ∂uα∂uβ ∂p1 ∂u ∂uj ∂pβ β1 ∂ ∂2L ∂ ∂2L ∂ U i = + + . (4.12) α ∂uα α β j ∂uα∂uβ ∂p1 i ∂ui ∂uj ∂pβ i β1 It follows from Equation (4.7) that

1 1 ker Ω(j ω) ⊆ T 1 S , ∀j ω ∈ S ; e x jxω L x L and, from Definitions 2.8 and 2.9, we must prove that !⊥,m+1 T 1 S T 1 S jxω L jxω L 1 = , ∀j ω ∈ SL . 1 1 x ker Ω(e jxω) ker Ω(e jxω) 1 If iSL : SL −→ J (π ◦ ν) is the natural inclusion then, using (4.6) and (4.8), we deduce that  ∂L ∂L   ∂L  i∗ Ω = −d duα + duα ∧ dmx = d dxi ∧ dmx = 0 SL e α α i i ∂u ∂ui ∂x which implies that !⊥,m+1 Tj1 ωSL Tj1 ωSL x ⊆ x . 1 1 ker Ω(e jxω) ker Ω(e jxω) Instead of showing the converse inclusion, we will show that there is no gap in between. For this, we first note that

T 1 S jxω L  i = Xi,Uα,U , 1 α ker Ω(e jxω) i where Xi, Uα, and Uα are given by Equations (4.10), (4.11) and (4.12). Besides, *( )+ 1 i ∂ ∂ ∂ ∂ T 1 J (π ◦ ν) = X ,U ,U , , , , , jxω i α α i i ∂p ∂pα ∂pj ∂pαj 16 CEDRIC´ M. CAMPOS, ELISA GUZMAN´ AND JUAN CARLOS MARRERO therefore 1 T 1 J (π ◦ ν)   jxω i ∂ ∂ = Xi,Uα,U , , , 1 α ∂pi ∂pi ker Ω(e jxω) α αi where i goes from 1 to m and the Einstein notation is not considered. Now, if ⊥,m+1  1   1  1 1 T 1 S / ker Ω(j ω) \ T 1 S / ker Ω(j ω) ⊂ T 1 J (π ◦ ν)/ ker Ω(j ω) jxω L e x jxω L e x jxω e x is not empty and v is a vector belonging to this set, by linearity, we may assume Dn ∂ ∂ oE i ∂ ∂ that v is a non-zero vector in i , i . Suppose that v = vα i + vα i . ∂pα ∂pαi ∂pα ∂pαi It follows that ˜ i  i ˜  0 = Ω v, Uα,X1,...,Xm = −vα and 0 = Ω v, Uα,X1,...,Xm = −vα , So v is null, which is a contradiction. Next, we present some examples. Example 4.3 (Affine Lagrangian densities). Let γ : E −→ Mπ =∼ (J 1π)† be a section of the fiber bundle ν : Mπ −→ E and consider the affine Lagrangian density given by its action on J 1π, that is, 1 1 Lγ (jxφ) := γ(φ(x)), jxφ η . i α i 1 i α α In adapted coordinates, if γ = (x , u , γ0(x, u), γα(x, u)) and jxφ = (x , u , ui ), then i α α i α m Lγ (x , u , ui ) = γ0(x, u) + γα(x, u)ui d x . Note that this Lagrangian is obviously degenerate since its Hessian matrix is null. Examples of this type of Lagrangians appear, for instance, in the -affine gravitation theory with the Hilbert-Einstein Lagrangian density. The Lagrangian density of Dirac fermion fields in the presence of a background tetrad field and a background spin connection is also affine (for more details, see [10]). η −1 1 The (m + 1)–Lagrangian submanifold SLγ = (Aπ) (dLγ (J π)) of the premul- tisymplectic manifold (J 1(π ◦ ν), Ω)e is then given from the local expression (4.4) of η Aπ by ( j ) ∂γ ∂γ S = (xi, uα, p, pi , uα, p , pi ): pi = γi , pi = ◦ + β uβ . Lγ α j j αj α α αi ∂uα ∂uα j C Example 4.4 (Quadratic Lagrangian densities). Let [: J 1π −→ Mπ =∼ (J 1π)† be a morphism of affine bundles over the identity of E. Then we define the quadratic Lagrangian function 1 L (z) := h[(z), (z)i , ∀z ∈ J 1π . [ 2 If we suppose that [ is locally given by i α α i α i α ˜i ij β [(x , u , ui ) = (x , u ,[◦(x, u) + [α(x, u)ui , [α(x, u) + [αβ(x, u)uj ), then the Lagrangian function L[ is locally given by 1   L (xi, uα, uα) = [ + ([i + ˜[i )uα + [ij uαuβ . [ i 2 ◦ α α i αβ i j 1ST ORDER CFT AND LAGRANGIAN SUBMANIFOLDS 17

1 ij ji Note that, in this case, the Hessian velocity matrix of L[ is 2 ([αβ +[βα). Therefore, ij ji if [ is assumed to be symmetric ([αβ = [βα), then L[ will be regular if and only if the composition µ ◦ [ : J 1π −→ J 1π◦ is an affine bundle isomorphism. Examples of this type of Lagrangian functions appear, for instance, in the theory of electromagnetic fields and Proca fields (refer again to [10]). η −1 1 The (m + 1)–Lagrangian submanifold SL[ = (Aπ) (dL[(J π)) of the premul- 1 tisymplectic manifold (J (π ◦ ν), Ω),e where L[ = L[η, is then given from the local η expression (4.4) of Aπ by ( 1   S = (xi, uα, p, pi , uα, p , pi ): pi = [ij + [ji uβ , L[ α j j αj α 2 αβ βα j ij !) 1 ∂[ ∂([i + ˜[i ) ∂[ pi = ◦ + β β uβ + βγ uβuγ . αi 2 ∂uα ∂uα i ∂uα i j

C We return to the general case.

Theorem 4.5. Given a Lagrangian density L, we have that: 1. A (local) section σ ∈ Sec(π) is a solution of the Euler-Lagrange equations if and only if η −1 1 1 1 (Aπ) ◦ dL ◦ j σ = j (LegL ◦j σ).

2. The local equations defining SL as an (m + 1)–Lagrangian submanifold of J 1(π ◦ ν) are just the Euler-Lagrange equations for L.

Proof. A local computation, using (3.5), (3.14) and (4.4), proves the result.

Figure3 illustrates the above situation

SL

Aη m+1 1 π 1$ Λ2 (J π) o J (π ◦ ν) o e

dL % z LegL ' J 1π / Mπ =∼ J 1π† S H π1,0 ν 1 1 j (LegL ◦j σ) π π◦ν 1 % E w 1 O j σ 1 σ π LegL ◦j σ    M

Figure 3. The Lagrangian formalism in the Tulczyjew’s triple 18 CEDRIC´ M. CAMPOS, ELISA GUZMAN´ AND JUAN CARLOS MARRERO

4.2. Hamiltonian formalism. The construction of the Hamiltonian part of the Tulczyjew’s triple for Classical Field Theory involves the 1st-jet bundle J 1(π ◦ ν), which has already been introduced in the previous section, and the bundle of forms m+1 m+1 Λ2 Mπ. An arbitrary elementω ¯ of Λ2 Mπ is locally written α m m α i m p¯αdu ∧ d x +pdp ¯ ∧ d x +p ¯i dpα ∧ d x , i α i α m+1 so we may consider natural local coordinates (x , u , p, pα, p¯α, p,¯ p¯i ) on Λ2 Mπ. 1 i α i We recall that, as usual, adapted coordinates on J (π◦ν) are denoted by (x , u , p, pα, α i uj , pj, pαj). 1 1 In general, for any fiber bundle π : E −→ M, a jet jxφ ∈ J π defines a horizontal projector in T E, h 1 := T φ ◦ T π. In local coordinates, φ(x) jxφ x φ(x)   i ∂ α ∂ hj1 φ = dx ⊗ + u . x ∂xi i ∂uα In the particular case of π ◦ ν : Mπ −→ M,   j ∂ α ∂ ∂ i ∂ hz¯ = dx ⊗ j + uj α + pj + pαj i , ∂x ∂u ∂p ∂pα i α i α i 1 for anyz ¯ = (x , u , p, pα, uj , pj, pαj) ∈ J (π ◦ ν). 1 m+1 We define the affine bundle morphism [Ω : J (π ◦ ν) −→ Λ2 (Mπ)

[Ω(¯z) = ihz¯ Ω(ω) − (m − 1)Ω(ω) (4.13) where ω = (π ◦ ν)1,0(¯z), Ω is the canonical multisymplectic form of Mπ and m+1 X (ihz¯ Ω(ω)) (X1,...,Xm+1) = Ω(ω)(X1,..., hz¯(Xi),...,Xm+1) i=1 for X1,...,Xm+1 ∈ TωMπ. Locally, we have that

i α i α i  i α i j α [Ω(x , u , p, pα, uj , pj, pαj) = x , u , p, pα, pαj, −1, −uj . (4.14)

m+1 According to Example 2.7, we know that Λ2 Mπ has a canonical multisym- plectic structure. Namely, α m m α i m Ω m+1 = −dp¯α ∧ du ∧ d x − dp¯ ∧ dp ∧ d x − dp¯ ∧ dp ∧ d x . (4.15) Λ2 Mπ i α

We pullback this structure by [Ω, obtaining a premultisymplectic (m + 2)–form on J 1(π ◦ ν). From Equations (4.14) and (4.15), it follows that ∗ j α m i α m [ (Ω m+1 ) = −dp ∧ du ∧ d x − dp ∧ du ∧ d x . (4.16) Ω Λ2 Mπ αj α i ∗ 1 Thus, [ (Ω m+1 ) turns out to be the (m + 2)–form Ωe on J (π ◦ ν) introduced Ω Λ2 Mπ in Equation (4.6) of Section 4.1.

1 m+1 Theorem 4.6. The morphism [Ω : J (π ◦ ν) −→ Λ2 (Mπ), locally given by Equation (4.14), is an affine bundle epimorphism over the identity of Mπ. More- over, it is canonical in the sense that it only depends on the original fiber bundle π : E −→ M.

m m+1 Let H: Mπ −→ Λ M be a Hamiltonian density, then dH: Mπ −→ Λ2 E. Using (3.8) and (4.14), it follows that 1 −dH(Mπ) ⊆ [Ω(J (π ◦ ν)) . 1ST ORDER CFT AND LAGRANGIAN SUBMANIFOLDS 19

i α i i α i m In fact, if H(x , u , p, pα) = (p + H(x , u , pα))d x, we deduce that

∂H α m m ∂H j m − dH = − α du ∧ d x − dp ∧ d x − j dpα ∧ d x . (4.17) ∂u ∂pα −1 Proposition 2. SH = [Ω (−dH(Mπ)) is an (m + 1)–Lagrangian submanifold of the premultisymplectic manifold (J 1(π ◦ ν), Ω)e . Proof. The proof is analogous to that of Proposition1. Nonetheless, it is worth noting that, from (4.14) and (4.17), we obtain that the submanifold SH is locally given by

  i α i α i α ∂H j ∂H SH = (x , u , p, pα, uj , pj, pαj): uj = j , pαj = − α . (4.18) ∂pα ∂u

Next, we present an example. Example 4.7 (Quadratic Hamiltonian densities). Let ]: J 1π◦ =∼ M◦π −→ J 1π be a morphism of affine bundles over the identity of E. Then, we may define ∼ 1 † m the Hamiltonian density H] : Mπ = J π −→ Λ M given by

H](ω) := hω, ](µ(ω))i η , ∀ω ∈ Mπ . If ] is locally given by

i α i i α α αβ j ](x , u , pα) = (x , u ,]i (x, u) + ]ij (x, u)pβ) then H] is locally given by

i α i  α i αβ i j  m H](x , u , p, pα) = p + ]i (x, u)pα + ]ij (x, u)pαpβ d x , with associated Hamiltonian function i α i α i αβ i j H](x , u , pα) = ]i (x, u)pα + ]ij (x, u)pαpβ .

From these expressions, we see that the Hamiltonian H] is quadratic. Examples of this type of Hamiltonian densities may be obtained from hyper- regular quadratic Lagrangian functions (see Example 4.4). −1 The (m + 1)–Lagrangian submanifold SH] = [Ω (−dH](Mπ)) of the premulti- (J 1(π ◦ ν), Ω)e is then given from the local expression (4.14) of [Ω by ( i α i α i α α  αβ βα  j SH] = (x , u , p, pα, uj , pj, pαj): ui = ]i (x, u) + ]ij (x, u) + ]ji (x, u) pβ ,

∂]β(x, u) ∂]βγ (x, u) ) pi = − j pj − jk pj pk . αi ∂uα β ∂uα β γ

C We return to the general case. Theorem 4.8. Given a Hamiltonian section h ∈ Sec(µ), let H: Mπ −→ ΛmM be the associated Hamiltonian density. We have that 20 CEDRIC´ M. CAMPOS, ELISA GUZMAN´ AND JUAN CARLOS MARRERO

1. A section τ : M −→ M◦π is a solution of the Hamilton-De Donder-Weyl equation if and only if 1 [Ω ◦ j (h ◦ τ) = −dH ◦ (h ◦ τ).

2. The local equations defining SH as an (m + 1)–Lagrangian submanifold of J 1(π ◦ ν) are just the Hamilton’s equations. Proof. A local computation, using (3.8) and (4.14), proves the result. Figure4 illustrates the above situation

SH

1 y [Ω m+1 J (π ◦ ν) / Λ2 Mπ O 9 (π◦ν) 1,0 % y −dH Mπ O h µ  j1(h◦τ) M◦π ν B π◦ν ∗ π1,0 & τ E

π Ö M t

Figure 4. The Hamiltonian formalism in the Tulczyjew’s triple

4.3. Equivalence between both formalisms. In section 3.3, we already saw how the Lagrangian and Hamiltonian formalisms are interrelated by means of the Legendre transform. Here, we recover this relation within the Tulczyjew’s triple for m+1 1 m+1 Classical field theory by pulling the dynamical structures of Λ2 J π and Λ2 Mπ to J 1(π ◦ ν) as Lagrangian submanifolds. Let L: J 1π −→ ΛmM be an hyperregular Lagrangian density and H the asso- ciated Hamiltonian density (see Equation (3.17)). A simple computation in local coordinates using Equations (4.8), (4.18) and (3.20) shows the following result. η −1 1 Theorem 4.9. The (m + 1)–Lagrangian submanifolds SL = (Aπ) (dL(J π)) and −1 1 SH = [Ω (−dH(Mπ)) of the premultisymplectic manifold (J (π ◦ ν), Ω)e are equal. Figure5 illustrates this situation.

5. Conclusions and future work. A Tulczyjew’s triple associated with a fibra- tion is introduced. This construction allows us to describe Euler-Lagrange and Hamilton-De Donder-Weyl equations for classical field theories as the local equa- tions defining Lagrangian submanifolds of a premultisymplectic manifold. It would be interesting to extend these results for the more general case of classical field theory on Lie algebroids using a multisymplectic formalism. The first steps 1ST ORDER CFT AND LAGRANGIAN SUBMANIFOLDS 21

/ SL = SH o

Aη m+1 1 π 1$ z bΩ m+1 Λ2 J π o J (π ◦ ν) / Λ2 Mπ d :

dL −dH $ z LegL $ z J 1π / Mπ Q 1 1 e µ 1 j (LegL ◦j σ) j (h◦τ) h π1,0 % ν M◦π ∗ 9 π1,0 j1σ  Ò E t O τ σ π  M

Figure 5. The Tulczyjew’s triple for Classical Field Theories have been done in this direction (see [19]). We remark that these ideas could be applied to the of classical field theories which are under the action of a Lie . Another interesting goal is to develop a geometric formulation of vakonomic (nonholonomic) classical field theories using our Tulczyjew’s triple. Extensions of this construction for vakonomic (nonholonomic) classical theories with (using the setting) could also be discussed.

Appendix . Volume independent formulation of the Tulczyjew’s triple. Even though the Euler-Lagrange (and Hamilton-De Donder-Weyl) equations of Classical Field Theory are usually given using a volume form on the base space of the configuration bundle, they can be obtained without the need of that volume form. In fact, one may establish such equations in the more general case where the base manifold is not necessarily orientable (see, for instance, [2,3, 10]). In this sense, it is important to be able to give a Tulczyjew’s triple independent of any chosen basic volume form. This is the aim of this Appendix. To do so, we construct the left Tulczyjew’s map in three steps: First, we transform the 1st-jet m m bundle of Λ πM :Λ M −→ M into a “simpler” space. Second, we compute (in some sense) the core of the Tulczyjew’s morphism from the “tangent” pairing and the involution. Third, we combine the previous two steps in order to obtain the desired map. Before we start, we present a short review of the multisymplectic for- mulation of Classical Field Theory without the need of the orientability assumption (compare with Section3). Along the rest of the paper, local coordinates on M will no longer be required compatible with any volume form.

Volume independent formulation of Classical Field Theory: Given a La- grangian density L: J 1π −→ ΛmM, the associated Lagrangian function L is now 22 CEDRIC´ M. CAMPOS, ELISA GUZMAN´ AND JUAN CARLOS MARRERO locally defined such that L = Ldmx. Of course, in presence of a volume form η on M (if such is the case), it will coincide on a compatible chart with the globally defined Lagrangian function L¯ : J 1π −→ R such that L = Lη¯ . In this setting, the Poincar´e-Cartan m–form is given by the expression

ΘL = L + hS, dLi , (A.1)

1 ∗ where S : TJ π −→ π1 (TM)⊗J 1πVert(π1,0) is the canonical vertical endomorphism, which has the local representation

α α j ∂ ∂ S = du − uj dx ⊗ i ⊗ α . (A.2) ∂x ∂ui This newly defined Poincar´e-Cartan m–form coincides in fact with the previously defined one, Equation (3.1), and therefore the Euler-Lagrange equations (3.5) follow. In the other side of the picture, we redefine the dual bundles of J 1π. Instead of considering as extended dual of J 1π the affine maps on J 1π with values in R, we consider the affine maps with values in ΛmM. The reduced dual of J 1π will be the quotient of the extended dual by fiberwise constant affine maps on J 1π with values in ΛmM. Resuming,

1 † 1 m n 1 m o (J π) := Affπ(J π, Λ M) = A ∈ Aff(Juπ, Λπ(u)M): u ∈ E , (A.3) 1 ◦ 1 m m (J π) := Affπ(J π, Λ M)/MorphM (E, Λ M) . (A.4) The improvement of considering ΛmM as the target space of the affine maps is that the identifications J 1π† =∼ Mπ and J 1π◦ =∼ M◦π (A.5) are now canonical, in contrast with those of (3.6), which depend on a volume form on the base. In order to emphasize this and avoid confusion with Section4, we will use the dual affine bundle notation J 1π† and J 1π◦ in spite of the the form bundle one Mπ and M◦π. The overall notation and objects (coordinates, forms, etc.) will remain unaltered, but the natural pairing is now slightly different:

i α i i α α i α m (x , u , p, pα), (x , u , ui ) = (p + pαui )d x . Another technical issue due to the lack of the volume form is that, the line bundle µ: J 1π† −→ J 1π◦ cannot be seen as an R–principal bundle. Therefore, the definition of a Hamiltonian density H: J 1π† −→ ΛmM must be reformulated by the condition

iV (Ω + dH) = 0 , ∀V ∈ Vert(µ) . From here, Hamilton’s equations (3.11) follow immediately.

The 1st-jet bundle of the bundle of m–forms: The 1st-jet bundle J 1π of an arbitrary fiber bundle π : E −→ M is, in general, only an affine bundle over E. But, in some particular cases, it can be can be seen as a vector bundle over M; for instance when E is a tensor bundle of M, i.e. E = TM,T ∗M, ΛmM,... , and M itself is provided with a linear connection ∇. For obvious reasons, we expose the particular case of the tensor bundle of m–forms ΛmM. 1ST ORDER CFT AND LAGRANGIAN SUBMANIFOLDS 23

1 m m If we think of the elements of J Λ πM as linear maps from TM to T (Λ πM ) (see Section 3.1), we may define the bundle isomorphism 1 m m ∗ ∗ m J Λ πM −→ (Λ πM ) (T M) ⊗ΛmM Vert(Λ πM ) 1 1 1 jxα = Txα 7−→ jxα − j α˜ = Txα − Txα˜ , m m whereα ˜ is any (local) section of Λ πM :Λ M −→ M such that

α˜(x) = α(x) and (∇X α˜)(x) = 0 , ∀X ∈ X(M) . It is easy to prove that the above defined morphism does not depend on the chosen m m sectionα ˜. In fact, if α(x) = ad x|x andα ˜ =ad ˜ x, the previous conditions imply i j that ∂a/∂x˜ |x = a · Γij(x), which determines the 1st-jet ofα ˜. Therefore, for local i 1 m coordinates (x , a, ai) on J Λ πM , we have     ∂ j1α − j1α˜ = xi, a, a − a · Γj = a − a · Γj · dxi ⊗ . x x i ij i ij ∂a m Besides of this, we have that the vertical bundle Vert(Λ πM ) is isomorphic to m m Λ M ×M Λ M by the vertical lift. Namely, v m m m (·)· :Λ M × Λ M −→ Vert(Λ πM ) v (α, β) 7−→ (β)α i m m i ∂ (x , a, b) = (ad x, bd x) 7−→ (x , a, b) = b ∂a admx . 1 m Therefore, we have for the the vector bundle associated to J Λ πM m ∗ ∗ m ∼ m ∗ ∗ m m (Λ πM ) (T M) ⊗ Vert(Λ πM ) = (Λ πM ) (T M) ⊗ (Λ M × Λ M) =∼ ΛmM × (T ∗M ⊗ ΛmM) , which in local coordinates reads m ∗ ∗ m m ∗ m (Λ πM ) (T M) ⊗ Vert(Λ πM ) −→ Λ M × (T M ⊗M Λ M) i i (x , a, ai) 7−→ (x , a, ai) i ∂ m i m aidx ⊗ ∂a admx 7−→ (ad x, aidx ⊗ d x) . Summing up, we obtain an affine bundle transformation Φ∇ over ΛmM from 1 m m ∗ m J Λ πM to Λ M ×M (T M ⊗ Λ M) which depends on the linear connection ∇. Locally, we have ∇ 1 m m ∗ m Φ : J Λ πM −→ Λ M ×M (T M ⊗ Λ M) i i j (x , a, ai) 7−→ (x , a, ai − a · Γij) i ∂ ∂ m j i m dx ⊗ ( ∂xi + ai ∂a admx) 7−→ (ad x, (ai − a · Γij)dx ⊗ d x) .

The core map: The natural pairing between the elements of J 1π† and J 1π is the fibered map 1 † 1 m h · , · i : J π ×E J π −→ Λ M 1 1 (ω, jxφ) 7−→ ω, jxφ i α i i α α i α m ((x , u , p, pα), (x , u , ui )) 7−→ (p + pαui )d x . We lift it by 1st-prolongation to the pairing 1 1 † 1 1 1 1 m j h · , · i : J π ×J 1π (J π1, j (π1,0),J π) −→ J Λ πM 1 1 1 (jxω, jxσ) 7−→ j (hω, σi) i α i α i i α α α α i i α ((x , u , p, pα, u¯j , pj, pαj), (x , u , ui , u¯j , uij)) 7−→ (x , p + pαui , i α i α pk + pαkui + pαuik) , 24 CEDRIC´ M. CAMPOS, ELISA GUZMAN´ AND JUAN CARLOS MARRERO

1 1 where the second argument fibers over J π by j (π1,0). Composing it with the exchange transformation, we get a map whose coordinate representation is

1 1 † 1 1 1 m j h·, ex ∇(·)i : J (π1) ×J 1π (J π1, (π1)1,0,J π) −→ J Λ πM 1 1 1 1 1 (jxω, jxσ) 7−→ j jxω, ex ∇(jxσ) i α i α i i α α α α i ((x , u , p, pα, ui , pj, pαj), (x , u , ui , u¯j , uij)) 7−→ (x , a, ak) , 1 i α where now the second argument fibers over J π by (π1)1,0, and where a = p + pαu¯i i α i α α α j and ak = pk + pαku¯i + pα(uki + (¯uj − uj )Γki).

The Tulczyjew’s map: Composing the transformations defined previously ∇ 1 Φ and j h·, ex ∇(·)i together with the 2nd component projection m ∗ m ∗ m pr 2 :Λ M ×M (T M ⊗ Λ M) −→ T M ⊗ Λ M, we obtain the map ∇ 1 1 † 1 1 ∗ m pr 2 ◦Φ ◦ j h·, ex ∇(·)i : J (π1) ×J 1π (J π1, (π1)1,0,J π) −→ T M ⊗ Λ M 1 1 1 1 1 (jxω, jxσ) 7−→ j jxω, ex ∇(jxσ) i α i α i i α α α α k m ((x , u , p, pα, ui , pj, pαj), (x , u , ui , u¯j , uij)) 7−→ akdx ⊗ d x ,

i α i α α α j i α j where ak = pk + pαku¯i + pα(uki + (¯uj − uj )Γki) − (p + pαu¯i )Γkj. Note that this map is a vector valued morphism that takes values from the affine bundles 1 † 1 † 1 1 1 (J (π1), j (π1,0),J π) and (J π1, (π1)1,0,J π). Hence it induces a vector valued affine bundle morphism over the identity of J 1π

∇ 1 † 1 ∗ m Agπ : J (π1) −→ Affπ1 (J π1,T M ⊗ Λ M) , where 1 ∗ m 1 ∗ m Aff (J π ,T M ⊗ Λ M) = ∪ 1 Aff(J π ,T M ⊗ Λ M) . π1 1 z∈J π z 1 π1(z) π1(z)

i α α j ij Considering the adapted coordinates (x , u , ui , p¯k, p¯αk, p¯αk) on the vector bundle 1 ∗ m Affπ1 (J π1,T M ⊗ Λ M),

∇ i α i α i Agπ (x , u , p, pα, uj , pj, pαj) = i α α i α l j i i l i i j ij j i = (x , u , ui , p¯k = pk −pαul Γki −pΓkj, p¯αk = pαk +pαΓkl −pαΓkj, p¯αk = pαδk) . By Lemma 4.1, we have 1 ∗ m ∼ 1 m ∗ ∼ m 1 ∗ m+1 1 Aff(J π1,T M ⊗Λ M) = Aff(J π1, Λ M)⊗T M = Λ2 J π⊗T M,→ Λ2 J π . where the last “inclusion” is the natural morphism given by the wedge product, namely m 1 ∗ m+1 1 i: ω ⊗ α ∈ Λ2 J π ⊗ T M 7−→ α ∧ ω ∈ Λ2 J π . Therefore, if we assume ∇ symmetric, we end up with the map

1 † m+1 1 Aπ : J (π1) −→ Λ2 J π 1 ∇ 1 1 jxω 7−→ (i ◦ pr 2 ◦Φ )(j jxω, ex ∇(·) ) i α i α i i α i α m (x , u , p, pα, ui , pj, pαj) 7−→ (pαidu + pαdui ) ∧ d x , which we call the (left) Tulczyjew’s morphism. It is important to note that, even though a symmetric linear connection ∇ was necessary for the construction of Aπ, actually the latter does not depend on the former. Hence, the Tulczyjew’s morphism 1 † m+1 1 Aπ is a of J (π1) on Λ2 J π. 1ST ORDER CFT AND LAGRANGIAN SUBMANIFOLDS 25

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