Symmetries in Lagrangian Field Theory

Symmetries in Lagrangian Field Theory

Vol. 75 (2015) REPORTS ON MATHEMATICAL PHYSICS No. 3 SYMMETRIES IN LAGRANGIAN FIELD THEORY ´ L UCIA B UA Departamento de Xeometr´ıa e Topolox´ıa, F...

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Vol. 75 (2015)

REPORTS ON MATHEMATICAL PHYSICS

No. 3

SYMMETRIES IN LAGRANGIAN FIELD THEORY ´ L UCIA B UA Departamento de Xeometr´ıa e Topolox´ıa, Facultad de Matem´aticas, USC, 15782-Santiago, Spain (e-mail: [email protected])

I OAN B UCATARU Faculty of Mathematics, Alexandru Ioan Cuza University, Ias¸i, Romania (e-mail: [email protected])

M ANUEL

DE

´ L E ON

ICMAT CSIC, Madrid, Spain (e-mail: [email protected])

M ODESTO S ALGADO Departamento de Xeometr´ıa e Topolox´ıa Facultad de Matem´aticas, USC, 15782-Santiago, Spain (e-mail: [email protected] )

and ˜ S ILVIA V ILARI NO Centro Universitario de la Defensa de Zaragoza & I.U.M.A. Academia General Militar. Carretera de Huesca s/n. 50090 Zaragoza, Spain (e-mail: [email protected]) (Received August 25, 2014 – Revised January 7, 2015) By generalising the cosymplectic setting for time-dependent Lagrangian mechanics, we propose a geometric framework for the Lagrangian formulation of classical field theories with a Lagrangian depending on the independent variables. For that purpose we consider the first-order jet bundles J 1 π of a fiber bundle π : E → Rk where Rk is the space of independent variables. Generalized symmetries of the Lagrangian are introduced and the corresponding Noether theorem is proved. Keywords: symmetries, Cartan theorem, Noether theorem, conservation laws, jet bundles.

1.

Introduction

As it is well known, the natural arena for studying mechanics is symplectic geometry. One interesting problem is to extend this geometric framework for the case of classical field theories. Several different geometric approaches are well known: the polysymplectic formalisms developed by Sardanashvily et al. [11, 12, 35] and [333]

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´ ´ ˜ L. BUA, I. BUCATARU, M. DE LEON, M. SALGADO and S. VILARINO

by Kanatchikov [17], as well as the n-symplectic formalism of Norris [27, 29] and the k-cosymplectic of de Le´on et al. [22, 23]. Let us remark that the multisymplectic formalism is the most ambitious program for developing the classical field theory (see for example [2, 9, 10, 13, 14, 18], and references quoted therein). The aims of this paper are: • To give a new Lagrangian description of first-order classical field theory, by considering a fibration E → Rk , which has as particular cases the cosymplectic setting for time-dependent Lagrangian mechanics, and it is related with the k-cosymplectic [23] and multisymplectic formalisms. Let us observe that although every fiber bundle over Rk is a trivial bundle since Rk is contractible (Steenrod 1951 [36]), we do not use this fact to develop our Lagrangian description. • To introduce and to study the generalized symmetries in first-order Lagrangian field theory. For time-dependent Lagrangian mechanics this was done by J. F. Cari˜nena et al. [5]. In the present paper we present a new approach for Lagrangian field theory, working with the first-order jet bundle J 1 π of a fiber bundle π : E → Rk , where E is (n + k)-dimensional. The crucial point is that each 1-form on Rk defines a tensor field of type (1, 1) on J 1 π, see Saunders [38]. The paper is organized as follows. The main tools to be used are those of vector fields, k-vector fields and forms along maps, the general definitions of which are given in Section 2. In Section 3 we introduce the geometric elements on J 1 π necessary to develop the geometric formulation of the Euler–Lagrange field equations in Section 4 and to the study of symmetries and conservations laws. The principal tools, here described, are the canonical vector fields, the k vertical endomorphisms and a kind of k-vector fields, known as SOPDEs, which describe systems of second-order partial differential equations. This machinery is later used to discuss symmetries in this context, extending some previous results (see [1, 26]). The geometric formulation of the Euler–Lagrange field equations is given in Section 4, see Theorem 1. For this purpose we introduce the k Poincar´e–Cartan 1-forms using the Lagrangian and the k vertical endomorphisms. Our formulation is a natural extension of the k-cosymplectic formalism developed in [23] as we show in Section 4.3. Section 5 is devoted to discussing symmetries and conservation laws. We introduce symmetries of the Lagrangian and we give a Noether theorem. 2.

Preliminaries

2.1. k -vector fields

A system of first-order ordinary differential equations, on a manifold M, can be geometrically described as a vector field on M. Accordingly, a system of first-order

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partial differential equations on M can be geometrically described as a k-vector field on M, for some k > 1. Particularly, we can identify a system of second-order partial differential equations (SOPDE) with some special k-vector fields on the manifold J 1 π, for some k > 1. We briefly recall the correspondence between systems of first-order partial differential equations and k-vector fields. Let us denote by Tk1 M the Whitney sum T M⊕ . k. . ⊕T M of k copies of T M and τM : Tk1 M → M the canonical projection. DEFINITION 1. A k-vector field on an arbitrary manifold M is a section X : M −→ Tk1 M of the canonical projection τM : Tk1 M → M. Each k-vector field X defines a family of k vector fields X1 , . . . , Xk ∈ X(M) by projecting X onto every factor; that is, Xα = τα ◦ X, where τα : Tk1 M → T M is the canonical projection on the α-th copy T M of Tk1 M. DEFINITION 2. An integral section of the k-vector field X = (X1 , . . . , Xk ), passing through a point x ∈ M, is a map ψ: U ⊂ Rk → M, defined on some open neighbourhood U of 0 ∈ Rk , such that   ∂ ψ(0) = x, ψ∗ (x) = Xα (ψ(x)) ∈ Tψ(x) M, ∀x ∈ U, 1 ≤ α ≤ k. (1) ∂x α x

A k-vector field X = (X1 , . . . , Xk ) on M is said to be integrable if there is an integral section of X passing through every point of M. From Definition 2 we deduce that ψ is an integral section of X = (X1 , . . . , Xk ) if, and only if, ψ is a solution to the system of first-order partial differential equations ∂ψ i Xαi (ψ(x)) = , 1 ≤ α ≤ k, 1 ≤ i ≤ dim M, ∂x α x

where Xα = Xαi ∂/∂y i on a coordinate system (U, y i ) on M, y i ◦ ψ = ψ i , and x α are coordinates on Rk . For a k-vector field X = (X1 , . . . , Xk ) on M, one can require the integrability condition [Xα , Xβ ] = 0, ∀α, β ∈ {1, . . . , k}, as it has been considered in [25], see also [19, 20]. 2.2.

Sections along a map

Given a fiber bundle π : B → M and a differentiable mapping f : N → M, a section of π along f is a differentiable map σ : N → B such that π ◦ σ = f (see e. g. [32]). When π is a vector bundle, then the set of such sections can be endowed with a structure of C ∞ (N)-module. In the case of π : B → M being the tangent bundle of M, τM : T M → M, or the cotangent bundle, πM : T ∗ M → M, the sections along f will be called vector fields along f and 1-forms along f , respectively.

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The notion of sections along a map has been shown to be very fruitful. Many objects commonly used in physics find their suitable geometric representative by means of this concept and a related one of f -derivations ([31, 38]), but they have only recently been introduced in physics ([4, 6, 15]). Let X be a vector field along f : N → M, say T== M X f

N

τ



// M

then we can define an f -derivation iX : 3p (M) → 3p−1 (N ) of degree −1 (and type i∗ ) as follows: iX g = 0 for all g ∈ C ∞ (M) and   (iX ω)(x) v1x , . . . , v(p−1)x = ω (f (x)) X(x), f∗ (x)v1x , . . . , f∗ (x)v(p−1)x , (2) where v1x , . . . , v(p−1)x ∈ Tx N. There is another related f -derivation dX defined by dX = iX ◦ d + d ◦ iX ,

(3)

where d stands for the operator of exterior differentiation. This derivation is of degree 0 (and type d∗ ), i.e. dX ◦ d(M) = d(N ) ◦ dX . Note that when X ∈ X(M), then the idM -derivations iX and dX are nothing but the inner product or contraction iX and the Lie derivative LX , respectively. Let us observe that dX is an f ∗ -derivation associated to X in the sense of Pidello and Tulczyjew [31]. If X is a vector field along f : N → M, from (2) and (3) we deduce that the map dX : C ∞ (M) → C ∞ (N ), F

→ dX F,

is given by (dX F )(x) = (iX dF )(x) = dF (f (x))(X(x)) = X(x)(F ),

∀x ∈ N.

(4)

Finally, if π : B → M is a differentiable fibre bundle, we associate to each π -semi-basic p-form α on B a p-form α V along π , as α V (b)(v1π(b) , . . . , vp π(b) ) = α(b)(w1b , . . . , wp b ),

b ∈ B,

(5)

where vi π(b) ∈ Tπ(b) M, i = 1, . . . , p, and wi b ∈ Tb B are such that π∗ (b)wi b = vi π(b) . This type of form α V will be used in Section 5.1. 3.

The geometry of jet bundles

In this paper, we work with the first- and second-order jet bundles J 1 π and J π of a fiber bundle π : E → Rk , where E is an (n + k)-dimensional manifold. 2

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SYMMETRIES IN LAGRANGIAN FIELD THEORY

If (x α ) are local coordinates on Rk and (x α , q i ) are local fiber coordinates on E, we consider the induced standard jet coordinates (x α , q i , vαi ) on J 1 π and i (x α , q i , vαi , vαβ ) on J 2 π , where 1 ≤ i ≤ n, 1 ≤ α, β ≤ k. The induced jet coordinates are given by x α (jx1 φ) = x α (x) = x α , ∂φ i i 1 vα (jx φ) = α , ∂x

q i (jx1 φ) = q i (φ(x)), ∂ 2 φ i i 2 vαβ (jx φ) = α β . ∂x ∂x

x

jx1 φ,

x

jx2 φ

are the 1-jet and 2-jet on x, respectively. Here, φ is a section for π and For the canonical projections, we use the usual notation π2,1

π1,0

π1

J 2 π −→ J 1 π −→ E,

J 1 π −→ Rk ,

jx2 φ → jx1 φ → φ(x),

3.1.

jx1 φ → x.

Canonical vector fields along the projections π1,0 and π2,1

The vector field Tα(0) on E along π1,0 and the vector field Tα(1) on J 1 π along π2,1

T (J 1 π) ;;

(1) Tα

J 2π

(0)



;; T E

τ 1 J π

π2,1

are defined respectively by



τE π1,0

// J 1 π



// E

 ∂ ∈ Tφ(x) E, ∂x α x   ∂ (1) 2 1 Tα (jx φ) = (j φ)∗ (x) ∈ Tjx1 φ (J 1 π ), ∂x α x Tα(0) (jx1 φ) = φ∗ (x)



as we can see from the above diagram. Their local expressions are given by ∂ ∂ Tα(0) = α ◦ π1,0 + vαi i ◦ π1,0 , ∂x ∂q ∂ ∂ ∂ i i Tα(1) = α ◦ π2,1 + vα i ◦ π2,1 + vαβ ◦ π2,1 . ∂x ∂q ∂vβi Using (4) we have the maps dT (0) and dT (1) α



dT (0) : C (E) α

α

−→ C ∞ (J 1 π ),

dT (1) : C ∞ (J 1 π) −→ C ∞ (J 2 π ) α

defined by Tα(0) , Tα(1) , respectively.

(6) (7)

(8)

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´ ´ ˜ L. BUA, I. BUCATARU, M. DE LEON, M. SALGADO and S. VILARINO

From (4) and (8) one obtains ∂F ∂F ◦ π1,0 + vαi ◦ π1,0 , α ∂x ∂q i

(9)

∂G ∂G i ∂G ◦ π2,1 + vαi ◦ π2,1 + vαβ ◦ π2,1 , α i ∂x ∂q ∂vβi

(10)

dT (0) F = α

and dT (1) G = α

where F ∈ C ∞ (E) and G ∈ C ∞ (J 1 π). REMARK 1. Let us observe that the vector fields Tα(0) and Tα(1) are (π2,1 , π1,0 )related in the following sense (π1,0 )∗ ◦ Tα(1) = Tα(0) ◦ π2,1 . 3.2.

Prolongations of vector fields

We recall the prolongations of vector fields from E to J 1 π and the prolongation of a vector field along π1,0 , see Saunders [38] (Sections 4.4 and Section 6.4) . Let X be a vector field on E locally given by X = Xα (q, v)

∂ ∂ + Xi (q, v) i , α ∂x ∂q

then its prolongation X 1 is the vector field on J 1 π whose local expression is  i β ∂ dX 1 α ∂ i ∂ i dX , (11) X =X +X + − vβ α α i α ∂x ∂q dx dx ∂vαi where d/dx α denotes the total derivative, that is, d ∂ ∂ = α + vαj j , α dx ∂x ∂q

1 ≤ α ≤ k.

dF = Tα(0) (F ) where F ∈ C ∞ (J 1 π ). dx α Let X be a vector field along π1,0 and X(1) its first prolongation along π2,1 , which means T (J 1 π) ;; T E

Let us observe that

X(1)

J 2π

π2,1

;;

τ 1 J π

 // J 1 π

X π1,0

τE

 // E

see Saunders [38], Section 6.4. If X has the local expression X = Xα (x, q, v)

∂ ∂ ◦ π1,0 + Xi (x, q, v) i ◦ π1,0 α ∂x ∂q

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then X(1) is locally given by ∂ ∂ X(1) = Xα ◦ π2,1 α ◦ π2,1 + Xi ◦ π2,1 i ◦ π2,1 ∂x ∂q   ∂ i β ◦ π2,1 . + dT (1) (X ◦ π2,1 ) − dT (1) (X ◦ π2,1 )vβi α α ∂vαi If X is a vector field along π1,0 and π -vertical then it is locally given by    i  i i ∂X ∂ j ∂X (1) i ∂ j ∂X X = X ◦π2,1 + ◦π2,1 +vα j ◦π2,1 +vαβ j ◦π2,1 ◦π2,1 . (12) i α ∂q ∂x ∂q ∂vαi ∂vβ According to (4), we know that the vector field X(1) along π2,1 defines a map dX(1) : C ∞ (J 1 π) → C ∞ (J 2 π ), → dX(1) F,

F with local expression dX(1) F = Xα ◦ π2,1

  ∂F ∂F i i i β ∂F ◦ π + X ◦ π ◦ π + d ◦ π2,1 . X − v d X (1) (1) 2,1 2,1 2,1 β Tα Tα ∂x α ∂q i ∂vαi (13)

REMARK 2. Let X be a π-vertical vector field on E. The vector field X ◦ π1,0 along π1,0 <
J 1π satisfies that

3.3.

π1,0

 // E

(X ◦ π1,0 )(1) = X 1 ◦ π2,1 .

(14)

Vertical endomorphisms

Each 1-form dx α , 1 ≤ α ≤ k, defines a canonical tensor field Sdx α on J 1 π of type (1, 1), see Saunders [38], page 156, with local expression Sdx α ≡ (dq i − vβi dx β ) ⊗

∂ . ∂vαi

(15)

Throughout the paper, we denote Sdx α by S α . The vector-valued k-form S on J 1 π, defined in [37, 38], whose values are vertical vectors over E, is given in coordinates by  ∂ S = (dq i − vβi dx β ) ∧ d k−1 xα ⊗ i , (16) ∂vα

´ ´ ˜ L. BUA, I. BUCATARU, M. DE LEON, M. SALGADO and S. VILARINO

340 where

d k−1 xα = i∂/∂x α d k x = (−1)α−1 dx 1 ∧ · · · ∧ dx α−1 ∧ dx α+1 ∧ · · · ∧ dx k and d k x = dx 1 ∧ · · · ∧ dx k is the standard volume form on Rk . From (15) and (16) we deduce that S and {Sdx 1 , . . . , Sdx k } are related by the formula S = S α ∧ d k−1 xα . We also have dx α ∧ S = −S α ∧ d k x. 3.4.

Contact structures and second order partial differential equations

Let us consider the Cartan distribution, which is the (k + nk)-dimensional distribution, given by C(J 1 π) = Ker S 1 ∪ . . . ∪ Ker S k . From (15) we deduce that X ∈ C(J 1 π) if, and only if, (dq i − vαi dx α )(X) = 0,

and thus X is locally given by X=X

α



∂ ∂ + vαi i α ∂x ∂q



+ Xαi

∂ . ∂vαi

Therefore, a local basis for C(J 1 π) is given by the local k + nk vector fields ∂ ∂ + vαi α ∂x ∂q i

,

∂ . ∂vαi

We also consider the contact codistribution, which is the n-dimensional distribution that represents the annihilator of the Cartan distribution and is given by 31C (J 1 π) = {θ ∈ 31 (J 1 π), (j 1 φ)∗ θ = 0, ∀φ ∈ Sec(π )}. A local basis for 31C (J 1 π) is the set of canonical 1-forms δq i = dq i − vβi dx β ,

i = 1, . . . , n.

DEFINITION 3. A k-vector field Ŵ = (Ŵ1 , . . . , Ŵk ) on J 1 π is said to be a secondorder partial differential equation (SOPDE for short) if

or equivalently,

dx α (Ŵβ ) = δβα ,

S α (Ŵβ ) = 0,

dx α (Ŵβ ) = δβα ,

δq i (Ŵβ ) = 0,

for all i = 1 . . . n, α, β = 1 . . . k. Every vector field Ŵα of a SOPDE belongs to the Cartan distribution.

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From Definition 3 and the expression (15) of S α , we obtain that the local expression of a SOPDE (Ŵ1 , . . . , Ŵk ) is ∂ ∂ ∂ i Ŵα (x β , q i , vβi ) = α + vαi i + Ŵαβ , 1 ≤ α ≤ k, (17) ∂x ∂q ∂vβi i where Ŵαβ are functions locally defined on J 1 π . As a direct consequence of the above local expressions, we deduce that the family of vector fields {Ŵ1 , . . . , Ŵk } is linearly independent.

DEFINITION 4. Let φ : Rk → E be a section of π, locally given by φ(x α ) = (x α , φ i (x α )), then the first prolongation j 1 φ of φ is the map j 1 φ : Rk −→ J 1 π, x

−→ jx1 φ



 ∂φ i 1 k ≡ x , . . . , x , φ (x , . . . , x ), α (x , . . . , x ) , ∂x 1

k

i

1

k

(18)

for all α = 1, . . . , k and for all x ∈ Domain φ. We will see that the integral sections of a SOPDE are prolongations of sections. The following proposition has been also proved in Saunders [38]. PROPOSITION 1. A section ψ of π1 is the 1-jet prolongation of a section of π (in other words it is a holonomic field) if, and only, if ψ ∗ θ = 0 for all θ ∈ 31c (J 1 π ). Proof : Consider ψ : Rk → J 1 π, ψ(x α ) = (x α , ψ i (x α ), ψβi (x α )) a section of π1 and θ = θi (dq i − vβi dx β ) ∈ 31c (J 1 π). It follows that  i  ∂ψ i β − ψ ψ ∗ θ = θi ◦ ψ β dx . ∂x β

Therefore, ψ ∗ θ = 0 for all θ ∈ 31c (J 1 π) if and only if ψβi = ∂ψ i /∂x β .



Now we characterize the integral sections of a SOPDE.

PROPOSITION 2. Let Ŵ be an integrable k-vector field. Then the following three properties are equivalent: i) Ŵ is a SOPDE ii) the integral sections of Ŵ are 1-jets prolongations of sections of π . iii) there exists a section γ for π2,1 such that Ŵα = Tα(1) ◦ γ . Proof : i) ⇒ ii) Let Ŵ be an integrable k-vector field and ψ: Rk → J 1 π an integral section for Ŵ. Then from (1) and the local expression (17) of Ŵα , we obtain      ∂ ∂ = (π1 )∗ (ψ(x)) ψ∗ (x) = (π1 )∗ (ψ(x))Ŵα (ψ(x)) (π1 ◦ ψ)∗ (x) ∂x α x ∂x α x ∂ = α , ∂x π1 (ψ(x))

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´ ´ ˜ L. BUA, I. BUCATARU, M. DE LEON, M. SALGADO and S. VILARINO

which means

∂ β (x ◦ π1 ◦ ψ) = δαβ , ∂x α x

so π1 ◦ ψ = idRk ,i.e. ψ is a local section for π1 . We must prove that ψ = j 1 φ where φ is a section of π . To this end we use Proposition 1, showing that ψ ∗ θ = 0 for all θ ∈ 31C (J 1 π ). Let us assume that ψ is an integral section passing through jx1 ϕ, that is ψ(x) = jx1 ϕ. Now, since Ŵ is a SOPDE and θ ∈ 31C (J 1 π ), then iŴα θ = 0. Thus, 0 = iŴα θ(jx1 ϕ) = θ(jx1 ϕ)Ŵα (jx1 ϕ) = θ (jx1 ϕ)Ŵα (ψ(x))      ∂ ∂ ∗ = θ(ψ(x)) ψ∗ (x) = (ψ θ )(x) . ∂x α x ∂x α x

So, we have proved ψ ∗ θ = 0 for all θ ∈ 31C (J 1 π ). ii) ⇒ iii) We define γ : J 1 π → J 2 π,

jx1 σ → γ (jx1 σ ) = jx2 ϕ,

where j 1 ϕ is an integral section of Ŵ passing through jx1 σ (i.e. jx1 ϕ = j 1 ϕ(x) = jx1 σ ). Then (π2,1 ◦ γ )(jx1 σ ) = π2,1 (γ (jx1 σ )) = π2,1 (jx2 ϕ) = jx1 ϕ = jx1 σ. It follows that the map γ so defined is a section for π2,1 . Moreover, the vector field Ŵα can be expressed as Ŵα = Tα(1) ◦ γ , in fact,   ∂ 1 1 Ŵα (jx σ ) = (j ϕ)∗ (x) = Tα(1) (jx2 ϕ) = (Tα(1) ◦ γ )(jx1 σ ). ∂x α x

iii) ⇒ i) Let γ be a section for π2,1 . We must prove that γ defines a SOPDE by composition with Tα(1) . Since τJ 1 π ◦ Ŵα = τJ 1 π ◦ Tα(1) ◦ γ = π2,1 ◦ γ = idJ 1 π ,

where τJ 1 π : T (J 1 π) → J 1 π is the canonical projection, then Ŵα = Tα(1) ◦ γ is a vector field on J 1 π . Moreover, Ŵ is a SOPDE dx β (Ŵα )(jx1 σ ) = dx β (Tα(1) ◦ γ )(jx1 σ ) = δαβ and

  δq i (Ŵα )(jx1 σ ) = δq i (jx1 σ ) (Tα(1) ◦ γ )(jx1 σ ) = δq i (jx1 σ ) Tα(1) (jx2 ϕ) = vαi (jx2 ϕ) − vαi (jx1 σ ) = 0,

where the last identity is true because γ is a section for π2,1 so jx1 σ = (π2,1 ◦ γ )(jx1 σ ) = π2,1 (jx2 ϕ) = jx1 ϕ. 

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If j 1 φ is an integral section of a SOPDE Ŵ, then φ is called a solution of Ŵ. From (1) and (17), we deduce that φ is a solution of Ŵ if, and only if, q i ◦φ = φ i is a solution to the following system of second-order partial differential equations   ∂φ i ∂ 2 φ i i i = Ŵαβ x, φ (x), γ , (19) ∂x α ∂x β ∂x x

where 1 ≤ i ≤ n and 1 ≤ α, β ≤ k. The integrability conditions for the system (19) requires that the k-dimensional distribution induced by the SOPDE Ŵ, H0 = span{Ŵ1 , . . . , Ŵk } is integrable. The γ local expression (17) for a SOPDE Ŵ shows that [Ŵα , Ŵβ ] = Aαβ Ŵγ if, and only if, γ Aαβ = 0. Therefore, for a SOPDE Ŵ, we assume throughout the paper the following integrability conditions i i = Ŵβα , Ŵαβ

i i Ŵα (Ŵβγ ) = Ŵβ (Ŵαγ ),

(20)

which are equivalent to the condition [Ŵα , Ŵβ ] = 0 for all α, β = 1, . . . , k. 4. 4.1.

Lagrangian field theory Poincar´e–Cartan 1-forms

Let L : J 1 π → R be a Lagrangian. For each α = 1, . . . , k we define the Poincar´e–Cartan 1-forms 2αL on J 1 π as the 1-forms 1 ≤ α ≤ k. 2αL = Ldx α + dL ◦ S α , Their local expressions are given by   ∂L i ∂L α α 2L = Lδβ − i vβ dx β + i dq i , ∂vα ∂vα

(21)

or equivalently 2αL =

∂L ∂L (dq i − vβi dx β ) + Ldx α = i δq i + Ldx α . ∂vαi ∂vα

(22)

Let us observe that the fact of working with a fiber bundle over Rk allows us to introduce the tensors S α and consequently the 1-forms 2αL . It is known that a Lagrangian L induces a k-form 2L , called the Cartan form in first-order field theories, Saunders [37, 38], which is given by 2L = Ld k x + dL ◦ S,

with local expression 2L =

∂L (dq i − vβi dx β ) ∧ (i ∂α ∧ d k x) + Ld k x. ∂x ∂vαi

The relationship between the Cartan form and the Poincar´e–Cartan 1-forms is given by the following identity 2L = 2αL ∧ d k−1 xα + (1 − k)Ld k x

.

(23)

344 4.2.

´ ´ ˜ L. BUA, I. BUCATARU, M. DE LEON, M. SALGADO and S. VILARINO

Euler–Lagrange field equations

Let Ŵ = (Ŵ1 , . . . , Ŵk ) be an integrable SOPDE. A direct computation, using (17), (20) and (22), gives the formula     ∂L ∂L − (dq i − vβi dx β ), (24) LŴα 2αL = dL + Ŵα ∂vαi ∂q i and proves the following lemma. LEMMA 1. Let Ŵ = (Ŵ1 , . . . , Ŵk ) be an integrable SOPDE satisfying   ∂L ∂L Ŵα − i = 0, i = 1, . . . , n. i ∂vα ∂q If j 1 φ is an integral section of Ŵ, then φ is a solution to the Euler–Lagrange equations ∂φ j ∂ 2 L ∂ 2 φ j ∂ 2 L ∂ 2 L ∂L (25) + + = i , ∂x α ∂vαi jx1 φ ∂x α x ∂q j ∂vαi jx1 φ ∂x α ∂x β x ∂vβj ∂vαi jx1 φ ∂q jx1 φ

where x ∈ Domain φ. Eqs. (25) are usually written as follows   ∂ ∂L ∂L 1 1 ≤ i ≤ k. ◦j φ = i , ∂x α ∂v i ∂q 1 α

x

jx φ

From (24) we deduce the following proposition.

PROPOSITION 3. Let Ŵ be an integrable SOPDE, then

LŴα 2αL = dL if and only if Ŵα



∂L ∂vαi





∂L = 0, ∂q i

(26)

i = 1, . . . , n.

Thus, we have the following result. COROLLARY 1. Let Ŵ be an integrable SOPDE; if j 1 φ is a solution to (26), that is LŴα 2αL (j 1 φ) = dL(j 1 φ), then φ is a solution to the Euler–Lagrange equations. From (26) and the identity iŴα 2αL = kL we deduce the following proposition.

PROPOSITION 4. Let Ŵ be a SOPDE, then Eq. (26) is equivalent to the equation iŴα αL = (k − 1)dL,

where αL = −d2αL will be called the Poincar´e–Cartan 2-forms. Taking into account the above results we are able to prove a theorem that gives us a new Lagrangian field formulation for a bundle E → Rk .

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First we recall that a Lagrangian L : J 1 π → R is said to be regular if the matrix  2  ∂ L , 1 ≤ α, β ≤ k, 1 ≤ j, i ≤ n, j ∂vα ∂vβi is not singular, where 1 ≤ α, β ≤ k, 1 ≤ j, i ≤ n.

THEOREM 1. Let X = (X1 , . . . , Xk ) be a k-vector on J 1 π such that dx α (Xβ ) = δβα ,

iXα αL = (k − 1)dL.

(27)

1. If L is regular then X = (X1 , . . . , Xk ) is a SOPDE. If X is integrable and φ: U0 ⊂ Rk → E is a solution of X then φ is a solution to the Euler–Lagrange equations (25). 2. Now, if X = (X1 , . . . , Xk ) is integrable and j 1 φ is an integral section of X, then φ is a solution to the Euler–Lagrange equations (25). Proof : (i) If we write each Xα in local coordinates as ∂ ∂ ∂ i Xα = α + Xαi i + Xαβ , 1 ≤ α ≤ k, ∂x ∂q ∂vβi then, from (21) and (27), we obtain   ∂L i ∂L ∂ 2L α (1) Xα Lδβ − i vβ + (vαi − Xαi ) β i = β , ∂vα ∂x ∂vα ∂x   ∂ 2L ∂L ∂L i i + (v − X ) (2) Xα = j, α α j j i ∂q ∂vα ∂q ∂vα ∂ 2L (3) (vαi − Xαi ) j = 0. ∂vα ∂vβi Using that L is regular we deduce from (3) that Xαi = vαi . Since X is a SOPDE and we assume it to be integrable, its integral sections are prolongations j 1 φ of sections φ of π . Then, from   ∂ 1 1 Xα (jx φ) = (j φ)∗ (x) ∂x α x we deduce

Xαi ◦ j 1 φ = vαi ◦ j 1 φ =

∂φ i , ∂x α

i Xαβ ◦ j 1φ =

∂ 2φi , ∂x α x β

and from (2), we obtain ∂ 2L ∂φ j ∂ 2 L ∂ 2φj ∂ 2L ∂L + + = i. ∂x α ∂vαi ∂x α ∂q j ∂vαi ∂x α ∂x β ∂vβj ∂vαi ∂q which means that φ is solution to the Euler–Lagrange equations (25).

(28)

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(ii) If j 1 φ is an integral section of X, then from the local expression (18) of j φ we obtain the equations (28). Thus, from equation (2) and (28) we deduce that φ is a solution to the Euler–Lagrange equations (25).  1

REMARK 3. For k = 1, that is the fibration is E → R, we obtain the well-known dynamical equation iŴ d2 = 0, where Ŵ is a second-order differential equation SODE, see [3]. In the case E = R×Q → R, Eqs. (27) can be written as the dynamical equations dt (X) = 1,

iX L = 0,

where L is the Poincar´e–Cartan 2-form defined by the Lagrangian L: R×T Q → R. These equations coincide with the cosymplectic formulation of the nonautonomous mechanics, see [7, 24]. The relationship between this formalism and the k-cosymplectic formalism, described in [23] (see also [8, 21, 28, 33]), will be analyzed in the next section. EXAMPLE 1. The Klein–Gordon equation. The equation of a scalar field φ (for instance the gravitational field) which acts on the four-dimensional space-time is [18] ( + m2 )φ = F ′ (φ),

(29)

where m is the mass of the particle over which the field acts, F is a scalar function such that F (φ) = − 21 m2 φ 2 is the potential energy of the particle of mass m, and  is the Laplace–Beltrami operator given by 1 ∂ √ ∂φ ( −gg αβ β ), φ := div gradφ = √ α −g ∂x ∂x

(gαβ ) being a pseudo-Riemannian metric tensor in the four-dimensional space-time of signature (− + ++). Now we consider the trivial bundle π : E = R4 × R → R4 , with coordinates 1 (x , . . . x 4 , q) on E, and (x 1 , . . . x 4 , q, v1 , . . . , v4 ) the induced coordinates on J 1 π. Let L be the Lagrangian L : J 1 π → R defined by √ 1 1 L(x 1 , . . . x 4 , q, v1 , . . . , v4 ) = −g(F (q) − m2 q 2 + g αβ vα vβ ), 2 2 which is regular. Let us assume that X = (X1 , X2 , X3 , X4 ) is an integrable 4-vector field on J 1 π solution to Eqs. (27), that is dx α (Xβ ) = δβα ,

From (30) we deduce

iX1 1L + iX2 2L + iX3 3L + iX4 4L = 3dL. Xα



∂L ∂vα



=

∂L , ∂q

which is equivalent to √ ∂L √ Xα ( −gg αβ vβ ) = = −g(F ′ (q) − m2 q). ∂q

(30)

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347

Since L is regular, then X is a SOPDE, and if j 1 ψ is an integral section of X,   then  √ √ ∂ 1 −gg αβ vβ = −g(F ′ (ψ(x)) − m2 ψ(x)), (j ψ)∗ (x) α ∂x x which can be written as   √ √ ∂ αβ ∂ψ − −g(F ′ (ψ) − m2 ψ), 0 = −g α g ∂x ∂x β

and thus we obtain that ψ is a solution to the scalar field equation (29). REMARK 4. Some particular cases of the scalar field equation (29) are: 1. If F = 0, we obtain the linear scalar field equation. 2. If F (q) = m2 q 2 , we obtain the Klein–Gordon equation, see [16], ( + m2 )φ = 0.

4.3.

Relationship with the k -cosymplectic formalism

We shall show the difference between the corresponding Poincar´e–Cartan forms in this new formalism and the Poincar´e–Cartan forms of the k-cosymplectic formalism. Let us remark that this new approach is closer to the multysimplectic description, see identity (23), than to the k-cosymplectic one. Throughout this section we consider the trivial bundle E = Rk × Q → Rk . In this case, the manifold J 1 π of 1-jets of sections of the trivial bundle π : Rk × Q → Rk is diffeomorphic to Rk × Tk1 Q, where Tk1 Q denotes the Whitney sum T Q⊕ . k. . ⊕T Q of k copies of T Q. The diffeomorphism is given by J 1 π → Rk × Tk1 Q,

jx1 φ = jx1 (I dRk , φQ ) → (x, v1 , . . . , vk ), φ

πQ

where φQ : Rk → Rk × Q → Q, and   ∂ , vα = (φQ )∗ (x) ∂x α x

1 ≤ α ≤ k.

Now we recall the k-cosymplectic Lagrangian formalism, beginning with the necessary geometric elements: • The Liouville vector field 1 on Rk × Tk1 Q is the infinitesimal generator of the following flow R × (Rk × Tk1 Q) → Rk × Tk1 Q,

(s, (t, v1q , · · · , vkq )) → (t, es v1q , · · · , es vkq ), and in local coordinates it has the form 1 = vαi

∂ . ∂vαi

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• The canonical vector field 1αβ , 1 ≤ α, β ≤ k, is the vector field on Rk × Tk1 Q defined by  d  α . . . , v , v + sv , v , . . . , v 1β (x, v1q , . . . , vk q ) = x, v , α−1 q αq βq α+1 q kq 1q ds 0

∂ . Let us observe that 1 = 1αα . ∂vαi • For a Lagrangian L : Rk × Tk1 Q → R, the energy function is defined as EL = 1(L) − L and its local expression is ∂L EL = vαi i − L. (31) ∂vα with local expression

1αβ

= vβi

• The canonical k-tangent structure on Tk1 Q is the family {J 1 , . . . , J k } of tensor fields locally given by ∂ Jα = i ⊗ dq i . ∂vα The natural extension J α of the tensor fields J α on Tk1 Q to Rk × Tk1 Q will be denoted by Jeα . • The Poincar´e–Cartan 1-forms introduced in [23] are defined as follows θLα = dL ◦ Jeα , 1 ≤ α ≤ k, and they have the local expression

θLα =

∂L dq i . ∂vαi

(32)

The corresponding Poincar´e–Cartan 2-forms are ωLα = −dθLα . From (21) and (32) we deduce that the relationship between the Poincar´e–Cartan 1-forms 2αL and θLα is given by the following equation  2αL = θLα + δβα L − 1αβ (L) dx β . (33)

As a consequence of (33), the solutions (X1 , . . . , Xk ) of our geometric field equations dx α (Xβ ) = δβα , iXα αL = (k − 1)dL, coincide with the solutions of the k-cosymplectic field equations dx α (Xβ ) = δβα ,

1 ≤ α, β ≤ k, ∂L iXα ωLα = dEL + α dx α , ∂x introduced in [23], and also the corresponding integral sections, if they exist. 5.

Symmetries and conservation laws The set of k-vector fields solution to Eq. (27) will be denoted by XkL (J 1 π ). As a consequence of Propositions 3 and 4 we have that an integrable SOPDE Ŵ belongs to XkL (J 1 π) if, and only if, LŴα 2αL = dL.

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349

DEFINITION 5. A conservation law (or a conserved quantity) for the Euler– Lagrange equations (25) is a map G = (G1 , . . . , Gk ): J 1 π → Rk such that the divergence of G ◦ j 1 φ = (G1 ◦ j 1 φ, . . . , Gk ◦ j 1 φ): U ⊂ Rk → Rk

is zero for every section φ : U ⊂ Rk → E, solution to the Euler–Lagrange equations (25), which means that for all x ∈ U ⊂ Rk we have  ∂  ∂(Gα ◦ j 1 φ) 1 α (1) 2 α 1 0 = [Div(G ◦ j φ)](x) = = (j φ∗ )(x) ∂x α (G ) = Tα (jx φ)(G ). ∂x α x x

We can characterize conservation laws for the Euler–Lagrange equations in terms of the SOPDEs in XkL (J 1 π).

PROPOSITION 5. The map G = (G1 , . . . , Gk ): J 1 π → Rk defines a conservation law for the Euler–Lagrange equations (25) if, and only if, for every integrable SOPDE Ŵ = (Ŵ1 , . . . , Ŵk ) ∈ XkL (J 1 π) we have that

LŴα Gα = 0.

Proof : Let jx1 φ be an arbitrary point of J 1 π . Since Ŵ is an integrable SOPDE, let us denote by j 1 ψ the integral section of Ŵ passing through by jx1 φ, which means   ∂ 1 1 1 1 1 x ∈ Domain ψ. Ŵα (jx ψ) = (j ψ)∗ (x) , j ψ(0) = jx ψ = jx φ, ∂x α x

XkL (J 1 π),

Since Ŵ ∈ and ψ is an integral section of Ŵ then ψ is a solution to the Euler–Lagrange equations (25). As G = (G1 , . . . , Gk ) is a conservation law, then by hypothesis ∂(Gα ◦ j 1 ψ) = 0, ∂x α 0

and therefore we deduce α

LŴα G

(jx1 φ)

=

Ŵα (j01 ψ)(Gα )

 ∂ ∂(Gα ◦ j 1 ψ) α = (j ψ)∗ (0) (G ) = = 0. ∂x α 0 ∂x α 0 1



Conversely, we must prove that

∂(Gα ◦ j 1 φ) = 0, ∂x α x

for all sections φ : W ⊂ Rk → E, which are solutions to the Euler–Lagrange equations (25). Since j 1 φ W : W ⊂ Rk → J 1 π is an injective immersion (j 1 φ is a section and hence its image is an embedded submanifold), we can define a k-vector field

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350

X = (X1 , . . . , Xk ) in j 1 φ(W ) as   ∂ ∂φ i ∂ ∂ 2 φ i ∂ ∂ 1 1 = α + α i + α β i , Xα (jx φ) = (j φ)∗ (x) ∂x α x ∂x jx1 φ ∂x ∂q jx1 φ ∂x ∂x ∂vβ jx1 φ

and so j 1 φ is an integral section of X and X it is an integrable SOPDE on j 1 φ(W ). Now we prove that X ∈ XkL (j 1 φ(W )). A direct computation shows that     ∂L ∂L − i = 0. Xα ∂vαi ∂q j 1 φ(W )

Now, since X is an integrable SOPDE, from Proposition 3 and Proposition 4 we deduce that X is a solution to Eq. (27), and then X ∈ XkL (j 1 φ(W )). The following identities finish the proof   ∂(Gα ◦ j 1 φ) ∂ α 1 1 α 1 α = (j φ)∗ (x) ∂x α (G ) = Xα (jx φ)(G ) = LXα G (jx φ) = 0.  ∂x α x x

5.1.

Generalized symmetries. Noether’s theorem

In this section, we introduce the (generalized) symmetries of the Lagrangian and we prove a Noether’s theorem which associates to each symmetry a conservation law. The following proposition can be seen as a motivation of the condition (39) in the definition of generalized symmetry, and it is also a generalization of Proposition 3.15 in [34]. PROPOSITION 6. Let X be a π -vertical vector field on E. If there exist functions g α : E → R, 1 ≤ α ≤ k, such that X1 (L) = dT (0) g α , α

α

∗ α

then the functions G = (π1,0 ) g −

2αL (X 1 )

define a conservation law.

Proof : Let us observe that locally Gα = g α ◦ π1,0 − (X i ◦ π1,0 )

∂L . ∂vαi

Then, taking into account (9) and (11), we deduce that for every solution φ of the Euler–Lagrange equations (25) we have    ∂ ∂L ∂(Gα ◦ j 1 φ) α i 1 = g ◦ φ − (X ◦ φ) ◦ j φ ∂x α ∂x α x ∂vαi x = [dT (0) g α − X1 (L)](jx1 φ) = 0, α

1

k

so (G , . . . , G ) defines a conservation law.



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SYMMETRIES IN LAGRANGIAN FIELD THEORY

The Euler–Lagrange form δL is the 1-form on J 2 π given by ∗ δL = dT (1) 2αL − π2,1 dL α

with local expression δL =



Tα(1)



∂L ∂vαi



 ∂L − i ◦ π2,1 (dq i − vβi dx β ). ∂q

This is a π2,0 -semi-basic form, and we consider its associated form (δL)V along π2,0 , see (5), with local expression   ∂L ∂L Tα(1) ( i ) − i ◦ π2,1 (dq i ◦ π2,0 − vβi dx β ◦ π2,0 ). (34) ∂vα ∂q The 1-forms 2αL are π1,0 -semi-basic 1-form and its associated forms (2αL )V along π1,0 are locally given by (2αL )V = (Lδβα −

∂L i ∂L v ) dx β ◦ π1,0 + i dq i ◦ π1,0 . ∂vαi β ∂vα

(35)

The following lemma will be useful in the study of generalized symmetries. LEMMA 2. Let X be a π-vertical vector field along π1,0 . Then 1. If there exist functions Gα : J 1 π → R, α = 1, . . . , k, such that dT (1) Gα (j 1 φ) = −(δL)V (X ◦ π2,1 )(j 1 φ) α

for any solution φ to the Euler–Lagrange equations, then (G1 , . . . , Gk ) is a conservation law. 2. The following identity holds

  dX(1) L = −(δL)V (X ◦ π2,1 ) + dT (1) (2αL )V (X) . α

Proof : 1. From (6) we have

α

dT (1) G α

for any jx2 φ ∈ J 2 π . Since X is locally given by

(jx2 φ)

∂(Gα ◦ j 1 φ) = ∂x α x

X = X i (x, q, v)

(36)

(37)

∂ ◦ π1,0 , ∂q i

then X ◦ π2,1 is locally given by   ∂ ∂ i X ◦ π2,1 = X (x, q, v) i ◦ π1,0 ◦ π2,1 = Xi (x, q, v) ◦ π2,1 i ◦ π2,0 . ∂q ∂q

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352

From the above local expressions of X ◦ π2,1 and the local expression (34) of (δL)V we obtain  !  ∂L ∂ ∂L 1 V 2 ◦j φ − i −(δL) (X ◦ π2,1 )(jx φ) = − Xi (jx1 φ). (38) ∂x α x ∂vαi ∂q jx1 φ

Now from (37) and (38) we obtain that if φ is a solution to the Euler–Lagrange equations then ∂(Gα ◦ j 1 φ) = 0. ∂x α x

2. A direct computation using (12), (13), (34) and (35) proves the identity (36). 

Some classes of symmetries depend only on the variables (coordinates) in E. In this section we consider vαi -dependent infinitesimal transformations which can be regarded as vector fields X along π1,0 . The following definition is motivated by Proposition 6. DEFINITION 6. A π -vertical vector field X along π1,0 is called a (generalized) symmetry if there exists a map (F 1 , . . . , F k ): J 1 π → Rk such that dX(1) L(jx2 φ) = dT (1) F α (jx2 φ) α

(39)

for every solution φ to the Euler–Lagrange equations. The following version of Noether’s theorem associates to each symmetry of the Lagrangian, in the sense given above, a conservation law. THEOREM 2. Let X be a symmetry of the Lagrangian L then the map G = (G1 , . . . , Gk ): J 1 π → Rk given by Gα = F α − (2αL )V (X) defines a conservation law. Proof : Let X be a symmetry of the Lagrangian L. Then from (36) we get   dT (1) F α − (2αL )V (X) = −(δL)V (X ◦ π2,1 ) α

and from Lemma 2, the functions Gα = F α − (2αL )V (X) define a conservation law.  EXAMPLE 2. Consider the homogeneous isotropic 2-dimensional wave equation 3

∂11 φ − c2 ∂22 φ − c2 ∂33 φ = 0,

(40)

where φ : R → R is a solution, and defines a section of the trivial bundle π : E = R3 × R → R3 .

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Since J 1 π = R3 ×T31 R, Eq. (40) can be described as the Euler–Lagrange equation for the Lagrangian L: R3 × T31 R → R given by

 1 (v1 )2 − c2 (v2 )2 − c2 (v3 )2 . 2 In this case, for simplicity we consider the case c = 1. ∂ ◦ π1,0 along π1,0 and the functions on J 1 π , With the vector field X = v1 ∂q L(x, q, v) =

F 1 (v1 , v2 , v3 ) = −c2 (v2 )2 −c2 (v3 )2 , F 2 (v1 , v2 , v3 ) = c2 v1 v2 , F 3 (v1 , v2 , v3 ) = c2 v1 v3 using Theorem 2, we deduce that the following functions: G1 = F 1 − (21L )V (X) = −c2 (v2 )2 − c2 (v3 )2 − (v1 )2 , G2 = F 2 − (22L )V (X) = 2c2 v1 v2 , G3 = F 3 − (23L )V (X) = 2c2 v1 v3 ,

define a conservation law. 5.2.

Variational symmetries

In this section we consider the trivial bundle π : E = Rk × Q → Rk , and we recall some results of variational symmetries of the Euler–Lagrange equations that can be found in Olver’s book [30]. Let us remember that the solution of the Euler–Lagrange equation (25) can be obtained as the extremals of the functional Z L(φ) = (L ◦ j 1 φ)(x)d k x, 0

where d k x = dx 1 ∧ · · · ∧ dx k is the volume form on Rk . Roughly speaking, a variational symmetry is a diffeomorphism that leaves the variational integral L unchanged. DEFINITION 7. 1. A variational symmetry is a diffeomorphism 8: E = Rk × Q → E = Rk × Q verifying the following conditions: (a) It is a fiber-preserving map for the bundle π: E → Rk ; that is, 8 induces a diffeomorphisms ϕ: Rk → Rk such that π ◦ 8 = ϕ ◦ π.

(b) If x˜ = ϕ(x) for each x ∈ Rk , Z Z 1 −1 k (L ◦ j (8 ◦ φ ◦ ϕ ))(x)d ˜ x˜ = (L ◦ j 1 φ)(x)d k x, ˜ 

˜ = ϕ(). where 



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2. An infinitesimal variational symmetry is a vector field X ∈ X(Rk × Q) whose local flows are variational symmetries. The following results can be seen in Theorem 4.12 and in Corollary 4.30 [30] . THEOREM 3. i) A vector field X on Rk × Q is a variational symmetry if, and only if, X1 (L) + L dT (0) Xα = 0, where Xα = dx α (X). α

ii) If X is a variational symmetry then 2αL (X 1 ) defines a conservation law.

EXAMPLE 3. We consider again the homogeneous isotropic 2-dimensional wave equation (40). ∂ ∂ The rotation group X = −x 3 2 + x 2 3 is a variational symmetry, and then the ∂x ∂x corresponding conservation law (21L (X1 ), 22L (X1 ), 23L (X1 )) is given by the functions   1 3 1 2 3 2 2 3 x v1 v2 − x v1 v3 , − x u + x v2 v3 , − x u − v3 v2 x , 2 2 where u = (v1 )2 + (v2 )2 + (v3 )2 . EXAMPLE 4. We consider again Q = R, and let   0 = 1 + (∂2 φ)2 ∂11 φ − 2∂1 φ ∂2 φ ∂12 φ + 1 + (∂1 φ)2 ∂22 φ

be the equation of minimal surfaces, which is the Euler–Lagrange p equations for the 1 2 1 2 Lagrangian L: R × T2 R → R defined by L(x , x , q, v1 , v2 ) = 1 + (v1 )2 + (v2 )2 . ∂ ∂ ∂ is a variational symmetry, and The vector field X = −q 1 − q 2 + (x 1 + x 2 ) ∂x ∂x ∂q then the corresponding conservation law (21L (X 1 ), 22L (X 1 )) is given by the functions   −q(1 + (v2 )2 − v1 v2 ) + (x 1 + x 2 )v1 −q(1 + (v1 )2 − v1 v2 ) + (x 1 + x 2 )v2 p p , . 1 + (v1 )2 + (v2 )2 1 + (v1 )2 + (v2 )2 Now we describe some relationships between the above symmetries.

THEOREM 4. i) Let X be a π-vertical variational symmetry. Then the vector field X ◦ π1,0 along π1,0 is a generalized symmetry. ii) The conservation laws induced by X and X ◦ π1,0 coincide. Proof : i) Since X is a π-vertical variational symmetry, then locally X = Xi (x, q) ∂q∂ i , and from Theorem 3 we know that X1 (L) = 0. From (14) we know

that (X ◦ π1,0 )(1) = X1 ◦ π2,1 . Then

d(X◦π1,0 )(1) L(jx2 φ) = (X ◦ π1,0 )(1) (jx2 φ)(L) = X1 (jx1 φ)(L) = 0,

and thus X is a generalized symmetry. ii) It is a consequence of 2αL (X1 ) = (2αL )V (X ◦ π1,0 ).



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5.3.

Noether symmetries

In the paper [26] we introduced the following definition. DEFINITION 8. A vector field Y ∈ X(Rk × Tk1 Q) is an infinitesimal Noether symmetry if LY ωLα = 0, iY dx α = 0, LY EL = 0. THEOREM 5. Let X be a π-vertical vector field on Rk × Q such that X1 is an infinitesimal Noether symmetry, then X = X ◦ π1,0 is a generalized symmetry. Proof : Using (11), (31), then from the local expression X = Xi

∂ and the ∂q i

condition LX1 EL = 0, we deduce that

  ∂L ∂L i 1 . (X ◦ π1,0 ) i = vα X ∂q ∂vαi i

(41)

From the condition LX1 ωLα = 0 we obtain d(LX1 θLα ) = 0 and so there exist (locally defined) functions F α : U ⊂ Rk × Tk1 Q → R such that

LX1 θLα = dF α

1 ≤ α ≤ k.

With these identities we obtain the following relations ∂F α ∂L ∂Xi = ◦ π1,0 , ∂x β ∂vαi ∂x β ∂F α j

∂vβ

  ∂F α ∂L ∂Xi ∂L 1 = i ◦ π1,0 − X , j ∂q j ∂vα ∂q j ∂vα

∂L ∂Xi = i ◦ π1,0 = 0. ∂vα ∂vβj

(42)

From (14), (41) and (42), we deduce that   i ∂L ∂X ∂Xi ∂L 2 i j 1 d(X◦π1,0 )(1) L(jx φ) = X (φ(x)) i + + vα (jx φ) j ∂q jx1 φ ∂x α φ(x) ∂q jx1 φ ∂vαi jx1 φ  α   ∂L ∂F α ∂F ∂L i j 1 1 1 = X (φ(x)) i + + vα (jx φ) + X (jx φ) j ∂q jx1 φ ∂x α jx1 φ ∂q j jx1 φ ∂vα ∂F α ∂F α 1 j φ) + v α(j = dT (1) F α (jx2 φ) = x α ∂x α jx1 φ ∂q j jx1 φ for any jx2 φ. This proves that X ◦ π1,0 is a generalized symmetry.



Conclusions In this paper, we have discussed a new geometric formalism for fiber bundles over Euclidean spaces; this new formalism allows us to understand better the similarities

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and differences between the multisymplectic and k-cosymplectic settings. Even if such a fiber bundle is topologically trivial, it has some interest from the geometric point of view. Indeed, it is a way to understand better the multisymplectic formalism (and, by the way, it is a usual case in continuun mechanics). The current paper is a first step to get more treatable ways to work with the field equations when the base space (the space-time manifold in the physical contexts) is not trivial or even a parallelizable manifold. We are considering more general situations and this paper will help very much for more generalizations. Acknowledgments We acknowledge the financial support of the Ministerio de Ciencia e Innovaci´on (Spain), projects MTM2011-22585 and MTM2011-15725-E, Ministerio de Econom´ıa y Competividad (Spain) project MTM2013-42870-P and MTM2014-54855-P; the European project IRSES-project “Geomech-246981”, the ICMAT Severo. REFERENCES [1] L. Bua, I. Bucataru and M. Salgado: Symmetries, Newtonoid vector fields and conservation laws on the Lagrangian k-symplectic formalism, Rev. Math. Phys. 24 (2012), 1250030. [2] J. F. Cari˜nena, M. Crampin and L. A. Ibort: On the multisymplectic formalism for first order field theories, Differ. Geom. Appl. 1 (1991), 345–374. [3] J. F. Cari˜nena and J. Fern´andez-N´un˜ ez: Geometric theory of time-dependent singular Lagrangians, Fortschr. Phys. 41 (1993), 517–552. [4] J. F. Cari˜nena, J. Fern´andez-N´un˜ ez and E. Mart´ınez: A geometric approach to Cartan’s Second Theorem in time-dependent Lagrangian Mechanics, Lett. Math. Phys. 23 (1991), 51. [5] J. F. Cari˜nena, J. Fern´andez-N´un˜ ez and E. Mart´ınez: Noether’s theorem in time-dependent Lagrangian Mechanics, Rep. Math. Phys. 31 (1992), 189–203. [6] J. F. Cari˜nena, C. L´opez and E. Mart´ınez: A new approach to the converse of Cartan’s theorem, J. Phys. A: Math. Gen. 22 (1989), 4777. [7] F. Cantrijn, M. de Le´on and E. A. Lacomba: Gradient vector fields on cosymplectic manifolds, J. Phys. A 25 (1992), 175–188. [8] A. Echeverr´ıa-Enr´ıquez, M. C. Mu˜noz-Lecanda and N. Rom´an-Roy: Geometrical setting of time-dependent regular systems: Alternative models, Rev. Math. Phys. 3 (1991), 301–330. [9] A. Echeverr´ıa-Enr´ıquez, M. C. Mu˜noz-Lecanda and N. Rom´an-Roy: Geometry of Lagrangian first-order classical field theories, Forts. Phys. 44 (1996), 235–280. [10] A. Echeverr´ıa-Enr´ıquez, M. C. Mu˜noz-Lecanda and Rom´an-Roy: Geometry of multisymplectic Hamiltonian first-order field theories, J. Math. Phys. 41 (2000), 7402–744. [11] G. Giachetta, L. Mangiarotti and G. Sardanashvily: New Lagrangian and Hamiltonian Methods in Field Theory, World Scientific, Singapore 1997. [12] G. Giachetta, L. Mangiarotti and G. Sardanashvily: Covariant Hamilton equations for field theory, J. Phys. A 32 (1999), 6629–6642. [13] M. J. Gotay, J. Isenberg, J. E. Marsden and R. Montgomery: Momentum Maps and Classical Relativistic Fields. Part I: Covariant Field Theory, arXiv:physics/9801019v2 (2004). [14] M. J. Gotay, J. Isenberg and J. E. Marsden: Momentun Maps and Classical Relativistic Fields. Part II: Canonical Analysis of Field Theories, arXiv:math-ph/0411032v1 (2004). [15] X. Gracia and J.M. Pons: On an evolution operator connecting Lagrangian and Hamiltonian formalisms, Lett. Math. Phys. 17 (1989), 175–180. [16] J. V. Jos´e and E. J. Saletan: Classical Dynamics, a Contemporary Approach, Cambridge University Press, Cambridge 1988.

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