A k -contact Lagrangian formulation for nonconservative field theories
Jordi Gaset, Xavier Gràcia, Miguel C. Muñoz-Lecanda, Xavier Rivas, Narciso Román-Roy
aa r X i v : . [ m a t h - ph ] F e b A k -contact Lagrangian formulationfor nonconservative field theories a Jordi Gaset, b Xavier Gr`acia, b Miguel C. Mu˜noz-Lecanda, b Xavier Rivas and b Narciso Rom´an-Roy ∗ a Department of Physics, Universitat Aut`onoma de Barcelona, Bellaterra, Catalonia, Spain b Department of Mathematics, Universitat Polit`ecnica de Catalunya, Barcelona, Catalonia, Spain
February 23, 2020
Abstract
We present a geometric Lagrangian formulation for first-order field theories with dis-sipation. This formulation is based on the k -contact geometry introduced in a previouspaper, and gathers contact Lagrangian mechanics with k -symplectic Lagrangian field the-ory together. We also study the symmetries and dissipation laws for these nonconservativetheories, and analyze some examples. Keywords: contact structure, field theory, Lagrangian system, dissipation, k -symplectic struc-ture, k -contact structure. MSC 2020 codes:
Contents k -tangent bundle, k -vector fields and geometric structures . . . . . . . . . . . . . . . . . . 32.2 k -symplectic manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.3 k -contact structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.4 k -contact Hamiltonian systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 k -contact Lagrangian field theory 9 k -contact Lagrangian systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93.2 The k -contact Euler–Lagrange equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113.3 k -contact canonical Hamiltonian formalism . . . . . . . . . . . . . . . . . . . . . . . . . . 13 ∗ emails: [email protected], [email protected], [email protected], [email protected],[email protected] . Gaset et al — A k -contact Lagrangian formulation for nonconservative field theories In the last years the methods of differential geometry have been used to develop an intrinsicframework to describe dissipative or damped systems, in particular using contact geometry[2, 17, 24]. It has been applied to give both the Hamiltonian and the Lagrangian descriptionsof mechanical systems with dissipation [3, 5, 7, 8, 9, 13, 16, 25, 27]. Contact geometry hasother physical applications, as for instance thermodynamics, quantum mechanics, circuit theory,control theory, etc (see [4, 8, 20, 24, 28], among others). All of them are described by ordinarydifferential equations to which some terms that account for the dissipation or damping havebeen added.These geometric methods have been also used to give intrinsic descriptions of the Lagrangianand Hamiltonian formalisms of field theory; in particular, those of multisymplectic and k -symplectic geometry (see, for instance, [6, 12, 14, 18, 29, 31] and references therein). Nev-ertheless, all these methods are developed, in general, to model systems of variational type; thatis, without dissipation or damping.In a recent paper [15] we have introduced a generalization of both contact geometry and k -symplectic geometry to describe field theories with dissipation, and more specifically theirHamiltonian (De Donder–Weyl) covariant formulation. This new formalism is inspired by con-tact Hamiltonian mechanics, where the addition of a “contact variable” s allows to describedissipation terms; geometrically this new variable comes from a contact form instead of theusual symplectic form of Hamiltonian mechanics. In the field theory case, if k is the numberof independent variables (usually space-time variables), we add k new dependent variables s α to introduce dissipation terms in the De Donder–Weyl equations. These new variables can beobtained geometrically from the notion of k -contact structure : a family of k differential 1-forms η α satisfying certain properties. Then a k -contact Hamiltonian system is a manifold endowedwith a k -contact structure and a Hamiltonian function H . With these elements we can statethe k -contact Hamilton equations, which indeed add dissipation terms to the usual Hamiltonianfield equations. The study of their symmetries also allows to obtain some dissipation laws. Thisformalism was applied to two relevant examples: the damped vibrating string and Burgers’equation.The aim of this paper is to extend the above study, developing the Lagrangian formalism offield theories with dissipation, mainly in the regular case. For this purpose, the aforementioned k -contact structure will be used to generalize the Lagrangian formalism of the contact mechanicspresented in [9, 16] and the Lagrangian k -symplectic formulation of classical field theories [12, 29].In this new formalism the phase bundle is ⊕ k T Q × R k = (T Q ⊕ k . . . ⊕ T Q ) × R k . Then, givena Lagrangian function L : ⊕ k T Q × R k → R , one defines k differential 1-forms η α L which, when L is regular , constitute a k -contact structure on the phase bundle. The k -contact Lagrangianfield equations are then defined as the k -contact Hamiltonian field equations for the Lagrangianenergy E L . When written in coordinates they are the Euler–Lagrange equations for L with someadditional terms which account for dissipation.We also study several types of symmetries for these Lagrangian field theories, as well astheir associated dissipation laws, which are characteristic of dissipative systems, and are theanalogous to the conservation laws for conservative systems.As examples of this formalism we study the construction of a k -contact Lagrangian formu- . Gaset et al — A k -contact Lagrangian formulation for nonconservative field theories k -contact formulation.The paper is organized as follows. Section 2 is devoted to briefly review several preliminaryconcepts on k -symplectic manifolds, k -contact geometry and k -contact Hamiltonian systems forfield theories with dissipation. In Section 3 we introduce the notion of k -contact Lagrangiansystem, and set the geometric framework for the Lagrangian formalism of field theories withdissipation, stating the geometric form of the contact Euler–Lagrange equations in several equiv-alent ways, as well as the Legendre transformation and the associated canonical Hamiltonianformalism. In Section 4 we study several types of Lagrangian symmetries and the relationsbetween them, as well as the corresponding dissipation laws. Finally, some examples are givenin Section 5.Throughout the paper all the manifolds and mappings are assumed to be smooth. Sum overcrossed repeated indices is understood. k -tangent bundle, k -vector fields and geometric structures (See [12, 29] for more details).Let Q be a manifold and consider ⊕ k T Q = T Q ⊕ k . . . ⊕ T Q (it is called the k -tangentbundle or bundle of k -velocities of Q ), which is endowed with the natural projections to eachdirect summand and to the base manifold: τ α : ⊕ k T Q → T Q , τ Q : ⊕ k T Q → Q .
A point of ⊕ k T Q is w q = ( v q , . . . , v kq ) ∈ ⊕ k T Q , where ( v i ) q ∈ T q Q .A k -vector field on Q is a section X : Q −→ ⊕ k T Q of the projection τ Q . It is specified bygiving k vector fields X , . . . , X k ∈ X ( Q ), obtained as X α = τ α ◦ X ; for 1 ≤ α ≤ k , and it isdenoted X = ( X , . . . , X k ).Given a map φ : D ⊂ R k → Q , the first prolongation of φ to ⊕ k T Q is the map φ ′ : D ⊂ R k → ⊕ k T Q defined by φ ′ ( t ) = (cid:18) φ ( t ) , T φ (cid:18) ∂∂t (cid:12)(cid:12)(cid:12) t (cid:19) , . . . , T φ (cid:18) ∂∂t k (cid:12)(cid:12)(cid:12) t (cid:19)(cid:19) ≡ ( φ ( t ); φ ′ α ( t )) , where t = ( t , . . . , t k ) are the canonical coordinates of R k . A map ϕ : D ⊂ R k → ⊕ k T Q is saidto be holonomic if it is the first prolongation of a map φ : D ⊂ R k → Q .A map φ : D ⊂ R k → Q is an integral map of a k -vector field X = ( X , . . . , X k ) when φ ′ = X ◦ φ . (1)Equivalently, T φ ◦ ∂∂t α = X α ◦ φ , for every α . A k -vector field X is integrable if every pointof Q is in the image of an integral map of X . . Gaset et al — A k -contact Lagrangian formulation for nonconservative field theories X α = X iα ∂∂x i , then φ is an integral map of X if, and only if, it is a solutionto the following system of partial differential equations: ∂φ i ∂t α = X iα ( φ ) . A k -vector field X = ( X , . . . , X k ) is integrable if, and only if, [ X α , X β ] = 0, for every α, β [26]; these are the necessary and sufficient conditions for the integrability of the above systemof partial differential equations.As in the case of the tangent bundle, local coordinates ( q i ) in U ⊂ Q induce natural coordi-nates ( q i , v iα ) in ( τ Q ) − ( U ) ⊂ ⊕ k T Q , with 1 ≤ i ≤ n and 1 ≤ α ≤ k .Given α and w q ∈ ⊕ k T Q , there exists a natural map (Λ w q q ) α : T q Q → T w q ( ⊕ k T Q ), calledthe α -vertical lift from q to w q , defined as(Λ w q q ) α ( u q ) = ddλ ( v q , . . . , v α − q , v αq + λu q , v α +1 q , . . . , v kq ) | λ =0 . In coordinates, if u q = a i ∂∂q i (cid:12)(cid:12)(cid:12) q , we have (Λ w q q ) α ( u q ) = a i ∂∂v iα (cid:12)(cid:12)(cid:12) w q . Observe that these α -verticallifts are τ Q -vertical vectors. These vertical lifts extend to vector fields in a natural way; that is,if X ∈ X ( Q ), then its α -vertical lift, Λ α ( X ) ∈ X ( ⊕ k T Q ), is given by (Λ α ( X )) w q := (Λ w q q ) α ( X q ).The canonical k -tangent structure on ⊕ k T Q is the set ( J , . . . , J k ) of tensor fields oftype (1 ,
1) in ⊕ k T Q defined as J α w q := (Λ w q q ) α ◦ T w q τ Q . In natural coordinates we have J α = ∂∂v iα ⊗ d q i .The Liouville vector field ∆ ∈ X ( ⊕ k T Q ) is the infinitesimal generator of the flow ψ : R ×⊕ k T Q −→ ⊕ k T Q , given by ψ ( t ; v q , . . . , v kq ) = ( e t v q , . . . , e t v kq ). Observe that ∆ = ∆ + . . . +∆ k , where each ∆ α ∈ X ( ⊕ k T Q ) is the infinitesimal generator of the flow ψ α : R × ⊕ k T Q −→⊕ k T Q ψ α ( s ; v q , . . . , v kq ) = ( v q , . . . , v ( α − q , e t v αq , v ( α +1) q , . . . , v kq ) . In coordinates, ∆ = v iα ∂∂v iα .Given a map Φ : M → N , there exists a natural extension ⊕ k TΦ : ⊕ k T M → ⊕ k T N , definedby ⊕ k TΦ( v q , . . . , v kq ) := (T q Φ( v q ) , . . . , T q Φ( v kq )) . By definition, a k -vector field Γ = (Γ , . . . , Γ k ) in ⊕ k T Q is a section of the projection τ ⊕ k T Q : T( ⊕ k T Q ) ⊕ k . . . ⊕ T( ⊕ k T Q ) → ⊕ k T Q .
Then, we say that Γ is a second order partial differential equation ( sopde ) if it is also asection of the projection ⊕ k T τ Q : T( ⊕ k T Q ) ⊕ k . . . ⊕ T( ⊕ k T Q ) → ⊕ k T Q ;that is, ⊕ k T τ Q ◦ Γ = Id ⊕ k T Q = τ ⊕ k T Q ◦ Γ . Notice that a k -vector field Γ in ⊕ k T Q is a sopde if, and only if, J α (Γ α ) = ∆. . Gaset et al — A k -contact Lagrangian formulation for nonconservative field theories k -vector field Γ = (Γ , . . . , Γ k ) in ⊕ k T Q is a sopde if, and onlyif, its integrable maps are holonomic.In natural coordinates, the expression of the components of a sopde is Γ α = v iα ∂∂q i +Γ iαβ ∂∂v iβ .Then, if ψ : R k → ⊕ k T Q , locally given by ψ ( t ) = ( ψ i ( t ) , ψ iβ ( t )), is an integral map of anintegrable sopde , from (1) we have that ∂ψ i ∂t α (cid:12)(cid:12)(cid:12) t = ψ iα ( t ) , ∂ψ iβ ∂t α (cid:12)(cid:12)(cid:12) t = Γ iαβ ( ψ ( t )) . Furthermore, ψ = φ ′ , where φ ′ is the first prolongation of the map φ = τ ◦ ψ : R k ψ → ⊕ k T Q τ → Q ,and hence φ is a solution to the system of second order partial differential equations ∂ φ i ∂t α ∂t β ( t ) = Γ iαβ (cid:18) φ i ( t ) , ∂φ i ∂t γ ( t ) (cid:19) . (2)Observe that, from (2) we obtain that, if Γ is an integrable sopde , then Γ iαβ = Γ iβα . k -symplectic manifolds (See [1, 10, 11, 12, 29] for more details.)Let M be a manifold of dimension N = n + kn . A k -symplectic structure on M isa family ( ω , . . . , ω k ; V ), where ω α ( α = 1 , . . . , k ) are closed 2-forms, and V is an integrable nk -dimensional tangent distribution on M such that( i ) ω α | V × V = 0 (for every α ) , ( ii ) k \ α =1 ker ω α = { } . Then (
M, ω α , V ) is called a k -symplectic manifold .For every point of M there exist a neighbourhood U and local coordinates ( q i , p αi ) (1 ≤ i ≤ n ,1 ≤ α ≤ k ) such that, on U , ω α = d q i ∧ d p αi , V = (cid:28) ∂∂p i , . . . , ∂∂p ki (cid:29) . These are the so-called
Darboux or canonical coordinates of the k -symplectic manifold [1].The canonical model for k -symplectic manifolds is ⊕ k T ∗ Q = T ∗ Q ⊕ k . . . ⊕ T ∗ Q , with naturalprojections π α : ⊕ k T ∗ Q → T ∗ Q , π Q : ⊕ k T ∗ Q → Q .
As in the case of the cotangent bundle, local coordinates ( q i ) in U ⊂ Q induce natural coordinates( q i , p αi ) in ( π Q ) − ( U ). If θ and ω = − d θ are the canonical forms of T ∗ Q , then ⊕ k T ∗ Q is endowedwith the canonical forms θ α = ( π α ) ∗ θ , ω α = ( π α ) ∗ ω = − ( π α ) ∗ d θ = − d θ α , (3)and in natural coordinates we have that θ α = p αi d q i and ω α = d q i ∧ d p αi . Thus, the triple( ⊕ k T ∗ Q, ω α , V ), where V = ker T π Q , is a k -symplectic manifold, and the natural coordinates in ⊕ k T ∗ Q are Darboux coordinates. . Gaset et al — A k -contact Lagrangian formulation for nonconservative field theories k -contact structures The definition of k -contact structure has been recently introduced in [15], where the reader canfind more details.Remember that, if M is a smoooth manifold of dimension m , a (generalized) distributionon M is a subset D ⊂ T M such that, for every x ∈ M , D x ⊂ T x M is a vector subspace. Thedistribution D is smooth when it can be locally spanned by a family of smooth vector fields, andis regular when it is smooth and has locally constant rank. A codistribution on M is a subset C ⊂ T ∗ M with similar properties. The annihilator D ◦ of a distribution D is a codistribution.A (smooth) differential 1-form η ∈ Ω ( M ) generates a smooth codistribution that we denoteby h η i ⊂ T ∗ M ; it has rank 1 at every point where η does not vanish. Its annihilator is adistribution h η i ◦ ⊂ T M ; it can be described also as the kernel of the vector bundle morphism b η : T M → M × R defined by η . This distribution has corank 1 at every point where η does notvanish.Now, given k differential 1-forms η , . . . , η k ∈ Ω ( M ), let: C C = h η , . . . , η k i ⊂ T ∗ M , D C = (cid:0) C C (cid:1) ◦ = ker b η ∩ . . . ∩ ker c η k ⊂ T M , D R = ker d d η ∩ . . . ∩ ker d d η k ⊂ T M , C R = (cid:0) D R (cid:1) ◦ ⊂ T ∗ M .
Definition 2.1. A k -contact structure on M is a family of k differential 1-forms η α ∈ Ω ( M ) such that, with the preceding notations,(i) D C ⊂ T M is a regular distribution of corank k ; or, what is equivalent, η ∧ . . . ∧ η k = 0 ,at every point.(ii) D R ⊂ T M is a regular distribution of rank k .(iii) D C ∩ D R = { } or, what is equivalent, k \ α =1 (cid:16) ker c η α ∩ ker d d η α (cid:17) = { } .We call C C the contact codistribution ; D C the contact distribution ; D R the Reeb distri-bution ; and C R the Reeb codistribution .A k -contact manifold is a manifold endowed with a k -contact structure. Remark 2.2.
If conditions (i) and (ii) hold, then (iii) is equivalent to( iii ′ ) T M = D C ⊕ D R . For k = 1 we recover the definition of contact structure. Theorem 2.3.
Let ( M, η α ) be a k –contact manifold.1. The Reeb distribution D R is involutive, and therefore integrable.2. There exist k vector fields R α ∈ X ( M ) , the Reeb vector fields , uniquely defined by therelations i ( R β ) η α = δ αβ , i ( R β )d η α = 0 . (4) . Gaset et al — A k -contact Lagrangian formulation for nonconservative field theories
3. The Reeb vector fields commute, [ R α , R β ] = 0 , and they generate D R . There are coordinates ( x I ; s α ) such that R α = ∂∂s α , η α = d s α − f αI ( x ) d x I , where f αI ( x ) are functions depending only on the x I , which are called adapted coordinates (to the k -contact structure). Example 2.4.
Given k ≥
1, the manifold ( ⊕ k T ∗ Q ) × R k has a canonical k -contact structuredefined by the 1-forms η α = d s α − θ α , where s α is the α -th cartesian coordinate of R k , and θ α is the pull-back of the canonical 1-formof T ∗ Q with respect to the projection ( ⊕ k T ∗ Q ) × R k → T ∗ Q to the α -th direct summand. Usingcoordinates q i on Q and natural coordinates ( q i , p αi ) on each T ∗ Q , their local expressions are η α = d s α − p αi d q i , from which d η α = d q i ∧ d p αi , and the Reeb vector fields are R α = ∂∂s α . The following result ensures the existence of canonical coordinates for a particular kind of k -contact manifolds: Theorem 2.5 ( k -contact Darboux theorem) . Let ( M, η α ) be a k –contact manifold of dimension n + kn + k such that there exists an integrable subdistribution V of D C with rank V = nk . Aroundevery point of M , there exists a local chart of coordinates ( U ; q i , p αi , s α ) , ≤ α ≤ k , ≤ i ≤ n ,such that η α | U = d s α − p αi d q i . In these coordinates, D R | U = (cid:28) R α = ∂∂s α (cid:29) , V| U = (cid:28) ∂∂p αi (cid:29) . These are the so-called canonical or Darboux coordinates of the k -contact manifold. This theorem allows us to consider the manifold presented in the example 2.4 as the canonicalmodel for these kinds of k -contact manifolds. k -contact Hamiltonian systems Together with k -contact structures, k -contact Hamiltonian systems have also been defined in[15].A k -contact Hamiltonian system is a family ( M, η α , H ), where ( M, η α ) is a k -contactmanifold, and H ∈ C ∞ ( M ) is called a Hamiltonian function . The k -contact Hamilton–deDonder–Weyl equations for a map ψ : D ⊂ R k → M are ( i ( ψ ′ α )d η α = (cid:0) d H − ( L R α H ) η α (cid:1) ◦ ψ ,i ( ψ ′ α ) η α = −H ◦ ψ . (5) . Gaset et al — A k -contact Lagrangian formulation for nonconservative field theories k -contact Hamilton–de Donder–Weyl equations for a k -vector field X = ( X , . . . , X k )in M are ( i ( X α )d η α = d H − ( L R α H ) η α ,i ( X α ) η α = −H . (6)Their solutions are called Hamiltonian k -vector fields . These equations are equivalent to ( L X α η α = − ( L R α H ) η α ,i ( X α ) η α = −H . (7)Solutions to these equations always exist, although they are neither unique, nor necessarilyintegrable.If X is an integrable k -vector field in M , then every integral map ψ : D ⊂ R k → M of X satisfies the k -contact equation (5) if, and only if, X is a solution to (6). Notice, however, thatequations (5) and (6) are not, in general, fully equivalent, since a solution to (5) may not be anintegral map of some integrable k -vector field in M solution to (6).An alternative, partially equivalent, expression for the Hamilton–De Donder–Weyl equations,which does not use the Reeb vector fields R α , can be given as follows. Consider the 2-formsΩ α = −H d η α + d H ∧ η α . On the open set O = { p ∈ M ; H ( p ) = 0 } , if a k -vector field X = ( X α )satisfies ( i ( X α )Ω α = 0 ,i ( X α ) η α = −H , (8)then X is a solution of the Hamilton–De Donder–Weyl equations (6)). Any integral map ψ ofsuch a k -vector field is a solution to ( i ( ψ ′ α )Ω α = 0 ,i ( ψ ′ α ) η α = −H ◦ ψ . (9) Remark 2.6.
If the family (
M, η α ) does not hold some of the conditions of Definition (2.1),then ( M, η α ) is called a k -precontact manifold and ( M, η α , H ) is said to be a k -precontactHamiltonian system . In this case, the Reeb vector fields are not uniquely defined. However,as it happens in other similar situations (precosymplectic mechanics, k -precosymplectic fieldtheories or precontact mechanics) [9, 23], it could be proved that equations (5) and (6) does notdepend on the used Reeb vector fields and, thus, the equations are still valid.In canonical coordinates, if ψ = ( q i ( t β ) , p αi ( t β ) , s α ( t β )), then ψ ′ α = (cid:16) q i , p αi , s α , ∂q i ∂t β , ∂p αi ∂t β , ∂s α ∂t β (cid:17) ,and these equations read ∂q i ∂t α = ∂ H ∂p αi ◦ ψ ,∂p αi ∂t α = − (cid:18) ∂ H ∂q i + p αi ∂ H ∂s α (cid:19) ◦ ψ ,∂s α ∂t α = (cid:18) p αi ∂ H ∂p αi − H (cid:19) ◦ ψ , (10) . Gaset et al — A k -contact Lagrangian formulation for nonconservative field theories X = ( X α ) is a k -vector field solution to (8) and in canonical coordinates we have that X α = X βα ∂∂s β + X iα ∂∂q i + X βαi ∂∂p βi , then X iα = ∂ H ∂p αi ,X ααi = − (cid:18) ∂ H ∂q i + p αi ∂ H ∂s α (cid:19) ,X αα = p αi ∂ H ∂p αi − H , (11) k -contact Lagrangian field theory k -contact Lagrangian systems Using the geometric framework introduced in Section 2.1, we are ready to deal with Lagrangiansystems with dissipation in field theories. First we need to enlarge the bundle in order to includethe dissipation variables. Then, consider the bundle ⊕ k T Q × R k with canonical projections¯ τ : ⊕ k T Q × R k → ⊕ k T Q , ¯ τ k : ⊕ k T Q × R k → T Q , s α : ⊕ k T Q × R k → R . Natural coordinates in ⊕ k T Q × R k are ( q i , v iα , s α ).As ⊕ k T Q × R k → ⊕ k T Q is a trivial bundle, the canonical structures in ⊕ k T Q (the canonical k -tangent structure and the Liouville vector field described above) can be extended to ⊕ k T Q × R k in a natural way, and are denoted with the same notation ( J α ) and ∆. Then, using thesestructures, we can extend also the concept of sode k -vector fields to ⊕ k T Q × R k as follows: Definition 3.1. A k -vector field Γ = (Γ α ) in ⊕ k T Q × R k is a second order partial differ-ential equation ( sopde ) if J α (Γ α ) = ∆ . The local expression of a sopde isΓ α = v iα ∂∂q i + Γ iαβ ∂∂v iβ + g βα ∂∂s β . (12) Definition 3.2.
Let ψ : R k → Q × R k be a section of the projection Q × R k → R k ; with ψ = ( φ, s α ) , where φ : R k → Q . The first prolongation of ψ to ⊕ k T Q × R k is the map σ : R k → ⊕ k T Q × R k given by σ = ( φ ′ , s α ) . The map σ is said to be holonomic . The following property is a straightforward consequence of the above definitions and theresults about sopdes in the bundle ⊕ k T Q given in Section 2.1: Proposition 3.3. A k -vector field Γ in ⊕ k T Q × R k is a sopde if, and only if, its integral mapsare holonomic. Now we can state the Lagrangian formalism of field theories with dissipation. . Gaset et al — A k -contact Lagrangian formulation for nonconservative field theories Definition 3.4. A Lagrangian function is a function
L ∈ C ∞ ( ⊕ k T Q × R k ) .The Lagrangian energy associated with L is the function E L := ∆( L ) − L ∈ C ∞ ( ⊕ k T Q × R k ) .The Cartan forms associated with L are θ α L = t ( J α ) ◦ d L ∈ Ω ( ⊕ k T Q × R k ) , ω α L = − d θ α L ∈ Ω ( ⊕ k T Q × R k ) . Finally, we can define the forms η α L = d s α − θ α L ∈ Ω ( ⊕ k T Q × R k ) , d η α L = ω α L ∈ Ω ( ⊕ k T Q × R k ) . The couple ( ⊕ k T Q × R k , L ) is said to be a k -contact Lagrangian system . In natural coordinates ( q i , v iα , s α ) of ⊕ k T Q × R k , the local expressions of these elements are E L = v iα ∂ L ∂v iα − L , η α L = d s α − ∂ L ∂v iα d q i . Before introducing the Legendre map, remember that, given a bundle map f : E → F between two vector bundles over a manifold B , the fibre derivative of f is the map F f : E → Hom(
E, F ) ≈ F ⊗ E ∗ obtained by restricting f to the fibres, f b : E b → F b , and computingthe usual derivative of a map between two vector spaces: F f ( e b ) = D f b ( e b ). This applies inparticular when the second vector bundle is trivial of rank 1, that is, for a function f : E → R ;then F f : E → E ∗ . This map also has a fibre derivative F f : E → E ∗ ⊗ E ∗ , which is usuallycalled the fibre Hessian of f . For every e b ∈ E , F f ( e b ) can be considered as a symmetricbilinear form on E b . It is easy to check that F f is a local diffeomorphism at a point e ∈ E if,and only if, the Hessian F f ( e ) is non-degenerate. (See [21] for details). Definition 3.5.
The
Legendre map associated with a Lagrangian
L ∈ C ∞ ( ⊕ k T Q × R k ) is thefibre derivative of L , considered as a function on the vector bundle ⊕ k T Q × R k → Q × R k ; thatis, the map F L : ⊕ k T Q × R k → ⊕ k T ∗ Q × R k given by F L ( v q , . . . , v kq ; s α ) = (cid:0) F L ( · , s α )( v q , . . . , v kq ) , s α (cid:1) ; ( v q , . . . , v kq ) ∈ ⊕ k T Q , where L ( · , s α ) denotes the Lagrangian with s α freezed. This map is locally given by
F L ( q i , v iα , s α ) = (cid:16) q i , ∂ L ∂v iα , s α (cid:17) . Remark 3.6.
The Cartan forms can also be defined as θ α L = F L ∗ θ α , ω α L = F L ∗ ω α , where θ α and ω α are given in (3). Proposition 3.7.
For a Lagrangian function L the following conditions are equivalent:1. The Legendre map F L is a local diffeomorphism.2. The fibre Hessian F L : ⊕ k T Q × R k −→ ( ⊕ k T ∗ Q × R k ) ⊗ ( ⊕ k T ∗ Q × R k ) of L is everywherenondegenerate. (The tensor product is of vector bundles over Q × R k .) . Gaset et al — A k -contact Lagrangian formulation for nonconservative field theories ( ⊕ k T Q × R k , η α L ) is a k -contact manifold.Proof. The proof can be easily done using natural coordinates, bearing in mind that F L ( q i , v iα , s α ) = (cid:16) q i , ∂ L ∂v iα , s α (cid:17) , F L ( q i , v iα , s α ) = ( q i , W αβij , s α ) , with W αβij = ∂ L ∂v iα ∂v jβ ! . Then the conditions in the proposition mean that the matrix W = ( W αβij ) is everywhere non-singular. Definition 3.8.
A Lagrangian function L is said to be regular if the equivalent conditions inProposition 3.7 hold. Otherwise L is called a singular Lagrangian. In particular, L is said tobe hyperregular if F L is a global diffeomorphism.
Given a regular k -contact Lagrangian system ( ⊕ k T Q × R k , L ), from (4) we have that the Reeb vector fields ( R L ) α ∈ X ( ⊕ k T Q × R k ) for this system are the unique solution to i (( R L ) α )d η β L = 0 , i (( R L ) α ) η β L = δ βα . If L is regular, then there exists the inverse W ijαβ of the Hessian matrix, namely W ijαβ ∂ L ∂v jβ ∂v kγ = δ ik δ γα , and then a simple calculation in coordinates leads to( R L ) α = ∂∂s α − W jiγβ ∂ L ∂s α ∂v jγ ∂∂v iβ . k -contact Euler–Lagrange equations As a result of the preceding definitions and results, every regular contact Lagrangian system hasassociated the k -contact Hamiltonian system ( ⊕ k T Q × R , η α L , E L ). Then: Definition 3.9.
Let ( ⊕ k T Q × R k , L ) be a k -contact Lagrangian system.The k -contact Euler–Lagrange equations for a holonomic maps σ : R k → ⊕ k T Q × R k are i ( σ ′ α )d η α L = (cid:16) d E L − ( L ( R L ) α E L ) η α L (cid:17) ◦ σ ,i ( σ ′ α ) η α L = − E L ◦ σ . (13) The k -contact Lagrangian equations for a k -vector field X L = (( X L ) α ) in ⊕ k T Q × R k are ( i (( X L ) α )d η α L = d E L − ( L ( R L ) α E L ) η α L ,i (( X L ) α ) η α L = − E L . (14) A k -vector field which is solution to these equations is called a Lagrangian k -vector field . A first relevant result is: . Gaset et al — A k -contact Lagrangian formulation for nonconservative field theories Proposition 3.10.
Let ( ⊕ k T Q × R k , L ) be a k -contact regular Lagrangian system. Then, the k -contact Euler–Lagrange equations (14) admit solutions. They are not unique if k > .Proof. The proof is the same as that of Proposition 4.3 in [15].In a natural chart of coordinates of ⊕ k T Q × R k , equations (13) read ∂∂t α (cid:18) ∂ L ∂v iα ◦ σ (cid:19) = (cid:18) ∂ L ∂q i + ∂ L ∂s α ∂ L ∂v iα (cid:19) ◦ σ , ∂s α ∂t α = L ◦ σ , (15)meanwhile, for a k -vector field X L = (( X L ) α ) with ( X L ) α = ( X L ) iα ∂∂q i +( X L ) iαβ ∂∂v iβ +( X L ) βα ∂∂s β ,the Lagrangian equations (14) are0 = (cid:16) ( X L ) jα − v jα (cid:17) ∂ L ∂v jα ∂s β , (16)0 = (cid:16) ( X L ) jα − v jα (cid:17) ∂ L ∂v iβ ∂v jα , (17)0 = (cid:16) ( X L ) jα − v jα (cid:17) ∂ L ∂q i ∂v jα + ∂ L ∂q i − ∂ L ∂s β ∂v iα ( X L ) βα − ∂ L ∂q j ∂v iα ( X L ) jα − ∂ L ∂v jβ ∂v iα ( X L ) jαβ + ∂ L ∂s α ∂ L ∂v iα , (18)0 = L + ∂ L ∂v iα (cid:16) ( X L ) jα − v jα (cid:17) − ( X L ) αα . (19)If L is a regular Lagrangian, equations (17) lead to v iα = ( X L ) iα , which are the sopde conditionfor the k -vector field X . Then, (16) holds identically, and (19) and (18) give( X L ) αα = L , − ∂ L ∂q i + ∂ L ∂s β ∂v iα ( X L ) βα + ∂ L ∂q j ∂v iα v jα + ∂ L ∂v jβ ∂v iα ( X L ) jαβ = ∂ L ∂s α ∂ L ∂v iα . Notice that, if this sopde X L is integrable, these last equations are the Euler–Lagrange equations(15) for its integral maps. In this way, we have proved that: Proposition 3.11. If L is a regular Lagrangian, then the corresponding Lagrangian k -vectorfields X L (solutions to the k -contact Lagrangian equations (14) ) are sopde ’s and if, in addi-tion, X L is integrable, then its integral maps are solutions to the k -contact Euler–Lagrange fieldequations (13) .This sopde X L ≡ Γ L is called the Euler–Lagrange k -vector field associated with theLagrangian function L . Remark 3.12.
It is interesting to point out how, in the Lagrangian formalism of dissipativefield theories, the second equation in (15) relates the variation of the “dissipation coordinates” s α to the Lagrangian function. Remark 3.13. If L is not regular then ( ⊕ k T Q × R k , η α L , E L ) is a k -precontact system and, ingeneral, equations (13) and (14) have no solutions everywhere in ⊕ k T Q × R k but, in the most . Gaset et al — A k -contact Lagrangian formulation for nonconservative field theories ⊕ k T Q × R k which is obtained by applyinga suitable constraint algorithm. Nevertheless, solutions to equations (14) are not necessarily sopde (unless it is required as the additional condition J α ( X α ) = ∆) and, as a consequence, ifthey are integrable, their integral maps are not necessarily holonomic. Remark 3.14.
Observe that the particular case k = 1 gives the Lagrangian formalism formechanical systems with dissipation [9, 16]. k -contact canonical Hamiltonian formalism In the regular or the hyper-regular cases we have that
F L is a (local) diffeomorphism between( ⊕ k T Q × R k , η α L ) and ( ⊕ k T ∗ Q × R k , η α ), where F L ∗ η α = η α L . Furthermore, there exists (maybelocally) a function H ∈ C ∞ ( ⊕ k T ∗ Q × R ) such that H = E L ◦ F L − ; then we have the k -contactHamiltonian system ( ⊕ k T ∗ Q × R k , η α , H ), for which F L ∗ ( R L ) α = R α . Therefore, if Γ L is anEuler–Lagrange k -vector field associated with L in ⊕ k T Q × R k , then F L ∗ Γ L = X H is a contactHamiltonian k -vector field associated with H in ⊕ k T ∗ Q × R k , and conversely.For singular Lagrangians, following [19] we define: Definition 3.15.
A singular Lagrangian L is almost-regular if1. P := F L ( ⊕ k T Q × R k ) is a closed submanifold of ⊕ k T ∗ Q × R k .2. F L is a submersion onto its image.3. The fibres
F L − ( p ) , for every p ∈ P , are connected submanifolds of ⊕ k T Q × R k . If L is almost-regular and : P ֒ → ⊕ k T ∗ Q × R k is the natural embedding, denoting by F L : ⊕ k T Q × R k → P the restriction of F L given by ◦ F L = F L ; then there exists H ∈ C ∞ ( P ) such that ( F L ) ∗ H = E L . Furthermore, we can define η α = ∗ η α , and then, the triple( P , η α , H ) is the k -precontact Hamiltonian system associated with L , and the correspondingHamiltonian fields equations are (8) or (9) (in P ). In general, these equations have no solutionseverywhere in P but, in the most favourable situations, they do in a submanifold P f ֒ → P ,which is obtained applying a suitable constraint algorithm, and where there are Hamiltonian k -vector fields in P , tangent to P f . As in [15], we introduce different concepts of symmetry of the system, depending on whichstructure is preserved, putting the emphasis on the transformations that leave the geometricstructures invariant, or on the transformations that preserve the solutions of the system (see, forinstance [22, 32]). In this way, the following definitions and properties are adapted from thosestated for generic k -contact Hamiltonian systems to the case of a k -contact regular Lagrangiansystem ( ⊕ k T Q × R k , L ); that is, for the system ( ⊕ k T Q × R k , η α L , E L ). The proofs of the resultsfor the general case are given in [15]. . Gaset et al — A k -contact Lagrangian formulation for nonconservative field theories Definition 4.1.
Let ( ⊕ k T Q × R k , L ) be a k -contact regular Lagrangian system. • A Lagrangian dynamical symmetry is a diffeomorphism
Φ : ⊕ k T Q × R k → ⊕ k T Q × R k such that, for every solution σ to the k -contact Euler–Lagrange equations (13) , Φ ◦ σ isalso a solution. • An infinitesimal Lagrangian dynamical symmetry is a vector field Y ∈ X ( ⊕ k T Q × R k ) whose local flow is made of local symmetries. The following results give characterizations of symmetries in terms of k -vector fields: Lemma 4.2.
Let
Φ : ⊕ k T Q × R k → ⊕ k T Q × R k be a diffeomorphism and X = ( X , . . . , X k ) a k -vector field in ⊕ k T Q × R k . If ψ is an integral map of X , then Φ ◦ ψ is an integral map of Φ ∗ X = (Φ ∗ X α ) . In particular, if X is integrable then Φ ∗ X is also integrable. Proposition 4.3. If Φ : ⊕ k T Q × R k → ⊕ k T Q × R k is a Lagrangian dynamical symmetry then,for every integrable k -vector field X solution to the k -contact Lagrangian equations (14) , Φ ∗ X is another solution.On the other side, if Φ transforms every k -vector field X L solution to the k -contact La-grangian equations (14) into another solution, then for every integral map ψ of X L , we havethat Φ ◦ ψ is a solution to the k -contact Euler–Lagrange equations (13) . Among the most relevant symmetries are those that leave the geometric structures invariant:
Definition 4.4. A Lagrangian k -contact symmetry is a diffeomorphism Φ : ⊕ k T Q × R k →⊕ k T Q × R k such that Φ ∗ η α L = η α L , Φ ∗ E L = E L . An infinitesimal Lagrangian k -contact symmetry is a vector field Y ∈ X ( ⊕ k T Q × R k ) whose local flow is a Lagrangian k -contact symmetry; that is, L ( Y ) η α L = 0 , L ( Y ) E L = 0 . Proposition 4.5.
Every (infinitesimal) Lagrangian k -contact symmetry preserves the Reeb vec-tor fields, that is; Φ ∗ ( R L ) α = ( R L ) α (or [ Y, ( R L ) α ] = 0 ). And, as a consequence of these results, we obtain the relation between these kinds of sym-metries:
Proposition 4.6. (Infinitesimal) Lagrangian k -contact symmetries are (infinitesimal) Lagrangiandynamical symmetries. Definition 4.7.
A map F : M → R k , F = ( F , . . . , F k ) , is said to satisfy: . Gaset et al — A k -contact Lagrangian formulation for nonconservative field theories
1. The dissipation law for maps if, for every map σ solution to the k -contact Euler–Lagrange equations (13) , the divergence of F ◦ σ = ( F α ◦ σ ) : R k → R k , which is definedas usual by div( F ◦ σ ) = ∂ ( F α ◦ σ ) /∂t α , satisfies that div( F ◦ σ ) = − h ( L ( R L ) α E L ) F α i ◦ σ . (20)
2. The dissipation law for k -vector fields if, for every k -vector field X L solution to the k -contact Lagrangian equations (14) , the following equation holds: L ( X L ) α F α = − ( L ( R L ) α E L ) F α . (21)Both concepts are partially related by the following property: Proposition 4.8. If F = ( F α ) satisfies the dissipation law for maps then, for every integrable k -vector field X L = (( X L ) α ) which is a solution to the k -contact Lagrangian equations (14) , wehave that the equation (21) holds for X L .On the other side, if (21) holds for a k -vector field X , then (20) holds for every integral map ψ of X . Proposition 4.9. If Y is an infinitesimal dynamical symmetry then, for every solution X L =(( X L ) α ) to the k -contact Lagrangian equations (14) , we have that i ([ Y, ( X L ) α ]) η α L = 0 , i ([ Y, ( X L ) α ])d η α L = 0 . Finally, we have the following fundamental result which associates dissipated quantities withsymmetries:
Theorem 4.10. (Dissipation theorem) . If Y is an infinitesimal dynamical symmetry, then F α = − i ( Y ) η α L satisfies the dissipation law for k -vector fields (21) . Consider a k -contact regular Lagrangian system ( ⊕ k T Q × R k , L ).First, remember that, if ϕ : Q → Q is a diffeomorphism, we can construct the diffeomorphismΦ := (T k ϕ, Id R k ) : ⊕ k T Q × R k −→ ⊕ k T Q × R k , where T k ϕ : ⊕ k T Q → ⊕ k T Q denotes thecanonical lifting of ϕ to ⊕ k T Q . Then Φ is said to be the canonical lifting of ϕ to ⊕ k T Q × R k .Any transformation Φ of this kind is called a natural transformation of ⊕ k T Q × R k .Moreover, given a vector field Z ∈ X ( ⊕ k T Q × R k ) we can define its complete lifting to ⊕ k T Q × R k as the vector field Y ∈ X ( ⊕ k T Q × R k ) whose local flow is the canonical lifting ofthe local flow of Z to ⊕ k T Q × R k ; that is, the vector field Y = Z C , where Z C denotes thecomplete lifting of Z to ⊕ k T Q , identified in a natural way as a vector field in ⊕ k T Q × R k . Anyinfinitesimal transformation Y of this kind is called a natural infinitesimal transformation of ⊕ k T Q × R k .It is well-known that the canonical k -tangent structure ( J α ) and the Liouville vector field∆ in ⊕ k T Q are invariant under the action of canonical liftings of diffeomorphisms and vectorfields from Q to ⊕ k T Q . Then, taking into account the definitions of the canonical k -tangent . Gaset et al — A k -contact Lagrangian formulation for nonconservative field theories J α ) and the Liouville vector field ∆ in ⊕ k T Q , it can be proved that canonical liftingsof diffeomorphisms and vector fields from Q to ⊕ k T Q preserve these canonical structures as wellas the Reeb vector fields ( R L ) α .Therefore, as an immediate consequence, we obtain a relationship between Lagrangian-preserving natural transformations and contact symmetries: Proposition 4.11. If Φ ∈ Diff( ⊕ k T Q ) (resp. Y ∈ X ( ⊕ k T Q ) ) is a canonical lifting to ⊕ k T Q of a diffeomorphism ϕ ∈ Diff( Q ) (resp. of a vector field Z ∈ X ( Q ) ) that leaves the Lagrangian L invariant, then it is a (infinitesimal) contact symmetry, i.e., Φ ∗ η α L = η α L , Φ ∗ E L = E L (resp . L Y η α L = 0 , L Y E L = 0 ) . As a consequence, it is a (infinitesimal) Lagrangian dynamical symmetry.
As an immediate consequence we have the following momentum dissipation theorem : Proposition 4.12. If ∂ L ∂q i = 0 , then ∂∂q i is an infinitesimal contact symmetry and its associateddissipation law is given by the “momenta” (cid:18) ∂ L ∂v iα (cid:19) ; that is, for every k -vector field X L = (( X L ) α ) solution to the k -contact Lagrangian equations (14) , then L ( X L ) α (cid:18) ∂ L ∂v iα (cid:19) = − ( L ( R L ) α E L ) ∂ L ∂v iα = ∂ L ∂s α ∂ L ∂v iα . A generic second-order linear PDE in R is Au xx + 2 Bu xy + Cu yy + Du x + Eu y + F u + G = 0 , where A, B, C, D, E, F, G are functions of ( x, y ), with
A >
0. If B − AC > B − AC < B − AC = 0 is parabolic. In R n weconsider the equation A αβ u αβ + D α u α + G ( u ) = 0 , (22)where 1 ≤ α, β ≤ n ; and now we consider the following case: A αβ is constant and invertible (notparabolic), D α is constant and G is an arbitrary function in u .In order to find a Lagrangian k-contact formulation of these kind of PDE’s, consider ⊕ n T R × R n , with coordinates ( u, u α , s α ) and a generic Lagrangian of the form L = 12 a αβ ( u ) u α u β + b ( u ) u α s α + d ( u, s ) . The associated k-contact structure is given by η α = d s α − ∂L∂u α d u = d s α − ( a αβ u β + bs α + c α )d u . . Gaset et al — A k -contact Lagrangian formulation for nonconservative field theories L are a αβ u αβ + (cid:18) ∂a αβ ∂u − ba αβ (cid:19) u α u β − ∂d∂s β a βα u α + (cid:18) − ∂d∂s α bs α + bd − ∂d∂u (cid:19) = 0 . (23)If this equation has to match (22) then a αβ = A αβ , b = 0 , d = − ( a − ) αβ D β s α − g, where a = ( a αβ ) and ∂g∂u = G . Damped vibrating membrane
As a particular example consider the damped vibratingmembrane, which is described by the PDE u tt − µ ( u xx + u yy ) + γu t = 0 ;then A αβ = − µ
00 0 − µ , D α = γ , G = 0 , and therefore a αβ = − µ
00 0 − µ , b = 0 , d = − γs t . Then, a Lagrangian that leads to this equation is L = 12 u t − µ u x + u y ) − γs t , for which η t = d s t − u t d u , η x = d s x + µ u x d u , η y = d s y + µ u y d u . In this case, we have the contact symmetry ∂∂u and the associated map F = ( F t , F x , F y ) thatsatisfies the dissipation law for 3-vector fields is F t = − i ( Y ) η t = u t , F x = − i ( Y ) η x = − µ u x , F y = − i ( Y ) η y = − µ u y . Terms linear in velocities can be found in Euler–Lagrange equations of symplectic systems.However, they have a specific form, arising from the coefficients of a closed 2-form in the con-figuration space. The canonical example is the force of a magnetic field acting on a movingcharged particle; such forces do not dissipate energy. By contrast, other forces linear in thevelocities do dissipate energy; for instance, damping forces. To illustrate the difference betweenthe equations arising from magnetic-like terms in the Lagrangian and the equations given bythe k -contact formulation of a linear dissipation, we analyze the following academic example.Consider an infinite string aligned with the z -axis, each of whose points can vibrate in ahorizontal plane. So, the independent variables are ( t, z ) ∈ R , and the phase space is the . Gaset et al — A k -contact Lagrangian formulation for nonconservative field theories ⊕ T R with coordinates ( x, y, x t , x z , y t , y z ). Let’s imagine that the string isnon-conducting, but charged with linear density charge λ . Then, inspired by the Lagrangianformulation of the Lorentz force, we set the Lagrangian L o = 12 ρ ( x t + y t ) − τ ( x z + y z ) − λ ( φ − A x t − A y t )depending on some fixed functions A ( x, y ), A ( x, y ) and φ ( x, y ). The resulting Euler–Lagrangeequations are ρx tt − τ x zz = − λ (cid:18) ∂A ∂x − ∂A ∂y (cid:19) y t + λ ∂φ∂x ,ρy tt − τ y zz = λ (cid:18) ∂A ∂x − ∂A ∂y (cid:19) x t + λ ∂φ∂y . (24)The left-hand side is the string equation with two modes of vibration in the plane XY and inthe right-hand side we have an electromagnetic-like term.Now, consider the contact phase space ⊕ T R × R , with coordinates ( x, y, x t , x z , y t , y z , s t , s z ).We add a simple dissipation term in the preceding Lagrangian: L = L o + γ s t = 12 ρ ( x t + y t ) − τ ( x z + y z ) − λ ( φ − A x t − A y t ) + γs t . The induced 2-contact structure is η t = d s t − ( ρx t + λA ) d x − ( ρy t + λA ) d y ; η z = d s z + τ x z d x + τ y z d y . The 2-contact Euler–Lagrange equations are ρx tt − τ x zz = − λ (cid:18) ∂A ∂x − ∂A ∂y (cid:19) y t + λ ∂φ∂x + γρx t + γλA ,ρy tt − τ y zz = λ (cid:18) ∂A ∂x − ∂A ∂y (cid:19) x t + λ ∂φ∂y + γρy t + γλA . (25)Comparing equations (24) and (25) we observe that the dissipation originates two new terms: adissipation force proportional to the velocity, and an extra term proportional to ( A , A ). Thislast term comes from the non-linearity of the 2-contact Euler–Lagrange equations with respectto the Lagrangian.This system has the Lagrangian 2-contact symmetry Y = ∂A ∂x ∂∂x + ∂A ∂y ∂∂y . The associated map F = ( F t , F z ) that satisfies the dissipation law for 2-vector fields is F t = − i ( Y ) η t = ρx t ∂A ∂x + λ ∂A ∂x A + ρy t ∂A ∂y + λ ∂A ∂y A ,F z = − i ( Y ) η z = − τ x z ∂A ∂x − τ y z ∂A ∂y . . Gaset et al — A k -contact Lagrangian formulation for nonconservative field theories In a previous paper [15] we introduced the notion of k -contact structure to describe Hamil-tonian (De Donder–Weyl) covariant field theories with dissipation, bringing together contactHamiltonian mechanics and k -symplectic field theory.In this paper, we have developed the Lagrangian counterpart of this theory, basing on contactLagrangian and k -contact Hamiltonian formalisms. Thus, we have obtained and analyzed theLagrangian (Euler–Lagrange) equations of dissipative field theories. It should be pointed outthat the regularity of the Lagrangian is required to obtain a k -contact structure.We have also studied several kinds of symmetries: dynamical symmetries (those preservingsolutions), k -contact symmetries (those preserving the k -contact structure and the energy) andsymmetries of the Lagragian function. We have showed how to associate a dissipation law withany dynamical symmetry.As interesting examples, we have constructed Lagrangian functions for certain classes ofelliptic and hyperbolic partial differential equations; in particular, we have analyzed the exampleof the damped vibrating membrane. Another example has shown the difference between theequations of the k -contact formulation of a linear dissipation and the equations arising frommagnetic-like terms appearing in some Lagrangian functions of field theories.Among future lines of research, the case of singular Lagrangians seems especially interesting,though it would require to define the notions of k -precontact structure and k -precontact Hamil-tonian system , and to develop a constraint analysis to check the consistency of field equations. Acknowledgments
We acknowledge the financial support from the Spanish Ministerio de Ciencia, Innovaci´on yUniversidades project PGC2018-098265-B-C33 and the Secretary of University and Research ofthe Ministry of Business and Knowledge of the Catalan Government project 2017–SGR–932.
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