On the closure property of Lepage equivalents of Lagrangians
aa r X i v : . [ m a t h - ph ] F e b On the closure property of Lepage equivalents of Lagrangians
Nicoleta VOICU, Stefan GAROIU, Bianca VASIANTransilvania University of Brasov, Romania
Abstract
For field–theoretical Lagrangians λ of arbitrary order, we construct local Lepage equivalentsΦ λ with the so-called closure property: Φ λ is a closed differential form if and only if λ hasvanishing Euler-Lagrange expressions. A variant of this construction, which is advantageous inthe cases when λ is locally equivalent to a lower order Lagrangian, is also presented. Keywords:
Lepage equivalent of a Lagrangian, Vainberg-Tonti Lagrangian
MSC2020:
The language of differential forms gives rise to an elegant, coordinate-free formulation of variationalcalculus on manifolds. In this approach, a Lagrangian is regarded as a differential form on a certainjet bundle of a fibered manifold (
Y, π, X ), where dim X = n, dim Y = m + n ; the total space Y of the respective fibered manifold is physically interpreted as the configuration space of a givensystem, the base manifold X is typically interpreted as space (or spacetime) and sections of π areinterpreted as fields . The case n = 1 corresponds to mechanics (and, in this case, sections of Y arecurves).A central concept in this theory - actually, the very concept which makes it possible to describebasically any notion of the calculus of variations in a coordinate-free manner - is the one of Lepageequivalent of a Lagrangian, [7], [9], [3] . This is a field-theoretical generalization of the notion ofPoincar´e-Cartan form from mechanics and it is a differential form obtained in the same manner: byadding specific contact forms to the given Lagrangian.But, whereas the Poincar´e-Cartan form in mechanics is unique, in field theory, a Lagrangian λ admits multiple Lepage equivalents θ λ ; moreover, these Lepage equivalents θ λ do not generallyshare the same symmetries as the corresponding Lagrangians λ . This drawback can be removed,[1], if we can find a mapping λ θ λ with the so-called closure property: λ is variationally trivial ⇒ θ λ is a closed differential form.Yet, Lepage equivalents with the closure property appeared to be hard to find - actually, inthe general case, this has been for several decades an open problem; up to now, mappings λ θ λ obeying it are only known up to now in some specific situations:- Mechanics ( n = 1) and scalar field theories ( m = 1) where the property is satisfied, [1], by the principal Lepage equivalent Θ λ (also called the Poincar´e-Cartan form) - which is a minimalist one,in the sense that only adds a 1-contact component to the Lagrangian.1 First order Lagrangians. In this case, a Lepage equivalent with the desired feature, called the fundamental Lepage equivalent ρ λ , was introduced by Krupka, [8] and re-discovered by Bethounes,[1]; it adds to the Lagrangian λ contact components up to degree k = min( m, n ); for homogeneousLagrangians, a similar notion was introduced by Urban and Brajercik, [14].- In the higher order case, an extension of the fundamental Lepage equivalent was only deter-mined in the particular case when the Lagrangian λ is homogeneous and n = 2, [2].In the present paper, we propose a general procedure which solves this problem locally , forLagrangians λ of arbitrary order r . Our construction uses as a raw material, the principal Lepageequivalent mapping Θ, as follows.To any Lagrangian λ defined on a given fibered chart, one can canonically associate on therespective chart, an equivalent Lagrangian: the so-called
Vainberg-Tonti Lagrangian λ V T of theEuler-Lagrange form of λ, [6]. The difference between λ and λ V T (which is a trivial Lagrangian) isthen expressed, up to a pullback by the corresponding jet projections, as: λ = λ V T + hdα, (1)where the ( n − α is uniquely determined, via fibered homotopy operators.Using the above decomposition, the relation:Φ λ := Θ λ V T + dα, (2)where Θ λ V T is the principal Lepage equivalent of λ V T , defines a Lepage equivalent of λ, which wecall canonical. As for variationally trivial Lagrangians λ, the associated Vainberg-Tonti Lagrangian λ V T vanishes, the above recipe guarantees that, in this case, Φ λ is closed.The canonical Lepage equivalent is 1-contact; yet, the closure property comes at a price: theorder of Φ λ is generally higher than the one of Θ λ . In particular, for first order Lagrangians, Φ λ isof order 2.A variant of the above construction, which is advantageous in the case when λ is locally equiva-lent to a lower order Lagrangian λ ′ , is to consider in (2), instead of the Vainberg-Tonti Lagrangian λ V T (which is in general of higher order than λ itself), a reduced Lagrangian λ ′ . This leads to(generally, non-unique) Lepage equivalents φ λ , which we will call reduced Lepage equivalents ; and,if we can ensure that λ ′ is truly of minimal order, the obtained reduced Lepage equivalent will stillpossess the closure property. In particular, for reducible second order Lagrangians, any reducedLepage equivalent will be of order 1.The article is structured as follows. Section 2 presents the known results and notions to be usedin the following. In Section 3, we introduce the canonical Lepage equivalent and in Section 4, wediscuss the case of reducible Lagrangians and introduce reduced Lepage equivalents φ λ . In the following, we present the technical ingredients to be used in our construction. The notionsand results presented in this section can be found in more detail, e.g., in the book by Krupka, [6]. Two Lagrangians are deemed equivalent if they produce the same Euler-Lagrange equations. .1 Lagrangians, Lepage equivalents and first variation formula A fibered manifold is a triple (
Y, π, X ) , where X , Y are smooth manifolds, with dim X = n, dim Y = m + n, and π : Y → X is a surjective submersion. On a fibered manifold, there existsan atlas consisting of fibered charts ( V, ψ ), ψ = ( x i , y σ ), such that π is locally represented as π : (cid:0) x i , y σ (cid:1) (cid:0) x i (cid:1) . We will denote by Γ( Y ) the set of local sections γ : U → Y, π ◦ γ = id U (where U ⊂ X is open); elements γ ∈ Γ( Y ) are represented in a fibered chart as γ : (cid:0) x i (cid:1) ( x i , y σ (cid:0) x i (cid:1) ) . Each fibered chart (
V, ψ ) induces a fibered chart ( V r , ψ r ) , ψ r = ( x i , y σ , y σj , ..., y σj j ...j r )on the r -jet prolongation J r Y. We denote by π r,s the canonical projections π r,s : J r Y → J s Y, ( x i , y σ , y σj , ..., y σj j ...j r ) ( x i , y σ , y σj , ..., y σj j ...j s ) ( r > s, J Y := Y ) and by π r , theprojection π r : J r Y → X, ( x i , y σ , y σj , ..., y σj j ...j r ) ( x i ) . By Ω k ( W r ) and Ω( W r ) we will denotethe set of k -forms (respectively, of all differential forms) defined on some open subset W r ⊂ J r Y. For the simplicity of writing, in relations between differential forms, if the precise derivativeorder is not essential, we may omit the pullback by the corresponding jet projections, i.e., we willautomatically identify forms ρ with their pullbacks ( π s,r ) ∗ ρ, s ≥ r, to any higher order jet bundle;that is, instead of ( π s,r ) ∗ ρ = θ, we may simply write ρ = θ. Horizontal forms and contact forms on J r Y . A differential form ρ ∈ Ω k ( W r ) is called π r -horizontal, if ρ (Ξ , ..., Ξ k ) = 0 whenever one of the vector fields Ξ i , i = 1 , k, is π r -vertical(i.e., T π r (Ξ i ) = 0); in a fibered chart, any horizontal form on Ω k ( W r ) is expressed as a linearcombination of the wedge products dx i ∧ dx i ∧ ... ∧ dx i k . Yet, for a more compact writing, it isadvantageous to use the following locally defined forms: ω := dx ∧ ... ∧ dx n , ω i := i ∂ i ω = ( − i +1 dx ∧ ... ∧ c dx i ∧ ... ∧ dx n , (3) ω i ...i k := i ∂ ik i ∂ ik − ... i ∂ i ω , (4)where i denotes interior product. This way, any π r -horizontal k -form on W r ( k ≤ n ) will have acoordinate expression: ρ = 1 k ! A i ...i k ω i ...i k , (5)where A i ...i k are smooth functions of the coordinates x i , y σ , y σj , ..., y σj j ...j r . The set of π r -horizontal k -forms over W will be denoted by Ω k,X ( W r ) . Similarly, one can define π r,s -horizontal forms, 0 ≤ s ≤ r ; locally, these are generated by wedgeproducts of dx i , dy σ , ..., dy σj ...j s . A form θ ∈ Ω( W r ) is called a contact form if it vanishes along all prolonged sections, i.e., J r γ ∗ θ = 0 ∀ γ ∈ Γ( Y ) . For instance: ω σ = dy σ − y σj dx j , (6) ω σi = dy σi − y σi j dx j , ..., ω σi i ...i r − = dy σi i ...i r − − y σi i ...i r − j dx j , (7)represent contact forms on W r ; these contact forms are elements of a local basis of Ω ( J r Y ) , calledthe contact basis: { dx i , ω σ , ...., ω σi ...i r − , dy σi ...i r } . A k -form θ ∈ Ω k ( J r Y ) is l -contact ( l ≤ k ) if, for any π r -vertical vector field Ξ , the interiorproduct i Ξ θ is ( l − J r +1 Y , l -contact forms are characterized by the fact that, in the expression of3 π r +1 ,r ) ∗ θ ∈ Ω k ( J r +1 Y ) in the contact basis, each term contains exactly l of the contact 1-forms(6)-(7).Rising to J r +1 Y, any ρ ∈ Ω k ( W r ) can be uniquely split into a π r -horizontal part hρ ∈ Ω k ( W r +1 )and a contact part pρ ∈ Ω k ( W r +1 ): ( π r +1 ,r ) ∗ ρ = hρ + pρ ; (8)the contact part can be in its turn decomposed as pρ = p ρ + ... + p k ρ, where the form p l ρ is l -contact, l = 1 , ..., k . On the other hand, the mapping h defined by (8), h : Ω( W r ) → Ω( W r +1 ) , ρ hρ is a morphism of exterior algebras, called horizontalization; it acts on functions f : J r Y → R as: hf = f ◦ π r +1 ,r and hdf = d i f dx i , where d i denotes total x i -derivative (of order r + 1): d i f := ∂ i f + ∂f∂y σ y σi + ... ∂f∂y σj ...j r y σj ...j r i and on the natural basis 1-forms, as: hdx i := dx i , hdy σ = y σi dx i , ..., hdy σj ...j k = y σj ...j k i dx i , k = 1 , r ; (9)the horizontal part of ρ is the only one that survives when pulled back by prolonged sections J r γ ;more precisely, for any γ ∈ Γ( Y ) , we can write: J r γ ∗ ρ = J r +1 γ ∗ ( hρ ) . Here are two important classes of differential forms on W r ⊂ J r Y :1. Lagrangians : A Lagrangian is, by definition, a π r -horizontal form λ ∈ Ω n ( W r ) of rank n = dim X ; in fibered coordinates, a Lagrangian is expressed as: λ = L ω , L = L ( x i , y σ , ..., y σi ...i r ) , (10)where ω is as in (3).2. Source forms (or dynamical forms, [11] ) are defined as π r, -horizontal, 1-contact ( n + 1)-forms; in fibered coordinates, any source form ε ∈ Ω n +1 ( W r ) is represented as: ε = ε σ ω σ ∧ ω , ε σ = ε σ ( x i , y σ , ..., y σi ...i r ) , where ω σ are as in (6). The most prominent particular case of source forms are Euler-Lagrangeforms of Lagrangians, to be discussed below. Lepage equivalents of a Lagrangian.
Consider a Lagrangian λ as in (10). The actionattached to λ and to a compact domain D ⊂ X is the function S : Γ( Y ) → R , given by: S ( γ ) = Z D J r γ ∗ λ. (11)4he variation of S under the flow of a π -projectable vector field Ξ = ξ i ∂ i + Ξ σ ∂ σ on Y is then givenby: δ Ξ S ( γ ) = Z D J r γ ∗ L J r Ξ λ, (12)where the symbol L stands for Lie derivative.A Lepage equivalent of a Lagrangian λ ∈ Ω n ( W r ) is an n -form θ λ ∈ Ω n ( W s ) on some jetprolongation J s Y, with the following properties:(1) θ λ and λ define the same variational problem , i.e., up to the corresponding jet projections: hθ λ = λ. (13)(2) E λ := p dθ λ is a source form.Condition (1) should be understood as follows. For any section γ ∈ Γ( Y ) , there holds: J s γ ∗ θ λ = J r γ ∗ λ , which means that we can substitute J s γ ∗ θ λ for J r γ ∗ λ into the action (11). Condition (2)actually states that p dθ λ must be locally generated by ω σ and dx i only (no higher order elements ω σi ...i k of the contact basis).Given any Lepage equivalent θ λ ∈ Ω n ( W s ) of λ, Cartan’s formula L J r Ξ θ λ = d i J r Ξ θ λ + i J r Ξ dθ λ yields: J r γ ∗ ( L J r Ξ λ ) = J s +1 γ ∗ i J s +1 Ξ E λ + d ( J s γ ∗ J Ξ ) , (14)where: (i) E λ = p dθ λ ∈ Ω n ( W s +1 ) is the Euler-Lagrange form of λ , locally given by E λ = E σ ( λ ) ω σ ∧ ω , E σ ( λ ) = δ L δy σ = ∂ L ∂y σ − d i ∂ L ∂y σi + ... + ( − r d i ...d i r ∂ L ∂y σi ...i r . (15)The Euler-Lagrange form E λ does not depend on the choice of the Lepage equivalent θ λ . (ii) J Ξ := h i J s Ξ θ λ ∈ Ω n − ( J s +1 Y ) is interpreted as a Noether current.Every Lagrangian λ ∈ Ω n ( W r ) admits Lepage equivalents. The most frequently used one, calledthe principal Lepage equivalent, is a 1-contact form of order ≤ r − λ = L ω + r − X k =0 ( r − − k X l =0 ( − l d p ...d p l ∂ L ∂y σj ...j k p ...p l i ω σj ...j k ) ∧ ω i . (16)In particular, if the coordinate expression of λ = L ω is affine in the highest order variables y σi ...i r , the order of Θ λ is actually, at most 2 r − d p ...d p r − ∂ L ∂y σp ...p r − i ).Generally, the principal Lepage equivalent Θ λ is defined only locally; yet, for first and secondorder Lagrangians, it is globally defined whenever λ itself is globally defined (see also [11]) . θ λ of λ can be locally expressed as: θ λ = Θ λ + dν + µ, (17)where ν is 1-contact and µ is 2-contact. In particular, any Lepage equivalent of λ can beexpressed, [11], as: θ λ = Θ λ + p dν. (18) Fix an arbitrary fibered chart domain W r ⊂ J r Y. A Lagrangian λ ∈ Ω n,X ( W r ) is called variationally trivial (or null ) if its Euler-Lagrange form E λ vanishes identically; it is known, e.g., [6], p. 123, that, λ is trivial if and only if there exists an( n − α ∈ Ω n − ( W r − ) of order r − , such that: λ = hdα. (19)A (possibly multi-valued) function θ : Ω n,X ( W r ) → Ω n ( W s ) attaching to any Lagrangian λ ∈ Ω n,X ( J r Y ) , a set of Lepage equivalents θ λ , is said to have the closure property, [13], if: λ - variationally trivial ⇒ dθ λ = 0 , (20)for all θ λ in the image of λ. Remark.
The converse implication: dθ λ = 0 ⇒ λ - null , is true for any Lepage equivalent θ λ , since dθ λ = 0 implies E λ = p dθ λ = 0; hence, whenever it holds, (20) is actually an equivalence.In the case of first order Lagrangians λ , a globally defined, first order Lepage equivalent pos-sessing the closure property is the so-called fundamental Lepage equivalent (or the
Krupka form ),given, [13], [11], by: ρ λ = L ω + min { m,n } X k =1 k !) ∂ k L ∂y σ i ...∂y σ k i k ω σ ∧ ... ∧ ω σ k ∧ ω i ...i k ; (21)the degree of contactness of ρ λ is min { m, n } . Actually, the closure property of ρ is a consequenceof a stronger property: λ = hdα, α ∈ Ω n − ( Y ) ⇒ ρ λ = (cid:0) π , (cid:1) ∗ dα. (22) Given a source form ε defined on some fibered chart domain W r ⊂ J r Y, one can canonically attachto ε and to the chart ( W ψ ) a Lagrangian called the Vainberg-Tonti Lagrangian, with the followingproperty: if the source form ε is locally variational (i.e., ε can be expressed as ε = E λ for somelocally defined Lagrangian λ ), then the Vainberg-Tonti Lagrangian is a Lagrangian for ε. The Vainberg-Tonti Lagrangian is introduced (see, e.g., Section 4.9 in [6]), by means of a specialhomotopy operator acting on locally defined differential forms on J r Y. Consider a fibered chart(
W, ψ ) on Y . We assume that the image ψ ( W ) ⊂ R m + n is vertically star-shaped , i.e., for any6 x i , y µ ) ∈ ψ ( W ) , the whole segment (cid:0) x i , ty µ (cid:1) , t ∈ [0 ,
1] remains in ψ ( W ) . Under this assumption ,( x i , y σ , y σi , ..., y σi ...i r ) ∈ ψ r ( W r ) the mapping χ : [0 , × ψ r ( W r ) → ψ r ( W r ) , ( t, ( x i , y σ , y σi , ..., y σi ...i r )) ( x i , ty σ , ty σi , ..., ty σi ...i r )is well defined and gives rise to an operator I : Ω k ( W r ) → Ω k − ( W r ) , as follows. For any ρ ∈ Ω k ( W r ) , set: Iρ := Z ρ (0) ( t ) dt, (23)where ρ (0) ( t ) ∈ Ω k − ( W r ) is obtained from the decomposition: χ ∗ ρ = ρ (0) ( t ) ∧ dt + ρ ′ ( t ) (24)into a dt -term and a term ρ ′ ( t ) which does not contain dt. The mapping I is a homotopy operator ,in the sense that: ρ = Idρ + dIρ + ρ , (25)where ρ := ( π r ) ∗ ∗ ρ (26)and 0 denotes the zero section 0 : (cid:0) x i (cid:1) (cid:0) x i , , , ..., (cid:1) of W r . The form ρ ∈ Ω k ( W r ) is obviouslyprojectable onto a differential form on π ( W ) ⊂ X. The fibered homotopy operator obeys, [6], p. 70:- (cid:0) π r +1 ,r (cid:1) ∗ Iρ = I (cid:0) π r +1 ,r (cid:1) ∗ ρ ;- Ihρ = 0 , Ip k ρ = p k − Iρ, ≤ k ≤ q. Applying the above operator I to a source form ε = ε σ ω σ ∧ ω ∈ Ω n +1 ( W r ) , the obtained n -form λ ε := Iε (27)is a Lagrangian defined on ψ r ( W r ) , called the Vainberg-Tonti Lagrangian attached to ε. In coordi-nates, this is: λ ε = L ω , L = y σ Z ε σ ( x i , ty µ , ty µi , ...ty µi ...i r ) dt. (28)As already stated above, if the source form ε is locally variational, then λ ε is a local Lagrangianfor ε. In the following, for a given locally defined Lagrangian λ of arbitrary order r , we will build a localLepage equivalent Φ λ , possessing the closure property; the obtained Lepage equivalent is 1-contactand of order 4 r − . The construction can also be extended to cases when ψ ( W ) is not vertically star-shaped, see, e.g., [5]. W, ψ ) as above and arbitrary Lagrangian λ ∈ Ω n,X ( W r ) of order r ≥
2. As, by definition, λ is a Lagrangian for its own Euler-Lagrange form E λ = E σ ω σ ∧ ω , the Vainberg-Tonti Lagrangian λ V T := I E λ (29)(of order ≤ r ) is always equivalent to λ. The difference between λ and λ V T is thus a trivialLagrangian; more precisely, see [6], p. 130: λ = λ V T + hdα, (30)where α := I Θ λ + µ (31)and µ ∈ Ω n − ( W r − ) is such that Θ = dµ ; (32)here, we have denoted: Θ = (cid:0) π r − (cid:1) ∗ ∗ Θ λ . The form µ is guaranteed to exist on W r − , since0 ∗ Θ λ is a form of maximal rank n on X .We will call the Lagrangian λ V T , the Vainberg-Tonti Lagrangian associated to λ. Remark 1
1. Taking into account that Θ λ is 1-contact and (generally) of order r − , the ( n − -form α defined in (31) is a π r − -horizontal form on W r − , i.e., its coordinateexpression is: α = α i ω i , α i = α i ( x j , y σj , ..., y σj ...j r − ) . (33)
2. For a Lagrangian λ of order r, the Euler-Lagrange expressions (15) are of order ≤ r, buttheir dependence on the variables y σi ...i r (if any) is, in any fibered chart, affine. Hence, theassociated Vainberg-Tonti Lagrangian λ V T is also at most affine in y σi ...i r . Consequently,the order of the principal Lepage equivalent Θ λ V T does not exceed r − . With these preparations, we are now able to state:
Theorem 2
Let λ ∈ Ω n,X ( W r ) be an arbitrary Lagrangian of order r, over the vertically star-shaped fibered chart domain W and λ V T = I E λ ∈ Ω n,X ( W r ) , its associated Vainberg-Tonti La-grangian. Then:(i) The differential form Φ λ ∈ Ω n ( W r − ) given by: Φ λ := Θ λ V T + (cid:0) π r − , r − (cid:1) ∗ dα, (34) where α is given by (31)-32), is a Lepage equivalent of λ ; (ii) If λ is a trivial Lagrangian, then d Φ λ = 0 . roof. (i) Write λ as in (30). Then, since the horizontalization h is a linear mapping, we have, upto the corresponding jet projections: h Φ λ = h Θ λ V T + hdα = λ V T + hdα = λ ;moreover, taking the exterior derivative of (30), we obtain: d Φ λ = d Θ λ V T , therefore, p d Φ λ = p d Θ λ V T = E λ V T = E λ , which proves that Φ λ is a Lepage equivalent of λ. (ii) Assuming that λ is trivial, we have E λ = 0 , which implies λ V T = 0 and, accordingly,Θ λ V T = 0; as a consequence, Φ λ = (cid:0) π r − , r − (cid:1) ∗ dα is locally exact - therefore, closed.We will call the equivalent Φ λ in (30)-(34), the canonical Lepage equivalent of λ. Remarks. (Uniqueness of Φ λ ): Though the ( n − µ in (32) is not uniquely defined,in the expression of Φ λ , α only appears through its exterior derivative dα := dI Θ λ + dµ = dI Θ λ + Θ , which is uniquely defined.2) (Linearity of Φ): A quick reasoning using the R -linearity of the principal Lepage equivalentmapping λ Θ λ shows that the mapping:Φ : Ω n,X ( W r ) Ω n ( W r − ) , λ Φ λ is R -linear.In the following, let us point out the relation between Φ λ and Θ λ . Proposition 3
For a Lagrangian λ = λ V T + hdα as in (30), there holds, up to the correspondingjet projections: Φ λ = Θ λ + ( dα − Θ hdα ) . (35) Proof.
From the linearity of Θ , we have: Θ λ = Θ λ V T + Θ hdα . Adding to both hand sides dα andtaking into account that, up to jet projections, Φ λ = Θ λ V T + dα , this leads to (35).The term dα − Θ hdα in (35) is 1-contact. Therefore, using (18), there exists a 1-contact form ν such that, up to the corresponding jet projections: dα − Θ hdα = p dν. (36) Particular case: first order Lagrangians.
For first order Lagrangians, the canonical Lep-age equivalent will be of order 2. In the following, for this particular case, we will calculate thecoordinate expression of ν in (36) - which gives the difference between Φ λ and Θ λ or, equivalently,the ”failure” of Θ from having the closure property. Proposition 4
For an arbitrary first order Lagrangian λ ∈ Ω n,X ( W ) : (i) λ = λ V T + (cid:0) d i α i (cid:1) ω , for some first order functions α i = α i ( x j , y σ , y σj ); (ii) there holds: Φ λ = (cid:0) π , (cid:1) ∗ Θ λ + p dν, (37) where ν := 14 ( ∂α i ∂y σj − ∂α j ∂y σi ) ω σ ∧ ω ij . (38)9 roof. (i) The form α = I Θ λ + µ is of order 1 and π -horizontal, therefore it is locally expressedas α = α i ω i , with α i = α i ( x j , y σ , y σj ) only (the precise expressions of α i can be found from (23)).The statement then follows from hdα = (cid:0) d i α i (cid:1) ω . (ii) With α as above, we have: Φ λ = Θ λ V T + (cid:0) π , (cid:1) ∗ dα. Denoting: L := d i α i , we find from(16): Θ hdα = L ω + B iσ ω σ ∧ ω i + B ijσ ω σj ∧ ω i , where: B iσ = ∂ L ∂y σi − d j ( ∂ L ∂y σij ) , B ijσ = ∂ L ∂y σij . (39)That is, on the one hand: B ijσ = ∂ L ∂y σij = ∂∂y σij ( d k α k ) = 12 ( ∂α i ∂y σj + ∂α j ∂y σi ) . (40)On the other hand, using the relation ∂ L ∂y σi = ∂∂y σi ( d k α k ) = d k ( ∂α k ∂y σi ) + ∂α i ∂y σ and (40), we get: B iσ = ∂α i ∂y σ + 12 d j ( ∂α j ∂y σi − ∂α i ∂y σj ) . Substituting into Θ hdα and taking into account that (cid:0) π , (cid:1) ∗ dα = ( d k α k ) ω + ∂α i ∂y σ ω σ ∧ ω i + ∂α i ∂y σj ω σj ∧ ω i , we find, after a brief computation:Θ hdα − (cid:0) π , (cid:1) ∗ dα = 12 d j ( ∂α j ∂y σi − ∂α i ∂y σj ) ω σ ∧ ω i + 12 ( ∂α j ∂y σi − ∂α i ∂y σj ) ω σj ∧ ω i . It can then be checked directly that, for ν as in (38), the right hand side of the above equals − p dν. The statement then follows from Φ λ − (cid:0) π , (cid:1) ∗ Θ λ = (cid:0) π , (cid:1) ∗ dα − Θ hdα = p dν. Some special cases when Φ λ = Θ λ will be discussed below. Particular cases: Vainberg-Tonti Lagrangians . An interesting special class is represented by Lagrangians λ ∈ Ω n ( W r ) that coincide with their associated Vainberg-Tonti Lagrangians, i.e., λ = λ V T . (41)These Lagrangians can be regarded as ”maximally nontrivial”, in the sense that, in (30), thevariationally trivial component hdα vanishes. In this case, we obtainΦ λ = Θ λ and Φ λ is thus of order ≤ r − . As examples of second order Lagrangians satisfying the property(41), we mention:- the Hilbert Lagrangian of general relativity, see [15], [5];- the Lagrangian of Gauss-Bonnet gravity, see [5].2.
Mechanics ( n = 1) . In mechanics, it is known that the Poincar´e-Cartan form Θ λ is the uniqueLepage equivalent of λ ∈ Ω ,X ( W r ) (this can be also checked directly, as any Lepage equivalentmust be a 1-form, hence it is at most 1-contact; then, uniqueness follows from (18), taking intoaccount that there are no nonzero 0-contact forms). Hence, in this case, Φ λ = Θ λ . Reduced Lepage equivalents
In the following, we present an alternative construction, which is advantageous in the case whenthe Lagrangian λ can be order-reduced (see, e.g., [4] [11], [12], for the conditions such that λ isequivalent to a lower order Lagrangian).Consider a Lagrangian λ ∈ Ω n,X ( W r ) and pick any equivalent Lagrangian λ ′ ∈ Ω n,X ( W s ) to λ, of minimal order s ≤ r. Then, again, we can write λ = ( π r,s ) ∗ λ ′ + hdα, (42)for some α ∈ Ω n − ( W r − ). Note.
We can regard a trivial Lagrangian λ as being equivalent to either a nonzero, π r -projectable n -form λ ′ = f ( x i ) ω , or as being equivalent to λ ′ = 0 , which would then producequalitatively different results when substituted into (42). In order to sort out this situation, weconventionally set the order of the zero Lagrangian λ ′ = 0 as −∞ ; nonzero π r -projectable formswill be included in the wider class of π r, -projectable forms (i.e., their order is zero). With thisconvention at hand, for any trivial Lagrangian, the minimal order Lagrangian equivalent to a trivialLagrangian λ is uniquely defined as λ ′ = 0 . Proposition 5
Let λ ∈ Ω n,X ( W r ) be an arbitrary Lagrangian and λ ′ ∈ Ω n,X ( W s ) , a dynamicallyequivalent Lagrangian to λ, of minimal order s ≤ r. Then:(i) The n -form φ λ := Θ λ ′ + dα, (43) where λ ′ and α are as in (42) and the equality must be understood up to the corresponding jetprojections, is a Lepage equivalent of λ. (ii) If λ is variationally trivial, then any φ λ constructed as above is closed. Proof. (i)
The proof is similar to the one of Theorem 2. First, we note that hφ λ = h Θ λ ′ + hdα = λ ′ + hdα = λ ;moreover, dφ λ = d Θ λ ′ implies p dφ λ = p d Θ λ ′ = E λ ′ = E λ , which is a source form, that is, φ λ is aLepage equivalent of λ. (ii) If λ is trivial, then λ ′ = 0 and accordingly, φ λ = dα. Definition 6
We will call any Lepage equivalent built as in (42)-(43), a reduced Lepage equivalentof λ. Remarks. Non-uniqueness of φ λ : Unlike the Vainberg-Tonti Lagrangian λ V T , the reduced Lagrangian λ ′ of λ (if it exists) is non-unique. As a consequence, we may obtain multiple Lepage equivalents φ λ ; more precisely, the correspondence φ : Ω n,X ( W r ) → Ω n ( W r − ) , λ φ λ , is a multi-valued function. 11. The splitting (42) is, generally, only local - therefore, reduced Lepage equivalents are also,generally defined only locally.Order reducibility criteria for Lagrangians of general order r ≥ second order Lagrangians (which represent a vast majority of higherorder Lagrangians used in physical theories), a simple criterion is given in [6], p. 145: a Lagrangian λ = L ω is locally reducible to a first order Lagrangian if and only if its Euler-Lagrange expressions E σ are of second order (moreover, in this case, E σ are actually, affine in the second order derivatives y σij ). Proposition 7 (reducible second order Lagrangians): If a second order Lagrangian λ ∈ Ω n,X ( W ) has second order Euler-Lagrange equations, then any reduced Lepage equivalent of λ is of order 1. Proof.
Assuming that λ ∈ Ω n,X ( W ) has second order Euler-Lagrange expressions, according tothe above mentioned result, it can be written as λ = (cid:0) π , (cid:1) ∗ λ ′ + hdα, where both the Lagrangian λ ′ and the ( n − α are of order 1. But, in this case, Θ λ ′ is of order 1, consequently, thereduced Lepage equivalent φ λ = Θ λ ′ + dα is of first order, too. Example:
A first example that comes to one’s mind is the Hilbert Lagrangian λ g . It wasshown, see, e.g., [9], [16], that its principal Lepage equivalent Θ λ g is of order 1. In the following,we will explore the structure of this Lepage equivalent more in detail - and point out that Θ λ g isactually, a reduced Lepage equivalent for λ g (moreover, as pointed out in the previous section, italso coincides with the canonical one Φ λ g ).Let Y = M et ( X ) denote the bundle of nondegenerate tensors of type (0,2) over a smoothmanifold X ; using ( g ij ; g ij,k ; g ij,kl ) as coordinate functions in a fibered chart on J Y, the (formal)Christoffel symbols Γ ijk , Riemann tensor components R ij kl and Ricci scalar R are regarded asfunctions defined on a chart of J Y. The Hilbert Lagrangian reads: λ g = R p | det g | ω . In [16], we have noticed that the principal Lepage equivalent Θ λ g is expressible as:Θ λ g = Θ λ ′ g + dα, (44)where λ g = λ ′ g + (cid:0) d i α i (cid:1) ω and λ ′ g = g jk (Γ ijl Γ lki − Γ ijk Γ lil ) p | det g | ω =: L ′ g ω is the reduced (non-invariant) Lagrangian for λ g ; the form α is immediately obtained as: α = (Γ ijj − Γ jij ) p | det g | ω i . Let us explicitly determine Θ λ ′ g . We have:Θ λ ′ g = L ′ g ω + ∂ L ′ g ∂g pq,r ω pq ∧ ω r ω pq := dg pq − g pq,i dx i ); then, rewriting L ′ g as L ′ g = g ih (cid:0) g jk g lm − g jl g mk (cid:1) Γ hjl Γ mki and using the relation: ∂ Γ hjl ∂g pq,r = 12 ( δ ph δ qj δ rl + δ ph δ ql δ rj − δ pj δ ql δ rh ) , we get after a brief computation:Θ λ ′ g = L ′ g ω + (Γ rqp + Γ kpk g qr ) ω pq ∧ ω r . Acknowledgment.
We are extremely grateful to prof. Demeter Krupka for drawing ourattention towards this topic and for useful and extensive talks on it.
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