The fundamental Lepage form in two independent variables: A generalization using order-reducibility

  Citation: Urban, Z.; Volná, J. The Fundamental Lepage Form in Two Independent Variables: A Generalization Using Order-Reducibility. Mathematics 2022, 10, 1211. https://doi.org/10.3390/ math10081211 Academic Editor: Christopher Goodrich Received: 8 March 2022 Accepted: 6 April 2022 Published: 7 April 2022 Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. Copyright: © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). mathematics Article The Fundamental Lepage Form in Two Independent Variables: A Generalization Using Order-Reducibility Zbynˇek Urban * and Jana Volná Department of Mathematics, Faculty of Civil Engineering, VŠB-Technical University of Ostrava, Ludvíka Podéštˇe 1875/17, 708 33 Ostrava, Czech Republic; [email protected] *Correspondence: [email protected] Abstract: A second-order generalization of the fundamental Lepage form of geometric calculus of variations over fibered manifolds with 2-dimensional base is described by means of insisting on (i) an equivalence relation “Lepage differential 2-form is closed if and only if the associated Lagrangian is trivial” and (ii) the principal component of Lepage form, extending the well-known Poincaré–Cartan form, preserving order prescribed by a given Lagrangian. This approach completes several attempts of finding a Lepage equivalent of a second-order Lagrangian possessing condition (i), which is well-known for first-order Lagrangians in field theory due to Krupka and Betounes. Keywords: Lagrangian; Lepage equivalent; Poincaré–Cartan form; fundamental form; calculus of variations; field theory; jet; fibered manifold MSC: 58A10; 58A20; 58E30; 70S05 1. Introduction Lepage forms play a basic role in the calculus of variations of both simpleand multipleintegral problems over fibered manifolds and Grassmann fibrations. Among the well-known examples of Lepage forms, we note: the Cartan form of classical mechanics and its generalization in higher-order mechanics (Krupka [ 1 ]); in first-order field theory, the Poincaré–Cartan form (García [ 2 ]), the Carathéodory form (Carathéodory [ 3 ]), and the fundamental Lepage form (also known as the Krupka–Betounes form) (Krupka [ 4 ], Betounes [ 5 ]); in secondorder field theory, the generalized Poincaré–Cartan form (Krupka [ 6 ]), the generalized Carathéodory form (Crampin and Saunders [ 7 ], Urban and Volná [ 8 ]), the fundamental Lepage form for second-order, homogeneous Lagrangians (Saunders and Crampin [ 9 ]). See also Gotay [ 10 ], Goldschimdt and Sternberg [ 11 ], Rund [ 12 ], Dedecker [ 13 ], Horák and Koláˇr [14], Krupka [ 6 ], Krupka and Štˇepánková [ 15 ], Saunders [ 16 ], and Sniatycki [ 17 ]. Further recent attempts of generalization and study of the fundamental Lepage equivalent for firstand second-order Lagrangians also include [18–20]. For a review of basic properties and results, see Krupka, Krupková, and Saunders [21]. Replacing the initial Lagrangian by its Lepage equivalent, the corresponding variational functional is preserved, and in addition the basic variational properties as variations, extremals, or conservation laws can be formulated and studied using geometric operations (such as the exterior derivative, the Lie derivative) acting on the corresponding Lepage equivalent of a Lagrangian. Our aim in this note is to study a generalization of the fundamental Lepage equivalent Zλ of a second-order Lagrangian λ . This Lepage equivalent was introduced for first-order Lagrangians in variational theory over fibered manifolds with an n -dimensional base (see [4,5]), and it obeys the following crucial property: dZλ=0 ifandonlyif Eλ=0, (1) Mathematics 2022,10, 1211. https://doi.org/10.3390/math10081211 https://www.mdpi.com/journal/mathematics Mathematics 2022,10, 1211 2 of 14 that is, the Lepage equivalent of a Lagrangian is closed if and only if the Lagrangian is trivial (i.e., the corresponding Euler–Lagrange expressions vanish identically). For a 2-dimensional base (i.e., two independent variables), we show that a Lepage equivalent Zλ of a second-order Lagrangian λ , obeying the aforementioned equivalence property, does not exist in general when Zλ=Θλ plus a 2-contact part, where Θλ is the principal component (Lepage form) of Zλ . Nevertheless, we describe here the fundamental Lepage form, associated with a second-order Lagrangian that assures the principal component of a Lepage form (the generalized Poincaré–Cartan form) has the same order as the initial Lagrangian. This order reducibility assumption is also motivated by the first-order theory and includes, for instance, an important class of Lagrangians linear in second derivatives. Recent studies on the fundamental Lepage form include Saunders and Crampin [ 9 ] (for two independent variables, generalization of the fundamental form is given for higherorder, homogeneous Largangians on tangent bundles), and Palese, Rossi, and Zanello [ 18 ] (on the basis of integration by parts, possible generalization of the fundamental form for a second-order Lagrangian is discussed, which, however, differs from our result for n=2). Basic underlying geometric structures, well adapted to the present paper, can be found in book chapters of Volná and Urban [ 22 ] and Krupka [ 23 ]. Throughout, we use the standard geometric concepts: the exterior derivative d , the contraction iξρ of a differential form ρ with respect to a vector field ξ , and the pull-back operation ∗ acting on differential forms. If (U , ϕ) , ϕ= (xi) , is a chart on smooth manifold X , the local volume element is denoted by ω0=dx1∧. . . ∧dxn, and we put ωj=i∂/∂xjω0=1 (n−1)!εji2...indxi2∧. . . ∧dxin, where εi1i2...in is the Levi-Civita permutation symbol. We denote by Y a fibered manifold of dimension n+m over an n -dimensional base manifold X with projection π:Y→X the surjective submersion; JrY denotes the r -th order jet prolongation of Y whose elements are r -jets Jr xγ of sections γ of π with source at x∈X and target at γ(x)∈Y . The canonical jet bundle projection is denoted by πr,s:JrY→JsY . Every fibered chart (V , ψ) , ψ= (xi , yσ) , 1 ≤i≤n , 1 ≤σ≤m , on Y induces the associated chart (U , ϕ) , ϕ= (xi) on X , and the associated fibered chart (Vr , ψr) on JrY , where U=π(V) , Vr= (πr,0)−1(V) , and ψr= (xi,yσ,yσ j, . . . , yσ j1...jr), where yσ j1...jk(Jr xγ) = Dj1. . . Djk(yσγϕ−1)(ϕ(x)), 0 ≤k≤r. A tangent vector ξ∈TyY is called π -vertical,if Tπ·ξ= 0, and a differential form ρ on Y is called π -horizontal, if for every point y∈Y the contraction iξρ(y) vanishes whenever ξ∈TyYis π-vertical. We denote by Ωr qY the Ωr 0Y -module of smooth differential q -forms defined on JrY ; πr - horizontal q -forms on JrY constitute a submodule of Ωr qY , denoted by Ωr q,XY . For a fibered manifold π:Y→X there exists a unique morphism h:ΩrY→Ωr+1Y of exterior algebras of differential forms such that for any fibered chart (V , ψ) , ψ= (xi , yσ) , on Y , and any differentiable function f:JrY→R, h f =f◦πr+1,r,hd f = (dif)dxi, where di (resp. d0 i ) is the i -th formal derivative (resp. the cut i -th formal derivative) operator associated with (V,ψ), di=d0 i+∑ j1≤...≤jr ∂ ∂yσ j1...jr yσ j1...jri, Mathematics 2022,10, 1211 3 of 14 and d0 i=∂ ∂xi+ r−1 ∑ k=0 ∑ j1≤...≤jk ∂ ∂yσ j1...jk yσ j1...jki. (2) A differential form q -form ρ∈Ωr qY satisfying hρ= 0 is called contact, and every contact form ρis generated by contact 1-forms ωσ j1...js=dyσ j1...js−yσ j1...jsidxi, 0 ≤s≤r−1. Any differential q-form ρ∈Ωr qYhas a unique invariant decomposition, (πr+1,r)∗ρ=hρ+ q ∑ k=1 pkρ, where pkρ is the k -contact component of ρ , containing exactly k exterior product factors ωσ j1...js with respect to any fibered chart (V,ψ). 2. Lepage Equivalents in Field Theory We summarize basic facts about Lepage differential forms on finite-order jet prolongations of fibered manifolds and, in particular, we discuss distinguished examples of Lepage equivalents of firstand second-order Lagrangians; for more details see [6,8,20,22–24]. By a Lagrangian λ for a fibered manifold π:Y→X of order r , we mean an element of the submodule Ωr n,XY of πr -horizontal n -forms in the module of n -forms Ωr nY , defined on the r -th jet prolongation JrY . In a fibered chart (V , ψ) , ψ= (xi , yσ) , Lagrangian λ∈Ωr n,XY has an expression λ=Lω0, (3) where ω0=dx1∧. . . ∧dxn is the (local) volume element, and L:Vr→R is said to be the Lagrange function associated to λand (V,ψ). An n -form ρ∈Ωs nY on JsY is called a Lepage form, if one of the following equivalent conditions is satisfied: (i) p1dρis a πs+1,0-horizontal (n+1)-form, (ii) hiξdρ=0 for arbitrary πs,0-vertical vector field ξon JsY, (iii) For every fibered chart (V,ψ)on Y,ρsatisfies (πs+1,s)∗ρ=f0ω0+ s ∑ k=0 fi,j1...jk σωσ j1...jk∧ωi+η, where n-form ηhas order of contactness ≥2, and ∂f0 ∂yσ j1...jk −difi,j1...jk σ−fjk,j1...jk−1 σ=0 Sym(j1. . . jk),k≤s, ∂f0 ∂yσ j1...js+1 −fjs+1,j1...js σ=0 Sym(j1. . . js+1). Let λ∈Ωr n,XY be a Lagrangian for π:Y→X . A Lepage form ρ∈Ωs nY is called aLepage equivalent of λ , if hρ=λ (up to a canonical jet projection). The following theorem describes the structure of Lepage equivalents of a Lagrangian. Theorem 1. Let λ∈Ωr n,XY be a Lagrangian of order r for π:Y→X , locally expressed by (3) with respect to a fibered chart (V , ψ) . An n -form ρ∈Ωs nY is a Lepage equivalent of λ if and only if it obeys the following decomposition: (πs+1,s)∗ρ=Θλ+dµ+η, (4) Mathematics 2022,10, 1211 4 of 14 where n-form Θλis defined on V2r−1by Θλ=Lω0+ r−1 ∑ k=0 r−1−k ∑ l=0 (−1)ldp1. . . dpl ∂L ∂yσ j1...jkp1...pli!ωσ j1...jk∧ωi, (5) where µis a contact (n−1)-form, and an n-form ηhas the order of contactness ≥2. Proof. See [22,24]. The expression Θλ , given by (5) on V2r−1 , is called the principal Lepage equivalent of λ with respect to fibered chart (V , ψ) . This Lepage form is uniquely determined by imposing that a Lepage form is π2r−1,r−1 -horizontal and it has the order of contactness ≤ 1.We note that, in general, decomposition (4) is not uniquely determined with respect to contact forms µ and η , although the Lepage equivalent ρ satisfying (4) is a globally defined differential form on JsY . For a first-order Lagrangian λ , Θλ(5) is the well-known Poincaré–Cartan form defined on J1Y(cf. [2]), Θλ=Lω0+∂L ∂yσ j ωσ∧ωj. (6) For a second-order Lagrangian λ , Θλ(5) is the generalized Poincaré–Cartan form defined on J3Y(cf. [6,23]), Θλ=Lω0+ ∂L ∂yσ j −di ∂L ∂yσ ij !ωσ∧ωj+∂L ∂yσ ij ωσ i∧ωj. (7) We point out that for Lagrangians of order r≥ 3, local expressions (5) need not define differential forms on J2r−1Yglobally (cf. [6,14]). The well-known Euler–Lagrange mapping of the calculus of variations assigns to a Lagrangian λ∈Ωr n,XYthe Euler–Lagrange form Eλ=Eσ(L)ωσ∧ω0, (8) with coefficients Eσ(L) = r ∑ k=0 (−1)kdi1. . . dik ∂L ∂yσ i1...ik (9) the Euler–Lagrange expressions associated to L:Vr→R . Note that the 1-contact and πr,0 -horizontal (n+ 1 ) -form Eλ is defined by means of (8) and (9) on J2rY globally. The following theorem explains a crucial relation between a Lepage equivalent of Lagrangian λ on one side and the associated Euler–Lagrange form Eλon the other side. Theorem 2. Let λ∈Ωr n,XY be a Lagrangian of order r for π:Y→X and let ρ∈Ωs nY be a Lepage equivalent of λ. Then (πs+1,s)∗dρ=Eλ+F, where Eλ is the Euler–Lagrange form (8) associated to λ , and F is an (n+ 1 ) -form with order of contactness ≥ 2. In particular , Eλ coincides with the 1-contact component of the exterior derivative of a Lepage equivalent of Lagrangian λ, i.e., on J2rY, Eλ=p1dρ. (10) Proof. See [22–24]. In addition to the principal Lepage equivalent Θλ , given by (6) and (7) for a firstand second-order Lagrangian λ , respectively, we recall the other known examples of Lepage equivalents, determined by means of additional requirements. Mathematics 2022,10, 1211 5 of 14 Lemma 1. (a) Let λ∈Ω1 n,XY be a non-vanishing first-order Lagrangian for π:Y→X , locally expressed by (3). Then the local expression Λλ=1 Ln−1 n ^ j=1 Ldxj+∂L ∂yσ j ωσ!(11) defines a π1,0-horizontal differential n-form Λλ∈Ω1 nY, which is a Lepage equivalent of λ. (b) Let λ∈Ω2 n,XY be a non-vanishing second-order Lagrangian for π:Y→X , locally expressed by (3). Then the local expression Λλ=1 Ln−1 n ^ j=1 Ldxj+ ∂L ∂yσ j −di ∂L ∂yσ ij !ωσ+∂L ∂yσ ij ωσ i!(12) defines a π3,1-horizontal differential n-form Λλ∈Ω3 nY, which is a Lepage equivalent of λ. Proof. See [3,8]. The expression Λλ(11) is the well-known Carathéodory form, associated to a nonvanishing, first-order Lagrangian λ (cf. [ 3 ]),whereas Λλ(12) is its generalization for secondorder Lagrangians, recently studied in [8]. Remark 1. Note that the Carathéodory form Λλ(12) is decomposed as a sum of the generalized Poincaré–Cartan form Θλ(7) and a π3,1 -horizontal, 2-contact differential n -form. For further purpose, we give this decomposition explicitly for the dimension of base n =2: Λλ=Θλ +1 L∂L ∂yσ 1 −di ∂L ∂yσ i1∂L ∂yν 2 −dk ∂L ∂yν k2ωσ∧ων +1 L ∂L ∂yν j2∂L ∂yσ 1 −di ∂L ∂yσ i1−∂L ∂yν j1∂L ∂yσ 2 −di ∂L ∂yσ i2!ωσ∧ων j(13) +1 L ∂L ∂yσ i1 ∂L ∂yν j2 ωσ i∧ων j. Lemma 2. Let λ∈Ω1 n,XY be a first-order Lagrangian for π:Y→X , locally expressed by (3) . There exists a unique Lepage equivalent Zλ∈Ω1 nY of λ , which satisfies Zλ= (π1,0)∗ρ for any n -form ρ∈Ω0 nW on W such that hρ=λ . With respect to a fibered chart (V , ψ) , Zλ has an expression Zλ=Lω0+ n ∑ k=1 1 (n−k)! 1 (k!)2 ∂kL ∂yσ1 j1. . . ∂yσk jk εj1...jkik+1...in(14) ·ωσ1∧. . . ∧ωσk∧dxik+1∧. . . ∧dxin. Proof. See [4,5]. The expression Zλ(14) is known as the fundamental Lepage form, or Krupka–Betounes form, associatedtofirst-orderLagrangian λ ;originalsources are[ 4 , 5 ], andfurtherrecent contributions include [18–20]. Remark 2. One can directly verify that the Lepage equivalent Zλ(14) satisfies the equivalence relation “ Zλ is closed if and only if the associated Lagrangian λ=hZλ is trivial”, that is “ dZλ= 0 if and only if Eλ= 0”. As Eλ=p1dZλ(10) , it is immediate that λ∈Ω1 n,XY is trivial, provided Zλ is closed. However, the converse implication is a non-trivial one. A construction of Mathematics 2022,10, 1211 6 of 14 (local, 1-contact) Lepage equivalents of higher-order Lagrangians satisfying this equivalence relation has been recently described in [25]. Another remarkable property that Zλ(14) satisfies is the following: Zλ is π1,0 -projectable if and only Eλis π2,1-projectable. 3. Trivial Lagrangians A Lagrangian λ is said to be variationally trivial (or null) if the associated Euler– Lagrange form Eλ vanishes identically. The following theorem describes variationally trivial Lagrangians. Theorem 3. Let λ∈Ωr n,XY be a Lagrangian of order r for π:Y→X . The following conditions are equivalent: (i) λis trivial. (ii) For every fibered chart (V , ψ) on Y there exists an (n− 1 ) -form µ∈Ωr−1 n−1Y such that on Vr , λ=hdµ. (iii) For every fibered chart (V , ψ) on Y there exist functions gi:Vr→R such that λ=Lω0 on Vr, where L=digi. (15) Proof. See [26]; also cf. [22]. As a consequence of Theorem 3, we now summarize explicit chart conditions for a trivial Lagrangian of second-order, needed later in this paper. Let λ∈Ω2 n,XY be a Lagrangian for Y , locally expressed by λ=Lω0(3) with respect to a fibered chart (V , ψ) , ψ= (xi , yσ) , on Y , where L=L(xi , yσ , yσ j , yσ jk) , j≤k , is the associated Lagrange function defined on V2⊂J2Y, and Eσ(L)are the corresponding Euler–Lagrange expressions on V4⊂J4Y, Eσ(L) = ∂L ∂yσ−di ∂L ∂yσ i +didj ∂L ∂yσ ij . (16) Lemma 3. (a) λis trivial if and only if it satisfies the following conditions: ∂L ∂yσ−d0 i ∂L ∂yσ i +d0 id0 j ∂L ∂yσ ij =0, (17) ∂2L ∂yν p∂yσ iq −∂2L ∂yν pq∂yσ i +2d0 j ∂2L ∂yν pq∂yσ ij !=0 Sym(pqi), (18) ∂3L ∂yµ st∂yν pq∂yσ ij =0 Sym(stj), Sym(pqi), (19) ∂2L ∂yν pq∂yσ ij =0 Sym(pqij), (20) where d0 iis the cut formal derivative operator (2). (b) For every fibered chart (V , ψ) on Y there exist functions gi:V2→R such that λ=Lω0 on V2, where L=digi, and ∂gi ∂yσ jk +∂gj ∂yσ ki +∂gk ∂yσ ij =0. (21) Mathematics 2022,10, 1211 7 of 14 Proof. Consider the Euler–Lagrange expressions Eσ(L) , associated to a second-order Lagrangian λ=Lω0. We have di ∂L ∂yσ i =∂2L ∂xi∂yσ i +∂2L ∂yν∂yσ i yν i+∂2L ∂yν p∂yσ i yν pi +∂2L ∂yν pq∂yσ i yν pqi =d0 i ∂L ∂yσ i +∂2L ∂yν pq∂yσ i yν pqi, and didj ∂L ∂yσ ij =di d0 j ∂L ∂yσ ij +∂2L ∂yν pq∂yσ ij yν pqj! =d0 id0 j ∂L ∂yσ ij +d0 j ∂2L ∂yν pq∂yσ ij !yν pqi +∂2L ∂yν p∂yσ iq yν pqi +d0 i ∂2L ∂yν pq∂yσ ij !yν pqj +∂3L ∂yµ st∂yν pq∂yσ ij yµ stiyν pqj +∂2L ∂yν pq∂yσ ij yν pqji. Thus (16) now reads, Eσ(L) = ∂L ∂yσ−d0 i ∂L ∂yσ i +d0 id0 j ∂L ∂yσ ij + ∂2L ∂yν p∂yσ iq −∂2L ∂yν pq∂yσ i +2d0 j ∂2L ∂yν pq∂yσ ij !!yν pqi (22) +∂3L ∂yµ st∂yν pq∂yσ ij yµ stiyν pqj +∂2L ∂yν pq∂yσ ij yν pqij, hence Eσ(L)vanish if and only if conditions (18) hold, proving (a). Condition (b) follows from Theorem 3, (iii), where (21) is implied by Equation (15) satisfied identically on V2⊂J2Y. 4. The Fundamental Lepage Equivalent of a Second-Order Lagrangian: Order Reduction First, we present an order reduction condition, which allows us to construct a generalization of the fundamental Lepage equivalent Zλ(14) for a second-order Lagrangian λ . From Theorem 1on the structure of every Lepage equivalent of a Lagrangian, it is immediate that a differential n -form to be found must be decomposable as Zλ=Θλ+dµ+η (up to a canonical jet projection), where Θλ is the generalized Poincaré–Cartan form (7) , µ is a contact (n−1)-form, and an n-form ηhas the order of contactness ≥2. Let us assume that Θλ(7) is of the same order as the Lagrangian λ. Note that this condition is automatically satisfied for first-order Lagrangians but, nevertheless, for the second-order it restricts the class of Lagrangians under consideration. Lemma 4. Let λ∈Ω2 n,XY be a second-order Lagrangian for π:Y→X . Then the generalized Poincaré–Cartan form Θλ(7) is of second-order, if and only if for every chart (V , ψ) , ψ= (xi , yσ) , on Y, ∂2L ∂yν pq∂yσ ij +∂2L ∂yν ip∂yσ qj +∂2L ∂yν qi∂yσ pj =0, (23) where Lis the Lagrange function associated to λ(3). Mathematics 2022,10, 1211 8 of 14 For n =2,(23)read ∂2L ∂yν 11∂yσ 11 =0, ∂2L ∂yν 11∂yσ 12 =0, ∂2L ∂yν 22∂yσ 21 =0, ∂2L ∂yν 22∂yσ 22 =0, ∂2L ∂yν 11∂yσ 22 +2∂2L ∂yν 12∂yσ 12 =0. (24) Proof. The necessary and sufficient condition (23) follows immediately from the chart expression (7) of Θλ, by means of annihilating terms linear in coordinates yτ pqi. Lemma 5. Let λ∈Ω2 n,XY be a second-order Lagrangian for π:Y→X such that the generalized Poincaré–Cartan form Θλ(7) is of second-order. Then λ is variationally trivial, if and only if for every chart (V,ψ),ψ= (xi,yσ), on Y, ∂L ∂yσ−d0 i ∂L ∂yσ i +d0 id0 j ∂L ∂yσ ij =0, ∂2L ∂yν p∂yσ iq −∂2L ∂yν pq∂yσ i =0 Sym(pqi). (25) For n =2,(25)read ∂L ∂yσ−d0 i ∂L ∂yσ i +d0 id0 j ∂L ∂yσ ij =0, i,j=1,2, ∂2L ∂yν 1∂yσ 11 −∂2L ∂yν 11∂yσ 1 =0, ∂2L ∂yν 2∂yσ 22 −∂2L ∂yν 22∂yσ 2 =0, 2∂2L ∂yν 1∂yσ 12 −2∂2L ∂yσ 1∂yν 12 +∂2L ∂yν 2∂yσ 11 −∂2L ∂yσ 2∂yν 11 =0, (26) 2∂2L ∂yν 2∂yσ 12 −2∂2L ∂yν 12∂yσ 2 +∂2L ∂yν 1∂yσ 22 −∂2L ∂yν 22∂yσ 1 =0. Proof. Necessary and sufficient conditions (25) for a variationally trivial Lagrangian are nothing but the reduction of (17)–(20), Lemma 3, with the help of (23). In the next theorem we construct the fundamental Lepage equivalent Zλ of a secondorder Lagrangian λover fibered manifolds π:Y→X, where dim X=2. Analogously to the Carathéodory form Λλ(13) , associated to second-order Lagrangian λ∈Ω2 2,XY , suppose that Zλ is decomposed as a sum of the generalized Poincaré–Cartan form Θλ(7) and contact terms, generated by the wedge products ωσ∧ων , ωσ∧ων j , and ωσ i∧ων j, i.e., Zλ=Θλ+1 2Pσνωσ∧ων+Qj σ,νωσ∧ων j+1 2Ri,j σ,νωσ i∧ων j, (27) where Pσν , Qj σ,ν , and Ri,j σ,ν are real-valued functions on V3⊂J3Y such that Pσν is skewsymmetric in (σ,ν), and Ri,j σ,νis skew-symmetric in pairs (i,σ),(j,ν). Theorem 4. Let λ∈Ω2 2,XY be a second-order Lagrangian for π:Y→X such that (23) holds. The following two conditions are equivalent: (i) If λis variationally trivial, then Zλ(27)is closed. Mathematics 2022,10, 1211 9 of 14 (ii) For every chart (V , ψ) , ψ= (xi , yσ) , on Y , Zλ(27) is uniquely determined by means of real-valued functions Pσν, Qj σ,ν, and Ri,j σ,ν, defined on V2⊂J2Y as Pσν =1 2∂2L ∂yσ 1∂yν 2 −∂2L ∂yν 1∂yσ 2 +d0 1∂2L ∂yν 1∂yσ 12 −∂2L ∂yσ 1∂yν 12 +d0 2∂2L ∂yσ 2∂yν 12 −∂2L ∂yν 2∂yσ 12 , Q1 σ,ν=2∂2L ∂yσ 1∂yν 12 −∂2L ∂yν 1∂yσ 12 −∂2L ∂yν 2∂yσ 11 −2d0 2∂2L ∂yσ 12∂yν 12 , (28) Q2 σ,ν=−2∂2L ∂yσ 2∂yν 12 +∂2L ∂yν 1∂yσ 22 +∂2L ∂yν 2∂yσ 12 +2d0 1∂2L ∂yσ 12∂yν 12 , R1,2 σ,ν=−2∂2L ∂yσ 12∂yν 12 =−R2,1 σ,ν, R1,1 σ,ν=0, R2,2 σ,ν=0, Proof. Suppose that Lagrangian λ∈Ω2 2,XY is variationally trivial and the generalized Poincaré–Cartan form Θλ(7) is defined on J2Y . Thus, in abritrary fibered chart (V , ψ) , ψ= (xi , yσ) , on Y , the associated Lagrange function L:V2→R satisfies conditions (17) – (19) of Lemma 3and (23) of Lemma 4. Note first that condition (23) already implies (20), and using (23), di ∂L ∂yσ ij =d0 i ∂L ∂yσ ij . Hence, the exterior derivative of Θλreads, dΘλ=dL∧ω0+d ∂L ∂yσ j −di ∂L ∂yσ ij !∧ωσ∧ωj+ ∂L ∂yσ j −di ∂L ∂yσ ij !dωσ∧ωj +d ∂L ∂yσ ij !∧ωσ i∧ωj+∂L ∂yσ ij dωσ i∧ωj = ∂L ∂yσ−dj ∂L ∂yσ j +didj ∂L ∂yσ ij !ωσ∧ω0−∂ ∂yτ ∂L ∂yσ j −d0 i ∂L ∂yσ ij !ωσ∧ωτ∧ωj(29) + ∂2L ∂yσ∂yτ kj −∂ ∂yτ k ∂L ∂yσ j −d0 i ∂L ∂yσ ij !!ωσ∧ωτ k∧ωj −∂ ∂yτ kl ∂L ∂yσ j −d0 i ∂L ∂yσ ij !ωσ∧ωτ kl ∧ωj−∂2L ∂yτ k∂yσ ij ωσ i∧ωτ k∧ωj −∂2L ∂yτ kl∂yσ ij ωσ i∧ωτ kl ∧ωj. From (27), we have dZλ=dΘλ +1 2dPσν ∧ωσ∧ων+1 2Pσνd(ωσ∧ων)+dQj σ,ν∧ωσ∧ων j+Qj σ,νdωσ∧ων j +1 2dRi,j σ,ν∧ωσ i∧ων j+1 2Ri,j σ,νdωσ i∧ων j, and combining with (29) we get a decomposition of dZλ containing independent base terms, dZλ=Eλ+F0+F1+F2,