Báo cáo toán học: "A Presentation of the Elements of the Quotient Sheaves Ωk /Θk in Variational Sequences" pdf

Vietnam Journal of Mathematics 33:3 (2005) 271–281 A Presentation of the Elements of the Quotient Sheaves Ω k r /Θ k r in Variational Sequences Nong Quoc Chinh Thai Nguyen University, Thai Nguyen City, Vietnam Received Ferbuary 05, 2004 Revised Ferbuary 17, 2005 Abstract In this paper we give a concrete presentation of the elements of the quotient sheaves Ω k r /Θ k r in the variational sequence 0 → R → Ω 0 r E 0 −→ Ω 1 r /Θ 1 r E 1 −→ Ω 2 r /Θ 2 r E 2 −→ E P −2 −→ Ω P −1 r /Θ P −1 r → E P −1 −→ Ω P r /Θ P r E P −→ Ω P +1 r E P +1 −→ → Ω N r → 0. 1. Introduction The notion of variational bicomplexes was introduced in studying the problem of characterizing the kernel and image of Euler-Lagrange mapping in the calcu- lus of variations. This problem has been considered by Anderson [1], Duchamp [2], Dedecker [4], Tulczyvew [7], Takens [6] and Krupka [5]. The variational bi- complex was mainly studied on the infinite jet prolongation J ∞ Y of the fibered manifold Y with the base X and the projection π : Y → X, where dimX = n, dimY −n = m. The variational bicomplexes contain Euler-Lagrange mapping as one of its morphisms. Then its developments in the theory of variational bicom- plexes have made an important role in many problems in caculus of variations on manifolds, in diffrential geometry, in the theory of differential equations and in mathematical physics. Krupka [5] studied the sheaves of differential forms on finite r-jet prolonga- tions J r Y . Then he constructed the sequence of quotient sheaves Ω k r /Θ k r .This quotient sequence is called the variational sequence of order r over Y .Itisan acyclic resolution of the constant sheaf R over Y 272 Nong Quoc Chinh 0 → R → Ω 0 r E 0 −→ Ω 1 r /Θ 1 r E 1 −→ Ω 2 r /Θ 2 r E 2 −→ E P −2 −→ Ω P −1 r /Θ P −1 r → E P −1 −→ Ω P r /Θ P r E P −→ Ω P +1 r E P +1 −→ → Ω N r → 0. (1) In the sequence (1), E n is the Euler-Lagrange mapping and E n+1 is the Helmholtz- Sonin mapping. In the study of variational sequence, it is very important to give a concrete presentation of the elements [] ∈ Ω k r /Θ k r . It also has been shown in [5] that []=p k r,0  for all positive integers k satisfying 1 ≤ k ≤ n,  ∈ Ω k r ,where p k r,0  is the horizontal component of k-form . Then Krupka [5] gave a concrete presentation of []for1≤ k ≤ n. The main purpose of this paper is to give a concrete presentation of elements [] ∈ Ω k r /Θ k r for all positive integers k satisfying n +1≤ k ≤ P. For simplicity, we will solve this problem in the case of r =1andr = 2. Then the other cases follow by the same method. 2. Notations and Premilinaries Throughout this paper, the following notations will be used: (Y, π,X) is a fibered manifold with base X and the projection π : Y → X, where dimX = n,dimY − n = m. J r Y is the finite r-jet prolongation of the fibered manifold (Y,π,X).π r : J r Y → X, π r,s : J r Y → J s Y are the canonical jet projections. (V,Ψ) is a fiber chart on Y ,whereΨ=(x i ,y σ ), 1 ≤ i ≤ n, 1 ≤ σ ≤ m. (V r , Ψ r ) is the fiber chart on r-jet prolongation J r Y associated with (V,Ψ), where V r = π −1 r,0 (V ), Ψ r = (x i ,y σ ,y σ j 1 , , y σ j 1 j 2 j r ), 1 ≤ i ≤ n, 1 ≤ σ ≤ m, 1 ≤ j 1 ≤ ≤ j r ≤ n. Ω k r is the sheave of k-forms over J r Y and π ∗ r+1,r is the pull-back of the mapping π r+1,r . We put ω 0 = dx 1 ∧ dx 2 ∧ ∧ dx n , ω i =(−1) i−1 dx 1 ∧ ∧ dx i−1 ∧ dx i+1 ∧···∧dx n , 1 ≤ i ≤ n, ω σ j 1 j 2 j k = dy σ j 1 j 2 j k − y σ j 1 j 2 j k i dx i , 1 ≤ k ≤ r − 1. Note that the forms dx i ,ω σ j 1 j 2 j k ,dy σ j 1 j 2 j r , for 0 ≤ k ≤ r − 1, define a basis of the space of linear forms on V r . Obviously we have dω σ j 1 j 2 j k ∧ ω i = −ω σ j 1 j 2 j k i ∧ ω 0 , for 0 ≤ k ≤ r − 2, dω σ j 1 j 2 j r−1 ∧ ω i = −dy σ j 1 j 2 j r−1 i ∧ ω 0 . Let N = n + m  n + r n  ,M= m  n + r − 1 n  ,P = m  n + r − 1 n  +2n −1. It is clear that N =dimJ r Y and M is the number of linear independent forms ω σ j 1 j 2 j k , where 0 ≤ k ≤ r − 1. A Presentation of the Elements of the Quotient Sheaves Ω k r /Θ k r 273 For any function f :V r → R we have h(df )=  n i=1 d i f.dx i , where d i f = ∂f ∂x i + m  σ=1 r  k=0 ∂f ∂y σ j 1 j k y σ j 1 j k . For any k-form ρ ∈ Ω k r ,wedenotebyρ 0 the horizontal component of ρ,and ρ q , 1 ≤ q ≤ k, is the q-contact component of ρ. Then for any ρ ∈ Ω k r there exists a unique decomposition π ∗ r+1,r ρ = ρ 0 + ρ 1 + ···+ ρ k . We denote by p k r,q :Ω k r → Ω k r+1 the morphism of sheaves defined by p k r,q ρ = ρ q , for 0 ≤ q ≤ k.Ω k r(c) = ker p k r,0 ,1≤ k ≤ n, is the sheave of contact k-forms. Ω k r(c) = ker p k r,k−n , n +1≤ k ≤ N, is the sheave of strongly contact k-forms. Let Θ k r = dΩ k−1 r(c) +Ω k r(c) . In [3] and [5] the authours proved the softness of the sheaves Θ k r , considered the quotient sheaves Ω k r /Θ k r , and they obtained the following short exact sequence 0 → Θ k r i k r →Ω k r τ k r →Ω k r /Θ k r → 0, where i k r is the canonical injective, τ k r is the canonical quotient mapping. Especially, Krupka [5] constructed the following variational sequence of order r over Y 0 → R → Ω 0 r E 0 −→ Ω 1 r /Θ 1 r E 1 −→ Ω 2 r /Θ 2 r E 2 −→··· E P −2 −→ Ω P −1 r /Θ P −1 r → E P −1 −→ Ω P r /Θ P r E P −→ Ω P +1 r E P +1 −→ ···→Ω N r → 0, where the sheaf morphism E k :Ω k r /Θ k r → Ω k+1 r /Θ k+1 r is defined by the formula E k ([]) = [d]. He proved that, the variational sequence of order r is an acyclic resolution of the constant sheaf R over Y ,and[]=p k r,0  for all 1 ≤ k ≤ n and for all  ∈ Ω k r . This is the horizontal component of k-form . Let 1 ≤ k ≤ n − 1and  ∈ Ω k r ,then []= 1 k! f i 1 ···i k dx i 1 ∧···∧dx i k , (2) where f i 1 i k are some functions on V r . Let k = n and  ∈ Ω n r .Then []=fω 0 , (3) where f is some function on V r .Inthiscase[] is called the Lagrange class of . Let n +1≤ k ≤ P. For every s>r,Krupka [5] proved that Ω k r /Θ k r ≈ Im(τ k s .π ∗ s,r+1 .p k r,k−n ), and this implies that for every  ∈ Ω k r []=τ k s (p k s−1,k−n .π ∗ s−1,r ). (4) 274 Nong Quoc Chinh Let k = n + 1. For each element  ∈ Ω n+1 r ,[] is called the Euler-Lagran ge class of (n +1)-forms. Let k = n + 2. For each element  ∈ Ω n+2 r ,[] is called the Helmiholtz-Sonin class of (n +2)-forms. Below we give a concrete presentation of the elements in Ω k r /Θ k r for all pos- itive integer k satisfying n +1≤ k ≤ P in the case of r =1andr =2. 3. TheCaseofr =1 Theorem 1. a) Let  ∈ Ω n+1 1 be a germ. Suppose that in the fiber chart (V,ψ),ψ=(x i ,y σ ), p n+1 1,1  = f σ w σ ∧ w 0 + f i σ w σ i ∧ w 0 , (5) where 1 ≤ σ ≤ m, 1 ≤ i ≤ n ,andf σ , f i σ are functions defined on V 2 ⊂ J 2 Y. Then we have []=(f σ − d i f i σ )w σ ∧ w 0 . (6) b) Let  ∈ Ω n+2 1 be a germ. Suppose that in the fiber chart (V,ψ),ψ=(x i ,y σ ), p n+2 1,2  = f σν w σ ∧ w ν ∧ w 0 + f i σν w σ i ∧ w ν ∧ w 0 + f ij σν w σ i ∧ w ν j ∧ w 0 , (7) where 1 ≤ σ, ν ≤ m, 1 ≤ i, j ≤ n,andf σν ,f i σν ,f ij σν are functions defined on V 2 ⊂ J 2 Y .Then []=((f σν w σ +(f i σν − d i f ij σν )w σ i ) − f ij σν w σ ij ) ∧ w ν ∧ w 0 . (8) c) Let n +3≤ k ≤ P and  ∈ Ω k 1 be a germ. Suppose that in the fiber chart (V,ψ),ψ =(x i ,y σ ), p k 1,k−n  = f σ 1 σ k−n w σ 1 ∧ w σ 2 ∧ ∧ ∧ w σ k−n ∧ w 0 + f i 1 σ 1 σ k−n w σ 1 i 1 ∧ w σ 2 ∧ ∧ ∧ w σ k−n ∧ w 0 + + f i 1 i k−n−1 σ 1 σ k−n w σ 1 i 1 ∧ w σ 2 i 2 ∧ ∧ w σ k−n−1 i k−n−1 ∧ w σ k−n ∧ w 0 + f i 1 i k−n σ 1 σ k−n w σ 1 i 1 ∧ w σ 2 i 2 ∧ ∧ w σ k−n−1 i k−n−1 ∧ w σ k−n i k−n ∧ w 0 , (9) where 1 ≤ σ 1 , , σ k−n ≤ m, 1 ≤ i 1 , i k−n ≤ n and each function defined on V 2 ⊂ J 2 Y. Then we have A Presentation of the Elements of the Quotient Sheaves Ω k r /Θ k r 275 []=  k−n−2  h=0 f i 1 i h σ 1 σ k−n w σ 1 i 1 ∧ ∧ w σ h i h ∧ w σ h+1 ∧ ∧ w σ k−n−1 +(f i 1 i k−n−1 σ 1 σ k−n − d i k−n f i 1 i k−n σ 1 σ k−n )w σ 1 i 1 ∧ ∧ w σ k−n−1 i k−n−1 − k−n−1  t=1 f i 1 i k−n σ 1 σ k−n w σ 1 i 1 ∧ ∧ w σ t−1 i t−1 ∧ w σ t i t i k−n ∧ w σ t+1 i t+1 ∧ ∧ w σ k−n−1 i k−n−1  ∧w σ k−n ∧ w 0 , (10) where each function is well defined on V 3 ⊂ J 3 Y . Proof. a) Considering all the factors in (5) which contains w σ i ,weget f i σ w σ i ∧ w 0 = −f i σ d(w σ ∧ w i )=−d(f i σ w σ ∧ w i )+df i σ ∧ w σ ∧ w i . (11) Since f i σ w σ ∧ w i ∈ Θ n 2 and π ∗ 3,2 (df i σ ∧ w σ ∧ w i )=h(df i σ ) ∧ w σ ∧ w i + p(df i σ ) ∧ w σ ∧ w i , we have []=τ n+1 3 .p n+1 2,1 .π ∗ 2,1 ()=f σ w σ ∧ w 0 + h(df i σ ) ∧ w σ ∧ w i =(f σ − d i f i σ )w σ ∧ w 0 . b) Considering all the factors in (7) which contains w σ i ∧ w ν j , we get f ij σν w σ i ∧ w ν j ∧ w 0 = −f ij σν w σ i ∧ d(w ν ∧ w j )= = f ij σν d(w σ i ∧ w ν ∧ w j ) − f ij σν dw σ i ∧ w ν ∧ w j = d(f ij σν w σ i ∧ w ν ∧ w j ) − df ij σν ∧ w σ i ∧ w ν ∧ w j + + n  l=1 f ij σν dy σ il ∧ dx l ∧ w ν ∧ w j . (12) Since f ij σν w σ i ∧ w ν ∧ w j ∈ Θ n+1 2 , we have []=τ n+2 3 .p n+2 2,2 .π ∗ 2,1 () = f σν w σ ∧ w ν ∧ w 0 + f i σν w σ i ∧ w ν ∧ w 0 − d j f ij σν w σ i ∧ w ν ∧ w 0 − f ij σν w σ ij ∧ w ν ∧ w 0 . (13) Therefore we have []=((f σν w σ +(f i σν − d i f ij σν )w σ i ) − f ij σν w σ ij ) ∧ w ν ∧ w 0 . c) In the formula (9), we consider all the factors containing w σ k−n i k−n .Theyare f i 1 i k−n σ 1 σ k−n w σ 1 i 1 ∧ ∧ w σ k−n i k−n ∧ w 0 . We get 276 Nong Quoc Chinh f i 1 i k−n σ 1 σ k−n w σ 1 i 1 ∧ ∧ w σ k−n i k−n ∧ w 0 = − f i 1 i k−n σ 1 σ k−n w σ 1 i 1 ∧ ∧ w σ k−n−1 i k−n−1 ∧ d(w σ k−n ∧ w i k−n ) =(−1) k−n f i 1 i k−n σ 1 σ k−n d(w σ 1 i 1 ∧ ∧ w σ k−n−1 i k−n−1 ∧ w σ k−n ∧ w i k−n ) + k−n−1  t=1 (−1) k−n+t f i 1 i k−n σ 1 σ k−n w σ 1 i 1 ∧ ∧ w σ t−1 i t−1 ∧ dw σ t i t ∧ ∧ w σ t+1 i t+1 ∧ ∧ w σ k−n−1 i k−n−1 ∧ w σ k−n ∧ w i k−n =(−1) k−n d(f i 1 i k−n σ 1 σ k−n w σ 1 i 1 ∧ ∧ w σ k−n−1 i k−n−1 ∧ w σ k−n ∧ w i k−n ) − (−1) k−n df i 1 i k−n σ 1 σ k−n ∧ w σ 1 i 1 ∧ ∧ w σ k−n−1 i k−n−1 ∧ w σ k−n ∧ w i k−n + k−n−1  t=1 n  l=1 (−1) k−n+t+1 f i 1 i k−n σ 1 σ k−n w σ 1 i 1 ∧ ∧ w σ t−1 i t−1 ∧ dy σ t i t l ∧ dx l ∧ ∧ w σ t+1 i t+1 ∧ ∧ w σ k−n−1 i k−n−1 ∧ w σ k−n ∧ w i k−n . (14) Since f i 1 i k−n σ 1 σ k−n w σ 1 i 1 ∧ ∧ w σ k−n−1 i k−n−1 ∧ w σ k−n ∧ w i k−n ∈ Θ k−1 2 , we have τ k 3 .p k 2,k−n (f i 1 i k−n σ 1 σ k−n w σ 1 i 1 ∧ ∧ w σ k−n i k−n ∧ w 0 ) = − d i k−n f i 1 i k−n σ 1 σ k−n w σ 1 i 1 ∧ ∧ w σ k−n−1 i k−n−1 ∧ w σ k−n ∧ w 0 − k−n−1  t=1 f i 1 i k−n σ 1 σ k−n w σ 1 i 1 ∧ ∧ w σ t−1 i t−1 ∧ w σ t i t i k−n ∧ w σ t+1 i t+1 ∧ ∧ w σ k−n−1 i k−n−1 ∧ w σ k−n ∧ w 0 . (15) Then [] can be presented in the following form []=  k−n−2  h=0 f i 1 i h σ 1 σ k−n w σ 1 i 1 ∧ ∧ w σ h i h ∧ w σ k+1 ∧ ∧ w σ k−n−1 +(f i 1 i k−n−1 σ 1 σ k−n − d i k−n f i 1 i k−n σ 1 σ k−n )w σ 1 i 1 ∧ ∧ w σ k−n−1 i k−n−1 − k−n−1  t=1 f i 1 i k−n σ 1 σ k−n w σ 1 i 1 ∧ ∧ w σ t−1 i t−1 ∧ w σ t i t i t−1 ∧ w σ t+1 i t+1 ∧ ∧ w σ k−n−1 i k−n−1  ∧w σ k−n ∧ w 0 , where each function is defined on V 3 ⊂ J 3 Y .  4. TheCaseofr =2 Theorem 2. a) Let  ∈ Ω n+1 2 be a germ. Suppose that in the fiber chart (V,ψ),ψ=(x i ,y σ ), p n+1 2,1  = f σ w σ ∧ w 0 + f i σ w σ i ∧ w 0 + f ij σ w σ ij ∧ w 0 , (16) A Presentation of the Elements of the Quotient Sheaves Ω k r /Θ k r 277 where 1 ≤ σ ≤ m, 1 ≤ i, j ≤ m, f σ ,f i σ ,f ij σ are functions well defined on V 3 ⊂ J 3 Y . Then we have []=(f σ − d i f i σ + d i d j f ij σ )w σ ∧ w 0 , (17) where each function is well defined on V 5 ⊂ J 5 Y . b) Let  ∈ Ω n+2 2 be a germ. Suppose that in the fiber chart (V,ψ),ψ=(x i ,y σ ), p n+2 2,2  =(f J σν w σ J ∧ w ν ∧ w 0 + f Ji σν w σ J ∧ w ν i ∧ w 0 + f Jij σν w σ J ∧ w ν ij ∧ w 0 , (18) where 0 ≤|J|≤2, 1 ≤ σ, ν ≤ m, 1 ≤ i, j ≤ n, and functions f J σν ,f Ji σν ,f Jij σν are wel l defined on V 3 ⊂ J 3 Y . Then we have []=((f J σν − d i f Ji σν + d i d j f Jij σν )w σ J +(−f Ji σν +2d j f Jij σν )w σ Ji + f Jij σν w σ Jij ) ∧ w ν ∧ w 0 , (19) where each function is well defined on V 5 ⊂ J 5 Y . c) Let n +3≤ k ≤ P and  ∈ Ω k 2 be a germ. Suppose that in the fiber chart (V,ψ),ψ =(x i ,y σ ) we have p k 2,k−n  = 2  q=0 f J 1 J k−n−1 j 1 j q σ 1 σ k−n w σ 1 J 1 ∧ ∧ w σ k−n−1 J k−n−1 ∧ w σ k−n j 1 j q ∧ w 0 , (20) where 0 ≤|J 1 |, , |J k−n−1 |≤2, 1 ≤ σ 1 , , σ k−n ≤ m, 1 ≤ j 1 , , j q ≤ n, and every function f J 1 J k−n−1 j 1 j q σ 1 σ k−n is well defined on V 3 ⊂ J 3 Y .Thenwe have []=  (f J 1 J k−n−1 σ 1 σ k−n − d j 1 f J 1 J k−n−1 j 1 σ 1 σ k−n + d j 1 d j 2 f J 1 J k−n−1 j 1 j 2 σ 1 σ k−n )w σ 1 J 1 ∧ ∧ w σ k−n−1 J k−n−1 + k−n−1  t=1 (−f J 1 J k−n−1 j 1 σ 1 σ k−n +2d j 2 f J 1 J k−n−1 j 1 j 2 σ 1 σ k−n )w σ 1 J 1 ∧ ∧ w σ t−1 J t−1 ∧ w σ t J t j 1 ∧ w σ t+1 J t+1 ∧ ∧ w σ k−n−1 J k−n−1 (21) + k−n−1  h=1 h=t k−n−1  t=1 f J 1 J k−n−1 j 1 j 2 σ 1 σ k−n w σ 1 J 1 ∧ ∧ w σ h−1 J h−1 ∧ w σ h J h j 1 ∧ w σ h+1 J h+1 ∧ ∧ w σ t−1 J t−1 ∧ w σ t J t j 2 ∧ w σ t+1 J t+1 ∧ ∧ w σ k−n−1 J k−n−1 + k−n−1  t=1 f J 1 J k−n−1 j 1 j 2 σ 1 σ k−n w σ 1 J 1 ∧ ∧ w σ t−1 J t−1 ∧ w σ t J t j 1 j 2 ∧ w σ t+1 J t+1 ∧ ∧ w σ k−n−1 J k−n−1  ∧w σ k−n ∧ w 0 , where each function is well defined on V 5 ⊂ J 5 Y . 278 Nong Quoc Chinh Proof. a) In the formula (16) we consider the factors containing w σ i . They aref i σ w σ i ∧w 0 . We get f i σ w σ i ∧ w 0 = −f i σ d(w σ ∧ w i )=−d(f i σ w σ ∧ w i )+df i σ ∧ w σ ∧ w i . (22) Since f i σ w σ ∧ w i ∈ Ω n 3 =Θ n 3 and π ∗ 4,3 (df i σ ∧ w σ ∧ w i )=h(df i σ ) ∧ w σ ∧ w i + p(df i σ ) ∧ w σ ∧ w i , it implies that τ n+1 4 .p n+1 3,1 (f i σ w σ i ∧ w 0 )=−d i f i σ ∧ w σ ∧ w 0 . (23) Now we consider all the factors containing w σ ij in the formula (16). Then we have f ij σ w σ ij ∧ w 0 = −d(f ij σ w σ i ∧ w j )+df ij σ ∧ w σ i ∧ w j . (24) This implies that τ n+1 4 .p n+1 3,1 (f ij σ w σ ij ∧ w 0 )=−d j f ij σ w σ i ∧ w 0 = d(d j f ij σ .w σ ∧ w i ) − d(d j f ij σ ) ∧ w σ ∧ w i . (25) Therefore we have τ n+1 5 .p n+1 4,1 .π ∗ 4,3 (f ij σ w σ ij ∧ w 0 )=d i d j f ij σ w σ ∧ w 0 . (26) Since (16), (23) and (26) we get []=τ n+1 5 .p n+1 4,1 .π ∗ 4,2 ()=(f σ − d i f i σ + d i d j f ij σ )w σ ∧ w 0 , where each function is defined on V 5 ⊂ J 5 Y . b) In the formula (18) we consider all the factors containing w ν i . Then we get f Ji σν w σ J ∧ w ν i ∧ w 0 = −f Ji σν w σ J ∧ d(w ν ∧ w i ) = f Ji σν d(w σ J ∧ w ν ∧ w i ) − f Ji σν dw σ J ∧ w ν ∧ w i = d(f Ji σν w σ J ∧ w ν ∧ w i ) − df Ji σν ∧ w σ J ∧ w ν ∧ w i + n  l=1 df Ji σν dy σ Jl ∧ dx l ∧ w ν ∧ w i . Therefore τ n+2 4 .p n+2 3,2 (f Ji σν w σ J ∧ w ν i ∧ w 0 )=− d i f Ji σν w σ J ∧ w ν ∧ w 0 − f Ji σν w σ Ji ∧ w ν ∧ w 0 . (27) Now we consider all the factors containing w ν ij in the formula (16). Then we have A Presentation of the Elements of the Quotient Sheaves Ω k r /Θ k r 279 f Jij σν w σ J ∧ w ν ij ∧ w 0 = −f Jij σν w σ J ∧ d(w ν i ∧ w j ) = d(f Jij σν w σ J ∧ w ν i ∧ w j ) − df Jij σν ∧ w σ J ∧ w ν i ∧ w j + n  l=1 f Jij σν dy σ Jl dx l ∧ w ν i ∧ w j τ n+2 4 .p n+2 3,2 −→ − d j f Jij σν w σ J ∧ w ν i ∧ w 0 − f Jij σν w σ Jj ∧ w ν i ∧ w 0 τ n+2 5 .p n+2 4,2 −→ d i d j f Jij σν w σ J ∧ w ν ∧ w 0 + d j f Jij σν w σ Ji ∧ w ν ∧ w 0 + d i f Jij σν w σ Jj ∧ w ν ∧ w 0 + f Jij σν w σ Jij ∧ w ν ∧ w 0 , (28) where τ n+2 4 .p n+2 3,2 (resp.τ n+2 5 .p n+2 4,2 ) are morphisms of sheaves. Since the formu- las (18), (27), (28) and the symmetry of indexes i, j we have []=τ n+2 5 .p n+2 4,2 .π ∗ 4,2 ()=((f J σν − d i f Ji σν + d i d j f Jij σν )w σ J +(−f Ji σν +2d j f Jij σν )w σ Ji + f Jij σν w σ Jij ) ∧ w ν ∧ w 0 , where each function is defined on V 5 ⊂ J 5 Y . c) We consider the factors containing w σ k−n J 1 in the formula (20). We have f J 1 J k−n−1 j 1 σ 1 σ k−n w σ 1 J 1 ∧ ∧ w σ k−n−1 J k−n−1 ∧ w σ k−n j 1 ∧ w 0 = − f J 1 J k−n−1 j 1 σ 1 σ k−n w σ 1 J 1 ∧ ∧ w σ k−n−1 J k−n−1 ∧ d(w σ k−n ∧ w j 1 ) =(−1) k−n f J 1 J k−n−1 j 1 σ 1 σ k−n d(w σ 1 J 1 ∧ ∧ w σ k−n−1 J k−n−1 ∧ w σ k−n ∧ w j 1 ) + k−n−1  t=1 (−1) k−n+t f J 1 J k−n−1 j 1 σ 1 σ k−n w σ 1 J 1 ∧ ∧ w σ t−1 J t−1 ∧ ∧ dw σ t J t ∧ w σ t+1 J t+1 ∧ ∧ w σ k−n−1 J k−n−1 ∧ w σ k−n ∧ w j 1 =(−1) k−n d(f J 1 J k−n−1 j 1 σ 1 σ k−n w σ 1 J 1 ∧ ∧ w σ k−n−1 J k−n−1 ∧ w σ k−n ∧ w j 1 ) − (−1) k−n df J 1 J k−n−1 j 1 σ 1 σ k−n ∧ w σ 1 J 1 ∧ ∧ w σ k−n−1 J k−n−1 ∧ w σ k−n ∧ w j 1 − k−n−1  t=1 n  l=1 (−1) k−n+t f J 1 J k−n−1 j 1 σ 1 σ k−n w σ 1 J 1 ∧ ∧ w σ t−1 J t−1 ∧ ∧ dy σ t J t l ∧ dx l ∧ w σ t+1 J t+1 ∧ ∧ w σ k−n−1 J k−n−1 ∧ w σ k−n ∧ w j 1 τ k 4 .p k 3,k−n −→ − d j 1 f J 1 J k−n−1 σ 1 σ k−n w σ 1 J 1 ∧ ∧ w σ k−n−1 J k−n−1 ∧ w σ k−n ∧ w 0 − k−n−1  t=1 f J 1 J k−n−1 σ 1 σ k−n w σ 1 J 1 ∧ ∧ w σ t−1 J t−1 ∧ w σ t J t j 1 ∧ ∧ w σ t+1 J t+1 ∧ ∧ w σ k−n−1 J k−n−1 ∧ w σ k−n ∧ w 0 . (29) We consider all the factors containing w σ k−n j 1 j 2 in the formula (20). We have 280 Nong Quoc Chinh f J 1 J k−n−1 j 1 j 2 σ 1 σ k−n w σ 1 J 1 ∧ ∧ w σ k−n−1 J k−n−1 ∧ w σ k−n j 1 j 2 ∧ w 0 τ k 4 .p k 3,k−n −→ − d j 2 f J 1 J k−n−1 j 1 j 2 σ 1 σ k−n w σ 1 J 1 ∧ ∧ w σ k−n−1 J k−n−1 ∧ w σ k−n j 1 ∧ w 0 − k−n−1  t=1 f J 1 J k−n−1 j 1 j 2 σ 1 σ k−n w σ 1 J 1 ∧ ∧ w σ t−1 J t−1 ∧ w σ t J t j 2 ∧ w σ t+1 J t+1 ∧ ∧ w σ k−n−1 J k−n−1 ∧ w σ k−n j 1 ∧ w 0 τ k 5 .p k 4,k−n d j 1 −→ d j 2 f J 1 J k−n−1 j 1 j 2 σ 1 σ k−n w σ 1 J 1 ∧ ∧ w σ k−n−1 J k−n−1 ∧ w σ k−n ∧ w 0 + k−n−1  h=1 d j 2 f J 1 J k−n−1 j 1 j 2 σ 1 σ k−n w σ 1 J 1 ∧ ∧ w σ h−1 J h−1 ∧ w σ h J h j 1 ∧ w σ h+1 J h+1 ∧ ∧ w σ k−n−1 J k−n−1 ∧ w σ k−n ∧ w 0 + k−n−1  t=1 d j 1 f J 1 J k−n−1 j 1 j 2 σ 1 σ k−n w σ 1 J 1 ∧ ∧ w σ t−1 J t−1 ∧ w σ t J t j 2 ∧ w σ t+1 J t+1 ∧ ∧ w σ k−n−1 J k−n−1 ∧ w σ k−n ∧ w 0 + k−n−1  h=1 h=t k−n−1  t=1 f J 1 J k−n−1 j 1 j 2 σ 1 σ k−n w σ 1 J 1 ∧ ∧ w σ h−1 J h−1 ∧ w σ h J h j 1 ∧ w σ h+1 J h+1 ∧ ∧ w σ t−1 J t−1 ∧ w σ t J t j 2 ∧ w σ t+1 J t+1 ∧ ∧ w σ k−n−1 J k−n−1 ∧ w σ k−n ∧ w 0 + k−n−1  t=1 f J 1 J k−n−1 j 1 j 2 σ 1 σ k−n w σ 1 J 1 ∧ ∧ w σ t−1 J t−1 ∧ w σ t J t j 1 j 2 ∧ w σ t+1 J t+1 ∧ ∧ w σ k−n−1 J k−n−1 ∧ w σ k−n ∧ w 0 . (30) Since the formulas (20), (29), (30) and the symmetry of indexes j 1 ,j 2 we have []=τ k 5 .p k 4,k−n .π ∗ 4,2 () =  (f J 1 J k−n−1 σ 1 σ k−n − d j 1 f J 1 J k−n−1 j 1 σ 1 σ k−n + d j 1 d j 2 f J 1 J k−n−1 j 1 j 2 σ 1 σ k−n )w σ 1 J 1 ∧ ∧ w σ k−n−1 J k−n−1 + k−n−1  t=1 (−f J 1 J k−n−1 j 1 σ 1 σ k−n +2d j 2 f J 1 J k−n−1 j 1 j 2 σ 1 σ k−n )w σ 1 J 1 ∧ ∧ w σ t−1 J t−1 ∧ w σ t J t j 1 ∧ w σ t+1 J t+1 ∧ ∧ w σ k−n−1 J k−n−1 + k−n−1  h=1 h=t k−n−1  t=1 f J 1 J k−n−1 j 1 j 2 σ 1 σ k−n w σ 1 J 1 ∧ ∧ w σ h−1 J h−1 ∧ w σ h J h j 1 ∧ w σ h+1 J h+1 ∧ ∧ w σ t−1 J t−1 ∧ w σ t J t j 2 ∧ w σ t+1 J t+1 ∧ ∧ w σ k−n−1 J k−n−1 [...]... Presentation of the Elements of the Quotient Sheaves Ωk /Θk r r k−n−1 σ 281 σ σ1 σt t−1 t+1 J1 Jk−n−1 fσ1 σk−n j1 j2 wJ1 ∧ ∧ wJt−1 ∧ wJt j1 j2 ∧ wJt+1 ∧ + t=1 σ k−n−1 ∧ wJk−n−1 ∧wσk−n ∧ w0 , where each function is defined on V5 ⊂ J 5 Y References 1 I M Anderson, Aspect of the inverse problem to the calculus of variation, Arch Math 24 (1988) 181–202 2 I M Anderson and T Duchamp, On the existence of. .. existence of global variational principles, Amer Math J 102 (1980) 781–868 3 N Q Chinh, Sheaf of contact forms , East-West J Math 4 (2002) 41–55 4 P Dedecker and W M Tulczyjew, Spectral Sequences and the Inverse Problem of the Calculus of Variations, Internat Coll on Diff Geom Methods in Math Physics, Sept., 1979 5 D Krupka, Variational Sequences on Finite Order Jet Spaces, World Scientific, Singapore, (1990)... Variational Sequences on Finite Order Jet Spaces, World Scientific, Singapore, (1990) 236–254 6 F Takens, A global version of the inverse problem of the calculus of variations, J Diff Geom 14 (1979) 543–562 7 W M Tulczyjew, The Euler-Lagrange Resolution, Internat Coll on Diff Geom Methods in Math Physics, Sept., 1979 . Ω N r → 0. 1. Introduction The notion of variational bicomplexes was introduced in studying the problem of characterizing the kernel and image of Euler-Lagrange mapping in the calcu- lus of variations dΩ k−1 r(c) +Ω k r(c) . In [3] and [5] the authours proved the softness of the sheaves Θ k r , considered the quotient sheaves Ω k r /Θ k r , and they obtained the following short exact sequence 0. Helmiholtz-Sonin class of (n +2)-forms. Below we give a concrete presentation of the elements in Ω k r /Θ k r for all pos- itive integer k satisfying n +1≤ k ≤ P in the case of r =1andr =2. 3. TheCaseofr =1 Theorem