DIFFERENTIAL GEOMETRY AND APPLICATIONS Satellite Conference of ICM in Berlin, Aug. 10 – 14, 1998, Brno Masaryk University in Brno (Czech Republic) 1999, 493–502

A REPRESENTATION OF THE 1ST-ORDER VARIATIONAL SEQUENCE IN FIELD THEORY

JANA KAŠPAROVÁ (KRPCOVÁ)

Abstract. Variational sequence of order r introduced by Krupka is defined on the r-jet prolongation of a fibered manifold π : Y → X, as a quotient sequence of the de Rham sequence of sheaves. In this paper we describe explicitly the elements of the 1st-order variational sequence over an n-dimensional manifold X.

1. Introduction

The theory of variational bicomplexes and variational sequences is motivated by the idea to represent the Euler-Lagrange mapping as one of morphisms of a suitable ”variational bicomplex” or the ”variational sequence”. This morphism is derived from the exterior derivative operator acting on differential forms.The idea that the Euler-Lagrange mapping should be obtained from the exterior derivative operator is appeared in papers of Lepage and Dedecker for the first time and it has been developed by Anderson [1], Krupka [6], [8], [9], Vinogradov[13], Vitolo [14],

[15] and others. The results of this study are applied in the calculus of variations (see [9]), in the theory of differential equations (see [2]), in the differential geometry and in further branches of mathematics.

In this paper we investigate the variational sequence on the finite order jet space introduced by Krupka (see [12]), that is defined as a quotient sequence of de Rham sequence of sheaves over some special subsequence of de Rham sequence on the finite order jet space. This subsequence is formed by means of so called contact and strongly contact forms. The structure of contact and strongly contact forms was precisely described in [4], [12], where it is shown that they are generated by contact forms “ω” and “dω”. The exactness of the variational sequence is discussed in [12]; it is shown the variational sequence is an acyclic resolution of the constant sheaf using the abstract de Rham theorem.

1991 Mathematics Subject Classification. 58E99, 58A12, 58A15, 58A20.

Key words and phrases. Fibered manifold, contact form, strongly contact form, trace decomposition problem, q-equation, sheaf, variational sequence.

Research supported by the Grant No.201/98/0853 of the Czech Grant Agency, and Project No. VS 96003 (“Global Analysis”) of the Ministry of Education, Youth and Sports of the Czech Republic.

This paper is in final form and no version of it will be submitted for publication elsewhere.

In Section 2 we state propositions which are used in Section 3. All the statements are given without proofs. The reader can find them in [4], [11], the trace decomposition problem and to problems of q-equations are defined in [11].

Section 3 is the main part of this paper. It contains the explicit description of the elements (Theorem (3.2)) of the variational sequence on J^{1}Y in the field theory. This calculation has been recently done in mechanics and we can see it in [8].

Throughout this paper we consider a fibered manifold π : Y → X, where dim X = n, dim Y = n + m, the fibered chart (W, ψ), ψ = (x^{i}, y^{σ}), 1 ≤ i ≤ n, 1 ≤ σ ≤ m, on Y associated with a fibered chart (U, φ), φ = (x^{i}), 1 ≤ i ≤ n, and the fibered chart (W^{r}, ψ^{r}) on J^{r}Y associated with (W, ψ) such that for each J_{x}^{r}γ ∈ J^{r}Y, ψ^{r}(J_{x}^{r}γ) = (x^{i}, y^{σ}, y_{i_{1}}^{σ}, y_{i_{1}i_{2}}^{σ}, ..., y_{i_{1}i_{2}...i_{p}}^{σ}, ..., y_{i_{1}i_{2}...i_{r}}^{σ}), where ∀p = 1, 2, ..., r, 1 ≤ i_{1}, i_{2}, ..., i_{p} ≤ n, i_{1} ≤ i_{2} ≤ ... ≤ i_{p}, y_{i_{1}i_{2}...i_{p}}^{σ}(J_{x}^{r}γ) = D_{i_{1}i_{2}...i_{p}}(y^{σ}∘γ∘φ^{−1})(φ(x)), and W^{r} = (π^{r,0})^{−1}(W). A ring of smooth functions on W^{r} is denoted by Ω_{0}^{r}W, Ω_{q}^{r}W is the Ω_{0}^{r}W -module of smooth differential q-forms on W^{r}.The mapping horizontalization, defined on Ω_{q}^{r}W is denoted by h. The total derivative of f ∈ Ω_{0}^{r}W with respect to x^{i} is denoted by d_{i}f, using fibered system (W, ψ), ψ = (x^{i}, y^{σ}). A k-contact component of the form ρ ∈ Ω_{q}^{r}W is denoted by p_{k}ρ where k positive integer. We mean symetrization, resp. antisymetrization of the coeficient X_{j_{1},j_{2}}

in indices (j_{1}, j_{2}) if we write sym_{(j_{1},j_{2})} X_{j_{1},j_{2}}, resp. alt_{(j_{1},j_{2})}X_{j_{1},j_{2}}.

E_{q}^{r} is the vector space of systems of real numbers A = A_{i_{1}i_{2}...i_{q}}^{j_{1}j_{2}...j_{r}}, where 1 ≤ j_{1}, j_{2}, ..., j_{r}, i_{1}, i_{2}, ..., i_{q} ≤ n. The standard summation convention is used.

2. Prerequisites

It si well known that a q-form ρ ∈ Ω_{q}^{r}W is called contact if and only if J^{r}γ^{∗}ρ = 0 for every smooth section γ of Y defined on an open subset of π (W) and a q-form ρ ∈ Ω_{q}^{r}W is called strongly contact if p_{q−n}ρ = 0. The next statetment describes the structure of contact and strongly cantact forms on fibered manifolds.

2.1. Proposition. Let W ⊂ Y be an open set and let ρ ∈ Ω_{q}^{r}W be a form. Let (V, ψ), ψ = (x^{i}, y^{σ}) be a fibered chart such that V ⊂ W.

(a) Let q = 1. Then ρ is contact if and only if

ρ = Φ_{σ}^{J}ω_{J}^{σ}

for some functions Φ_{σ}^{J} : V^{r} → R, where |J| ≤ r − 1.

(b) Let 2 ≤ q ≤ n. Then ρ is contact if and only if

ρ = ω_{J}^{σ} ∧ Φ_{σ}^{J} + dω_{I}^{σ} ∧ Ψ_{σ}^{I}

where Φ_{σ}^{J} (resp. Ψ_{σ}^{I} ) are some (q − 1)-forms (resp. (q − 2)-forms on V^{r} and |J| ≤ r − 1, |I| = r − 1.

(c) Let n + 1 ≤ q ≤ dim J^{r}Y. Then ρ is strongly contact if and only if

ρ = ω_{J_{1}}^{σ_{1}} ∧ ω_{J_{2}}^{σ_{2}} ∧ ... ∧ ω_{J_{p}}^{σ_{p}} ∧ dω_{I_{1}}^{ν_{1}} ∧ dω_{I_{2}}^{ν_{2}} ∧ ... ∧ dω_{I_{s}}^{ν_{s}} ∧ Φ_{σ_{1}σ_{2}...σ_{p}ν_{1}ν_{2}...ν_{s}}^{J_{1}J_{2}...J_{p}I_{1}I_{2}...I_{s}}

for some (q − p − 2s)-forms Φ_{σ_{1}σ_{2}...σ_{p}ν_{1}ν_{2}...ν_{s}}^{J_{1}J_{2}...J_{p}I_{1}I_{2}...I_{s}} on V^{r} such that p + s ≥ q − n + 1, |J_{1}| , |J_{2}| , ..., |J_{p}| ≤ r − 1, |I_{1}| , |I_{2}| , ..., |I_{s}| = r − 1.

For more complete discussion of the theory of contact and strongly contact forms and for the proofs, the reader is referred to [10] and [12].

The exposition of the q − equation theory and proofs of the following propositions is given in [11], [12], as well as the definition of linear mappings tr_{m}, q_{m}, where m is some positive integer. You can also find in [12] the proof of the fact, that for every system of real number A = A_{i_{1}i_{2}...i_{q}}^{I_{1}I_{2}...I_{m−1}I_{m}I_{m+1}...I_{r}}, I_{m} = (j_{1}^{m}j_{1}^{m}...j_{r_{m}}^{m}), m = 1, 2, ..., p, 1 ≤ j_{1}^{m}, j_{2}^{m}, ..., j_{r_{m}}^{m} ≤ n, 1 ≤ i_{1}, i_{2}, ..., i_{q} ≤ n symmetric in superscripts entering the multiindex I_{m} of the length r_{m}, m = 1, 2, ..., p, and antisymmetric in subscripts, r_{1}, r_{2}, ..., r_{p}, q, n > 0 are integers, the following equations are satisfied:

A = q _{m}tr_{m}A + tr_{m} q_{m}A (1) q _{m}q_{m}A = 0 (2) tr_{m}tr_{m}A = 0 (3)

for any m = 1, 2, ..., p.

2.2. Proposition. (a) Let r, q, n > 0 be integers and let A_{i_{1}i_{2}...i_{q}}^{j_{1}j_{2}...j_{r}} ∈ E_{q}^{r} 1 ≤ j_{1}, j_{2}, ..., j_{r}, i_{1}, i_{2}, ..., i_{q} ≤ n be a system of real numbers symmetric in the superscripts, antisymmetric in the subscripts. Then the equation

qX = A (4)

has a solution X if and only if

qA = 0 (5)

(b) Let p, q, n, r > 0 be integers, 1 ≤ p ≤ q ≤ n. Let

Y = Y _{i_{p+1}i_{p+2}...i_{p+q}}^{J_{1}J_{2}...J_{m−1}J_{m}J_{m+1}...J_{p}},
1 ≤ i_{p+1}, i_{p+2}, ..., i_{p+q} ≤ n,
|J_{i}| = r, 1 ≤ i ≤ p

be a traceless system of real numbers symmetric in the superscripts entering each of the multi-indices J_{1}, J_{2}, ..., J_{m−1}, J_{m}, J_{m+1}, ..., J_{p}. If X satisfies the system of algebraic equations

sym_{(J_{1}j_{1})} sym_{(J_{2}j_{2})} . .. sym_{(J_{p}j_{p})} alt_{(i_{1},i_{2},...,i_{p+q})} (6)

X_{i_{p+1}i_{p+2}...i_{p+q}}^{J_{1}J_{2}...J_{m−1}J_{m}J_{m+1}...J_{p}}δ_{i_{1}}^{j_{1}}δ_{i_{2}}^{j_{2}}...δ_{i_{p}}^{j_{p}} = 0

then

X_{i_{p+1}i_{p+2}...i_{p+q}}^{J_{1}J_{2}...J_{m−1}J_{m}J_{m+1}...J_{p}} = 0 (7)

The trace decomposition problem is formulated and proved in [11]. Now we give the statement, which will be used in the next chapter.

2.3. Proposition. (a) Let r, q, n be a positive intgers, r ≤ q, and let A ∈ E_{q}^{r}, A = A_{i_{1}i_{2}...i_{q}}^{j_{1}j_{2}...j_{r}}, 1 ≤ j_{1}, j_{2}, ..., j_{r}, i_{1}, i_{2}, ..., i_{q} ≤ n be a system of real numbers. Then there exist a traceless system B = B_{i_{1}i_{2}...i_{q}}^{j_{1}j_{2}...j_{r}} and systems

B_{(s)}^{(p)} = B_{(s)i_{1}i_{2}...i_{q}}^{(p)j_{1}j_{2}...j_{r}}, where 1 ≤ p ≤ r, 1 ≤ s ≤ q, such that

(8)

A_{i_{1}i_{2}...i_{q}}^{j_{1}j_{2}...j_{r}} =

B_{i_{1}i_{2}...i_{q}}^{j_{1}j_{2}...j_{r}} + ∑_{a=1}^{q} ∑_{b=1}^{r} δ_{i_{a}}^{j_{b}}B_{(a)i_{1}i_{2}...i_{a−1}i_{a+1}...i_{q}}^{(b)j_{1}j_{2}...j_{b−1}j_{b+1}...j_{r}}

The system B is unique.

(b) Let r, q, n be a positive intgers, r ≤ q and let A ∈ E_{q}^{r}. There exist unique δ^{k}-generated, relatively traceless systems A^{(}r) ∈ E_{q}^{r}, where 0 ≤ k ≤ r − 1 and a δ^{r}-gererated system A^{(r)} ∈ E_{q}^{r}, such that

A = A^{(0)} + A^{(1)} + A^{(2)} + ... + A^{(r)} (9)

3. Variational sequence of the 1st order

The variational sequence on the finite order jet space is defined as a quotient sequence of de Rham sequence of sheaves on J^{r}Y. It has been shown in [7] the exactness of the variational sequence and we can find there also the detailed definition of this concept.

3.1. Definition. Let

0 → R_{Y} → Ω_{0}^{r} → Ω_{1}^{r} → Ω_{2}^{r} → ... → Ω_{n}^{r} → Ω_{n+1}^{r} → Ω_{n+2}^{r} → ... (10)

... → Ω_{M}^{r} → Ω_{M+1}^{r} → ... → Ω_{N}^{r} → Ω_{N+1}^{r} → 0

be de Rham sequence of sheaves defined on J^{r}Y, where

M = m ( \begin{matrix}n + r − 1 \\ n \end{matrix} ) + 2 (n − 1) + 1, (11)

N = dim J^{r}Y = n + m ( \begin{matrix}n + r \\ n \end{matrix} ) . (12)

Let

0 → Θ_{1}^{r} → Θ_{2}^{r} → ... → Θ_{n}^{r} → Θ_{n+1}^{r} → Θ_{n+2}^{r} → ... → Θ_{M}^{r} → 0 (13)

be the exact subsequence of (10), Θ_{q}^{r} ⊂ Ω_{q}^{r},, Θ_{q}^{r} = Ω_{q}^{r(c)} + dΩ_{q−1}^{r(c)}, where

Ω_{q}^{r(c)} = [ \begin{matrix}ker h & 1 ≤ q ≤ n \\ ker p_{q−n} & q ≥ n + 1 \end{matrix}

h, p_{q−n}, are morphisms of sheaves induced by operators h, p_{q−n} acting on the differential forms on fibered manifold, d is the morphism of sheaves induced by a exterior derivative operator d acting on the differential forms on a manifold. So here we admit a denotation abuse, but we prefer this denomination since it leads to no misunderstanding. So Ω_{q}^{r(c)} are just the sheaves of contact and strongly contact forms described by Proposition (2.1).

The variational sequence of order r is defined as a the quotient sequence of (10), where we consider (13) as a subsequence:

0 → R_{Y} → Ω_{0}^{r}→_{E_{0}}Ω_{1}^{r}/Θ_{1}^{r}→_{E_{1}}Ω_{2}^{r}/Θ_{2}^{r}→_{E_{2}}...

... →_{E_{n−1}}Ω_{n}^{r}/Θ_{n}^{r}→_{E_{n}}Ω_{n+1}^{r}/Θ_{n+1}^{r} →_{E_{n+1}}Ω_{n+2}^{r}/Θ_{n+2}^{r} →_{E_{n+2}}...

... → Ω_{M}^{r}/Θ_{M}^{r} →_{E_{M}}Ω_{M+1}^{r} →_{E_{M+1}}... → Ω_{N−1}^{r} →_{E_{N−1}}Ω_{N}^{r} → 0 (14)

Let us denote α_{q} : Ω_{q}^{r} → Ω_{q}^{r}/Θ_{q}^{r}, q = 0, 1, ..., N the quotient mappings between the sheaves, E_{q} : Ω_{q}^{r}/Θ_{q}^{r} → Ω_{q+1}^{r}/Θ_{q+1}^{r}, q = 0, 1, ..., N − 1 the sheaf morphisms between the quotient sheaves. We shall describe elements of quotient sheaves Ω_{q}^{r}/Θ_{q}^{r}, q = 0, 1, ..., M (it is clear,that other cases are trivial, the appropriate morphisms α_{q} being the identity morphisms, for every q ≥ M + 1).

3.2. Theorem. Let π : Y → X be a fibered manifold, dim X = n, dim Y = m, (V, ψ), ψ = (x^{i}, y^{σ}) be a fibered chart on Y.

(a) Let q ≤ n and let

(15) ρ = ∑_{k=0}^{q} \frac{1}{k! (q − k)!}A_{σ_{1}σ_{2}...σ_{k},i_{k+1}i_{k+2}...i_{q}}^{s_{1}s_{2}...s_{k}}dy_{s_{1}}^{σ_{1}} ∧ dy_{s_{2}}^{σ_{2}} ∧ ... ∧ dy_{s_{k}}^{σ_{k}}

∧dx^{i_{k+1}} ∧ dx^{i_{k+2}} ∧ ... ∧ dx^{i_{q}}

∑_{k=1}^{q} ∑_{l=0}^{q−k} \frac{1}{k!l! (q − k − l)!}B_{σ_{1}σ_{2}...σ_{k},ν_{k+1}ν_{k+2}...ν_{k+l},i_{k+l+1}i_{k+l+2}...i_{q}}^{s_{k+1}s_{k+2}...s_{k+l}}

ω^{σ_{1}} ∧ ω^{σ_{2}} ∧ ... ∧ ω^{σ_{k}} ∧ dy_{s_{k+1}}^{ν_{k+1}} ∧ dy_{s_{k+2}}^{ν_{k+2}} ∧ ... ∧ dy_{s_{k+l}}^{ν_{k+l}} ∧

dx^{i_{k+l+1}} ∧ dx^{i_{k+l+2}} ∧ ... ∧ dx^{i_{q}}

be a q-form defined on V^{1} ⊂ J^{1}Y. Then [ρ] can be represented by a horizontal q-form on V^{2} ⊂ J^{2}Y

498 J. KAŠPAROVÁ (KRPCOVÁ)

\frac{1}{q!}X_{i_{1}i_{2}...i_{q}}dx^{i_{1}} ∧ dx^{i_{2}} ∧ ... ∧ dx^{i_{q}} (16)

where

X_{i_{1}i_{2}...i_{q}} = alt_{(i_{1},i_{2},...,i_{q})} ∑_{k=0}^{q} \binom{q}{k}A_{σ_{1}σ_{2}...σ_{k},i_{k+1}i_{k+2}...i_{q}}^{s_{1}s_{2}...s_{k}}y_{s_{1}i_{1}}^{σ_{1}}y_{s_{2}i_{2}}^{σ_{2}}...y_{s_{k}i_{k}}^{σ_{k}} (17)

(b) Let

(18) ρ = ∑_{m=0}^{n} \frac{1}{m! (n − m)!}A_{σ_{1},ν_{1}ν_{2}...ν_{m},i_{m+1}i_{m+2}...i_{n}}^{s_{1}s_{2}...s_{m}}

ω^{σ_{1}} ∧ dy_{s_{1}}^{ν_{1}} ∧ dy_{s_{2}}^{ν_{2}} ∧ ... ∧ dy_{s_{m}}^{ν_{m}} ∧ dx^{i_{m+1}} ∧ dx^{i_{m+2}} ∧ ... ∧ dx^{i_{n}} +

∑_{m=0}^{n} \frac{1}{(m + 1)! (n − m)!}A_{σ_{1}ν_{l_{1}} ν_{l_{2}} ...ν_{lm} ,i_{m+1}i_{m+2}...i_{n}}^{s_{1}s_{l_{1}} s_{l_{2}} ...s_{lm}}

dy_{s_{1}}^{σ_{1}} ∧ dy_{s_{l_{1}}}^{ν_{l_{1}}} ∧ dy_{s_{l_{2}}}^{ν_{l_{2}}} ∧ ... ∧ dy_{s_{lm}}^{ν_{lm}} ∧ dx^{i_{m+1}} ∧ dx^{i_{m+2}} ∧ ... ∧ dx^{i_{n}} +

∑_{k=2}^{n+1} ∑_{m=0}^{n+1−k} \frac{1}{k!m! (n + 1 − k − m)!}B_{ν_{1}ν_{2}...ν_{k},σ_{1}σ_{2}...σ_{m},i_{m+1}i_{m+2}...i_{n+1−k}}^{s_{1}s_{2}...s_{m}}

ω^{ν_{1}} ∧ ω^{ν_{2}} ∧ ... ∧ ω^{ν_{k}} ∧ dy_{s_{1}}^{σ_{1}} ∧ dy_{s_{2}}^{σ_{2}} ∧ ... ∧ dy_{s_{m}}^{σ_{m}} ∧ dx^{i_{m+1}} ∧ dx^{i_{m+2}} ∧ ... ∧ dx^{i_{n+1−k}}

be a (n+1)-form defined on V^{1} ⊂ J^{1}Y. Then [ρ] can be represented by a 1-contact, π^{3,0}-horizontal (n + 1)-form on V^{3} ⊂ J^{3}Y

(19)  [ρ] = \frac{1}{n!} (X_{σ_{1},i_{1}i_{2}...i_{n}} − d_{j}X_{σ_{1},i_{1}i_{2}...i_{n}}^{j})
ω^{σ_{1}} ∧ dx^{i_{1}} ∧ dx^{i_{2}} ∧ ... ∧ dx^{i_{n−1}} ∧ dx^{i_{n}}

where X_{σ_{1},i_{1}i_{2}...i_{n}} = (20)

alt_{(i_{1},i_{2},...,i_{n})} ∑_{m=0}^{n} \binom{n}{m}A_{σ_{1},ν_{1}ν_{2}...ν_{m},i_{m+1}i_{m+2}...i_{n}}^{s_{1}s_{2}...s_{m}}y_{s_{1}i_{1}}^{ν_{1}}y_{s_{2}i_{2}}^{ν_{2}}...y_{s_{m}i_{m}}^{ν_{m}},

X_{σ_{1},i_{1}i_{2}...i_{n}}^{s} =

alt_{(i_{1},i_{2},...,i_{n})} ∑_{m=0}^{n} \binom{n}{m}A_{σ_{1}ν_{l_{1}} ν_{l_{2}} ...ν_{lm} ,i_{m+1}i_{m+2}...i_{n}}^{ss_{l_{1}} s_{l_{2}} ...s_{lm}}y_{s_{l_{1}} i_{1}}^{ν_{l_{1}}}y_{s_{l_{2}} i_{2}}^{ν_{l_{2}}}...y_{s_{lm} i_{m}}^{ν_{lm}}

(c) Let q = n + l where 2 ≤ l ≤ M − n. Let

ρ = ∑_{m=0}^{q−l} ∑_{k=0}^{l} \frac{1}{k! (m + l − k)! (n − m)!} (21)

A_{σ_{1}σ_{2}...σ_{k},ν_{k+1}ν_{k+2}...ν_{l}ν_{l_{1}} ν_{l_{2}} ...ν_{lm} ,i_{m+1}i_{m+2}...i_{n}}^{s_{k+1}s_{k+2}...s_{l}s_{l_{1}} s_{l_{2}} ...s_{lm}}

ω^{σ_{1}} ∧ ω^{σ_{2}} ∧ ... ∧ ω^{σ_{k}} ∧ dy_{s_{k+1}}^{ν_{k+1}} ∧ dy_{s_{k+2}}^{ν_{k+2}} ∧ ... ∧ dy_{s_{l}}^{ν_{l}}

∧dy_{s_{l_{1}}}^{ν_{l_{1}}} ∧ dy_{s_{l_{2}}}^{ν_{l_{2}}} ∧ ... ∧ dy_{s_{lm}}^{ν_{lm}} ∧ dx^{i_{m+1}} ∧ dx^{i_{m+2}} ∧ ... ∧ dx^{i_{n}} +

∑_{k=l+1}^{q} ∑_{m=0}^{q−k} \frac{1}{k!m! (q − k − m)!}B_{ν_{1}ν_{2}...ν_{k},σ_{1}σ_{2}...σ_{m},i_{m+1}i_{m+2}...i_{q−k}}^{s_{1}s_{2}...s_{m}}

ω^{ν_{1}} ∧ ω^{ν_{2}} ∧ ... ∧ ω^{ν_{k}} ∧ dy_{s_{1}}^{σ_{1}} ∧ dy_{s_{2}}^{σ_{2}} ∧ ... ∧ dy_{s_{m}}^{σ_{m}} ∧

dx^{i_{m+1}} ∧ dx^{i_{m+2}} ∧ ... ∧ dx^{i_{q−k}}

be a q-form defined on V^{1} ⊂ J^{1}Y. Then [ρ] can be represented by an l-contact, π^{3,0}-horizontal q-form on V^{3} ⊂ J^{3}Y

[ρ] = \frac{1}{l!n!} (X_{σ_{1}σ_{2}...σ_{l},i_{1}i_{2}...i_{n}} − d_{j} alt_{(σ_{1}σ_{2},...,σ_{l})} X_{σ_{1}σ_{2}...σ_{l−1},σ_{l},i_{1}i_{2}...i_{n}}^{j}) (22)

ω^{σ_{1}} ∧ ω^{σ_{2}} ∧ ... ∧ ω^{σ_{l}} ∧ dx^{i_{1}} ∧ dx^{i_{2}} ∧ ...dx^{i_{n}} +

\frac{1}{(l − 1)!n!} \frac{2}{l} sym_{(σ_{l−1},σ_{l})} X_{σ_{1}σ_{2}...σ_{l−1},σ_{l},i_{1}i_{2}...i_{n}}^{s_{l}}

ω^{σ_{1}} ∧ ω^{σ_{2}} ∧ ... ∧ ω^{σ_{l−1}} ∧ ω_{s_{l}}^{σ_{l}} ∧ dx^{i_{1}} ∧ dx^{i_{2}} ∧ ... ∧ dx^{i_{n}}

∑_{k=0}^{l−2} \frac{1}{k! (l − k)!n!}X_{σ_{1}σ_{2}...σ_{k},ν_{k+1}ν_{k+2}...ν_{l},i_{1}i_{2}...i_{n}}^{s_{k+1}s_{k+2}...s_{l}}

ω^{σ_{1}} ∧ ω^{σ_{2}} ∧ ... ∧ ω^{σ_{k}} ∧ ω_{s_{k+1}}^{ν_{k+1}} ∧ ω_{s_{k+2}}^{ν_{k+2}} ∧ ... ∧ ω_{s_{l}}^{ν_{l}} ∧ dx^{i_{1}} ∧ dx^{i_{2}} ∧ ... ∧ dx^{i_{n}}

where

X_{σ_{1}σ_{2}...σ_{k},ν_{k+1}ν_{k+2}...ν_{l},i_{1}i_{2}...i_{n}}^{s_{k+1}s_{k+2}...s_{l}} = (23)

alt_{(i_{1},i_{2},...,i_{n})} ∑_{m=0}^{n} ( \begin{matrix}n \\ m \end{matrix} ) A_{σ_{1}σ_{2}...σ_{k},ν_{k+1}ν_{k+2}...ν_{l}ν_{l_{1}} ν_{l_{2}} ...ν_{lm} ,i_{m+1}i_{m+2}...i_{n}}^{s_{k+1}s_{k+2}...s_{l}s_{l_{1}} s_{l_{2}} ...s_{lm}}

y_{s_{l_{1}} i_{1}}^{ν_{l_{1}}}y_{s_{l_{2}} i_{2}}^{ν_{l_{2}}}...y_{s_{lm} i_{m}}^{ν_{lm}}

1 ≤ k ≤ l

Proof. We prove the equivalence [ρ] = 0 ⇔ ρ ∈ Θ_{q}^{r} in parts (a), (b), and (c). This yields the fact, that the forms (16), (19), (22), represent just the elements of the quotient sheaves in question. This representation is considered up to an isomorphism.

(a) Putting [ρ] = 0, some coefficients become immediately equal to zero and at the same time we obtain a system of q-equation. We have used the fact that the equation [ρ] = 0 is a polynomial in “ y_{si}^{σ}” here. We solve the q-equation by means of Proposition (2.3) throwing off the δ-generated part of some coefficients “A”. We can do it, because δ^{1}-generated part of the coefficient “A” plays no role in qequation because of antisymmetry properties of coefficients. So only the traceless coefficients enter q-equations and Proposition (2.2)(b) can be applied since all its assumptions are satisfied. Finally, it is easy to see that ρ ∈ Θ_{q}^{r}, using Proposition (2.1). Further, let us have ρ ∈ Θ_{q}^{r}. Using Proposition (2.1) and the fact that the set of contact forms is endowed with a structure of the ideal (in the exterior algebra of forms on J^{1}Y) we obtain [ρ] = 0.

(b) The first part of the proof is obtained by the similar procedure as in (a). Proposition (2.2)(a) and equations (1), (2), (3) must be applied. We must solve the cases l + 1 ≤ n − m, l + 1 > n − m, apart.

Further, let us have an element ρ ∈ Θ_{n+1}^{1}. We use the structure of the sheaf Θ_{n+1}^{1} and the linearity of a sheaf morphism α_{n+1} here (see (19), (20)). First, let us have a strongly contact (n + 1)-form; applying α_{n+1} to it we get immediately zero as it is claimed. Second, let κ ∈ θ_{n}^{1} be a contact n-form. Proposition (2.1)(b) provides

dκ = dω^{σ} ∧ Φ_{σ} − ω^{σ} ∧ dΦ_{σ} + dω^{σ} ∧ dΨ_{σ},

where Φ_{σ} is some (n − 1)-form and Ψ_{σ} is some (n − 2)-form on V^{1}. We want to show that [dκ] = 0. Let us take the form dω^{σ} ∧ Φ_{σ} − ω^{σ} ∧ dΦ_{σ}. The coefficients of the form Φ_{σ} can be decomposed by Proposition (2.3) and thus the form Φ_{σ} is splitted into two parts. The first one, denoted Φ′_{σ}, is generated by forms ω and dω and the second one Φ′′_{σ} is of the form

Φ′′_{σ} = ∑_{k=0}^{n−1} Φ_{σ,σ_{1}σ_{2}...σ_{k},i_{k+1}i_{k+2}...i_{n−1}}^{s_{1}s_{2}...s_{k}}dy_{s_{1}}^{σ_{1}} ∧ dy_{s_{2}}^{σ_{2}} ∧ ... ∧ dy_{s_{k}}^{σ_{k}}

∧dx^{i_{k+1}} ∧ dx^{i_{k+2}} ∧ ... ∧ dx^{i_{n−1}}

where all coefficients Φ_{σ,σ_{1}σ_{2}...σ_{k},i_{k+1}i_{k+2}...i_{n−1}}^{s_{1}s_{2}...s_{k}} are traceless. It is clear that

dω^{σ} ∧ Φ′_{σ} − ω^{σ} ∧ dΦ′_{σ} ∈ ker p_{1}.

using Proposition (2.3) again. Further, we have

α_{n+1}(dω^{σ} ∧ Φ′′_{σ} − ω^{σ} ∧ dΦ′′_{σ}) = 0,

applying antisymmetry properties of some coefficients. Now let us concern with the form dω^{σ} ∧ dΨ_{σ}. By the same arguments as above we can decompose the form Ψ_{σ} to the part Ψ′_{σ}, which is generated by ω and dω and the second part Ψ′′_{σ} which has a form

Ψ′′_{σ} = ∑_{k=0}^{n−1} \frac{1}{k! (n − 1 − k)!}Ψ′′_{σ,σ_{1}σ_{2}...σ_{k},i_{k+1}i_{k+2}...i_{n−1}}^{s_{1}s_{2}...s_{k}}

dy_{s_{1}}^{σ_{1}} ∧ dy_{s_{2}}^{σ_{2}} ∧ ... ∧ dy_{s_{k}}^{σ_{k}} ∧ dx^{i_{k+1}} ∧ dx^{i_{k+2}} ∧ ... ∧ dx^{i_{n−1}}

where Ψ′′_{σ,σ_{1}σ_{2}...σ_{k},i_{k+1}i_{k+2}...i_{n−1}}^{s_{1}s_{2}...s_{k}} are traceless. It is obvious that α_{n+1}(dω^{σ} ∧ dΨ′_{σ}) = 0 by Proposition (2.3), the conclusion α_{n+1}(dω^{σ} ∧ Ψ′′_{σ}) = 0 is obtained by the direct calculation, applying antisymmetry properties of coefficients.

(c) The first part of the equivalence [ρ] = 0 ⇔ ρ ∈ Θ_{q}^{r} is obtained by the similar way as in (b). So let us have ρ ∈ Θ_{q}^{r}. We use the linearity of the sheaf morphism α_{q}, q = n+2, n+3, ..., M. First, let us have a strongly contact q-form, where q = n + 2, n + 3, ..., M. Its class is equal to zero as can be shown by a straightforward calculation. Second, let κ ∈ θ_{q−1}^{1} be a strongly contact (q − 1)- form. Using Proposition (2.1) we see that

κ = ω^{σ_{1}} ∧ ω^{σ_{2}} ∧ ... ∧ ω^{σ_{p}} ∧ dω^{ν_{1}} ∧ dω^{ν_{2}} ∧ ... ∧ dω^{ν_{k}}
∧Φ_{σ_{1}σ_{2}...σ_{p},ν_{1}ν_{2}...ν_{k}}

where p + k = l and Φ_{σ_{1}σ_{2}...σ_{p},ν_{1}ν_{2}...ν_{k}} are some (q − 1 − p − 2k)-forms on V^{1}. We show that [dκ] = 0 by the similar method as in (b), but we must solve the cases l = 2, p = 0, 1, 2; l = 3, p = 0, 1, 2, 3; l = 4, 5, ..., M − n, p = 0, p = 1, 2, ..., l − 3, p = l − 2, p = l − 1, p = l apart.

3.3. Remark. Elements of Ω_{n}^{1}/Θ_{n}^{1} represent some Lagrangians of the order two, affine in “y_{si}^{σ}”. We can see that the morphism of sheaves

E_{n} : Ω_{n}^{1}/Θ_{n}^{1} → Ω_{n+1}^{1}/Θ_{n+1}^{1}

can be locally represented by the following equation for any form ρ described by (15), q = n and X_{i_{1}i_{2}...i_{n}} described by (17):

E_{n}α_{n} (ρ) = α_{n+1} (dρ) = (24)

\frac{1}{n!} (\frac{∂X_{i_{1}i_{2}...i_{n}}}{∂y^{σ}} − d_{j} \frac{∂X_{i_{1}i_{2}...i_{n}}}{∂y_{j}^{σ}} + d_{j}d_{i} \frac{∂X_{i_{1}i_{2}...i_{n}}}{∂y_{ji}^{σ}} ) ω^{σ} ∧ dx^{i_{1}} ∧ dx^{i_{2}} ∧ ...dx^{i_{n}}

and thus the morphism of sheaves E_{n} can be represented as the Euler-Lagrange mapping which assigns to each Lagrangian \frac{1}{n!}X_{i_{1}i_{2}...i_{n}}dx^{i_{1}} ∧dx^{i_{2}} ∧...dx^{i_{n}} its Euler-Lagrange form.

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Institute of Mathematics, Silesian University at Opava Bezručovo nám. 13, 746 01 Opava 1, CZECH REPUBLIC E-mail: Jana.Kasparova@math.slu.cz