Proc. Conf. Opava (Czechoslovakia), August 24–28, 1992 Silesian University, Opava, 1993, 473–495

Topics in the calculus of variations: Finite order variational sequences

D. Krupka

Abstract: It is known that there exists a mapping Λ assigning to a first order lagrangian λ its Lepagean equivalent Λ(λ) in such a way that dΛ(λ) = 0 if and only if the Euler-Lagrange form E_{λ} vanishes identically, i.e., E_{λ} = 0. In this paper we discuss within the theory of finite order variational sequences an analogue of Λ for higher order lagrangians. It turns out that Λ(λ) is a class of forms rather than a differential form defined on the same domain as λ. The main use of Λ is to define the order of a lagrangian in a more adequate way than the usual one. We show how the order of a lagrangian can be determined. We find the order of a variationally trivial lagrangian. Applying these results to the classification problem of symmetry transformations we obtain a higher order analogue of the Noether-Bessel-Hagen equation.

Keywords: Fibered manifold, r-jet, lagrangian, contact form, variational sequence, order of a lagrangian, Euler-Lagrange mapping, Helmholtz-Sonin mapping, variationally trivial lagrangians.

MS classification: 58E30, 58A10, 58G05, 49F99.

1. Introduction

Let X be an n-dimensional manifold, Y a fibered manifold of dimension n + m with projection π : Y → X. The r-jet prolongation of Y will be written as J^{r}Y, and π_{r} : J^{r}Y → X, π_{r,s} : J^{r}Y → J^{s}Y, where 0 ≤ s ≤ r, will denote the canonical jet projections.

If W ⊂ Y is an open set, we write W_{r} = π_{r,0}^{−1}(W), and denote by Ω_{r}^{0}W the ring of smooth functions defined on W_{r}. For k ≥ 1 we denote by Ω_{r}^{k}W the Ω_{r}^{0}W -module of smooth k-forms on W_{r}. The submodule of Ω_{r}^{k}W consisting of π_{r}-horizontal (resp. π_{r,0}- horizontal, resp. π_{r,s}-horizontal) forms will be written as Ω_{r,X}^{k}W (resp. Ω_{r,Y}^{k} W resp. Ω_{r,Z}^{k}W, where Z = J^{s}Y).

The standard horizontalization mapping is denoted by h; h maps Ω_{r}^{k}W to Ω_{r+1,X}^{k}W, where 1 ≤ k ≤ n. Notice that restricting h to Ω_{r,Z}^{k}W ⊂ Ω_{r}^{k}W where Z = J^{r}Y, we obtain a mapping h : Ω_{r,Z}^{k}W → Ω_{r}^{k}W ⊂ Ω_{r+1,X}^{k}W. We also denote h by p_{0} and write p_{i}ρ for the i-contact component of a form ρ (the definitions of these concepts will be recalled in Sec. 2).

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

Let E_{λ} denote the Euler-Lagrange form of a lagrangian λ ∈ Ω_{1,X}^{n}Y. The following result concerns the first order variational calculus, and is taken from [8].

There exists an R-linear mapping Λ : Ω_{1,X}^{n}Y → Ω_{1,Y}^{n} Y satisfying the following three conditions :

(1) For every lagrangian λ ∈ Ω_{1,X}^{n}Y,

h(Λ(λ)) = λ. (1.1)

(2) If ρ ∈ Ω_{1}^{n}Y is a π_{1,0}-projectable form, then

Λ(hρ)) = ρ. (1.2)

(3) For every lagrangian λ ∈ Ω_{1,X}^{n}W the form E_{λ} = p_{1}dΛ(λ) ∈ Ω_{2}^{n+1}Y is π_{2,0}- horizontal, and E_{λ} = 0 if and only if

dΛ(λ) = 0. (1.3)

The first part of (3) means that Λ(λ) is a Lepagean form; condition (1) then means that Λ(λ) is a Lepagean equivalent of λ. The second part of (3) shows that Euler-Lagrange form E_{λ} vanishes if and only if dΛ(λ) = 0; in this sense Λ(λ) has a remarkable property to stand “closer” to the exterior derivative operator d as the well-known Poincaré-Cartan equivalent Θ(λ).

The mapping Λ was discovered by Krupka in connection with the study of lagrangians λ ∈ Ω_{1,X}^{n}Y of the form h(ρ), where ρ ∈ Ω_{0}^{n}Y (see [12]).

Recall that a π-projectable vector field ξ on Y is said to generate generalized invariant transformations of a lagrangian λ ∈ Ω_{1,X}^{n}Y, if

∂_{J^{2}ξ}E_{λ} = 0, (1.4)

i.e., if the Lie derivative of the Euler-Lagrange form E_{λ} by the 2-jet prolongation J^{2}ξ of ξ vanishes.

The following additional properties of Λ have been proved (see [11] and [10]):

(4) If λ ∈ Ω_{1,X}^{n}Y is such that E_{λ} = 0, then the form Λ(λ) is a unique form such that h(Λ(λ)) = λ and dΛ(λ) = 0. This form is π_{1,0}-projectable.

(5) A π-projectable vector field ξ on Y generates generalized invariant transformations of a lagrangian λ ∈ Ω_{1,X}^{n}Y if and only if there exists an n-form ρ on Y such that

∂_{J^{2}ξ}λ = h(ρ), dρ = 0. (1.5)

Clearly, applying (1.2) to (1.5) we get ρ = Λ(∂_{J^{2}ξ}λ), and (1.5) reduces to the condition

dΛ(∂_{J^{2}ξ}λ) = 0. (1.6)

Condition (1.6) which may be used to determine the generators ξ of the generalized invariant transformation, is a generalization of the classical Noether-Bessel-Hagen equation.

Practically the same theory as above was developed by Betounes [5] (see also [6]) who obtained, moreover, the following result [4]:

(6) For every isomorphism α : Y → Y of the fibered manifold Y,

(J^{1}α)^{∗}Λ(λ) = Λ((J^{1}α)^{∗}λ), (1.7)

where J^{1}α : J^{1}Y → J^{1}Y is the first jet prolongation of α.

Combining this result with (1.6) we obtain ∂_{J^{1}ξ}dΛ(λ) = 0 which is another version of the Noether-Bessel-Hagen equation.

A comparison of the papers [8] and [4] from the point of view of the Lepagean forms can be found in [9].

Denote

ω_{0} = dx^{1} ∧ dx^{2} ∧… ∧ dx^{n}, ω^{σ} = dy^{σ} − y_{i}^{σ}dx^{i}. (1.8)

If λ is expressed by

λ = Lω_{0}, (1.9)

then

{Λ(λ) = Lω_{0} + ∑_{q=1}^{n} \frac{∂^{q}L}{∂y_{j_{1}}^{σ_{1}}∂y_{j_{2}}^{σ_{2}} … ∂y_{j_{q}}^{σ_{q}}} dx^{1} ∧… ∧ dx^{j_{1}−1} ∧ ω^{σ_{1}} ∧ dx^{j_{1}+1} \atop ∧… ∧ dx^{j_{q}−1} ∧ ω^{σ_{q}} ∧ dx^{j_{q}+1} ∧… ∧ dx^{n}. (1.10)}

The aim of this paper is to discuss a possible extension of the mapping Λ to the higher order variational calculus. Our discussion is going on within the framework of the theory of finite order variational sequences [13].

Since the calculations in the general case of r-th order jet spaces are extremely difficult and long instead of presenting the details of the proofs we shall explain the main ideas of how the mapping Λ is introduced for the first order case; a complete exposition including proofs will be given elsewhere.

Let W ⊂ Y be an open set, and ρ ∈ Ω_{0}^{n}W. Write in a fibered chart (V, ψ), ψ = (x^{i}, y^{σ}), where V ⊂ W,

hρ = \frac{1}{n!} B_{i_{1}i_{2}...i_{n}}dx^{i_{1}} ∧… ∧ dx^{i_{n}},

p_{j}ρ = ∑ \frac{1}{j!(n − j)!} B_{σ_{1}...σ_{j}i_{j+1}...i_{n}}ω^{σ_{1}} ∧… ∧ ω^{σ_{j}} ∧ dx^{i_{j+1}} ∧… ∧ dx^{i_{n}},

1 ≤ j ≤ n − 1,

p_{n}ρ = \frac{1}{n!} B_{σ_{1}σ_{2}...σ_{n}}ω^{σ_{1}} ∧ ω^{σ_{2}} ∧… ∧ ω^{σ_{n}}. (1.11)

Using the decomposition

π_{1,0}^{∗}ρ = hρ + p_{1}ρ + … + p_{n}ρ

and the formula dπ_{1,0}^{∗}ρ = π_{1,0}^{∗}dρ, we obtain hdρ + p_{1}dρ + … + p_{n+1}dρ = dhρ + dp_{1}ρ + … + dp_{n}ρ. Hence

π_{2,1}^{∗}hdρ = hdhρ,
π_{2,1}^{∗}p_{j}dρ = p_{j}dp_{j−1}ρ + p_{j}dp_{j}ρ, 1 ≤ j ≤ n,

π_{2,1}^{∗}p_{n+1}dρ = p_{n+1}dp_{n}ρ (1.12)

(here p_{0} = h). Express both sides in terms of the basis of linear forms dx^{i}, ω^{σ}, ω_{j}^{σ}, dy_{ij}^{σ} on V_{2} ⊂ J^{2}Y. Then the forms on the left contain the exterior factors dx^{i}, ω^{σ} only, while the right-hand side expressions depend also on ω_{j}^{σ}. The sum of such terms should therefore vanish identically. We shall find the corresponding identities.

The first or the conditions (1.12) reduces to the identity 0 = 0. Consider the case 1 ≤ j ≤ n. We get

\frac{1}{(j − 1)!(n − j + 1)!} \frac{∂B_{σ_{1}...σ_{j−1}i_{j}...i_{n}}}{∂y_{p}^{ν}} ω_{p}^{ν} ∧ ω^{σ_{1}} ∧… ∧ ω^{σ_{j−1}} ∧ dx^{i_{j}} ∧… ∧ dx^{i_{n}}

+ \frac{1}{j!(n − j)!} j · (−1)^{j}B_{σ_{1}...σ_{j}i_{j+1}...i_{n}}ω^{σ_{1}} ∧… ∧ ω^{σ_{j−1}} ∧ ω_{i_{j}}^{σ_{j}}

∧ dx^{i_{j}} ∧ dx^{i_{j+1}} ∧. . . . . . ∧ dx^{i_{n}}

= (\frac{1}{(j − 1)!(n − j + 1)!} \frac{∂B_{σ_{1}...σ_{j−1}i_{j}...i_{n}}}{∂y_{p}^{ν}} − \frac{1}{(j − 1)!(n − j)!}δ_{i_{j}}^{p}B_{σ_{1}...σ_{j_{1}}νi_{j+1}...i_{n}})

· ω_{p}^{ν} ∧ ω^{σ_{1}} ∧… ∧ ω^{σ_{j−1}} ∧ dx^{i_{j}} ∧… ∧ dx^{i_{n}}. (1.13)

Therefore

\frac{1}{n − j + 1} \frac{∂B_{σ_{1}...σ_{j−1}i_{j}...i_{n}}}{∂y_{p}^{ν}} − B_{σ_{1}...σ_{j−1}νi_{j}...i_{n}}δ_{i_{j}}^{p} = 0. (1.14)

Contracting the left-hand side in p, i_{j} we finally obtain

\frac{1}{j} \frac{∂B_{σ_{1}...σ_{j−1}qi_{j+1}...i_{n}}}{∂y_{q}^{ν}} − B_{σ_{1}...σ_{j−1}νi_{j+1}...i_{n}} = 0, 1 ≤ j ≤ n. (1.15)

If j = n + 1, we get the term

\frac{1}{n!} \frac{∂B_{σ_{1}σ_{2}...σ_{n}}}{∂y_{p}^{ν}} ω_{p}^{ν} ∧ ω^{σ_{1}} ∧ ω^{σ_{2}} … ∧ ω^{σ_{n}}, (1.16)

which gives

\frac{∂B_{σ_{1}σ_{2}...σ_{n}}}{∂y_{p}^{ν}} = 0. (1.17)

Conditions (1.15), (1.16) and (1.17) determine p_{j}ρ with the help of hρ.

In particular, writing

B_{i_{1}i_{2}...I_{n}} = Lε_{i_{1}i_{2}...i_{n}}, (1.18)

we get

{p_{j}ρ = \frac{1}{j!(n − j)!} \frac{1}{j!} \frac{∂^{j}L}{∂y_{q_{1}}^{σ_{1}}∂y_{q_{2}}^{σ_{2}} … ∂y_{q_{j}}^{σ_{j}}} ε_{q_{1}q_{2}...q_{j}i_{j+1}...i_{n}}ω^{σ_{1}} ∧… ∧ ω^{σ_{j}} \atop ∧ dx^{i_{j+1}} ∧… ∧ dx^{i_{n}}. (1.19)}

Denote by (p_{1},…, p_{j}) the unique sequence of indices between 1 and n, complementary to (i_{j+1},…, i_{n}), such that p_{1} < … < p_{j}. Then

{ω^{σ_{1}} ∧… ∧ ω^{σ_{j}} ∧ dx^{i_{j+1}} ∧… ∧ dx^{i_{n}} = ε^{p_{1}...p_{j}i_{j+1}...i_{n}}dx^{1} ∧… ∧ dx^{p_{1}−1} ∧ ω^{σ_{1}} \atop ∧ dx^{p_{1}+1} ∧… ∧ dx^{p_{j}−1} ∧ ω^{σ_{j}} ∧ dx^{p_{j}+1} ∧… ∧ dx^{n}. (1.20)}

Since

ε_{q_{1}...g_{j}i_{j+1}...i_{n}}ε^{p_{1}...p_{j}i_{j+1}...i_{n}} = j!(n − j)!δ_{q_{1}}^{p_{1}} … δ_{q_{j}}^{p_{j}} antisym(p_{1} … p_{j}), (1.21)

we easily obtain the above expression (1.10).

We note that this paper is not intended to compare the techniques and result of the theory of finite order variational sequences and the variational bicomplexes which are based on finite jet constructions. For the theory of variational bicomplexes the reader is referred to [1–3], [7], and [14–16], where further references can be found.

2. Finite order variational sequences

In this section we recall basic notions of the theory of differential forms on jet prolongations of a fibered manifold which are necessary for next sections. (see [9], [12], [13]). We use the same notation as in Introduction.

Let r ≥ 1 be an integer. To every tangent vector ξ ∈ T J^{r+1}Y at a point J_{x}^{r+1}γ ∈ J^{r+1}Y one can assign a tangent vector hξ ∈ T J^{r}Y at the point π_{r+1,r}(J_{x}^{r+1}γ) = J_{x}^{r}γ, and a tangent vector pξ ∈ T J^{r}Y at J_{x}^{r}γ by

hξ = T_{x}J^{r}γ · T π_{r+1} · ξ, pξ = T π_{r+1,r} · ξ − hξ. (2.1)

Clearly, pξ is always a π_{r}-vertical vector. ξ is π_{r+1}-vertical if and only if hξ = 0; ξ is π_{r+1,r}-vertical if and only if hξ = 0, pξ = 0.

The mappings ξ → hξ, ξ → pξ are vector bundle homomorphisms of T J^{r+1}Y to T J^{r}Y over the jet projection π_{r+1,r}. We call h the horizontalization.

The horizontalization h : T J^{r+1}Y → T J^{r}Y induces a decomposition of the modules of differential forms on J^{r}Y in the following sense. Let W ⊂ Y be an open set, and let ρ ∈ Ω_{r}^{k}W, k ≥ 1. Let J_{x}^{r+1}γ ∈ W_{r+1} be a point, ξ_{1}, ξ_{2},…, ξ_{k} ∈ T J^{r+1}Y tangent vectors at this point. Write for each i = 1, 2,…, k

T π_{r+1,r} · ξ_{i} = hξ_{i} + pξ_{i}, (2.2)

and denote

p_{0}ρ(J_{x}^{r+1}γ)(ξ_{1}, ξ_{2},…, ξ_{k}) = ρ(J_{x}^{r}γ)(hξ_{1}, hξ_{2},…, hξ_{k}),

p_{1}ρ(J_{x}^{r+1}γ)(ξ_{1}, ξ_{2},…, ξ_{k}) = ρ(J_{x}^{r}γ)(pξ_{1}, hξ_{2},…, hξ_{k})

+ ρ(J_{x}^{r}γ)(hξ_{1}, pξ_{2}, hξ_{3},…, hξ_{k}) + … + ρ(J_{x}^{r}γ)(hξ_{1},…, hξ_{k−1}, pξ_{k}),

…

p_{k}ρ(J_{x}^{r}γ)(ξ_{1}, ξ_{2},…, ξ_{k}) = ρ(J_{x}^{r}γ)(pξ_{1}, pξ_{2},…, pξ_{k}). (2.3)

p_{i}ρ is a k-form on W_{r+1}, called the i-contact component of ρ. Sometimes we denote

hρ = p_{0}ρ, pρ = ∑_{i=1}^{k} p_{i}ρ (2.4)

and call hρ (resp. pρ) the horizontal (resp. contact) component of ρ.

Since by the definition of the pull-back,

{π_{r+1,r}^{∗}ρ(J_{x}^{r+1}γ)(ξ_{1}, ξ_{2},…, ξ_{k}) \atop = ρ(J_{x}^{r}γ)(T π_{r+1,r} · ξ_{1}, T π_{r+1,r} · ξ_{2},…, T π_{r+1,r} · ξ_{k}), (2.5)}

we easily obtain a decomposition formula

π_{r+1,r}^{∗}ρ = ∑_{i=0}^{k} p_{i}ρ = hρ + pρ · (2.6)

ρ is called 0-contact, or horizontal, if pρ = 0; ρ is called contact, if hρ = 0. We say that ρ is q-contact, if π_{r+1,r}^{∗}ρ = p_{q}ρ, or, which is the same, if p_{j}ρ = 0 for j ≠ q. We say that ρ is of order of contactness ≥ q, if hρ = 0, p_{1}ρ = 0,…, p_{q−1}ρ = 0.

In what follows we often use multiindices I = (i_{1}i_{2} … i_{p}), where 1 ≤ i_{1}, i_{2},…, i_{p} ≤ n. For such a multiindex I we denote p = |I|, and call this number the length of I. It will be convenient to denote by Ii the multiindex (i_{1}i_{2} … i_{p}i).

Let (V, ψ), ψ = (x^{i}, y^{σ}), be a fibered chart on Y. Denote for every multiindex I such that 0 ≤ |I| ≤ r, and every σ, 1 ≤ σ ≤ m,

ω_{I}^{σ} = dy_{I}^{σ} − y_{Ii}^{σ}dx^{i}. (2.7)

Then on V_{r+1}, with the obvious notational convention,

hdx^{i} = dx^{i}, hdy_{I}^{σ} = y_{Ii}^{σ}dx^{i}, pdx^{i} = 0, pdy_{I}^{σ} = ω_{I}^{σ}. (2.8)

Obviously, the forms

dx^{i}, ω_{I}^{σ}, dy_{J}^{σ}, 0 ≤ |I| ≤ r, |J| = r + 1, (2.9)

define a basis of linear forms on V_{r+1}. We have

{dω_{I}^{σ} = −ω_{Ii}^{σ} ∧ dx^{i}, 0 ≤ |I| ≤ r − 1, \atop dω_{I}^{σ} = −dy_{Ii}^{σ} ∧ dx^{i}, |I| = r. (2.10)}

If J_{x}^{r+1}γ ∈ V_{r+1} is a point and ξ ∈ T J^{r+1}Y a tangent vector at J_{x}^{r+1}γ,

ξ = ξ^{i} \frac{∂}{∂x^{i}} + ∑_{0≤|I|≤r+1} Ξ_{I}^{σ} \frac{∂}{∂y_{I}^{σ}} , (2.11)

then

dx^{i}(J_{x}^{r+1}γ)(ξ) = ξ^{i},
ω_{I}^{σ}(J_{x}^{r+1}γ)(ξ) = Ξ_{I}^{σ} − y_{I_{i}}^{σ}ξ^{i}, 0 ≤ |I| ≤ r,
dy_{J}^{σ}(J_{x}^{r+1}γ)(ξ) = Ξ_{J}^{σ}, |J| = r + 1. (2.12)

If f : V_{r} → R is a smooth function, we have

π_{r+1,r}^{∗}df = hdf + pdf, (2.13)

where

hdf = d_{i}f · dx^{i}, d_{i}f = \frac{∂f}{∂x^{i}} + ∑_{|I|≤r} \frac{∂f}{∂y_{I}^{σ}} y_{Ii}^{σ}, (2.14)

pdf = ∑_{|I|≤r} \frac{∂f}{∂y_{I}^{σ}} ω_{I}^{σ}. (2.15)

In particular,

d_{i}y_{I}^{σ} = y_{Ii}^{σ}. (2.16)

We also use the notation

d′f = \frac{df}{dx^{i}} dx^{i} + ∑_{|I|≤r−1} \frac{df}{dy_{I}^{σ}} dy_{I}^{σ}, (2.17)

and define a function d′_{i}f : V_{r} → R by

hd′f = d′_{i}f · dx^{i}. (2.18)

Then

df = d′f + ∑_{|I|=r} \frac{df}{dy_{I}^{σ}} dy_{I}^{σ}, (2.19)

d_{i}f = d′_{i}f + ∑_{|I|=r} \frac{df}{dy_{I}^{σ}} y_{Ii}^{σ}. (2.20)

Let r ≥ 1, k ≥ 0 be fixed. We denote by Ω_{r}^{k} the direct image of the sheaf of k-forms over J^{r}Y by the projection π_{r,0} : J^{r}Y → Y. If p_{i} : Ω_{r}^{k} → Ω_{r+1}^{k} is considered as a sheaf morphism, we denote

Ω_{r(c)}^{k} = \{ \begin{matrix}ker h, & 1 ≤ k ≤ n, \\ ker p_{k−n}, & k ≥ n + 1. \end{matrix} (2.21)

Then we set

dΩ_{r(c)}^{k} = im d_{k}, (2.22)

where d_{k} is the sheaf morphism defined by the exterior derivative d of k-forms, and

Θ_{r}^{k} = dΩ_{r(c)}^{k−1} + Ω_{r(c)}^{k}. (2.23)

We get a diagram

[figure: 0; 0; 0; 0; Θ_{r}^{1}; d; Θ_{r}^{2}; d; Θ_{r}^{3}; …; 0; R; Ω_{r}^{0}; d; Ω_{r}^{1}; d; Ω_{r}^{2}; d; Ω_{r}^{3}; …; Ω_{r}^{1}/Θ_{r}^{1}; E_{1}; Ω_{r}^{2}/Θ_{r}^{2}; E_{2}; Ω_{r}^{3}/Θ_{r}^{3}; …; 0; 0; 0] (2.24)

in which the sequence

0 → R → Ω_{r}^{0} → Ω_{r}^{1}/Θ_{r}^{1} → Ω_{r}^{2}/Θ_{r}^{2} → Ω_{r}^{3}/Θ_{r}^{3} → … (2.25) is an acyclic resolution of the constant sheaf R over Y. This sequence is called the variational sequence of order r over Y. The sheaf morphism E_{n} (resp. E_{n+1}) is the Euler-Lagrange mapping (resp. the Helmholtz-Sonin mapping).

If ρ ∈ Ω_{r}^{n} is a germ and [ρ] = λ = f_{0}ω_{0} its class in Ω_{r}^{n}/Θ_{r}^{n}, then

E_{n}(λ) = ε_{σ}ω^{σ} ∧ ω_{0}, (2.26)

where

ε_{σ} = ∑_{k=0}^{r+1}(−1)^{k}d_{j_{1}} … d_{j_{k}} \frac{∂f_{0}}{∂y_{j_{1}...j_{k}}^{σ}} . (2.27)

If ρ ∈ Ω_{r}^{n+1} is a germ and [ρ] = ε = ε_{σ}ω^{σ} ∧ ω_{0} its class in Ω_{r}^{n+1}/Θ_{r}^{n+1}, where ε_{σ} are defined on V_{s} ⊂ J^{s}Y, then

E_{n+1}(ε) = \frac{1}{2} ∑_{i=1}^{s}(\frac{∂ε_{ν}}{∂y_{j_{1}...j_{i}}^{σ}} − (−1)^{i} \frac{∂ε_{σ}}{∂y_{j_{1}...j_{i}}^{ν}} ; − ∑_{k=i+1}^{s}(−1)^{k} \binom{k}{i} d_{j_{i+1}} … d_{j_{k}} \frac{∂ε_{σ}}{∂y_{j_{1}...j_{k}}^{ν}} )ω_{j_{1}...j_{i}}^{σ} ∧ ω^{ν} ∧ ω_{0}. (2.28)

A well-known conclusion of these results for the calculus of variations consists in a cohomology description of the difference between the local and global triviality of global sections of Ω_{r}^{n} (or Ω_{r}^{n}/Θ_{r}^{n}), and between the local and global variationality of global sections of Ω_{r}^{n+1} (or Ω_{r}^{n+1}/Θ_{r}^{n+1}) (see [1], [2], [13–16]).

3. Projectable and π-decomposable forms

We begin by presenting three auxiliary algebraic assertions which can be used in the proofs of our main theorems. If in the formulas below a symbol like antisym(i_{1}i_{2} … i_{s}) (resp. sym(j_{1}j_{2} … j_{r})) is used, then the expression preceding this symbol should be considered as antisymmetrized in i_{1}, i_{2},…, i_{s} (resp. symmetrized in j_{1}, j_{2},…, j_{r}).

Lemma 1. Let r, s, and n be positive integers, and let A^{j_{1}...j_{r}}_{i_{1}...i_{s}}, 1 ≤ j_{1},…, j_{r}, i_{1},…, i_{s} ≤ n, be a system of real numbers symmetric in the superscripts and antisymmetric in the subscripts. Put for every p, q, 1 ≤ p, q ≤ n,

Q^{j_{1}...j_{r}p}_{i_{1}...i_{s}q} = A^{j_{1}...j_{r}}_{i_{1}...i_{s}}δ_{q}^{p} antisym(i_{1} … i_{s}q) sym(j_{1} … j_{r}p), (3.1)

B^{j_{2}...j_{r}}_{i_{2}...i_{s}} = \frac{rs}{h + r − s}A^{pj_{2}...j_{r}}_{pi_{2}...i_{s}}. (3.2)

(a) The system A^{j_{1}...j_{r}}_{i_{1}...i_{s}} obeys the identity

Q^{j_{1}...j_{r}p}_{i_{1}...i_{s}p} = \frac{n + r − s}{(r + 1)(s + 1)}(A^{j_{1}...j_{r}}_{i_{1}...i_{s}} − B^{j_{2}...j_{r}}_{i_{2}...i_{s}}δ_{i_{1}}^{j_{1}})

antisym(i_{1}i_{2} … i_{s}) sym(j_{1}j_{2} … j_{r}). (3.3)

(b) The system C^{j_{1}...j_{r}}_{i_{1}...i_{s}} defined by the decomposition

A^{j_{1}...j_{r}}_{i_{1}...i_{s}} = B^{j_{2}...j_{r}}_{i_{2}...i_{s}}δ_{i_{1}}^{j_{1}} + C^{j_{1}...j_{r}}_{i_{1}...i_{s}}

antisym(i_{1}i_{2} … i_{s}) sym(j_{1}j_{2} … j_{r}) (3.4)

is traceless, i.e., for all j_{2},… j_{r}, i_{2},…, i_{s}

C^{pj_{2}...j_{r}}_{pi_{2}...i_{s}} = 0. (3.5)

(c) Equation

Q^{j_{1}...j_{r}p}_{i_{1}...i_{s}q} = 0 (3.6)

is satisfied if and only if there exists a system B^{j_{2}...j_{r}}_{i_{2}...i_{s}} symmetric in the superscripts and antisymmetric in the subscripts such that

A^{j_{1}...j_{r}}_{i_{2}...i_{s}} = B^{j_{2}...j_{r}}_{i_{2}...i_{s}}δ_{i_{1}}^{j_{1}} antisym(i_{1}i_{2} … i_{s}) sym(j_{1}j_{2} … j_{r}). (3.7)

The system B^{j_{2}...j_{r}}_{i_{2}...i_{s}} is unique, and is traceless, i.e.

B^{pj_{3}...j_{r}}_{pi_{3}...i_{s}} = 0. (3.8)

Lemma 2. Let n, k, j be integers such that 2 ≤ k ≤ n, 1 ≤ j ≤ k−1. Let r_{1}, r_{2},…, r_{j} be positive integers, and let I_{1} (resp. I_{2},…, I_{j}) be a multiindex of length r_{1} (resp. r_{2},…, r_{j}) whose entries are arbitrary indices belonging to the set {1, 2,…, n}. Consider the system of linear, homogeneous equations

A^{I_{1}I_{2}...I_{j}}_{i_{j+1}i_{j+2}...i_{k}}δ_{i_{1}}^{p_{1}}δ_{i_{2}}^{p_{2}} … δ_{i_{j}}^{p_{j}} = 0

antisym(i_{1}i_{2} … i_{k}) sym(I_{1}p_{1}) sym(I_{2}p_{2}) … sym(I_{j}p_{j}), (3.9)

where 1 ≤ p_{1}, p_{2},…, p_{j}, i_{1}, i_{2},…, i_{k} ≤ n, and the unknown system A^{I_{1}I_{2}...I_{j}}_{i_{j+1}i_{j+2}...i_{k}} is antisymmetric in the subscripts, and symmetric in the superscripts entering every of the multiindices I_{1}, I_{2},…, I_{j}.

Then every solution of this system is of the form

A^{I_{1}I_{2}...I_{j}}_{i_{j+1}i_{j+2}...i_{k}}

= B^{Q_{1}I_{2}...I_{j}}_{i_{j+2}...i_{k}}δ_{i_{j+1}}^{q_{1}} + B^{I_{1}Q_{2}I_{3}...I_{j}}_{i_{j+2}...i_{k}}δ_{i_{j+1}}^{q_{2}} + B^{I_{1}...I_{j−1}Q_{j}}_{i_{j+2}...i_{k}}δ_{i_{j+1}}^{q_{j}}

antisym(i_{j+1}i_{j+2} … i_{k}) sym(Q_{1}q_{1}) … sym(Q_{j}q_{j}), (3.10)

where I_{1} = Q_{1}q_{1},…, I_{j} = Q_{j}q_{j}, and B^{Q_{1}I_{2}...I_{s}}_{i_{j+2}...i_{k}},…, B^{I_{1}...I_{j−1}Q_{j}}_{i_{j+2}...i_{k}} are arbitrary constants antisymmetric in the subscripts, and symmetric in the superscripts entering every of the multiindices Q_{1},…, Q_{j}, I_{1},…, I_{j}.

Lemma 3. Let n, k, j be integers such that 1 ≤ j ≤ n − 1, k ≥ n + 1. Let r be a positive integer, and let I_{1}, I_{2},…, I_{n−k+j} be multiindices of length r whose entries are arbitrary indices from the set {1, 2,…, n}. Consider the system of linear, homogeneous equations

A^{I_{1}...I_{k−n}I_{k−n+1}...I_{k−n+j}}_{i_{j+1}...i_{n}}δ_{i_{1}}^{p_{k−n+1}} … δ_{i_{j}}^{p_{k−n+j}} = 0

antisym(i_{1}i_{2} … i_{n}) sym(I_{k−n+1}p_{k−n+1}) … sym(I_{k−n+j}p_{k−n+j}) (3.11) for the unknowns A^{I_{1}...I_{k−n}I_{k−n+1}...I_{k−n+j}}_{i_{j+1}...i_{n}}, antisymmetric in the subscripts and in the multiindices I_{1},…, I_{k−n+j}, and symmetric in the superscripts entering every of the multiindices I_{1},…, I_{k−n+j}, where 1 ≤ p_{1},…, p_{j}, i_{1},…, i_{n} ≤ n. Every solution of this system has the form

A^{I_{1}...I_{k−n}I_{k−n+1}...I_{k−n+j}}_{i_{j+1}...i_{n}}
= C^{Q_{1}...Q_{k−n}Q_{k−n+1}I_{k−n+2}...I_{k−n+j}}_{i_{j+k−n+2}...i_{n}}δ_{i_{j+1}}^{p_{1}} … δ_{i_{j+k−n}}^{p_{k−n}}δ_{i_{j+k−n+1}}^{p_{k−n+1}}

+ … + C^{I_{1}...I_{j−1}Q_{j}Q_{j+1}...Q_{k−n+j}}_{i_{j+k−n+2}...i_{n}}δ_{i_{j+1}}^{p_{n}} … δ_{i_{j+k−n}}^{p_{k−n+j−1}}δ_{i_{j+k−n+1}}^{p_{k−n+j}}

antisym(i_{j+1} … i_{n}) sym(Q_{1}p_{1}) … sym(Q_{k−n+j}p_{k−n+j}),
j ≤ 2n − k − 1,

A^{I_{1}...I_{k−n}I_{k−n+1}...I_{k−n+j}}_{i_{j+1}...i_{n}} = 0, j > 2n − k − 1, (3.12)

where I_{1} = Q_{1}p_{1},…, I_{k−n+j} = Q_{k−n+j}p_{k−n+j}, and C^{Q_{1}...Q_{k−n+1}Q_{k−n+2}...I_{k−n+j}}_{i_{j+k−n+2}...i_{n}},…, C^{I_{1}...I_{n−1}Q_{n}...Q_{k−n+j}}_{i_{j+k−n+2}...i_{n}} are arbitrary constants antisymmetric in the subscripts and symmetric in the superscripts entering every of the multiindices Q_{1},…, Q_{k−n+j}, I_{1},…, I_{k−n+j}.

Let ρ ∈ Ω_{r}^{K}W be a form, and consider the pull-back π_{r+1,r}^{∗}ρ. We have an invariant decomposition of π_{r+1,r}^{∗}ρ into the summands p_{i}ρ (2.6). In the following theorem we derive some identities for the chart expressions of the coefficients in p_{i}ρ.

Theorem 1. Let k ≥ 1, and let ρ ∈ Ω_{r}^{k}W be a form. Let for every j, 0 ≤ j ≤ k, p_{j}ρ be expressed by

{p_{j}ρ = ∑_{s=0}^{j} \frac{1}{s!(j − s)!(k − j)!}B_{σ_{1}...}^{J_{1}}_{σ_{s}σ_{s+1}...}^{J_{s}I_{s+1}}_{σ_{j}i_{j+1}...i_{k}}^{I_{j}}ω_{J_{1}}^{σ_{1}} ∧… ∧ ω_{J_{s}}^{σ_{s}} \atop ∧ ω_{I_{s+1}}^{σ_{s+1}} ∧… ∧ ω_{I_{j}}^{σ_{j}} ∧ dx^{i_{j+1}} ∧… ∧ dx^{i_{k}}, (3.13)}

where J_{1},…, J_{s} (resp. I_{s+1},…, I_{j}) are multiindices of length < r (resp. = r). Denote for j < k

{C_{σ_{1}...}^{J_{1}}_{σ_{s}σ_{s+1}...}^{J_{s}I_{s+1}}_{σ_{j−1}σ_{j}}^{I_{j−1}p_{2}...p_{r}}_{i_{j+2}...i_{k}} \atop = \frac{r(k − j)}{n + r − k + j} B_{σ_{1}...}^{J_{1}}_{σ_{s}σ_{s+1}...}^{J_{s}I_{s+1}}_{σ_{j−1}σ_{j}}^{I_{j−1}pp_{2}...p_{r}}_{pi_{j+2}...i_{k}}. (3.14)}

Then the coefficients B_{σ_{1}...}^{J_{1}}_{σ_{s}σ_{s+1}...}^{J_{s}I_{s+1}}_{σ_{j}σ_{j+1}...i_{k}}^{I_{j}} obey the following conditions:

\frac{∂B_{σ_{1}...}^{J_{1}}_{σ_{s}σ_{s+1}...}^{J_{s}I_{s+1}}_{σ_{k}}^{I_{k}}}{∂y_{L}^{σ}} = 0, 0 ≤ s ≤ k, |L| = r + 1,

B_{σ_{1}...}^{J_{1}}_{σ_{s}σ_{s+1}...}^{J_{s}I_{s+1}}_{σ_{j−1}σ_{j}i_{j+1}...i_{k}}^{I_{j−1}I_{j}} − C_{σ_{1}...}^{J_{1}}_{σ_{s}σ_{s+1}...}^{J_{s}I_{s+1}}_{σ_{j−1}σ_{j}}^{I_{j−1}p_{2}...p_{r}}_{i_{j+2}...i_{k}}δ_{i_{j+1}}^{p_{1}}

= \frac{r + 1}{n + r − k + j} \frac{∂B_{σ_{1}...}^{J_{1}}_{σ_{s}σ_{s+1}...}^{J_{s}I_{s+1}}_{σ_{j−1}qi_{j+1}...i_{k}}^{I_{j−1}}}{∂y_{I_{j}q}^{σ_{j}}} antisym(i_{j+1} … i_{k})

sym(p_{1} … p_{r}), I_{j} = (p_{1} … p_{r}), 1 ≤ j ≤ k, 0 ≤ s ≤ j − 1,

\frac{∂B_{σ_{1}...}^{J_{1}}_{σ_{s}σ_{s+1}...}^{J_{s}I_{s+1}}_{σ_{j−1}i_{j}...i_{k}}^{I_{j−1}}}{∂y_{I_{j}q}^{σ_{j}}} − \frac{(r + 1)(k − j + 1)}{n + r − k + j} \frac{∂B_{σ_{1}...}^{J_{1}}_{σ_{s}σ_{s+1}...}^{J_{s}I_{s+1}}_{σ_{j−1}qi_{j+1}...i_{k}}^{I_{j−1}}}{∂y_{I_{j}q}^{σ_{j}}} δ_{i_{j}}^{p} = 0

antisym(i_{j} … i_{k}) sym(I_{j}p), 1 ≤ j ≤ k, 0 ≤ s ≤ j − 1. (3.15)

The proof of Theorem 1 is based on analogous identities as (1.12), and on Lemma 1,(a).

Let ρ be a k-form on an open set W ⊂ J^{r}Y. Let 1 ≤ k ≤ n. We say that ρ is π-decomposable if its horizontal (i.e. 0-contact) component hρ is π_{r+1,r}-projectable. If 1 ≤ k ≤ dim J^{r}Y we say that ρ is π-decomposable if its (k−n)-contact component p_{k−n}ρ is π_{r+1,r}-projectable.

If 1 ≤ k ≤ n and ρ is π-decomposable, then

ρ = ρ_{1} + ρ_{2}, (3.16)

where ρ_{1} is a π_{r}-horizontal form, and ρ_{2} is a contact form; in fact, ρ_{1} is the π_{r+1,r}- projection of hρ. Analogous decompositions arise for k-forms with k ≥ n + 1; in this case ρ_{1} is the (k−n)-contact component of ρ, and ρ_{2} is a form whose order of contactness is ≥ k − n + 1.

Our second result represents a criterion for a form to be π-decomposable.

Theorem 2. Let W ⊂ Y be an open set, ρ ∈ Ω_{r}^{k}W a form, and let (V, ψ), ψ = (x^{i}, y^{σ}), be any fibered chart such that V ⊂ W.

(a) Let k = 1. Then ρ is π-decomposable if and only if

ρ = ϕ_{i}dx^{i} + ϕ_{σ}^{J}ω_{J}^{σ} (3.17)

for some functions ϕ_{i}, ϕ_{σ}^{J} : V_{r} → R, where |J| ≤ r − 1.

(b) Let 2 ≤ k ≤ n. Then ρ is π-decomposable if and only if

ρ = ϕ_{i_{1}...i_{k}}dx^{i_{1}} ∧… ∧ dx^{i_{k}} + ω_{J}^{σ} ∧ ϕ_{σ}^{J} + dω_{I}^{σ} ∧ Ψ_{σ}^{I}, (3.18)

where ϕ_{i_{1}...i_{k}} (resp. ϕ_{σ}^{J}, resp. Ψ_{σ}^{I}) are some functions (resp. (k − 1)-forms, resp. (k − 2)- forms) on V_{r}, and |J| ≤ r − 1, |I| = r − 1.

(c) Let n + 1 ≤ k ≤ dim J^{r}Y. Then ρ is π-decomposable if and only if

{ρ = ϕ_{i_{1}...i_{k}} ∧ dx^{i_{1}} ∧… ∧ dx^{i_{n}} + ω_{J_{1}}^{σ_{1}} ∧… ∧ ω_{J_{p}}^{σ_{p}} ∧ dω_{I_{i}}^{ν_{1}} ∧… \atop ∧ dω_{I_{q}}^{ν_{q}} ∧ ϕ_{σ_{1}...}^{J_{1}}_{σ_{p}ν_{1}...}^{J_{p}I_{1}}_{ν_{q}}^{I_{q}}, p + q = k − n + 1, (3.19)}

for some (k − n)-forms ϕ_{i_{1}...i_{n}} and (k − p − 2q)-forms ϕ_{σ_{1}...}^{J_{1}}_{σ_{p}ν_{1}...}^{J_{p}I_{1}}_{ν_{q}}^{I_{q}} on V_{r} where |J_{1}|,…, |J_{p}| ≤ r − 1, |I_{1}|,…, |I_{q}| = r − 1.

The proof of (a) is straightforward.

To prove (b), suppose that 2 ≤ k ≤ n. Then ρ has an expression

ρ = ρ_{0} + A_{i_{1}i_{2}...i_{k}}dx^{i_{1}} ∧ dx^{i_{2}} ∧… ∧ dx^{i_{k}}
+ A_{σ_{1}i_{2}...i_{k}}^{I_{1}}dy_{I_{1}}^{σ_{1}} ∧ dx^{i_{2}} ∧… ∧ dx^{i_{k}} + …

+ A_{σ_{1}...}^{I_{1}}_{σ_{k−1}i_{k}}^{I_{k−1}}dy_{I_{1}}^{σ_{1}} ∧… ∧ dy_{I_{k−1}}^{σ_{k−1}} ∧ dx^{i_{k}} + A_{σ_{1}...}^{I_{1}}_{σ_{k}}^{I_{k}}dy_{I_{1}}^{σ_{1}} ∧… ∧ dy_{I_{k}}^{σ_{k}}, (3.20) where ρ_{0} belongs to the ideal of forms generated by the 1-forms ω_{J}^{σ}, where |J| ≤ r − 1. Since

hρ = (A_{i_{1}i_{2}...i_{k}} + A_{σ_{1}σ_{2}...i_{k}}^{I_{1}}y_{I_{1}i_{1}}^{σ_{1}} + … + A_{σ_{1}...}^{I_{1}}_{σ_{k−1}i_{k}}^{I_{k−1}}y_{I_{1}i_{1}}^{σ_{1}} … y_{I_{k−1}i_{k−1}}^{σ_{k−1}} ; + A_{σ_{1}...}^{I_{1}}_{σ_{k}}^{I_{k}}y_{I_{1}i_{1}}^{σ_{1}} … y_{I_{k}i_{k}}^{σ_{k}})dx^{i_{1}} ∧… ∧ dx^{i_{k}}, (3.21)

the form hρ is π_{r+1,r}-projectable if and only if

A_{σ_{1}...}^{I_{1}}_{σ_{j}i_{j+1}...i_{k}}^{I_{j}}δ_{i_{1}}^{p_{1}} … δ_{i_{j}}^{p_{j}} = 0 antisym(i_{1} … i_{j}i_{j+1} … i_{k})

sym(I_{1}p_{1}) … sym(I_{j}p_{j}), 1 ≤ j ≤ k. (3.22)

If j = k, this implies

A_{σ_{1}...}^{I_{1}}_{σ_{k}}^{I_{k}} = 0. (3.23)

If 1 ≤ j ≤ k − 1, we apply Lemma 2, (3.10).

To prove (c), first consider the case k = n + 1, and suppose that we have an (n + 1)- form ρ such that p_{1}ρ is π_{r+1,r}-projectable. ρ has a unique decomposition in the form

ρ = ρ_{0} + ρ′ + ρ′′, (3.24)

where ρ_{0} is a multiple of ω_{0}, ρ′ contains at most n − 1 factors dx^{i} and is generated by the forms ω_{J}^{ν}, |J| ≤ r − 1, and ρ′′ contains at most n − 1 factors dx^{i} and none of the factors ω_{J}^{ν}, |J| ≤ r − 1. We wish to show that the π_{r+1,r}-projectability of p_{1}ρ implies that both ρ′ and ρ′′ have the desired structure (3.19).

Notice that ρ′ and ρ′′ are of the form

ρ′ = ω_{J}^{ν} ∧ ϕ_{ν}^{J}, 0 ≤ |J| ≤ r − 1, (3.25)

ρ′′ = A_{νσ_{1}i_{2}...i_{n}}^{JI_{1}}dy_{J}^{ν} ∧ dy_{I_{1}}^{σ_{1}} ∧ dx^{i_{2}} ∧… ∧ dx^{i_{n}}
+ A_{νσ_{1}σ_{2}i_{3}...i_{n}}^{JI_{1}I_{2}}dy_{J}^{ν} ∧ dy_{I_{1}}^{σ_{1}} ∧ dy_{I_{2}}^{σ_{2}} ∧ dx^{i_{3}} ∧… ∧ dx^{i_{n}} + …
+ A_{νσ_{1}...}^{JI_{1}}_{σ_{n−1}i_{n}}^{I_{n−1}}dy_{J}^{ν} ∧ dy_{I_{1}}^{σ_{1}} ∧… ∧ dy_{I_{n−1}}^{σ_{n−1}} ∧ dx^{i_{n}}

+ A_{νσ_{1}...}^{JI_{1}}_{σ_{n}}^{I_{n}}dy_{J}^{ν} ∧ dy_{I_{1}}^{σ_{1}} ∧… ∧ dy_{I_{n}}^{σ_{n}}, |J|, |I_{1}|,…, |I_{n}| = r. (3.26)

We apply (b) to the form hϕ_{ν}^{J}, and Lemma 2(a) to p_{1}ρ′′, and get the required result after several steps.

If k ≥ n + 1, we proceed in the same way.

Remark 1. Taking ϕ_{i} = 0, ϕ_{i_{1}...i_{k}} = 0, and ϕ_{i_{1}...i_{n}} = 0 in Theorem 2 we obtain necessary and sufficient conditions for a k-form ρ to be contact, resp. strongly contact.

4. Order of a lagrangian

Usually, a lagrangian λ ∈ Ω_{r,X}^{n}W is said to be of order r, if it is not π_{r,r−1}-projectable. In this paper we adopt a little different terminology. We shall say that λ ∈ Ω_{r,X}^{n}W is of order r, if there is no form ρ ∈ Ω_{r−1}^{n}W for which hρ = λ or, which is the same, if hρ ≠ λ for every ρ ∈ Ω_{r−1}^{n}W. If λ is not of order r we call the order of λ the smallest integer s such that there exists a form ρ ∈ Ω_{s}^{n}W for which λ = hρ.

Clearly, if λ ∈ Ω_{r,X}^{n}W is π_{r,s}-projectable, and is not π_{r,s−1}-projectable, then the order of λ is s or s − 1; the latter case takes place if and only if λ = hρ with ρ ∈ Ω_{s−1}^{n}W (up to π_{r,s}^{∗}).

For example, the scalar curvature lagrangian (the Hilbert lagrangian for the Einstein equations in vacuo), is of the 1st order (see [9]).

Our problem to be considered in this section is to find the order of a lagrangian, that is, to solve the equation

λ = hρ (4.1)

with respect to ρ.

Let us consider the module of r-th order lagrangians Ω_{r}^{n}W. Our aim will be to describe the image of Ω_{r}^{n}W by the horizontalization h, i.e., the subspace hΩ_{r}^{n}W ⊂ Ω_{r+1,X}^{n}W. We have the canonical decomposition of h,

[figure: Ω_{r}^{n}W; h; Ω_{r+1,X}^{n}W; Ω_{r}^{n}W/Θ_{r}^{n}W; g; hΩ_{r}^{n}W] (4.2)

in which the right vertical arrow represents the canonical inclusion, and g is a bijection. We wish to construct the inverse mapping

Λ = g^{−1} (4.3)

in terms of coordinates.

To this purpose we specify the result of Theorem 1 to the case k = n.

Let us consider a form ρ ∈ Ω_{r}^{n}W, and a fibered chart (V, ψ), ψ = (x^{i}, y^{σ}), where V ⊂ W. ρ has a unique decomposition

ρ = ρ_{0} + ρ′, (4.4)

where ρ_{0} is a contact form, i.e., is generated by ω_{J}^{σ}, |J| ≤ r − 1 and dω_{I}^{σ}, |I| = r − 1 (see Sec. 3, Remark 1), and

ρ′ = ∑_{p=0}^{n} \frac{1}{p!(n − p)!}A_{σ_{1}...}^{I_{1}}_{σ_{p}i_{p+1}...i_{n}}^{I_{p}}dy_{I_{1}}^{σ_{1}} ∧… ∧ dy_{I_{p}}^{σ_{p}} ∧ dx^{i_{p+1}} ∧… ∧ dx^{i_{n}}, (4.5)

where |I_{1}|,…, |I_{p}| = r and all the coefficients A_{σ_{1},i_{2}... i_{n}}^{I_{1}},…, A_{σ_{1}...}^{I_{1}}_{σ_{n−1}i_{n}}^{I_{n−1}} are completely traceless; in particular, ρ′ does not contain any non-zero term of the form dω_{J}^{ν} ∧χ_{ν}^{J} with |J| = r − 1. This decomposition may be referred to as the canonical decomposition of ρ relative to the fibered chart (V, π).

Clearly, ρ′ may be viewed as a representative of the class [ρ] ∈ Ω_{r}^{n}W/Θ_{r}^{n}W in the chart (V, ψ). We have

hρ′ = (∑_{p=0}^{n} \frac{1}{p!(n − p)!}A_{σ_{1}...}^{I_{1}}_{σ_{p}i_{p+1}...i_{n}}^{I_{p}}y_{I_{1}i_{1}}^{σ_{1}} … y_{I_{p}i_{p}}^{σ_{p}})dx^{i_{1}} ∧… ∧ dx^{i_{n}}. (4.6)

If we write hρ′ in the form hρ′ = Bdx^{1} ∧… ∧ dx^{n}, or

hρ′ = \frac{1}{n!}B_{i_{1}...i_{n}}dx^{r_{1}} ∧… ∧ dx^{i_{n}}, (4.7)

it is clear that one can reconstruct ρ′ from B, or B_{i_{1}...i_{n}}, by a differentiation process (the same method has been used in [8], [12], [11]).

Consider the form (4.5) and write

{π_{r+1,r}^{∗}ρ′ \atop = ∑_{p=0}^{n} \frac{1}{p!(n − p)!}B_{σ_{1}...}^{I_{1}}_{σ_{p}i_{p+1}...i_{n}}^{I_{p}}ω_{I_{1}}^{σ_{1}} ∧…∧ω_{I_{p}}^{σ_{p}} ∧dx^{i_{p}+1} ∧…∧dx^{i_{n}}. (4.8)}

This form has a unique decomposition

π_{r+1,r}^{∗}ρ′ = ρ′_{0} + ρ′′, (4.9)

where ρ′_{0} is of the form dω_{J}^{ν} ∧ χ_{ν}^{J}, |J| = r − 1,

ρ′′ = ∑_{p=0}^{n} \frac{1}{p!(n − p)′} C_{σ_{1}...}^{I_{1}}_{σ_{p}i_{p+1}...i_{n}}^{I_{p}}ω_{I_{1}}^{σ_{1}} ∧… ∧ ω_{I_{p}}^{σ_{p}} ∧ dx^{i_{p+1}} ∧… ∧ dx^{i_{n}}, (4.10)

and the coefficients C_{σ_{1}i_{2}...i_{n}}^{I_{1}}, C_{σ_{1}...}^{I_{1}}_{σ_{n−1}i_{n}}^{I_{n−1}} are completely traceless. If ι : V_{r} → V_{r+1} is the section of J^{r+1}Y described by the equations

y_{I}^{σ} ∘ ι = 0, |I| = r + 1, (4.11)

then Theorem 1, (3.15) implies, since the coefficients in (4.6) are supposed to be traceless,

ι^{∗}ρ′_{0} = 0 (4.12)

hence

ρ′ = ι^{∗}ρ′′. (4.13)

On the other hand, we have the following result.

Lemma 4. If ρ′ is expressed by (4.5) and ρ′′ by (4.10), we have

C_{σ_{1}i_{2}...i_{n}}^{I_{1}} = \frac{∂C_{qi_{2}...i_{n}}}{∂y_{I_{1}q}^{σ_{1}}} ,

C_{σ_{1}σ_{2}i_{3}...i_{n}}^{I_{1}I_{2}} = \frac{r + 1}{r + 2} \frac{∂C_{σ_{1}qi_{3}...i_{n}}^{I_{1}}}{∂y_{I_{2}q}^{σ_{2}}} ,

…

C_{σ_{1}...}^{I_{1}}_{σ_{n−1}i_{n}}^{I_{n−1}} = \frac{r + 1}{r + n − 1} \frac{C_{σ_{1}...}^{I_{1}}_{σ_{n−2}q_{in}}^{I_{n−2}}}{∂y_{I_{n−1}q}^{σ_{n−1}}} ,

C_{σ_{1}...}^{I_{1}}_{σ_{n}}^{I_{n}} = \frac{r + 1}{r + n} \frac{C_{σ_{1}...}^{I_{1}}_{σ_{n−1}}^{I_{n−1}}q_{i_{n}}}{∂y_{I_{n}q}^{σ_{n}}} , (4.14)

and

\frac{C_{σ_{1}...}^{I_{1}}_{σ_{n}}^{I_{n}}}{∂y_{L}^{σ}} = 0, |L| = r + 1,

\frac{C_{σ_{1}...}^{I_{1}}_{σ_{j−1i_{j}...in}}^{I_{j−1}}}{∂y_{I_{j}p}^{σ_{j}}} − \frac{(r + 1)(n − j + 1)}{r + 1} \frac{C_{σ_{1}...}^{I_{1}}_{σ_{j−1qi_{j+1}...in}}^{I_{j−1}}}{∂y_{I_{j}q}^{σ_{j}}} δ_{I_{j}q}^{σ_{j}} = 0,

1 ≤ j ≤ n, antisym(i_{1}i_{2} … i_{n}) sym(I_{j}p). (4.15)

Formulas (4.14) and (4.15) follow from (3.15).

In particular, ρ′′ is completely determined by hρ′′ = hρ′ = hρ, where

hρ = \frac{1}{n!}C_{i_{1}...i_{n}}dx^{i_{1}} ∧… ∧ dx^{i_{n}}. (4.16)

Using (4.14) we get for every p, 1 ≤ p ≤ n,

C_{σ_{1}...}^{I_{1}}_{σ_{p}i_{p+1}...i_{n}}^{I_{p}} = \frac{(r + 1)^{p}}{(r + p) … (r + 2)(r + 1)} \frac{∂^{p}C_{q_{1}...q_{p}i_{p+1}...i_{n}}}{∂y_{I_{1}}^{σ_{1}}_{q_{1}} … ∂y_{I_{p}}^{σ_{p}}_{q_{p}}} . (4.17)

Now we shall extend this formula to arbitrary lagrangians belonging to Ω_{r+1,X}^{n}W. Let λ ∈ Ω_{r+1,X}^{n}W be a lagrangian. Write in a fibered chart (V, ψ), ψ = (x^{i}, y^{σ}),

{λ = Ldx^{1} ∧… ∧ dx^{n} = \frac{1}{n!}Lε_{i_{1}...i_{n}}dx^{i_{1}} ∧… ∧ dx^{i_{n}}, \atop L_{i_{1}...i_{n}} = Lε_{i_{1}...i_{n}}, (4.18)}

and put

{Λ_{(V,ψ)}(λ) \atop = ∑_{p=0}^{n} \frac{1}{p!(n − p)!}C_{σ_{1}...}^{I_{1}}_{σ_{p}}^{I_{p}}_{i_{p+1}...i_{n}}ω_{I_{1}}^{σ_{1}} ∧… ∧ ω_{I_{p}}^{σ_{p}} ∧ dx^{i_{p+1}} ∧… ∧ dx^{i_{n}}, (4.19)}

where

{C_{σ_{1}...}^{I_{1}}_{σ_{p}}^{I_{p}}_{i_{p+1}...i_{n}} = \frac{(r + 1)^{p}}{(r + p) … (r + 2)(r + 1)} \frac{∂^{p}L_{q_{1}...q_{p}i_{p+1}...i_{n}}}{∂y_{I_{1}q_{1}}^{σ_{1}} … ∂y_{I_{p}q_{p}}^{σ_{p}}} , \atop |I_{1}|,…, |I_{p}| = r. (4.20)}

Theorem 3. Let λ ∈ Ω_{r+1,X}^{n}W. The following two conditions are equivalent: (1) There exists a form ρ ∈ Ω_{r}^{n}W such that λ = hρ.

(2) For every fibered chart (V, ψ), ψ = (x^{i}, y^{σ}), the form Λ_{(V,ψ)}(λ) is π_{r+1,r}- projectable.

Suppose that (1) is satisfied. Then we have a decomposition

ρ = ρ_{0} + Λ_{(V,ψ)}(hρ) = ρ_{0} + Λ_{(V,ψ)}(λ) (4.21)

in which ρ_{0} is a contact form, and both ρ_{0}, Λ_{(V,ψ)}(λ) are defined on V_{r} ⊂ W.

Conversely, suppose that (2) holds. Is (V_{ι}, ψ_{ι}), ι ∈ I, is a system of fibered charts such that V_{ι} ⊂ W for every ι ∈ I and ∪V_{ι} = W, and if (ζ_{ι}) is a partition of unity subordinate to the covering (V_{ι}) of W, we put

ρ = ∑ ζ_{ι}Λ_{(V_{ι},ψ_{ι})}(λ). (4.22)

Then hρ = λ.

Recall that a lagrangian λ on a fibered manifold π : Y → X is called variationally trivial, if its Euler-Lagrange form E_{λ} = E_{n}(λ) vanishes, i.e., E_{λ} = 0. Our goal now is to discuss the local and global structure of variationally trivial lagrangians.

Theorem 4. Let λ ∈ Ω_{r,X}^{n}Y be a variationally trivial lagrangian on a fibered manifold π : Y → X, n = dim X. Then to each point J_{x}^{r}γ ∈ J^{r}Y there exist a neighborhood V of γ(x) ∈ Y and an n-form ρ_{V} defined on V_{r−1} ⊂ J^{r−1}Y such that (1) λ = hρ_{V} on V_{r}, and (2) dρ_{V} = 0.

To prove Theorem 4 we may suppose that V = U × W, where U ⊂ R^{n} is an open set and W ⊂ R^{m} is an open ball with centre 0. Let x^{i}, y^{σ}, 1 ≤ i ≤ n, 1 ≤ σ ≤ m, be the canonical coordinate on V. We define a mapping χ : R × V_{r} → V_{r} by the formula

χ(s, (x^{i}, y^{σ}, y_{j_{1}}^{σ},…, y_{j_{1}...j_{r}}^{σ})) = (x^{i}, y^{σ}, y_{j_{1}}^{σ},…, y_{j_{1}...j_{r−1}}^{σ}, sy_{j_{1}...j_{r}}^{σ}). (4.23)

Let η be any (n − 1)-form on V_{r} such that hdη is π_{r+1,r}-projectable. Write

η = η_{0} + η_{1}, (4.24)

where η_{0} is generated by the forms ω^{σ}, ω_{j_{1}},…, ω_{j_{1}...j_{r−1}}^{σ} and

η_{1} = B_{i_{2}...i_{n}}dx^{i_{2}} ∧… ∧ dx^{i_{n}} + B_{σ_{2}i_{3}...i_{n}}^{I_{2}}dy_{I_{2}}^{σ_{2}} ∧ dx^{i_{3}} ∧… ∧ dx^{i_{n}}
+ B_{σ_{2}σ_{3}i_{4}...i_{n}}^{I_{2}I_{3}}dy_{I_{2}}^{σ_{2}} ∧ dy_{I_{3}}^{σ_{3}} ∧ dx^{i_{4}} ∧… ∧ dx^{i_{n}} + …
+ B_{σ_{2}...}^{I_{2}}_{σ_{n−1}i_{n}}^{I_{n−1}}dy_{I_{2}}^{σ_{2}} ∧… ∧ dy_{I_{n−1}}^{σ_{n−1}} ∧ dx^{i_{n}} + B_{σ_{2}...}^{I_{2}}_{σ_{n}}^{I_{n}}dy_{I_{2}}^{σ_{2}} ∧… ∧ dy_{I_{n}}^{σ_{n}}. (4.25)

Clearly,

hdη = hdη_{1}. (4.26)

In particular, hdη_{1} is π_{r+1,r}-projectable. By a direct computation

hdη_{1} = [d′_{i_{1}}B_{i_{2}...i_{n}} + (\frac{∂B_{i_{2}...i_{n}}}{∂y_{I_{1}}^{σ_{1}}} − d′_{i_{1}}B_{σ_{1}i_{3}...i_{n}}^{I_{1}})

+ (\frac{∂B_{σ_{2}i_{3}...i_{n}}^{I_{2}}}{∂y_{I_{1}}^{σ_{1}}} + d′_{i_{3}}B_{σ_{1}σ_{2}i_{4}...i_{n}}^{I_{1}I_{2}})y_{I_{1}i_{1}}^{σ_{1}}y_{I_{2}i_{2}}^{σ_{2}} + …

+ (\frac{∂B_{σ_{2}...}^{I_{2}}_{σ_{n−1}i_{n}}^{I_{n−1}}}{∂y_{I_{1}}^{σ_{1}}} + (−1)^{n−1}d′_{i_{n}}B_{σ_{1}...}^{I_{1}}_{σ_{n−1}}^{I_{n−1}}) · y_{I_{1}i_{1}}^{σ_{1}} … y_{I_{n−1}i_{n−1}}^{σ_{n−1}}

+ \frac{∂B_{σ_{2}...}^{I_{2}}_{σ_{n}}^{I_{n}}}{∂y_{I_{1}}^{σ_{1}}} y_{I_{1}i_{1}}^{σ_{1}} … y_{I_{n}i_{n}}^{σ_{n}}]dx^{i_{1}} ∧… ∧ dx^{i_{n}}. (4.27)

This formula enables us to express the conditions of π_{r+1,r}-projectability of hdη_{1} explicitly.

Put

χ_{1} = C_{σ_{3}...}^{I_{3}}_{σ_{n}}^{I_{n}}dy_{I_{3}}^{σ_{3}} ∧… ∧ dy_{I_{n}}^{σ_{n}}, (4.28)

where

C_{σ_{j}...}^{I_{3}}_{σ_{n}}^{I_{n}} = y_{J}^{ν} ∫ B_{νσ_{3}...}^{JI_{3}}_{σ_{n}}^{I_{n}} ∘ χ · s^{n−2}ds (4.29)

(integration from 0 to 1). Write dχ_{1} in the form

{dχ_{1} = d′C_{σ_{3}...}^{I_{3}}_{σ_{n}}^{I_{n}} ∧ dy_{I_{3}}^{σ_{3}} ∧… ∧ dy_{I_{n}}^{σ_{n}} \atop + \frac{∂C_{σ_{3}...}^{I_{3}}_{σ_{n}}^{I_{n}}}{∂y_{I_{2}}^{σ_{2}}} dy_{I_{2}}^{σ_{2}} ∧ dy_{I_{3}}^{σ_{3}} ∧… ∧ dy_{I_{n}}^{σ_{n}}, (4.30)}

where

\frac{∂C_{σ_{3}...}^{I_{3}}_{σ_{n}}^{I_{n}}}{∂y_{I_{2}}^{σ_{2}}} dy_{I_{2}}^{σ_{2}} ∧… ∧ dy_{I_{n}}^{σ_{n}} = (∫ B_{σ_{2}...}^{I_{2}}_{σ_{n}}^{I_{n}} ∘ χ · s^{n−2}ds

+ y_{J}^{ν} ∫ \frac{∂B_{νσ_{3}...}^{JI_{3}}_{σ_{n}}^{I_{n}}}{∂y_{I_{2}}^{σ_{2}}} ∘ χ · s^{n−1}ds)dy_{I_{2}}^{σ_{2}} ∧… ∧ dy_{I_{n}}^{σ_{n}}

= [∫ B_{σ_{2}...}^{I_{2}}_{σ_{n}}^{I_{n}} ∘ χ · s^{n−2}ds + \frac{1}{n − 1} ∫ ((\frac{∂B_{νσ_{3}...}^{JI_{3}}_{σ_{n}}^{I_{n}}}{∂y_{I_{2}}^{σ_{2}}} − \frac{∂B_{νσ_{2}σ_{4}...}^{JI_{2}I_{4}}_{σ_{n}}^{I_{n}}}{∂y_{I_{3}}^{σ_{3}}} −…

− \frac{∂B_{νσ_{3}...}^{JI_{3}}_{σ_{n−1}σ_{2}}^{I_{n−1}I_{2}}}{∂y_{I_{n}}^{σ_{n}}} ) · y_{J}^{ν}) ∘ χ · s^{n−2}ds]dy_{I_{2}}^{σ_{2}} ∧… ∧ dy_{I_{n}}^{σ_{n}}. (4.31)

By (4.27)

\frac{∂B_{σ_{2}σ_{3}...}^{I_{2}I_{3}}_{σ_{n}}^{I_{n}}}{∂y_{J}^{ν}} = \frac{∂B_{νσ_{3}...}^{JI_{3}}_{σ_{n}}^{I_{n}}}{∂y_{I_{2}}^{σ_{2}}} − \frac{∂B_{νσ_{2}σ_{4}...}^{JI_{2}I_{4}}_{σ_{n}}^{I_{n}}}{∂y_{I_{3}}^{σ_{3}}} − ⋯ − \frac{∂B_{νσ_{3}...}^{JI_{3}}_{σ_{n−1}σ_{2}}^{I_{n−1}I_{2}}}{∂y_{I_{n}}^{σ_{n}}} , (4.32)

from which it follows that

\frac{∂C_{σ_{3}...}^{I_{3}}_{σ_{n}}^{I_{n}}}{∂y_{I_{2}}^{σ_{2}}} dy_{I_{2}}^{σ_{2}} ∧… ∧ dy_{I_{n}}^{σ_{n}} = (∫ B_{σ_{2}...}^{I_{2}}_{σ_{n}}^{I_{n}} ∘ χ · s^{n−2}ds

+ \frac{1}{n − 1} ∫ (\frac{∂B_{σ_{2}...}^{I_{2}}_{σ_{n}}^{I_{n}}}{∂y_{I_{2}}^{σ_{2}}} y_{J}^{ν}) ∘ χ · s^{n−2}ds)dy_{I_{2}}^{σ_{2}} ∧… ∧ dy_{I_{n}}^{σ_{n}}

= ∫ \frac{d}{ds}(σ_{2} …^{I_{2}} _{σ_{n}}^{I_{n}} ∘ χ · \frac{1}{n − 1}s^{n−1})ds · dy_{I_{2}}^{σ_{2}} ∧… ∧ dy_{I_{n}}^{σ_{n}}

= B_{σ_{2}...}^{I_{2}}_{σ_{n}}^{I_{n}}dy_{I_{2}}^{σ_{2}} ∧… ∧ dy_{I_{n}}^{σ_{n}}. (4.33)

Consequently

η_{1} = η_{2} + dχ_{1} − p(d′C_{σ_{3}...}^{I_{3}}_{σ_{n}}^{I_{n}}) ∧ dy_{I_{3}}^{σ_{3}} ∧… ∧ dy_{I_{n}}^{σ_{n}}, (4.34) where

η_{2} = B_{i_{2}...i_{n}}dx^{i_{2}} ∧… ∧ dx^{i_{n}} + B_{σ_{2}i_{3}...i_{n}}^{I_{2}}dy_{I_{2}}^{σ_{2}} ∧ dx^{i_{3}} ∧… ∧ dx^{i_{n}}
+ B_{σ_{2}σ_{3}i_{4}...i_{n}}^{I_{2}I_{3}}dy_{I_{2}}^{σ_{2}} ∧ dy_{I_{3}}^{σ_{3}} ∧ dx^{i_{4}} ∧… ∧ dx^{i_{n}} + …
+ (B_{σ_{2}...}^{I_{2}}_{σ_{n−1}i_{n}}^{I_{n−1}} − (−1)^{n−2}d′_{i_{n}}C_{σ_{2}...}^{I_{2}}_{σ_{n−1}i_{n}}^{I_{n−1}})dy_{I_{2}}^{σ_{2}} ∧… ∧ dy_{I_{n−1}}^{σ_{n−1}} ∧ dx^{i_{n}}. (4.35)

Clearly

hdη_{1} = hdη_{2}, (4.36)

Now we can use (4.27) again replacing η_{1} by η_{2}. After several steps we conclude that η is expressible in the form

η = ν + dφ + ψ, (4.37)

where ν is π_{r}-horizontal,

ν = B̄_{i_{2}...i_{n}}dx^{i_{2}} ∧… ∧ dx^{i_{n}}, (4.38)

and ψ is contact; the form hdη = hdν is π_{r+1,r}-projectable which means that

\frac{∂B̄_{i_{2}...i_{n}}}{∂y_{I_{1}}^{σ_{1}}} = \frac{r(n − 1)}{r + 1}δ_{i_{2}}^{j_{1}} \frac{∂B̄_{qi_{3}...i_{n}}}{∂y_{qj_{2}...j_{r}}^{σ}}

antisym(i_{2}i_{3} … i_{n}) sym(j_{1}j_{2} … j_{r}); I_{1} = (j_{1}j_{2} … j_{r}). (4.39)

Let U ⊂ R^{n} be an open set, W ⊂ R^{m} an open ball with centre 0, and denote by (x^{i}, y^{σ}) the canonical coordinates on U × W. Put for every (s, (x^{i}, y^{σ})) ∈ R × (U × W)

χ(s, (x^{i}, y^{σ})) = (x^{i}, sy^{σ}). (4.40)

χ is a mapping from R × (U × W) into U × W. Let k ≥ 1, and let ρ be a k-form on U × W. There exists a unique decomposition.

χ^{∗}ρ = ds ∧ ρ_{0}(s) + ρ′(s), (4.41)

where the form ρ_{0}(s), ρ′(s) do not contain ds. We set

Iρ = ∫ ρ_{0}(s), (4.42)

where the right-hand side means that we integrate the coefficients in the form ρ_{0}(s) over s from 0 to 1. Let π : U × W → U be the canonical projection, and let ι : U → U × W be the zero section. Then it is standart to show that

ρ = Idρ + dIρ + π^{∗}ι^{∗}ρ. (4.43)

In what follows we apply this formula to the case of the jet projection π_{r,r−1} = V_{r} → V_{r−1}, where V = U × W.

Let λ be a variationally trivial lagrangian on V_{r}. Using the variational sequence (2.25) we get π_{r+1,r}^{∗}λ = E_{n−1}hν = hdν, where ν is an (n − 1)-form on V_{r}. Since hdν is π_{r+1,r}-projectable we may suppose, by the first part of the proof, that ν is π_{r}-horizontal. Put

ρ = dν, η = Idν, (4.44)

where I is defined by (4.42) (with respect by the projection π_{r,r−1} : V_{r} → V_{r−1}. Is ι : V_{r−1} → V_{r}). By (4.43)

ρ = dη + π_{r,r−1}^{∗}ι^{∗}ρ. (4.45)

Clearly,

ι^{∗}dη = dι^{∗}η = 0, (4.46)

and it is easily verified with the help of (4.41) and (4.42) that η is π_{r}-horizontal.

Let Ω ⊂ U be any compact, n-dimensional submanifold with boundary ∂Ω. Let us consider a section w : V_{r−1} → V_{r} satisfying

supp w ⊂ V_{r−1} ∩ π_{r−1}^{−1}(Ω). (4.47)

If γ : U → V is a section and x ∈ ∂Ω, then

(w ∘ J^{r−1}γ)(x) = (w ∘ J^{r−1}γ)(x). (4.48)

Now we integrate both sides of the equality

w^{∗}ρ = dw^{∗}η + ι^{∗}ρ (4.49)

arising from (4.45). We obtain by (4.48) and (4.46)

∫_{Ω} J^{r−1}γ^{∗}dw^{∗}η = ∫_{∂Ω} J^{r−1}γ^{∗}w^{∗}η = ∫_{∂Ω}(w ∘ J^{r_{1}}γ)^{∗}η ; = ∫_{∂Ω}(ι ∘ J^{r_{1}}γ)^{∗}η = ∫_{∂Ω} J^{r−1}γ^{∗}i^{∗}η = ∫_{Ω} J^{r−1}γ^{∗}di^{∗}η = 0 (4.50)

so that

∫_{Ω} J^{r−1}γ^{∗}(w^{∗}ρ − ι^{∗}ρ) = 0. (4.51)

Since this condition holds for every γ we get on Ω

hw^{∗}ρ = hi^{∗}ρ (4.52)

and (4.49) gives

hdw^{∗}η = 0. (4.53)

Writing

η = B_{i_{2}...i_{n}}dx^{i_{2}} ∧… ∧ dx^{i_{n}} (4.54)

we get

hdw^{∗}η = d_{i_{1}}(B_{i_{2}...i_{n}} ∘ w)dx^{i_{1}} ∧… ∧ dx^{i_{n}} = 0, (4.55) i.e.,

\frac{∂B_{i_{2}...i_{n}} ∘ w}{∂x^{i_{1}}} + \frac{∂B_{i_{2}...i_{n}} ∘ w}{∂y^{σ}} y_{i_{1}}^{σ} + … + \frac{∂B_{i_{2}...i_{n}} ∘ w}{∂y_{q_{1}...q_{r−1}i_{1}}^{σ}} y_{q_{1}...q_{r−1}i_{1}}^{σ}

= \frac{∂B_{i_{2}...i_{n}}}{∂x^{i_{1}}} ∘ w + \frac{∂B_{i_{2}...i_{n}}}{∂y_{j_{1}...j_{r}}^{ν}} ∘ w · \frac{∂y_{j_{1}...j_{r}}^{ν} ∘ w}{∂x^{i_{1}}}

+ (\frac{∂B_{i_{2}...i_{n}}}{∂y^{σ}} ∘ w + \frac{∂B_{i_{2}...i_{n}}}{∂y_{j_{1}...j_{r}}^{ν}} ∘ w · \frac{∂y_{j_{1}...j_{r}}^{ν} ∘ w}{∂y^{σ}} )y_{i_{1}}^{σ} + …

+ (\frac{∂B_{i_{2}...i_{n}}}{∂y_{q_{1}...q_{r−1}}^{σ}} ∘ w + \frac{∂B_{i_{2}...i_{n}}}{∂y_{j_{1}...j_{r}}^{ν}} ∘ w · \frac{∂y_{j_{1}...j_{r}}^{ν} ∘ w}{∂y_{q_{1}...q_{r_{1}}}^{σ}} )y_{q_{1}...q_{r−1}i_{1}}^{σ}

= \frac{∂B_{i_{2}...i_{n}}}{∂x^{i_{1}}} ∘ w + \frac{∂B_{i_{2}...i_{n}}}{∂y^{σ}} ∘ w · y_{i_{1}}^{ν} + …

+ \frac{∂B_{i_{2}...i_{n}}}{∂y_{q_{1}...q_{r−1}}^{σ}} ∘ w · y_{q_{1}...q_{r−1}i_{1}}^{σ} + \frac{∂B_{i_{2}...i_{n}}}{∂y_{j_{1}...j_{r}}^{σ}} ∘ w · d_{i_{1}}(y_{j_{1}...j_{r}}^{ν} ∘ w)

= 0. (4.56)

Since the functions y_{j_{1}...j_{r}}^{ν} ∘ w are arbitrary, we have

\frac{∂B_{i_{2}...i_{n}}}{∂y_{j_{1}...j_{r}}^{ν}} ∘ w = 0, (4.57)

and

\frac{∂B_{i_{2}...i_{n}}}{∂y_{j_{1}...j_{r}}^{ν}} = 0. (4.58)

In particular, η is defined on V_{r−1}, and so must be ρ (4.43) as required by Theorem 4.

Using the variational sequence it is easy to obtain the following slightly different version of Theorem 4 refining the results known so far on the kernel of the higher order Euler-Lagrange mapping (see [1]).

Theorem 5. Let λ ∈ Ω_{r,X}^{n}Y be a lagrangian on a fibered manifold π : Y → X. λ is variationally trivial if and only if there exists an n-form ρ on J^{r−1}Y such that (1) λ = hρ, (2) dρ ∈ Θ_{r−1}^{n+1}Y.

Only necessity of conditions (1), (2) needs proof. We have an exact sequence

0 → Θ_{r−1}^{n} → Ω_{r−1}^{n} → Ω_{r−1}^{n}/Θ_{r−1}^{n} → 0 (4.59)

in which Θ_{r−1}^{n}, Ω_{r−1}^{n} are soft sheaves. Hence Ω_{r−1}^{n}/Θ_{r−1}^{n} is also soft, and the induced sequence of global sections must be exact. But by Theorem 4 a variationally trivial lagrangian λ ∈ Ω_{r,X}^{n}Y belongs to Ω_{r−1}^{n}/Θ_{r−1}^{n}. This ensures the existence of a form ρ on J^{r−1}Y such that hρ = λ. Then obviously dρ ∈ Θ_{r−1}^{n+1}.

Corollary 1. A lagrangian λ ∈ Ω_{1,X}^{n}Y is variationally trivial if and only if there exists an n-form ρ on Y such that (1) λ = hρ, (2) dρ = 0.

By Theorem 4, if E_{n}(λ) = 0, then λ = hρ for an n-form on J^{0}Y = Y such that dρ ∈ Θ_{0}^{n+1}. But Θ_{0}^{n+1} = {0}.

Remark 2. Condition (1) of Theorem 5 completely describes the polynomial structure of variationally trivial lagrangians in the variables y_{j_{1}...j_{r}}^{σ}; the polynomial structure is therefore defined by the horizontalization h. Condition (2) of Theorem 4 then determines relations among the coefficients of the polynomials.

In the following theorem we summarize our result on the structure of variationally trivial lagrangians; in its second part we use a global hypothesis and derive global results, based on the variational sequence (2.25). We denote by H^{n}Y the n-th cohomology group of Y.

Theorem 6. Let λ ∈ Ω_{r,X}^{n}Y be a lagrangian, n = dim X. λ is variationally trivial if and only if to each point J_{x}^{r}γ ∈ J^{r}Y there exists a neighborhood V of γ(x) ∈ Y and an (n − 1)-form η_{V} defined on V_{r−1} ⊂ J^{r−1}Y such that on V_{r},

λ = hdη_{V} . (4.60)

(b) Suppose that λ is variationally trivial and moreover, H^{n}Y = {0}. Then there exists an (n − 1)-form η defined on J^{r−1}Y such that

λ = hdη. (4.61)

To prove (a) we use the identity E_{n−1}hη_{V} = hdη_{V}, where λ = E_{n−1}hη_{V}.

Suppose that λ is variationally trivial and H^{n}Y = {0}. Then λ = E_{n−1}τ, where τ is a global section of Ω_{r−1}^{n−1}/Θ_{r−1}^{n−1}. Since the induced sequence of global section associated to the exact sequence

0 → Θ_{r−1}^{n} → Ω_{r−1}^{n} → Ω_{r−1}^{n}/Θ_{r−1}^{n} → 0 (4.62)

is exact, τ = hη for some global section η of Ω_{r−1}^{n}. Then λ = E_{n−1}hη = hdη.

If a lagrangian λ is variationally trivial and there exists a form η such that λ = hdη, we say that λ is globally variationally trivial. The condition H^{n}Y = {0} guarantees that every variationally trivial lagrangian λ ∈ Ω_{r,X}^{n}Y is globally variational, and is of the form λ = hdη, where η ∈ Ω_{r−1}^{n}Y.

As mentioned in Introduction, a complete characterization of the so called generalized invariant transformations of a lagrangian λ requires the knowledge of the kernel of the Euler-Lagrange mapping. The definition of a generalized invariant transformation of λ, saying that the transformed lagrangian should belong to the kernel of the Euler-Lagrange mapping, can then effectively be applied.

Having now a proper theorem on the structure of the variationally trivial lagrangians (Theorem 6), we are in a position to give a precise description of generators of the generalized invariants transformations, quite in the fashion of the first order case [10]. Recall that a π-projectable vector field ξ on Y is said to generate generalized invariant transformations of a lagrangian λ ∈ Ω_{r,X}^{n}Y, if its one-parameter group preserves the Euler-Lagrange form E_{λ} = E_{n}(λ) i.e.,

∂_{J^{2r}ξ}E_{λ} = 0, (4.63)

where J^{2r}ξ is the 2r-jet prolongation of ξ. The following result gives a complete description of the generators of generalized invariants transformations.

Theorem 7. Let λ ∈ Ω_{r,X}^{n}Y be a lagrangian, and let ξ be a π-projectable vector field. The following three conditions are equivalent:

(1) ξ generates generalized invariants transformations of λ.

(2) There exists an n-form ρ on J^{r−1}Y such that

∂_{J^{r}ξ}λ = hρ, dρ ∈ Θ_{r−1}^{n+1}Y. (4.64)

(3) For every fibered chart (V, ψ) on Y, dΛ_{(V,ψ)}∂_{J^{r}ξ}λ is π_{r,r−1}-projectable, and

dΛ_{(V,ψ)}∂_{J^{r}ξ}λ ∈ Θ_{r−1}^{n+1}V. (4.65)

Suppose that ξ generates generalized invariants transformations of λ. Then

∂_{J^{2r}ξ}E_{λ} = E_{∂_{J^{2r}ξ}λ} = 0, (4.66)

and we apply Theorem 6.

Suppose that (2) is satisfied, and choose a fibered chart (V, ψ) on Y. Write

ρ = ρ_{0} + Λ_{(V,ψ)}hρ = ρ_{0} + Λ_{(V,ψ)}∂_{J^{r}ξ}λ, (4.67)

where ρ_{0} is contact. Since by hypothesis, dρ ∈ Θ_{r−1}^{n+1}Y, and obviously dρ_{0} ∈ Θ_{r−1}^{n+1}V, we have

dΛ_{(V,ψ)}∂_{J^{r}ξ}λ ∈ Θ_{r−1}^{n+1}V (4.68)

as required.

Suppose that (3) holds, and consider the lagrangian

hΛ_{(V,ψ)}∂_{J^{r}ξ}λ = ∂_{J^{r}ξ}λ. (4.69)

The Euler-Lagrange form E_{n}(∂_{J^{r}ξ}λ) is the class of the form dΛ_{(V,ψ)}∂_{J^{r}ξ}λ. By hypothesis this class is equal to 0.

Equation (4.65) replaces the generalized Noether-Bessel-Hagen equation (1.6) on a higher order jet space J^{r}Y. If r = 1 it reduces to (1.6).

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[13] D. Krupka, Variational sequences on finite order jet spaces, Diff. Geom. and Appl., Proc. Conf., Aug. 27–Sept. 2, 1989, Brno, Czechoslovakia, World Scientific, Singapore, 1990, 236–254.

[14] F. Takens, A global version of the inverse problem of the calculus of variations, J. Differential Geometry 14 (1979) 543–562.

[15] W. M. Tulczyjew, The Euler-Lagrange resolution, Internat. Coll. on Diff. Geom. Methods in Math. Physics, Aix-en-Provence, Sept. 1979; Lecture Notes in Math. 836, Springer, Berlin, 1980, 22–48.

[16] A. M. Vinogradov, The C-spectral sequence, Lagrangian formalism, and conservation laws. I. The linear theory, II. The nonlinear theory, Math. Anal. Appl. 100 (1984) 1–40, 41–129.

D. Krupka Department of Mathematics Silesian University Opava Bezručovo nám. 17 746 01 Opava Czech Republic

Received 28 October 1992