DIFFERENTIAL GEOMETRY AND APPLICATIONS Satellite Conference of ICM in Berlin, Aug. 10 – 14, 1998, Brno Masaryk University in Brno (Czech Republic) 1999, 525–531

VARIATIONAL SEQUENCES AND VARIATIONAL BICOMPLEXES

DEMETER KRUPKA

Abstract. We compare basic features of the variational sequence, and the variational bicomplex theories on a fibered manifold. We show that these two theories, based on different underlying geometric structures, give different results.

1. Introduction

The idea that there should exist a close relationship between the exterior derivative operator on one side, and the Euler-Lagrange expressions on the other side, first appeared in the works of Lepage and Dedecker. Later, this idea led to introducing Lepage forms in field theory (Krupka [10], [12]). Let π : Y → X be a fibered manifold over an n-dimensional base X, and let J^{r}Y be the r-jet prolongation of Y. The concept of a Lepage form allowed to construct a commutative diagram

Lep ⟶^{d} Ω_{n+1}^{2r−1}Y
h ↓ ↓ h̃
Lagr ⟶^{E} Ω_{n+1}^{2r}Y

in which Lagr stands for lagrangians of order r on Y (i.e., for horizontal n-forms on J^{r}Y), Lep denotes Lepage equivalents of lagrangians of order r, Ω_{n+1}^{2r−1}Y (resp. Ω_{n+1}^{2r}Y) denotes the module of (n + 1)-forms on J^{2r−1}Y (resp. J^{2r}Y), and h and h̃ are horizontalizations of differential forms, induced by the fibered structure of Y; E is the Euler-Lagrange mapping, assigning to a lagrangian the corresponding Euler-Lagrange form.

To the end of the seventieth, Anderson and Duchamp [2], Dedecker and Tulczyjew [4], Takens [18], Tulczyjew [19], and Vinogradov [20], observed, on the basis of different assumptions, that one can include the Euler-Lagrange operator into an exact sequence, constructed by means of a variational bicomplex. To this purpose, they used an invariant decomposition of the exterior derivative operator on the infinite prolongation J^{∞}Y into its horizontal and contact components.

1991 Mathematics Subject Classification. 49F05, 58A15, 58E99.

Key words and phrases. Fibered manifold, lagrangian, Euler-Lagrange mapping, variational sequence, variational bicomplex.

Research supported by Grant No 201/98/0853 of the Czech Grant Agency, and Project VS 96003 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.

A different modification of similar ideas was used by Krupka [11], [13], who considered the De Rham sequence of differential forms on J^{r}Y, with r fixed, and a subsequence of the De Rham sequence, such that the corresponding quotient sequence included the Euler-Lagrange mapping as one of its quotient mappings.

Pommaret [15] and Vitolo [21] compared and discussed the relationship of these two theories.

In this short note, we briefly recall basic concepts of the variational sequence theory, and on the differences between this theory and the variational bicomplex theory. For the variational bicomplex theory, the reader is referred to [1], [2], [3], [4], [6], [8], [16], [18], [19], [20].

2. The variational sequence

Let π : Y → X be a fibered manifold, n = dim X, n + m = dim Y. We denote by J^{r}Y the r-jet prolongation of Y, and by π^{r} : J^{r}Y → X, π^{r,s} : J^{r}Y → J^{s}Y the canonical jet projections. J^{r}γ : X → J^{r}Y denotes the r-jet prolongation of a section γ : X → Y.

An n-form λ on J^{r}Y, horizontal with respect to the projection π^{r}, is called a lagrangian of order r for Y. If (V, ψ), ψ = (x^{i}, y^{σ}), is a fibered chart on Y, and (V^{r}, ψ^{r}), ψ^{r} = (x^{i}, y^{σ}, y_{j_{1}}^{σ}, y_{j_{1}j_{2}}^{σ}, …, y_{j_{1}j_{2}...j_{r}}^{σ}), is the associated chart on J^{r}Y, then

λ = L(x^{i}, y^{σ}, y_{j_{1}}^{σ}, y_{j_{1}j_{2}}^{σ}, …, y_{j_{1}j_{2}...j_{r}}^{σ}) dx^{1} ∧ dx^{2} ∧ … ∧ dx^{n}.

Given λ, we get for any compact, n-dimensional submanifold Ω ⊂ X with boundary, a real-valued function

γ → ∫_{Ω} J^{r}γ^{∗}λ (1)

defined on sections of Y (the action function associated with λ and Ω). The first variation formula for (1) can be derived in an intrinsic way by means of the Lepage forms, the Lie derivative, and the exterior derivative d. This involves introducing a global Euler-Lagrange form associated with λ,

E_{λ} = E_{σ}(L) dy^{σ} ∧ dx^{1} ∧ dx^{2} ∧ … ∧ dx^{n},

where

E_{σ}(L) = \frac{∂L}{∂y^{σ}} + ∑_{k=1}^{r}(−1)^{k}d_{i_{1}}d_{i_{2}} … d_{i_{k}} \frac{∂L}{∂y_{i_{1}i_{2}...i_{k}}^{σ}}

are the Euler-Lagrange expressions. The mapping λ → E_{λ} is called the Euler-Lagrange mapping. The kernel of this mapping describes trivial, or null lagrangians, its image are variational (n + 1)-forms.

If (V, ψ), ψ = (x^{i}, y^{σ}), is a fibered chart on Y, we define

ω_{j_{1}j_{2}...j_{k}}^{σ} = dy_{j_{1}j_{2}...j_{k}}^{σ} − y_{j_{1}j_{2}...j_{k}i}^{σ}dx^{i}, 1 ≤ k ≤ r. (2)

These forms define a basis of 1-forms on J^{r+1}Y, dx^{i}, ω^{σ}, ω_{j_{1}}^{σ}, …, ω_{j_{1}j_{2}...j_{r}}^{σ}, dy_{j_{1}j_{2}...j_{r+1}}^{σ}. If ρ is any k-form on J^{r}Y, the forms (2) define a decomposition

(π^{r+1,r})^{∗}ρ = hρ + p_{1}ρ + p_{2}ρ + ⋯ + p_{k}ρ

as follows: We take for the chart expression of hρ, the terms in (π^{r+1,r})^{∗}ρ which contain no factor (2), and for p_{k}ρ the terms containing exactly k factors (2). Then h is a homomorphism of exterior algebras called the horizontalization. We say that ρ is contact if hρ = 0. If k ≥ n + 1, ρ is always contact; we say that ρ is strongly contact if p_{k−n}ρ = 0. In a fibered chart,

hf = (π^{r+1,r})^{∗}f, hdx^{i} = dx^{i}, hdy_{j_{1}j_{2}...j_{k}}^{σ} = y_{j_{1}j_{2}...j_{k}i}^{σ}dx^{i}.

Now we introduce some sheaves of forms on J^{r}Y. Let Ω_{k}^{r} be the sheaf of smooth k-forms over J^{r}Y. If 1 ≤ k ≤ n (resp. k ≥ n + 1), denote by Ω_{k,c}^{r}, the subsheaf of contact (resp. strongly contact) forms (Ω_{0,c}^{r} = {0}). Define

Θ_{k}^{r} = Ω_{k,c}^{r} + dΩ_{k−1,c}^{r},

where k ≥ 1, and dΩ_{k−1,c}^{r} is the image sheaf of Ω_{k−1,c}^{r} by d. We get a subsequence 0 → Θ_{1}^{r} → Θ_{2}^{r} → Θ_{3}^{r} → ⋯ of the DeRham sequence

0 → R_{Y} → Ω_{1}^{r} → Ω_{2}^{r} → Ω_{3}^{r} → ⋯

and the quotient sequence

0 → R_{Y} → Ω_{1}^{r}/Θ_{1}^{r} → Ω_{2}^{r}/Θ_{2}^{r} → Ω_{3}^{r}/Θ_{3}^{r} → ⋯ (3)

which is called the variational sequence. The quotient mapping, denoted by E, is defined by E([ρ]) = [dρ], where [ρ] is the class of a form ρ. It can be shown that the variational sequence is an acyclic resolution of the constant sheaf R_{Y}. Consequently, by the abstract DeRham theorem, the complex of global sections of the variational sequence has the same cohomology as the manifold Y. Denoting the sequence (3) by 0 → R_{Y} → V^{r}, and denoting the corresponding cochain complex of global sections by

0 → Γ(Y, R_{Y}) → Γ(Y, Ω_{0}^{r}) → Γ(Y, Ω_{1}^{r}/Θ_{1}^{r}) → Γ(Y, Ω_{2}^{r}/Θ_{2}^{r}) → ⋯

or simply by Γ(Y, V^{r}), we have H^{k}(Γ(R_{Y} , V^{r})) = H^{k}(Y, R).

It should be pointed out that each term in the variational sequence, which is, as a quotient, formed by classes of forms, is determined up to an isomorphism. This means, however, that we may ask whether there exists a representation of a term by means of a module of forms. A basic observation connecting the variational sequence with the calculus of variations, comes from an analysis of the (n − 1), n, (n + 1)- terms in (3) (the variational terms). Describing the sheaf Ω_{n}^{r}/Θ_{n}^{r} (resp. Ω_{n+1}^{r}/Θ_{n+1}^{r}) as a certain subsheaf of the sheaf of forms Ω_{k}^{r+1} (resp. Ω_{k}^{2r−1}), one can easily see that the corresponding representation of the qoutient mapping E : Ω_{n}^{r}/Θ_{n}^{r} → Ω_{n+1}^{r}/Θ_{n+1}^{r} coincides with the Euler-Lagrange mapping λ → E_{λ}.

Representations of variational sequences by forms in higher order mechanics and in the first order field theory, as well as different computational techniques, are discussed by Kašparová [7], Krbek and Musilová [9], Musilová [14], Francaviglia, Palese and Vitolo [5], and Štefánek [17].

3. The variational bicomplex

Now we briefly recall some concepts of the variational bicomplex theory. There are several approaches to the subject; for the discussion in this paper, we choose [16]. As before, π : Y → X denotes a fibered manifold, n = dim X, n+m = dim Y.

The infinite jet prolongation of Y, needed in the variational bicomplex theory, is denoted by J^{∞}Y; J^{∞}Y is introduced by various authors in slightly different ways, with different (topological, algebraic, smooth) structures understood on the set J^{∞}Y (compare with Dedecker and Tulczyjew, Saunders, Takens, Vinogradov). The module of (p + q)-forms on J^{∞}Y, constructed by the inverse limit operation with respect to the jet projections π^{∞,r} : J^{∞}Y → J^{r}Y can canonically be decomposed into the direct sum of submodules containing p contact, and q horizontal factors. These submodules are denoted by Φ_{q}^{p}.

Correspondingly, the exterior derivative d admits a decomposition into its horizontal and vertical components as well. We write for this decomposition

d = d_{h} + d_{v}.

We have the following commutative diagram with locally exact rows and columns:

0  0
↓  ↓
R  R  0  0  0
↓  ↓  ↓  ↓  ↓
0 ⟶ Λ^{0}X ⟶ Φ_{0}^{0}  ⟶^{d_{ν}} Φ_{1}^{0} ⋯ Φ_{s}^{0}  ⟶^{d_{ν}} Φ_{s+1}^{0} ⋯
↓  d_{h}↓  −d_{h}↓  ↓  ↓
0 ⟶ Λ^{1}X ⟶ Φ_{0}^{1}  ⟶^{d_{ν}} Φ_{1}^{1} ⋯ Φ_{s}^{1}  ⟶^{d_{ν}} Φ_{s+1}^{1} ⋯
⋮  ⋮  ⋮  ⋮  ⋮
0 ⟶ Λ^{n−1}X ⟶ Φ_{0}^{n−1} ⟶^{d_{ν}} Φ_{1}^{n−1} ⋯ Φ_{s}^{n−1} ⟶^{d_{ν}} Φ_{s+1}^{n−1} ⋯
↓  d_{h}↓  −d_{h}↓  ↓  ↓
0 ⟶ Λ^{n}X ⟶ Φ_{0}^{n}  ⟶^{d_{ν}} Φ_{1}^{n} ⋯ Φ_{s}^{n}  ⟶^{d_{ν}} Φ_{s+1}^{n} ⋯
↓  ↓  ↓  ↓  ↓
0  ⟶ Ξ_{0} ⟶ Ξ_{1} ⋯ Ξ_{s} ⟶ Ξ_{s+1} ⋯
↓  ↓  ↓  ↓
0  0  0  0

In this diagram, the first two horizontal arrows in each row are natural inclusions, and the firs column consists of exterior derivatives. Ξ_{k} stands for the corresponding quotient space, and the mappings in the last row stand for the quotient morphisms. In order to ensure commutativity, the horizontal component d_{h} of d is taken with the minus sign in each odd column. The Euler-Lagrange mapping in this context is the mapping E : Ξ_{n} → Ξ_{n+1}.

4. Remarks

The reader can easily see that the theory of finite order variational sequences differs from the variational bicomplex theory in many respects. Now we discuss some points where differences appear.

Differences in formulation of the theories are apparent, and consist in the choice of basic structures serving as domains of definition of the exterior derivative operator d. Both theories are based on an analysis of d. In the first case, d acts on the modules Ω_{p}^{r}W of differential forms on jet prolongations J^{r}Y, while the basic geometric structures for the variational bicomplex are J^{∞}Y, and the corresponding spaces Φ_{q}^{p} of forms on J^{∞}Y, constructed by the inverse limit operation.

The results differ in several points.

(a) The main difference is that the finite order theory keeps the order of all sequence operations fixed. This gives us a deeper understanding of the role of the order and of the analytic structure of basic variational notions: order increasing in operators as the Euler-Lagrange operator appears due to a different representation of the same class of forms.

Namely, if [ρ] is an element of Ω_{k}^{r}/Θ_{k}^{r} in (3), then its image E([ρ]) is again of order r.

As we have pointed out, classes of forms in the quotient Ω_{k}^{r}/Θ_{k}^{r} may also be expressed in equivalent ways, in particular, as differential forms defined, formally, on jet prolongations of higher order than r. For the discussion of basic ideas, as well as for concrete formulas, we refer to the simplest cases n = 1, r = 1 (first order mechanics [9], [13], [17]), and n = 1, r arbitrary (first order field theory [7]).

(b) Classes of forms in (3) have a certain algebraic structure, which determines the meaning of the cohomology of the sequence of global sections associated with (3): the cohomology concerns structured objects. For example, if ε ∈ Ω_{n+1}^{r}/Θ_{n+1}^{r} is a locally variational form, and H^{n+1}Y = 0, then there exists a lagrangian λ for ε in Ω_{n}^{r}/Θ_{n}^{r}, i.e., a lagrangian of a specific form defined by the structure of Ω_{n}^{r}/Θ_{n}^{r}. Therefore, the cohomology conditions have a different meaning than in the bicomplex theory, where only existence of a lagrangian for ε is guaranteed, without specifying its order, eventually its further properties.

(c) Now assume that n = 1, r = 1; this choice refers to the (first order) mechanics. Recall that a form α ∈ Ω_{2}^{1}Y is called locally variational, if dα ∈ Θ_{3}^{1}Y [13].

To a locally variational form α ∈ Ω_{2}^{1}Y one can associate two distributions. The associated distribution of α, denoted by D_{α}, is generated by 1-forms i_{ξ}α with ξ running through all vector fields on J^{1}Y. The distribution, locally generated by 1-forms i_{ξ}α with ξ running through all π^{1}-vertical vector fields, is called the Euler-Lagrange distribution associated with α, and is denoted by E_{α}. D_{α} is a subdistribution of E_{α}.

Integral sections of these distributions have a close relation to the solutions of the Euler-Lagrange equations. In particular, holonomic integral sections of the Euler-Lagrange distribution E_{α} coincide with the solutions of the Euler-Lagrange equations defined by α ∈ Ω_{2}^{1}Y.

(d) Since the lagrangians and the Euler-Lagrange forms arising on a fixed order jet bundles are a priory of a certain algebraic structure, differences also appear in the form of the theorems giving necessary and sufficient local variationality conditions.

As mentioned in (b), fixing order in the variational sequence requires appearance of classes of forms, represented by forms of a certain algebraic structure. These structured expressions imply that the variational triviality conditions, and the Helmholtz-Sonin variationality conditions, have a different form, and a different meaning in the variational sequence, and the variational bicomplex theories.

(e) An important difference consists in possibilities of a generalization of the variational sequence theory. The idea of constructing the Euler-Lagrange mapping as a quotient mapping, can be generalized to any subsequence of the DeRham sequence.

(f) Another difference consists in the fact that no horizontal Poincaré lemma is needed in the variational sequence theory. In fact, the need of the horizontal Poincaré lemma in the bicomplex theory is replaced by the exactness of the variational sequence.

References

[1] Anderson, I., Introduction to the variational bicomplex, Contemporary Mathematics 132 (1992), 51–73.

[2] Anderson, I., Duchamp, T., On the existence of global variational principles, Am. J. Math. 102 (1980), 781–867.

[3] Chrastina, J., Formal calculus of Variations on Fibered Manifolds, J. E. Purkyně University, Brno, 1989.

[4] Dedecker, P. and Tulczyjew, W. M., Spectral sequences and the inverse problem of the calculus of variations, Internat. Coll., Aix-en-Provence, 1979; in Differential Geometric Methods in Mathematical Physics, Lecture Notes in Math. 836, Springer, Berlin, 1980, 498–503.

[5] Francaviglia, M., Palese, M. and Vitolo, R., Superpotentials in variational sequences, in: Differential Geometry and Applications, Proc. Conf., Brno, Czech Republic (1998), Masaryk University (1999), 469–480; Differential Geometry and its Applications 10 (1999), 43–47.

[6] Grigore, D. R., The variational sequence on finite jet bundle extensions and the lagrangian formalism, preprint, 1997.

[7] Kašparová, J., A representation of the 1st order variational sequence in field theory, in: Differential Geometry and Applications, Proc. Conf., Brno, Czech Republic (1998), Masaryk University (1999), 493–502.

[8] Krasilshik, I. S., Lychagin, V. V., Vinogradov, A. M., Geometry of Jet Spaces and Nonlinear Partial Differential Equations, Advanced Studies in Contemporary Mathematics, 1, Gordon & Breach, New York, 1986.

[9] Krbek, M., Musilová, J., A note to the representation of the variational sequence in mechanics, Preprint GA 5/1998, Department of Mathematics, Silesian University Opava (Czech Republic), November 1998.

[10] Krupka, D., A geometric theory of ordinary first order variational problems on fibered manifolds, I. Critical sections, J. Math. Anal. Appl. 49 (1975), 180–206.

[11] Krupka, D., Variational sequences on finite order jet spaces, in: Differential Geometry and Its Applications, Proc. Conf., J. Janyška and D. Krupka, Editors, Brno, Czechoslovakia, 1989; World Scientific, Singapore, 1990, 236–254.

[12] Krupka, D., Some Geometric Aspects of the Calculus of Variations in Fibered Manifolds, Folia Fac. Sci. Nat. UJEP Brunensis 14 (1973).

[13] Krupka, D., Variational sequences in Mechanics, Calc. Var. 5 (1997), 557–583.

[14] Musilová, J., Variational sequence in higher order mechanics, in Differential Geometry and Applications, Proc. Conf., August 1995, Brno, Czech Republic; Masaryk University, Brno, 1996, 611–624.

[15] Pommaret, J. F., Partial Differential Equations and Group Theory, Kluwer Academic Publishers, Dordrecht, 1994.

[16] Saunders, D., The Geometry of Jet Bundles, London Mathematical Society Lecture Note Series 142, Cambridge University Press, Cambridge, 1989.

[17] Štefánek, J., A representation of the variational sequence in higher order mechanics, in Differential Geometry and Applications, Proc. Conf., August 1995, Brno, Czech Republic; Masaryk University, Brno, 1996, 469–478.

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

[19] Tulczyjew, W. M., The Euler-Lagrange resolution, Internat. Coll., Aix-en-Provence, 1979; in Differential Geometric Methods in Mathematical Physics, Lecture Notes in Math. 836, Springer, Berlin, 1980, 22–48.

[20] Vinogradov, A. M., A spectral sequence associated with a non-linear differential equation, and the algebro-geometric foundations of Lagrangian field theory with constraints, Soviet Math. Dokl. 19 (1978), 790–794.

[21] Vitolo, R., On different geometric formulations of lagrangian formalism, Differential Geometry and its Applications 10 (1999), 225–255.

Department of Mathematics , Silesian University at Opava 74601 Opava, CZECH REPUBLIC E-mail: demeter.krupka@slu.fpf.cz