DIFFERENTIAL GEOMETRY AND APPLICATIONS Satellite Conference of ICM in Berlin, Aug. 10 – 14, 1998, Brno Masaryk University in Brno (Czech Republic) 1999, 481–491

FOCK SPACE METHODS AND THE LAGRANGIAN FORMALISM ON FINITE JET BUNDLE EXTENSIONS

D. R. GRIGORE

Abstract. The geometric Lagrangian theory is formulated in the language of jet bundle extensions. The use of finite jet bundle extensions corresponds more closely to spirit of the original Lagrangian theory as studied in physical literature corresponding to a Lagrangian theory of arbitrary, but finite order.

The underlying structure is based on the analysis of some basic mathematical objects such as: the contact ideal and the (exact) variational sequence. In this paper we give new and much simpler proofs for the whole theory using Fock space methods. For instance, in a given chart, one finds out quite naturally that some global differential forms involved in the description of the variational sequence (the so-called contact forms) can be locally written in terms of expressions having some symmetry and/or antisymmetry properties. These expressions can be regarded as tensors in some Fock space and some very complicated identities become very simple if one uses the creation and annihilation operators.

Using these results we give the most general expression for a variationally trivial Lagrangian.

1. Introduction

The modern Lagrangian theory is most suitable analyzed in the language of finite order jet bundle extensions [11] - [13]. Some very hard combinatorial problems must be solved and methods inspired by quantum mechanics i.e. Fock space methods proved to be extremely helpful. The central observation is that various function coefficients of the various differential forms appearing in the proofs have some symmetry or antisymmetry properties which make them as elements in some Fock space so, most of the hard combinatorial relations can be written with the help of the creation and annihilation operators and solved using elementary properties of these Fock space operators. In [8] we have used this idea to lay the foundations of the whole theory i.e. we have given new proofs for the structure formula of the contact and respectively the strong contact forms, for the existence

1991 Mathematics Subject Classification. 49S05, 70G50, 70H35. Key words and phrases. Lagrangian Formalism, Variational Sequence. This paper is in final form and no version of it will be submitted for publication elsewhere.

of the Euler-Lagrange and of the Helmholtz-Sonin forms and for the exactness of the variational sequence. In [9] the analysis of the most general form of a variationally trivial Lagrangian of arbitrary order and the generic form of a locally variational differential equation was provided. (We mention that an independent proof recently appeared in [15].) In both cases the dependence on the highest order derivatives is through some polynomial expressions, called hyper-Jacobians [4], [10], but in the first case we are also able to explicitate the form of the coefficients of these polynomial expressions. These results generalize results from the previous references (see for instance [5], [5], [7]).

2. Jet Bundles and the Lagrangian Formalism

2.1. Basic Facts about Jet Bundle Extensions. The kinematical structure of a classical field theory is described by a fibred bundle structures π : Y ↦ X where X , Y are differentiable manifolds of dimensions dim(X) = n, dim(Y) = m + n and π is the canonical projection of the fibration. We use standard notations for the adapted charts (V, ψ) where ψ = (x^{i}, y^{σ}) (i = 1, ..., n; σ = 1, ..., m) and on the r-jet bundle extension J_{n}^{r}Y ↦ X (r ∈ ℕ) we use the adapted charts (V^{r}, ψ^{r}), where ψ = (x^{i}, y^{σ}, y_{j}^{σ}, ..., y_{j_{1},...,j_{k}}^{σ}, ..., y_{j_{1},...,j_{r}}^{σ}), j_{1} ≤ j_{2} ≤ ⋯ ≤ j_{k}, k = 1, ..., r. Some other standard notations [2] are useful: ∂_{σ}^{j_{1},...,j_{k}} ≡ \frac{r_{1}!...r_{n}!}{k!} _{\frac{∂}{∂y_{j_{1},...,j_{k}}^{σ}}} , k =

1, ..., r and the formal derivatives d_{i}^{r} ≡ \frac{∂}{∂x^{i}} + ∑_{k=0}^{r−1} y_{i,j_{1},...,j_{k}}^{σ}∂_{σ}^{j_{1},...,j_{k}} = \frac{∂}{∂x^{i}} + ∑_{|J|≤r−1} y_{iJ}∂_{σ}^{J}; (multi-index notation will be frequently used.)

We denote the forms of degree q on J^{r}Y by Ω_{q}^{r}. A form ρ ∈ Ω_{q}^{r} is called π^{r}- horizontal (or basic) if it has the following local expression: ρ = B_{i_{1},...,i_{q}}dx^{i_{1}} ∧⋯∧ dx^{i_{q}} with B_{i_{1},...,i_{q}} smooth skew-symmetric functions on V^{r}. We denote the set of basic forms of degree q by Ω_{q,X}^{r}. The elements of λ ∈ Ω_{n,X}^{r} are called Lagrange forms. They have the local expression λ = Lθ_{0} where L is a smooth function on V^{r} and θ_{0} ≡ dx^{1} ∧ ⋯ ∧ dx^{n}.

By a contact form we mean any form ρ ∈ Ω_{q}^{r} verifying (j^{r}γ)^{∗} ρ = 0 for any section γ. We denote by Ω_{q(c)}^{r} the set of contact forms of degree q ≤ n. By elementary computations one finds out that for any chart (V, ψ) on Y, every element of the set Ω_{1(c)}^{r}(V) is a linear combination of the following expressions:

ω_{j_{1},...,j_{k}}^{σ} ≡ dy_{j_{1},...,j_{k}}^{σ} − y_{i,j_{1},...,j_{k}}^{σ}dx^{i}, k = 0, ..., r − 1 . (2.1)

If q = n + 1, ..., N ≡ dim(J^{r}Y) = m\binom{n+r}{n} then ρ ∈ Ω_{q}^{r} is a strongly contact form iff its contact component of order q − n vanishes i.e. p_{q−n}ρ = 0. For more details, we refer to [8], [9] and [13].

2.2. Fock Space methods. The rôle of the Fock space methods emerges from a canonical expression for any form ρ ∈ Ω_{q}^{r}. We start from the fact that the forms dx^{i}, ω_{J}^{σ} (|J| ≤ r − 1) and dy_{I}^{σ} (|I| = r) are a basis in the linear space of 1forms. The form ρ is a polynomial of degree q in these forms (with respect to the wedge product ∧). We separate all the terms containing at least a factor ω_{J}^{σ} and get a decomposition ρ = ρ_{0} +ρ′ where ρ_{0} has the structure ρ_{0} = ∑_{|J|≤r−1} ω_{J}^{σ} ∧Φ_{σ}^{J} and ρ′ is a polynomial of degree q only in dx^{i} and dy_{I}^{σ} (|I| = r). It is clear the one can write it as follows:

ρ′ = ∑_{s=0}^{q} \frac{1}{s!(q − s)!} ∑_{|I_{1}|,...,|I_{s}|=r} A_{σ_{1},...,σ_{s},i_{s+1},...,i_{q}}^{I_{1},...,I_{s}}

dy_{I_{1}}^{σ_{1}} ∧ ⋯ ∧ dy_{I_{s}}^{σ_{s}} ∧ dx^{i_{s+1}} ∧ ⋯ ∧ dx^{i_{q}} , (2.2)

where A_{σ_{1},...,σ_{s},i_{s+1},...,i_{q}}^{I_{1},...,I_{s}} are smooth functions on V^{r} and can be assumed to verify the following symmetry property:

A_{σ_{P (1)},...,σ_{P (s)},i_{Q(s+1)},...,i_{Q(q)}}^{I_{P (1)},...,I_{P (s)}} = (−1)^{|P |+|Q|}A_{σ_{1},...,σ_{s},i_{s+1},...,i_{q}}^{I_{1},...,I_{s}} (2.3)

where P and Q are permutations and |P |, |Q| are their signature.

Let us define the following Hilbert spaces of Fock type:

H_{s} ≡ F^{(−)}(ℝ^{n}) ⊗ \underbrace{F^{(+)}(ℝ^{n}) ⊗ ⋯ ⊗ F^{(+)}(ℝ^{n})}_{s−times}

where F^{(±)}(ℝ^{n}) are the symmetric (corresp.. +) and the antisymmetric (corresp.. −) Fock spaces. Then we have the well known decomposition in subspaces with fixed number of “Bosons” and “Fermions”: H_{s} = ⊕H_{k,l_{1},...,l_{s}} with the convention: H_{k,l_{1},...,l_{s}} ≡ 0 if any one of the indices k, l_{1}, ..., l_{q} is negative or, if k > n. Then we can consider A_{σ_{1},....,σ_{s},i_{s+1},...,i_{q}}^{I_{1},...,I_{s}} as the components of a tensor

A_{σ_{1},...,σ_{s}} ∈ H_{q−s,\underbrace{r, ..., r}_{s−times}}.

Let us denote the creation and the annihilation Fermionic operators a^{∗i}, a_{i}, (i = 1, ..., n) and the corresponding creation and annihilation Bosonic operators b_{(α)i}^{∗}, b_{(α)}^{i} (α = 1, ..., s; i = 1, ..., n).

One introduces the operators

B_{α} : H_{k,l_{1},...,l_{s}} → H_{k+1,l_{1},...,l_{α−1},l_{α}+1,l_{α+1},...,l_{s}} (α = 1, ..., s)

according to the formula

B_{α} ≡ b_{(α)i}^{∗}a^{∗i} , (α = 1, ..., s) . (2.4)

It is convenient to rescale these operators (compare with [8]) such that we have for instance:

(B_{1}A)_{σ_{1},...,σ_{s},i_{0},...,i_{l}}^{{j_{0},...,j_{k}},I_{2},...,I_{s}} = (−1)^{s−1}S_{j_{0},...,j_{k}}^{+}S_{i_{0},...,i_{l}}^{−}A_{σ_{1},...,σ_{s},i_{1},...,i_{l}}^{{j_{1},...,j_{k}},I_{2},...,I_{s}}δ_{i_{0}}^{j_{0}} (2.5)

and similarly for B_{2}, ..., B_{s}; here S_{J}^{±} is the symmetrization (resp. antisymmetrization) operator in the indices J. The central property of these operators is (see [8], lemma 3.6):

Lemma 2.1. Let X ∈ H_{k,r_{1},...,r_{s}} Then X verifies the equation

B_{1} ⋯ B_{s}X = 0 (2.6) iff it is of the form

X = ∑_{α=1}^{s} B_{α}X_{α} (2.7)

for some X_{α} ∈ H_{k−1,r_{1},...r_{α−1},r_{α}−1,r_{α+1},...,r_{s}}.

This key result can be used (see [8]) to obtain a structure formula for an arbitrary contact form generalizing the expression (2.1) for contact forms of degree 1. Explicitly, let (V, ψ) an adapted chart on the fibre bundle Y and let ρ ∈ Ω_{q}^{r}(Y), q = 2, ..., n. Then ρ is contact iff in the associated chart is sum a sum of terms each of them having at least a factor which is a contact form of degree 1 i.e. ω_{J}^{σ}, |J| ≤ r −1 or an exterior differential of the form dω_{I}^{σ}, |I| = r. Applying the same methods one obtains the structure theorem for strongly contact forms. Let n + 1 ≤ q ≤ N and ρ ∈ Ω_{q}^{r}. Let (V, ψ) be a chart on Y Then ρ is a strongly contact form iff in the associated chart it is sum of terms each of one having at least q − n + 1 factors of the type ω_{J}^{σ}, |J| ≤ r − 1 or dω_{I}^{σ}, |I| = r. We mention that the original proof of this structure formula, based on a different treatment of the combinatorial aspects, appears in [14].

We also have a dual result of the lemma (2.1), namely:

Lemma 2.2. The tensor X ∈ H_{s,\underbrace{r, ..., r}_{k−times}} verifies the system of equations

B_{α}X = 0 (α = 1, ..., k). (2.8)

iff is of the following form:

X = (∏_{α=1}^{k} B_{α}) X_{0} (2.9)

where X_{0} is arbitrary.

2.3. Euler-Lagrange and Helmholtz-Sonin forms. An element T ∈ Ω_{n+1,X}^{s} is called a differential equation (or a source form in the terminology of [1]) if it has the following local expression in every associated chart:

T = T_{σ}ω^{σ} ∧ θ_{0} . (2.10)

Let us take λ ∈ Ω_{n,X}^{r} with the local expression (2.1). Then one defines on any chart (V^{s}, ψ^{s}), (s ≥ 2r) the so-called Lie-Euler expressions [1] according to:

E_{σ}^{I}(L) ≡ ∑_{|J|≤r−|I|}(−1)^{|J|}\binom{|I| + |J|}{|I|}d_{J}∂_{σ}^{IJ}L . (2.11)

In particular, for I = ∅ we have the Euler-Lagrange expressions:

E_{σ}(L) ≡ ∑_{|J|≤r}(−1)^{|J|}d_{J}∂_{σ}^{J}L . (2.12)

Then there exists a globally defined (n + 1)-form, denoted by E(λ) such that in V^{s}:

E(λ) = E_{σ}(L)ω^{σ} ∧ θ_{0} . (2.13)

One calls this form the Euler-Lagrange form associated to λ and notes that it is a differential equation (see formula (2.10)). In general, a differential equation T ∈ Ω_{n+1,X}^{s} is called (locally) variational or of Euler-Lagrange type iff there exists a (local) Lagrange form λ ∈ Ω_{n,X}^{r} such that

(2.14) T = E(λ), up to a pull − back .

One notices that in this case the general form of a differential equation coincides with the well-known form of the Euler-Lagrange equations.

A similar construction gives the so-called Helmholtz-Sonin form. Let T ∈ Ω_{n+1,X}^{s} be a differential equation with the local form given by (2.10). We define the following expressions in any chart V^{t}, t ≥ 2s:

H_{σν}^{J} ≡ ∂_{ν}^{J}T_{σ} − (−1)^{|J|}E_{σ}^{J}(T_{ν}), |J| ≤ s . (2.15)

Then there exists a globally defined (n + 2)-form, denoted by H(T) such that on V^{t}:

H(T) = ∑_{|J|≤s} H_{σν}^{J}ω_{J}^{ν} ∧ ω^{σ} ∧ θ_{0} . (2.16)

2.4. The Exact Variational Sequence. The basic constructions goes along the following lines. If π : Y → X is a fibre bundle and U, V ⊂ Y are charts such that U ⊂ V, we denote by i_{U,V} : U^{r} → V^{r} the canonical inclusion. Then, the collection {Ω_{q}^{r}(V)} (q ≥ 0), {i_{U,V}^{∗} } is a presheaf denoted by Ω_{q}^{r}. One also needs the subspaces θ_{q}^{r} ∈ Ω_{q(c)}^{r} by:

θ_{1}^{r} ≡ Ω_{1(c)}^{r}, θ_{q}^{r} ≡ dΩ_{q−1(c)}^{r} + Ω_{q(c)}^{r} (q = 2, ..., N = dim(J^{r}Y)) . (2.17)

Next, one introduces the so-called contact homotopy operator. The construction is the following. Let U ⊂ ℝ^{n} (resp. V ⊂ ℝ^{m}) an open set (resp. a ball centred in 0 ∈ ℝ^{m}) and W = U × V. One considers the operator χ_{r} as a map χ_{t}^{r} : [0, 1] × J^{r}W → J^{r}W given by:

χ_{r}(t, (x^{i}, y^{σ}, y_{j}^{σ}, ..., y_{j_{1},...,j_{s}}^{σ})) = (x^{i}, ty^{σ}, ty_{j}^{σ}, ..., ty_{j_{1},...,j_{s}}^{σ}) . (2.18)

Then for any ρ ∈ Ω_{q}^{r}(W) we have the unique decomposition

(χ_{t}^{r})^{∗}ρ = dt ∧ ρ_{0}(t) + ρ_{1}(t) (2.19)

where ρ_{0}(t) (resp. ρ_{1}(t)) are q − 1 (resp. q) forms which do not contain the differential dt. Then the contact homotopy operator is by definition the map A : Ω_{q}^{r}(V) → Ω_{q−1}^{r}(V) given by:

Aρ ≡ ∫_{0}^{1} ρ_{0}(t) . (2.20)

This definition is justified by the following result resembling the key idea of the Poincaré lemma. Let ρ ∈ Ω^{r}W be arbitrary. The following formula is true:

ρ = Adρ + dAρ + ρ_{1}(0) . (2.21)

The central result of [12] (the contact homotopy is essential for the proof) is the following one.

Theorem 2.3. Let E_{q} : Ω_{q}^{r}/θ_{q}^{r} → Ω_{q+1}^{r}/θ_{q+1}^{r} be given by

E_{q}([ρ]) ≡ [dρ] (2.22)

where [ρ] is the class of ρ modulo θ_{q}^{r}. Then the quotient sequence

0 ⟶ ℝ ⟶ Ω_{0}^{r}⟶^{E_{0}}Ω_{1}^{r} /θ_{1}^{r} ⟶^{E_{1}} ⋯ ⟶^{E_{P −1}}Ω_{P}^{r} /θ_{P}^{r} ⟶^{E_{P}}

Ω_{P +1}^{r}⟶^{d} ⋯ ⟶^{d}Ω_{N}^{r} ⟶ 0 (2.23)

is a acyclic resolution of the constant sheaf ℝ. In particular it is exact.

One calls this the variational sequence of order r over Y and denotes for simplicity: V_{q}^{r} ≡ Ω_{q}^{r}/θ_{q}^{r}. Some special classes have distinct names. So, if λ ∈ Ω_{n}^{r} then the class [λ] ∈ Ω_{n+1}^{r}/θ_{n+1}^{r} is called the Euler-Lagrange class of λ. If T ∈ Ω_{n+1}^{r} then [T] ∈ Ω_{n+2}^{r}/θ_{n+2}^{r} is called the Helmholtz-Sonin class of T.

One can give characterizations of V_{q}^{r}, q = n, n + 1 using Euler-Lagrange and Helmholtz-Sonin forms.

3. Variationally Trivial Lagrangians

A variationally trivial Lagrange form of order r is any λ ∈ Ω_{n,X}^{r} such that E(λ) = 0. In other words, the corresponding Euler-Lagrange expressions are identically zero, or the Euler-Lagrange equations are identities for any section γ. Here we give the general form of such a Lagrange form. We follow, basically, the line of argument from [13]; as before, we simplify considerably and complete the proofs from this reference using the techniques developed in [8]. The main reward is the possibility to obtain a very explicit expression for a locally trivial Lagrangian.

First, we remind that ρ ∈ Ω_{q}^{r+1} is called π^{r+1,r}-projectable iff there exists ρ′ ∈ Ω_{q}^{r} such that

ρ = (π^{r+1,r})^{∗}ρ′ . (3.1)

One can easily see that locally this amounts to the condition that if ρ is written as a polynomial in dx^{i} and dy_{J}^{σ} (|J| ≤ r + 1) then, the differentials dy_{J}^{σ} (|J| = r + 1) must be absent and moreover, the coefficient functions must not depend on y_{J}^{σ} (|J| = r + 1).

We start with the following result [12]:

Proposition 3.1. Let η ∈ Ω_{q}^{r} (q = 1, ..., n − 1) such that hdη is a π^{r+1,r}- projectable (q + 1)-form. Then one can write η as follows:

η = ν + dϕ + ψ (3.2)

where ν ∈ Ω_{q,X}^{r} is a basic q-form and ψ ∈ Ω_{q(c)}^{r} is a contact q-form.

Proof. Let (V, ψ) be a chart on Y. Then in the associated chart (V^{r}, ψ^{r}) one can write η in the standard form

η = η_{0} + η_{1} (3.3)

where in η_{0} we collect all terms containing at least one factor ω_{J}^{σ} (|J| ≤ r − 1) and η_{1} is a polynomial only in dx^{i} and dy_{J}^{σ} (|J| = r + 1):

η_{1} = ∑_{s=0}^{q} ∑_{|I_{1}|,...,|I_{s}|=r} A_{σ_{1},...,σ_{s},i_{s+1},...,i_{q}}^{I_{1},...,I_{s}}dy_{I_{1}}^{σ_{1}} ∧ ⋯ ∧ dy_{I_{s}}^{σ_{s}} ∧ dx^{i_{s+1}} ∧ ⋯ ∧ dx^{i_{q}}, (3.4)

where the coefficients A_{σ_{1},...,σ_{s},i_{s+1},...,i_{q}}^{I_{1},...,I_{s}} have antisymmetry properties of type (2.3).

In particular η_{0} is a contact form so we have hdη = hdη_{1} i.e. the form hdη_{1} is, by hypothesis, π^{r+1,r}-projectable. One first computes in general:

dη_{1} = ∑_{s=0}^{q} ∑_{|I_{1}|,...,|I_{s}|=r} ∑_{|J|≤r−1} (∂_{ν}^{J}A_{σ_{1},...,σ_{s},i_{s+1},...,i_{q}}^{I_{1},...,I_{s}})

ω_{J}^{ν} ∧ dy_{I_{1}}^{σ_{1}} ∧ ⋯ ∧ dy_{I_{s}}^{σ_{s}} ∧ dx^{i_{s+1}} ∧ ⋯ ∧ dx^{i_{q}}

+ ∑_{s=0}^{q+1} ∑_{|I_{1}|,...,|I_{s}|=r} Ã_{σ_{1},...,σ_{s},i_{s+1},...,i_{q+1}}^{I_{1},...,I_{s}}

dy_{I_{1}}^{σ_{1}} ∧ ⋯ ∧ dy_{I_{s}}^{σ_{s}} ∧ dx^{i_{s+1}} ∧ ⋯ ∧ dx^{i_{q+1}},

where

Ã_{σ_{1},...,σ_{s},i_{s+1},...,i_{q+1}}^{I_{1},...,I_{s}} ≡ S_{(I_{1},σ_{1}),...,(I_{s}σ_{s})}^{−}∂_{σ_{1}}^{I_{1}}A_{σ_{2},...,σ_{s},i_{s+1},...,i_{q+1}}^{I_{2},...,I_{s}} +

(−1)^{s}S_{i_{s+1},...,i_{q+1}}^{−}d_{i_{s+1}}A_{σ_{1},...,σ_{s},i_{s+2},...,i_{q+1}}^{I_{1},...,I_{s}}; (3.5)

here S_{(I_{1},σ_{1}),...,(I_{s}σ_{s})}^{−} is the antisymmetrization projector in the corresponding couples of indices.

Then one gets

hdη = hdη_{1} = ∑_{s=0}^{q+1} ∑_{|I_{1}|,...,|I_{s}|=r} Ã_{σ_{1},...,σ_{s},i_{s+1},...,i_{q+1}}^{I_{1},...,I_{s}}y_{I_{1}i_{1}}^{σ_{1}} ⋯ y_{I_{s}i_{s}}^{σ_{s}}dx^{i_{1}} ∧ ⋯ ∧ dx^{i_{q+1}}

which is π^{r+1,r}-projectable iff the coefficient functions do not depend on y_{I}^{σ} (|I| = r+1). By repeatedly applying derivative operators, one obtains that the condition of π^{r+1,r}-projectability is equivalent to

B_{1} ⋯ B_{s}Ã_{σ_{1},...,σ_{s}} = 0, (s = 1, ..., q + 1). (3.6)

It is possible to apply lemma 2.1 and one obtains the following generic expression: Ã_{σ_{1},...,σ_{s}} = ∑_{α=1}^{s} B_{α}Ã_{σ_{1},...,σ_{s}}^{α} or, in index notations, this means the function Ã_{σ_{1},...,σ_{s},i_{s+1},...,i_{q+1}}^{I_{1},...,I_{s}} is a sum of terms containing at least a factor of the type δ_{i}^{j} where j belongs to one of the multi-indices I_{p}, p = 1, …, s and i ∈ {i_{s+1}, ..., i_{q+1}}.

Inserting back into the expression of dη_{1} we get after minor prelucrations that

dη_{1} = ν_{1} + dϕ + ∑_{|J|≤r−1} ω_{J}^{σ} ∧ Φ_{σ}^{J} (3.7)

with ν_{1} ∈ Ω_{q,X}^{r} a basic q-form, ϕ ∈ Ω_{q(c)}^{r} and Φ_{σ}^{J} a polynomial of degree q in dx^{i} and dy_{I}^{σ} |I| = r.

Now one continues by induction and finally obtains:

dη_{1} = ν_{1} + dϕ_{1} + ∑_{|J|≤r−1} Φ_{σ,i_{1},...,i_{q}}^{J}ω_{J}^{σ} ∧ dx^{i_{1}} ∧ ⋯ ∧ dx^{i_{q}} + ψ (3.8)

where ϕ_{1} (resp. ψ) is some contact (resp. 2-contact) form of degree q (resp. q +1.)

(iii) The last step of our proof consists in using the contact homotopy operator A (see the definition contained in the relations (2.18) - (2.20)). We apply the relation (2.21) to the form ρ = η_{1} − ϕ_{1}:

η_{1} − ϕ_{1} = A (ν_{1} + ∑_{|J|≤r−1} Φ_{σ,i_{1},...,i_{q}}^{J}ω_{J}^{σ} ∧ dx^{i_{1}} ∧ ⋯ ∧ dx^{i_{q}} + ψ) +

dA(η_{1} − ϕ_{1}) + η_{1}(0) − ϕ_{1}(0). (3.9)

One proves by direct computation that: Aν_{1} = 0; if ψ is 2-contact form, then Aψ is a contact form; A(∑ ⋯) and η_{1}(0) are basic forms; if ϕ_{1} is a contact form, then ϕ_{1}(0) is also a contact form.

Inserting this information in the preceding relation one obtains that the result from the statement of the proposition is true for the form η_{1}. Taking into account (3.3) we obtain the same result for the form η. □

Let us note that the converse of this statement is not true. In fact the condition of π^{r+1,r}-projectability imposes additional constraints on the basic form ν which will be analyzed in the next lemma. A complete proof of the following result appears in [2]; here we offer an alternative proof which seems to be much more simpler.

Proposition 3.2. Let ν ∈ Ω_{q,X}^{r} (q = 1, ..., n − 1) Then the basic form hdν is π^{r+1,r}-projectable iff there exists for any chart (V, ψ) a form ν̃_{V} ∈ Ω_{q}^{r−1} such that we have in the associated chart (V^{r}, ψ^{r}) the equality ν = hν̃_{V}.

Proof. According to (2.1) we have in the associated chart (V^{r}, ψ^{r}):

ν = A_{i_{1},...,i_{q}}dx^{i_{1}} ∧ ⋯ ∧ dx^{i_{q}} .

Then one obtains by direct computation that:

hdν = [(d_{i_{0}}A_{i_{1},...,i_{q}}) + ∑_{|I|=r} (∂_{σ}^{I}A_{i_{1},...,i_{q}}) y_{Ii_{0}}^{σ} ] dx^{i_{0}} ∧ ⋯ ∧ dx^{i_{q}}.

(here d_{j} is the formal derivative on V^{r}).

This form is π^{r+1,r}-projectable iff the square bracket does not depend on y_{I}^{σ} (|I| = r + 1), i.e.

S_{i_{0},...,i_{q}}^{−}S_{Ip}^{+}δ_{i_{0}}^{p} (∂_{σ}^{I}A_{i_{1},...,i_{q}}) = 0 . (3.10)

After some work (using some ideas from [3]) one obtains

A_{i_{1},...,i_{q}} = ∑_{s=0}^{q} \frac{1}{s!(q − s)!} ∑_{|J_{1}|,...,|J_{s}|=r−1} S_{i_{1},...,i_{q}}^{−}A_{σ_{1},...,σ_{s},i_{1},...,i_{q}}^{J_{1},...,J_{s}}y_{J_{1}i_{1}}^{σ_{1}} ⋯ y_{J_{s}i_{s}}^{σ_{s}} (3.11)

and we have the formula from the statement with:

ν̃_{V} ≡ ∑_{s=0}^{q} \frac{1}{s!(q − s)!} ∑_{|J_{1}|,...,|J_{s}|=r−1} A_{σ_{1},...,σ_{s},i_{s+1},...,i_{q}}^{J_{1},...,J_{s}}

dy_{J_{1}}^{σ_{1}} ∧ ⋯ ∧ dy_{J_{s}}^{σ_{s}} ∧ dx^{i_{s+1}} ∧ ⋯ ∧ dx^{i_{q}}. (3.12) □

Now we are ready to obtain the most general form of a variationally trivial Lagrangian. First we note that we have

Theorem 3.3. Let λ ∈ Ω_{n,X}^{r} be a variationally trivial Lagrange form. Then for every point j_{x}^{r}γ ∈ J^{r}Y there exists a neighborhood V of γ(x) ∈ Y and a n-form ρ_{V} defined in the chart (V^{r−1}, ψ^{r−1}) such that we have: (1) λ = hρ_{V} in the chart (V^{r}, ψ^{r}); (2) dρ_{V} = 0. Conversely, if such a local form ρ_{V} exists, then the form λ is variationally trivial.

One can now obtain the most general form of a variationally trivial local Lagrangian.

Theorem 3.4. Any variationally trivial local Lagrangian of order r has the following form in the chart (V^{r}, ψ^{r}):

L = ∑_{s=0}^{n} \frac{1}{s!(n − s)!} ∑_{|I_{1}|,...,|I_{s}|=r−1} L_{σ_{1},...,σ_{s},i_{s+1},...,i_{n}}^{I_{1},...,I_{s}}J_{I_{1},...,I_{s}}^{σ_{1},...,σ_{s},i_{s+1},...,i_{n}} . (3.13)

Here we have defined (see also [4] and [10]):

J_{I_{1},...,I_{s}}^{σ_{1},...,σ_{s},i_{s+1},...,i_{n}} ≡ ε^{i_{1},...,i_{n}} ∏_{l=1}^{s} y_{I_{l}i_{l}}^{σ_{l}} (s = 0, ..., n) (3.14)

and the function L_{...}^{...} are given by

L_{σ_{1},...,σ_{s}i_{s+1},...,i_{n}}^{I_{1},...,I_{s}} ≡ ∑_{k=1}^{s}(−1)^{k−1}∂_{σ_{k}}^{I_{k}}A_{σ_{1},...,σ̂_{k},...,σ_{s},i_{s+1},...,i_{n}}^{I_{1},...,Î_{k},...,I_{s}} +

∑_{k=s+1}^{q}(−1)^{k−1}d′_{i_{k}}A_{σ_{1},...,σ_{s},i_{s+1},...,î_{k},...,i_{n}}^{I_{1},...,I_{s}} ; (3.15)

the expressions A_{σ_{1},...,σ_{s}i_{s+1},...,i_{n−1}}^{I_{1},...,I_{s}} are smooth functions on V^{r−1} verifying the symmetry property (2.3) and d′_{j} = d_{j}^{r−1} is the corresponding formal derivative on V^{r−1}.

Proof. It is convenient to introduce the following forms:

A ≡ ∑_{s=0}^{n−1} \frac{1}{s!(n − 1 − s)!} ∑_{|I_{1}|,...,|I_{s}|=r−1} A_{σ_{1},...,σ_{s},i_{s+1},...,i_{n−1}}^{I_{1},...,I_{s}}

dy_{I_{1}}^{σ_{1}} ∧ ⋯ ∧ dy_{I_{s}}^{σ_{s}} ∧ dx^{i_{s+1}} ∧ ⋯ ∧ dx^{i_{n−1}}

and

F ≡ ∑_{s=0}^{n} \frac{1}{s!(n − s)!} ∑_{|I_{1}|,...,|I_{s}|=r−1} L_{σ_{1},...,σ_{s},i_{s+1},...,i_{n}}^{I_{1},...,I_{s}}

dy_{I_{1}}^{σ_{1}} ∧ ⋯ ∧ dy_{I_{s}}^{σ_{s}} ∧ dx^{i_{s+1}} ∧ ⋯ ∧ dx^{i_{n}}.

Then one finds out by direct computation that

F = dA + contact terms .

Next, one takes in theorem 3.3 ν̃_{V} = A and it follows that ρ_{V} = F + contact terms. Finally, by direct computation one discovers that λ = hρ_{V} has the expression (3.13). □

This theorem generalizes some results obtained in particular cases (see [9] for the references.)

One can analyze in the same way the generic form of a locally variational differential equation but the results are not of the if and only if type as above.

References

[1] Anderson, I. M., The Variational Bicomplex, Utah State Univ. preprint, 1989, Academic Press, Boston, to appear.

[2] Anderson, I. M., Duchamp, T., On the Existence of Global Variational Principles, American Journ. Math. 102 (1980), 781-868.

[3] Anderson, I. M., Pohjanpetlo, J., Symmetries, Conservation Laws and Variational Principles for Vector Field Theories, preprint, 1995.

[4] Ball, J. M., Currie, J. C., Olver, P. J., Null Lagrangians, Weak Continuity, and Variational Problems of Arbitrary Order, Journ. Functional Anal. 41 (1981), 135-174.

[5] Grigore, D. R., Variational Equations and Symmetries in the Lagrangian Formalism, J. Phys. A 28 (1995), 2921-2937.

[5] Grigore, D. R., Variational Equations and Symmetries in the Lagrangian Formalism; Arbitrary Vector Fields, hep-th/9512176, Fortschr. der Physik 45 (1997), 727-751.

[7] Grigore, D. R., Trivial Second-Order Lagrangians in Classical Field Theory, hepth/9609199, to appear in Fortschr. der Physik.

[8] Grigore, D. R., The Variational Sequence on Finite Jet Bundle Extensions and the Lagrangian Formalism, dg-ga/9702016, Differential Geometry and Applications 10 (1999), 43-77.

[9] Grigore, D. R., Variationally Trivial Lagrangians and Locally Variational Differential Equations of Arbitrary Order, Differential Geometry and Applications 10 (1999), 79-105.

[10] Olver, P. J., Hyperjacobians, Determinant Ideals and the Weak Solutions to Variational Problems, Proc. Roy. Soc. Edinburgh 95A (1983), 317-340.

[11] Krupka, D., Geometry of Lagrangian Structures.3, in Proceedings of the 14th Winter School on Abstract Analysis, SRNI, 1986, Suppl. ai Rendiconti del Circolo Matematico di Palermo, Serie II, no.14, 1987, 187-224.

[12] Krupka, D., Variational Sequence on Finite Order Jet Spaces, in: Differential Geometry and its Applications, Proceedings of the Conference (Brno, August, 1989), World Scientific, Singapore, 1990, 236-254.

[13] Krupka, D., Topics in the Calculus of Variation: Finite Order Variational Sequences, in: Differential Geometry and its Applications, Proceedings Conf. Opava, 1992, (Opava: Open Univ. Press), 437-495.

[14] Krupka, D., The Contact Ideal, Differential Geom. and Appl. 5 (1995), 257-276.

[15] Krupka, D., Musilova, J., Trivial Lagrangians in Field Theory, Differential Geom. and Appl. 9 (1998), 293-505.

Department of Theor. Phys., Inst. Atomic Phys. Bucharest-Măgurele, P. O. Box MG 6, ROMÂNIA E-mail: grigore@theory.nipne.ro, grigore@roifa.ifa.ro