DIFFERENTIAL GEOMETRY AND APPLICATIONS Proc. Conf., Aug. 28 – Sept. 1, 1995, Brno, Czech Republic Masaryk University, Brno 1996, 469–478

A REPRESENTATION OF THE VARIATIONAL

SEQUENCE IN HIGHER ORDER MECHANICS

JAROSLAV ŠTEFÁNEK

Abstract. Let Y be a fibered manifold over a base manifold X. A differential form ρ defined on the r-jet prolongation J^{r}Y of Y is said to be contact if it vanishes along the r-jet prolongation J^{r}γ of every section γ of Y. The contact forms define a subsequence of the de Rham sequence on J^{r}Y. The corresponding quotient sequence is known as the r-th order variational sequence. We consider the case of 1-dimensional base X (mechanics).

We construct an isomorphism between the variational sequence and another sequence whose elements are spaces of forms rather than spaces of classes of forms. In this way we determine explicitly the first two mappings of the variational sequence.

1. Introduction

The subject of this paper is the variational sequence of a fibered manifold which was defined by Krupka in [3], especially the case of mechanics i.e. the case of fibered manifolds with 1-dimensional base manifold. Our aim is to give a description of such variational sequence in some form convenient for practical computation.

The variational sequence is defined as the factor complex Ω/Θ of the de Rham complex Ω, where Θ is a certain subcomplex of Ω, so it is natural to try to find some mapping Φ : Ω → A such that Ker Φ = Θ and to express Ω/Θ with help of the corresponding isomorphism Ω/Θ → Im Φ. Actually, in Definition 1 we construct such a mapping Φ which assigns to every class from Ω/Θ a differential form and in Theorem 1 we prove that Ker Φ = Θ. Finally, in Theorem 2 we give the explicit coordinate description of the first three elements of the variational sequence in our representation as well as the coordinate form of first three mappings of Φ. This description enables us to see the connection between the first two mappings of the variational sequence and the Euler–Lagrange and the Helmholtz–Sonin mapping.

Let us note that the problem of the representation of the variational sequence has been solved by Krupka in [4] for the first order mechanics and by Musilová in [2] for the first three elements of the variational sequence of general order. The results of this article coincide with those from [2] and [3] for the first two members of the variational sequence. For the third member Ω^{3}/Θ^{3} of the variational sequence our representation is different, but the corresponding systems of equations (Helmholtz– Sonin equations) are equivalent.

1991 Mathematics Subject Classification. 49F05, 58A15, 58A20, 58E99.

Key words and phrases. Fibered manifold, jet, contact form, variational sequence.

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

Research supported by grant No. 201/93/2245 from the Czech Grant Agency.

2. Notation

Through the paper we use the same notation as in [2, 3]. Let us briefly recall the main facts. π : Y → X denotes the fibered manifold pr_{1} : ℝ^{m+1} → ℝ, where pr_{1} is the standard projection onto the first coordinate. The canonical coordinates on Y are denoted by (t, q^{1}, …, q^{m}), the corresponding coordinates on the r-jet prolongation J^{r}Y of Y are denoted by (t, q^{σ_{0}}, q_{1}^{σ_{1}}, …, q_{r}^{σ_{r}}). The canonical jet projection of J^{r}Y onto J^{s}Y, where 0 ≤ s ≤ r, (resp. onto X) is denoted by π_{r,s} (resp. π_{r}). The formal derivative of a function f ∈ C^{∞}(J^{r}Y) in the canonical coordinate system (t, q^{1}, …, q^{m}) is denoted by d_{1} f. By d_{k} f we denote the k-th derivative d_{1}(d_{1}(… d_{1} f)) of f. The exterior derivative is denoted by d. The set of all multiindices of the length k over a set A is denoted by multi_{k} (A). For a system (F_{Σ}^{I}) where I = (i_{1}, …, i_{k}) ∈ multi_{k}(A), Σ = (σ_{1}, …, σ_{k}) ∈ multi_{k} (B), we denote by (F_{Σ}^{I})_{anti} the antisymmetrisation of (F_{Σ}^{I}) with respect to the indices (_{σ_{1}}^{i_{1}}), …, (_{σ_{k}}^{i_{k}}). By ω_{i}^{σ} we denote the contact form ω_{i}^{σ} = d q_{i}^{σ} − q_{i+1}^{σ} dt. For I = (i_{1}, …, i_{k}) ∈ multi_{k} (0, …, r), Σ = (σ_{1}, …, σ_{k}) ∈ multi_{k} (0, …, m) we define ω_{I}^{Σ} = ω_{i_{1}}^{σ_{1}} ∧ … ∧ ω_{i_{k}}^{σ_{k}}. Whenever we use the coordinate representation ρ = A_{Σ}^{I} ω_{I}^{Σ}, ρ = A_{Σ}^{I} dt ∧ ω_{I}^{Σ} or ρ = A_{σ_{1},... ,σ_{k}}^{i_{1},... ,i_{k}} dq_{i_{1}}^{σ_{1}} ∧ … ∧ dq_{i_{k}}^{σ_{k}} (with the usual Einstein summation) we suppose the antisymmetry of the system (A_{Σ}^{I})_{I_{Σ}} with respect to the indices (_{σ_{1}}^{i_{1}}), …, (_{σ_{k}}^{i_{k}}). By 𝔍, ℨ, etc., we denote multiindices of finite length with indices of the form 𝔍_{i} = (0, …, 0, 1, 0, …, 0). (All indices 𝔍_{i} of such multiindex 𝔍 are supposed to be of the same length.) If I ∈ multi_{k} (ℤ) and 𝔍 = (𝔍_{1}, …, 𝔍_{p}), where 𝔍_{i} = (0, …, 1, …, 0) ∈ multi_{k} ({0, 1}), then we define

|𝔍| = p, I − 𝔍 = I − 𝔍_{1} − ⋯ − 𝔍_{p}. By A^{i_{1},... ,i_{p}^{\underline{p}},... ,i_{k}} we denote the fact that i_{p} is

the p-th index of the multiindex (i_{1}, …, i_{k}).

3. A representation of the variational sequence by forms

Definition 1. Let k > 0, let η ∈ Ω_{r}^{k+1}, let p_{k}η = ∑_{I∈multi_{k}(ℤ)} A_{Σ}^{I}dt ∧ ω_{I}^{Σ}, where A_{Σ}^{I} = 0 for I ∉ multi_{k}(0, ..., r). We denote

ηF_{Σ}^{(α,K)} = ∑_{|𝔍|≥0}(−1)^{|𝔍|} ∑_{λ=0}^{∞}(−1)^{λ} ( \begin{matrix}λ + |𝔍| \\ |𝔍| \end{matrix} ) d_{λ}A_{Σ}^{(α+1+|𝔍|+λ,K−𝔍)} (1)

for (α, K) ∈ multi_{k}(ℤ), Σ ∈ multi_{k}(1, ..., m),

ηG_{Σ}^{I} = (^{η}F_{Σ}^{I})_{anti} (2)

A REPRESENTATION OF THE VARIATIONAL SEQUENCE 471

for I ∈ multi_{k}(ℤ), Σ ∈ multi_{k}(1, …, m),

Ψ_{r}^{k+1}(η) = ∑_{I∈multi_{k}(0,... ,r)}^{η}G_{Σ}^{I} ω_{I}^{Σ}, (3)

Φ_{r}^{k+1}(η) = p_{k}(η − d Ψ_{r}^{k+1}(η)). (4) For k = 0 we set Φ_{r}^{1} = p_{0} : Ω_{r}^{1} → Ω_{r+1}^{1}.

Remark. (1) All the sums in Definition 1 are in actual fact finite. (2) The formulas in Definition 1 define linear mappings

Φ_{r}^{k+1} : Ω_{r}^{k+1} → Ω_{2r+2}^{k+1}, Ψ_{r}^{k+1} : Ω_{r}^{k+1} → Ω_{2r+1 (c)}^{k}. Lemma 1. Let ρ = ∑_{I∈multi_{k}(ℤ)} H_{Σ}^{I} ω_{I}^{Σ} ∈ Ω_{s (c)}^{k}, where H_{Σ}^{I} = 0 for I ∉ multi_{k} (0, …, r). Then

p_{k}dρ = ∑_{I∈multi_{k}(ℤ)}(d_{1}H_{Σ}^{I} + ∑_{|𝔍|=1} H_{Σ}^{I−𝔍}) dt ∧ ω_{I}^{Σ}. (5)

Proof. In the usual notation we can write

ρ = ∑_{0≤l_{1},... ,l_{k}≤r} H_{σ_{1},... ,σ_{k}}^{l_{1},... ,l_{k}} ω_{l_{1}}^{σ_{1}} ∧ … ∧ ω_{l_{k}}^{σ_{k}}.

Then

dρ = ∑_{0≤l_{1},... ,l_{k}≤r} \frac{∂H_{σ_{1},... ,σ_{k}}^{l_{1},... ,l_{k}}}{∂t} dt ∧ ω_{l_{1}}^{σ_{1}} ∧ … ∧ ω_{l_{k}}^{σ_{k}}

+ ∑_{\begin{matrix}0≤l≤s \\ 0≤l_{1},... ,l_{k}≤r \end{matrix}} \frac{∂H_{σ_{1},... ,σ_{k}}^{l_{1},... ,l_{k}}}{∂q_{l}^{σ}} dq_{l}^{σ} ∧ ω_{l_{1}}^{σ_{1}} ∧ … ∧ ω_{l_{k}}^{σ_{k}}

+ ∑_{\begin{matrix}1≤i≤k \\ 0≤l_{1},... ,l_{k}≤r \end{matrix}} (−1)^{i−1} H_{σ_{1},... ,σ_{k}}^{l_{1},... ,l_{k}} ω_{l_{1}}^{σ_{1}} ∧ … ∧ dω_{l_{i}}^{σ_{i}} ∧ … ∧ ω_{l_{k}}^{σ_{k}},

p_{k}dρ = ∑_{0≤l_{1},... ,l_{k}≤r} \frac{∂H_{σ_{1},... ,σ_{k}}^{l_{1},... ,l_{k}}}{∂t} dt ∧ ω_{l_{1}}^{σ_{1}} ∧ … ∧ ω_{l_{k}}^{σ_{k}}

+ ∑_{\begin{matrix}0≤l≤s \\ 0≤l_{1},... ,l_{k}≤r \end{matrix}} q_{l+1}^{τ} \frac{∂H_{σ_{1},... ,σ_{k}}^{l_{1},... ,l_{k}}}{∂q_{l}^{τ}} dt ∧ ω_{l_{1}}^{σ_{1}} ∧ … ∧ ω_{l_{k}}^{σ_{k}}

+ ∑_{\begin{matrix}1≤i≤k \\ 0≤l_{1},... ,l_{k}≤r \end{matrix}} (−1)^{i} H_{σ_{1},... ,σ_{k}}^{l_{1},... ,l_{k}} ω_{l_{1}}^{σ_{1}} ∧ … ∧ ω_{l_{i}+1}^{σ_{i}} ∧ dt ∧ … ∧ ω_{l_{k}}^{σ_{k}}

= ∑_{I∈multi_{k}(0,... ,r)} d_{1}H_{Σ}^{I} dt ∧ ω_{I}^{Σ}

+ ∑_{\begin{matrix}1≤i≤k \\ 0≤l_{1},... ,l_{k}≤r \end{matrix}} H_{σ_{1},... ,σ_{k}}^{l_{1},... ,l_{k}} dt ∧ ω_{l_{1}}^{σ_{1}} ∧ … ∧ ω_{l_{i}+1}^{σ_{i}} ∧ … ∧ ω_{l_{k}}^{σ_{k}} .

472 JAROSLAV ŠTEFÁNEK

Now, we can use the assumption H_{Σ}^{I} = 0 for I ∉ multi_{k} (0, ..., r) and write

p_{k}dρ = ∑_{I∈multi_{k}(ℤ)} d_{1}H_{Σ}^{I} dt ∧ ω_{I}^{Σ}

+ ∑_{\begin{matrix}1≤i≤k \\ 0≤l_{1},... ,l_{k}∈ℤ \end{matrix}} H_{σ_{1},... ,σ_{k}}^{l_{1},... ,l_{i}−1,... ,l_{k}} dt ∧ ω_{l_{1}}^{σ_{1}} ∧ … ∧ ω_{l_{k}}^{σ_{k}}

= ∑_{I∈multi_{k}(ℤ)} (d_{1}H_{Σ}^{I} + ∑_{|𝔍|=1} H_{Σ}^{I−𝔍}) dt ∧ ω_{I}^{Σ}. □

Lemma 2. Let (H_{Σ}^{I})_{I,Σ∈multi_{k}(ℤ)} be an arbitrary system of functions. Then

( ∑_{|𝔍|=1} H_{Σ}^{I−𝔍})_{anti} = ∑_{|𝔍|=1} (H_{Σ}^{I−𝔍})_{anti}

for each I, Σ ∈ multi_{k}(ℤ). Remark. The immediate consequence of Lemma 2 is that the expression (5) is coordinate representation of p_{k}dρ with antisymmetric coefficients. (The antisymmetry of the coefficients in the representation ρ = H_{Σ}^{I} ω_{I}^{Σ} is guaranteed by our general convention.) Lemma 3. Let (^{η}G_{Σ}^{I})_{I∈multi_{k}(ℤ),Σ∈multi_{k}(1,... ,m)} be the system (2) from Definition 1 Then

d_{1}^{η}G_{Σ}^{I} + ∑_{|𝔍|=1}^{η}G_{Σ}^{I−𝔍} = A_{Σ}^{I}

for each I ∈ multi_{k} (0, …, r), Σ ∈ multi_{k} (1, …, m). Proof. Let I = (i_{1}, K) ∈ multi_{k} (0, …, r), Σ ∈ multi_{k} (1, …, m). Then

d_{1}^{η}G_{Σ}^{I} + ∑_{|𝔍|=1}^{η}G_{Σ}^{I−𝔍} = d_{1} (^{η}F_{Σ}^{I})_{anti} + ∑_{|𝔍|=1} (^{η}F_{Σ}^{I−𝔍})_{anti}

= (d_{1}^{η}F_{Σ}^{I} + ∑_{|𝔍|=1}^{η}F_{Σ}^{I−𝔍})_{anti}

= ( ∑_{|𝔍|≥0}(−1)^{|𝔍|} ∑_{λ=0}^{∞}(−1)^{λ} ( \begin{matrix}λ + |𝔍| \\ |𝔍| \end{matrix} ) d_{λ+1}A_{Σ}^{i_{1}+1+|𝔍|+λ,K−𝔍}

+ ∑_{|𝔍|≥0}(−1)^{|𝔍|} ∑_{λ=0}^{∞}(−1)^{λ} ( \begin{matrix}λ + |𝔍| \\ |𝔍| \end{matrix} ) d_{λ}A_{Σ}^{i_{1}+|𝔍|+λ,K−𝔍}

+ ∑_{|𝔏|=1} ∑_{|𝔍|≥0}(−1)^{|𝔍|} ∑_{λ=0}^{∞}(−1)^{λ} ( \begin{matrix}λ + |𝔍| \\ |𝔍| \end{matrix} ) d_{λ}A_{Σ}^{i_{1}+1+|𝔍|+λ,(K−𝔏)−𝔍})_{anti}

A REPRESENTATION OF THE VARIATIONAL SEQUENCE 473

= (− ∑_{|𝔍|≥0}(−1)^{|𝔍|} ∑_{λ=1}^{∞}(−1)^{λ} ( \begin{matrix}λ + |𝔍| − 1 \\ |𝔍| \end{matrix} ) d_{λ}A_{Σ}^{i_{1}+|𝔍|+λ,K−𝔍}

+ ∑_{|𝔍|≥0}(−1)^{|𝔍|} ∑_{λ=0}^{∞}(−1)^{λ} ( \begin{matrix}λ + |𝔍| \\ |𝔍| \end{matrix} ) d_{λ}A_{Σ}^{i_{1}+|𝔍|+λ,K−𝔍}

− ∑_{|𝔍|≥1}(−1)^{|𝔍|} ∑_{λ=0}^{∞}(−1)^{λ} ( \begin{matrix}λ + |𝔍| − 1 \\ |𝔍| − 1 \end{matrix} ) d_{λ}A_{Σ}^{i_{1}+|𝔍|+λ,K−𝔍})_{anti}

= ((−1)^{0}(−1)^{0} ( \begin{matrix}0 + 0 \\ 0 \end{matrix} ) d_{0}A_{Σ}^{i_{1}+0+0,K−0}

+ (−1)^{0} ∑_{λ=1}^{∞}(−1)^{λ} ( \begin{matrix}l + 0 \\ 0 \end{matrix} ) d_{λ}A_{Σ}^{i_{1}+0+λ,K−0}

− (−1)^{0} ∑_{λ=1}^{∞}(−1)^{λ} ( \begin{matrix}λ + 0 − 1 \\ 0 \end{matrix} ) d_{λ}A_{Σ}^{i_{1}+0+λ,K−0}

+ ∑_{|𝔍|≥1}(−1)^{|𝔍|}(−1)^{0} ( \begin{matrix}0 + |𝔍| \\ |𝔍| \end{matrix} ) d_{0}A_{Σ}^{i_{1}+|𝔍|+0,K−𝔍}

− ∑_{|𝔍|≥1}(−1)^{|𝔍|}(−1)^{0} ( \begin{matrix}0 + |𝔍| − 1 \\ |𝔍| − 1 \end{matrix} ) d_{0}A_{Σ}^{i_{1}+|𝔍|+0,K−𝔍}

+ ∑_{|𝔍|≥1}(−1)^{|𝔍|} ∑_{λ=1}^{∞}(−1)^{λ}[( \begin{matrix}λ + |𝔍| \\ |𝔍| \end{matrix} ) − ( \begin{matrix}λ + |𝔍| − 1 \\ |𝔍| \end{matrix} )

− ( \begin{matrix}λ + |𝔍| − 1 \\ |𝔍| − 1 \end{matrix} )]d_{λ}A_{Σ}^{i_{1}+|𝔍|+λ,K−𝔍})_{anti}

= (A_{Σ}^{I})_{anti} = A_{Σ}^{I}.

(Due to our convention the system (A_{Σ}^{I}) is antisymmetric.) □ Lemma 4. Let η ∈ Ω_{r}^{k+1}. Then

Φ_{r}^{k+1}(η) = ∑_{p=1,... ,k} ∑_{0≤a_{1}<···<a_{p}≤k} ∑_{I=(i_{1},...,i_{k})∈multi_{k}(0,...,r) i_{a_{1}} =···=i_{ap} =0 i_{a}≠0 for a∉{a_{1},... ,a_{p}}}

(^{η}G_{Σ}^{i_{1},... ,−1^{\underline{a_{1}}},... ,i_{k}} + ⋯ +^{η}G_{Σ}^{i_{1},... ,−1^{\underline{ap}},... ,i_{k}}) dt ∧ ω_{I}^{Σ}

− ∑_{0≤a_{1}≤k} ∑_{I∈multi_{k}(0,...,r+1)\ multi_{k}(0,...,r) i_{a_{1}} =r+1 i_{a}≠r+1 for a≠a_{1}} ^{η}G_{Σ}^{i_{1},... , r^{\underline{a_{1}}},... ,i_{k}} dt ∧ ω_{I}^{Σ}.

Proof. Let p_{k}η = ∑_{I∈multi_{k}(ℤ)} A_{Σ}^{I} dt ∧ ω_{I}^{Σ}, where A_{Σ}^{I} = 0 for I ∉ multi_{k} (0, …, r). Let us denote H_{Σ}^{I} = ^{η}G_{Σ}^{I} for I ∈ multi_{k} (0, …, r) and put H_{Σ}^{I} = 0 for I ∈ multi_{k} (ℤ) \ multi_{k} (0, …, r). Ψ_{r}^{k+1} (η) = ∑_{I∈multi_{k}(ℤ)} H_{Σ}^{I} ω_{I}^{Σ}. According to Lemma 1 we can write

Φ_{r}^{k+1} (η) = ∑_{I∈multi_{k}(ℤ)} (A_{Σ}^{I} − d_{1}H_{Σ}^{I} − ∑_{|𝔍|=1} H_{Σ}^{I−𝔍}) dt ∧ ω_{I}^{Σ}.

First let us suppose that I ∈ multi_{k} (0, …, r), Σ ∈ multi_{k} (1, …, m). Let I = (i_{1}, …, i_{k}), where i_{a_{1}} = ⋯ = i_{a_{p}} = 0, i_{a} ≠ 0 for a ∉ {a_{1}, …, a_{p}}. Then

A_{Σ}^{I} − d_{1}H_{Σ}^{I} − ∑_{|𝔍|=1} H_{Σ}^{I−𝔍} = A_{Σ}^{I} − d_{1}^{η}G_{Σ}^{I} − ∑_{\begin{matrix}|𝔍|=1 \\ 𝔍_{a_{1}} =...=𝔍_{ap} =0 \end{matrix}}^{η}G_{Σ}^{I−𝔍}

= A_{Σ}^{I} − (d_{1}^{η}G_{Σ}^{I} + ∑_{|𝔍|=1}^{η}G_{Σ}^{I−𝔍}) + ∑_{l=1...p} G_{σ_{1},... ,σ_{k}}^{i_{1},... ,−1^{\underline{a_{l}}},... ,i_{k}}

= ∑_{l=1...p} G_{σ_{1},... ,σ_{k}}^{i_{1},... ,−1^{\underline{a_{l}}},... ,i_{k}}.

(The last step follows from Lemma 3.)

Now, let us suppose that I ∈ multi_{k} (0, …, r + 1) \ multi_{k} (0, …, r) and that Σ ∈ multi_{k} (1, …, m). Let I = (i_{1}, …, i_{k}), let i_{a_{1}} = ⋯ = i_{a_{p}} = r+1, let i_{a} ≠ r+1 for a ∉ {a_{1}, …, a_{p}}. If p > 1 then each upper multiindex in the expression A_{Σ}^{I} − d_{1}H_{Σ}^{I} − ∑_{|𝔍|=1} H_{Σ}^{I−𝔍} contains r + 1 which implies A_{Σ}^{I} − d_{1}H_{Σ}^{I} − ∑_{|𝔍|=1} H_{Σ}^{I−𝔍} = 0.

Let p = 1. Then

{A_{Σ}^{I} − d_{1}H_{Σ}^{I} − ∑_{|𝔍|=1} H_{Σ}^{I−𝔍} = −H_{σ_{1},... ,σ_{k}}^{i_{1},... ,i_{ap} −1,... ,i_{k}} = −H_{σ_{1},... ,σ_{k}}^{i_{1},... , r^{\underline{ap}},... ,i_{k}} \atop = −^{η}G_{σ_{1},... ,σ_{k}}^{i_{1},... , r^{\underline{ap}},... ,i_{k}}.}

Finally, for I ∈ multi_{k} (ℤ) \ multi_{k} (0, …, r + 1), Σ ∈ multi_{k} (1, …, m) we have A_{Σ}^{I} − d_{1}H_{Σ}^{I} − ∑_{|𝔍|=1} H_{Σ}^{I−𝔍} = 0 because none of the upper multiindices in this expression belongs to multi_{k} (0, …, r). The combination of these results gives us the desired formula. □

Corollary 1. Im Φ_{r}^{k+1} ⊆ Ω_{2r+1}^{k+1} for each k > 0.

Theorem 1. Ker Φ_{r}^{k} = Θ_{r}^{k} for each k > 0.

Proof. For k = 1 the relation follows directly from the definition of Φ_{r}^{1} and Θ_{r}^{1}. Now, let us prove that Ker Φ_{r}^{k+1} = Θ_{r}^{k+1} for each k > 0. Let k > 0, let η ∈ Θ_{r}^{k+1} be an arbitrary form. Let us prove that η ∈ Ker Φ_{r}^{k+1}. η ∈ Θ_{r}^{k+1} implies η = dα+β for some forms α ∈ Ω_{r(c)}^{k}, β ∈ Ω_{r(c)}^{k+1}. Let α = ∑_{I∈multi_{k}(ℤ)} H_{Σ}^{I} ω_{I}^{Σ}, where H_{Σ}^{I} = 0 for I ∉ multi_{k} (0, …, r − 1). According to Lemma 1

p_{k}η = p_{k}dα = ∑_{I∈multi_{k}(ℤ)} (d_{1}H_{Σ}^{I} + ∑_{|𝔍|=1} H_{Σ}^{I−𝔍}) dt ∧ ω_{I}^{Σ}. (6)

(Let us notice that d_{1}H_{Σ}^{I} + ∑_{|𝔍|=1} H_{Σ}^{I−𝔍} = 0 for I ∉ multi_{k} (0, …, r) as required in Definition 1 and that the expression (6) is coordinate representation of p_{k}η with antisymmetric coefficients.) Then for each I = (i_{1}, K) ∈ multi_{k} (ℤ), Σ ∈ multi_{k} (1, …, m) we have

ηF_{Σ}^{I} = ∑_{|𝔍|≥0} (−1)^{|𝔍|} ∑_{λ=0}^{∞} (−1)^{λ} ( \begin{matrix}l + |𝔍| \\ |𝔍| \end{matrix} ) d_{λ}(d_{1}H_{Σ}^{i_{1}+1+|𝔍|+λ,K−𝔍} ; + H_{Σ}^{i_{1}+|𝔍|+λ,K−𝔍} + ∑_{|ℨ|=1} H_{Σ}^{i_{1}+1+|𝔍|+λ,K−ℨ−𝔍}) = H_{Σ}^{I}.

(The computation is similar to that one in Lemma 3.) Now, according to Lemma 4 we get Φ_{r}^{k+1} (η) = 0 because ^{η}G_{Σ}^{I} = (^{η}F_{Σ}^{I})_{anti} = (H_{Σ}^{I})_{anti} = 0 for each I ∉ multi_{k}(0, …, r − 1).

It remains to prove the inclusion Ker Φ_{r}^{k+1} ⊆ Θ_{r}^{k+1}. Let η ∈ Ker Φ_{r}^{k+1}. Then p_{k} (η − dΨ_{r}^{k+1} (η)) = 0 which means that both the forms Ψ_{r}^{k+1} (η), η − dΨ_{r}^{k+1} (η) are strongly contact. If we prove that these forms are defined on J^{r}Y we can write η in the form η = dΨ_{r}^{k+1} (η) + (η − dΨ_{r}^{k+1} (η)) as an element of dΩ_{r(c)}^{k} + Ω_{r(c)}^{k+1} = Θ_{r}^{k+1}. Clearly, it is sufficient to prove that Ψ_{r}^{k+1} (η) ∈ Ω_{r}^{k}. Let p_{k}η = ∑_{I∈multi_{k}(ℤ)} A_{Σ}^{I} dt ∧ ω_{I}^{Σ}, where A_{Σ}^{I} = 0 for I ∉ multi_{k} (0, …, r). Let us denote H_{Σ}^{I} = ^{η}G_{Σ}^{I} for I ∈ multi_{k} (0, …, r), H_{Σ}^{I} = 0 for I ∉ multi_{k} (0, …, r). Then Ψ_{r}^{k+1} (η) = ∑_{I∈multi_{k}(0,... ,r)} H_{Σ}^{I} ω_{I}^{Σ}. In order to prove that Ψ_{r}^{k+1} (η) ∈ Ω_{r}^{k} we have to prove that all the functions H_{Σ}^{I} are defined on J^{r}Y and that H_{Σ}^{I} = 0 whenever I contains at least one r. The second condition follows immediately from the coordinate representation of the equation Φ_{r}^{k+1} (η) = 0. (See Lemma 4.) Let us prove the first one. In the coordinates the equation Φ_{r}^{k+1} (η) = 0 has the form

∑_{I∈multi_{k}(ℤ)} A_{Σ}^{I} dt ∧ ω_{I}^{Σ} = ∑_{I∈multi_{k}(ℤ)} (d_{1}H_{Σ}^{I} + ∑_{|𝔍|=1} H_{Σ}^{I−𝔍}) dt ∧ ω_{I}^{Σ}. (7)

Both of the systems

(A_{Σ}^{I})\begin{matrix}I \\ Σ \end{matrix} , (d_{1}H_{Σ}^{I} + ∑_{|𝔍|=1} H_{Σ}^{I−𝔍})\begin{matrix}I \\ Σ \end{matrix}

are antisymmetric and for each I ∉ multi_{k}(0, …, r) we have A_{Σ}^{I} = 0, d_{1}H_{Σ}^{I} + ∑_{|𝔍|=1} H_{Σ}^{I−𝔍} = 0 so that the equation (7) is equivalent to the following system of equations

d_{1}H_{Σ}^{I} + ∑_{|𝔍|=1} H_{Σ}^{I−𝔍} = A_{Σ}^{I}

I ∈ multi_{k}(0, …, r), Σ ∈ multi_{k}(1, …, m).

(8)

Let us prove by the mathematical induction that all the functions H_{Σ}^{I} are defined on J^{r}Y. For |I| = 0, Σ ∈ multi_{k}(1, …, m) we get from the system (8) the equation d_{1}H_{Σ}^{0,... ,0} = A_{Σ}^{0,... ,0} which shows that H_{Σ}^{0,... ,0} ∈ Ω_{r}^{0} because A_{Σ}^{0,... ,0} ∈ Ω_{r+1}^{0}. Let us suppose that for all I ∈ multi_{k}(0, …, r) such that |I| ≤ p and for all Σ ∈ multi_{k}(1, …, m) the functions H_{Σ}^{I} belong to Ω_{r}^{0} . Let I ∈ multi_{k}(0, …, r) be a multiindex such that |I| = p + 1, let Σ ∈ multi_{k}(1, …, m). The corresponding equation from the system (8) gives us

d_{1}H_{Σ}^{I} = A_{Σ}^{I} − ∑_{|𝔍|=1} H_{Σ}^{I−𝔍}. (9)

The function on the right side of (9) belongs to Ω_{r+1}^{0} which means that H_{Σ}^{I} is defined on J^{r}Y. □

The mappings Φ allow us to represent the variational sequence in the following way

[figure: 0  R  Ω_{0}  Ω^{1}/Θ^{1}  Ω^{2}/Θ^{2}  Ω^{3}/Θ^{3}  … Im Φ_{r}^{1}  Im Φ_{r}^{2}  Im Φ_{r}^{3}  …; E_{0}; _{\overline{E}_{0}}; ≅; Φ_{r}^{1}=p_{0}; E_{1}; ≅; Φ_{r}^{2}; E_{2}; ≅; Φ_{r}^{3}; _{\overline{E}_{1}}; _{\overline{E}_{2}}; _{\overline{E}_{3}}] (10)

where Im Φ_{r}^{1} ⊆ Ω_{r+1,X}^{1}, Im Φ_{r}^{i} ⊆ Ω_{2r+1}^{i} for i > 1. (We use the same notation for the mapping Φ_{r}^{k} : Ω_{r}^{k} → Ω_{s}^{k} and for the corresponding mapping Ω_{r}^{k}/ Ker Φ_{r}^{k} = Ω_{r}^{k}/Θ_{r}^{k} ⟶ Ω_{s}^{k}.) For applications of this representation it would be convenient to know the coordinate representation of the mappings from the diagram

Ω_{1}  Ω_{2}  Ω_{3}

Im Φ_{r}^{1}  Im Φ_{r}^{1}  Im Φ_{r}^{1}.

Φ_{r}^{1}

[figure]

Φ_{r}^{2}

[figure]

Φ_{r}^{3}

[figure]

\overline{E}_{1}

[figure]

\overline{E}_{2}

[figure]

Notation. For a function L : J^{r}Y → ℝ we denote by ε_{σ}(L) the Euler–Lagrange expressions

ε_{σ}(L) = ∑_{l=0}^{r}(−1)^{l} d_{l} \frac{∂L}{∂q_{l}^{σ}} .

A REPRESENTATION OF THE VARIATIONAL SEQUENCE 477

For a system (ε_{σ})_{σ=1,... ,m} of functions on J^{r}Y we denote by H_{σ_{1},σ_{2}}^{i}(ε_{σ}) the Helmholtz–Sonin expressions

H_{σ_{1},σ_{2}}^{i}(ε_{σ}) = \frac{∂ε_{σ_{1}}}{∂q_{i}^{σ_{2}}} − (−1)^{i} \frac{∂ε_{σ_{2}}}{∂q_{i}^{σ_{1}}} − ∑_{k=i+1}^{∞}(−1)^{k} ( \begin{matrix}k \\ i \end{matrix} ) d_{k−i} \frac{∂ε_{σ_{2}}}{∂q_{k}^{σ_{1}}} .

(For further information see [3, 4].)

Theorem 2. (1) Let η = A dt + ∑\begin{matrix}r−1 \\ i=0 \end{matrix} B_{σ}^{i} ω_{i}^{σ} + C_{σ} dq_{r}^{σ} ∈ Ω_{r}^{1}. Then

Φ_{r}^{1} (η) = p_{0} (η) = (A + q_{r+1}^{τ}C_{τ} ) dt.

(2) Let η ∈ Ω_{r}^{2}, let p_{1} (η) = ∑\begin{matrix}r \\ i=0 \end{matrix} A_{σ}^{i} dt ∧ ω_{i}^{σ}. Then

Φ_{r}^{2}(η) = (∑_{l=0}^{r}(−1)^{l}d_{l} A_{σ}^{l}) dt ∧ ω^{σ}.

(3) Let η ∈ Ω_{r}^{3}, let p_{2} (η) = ∑\begin{matrix}r \\ i_{1},i_{2}=0 \end{matrix} A_{σ_{1},σ_{2}}^{i_{1},i_{2}} dt ∧ ω_{i_{1}}^{σ_{1}} ∧ ω_{i_{2}}^{σ_{2}}. Then

Φ_{r}^{3} (η) = \frac{1}{2} ∑_{l=0}^{r}(−1)^{l}d_{l} (A_{σ_{1},σ_{2}}^{l , 0} + A_{σ_{1},σ_{2}}^{0 , l}) dt ∧ ω^{σ_{1}} ∧ ω^{σ_{2}}

+ ∑_{i_{2}=1}^{r} [∑_{j=0}^{i_{2}}(−1)^{j} ∑_{l=0}^{r−j}(−1)^{l} ( \begin{matrix}l + j \\ j \end{matrix} ) d_{l} A_{σ_{1},σ_{2}}^{j+l,i_{2}−j}] dt ∧ ω^{σ_{1}} ∧ ω_{i_{2}}^{σ_{2}}

+ ∑_{i_{1}=0}^{r−1} [∑_{j=0}^{r}(−1)^{j} ∑_{l=0}^{r−i_{1}−1−j}(−1)^{l} ( \begin{matrix}l + j \\ j \end{matrix} ) d_{l} A_{σ_{1},σ_{2}}^{i_{1}+1+j+l,r−j}] dt ∧ ω_{i_{1}}^{σ_{1}}

∧ ω_{r+1}^{σ_{2}}.

(4) Let ρ = L dt ∈ Im Φ_{r}^{1}. Then \overline{E_{1}} = −ε_{σ}(L) dt ∧ ω^{σ}.

(5) Let ρ = ε_{σ} dt ∧ ω^{σ} ∈ Im Φ_{r}^{2}. Then

\overline{E}_{2}(ρ) = \frac{1}{2} ∑_{i=0}^{r} H_{σ_{1},σ_{2}}^{i}(ε_{σ}) dt ∧ ω^{σ_{1}} ∧ ω^{σ_{2}}

− \frac{1}{2} [(−1)^{0}dt ∧ ω_{0}^{σ_{1}} ∧ ω_{r+1}^{σ_{2}}, …, (−1)^{r−1} dt ∧ ω_{r−1}^{σ_{1}} ∧ ω_{r+1}^{σ_{2}}]

· P^{−1} [ H_{σ_{1},σ_{2}}^{r+1}(ε_{σ}) ; ⋮ ; H_{σ_{1},σ_{2}}^{2r}(ε_{σ}) ] ,

where P = [( \begin{matrix}j − 1 \\ i − 1 \end{matrix} ) d_{j−i}]_{1≤i,j≤r}.

Remark. The matrix P is triangular and all the elements on its diagonal are identical mappings. That means that det P is also the identical mapping and, consequently, P^{−1} can be defined by the usual formula.

Proof. (1),(2),(3) can be proved by putting k = 0, 1, 2 into Definition 1. In order to prove (4),(5) we have to express the coordinate representation of Φ_{r}^{2} ∘ d, Φ_{r}^{3} ∘ d in the required form which is trivial for Φ_{r}^{2} ∘ d and a long and tedious computation for Φ_{r}^{3} ∘ d. □

Remark. The expression in (4) shows that \overline{E_{1}} is the Euler–Lagrange mapping. The assertion (5) shows that the representation of E_{2} is not the same as in [2] or [4]. On the other hand, the equation \overline{E_{2}} Φ_{r}^{2} (η) = 0 is equvalent to the Helmholtz– Sonin equations from [2] and [4] because the operator defined by the matrix P is bijective.

Acknowledgements

The author would like to thank Friedrich Hirzebruch for his invitation to the Max Planck Institut für Mathematik, Bonn, where part of this research was done. The author also wishes to thank Demeter Krupka for many helpful discussions and suggestions.

References

1. I. Anderson, The Variational Bicomplex, Preprint (Utah State University, 1989).

2. J. Musilová, Variational Sequence in Higher Order Mechanics, in: Differential Geometry and its Applications, Proc. Conf., J. Janyška, I. Kolář and J. Slovák, Editors, Brno, Czech Republic, 1995, to appear.

3. D. Krupka, 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.

4. D. Krupka, Lectures on Variational Sequence (Mechanics) (Open Education and Sciences, Opava, 1995).

5. D. Saunders, The Geometry of Jet Bundles, London Mathematical Society Lecture Notes Series 142 (Cambridge University Press, New York, 1989).

Department of Mathematics, Silesian University Opava Bezručovo nám 13, 746 01 Opava, CZECH REPUBLIC