DIFFERENTIAL GEOMETRY AND APPLICATIONS Proc. Conf., Aug. 28 – Sept. 1, 1995, Brno, Czech Republic Masaryk University, Brno 1996, 611–624

VARIATIONAL SEQUENCE IN HIGHER ORDER MECHANICS

JANA MUSILOVÁ

Abstract. The variational sequence in higher order mechanics is considered. Explicit coordinate formulas for factorization mappings and quotient mappings for the first three columns are determined which includes the total derivative, Euler – Lagrange and Helmholtz – Sonin mappings.

1. Introduction

One of the key problems of the calculus of variations is characterization of the Euler – Lagrange mapping, particularly its kernel and image. These problems can be solved very effectively by means of the so called variational sequence constructed as the quotient sequence obtained by factorization of well–known De Rham sequence of modules of differential forms defined on a finite jet prolongation of an underlying fibered manifold with respect to appropriately chosen subsequence.

The general concept of the variational sequence was presented by Krupka in his earlier works (see e.g. [1]). The problem was then completely solved by the same author for the first order mechanics [2]. The suitable subsequence of De Rham sequence was found and the corresponding quotient sequence was completely constructed. It was shown that first three quotient mappings are very closely related to the problem of triviality of a lagrangian, Euler – Lagrange mapping and variationality conditions, respectively.

In our paper the more general problem is treated: The properties of the variational sequence in higher order mechanics are studied. The calculation technique for obtaining the chart expressions representing the classes of differential forms in the factorization procedure is developed and the explicit coordinate formulas for factorization as well as for quotient mappings are obtained for first three columns of the corresponding sequence diagram. Finally, it is shown that in the general (i.e. higher order) case the first three quotient mappings can be interpreted in the same way as in the first–order case: For the first of them, E_{0}, the image of a function is the corresponding trivial lagrangian, E_{1} is the well–known Euler – Lagrange mapping, and E_{2} is very closely connected with the variationality conditions. The E_{2}–image of the class of variational 2–forms is represented by the zero form and the coefficients of the chart expression of the representative of each class belonging to Im E_{2} are directly the left–hand sides of well–known Helmholtz – Sonin conditions.

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

Key words and phrases. Fibered manifold, r–jet prolongation, contact form, variational sequence, Lagrangian, Euler – Lagrange mapping, Helmholtz – Sonin conditions of variationality.

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

The author was supported by grant 201/93/2245 of the Czech Grant Agency.

Let us note that the same problem was studied by Štefánek [3] whose results on variationality are equivalent to ours.

2. Underlying structures and basic notations

We use the following standard notation through the paper. Y is a fibered manifold with the one–dimensional base X and projection π. Let us denote m = dim Y −1. For an arbitrary integer s ≥ 0, j^{s}Y is the s – jet prolongation of Y, π_{s} and π_{s,r} for s ≥ r ≥ 0 are the canonical projections of j^{s}Y on X and j^{r}Y, respectively; dim j^{s}Y = (s + 1)m + 1. Let 1 ≤ σ ≤ m and (V, ψ), ψ = (t, q_{0}^{σ}), be a fibered chart on Y. Then we denote by (U, φ) and (V_{s}, ψ_{s}) the associated chart on X and associated fibered chart on j^{s}Y, respectively, where U = π(V), φ = (t), V_{s} = π_{s,0}^{−1} (V), ψ_{s} = (t, q_{0}^{σ}, q_{1}^{σ}, …, q_{s}^{σ}).

Let Ω_{s}^{0}V be the ring of smooth functions on V_{s}. We denote by Ω_{s}^{k}V the Ω_{s}^{0}V –module of smooth differential k–forms on V_{s}. The following definitions are adapted for the practical purpose.

Let (V, ψ) be a fibered chart on Y and ϱ ∈ Ω_{s}^{k}V an arbitrary k–form with the following chart expression:

ϱ = \frac{1}{(k − 1)!} ∑_{0≤i_{1},..., i_{k−1}≤s} A _{σ_{1}...σ_{k−1}}^{i_{1}...i_{k−1}}dq_{i_{1}}^{σ_{1}} ∧ ⋯ ∧ dq_{i_{k−1}}^{σ_{k−1}} ∧ dt +

+ \frac{1}{k!} ∑_{0≤i_{1},..., i_{k}≤s} B _{σ_{1}...σ_{k}}^{i_{1}...i_{k}}dq_{i_{1}}^{σ_{1}} ∧ ⋯ ∧ dq_{i_{k}}^{σ_{k}}. (1)

Coefficients A _{σ_{1}...σ_{k−1}}^{i_{1}...i_{k−1}}, B _{σ_{1}...σ_{k}}^{i_{1}...i_{k}} are completely antisymmetric in multiindices, 0 ≤ i_{1}, …, i_{k} ≤ s, 1 ≤ σ_{1}, …, σ_{k} ≤ m.

It can be easily shown (see e.g. [2]) that π_{s+1, s}^{∗}ϱ can be written in the form

π_{s+1, s}^{∗}ϱ = p_{k−1}ϱ + p_{k}ϱ,

where

p_{k−1}ϱ = \frac{1}{(k − 1)!} ∑_{0≤i_{1},..., i_{k−1}≤s} \overline{A _{σ_{1}...σ_{k−1}}^{i_{1}...i_{k−1}}} ω_{i_{1}}^{σ_{1}} ∧ ⋯ ∧ ω_{i_{k−1}}^{σ_{k−1}} ∧ dt , (2)

\overline{A _{σ_{1}...σ_{k−1}}^{i_{1}...i_{k−1}}} = A _{σ_{1}...σ_{k−1}}^{i_{1}...i_{k−1}} + ∑_{j=0}^{s} B _{σ_{1}...σ_{k−1} ν}^{i_{1}...i_{k−1} j} q_{j+1}^{ν} ,

p_{k}ϱ = \frac{1}{k!} ∑_{0≤i_{1},..., i_{k}≤s} B _{σ_{1}...σ_{k}}^{i_{1}...i_{k}} ω_{i_{1}}^{σ_{1}} ∧ ⋯ ∧ ω_{i_{k}}^{σ_{k}} ,

ω_{j}^{σ} = dq_{j}^{σ} − q_{j+1}^{σ} dt for 0 ≤ j ≤ s.

Coefficients \overline{A _{σ_{1}...σ_{k−1}}^{i_{1}...i_{k−1}}} are again completely antisymmetric in multiindices. p_{k−1}ϱ and p_{k}ϱ represent the (k − 1)–contact and k–contact part of the form ϱ, respectively. The 0–contact part of a form is usually called the horizontal one and the mapping h : Ω_{s}^{k} ∋ ϱ ⟶ p_{0}ϱ = hϱ ∈ Ω_{s+1}^{k} is called the horizontalization mapping. The (nonzero) k–form ϱ is called k–contact if p_{k−1}ϱ = 0, or (k − 1)–contact if p_{k}ϱ = 0. The 0–contact form is called horizontal.

3. Variational sequence

In the following we denote Ω_{s, c}^{k} ⊂ Ω_{s}^{k} the subspace of k–contact k–forms and Θ_{s}^{k} = Ω_{s, c}^{k} + d Ω_{s, c}^{k−1} for k ≥ 1. d Ω_{s, c}^{k−1} is the image of Ω_{s, c}^{k−1} by the exterior derivative operator d. For any open set V ⊂ Y there is a unique decomposition of any form ϱ ∈ Θ_{s}^{k} V such that ϱ = ϱ_{0} +dϱ′, ϱ_{0} ∈ Ω_{s, c}^{k} V, ϱ′ ∈ Ω_{s, c}^{k−1} V (see e.g. [2]).

Now let us construct the variational sequence. The diagram

0 ⟶ R ⟶ Ω_{s}^{0} ⟶ Ω_{s}^{1} ⟶ Ω_{s}^{2} ⟶ ⋯ ⟶ Ω^{(s+1)m+1} ⟶ 0 , (3)

in which the first two arrows denote the canonical injections and the remaining ones mean the exterior derivative d, represents the well–known (exact) De Rham sequence. Consider its subsequence

0 ⟶ Θ_{s}^{1} ⟶ Θ_{s}^{2} ⟶ ⋯ ⟶ Θ_{s}^{sm+1} ⟶ 0 . (4)

It can be immediately seen that it is exact too. The corresponding quotient sequence

0 ⟶ R ⟶ Ω_{s}^{0} ⟶^{E_{0}} Ω_{s}^{1}/Θ_{s}^{1} ⟶^{E_{1}} Ω_{s}^{2}/Θ_{s}^{2} ⟶^{E_{2}} ⋯

⋯ ⟶ Ω_{s}^{sm+1}/Θ_{s}^{sm+1} ⟶ Ω_{s}^{sm+2} ⟶ ⋯ ⟶ Ω_{s}^{(s+1)m+1} ⟶ 0 (5)

is exact as well and is called the variational sequence of the s–th order. The quotient mappings E_{k} : Ω_{s}^{k}/Θ_{s}^{k} ⟶ Ω_{s}^{k+1}/Θ_{s}^{k+1} are defined by

E_{k} ( [ϱ] ) = [dϱ] , (6)

where [ϱ] denotes the class of the form ϱ. The general properties of these mappings were studied elsewhere [1], [2]. The main purpose of our calculations is to obtain the explicit coordinate formulas for [ϱ] and E_{k} ( [ϱ] ). From the point of view of the variational theory the meaning of first three quotient mappings E_{0}, E_{1}, E_{2} is evident. On the other hand, it appears that solving the problem of finding the coordinate expressions for factorizations and quotient mappings in the r–th order variational sequence, we must work on s = (2r + 1)–jet prolongation of the underlying fibered manifold Y. Thus we examine the factorization mappings F_{k} and quotient mappings E_{k} generally in the following diagram (see the final note of Chapter 4) and we obtain the explicit coordinate formulas for the appropriately chosen representatives of classes [ϱ] = F_{k}(ϱ) and E_{k} ( [ϱ] ) = [dϱ] especially for a k–form ϱ defined primarily on j^{r}Y, i.e. [π_{2r+1, r}^{∗}ϱ] and [π_{2r+1, r}^{∗}dϱ]. In the following calculations we write ϱ ∼ ω for [ϱ] = [ω]. The fact that a form η is a representative of the class [ϱ] we express by [ϱ] ≅ η (the class [ϱ] is represented by η).

0  0  0
↓ i_{0,1}  ↓ i_{0,2}  ↓ i_{0,3}
0  →^{i_{0,1}}  Θ_{2r+1}^{1}  →^{d_{1,c}} Θ_{2r+1}^{2}  →^{d_{2,c}} Θ_{2r+1}^{3}  →⋯ 0
↓ i_{0}  ↓ i_{1}  ↓ i_{2}  ↓ i_{3}
0→^{i_{0}}R→^{i_{1}} Ω_{2r+1}^{0} →^{d_{0}}  Ω_{2r+1}^{1}  →^{d_{1}} Ω_{2r+1}^{2}  →^{d_{2}} Ω_{2r+1}^{3}  →⋯ 0
E_{0} ↘ ↓ F_{1}  ↓ F_{2}  ↓ F_{3}
Ω_{2r+1}^{1}/Θ_{2r+1}^{1} →^{E_{1}} Ω_{2r+1}^{2}/Θ_{2r+1}^{2} →^{E_{2}} Ω_{2r+1}^{3}/Θ_{2r+1}^{3} →⋯ 0
↓ o_{1}  ↓ o_{2}  ↓ o_{3}
0  0  0

(7)

d … exterior derivative  i … canonical injections
o … zero mappings  F … factorization mappings
E … quotient mappings  Θ_{2r+1}^{k} = Ker F_{k}

4. Factorization mappings

The idea of the calculation procedure which leads to the chart expresions of the representative of the class [ϱ] of a k–form ϱ ∈ Ω_{r}^{k}V lies in finding the most suitable decomposition of the form π_{s, r}^{∗}ϱ with appropriately chosen s ≥ r as follows: π_{s, r}^{∗}ϱ = ϱ_{0} + ϱ_{Θ} where ϱ_{Θ} ∈ Θ_{s}^{k} and ϱ_{0} is the representative of the class of ϱ, i.e. [ϱ] ≅ ϱ_{0}. The mentioned procedure is used in the proofs of the following theorem.

Theorem 1. Let (V, ψ), ψ = (t, q_{0}^{σ}), 1 ≤ σ ≤ m, be a fibered chart on Y.

(a) Let ϱ ∈ Ω_{r}^{1}V have the chart expression

ϱ = A dt + ∑_{j=0}^{r} B_{σ}^{j} dq_{j}^{σ} .

Then the class of the form ϱ, [ϱ] = F_{1}(ϱ), is defined on the s = (r + 1)–jet prolongation of Y, i.e. [ϱ] ∈ Ω_{r+1}^{1}V/Θ_{r+1}^{1}V, by the coordinate formula for its representative:

[ϱ] ≅ \overline{A} dt = (A + ∑_{j=0}^{r} B_{σ}^{j} q_{j+1}^{σ}) dt .

(b) Let ϱ ∈ Ω_{r}^{2}V have the chart expression

ϱ = ∑_{j=0}^{r} A_{σ}^{j} dq_{j}^{σ} ∧ dt + \frac{1}{2} ∑_{i, j=0}^{r} B_{σν}^{ji} dq_{j}^{σ} ∧ dq_{i}^{ν} , B_{σν}^{ji} + B_{νσ}^{ij} = 0 .

Then the class [ϱ] = F_{2}(ϱ) is defined on s = (2r + 1)–jet prolongation of Y, i.e. [ϱ] ∈ Ω_{2r+1}^{2}V/Θ_{2r+1}^{2}V, and its representative is given by the following coordinate formula:

[ϱ] ≅ ∑_{j=0}^{r}(−1)^{j} \frac{d^{j}\overline{A_{σ}^{j}}}{dt^{j}} ω_{0}^{σ} ∧ dt , \overline{A_{σ}^{j}} = A_{σ}^{j} + ∑_{i=0}^{r} B_{σν}^{ji} q_{i+1}^{ν} ,

in which d/dt denotes the total (formal) derivative.

(c) Let ϱ ∈ Ω_{r}^{3}V have the chart expression

ϱ = \frac{1}{2} ∑_{i, j=0}^{r} A_{σν}^{ji} dq_{j}^{σ} ∧ dq_{i}^{ν} ∧ dt + \frac{1}{3!} ∑_{i, j, k=0}^{r} B_{σνλ}^{jik} dq_{j}^{σ} ∧ dq_{i}^{ν} ∧ dq_{k}^{λ} ,

A_{σν}^{ji} + A_{νσ}^{ij} = 0 and B_{σνλ}^{jik} are completely antisymmetric in multiindices. Then the class [ϱ] = F_{3}(ϱ) is defined on s = (2r + 1)–jet prolongation of Y, i.e. [ϱ] ∈ Ω_{2r+1}^{3}V/Θ_{2r+1}^{3}V, and its representative is given by the coordinate formula

[ϱ] ≅ ∑_{k=0}^{2r} B_{σν}^{k}ω_{k}^{σ} ∧ ω_{0}^{ν} ∧ dt ,

where

B_{σν}^{k} = \frac{1}{2} ∑_{m=p}^{q} ∑_{i=m}^{r}(−1)^{i} \binom{i}{m}\frac{d^{i−m}\overline{A_{σν}^{k−m, i}}}{dt^{i−m}} , \overline{A_{σν}^{ji}} = A_{σν}^{ji} + ∑_{l=0}^{r} B_{σνλ}^{jil} q_{l+1}^{λ}

and p = 0, q = k for 0 ≤ k ≤ r, p = k − r, q = r for r < k ≤ 2r.

Proof. Let ϱ ∈ Ω_{r}^{k}V. We obtain the relation [ϱ] = [p_{k−1}ϱ] as the immediate consequence of the definition of the class [ϱ].

(a) For k = 1 we have [ϱ] = [p_{0}ϱ] = [hϱ]. For ϱ = A dt + ∑_{j=0}^{r} B_{σ}^{j} dq_{j}^{σ} the relations (2) give the coordinate expression for hϱ ∈ Ω_{r+1}^{k}V which is simultaneously the representative of the class [ϱ]:

[ϱ] = [hϱ] ≅ \overline{A} dt = (A + ∑_{j=0}^{r} B_{σ}^{j} q_{j+1}^{σ}) dt . (8)

We can see that [ϱ] ∈ Ω_{r+1}^{1}V/Θ_{r+1}^{1}V, because \overline{A} ∈ Ω_{r+1}^{0}V.

(b) For k = 2 and ϱ = A_{σ}^{j} dq_{j}^{σ} ∧ dt + ∑_{i=0}^{r} B_{σν}^{ji} dq_{j}^{σ} ∧ dq_{i}^{ν} ∈ Ω_{r}^{2}V we obtain by using the formulas (2):

p_{1}ϱ = ∑_{j=0}^{r} \overline{A_{σ}^{j}} ω_{j}^{σ} ∧ dt , \overline{A_{σ}^{j}} + ∑_{i=0}^{r} B_{σν}^{ji} q_{i+1}^{ν} ∈ Ω_{r+1}^{0}V .

For obtaining the representative of the class [ϱ] = [p_{1}ϱ] we look for the appropriate decomposition of π_{s, r+1}^{∗}p_{1}ϱ:

p_{1}ϱ = \overline{A_{σ}^{0}} ω_{0}^{σ} ∧ dt + ∑_{j=1}^{r} \overline{A_{σ}^{j}} ω_{j}^{σ} ∧ dt = \overline{A_{σ}^{0}} ω_{0}^{σ} ∧ dt + ∑_{j=1}^{r}(−\overline{A_{σ}^{j}}) dω_{j−1}^{σ} .

The last relation follows from the fact that dω_{j−1}^{σ} = π_{r+1, r}^{∗}d(dq_{j−1}^{σ} − −q_{j}^{σ} dt) = −π_{r+1, r}^{∗}(dq_{j}^{σ} ∧ dt) = −ω_{j}^{σ} ∧ dt . The next calculation procedure is based on repeating the appropriate number of steps of the following type:

π_{r+2, r+1}^{∗}(−\overline{A_{σ}^{j}} dω_{j−1}^{σ}) = π_{r+2, r+1}^{∗} [(d\overline{A_{σ}^{j}} ∧ ω_{j−1}^{σ}) − d(\overline{A_{σ}^{j}} ω_{j−1}^{σ})] =

= −\frac{d\overline{A_{σ}^{j}}}{dt} ω_{j−1}^{σ} ∧ dt + p_{1}(d\overline{A_{σ}^{j}}) ∧ ω_{j−1}^{σ} − π_{r+2, r+1}^{∗} d(\overline{A_{σ}^{j}}ω_{j−1}^{σ}) .

The form π_{r+2, r+1}^{∗}(−\overline{A_{σ}^{j}} dω_{j−1}^{σ}) is expressed as the sum of forms −\frac{d\overline{A_{σ}^{j}}}{dt} ω_{j−1}^{σ} ∧dt ∈

Ω_{r+2}^{2}V, p_{1}(d\overline{A_{σ}^{j}})∧ω_{j−1}^{σ} ∈ Ω_{r+2, c}^{2}V and −π_{r+2, r+1}^{∗} d(\overline{A_{σ}^{j}}ω_{j−1}^{σ}) ∈ π_{r+2, r+1}^{∗}dΩ_{r+1, c}^{1}V ⊂ dΩ_{r+2, c}^{1}V and thus

p_{1}ϱ ∼ \overline{A_{σ}^{0}} ω_{0}^{σ} ∧ dt + (−1)^{1} ∑_{j=1}^{r} \frac{d\overline{A_{σ}^{j}}}{dt} ω_{j−1}^{σ} ∧ dt =

= (\overline{A_{σ}^{0}} + (−1)^{1} \frac{d\overline{A_{σ}^{1}}}{dt} ) ω_{0}^{σ} ∧ dt + (−1)^{1} ∑_{j=2}^{r} \frac{d\overline{A_{σ}^{j}}}{dt} ω_{j−1}^{σ} ∧ dt =

= (\overline{A_{σ}^{0}} + (−1)^{1} \frac{d\overline{A_{σ}^{1}}}{dt} ) ω_{0}^{σ} ∧ dt + (−1)^{2} ∑_{j=2}^{r} \frac{d\overline{A_{σ}^{j}}}{dt} dω_{j−2}^{σ} .

Repeating the preceding procedure without writing the necessary pullbacks we finally obtain

[ϱ] ≅ (\overline{A_{σ}^{0}} − \frac{d\overline{A_{σ}^{1}}}{dt} + \frac{d^{2}\overline{A_{σ}^{2}}}{dt^{2}} + ⋯ + (−1)^{r} \frac{d^{r}\overline{A_{σ}^{r}}}{dt^{r}} ) ω_{0}^{σ} ∧ dt = ∑_{j=0}^{r}(−1)^{j} \frac{d^{j}\overline{A_{σ}^{j}}}{dt^{j}} ω_{0}^{σ} ∧ dt . (9)

It is evident that because of the expression d^{r}(\overline{A_{σ}^{r}})/dt^{r} ∈ Ω_{2r+1}^{0}V we get the resulting chart expression defined on the s = (2r + 1)–jet prolongation of Y, i.e. [ϱ] ∈ Ω_{2r+1}^{2}V/Θ_{2r+1}^{2}V .

(c) Let k = 3 and ϱ = \frac{1}{2} ∑_{i, j=0}^{r} A_{σν}^{ji} dq_{j}^{σ} ∧ dq_{i}^{ν} ∧ dt + \frac{1}{3!} ∑_{i, j, k=0}^{r} B_{σνλ}^{jik} dq_{j}^{σ} ∧

∧ dq_{i}^{ν} ∧ dq_{k}^{λ} . Then the relations (2) give p_{2}ϱ = \frac{1}{2} ∑_{i, j=0}^{r} \overline{A_{σν}^{ji}} ω_{j}^{σ} ∧ ω_{i}^{ν} ∧ dt , where

\overline{A_{σν}^{ji}} = A_{σν}^{ji} + ∑_{k=0}^{r} B_{σνλ}^{jik}q_{k+1}^{λ} and \overline{A_{σν}^{ji}} + \overline{A_{νσ}^{ij}} = 0 . We try to write the representative of [ϱ] in the form

[ϱ] ≅ ∑_{k=0}^{s} B_{σν}^{k}ω_{k}^{σ} ∧ ω_{0}^{ν} ∧ dt , (10)

in which s is the appropriate order of the jet prolongation of Y and B_{σν}^{k}, 1 ≤ σ, ν ≤ m , 0 ≤ k ≤ s , are unknown coefficients. Let F be a function and ω and η contact 1–forms on j^{s}Y. Then we have, again without writing the necessary pullbacks,

d(F ω ∧ η) = dF ∧ ω ∧ η + F dω ∧ η − F ω ∧ dη =

= \frac{dF}{dt} ω ∧ η ∧ dt + p_{1}(dF) ∧ ω ∧ η + F dω ∧ η − F ω ∧ dη

and thus

F ω∧dη = \frac{dF}{dt} ω∧η∧dt+F dω∧η−d(F ω∧η)+p_{1}(dF)∧ω∧η ∼ \frac{dF}{dt} ω∧η∧dt+F dω∧η ,

because −d(F ω ∧ η) ∈ dΩ_{s, c}^{2} and p_{1}(dF) ∧ ω ∧ η ∈ Ω_{s+1, c}^{3}.

By using this relation for the expression \overline{A_{σν}^{ji}}ω_{j}^{σ} ∧ ω_{i}^{ν} ∧ dt = −\overline{A_{σν}^{ji}}ω_{j}^{σ} ∧ dω_{i−1}^{ν} in

which we denote \overline{A_{σν}^{ji}} = F, ω_{j}^{σ} = ω and ω_{i−1}^{ν} = η for i ≥ 1 , we obtain

\overline{A_{σν}^{ji}}ω_{j}^{σ} ∧ ω_{i}^{ν} ∧ dt = −\overline{A_{σν}^{ji}}ω_{j}^{σ} ∧ dω_{i−1}^{ν} ∼ −\frac{d\overline{A_{σν}^{ji}}}{dt} ω_{j}^{σ} ∧ ω_{i−1}^{ν} ∧ dt − \overline{A_{σν}^{ji}}dω_{j}^{σ} ∧ ω_{i−1}^{ν} =

=−\frac{d\overline{A_{σν}^{ji}}}{dt} ω_{j}^{σ}∧ω_{i−1}^{ν}∧dt−\overline{A_{σν}^{ji}}ω_{j+1}^{σ}∧ω_{i−1}^{ν}∧dt =−(\frac{d\overline{A_{σν}^{ji}}}{dt} ω_{j}^{σ} + \overline{A_{σν}^{ji}}ω_{j+1}^{σ})∧ω_{i−1}^{ν}∧dt =

= \{ \begin{matrix}− (\frac{d\overline{A_{σν}^{j1}}}{dt} ω_{j}^{σ} + \overline{A_{σν}^{j1}}ω_{j+1}^{σ}) ∧ ω_{0}^{ν} ∧ dt & for i = 1 \\ \frac{d\overline{A_{σν}^{ji}}}{dt} ω_{j}^{σ} ∧ dω_{i−2}^{ν} + \overline{A_{σν}^{ji}}ω_{j+1}^{σ} ∧ dω_{i−2}^{ν} & for i ≥ 2 \end{matrix}

By repeating this procedure step by step we get

\overline{A_{σν}^{ji}}ω_{j}^{σ} ∧ ω_{i}^{ν} ∧ dt ∼ ∑_{l=0}^{i}(−1)^{i}\binom{i}{l}\frac{d^{i−l} \overline{A_{σν}^{ji}}}{dt^{i−l}} ω_{j+l}^{σ} ∧ ω_{0}^{σ} ∧ dt

and thus

[ϱ] = [p_{2}ϱ] ≅ ∑_{i, j=0}^{r} \frac{1}{2}(−1)^{i} ∑_{l=0}^{i} \binom{i}{l}\frac{d^{i−l}\overline{A_{σν}^{ji}}}{dt^{i−l}} ω_{j+l}^{σ} ∧ ω_{0}^{ν} ∧ dt (11)

for j + l ∈ {0, 1, …, 2r} .

For j + l = k ≤ r the following possibilities realize: j = 0, l = k for i ∈ ∈ {k, …, r}; j = 1, l = k − 1 for i ∈ {k − 1, …, r}; …; j = m, l = k − m for i ∈ {k − m, …, r}; …; j = k, l = 0 for i ∈ {0, …, r} . Then

B_{σν}^{k} = \frac{1}{2} ∑_{m=0}^{k} ∑_{i=m}^{r}(−1)^{i}\binom{i}{m}\frac{d^{i−m}\overline{A_{σν}^{k−m, i}}}{dt^{i−m}} for 0 ≤ k ≤ r , (12)

and, by the completely analogous way,

B_{σν}^{k} = \frac{1}{2} ∑_{m=k−r}^{r} ∑_{i=m}^{r}(−1)^{i}\binom{i}{m}\frac{d^{i−m}\overline{A_{σν}^{k−m, i}}}{dt^{i−m}} for r + 1 ≤ k ≤ 2r . (13)

It can be again seen that B_{σν}^{k} are defined on j^{2r+1}Y. The resulting coordinate expression for the representative of [ϱ] ∈ Ω_{2r+1}^{3}V/Θ_{2r+1}^{3}V is as follows:

[ϱ] ≅ ∑_{k=0}^{r} (\frac{1}{2} ∑_{m=0}^{k} ∑_{i=m}^{r}(−1)^{i}\binom{i}{m}\frac{d^{i−m}\overline{A_{σν}^{k−m, i}}}{dt^{i−m}} ) ω_{k}^{σ} ∧ ω_{0}^{ν} ∧ dt + (14)

+ ∑_{k=r+1}^{2r} (\frac{1}{2} ∑_{m=k−r}^{r} ∑_{i=m}^{r}(−1)^{i}\binom{i}{m}\frac{d^{i−m}\overline{A_{σν}^{k−m, i}}}{dt^{i−m}} ) ω_{k}^{σ} ∧ ω_{0}^{ν} ∧ dt .

It is evident that the above used calculation procedure ensures [ϱ] ≅ 0 if and only if ϱ ∈ Θ_{r}^{k} .

Note. Let in general η ∈ Ω_{2r+1}^{k}V. Then there is a unique decomposition η = ϱ+η′ in which ϱ ∈ π_{2r+1, r}^{∗}Ω_{r}^{k} and

η′ = ∑_{0≤j_{1},...,j_{k−1}≤2r+1} C _{σ_{1}...σ_{k−1}}^{j_{1}...j_{k−1}} dq_{j_{1}}^{σ_{1}} ∧ ⋯ ∧ dq_{j_{k−1}}^{σ_{k−1}} ∧ dt +

+ ∑_{0≤j_{1},...,j_{k}≤2r+1} D _{σ_{1}...σ_{k}}^{j_{1}...j_{k}} dq_{j_{1}}^{σ_{1}} ∧ ⋯ ∧ dq_{j_{k}}^{σ_{k}} ,

where none of the forms C _{σ_{1}...σ_{k−1}}^{j_{1}...j_{k−1}} dq_{j_{1}}^{σ_{1}} ∧ ⋯ ∧ dq_{j_{k−1}}^{σ_{k−1}} ∧ dt, D _{σ_{1}...σ_{k}}^{j_{1}...j_{k}} dq_{j_{1}}^{σ_{1}} ∧ ∧ ⋯∧dq_{j_{k}}^{σ_{k}} is π_{2r+1, r}^{∗}–projectable. Let η_{0} be an arbitrarily choosen representative of the class [η′] ∈ Ω_{2r+1}^{k}V/Θ_{2r+1}^{k}V. We define F_{k} (η) = [η] = [ϱ] + [η′] or [η] ≅ ≅ ϱ_{0} + η_{0} , where ϱ_{0} is the representative of the class [ϱ] given for k = 1, 2, 3 by Theorem 1. It can be cleary seen that Θ_{2r+1}^{k}V = Ker F_{k} .

5. Quotient mappings

The explicit coordinate formulas for quotient mappings E_{0}, E_{1}, E_{2} are obtained from the definition of E_{k} (6) by the direct application of relations (8), (9) and (10) on the form dϱ for ϱ = f ∈ Ω_{r}^{0}V, ϱ ∈ Ω_{r}^{1}V and ϱ ∈ Ω_{r}^{2}V , respectively. The results are summarized in the following theorem:

Theorem 2. Let (V, ψ), ψ = (t, q_{0}^{σ}), 1 ≤ σ ≤ m, be a fibered chart on Y.

(a) Let f ∈ Ω_{r}^{0}V. Then E_{0} is defined on j^{r+1}Y , i.e. E_{0} : Ω_{r}^{0}V ∋ [f] → → E_{0} ([f]) = [df] ∈ Ω_{r+1}^{1}V/Θ_{r+1}^{1}V and for the representative of E_{0} ([f]) it holds

E_{0} ([f]) = [df] ≅ (\frac{df}{dt} ) dt .

(b) Let ϱ ∈ Ω_{r}^{1}V , ϱ = P dt + ∑_{j=0}^{r} Q_{σ}^{j}dq_{j}^{σ}. Then E_{1} is defined on j^{2r+1}Y i.e. E_{1} : Ω_{2r+1}^{1}V/Θ_{2r+1}^{1}V ∋ [ϱ] → E_{1} ([ϱ]) = [dϱ] ∈ Ω_{2r+1}^{2}V/Θ_{2r+1}^{2}V , and for the representative of E_{1} ([ϱ]) it holds

E_{1} ([ϱ]) ≅ ε_{σ}ω_{0}^{σ} ∧ dt = ∑_{j=0}^{r}(−1)^{i} \frac{d^{j}}{dt^{j}} (\frac{∂\overline{P}}{∂q_{j}^{σ}} − \frac{dQ_{σ}^{j}}{dt} − Q_{σ}^{j−1}) =

= (∑_{j=0}^{r} [(−1)^{j} \frac{d^{j}}{dt^{j}} (\frac{∂\overline{P}}{∂q_{j}^{σ}} )] + (−1)^{r+1} \frac{d^{r+1}Q_{σ}^{r}}{dt^{r+1}} ) ω_{0}^{σ} ∧ dt ,

where \overline{P} = P + ∑_{j=0}^{r} Q_{σ}^{j}q_{j+1}^{σ} and we put Q_{σ}^{j−1} = 0 for j = 0.

(c) Let ϱ ∈ Ω_{r}^{2}V, ϱ = ∑_{j=0}^{r} P_{σ}^{j}dq_{j}^{σ}∧dt+ \frac{1}{2} ∑_{i,j=0}^{r} Q_{σν}^{ji}dq_{j}^{σ}∧dq_{i}^{ν} . Then E_{2} is defined on j^{2r+1}Y, i.e. E_{2} : Ω_{2r+1}^{2}V/Θ_{2r+1}^{2}V ∋ [ϱ] → E_{2} ([ϱ]) = [dϱ] ∈ Ω_{2r+1}^{3}V/Θ_{2r+1}^{3}V , and for the representative of E_{2} ([ϱ]) it holds:

E_{2} ([ϱ]) ≅ ∑_{k=0}^{2r} B_{σν}^{k}ω_{k}^{σ} ∧ ω_{0}^{ν} ∧ dt ,

where

B_{σν}^{k}= \frac{1}{2} ∑_{j=p}^{q} ∑_{i=k−j}^{r}(−1)^{i}\binom{i}{k − j}\frac{d^{i+j−k}}{dt^{i+j−k}} [\frac{∂\overline{P_{ν}^{i}}}{∂q_{j}^{σ}} − \frac{∂\overline{P_{σ}^{j}}}{∂q_{i}^{ν}} + \frac{dQ_{σν}^{ji}}{dt} + Q_{σν}^{j, i−1}− Q_{νσ}^{i, j−1}] ,

p = 0 and q = k for 0 ≤ k ≤ r, p = k − r and q = r for r + 1 ≤ k ≤ 2r, \overline{P_{σ}^{j}} = P_{σ}^{j} + ∑_{i=0}^{r} Q_{σν}^{ji}q_{i+1}^{ν} and we put Q_{σν}^{u, v−1} = 0 for v = 0.

Proof. (a) For f ∈ Ω_{r}^{0}V it holds E_{0} ([f]) = [df] = [hdf] and thus, by using the formula (8) and taking into account that [f] ≅ f we obtain

E_{0} ([f]) ≅ (\frac{df}{dt} ) dt . (15)

It is evident that the representative of E_{0} ([f]) is defined on j^{r+1}Y .

(b) For ϱ ∈ Ω_{r}^{1}V and ϱ = P dt + ∑_{j=0}^{r} Q_{σ}^{j}dq_{j}^{σ} the representative of [ϱ] is given by

[ϱ] ≅ \overline{P} dt = (P + ∑_{j=0}^{r} Q_{σ}^{j}q_{j+1}^{σ}) dt . On the other hand, the exterior derivative of ϱ is as follows

dϱ = ∑_{j=0}^{r} (\frac{∂P}{∂q_{j}^{σ}} − \frac{∂Q}{∂t}) dq_{j}^{σ} ∧ dt + ∑_{i,j=0}^{r} \frac{∂Q_{ν}^{i}}{∂q_{j}^{σ}} dq_{j}^{σ} ∧ dq_{i}^{ν} .

After antisymmetrization of the second sum we use the relations (2) with k = 2 (for obtaining p_{1}dϱ) and the formula (9). This gives the following result:

E_{1}([ϱ]) = [dϱ] ≅ ∑_{j=0}^{r}(−1)^{j} \frac{d^{j}}{dt^{j}} (\frac{∂\overline{P}}{∂q_{j}^{σ}} − \frac{dQ_{σ}^{j}}{dt} − Q_{σ}^{j−1}) ω_{0}^{σ} ∧ dt =

= [∑_{j=0}^{r}(−1)^{j} \frac{d^{j}}{dt^{j}} (\frac{∂\overline{P}}{∂q_{j}^{σ}} ) + (−1)^{r+1} \frac{d^{r+1}Q_{σ}^{r}}{dt^{r+1}} ] ω_{0}^{σ} ∧ dt , (16)

where we have put Q_{σ}^{j−1} = 0 for j = 0. The representative of E_{1}([ϱ]) is defined od j^{2r+1}Y .

(c) Let ϱ ∈ Ω_{r}^{2}V is expressed by ϱ = ∑_{j=0}^{r} P_{σ}^{j}dq_{j}^{σ} ∧ dt + \frac{1}{2} ∑_{i,j=0}^{r} Q_{σν}^{ji}dq_{j}^{σ} ∧ dq_{i}^{ν} . Then

[ϱ] ≅ (∑_{j=0}^{r}(−1)^{j} \frac{d^{j}\overline{P_{σ}^{j}}}{dt^{j}} ) ω_{0}^{σ} ∧ dt , \overline{P_{σ}^{j}} = P_{σ}^{j} + ∑_{i=0}^{r} Q_{σν}^{ji}q_{i+1}^{ν} .

It holds, after antisymmetrization:

dϱ = \frac{1}{2} ∑_{i,j=0}^{r} (\frac{∂P_{ν}^{i}}{∂q_{j}^{σ}} − \frac{∂P_{σ}^{j}}{∂q_{i}^{ν}} + \frac{1}{2} \frac{∂Q_{σν}^{ji}}{∂t} ) dq_{j}^{σ} ∧ dq_{i}^{ν} ∧ dt +

+ \frac{1}{3!} ∑_{i,j,k=0}^{r} (\frac{∂Q_{νλ}^{ik}}{∂q_{j}^{σ}} + \frac{∂Q_{σν}^{ji}}{∂q_{k}^{λ}} + \frac{∂Q_{λσ}^{kj}}{∂q_{i}^{ν}} ) dq_{j}^{σ} ∧ dq_{i}^{ν} ∧ dq_{k}^{λ} .

By denoting A_{σν}^{ji} = (\frac{∂P_{ν}^{i}}{∂q_{j}^{σ}} − \frac{∂P_{σ}^{j}}{∂q_{i}^{ν}} + \frac{1}{2} \frac{∂Q_{σν}^{ji}}{∂t} ), B_{σνλ}^{jik} = (\frac{∂Q_{νλ}^{ik}}{∂q_{j}^{σ}} + \frac{∂Q_{σν}^{ji}}{∂q_{k}^{λ}} + \frac{∂Q_{λσ}^{kj}}{∂q_{i}^{ν}} ) and

using (2) for k = 3 and (11), we get

E_{2} ([ϱ]) = [dϱ] ≅ ∑_{k=0}^{2r} B_{σν}^{k}ω_{k}^{σ} ∧ ω_{0} ∧ dt ,

with

B_{σν}^{k}= \frac{1}{2} ∑_{j=p}^{q} ∑_{i=k−j}^{r}(−1)^{i}\binom{i}{k − j}\frac{d^{i+j−k}}{dt^{i+j−k}} [\frac{∂\overline{P_{ν}^{i}}}{∂q_{j}^{σ}} − \frac{∂\overline{P_{σ}^{j}}}{∂q_{i}^{ν}} + \frac{dQ_{σν}^{ji}}{dt} + Q_{σν}^{j, i−1}− Q_{νσ}^{i, j−1}] , (17)

where p = 0, q = k for 0 ≤ k ≤ r; p = k − r, q = r for r + 1 ≤ k ≤ 2r and Q_{σν}^{u,v−1} = 0 for v = 0. We can see that the representative of E_{2}([ϱ]) is defined again on j^{2r+1}Y.

The relation of our results obtained for the quotient mappings to the commonly used objects of higher order mechanics is clear from the following theorem:

Theorem 3. Let (V, ψ), ψ = (t, q_{0}^{σ}), 1 ≤ σ ≤ m, be a fibered chart on Y.

(a) Let ϱ ∈ Ω_{r−1}^{0}V. Then the representative of E_{0}([f]) is the trivial lagrangian of the order r.

(b) Let ϱ ∈ Ω_{r}^{1}V and let the representative of [ϱ] be a lagrangian of the order r. Then the representative E_{1}([ϱ]) is its Euler – Lagrange form.

(c) Let ϱ ∈ Ω_{r}^{2}V and [ϱ] ∈ Ω_{2r}^{2}V/Θ_{2r}^{2}V, i.e. the representative of the class [ϱ] is the π_{2r, 0}–horizontal form defined on j^{2r}Y. Then ϱ is locally variational iff E_{2}([ϱ]) = 0.

Proof. (a) Let f ∈ Ω_{r−1}^{0}V. Then, after (15), E_{0}([f]) ≅ \frac{df}{dt} dt, where \overline{P} = \frac{df}{dt} ∈ Ω_{r}^{0}. Thus λ = \frac{df}{dt} dt is the trivial lagrangian because its Lagrange function has the form of the total derivative of a function defined on j^{r−1}Y.

(b) Let ϱ ∈ Ω_{r}^{1}V and ϱ = P dt + ∑_{j=0}^{r} Q_{σ}^{j}dq_{j}^{σ}. The representative of [ϱ] is the lagrangian of the r–th order iff Q_{σ}^{r} = 0. Then [ϱ] ≅ λ = \overline{P} dt = = (P + ∑_{j=0}^{r−1} Q_{σ}^{j}q_{j+1}^{σ}) dt. The formula (16) gives

E_{1}([ϱ]) ≅ [∑_{j=0}^{r}(−1)^{j} \frac{d^{j}}{dt^{j}} (\frac{∂\overline{P}}{∂q_{j}^{σ}} )] ω_{0}^{σ} ∧ dt ,

which is the usual expression for the Euler – Lagrange form of λ.

(c) Let ϱ ∈ Ω_{r}^{2}V and ϱ = ∑_{j=0}^{r} P_{σ}^{j}dq_{j}^{σ} ∧ dt + \frac{1}{2} ∑_{i,j=0}^{r} Q_{σν}^{ji}dq_{j}^{σ} ∧ dq_{i}^{ν}. The relation (9) leads to the following result:

[ϱ] ≅ ∑_{j=0}^{r}(−1)^{j} \frac{d^{j}\overline{P_{σ}^{j}}}{dt^{j}} ω_{j}^{σ} ∧ dt , where \overline{P_{σ}^{j}} = P_{σ}^{j} + ∑_{i=0}^{r} Q_{σν}^{ji}q_{i+1}^{ν} .

It can be easily seen that [ϱ] ∈ Ω_{2r}^{2}V/Θ_{2r}^{2}V iff Q_{σν}^{rr} = 0 for 1 ≤ σ, ν ≤ m. Let us now denote

ε_{σ} = ∑_{j=0}^{r}(−1)^{j} \frac{d^{j}\overline{P_{σ}^{j}}}{dt^{j}} (18)

and

H_{σν}^{k} = \frac{∂ε_{ν}}{∂q_{k}^{σ}} − (−1)^{k} \frac{∂ε_{σ}}{∂q_{k}^{ν}} − ∑_{l=k+1}^{2r}(−1)^{l}\binom{l}{k}\frac{d^{l−k}}{dt^{l−k}} (\frac{∂ε_{σ}}{∂q_{l}^{ν}} ) (19)

the Helmholtz – Sonin expressions commonly used in higher order mechanics. By using the expression (17) for cooeficients B_{σν}^{k} and the relation

\frac{∂}{∂q_{i}^{ν}} (\frac{d^{k}f}{dt^{k}} ) = ∑_{l=0}^{k} \binom{k}{l}\frac{d^{l}}{dt^{l}} (\frac{∂f}{∂q_{i−k+l}^{ν}} ) ,

in which f is a function and \frac{∂f}{∂q_{i−k+l}^{ν}} ≡ 0 formally for i − k + l < 0, we obtain, after

some rather technical calculations, B_{σν}^{k} = \frac{1}{2}H_{σν}^{k} .

Example. For making our results more transparent we use them for the first order case. Let r = 1 and (V, ψ), ψ = (t, q_{0}^{σ}), (π_{0}^{−1}(V), ψ_{1}), ψ_{1} = (t, q_{0}^{σ}, q_{1}^{σ}) are the fibered chart on Y and the associated fibered chart on j^{1}Y, respectively.

(a) Let f = f(t, q_{0}^{σ}, q_{1}^{σ}) ∈ Ω_{1}^{0}V. Then

E_{0} ([f]) ≅ (\frac{∂f}{∂t} + \frac{∂f}{∂q_{0}^{σ}} q_{1}^{σ} + \frac{∂f}{∂q_{1}^{σ}} q_{2}^{σ}) dt

(b) Let ϱ = P dt + ∑_{j=0}^{1} Q_{σ}^{j}dq_{j}^{σ} = P dt + Q_{σ}^{0}dq_{0}^{σ} + Q_{σ}^{1}dq_{1}^{σ}. [ϱ] ≅ \overline{P} dt = = (P + Q_{σ}^{0}q_{1}^{σ} + Q_{σ}^{1}q_{2}^{σ}) dt has the form of the lagrangian of the order 2.

E_{1} ([f]) ≅ [\frac{∂\overline{P}}{∂q_{0}^{σ}} − \frac{d}{dt} (\frac{∂\overline{P}}{∂q_{1}^{σ}} ) + \frac{d^{2}Q_{σ}^{1}}{dt^{2}} ] ω_{0}^{σ} ∧ dt =

= [\frac{∂\overline{P}}{∂q_{0}^{σ}} − \frac{d}{dt} (\frac{∂\overline{P}}{∂q_{1}^{σ}} ) + \frac{d^{2}}{dt^{2}} (\frac{∂\overline{P}}{∂q_{2}^{σ}} )] ω_{0}^{σ} ∧ dt .

(c) Let ϱ = P_{σ}^{0}dq_{0}^{σ} ∧ dt + P_{σ}^{1}dq_{1}^{σ} ∧ dt + \frac{1}{2}Q_{σν}^{00}dq_{0}^{σ} ∧ dq_{0}^{ν} + \frac{1}{2}Q_{σν}^{10}dq_{1}^{σ} ∧ dq_{0}^{ν} + +\frac{1}{2}Q_{σν}^{01}dq_{0}^{σ} ∧ dq_{1}^{ν} + \frac{1}{2}Q_{σν}^{11}dq_{1}^{σ} ∧ dq_{1}^{ν} .

Denote \overline{P_{σ}^{0}} = P_{σ}^{0} + Q_{σν}^{00}q_{1}^{ν} + Q_{σν}^{01}q_{2}^{ν} , \overline{P_{σ}^{1}} = P_{σ}^{1} + Q_{σν}^{10}q_{1}^{ν} + Q_{σν}^{11}q_{2}^{ν}. Then

[ϱ] ≅ (\overline{P_{σ}^{0}} − \frac{d}{dt}\overline{P_{σ}^{1}}) ω_{0}^{σ} ∧ dt = [(P_{σ}^{0} + Q_{σν}^{00}q_{1}^{ν} + Q_{σν}^{01}q_{2}^{ν}) −

− (\frac{dP_{σ}^{1}}{dt} + \frac{dQ_{σν}^{10}}{dt} q_{1}^{ν} + \frac{dQ_{σν}^{11}}{dt} q_{2}^{ν}) − Q_{σν}^{10}q_{2}^{ν} − Q_{σν}^{11}q_{3}^{ν}] ω_{0}^{σ} ∧ dt

is defined on j^{3}Y, and

ε_{σ} = (P_{σ}^{0} − \frac{dP_{σ}^{1}}{dt} ) + (Q_{σν}^{00} − \frac{dQ_{σν}^{10}}{dt} ) q_{1}^{ν} + (Q_{σν}^{01} − Q_{σν}^{10} − \frac{dQ_{σν}^{11}}{dt} ) q_{2}^{ν} − Q_{σν}^{11}q_{3}^{ν} .

Then

E_{2}([ϱ]) = B_{σν}^{0}ω_{0}^{σ} ∧ ω_{0}^{ν} ∧ dt + B_{σν}^{1}ω_{1}^{σ} ∧ ω_{0}^{ν} ∧ dt + B_{σν}^{2}ω_{2}^{σ} ∧ ω_{0}^{ν} ∧ dt ,

where (see (17))

B_{σν}^{0} = \frac{1}{2} ∑_{i=0}^{1}(−1)^{i} \frac{d^{i}}{dt^{i}} (\frac{∂\overline{P_{ν}^{i}}}{∂q_{0}^{σ}} − \frac{∂\overline{P_{σ}^{0}}}{∂q_{i}^{ν}} ) − \frac{1}{2} \frac{d^{2}Q_{σν}^{01}}{dt^{2}} ,

B_{σν}^{1} = \frac{1}{2} ∑_{i=0}^{1}(−1)^{i} \frac{d^{i}}{dt^{i}} (\frac{∂\overline{P_{ν}^{i}}}{∂q_{1}^{σ}} − \frac{∂\overline{P_{σ}^{1}}}{∂q_{i}^{ν}} ) − \frac{1}{2} (\frac{∂\overline{P_{ν}^{1}}}{∂q_{0}^{σ}} − \frac{∂\overline{P_{σ}^{0}}}{∂q_{1}^{ν}} ) − \frac{dQ_{σν}^{01}}{dt} − \frac{1}{2} \frac{d^{2}Q_{σν}^{11}}{dt^{2}} ,

B_{σν}^{2} = −\frac{1}{2} [(\frac{∂\overline{P_{ν}^{1}}}{∂q_{1}^{σ}} − \frac{∂\overline{P_{σ}^{1}}}{∂q_{1}^{ν}} ) + Q_{σν}^{01} + Q_{σν}^{10} + \frac{dQ_{σν}^{11}}{dt} ] .

These results are the same as those obtained for r = 1 by Krupka [2]. For Q_{σν}^{11} = 0, i.e. ε_{σ} ∈ Ω_{2}^{0}V, it can be seen after some calculations that B_{σν}^{k} = \frac{1}{2}H_{σν}^{k} for k = 0, 1, 2 .

6. Conclusions

From the Theorem 3 it follows that the variational sequence is in a very close relation to the commonly used objects of calculus of variations in higher order mechanics. Thus we can formulate the following generalization of usual definitions of calculus of variations:

representative of [ϱ] ∈ Ω_{r}^{1} … lagrangian associated with ϱ
representative of [ϱ] ∈ Ω_{r}^{2} … Euler – Langrange form of ϱ
representative of [ϱ] ∈ Ω_{r}^{3} … Helmholtz – Sonin form of ϱ
E_{1} . . . . . . . . . . . . . . . . . . . . . . . . . Euler – Lagrange mapping
E_{2} . . . . . . . . . . . . . . . . . . . . . . . . . Helmholtz – Sonin mapping

There are some open problems:

• the application of the presented calculation technique for obtaining the mappings F_{k} and E_{k−1} for k ≥ 4,

• the invariance of the obtained coordinate expressions,

• the problem of the interpretation of mappings F_{k} and F_{k−1} for k ≥ 4 from the point of view of calculus of variations.

I would like to thank Professor Demeter Krupka for his interest in my work and for many helpful discussions.

References:

[1] 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.

[2] Krupka, D.: Lectures on Variational Sequence (Mechanics), Open Education & Sciences, Opava, 1995, 1–92.

[3] Štefánek, J.: Variational Sequences in Higher Order Mechanics, in: Differential Geometry and its Applications, Proc. Conf., J. Janyška and I. Kolář, Editors, Brno, Czech Republic, 1995.

Department of General Physics Faculty of Science, Masaryk University Brno Kotlářská 2, 61137 Brno, CZECH REPUBLIC