DIFFERENTIAL GEOMETRY AND APPLICATIONS Proc. Conf., Aug. 28 – Sept. 1, 1995, Brno, Czech Republic Masaryk University, Brno 1996, 487–494

SOME ASPECTS OF VARIATIONAL SEQUENCES IN MECHANICS

RAFFAELE VITOLO

Abstract. We discuss intrinsic aspects of Krupka’s approach to finite–order variational sequences. We recover in an intrinsic way the second–order variational calculus for affine Lagrangians by means of a natural generalisation of first–order theories. Moreover, we find an intrinsic expression for the Helmholtz morphism using a technique introduced by Kolář that we have adapted to our context.

Introduction

The theory of variational bicomplexes was established at the end of the seventies by several authors [AnDu80], [OlSh78], [Tak79], [Tul77], [Vin77]. In these works there is the idea that one can give a geometric formulation of the calculus of variations without using integrals. But, except of [AnDu80], in these works the variational bicomplex is built over the space of infinite jets of a fibred manifold. This procedure is suggested by the relatively simple structure of such spaces.

In this paper, we start from Krupka’s setting of variational sequences on finite– order jet spaces [Kru90]. A finite–order bicomplex is produced when one quotients the de Rham sequence on a finite–order jet space by means of an intrinsically defined subsequence. This finite–order approach has been fruitfully applied to a concrete relativistic theory in [MoVi95]; the idea of a study of the variational bicomplex by means of intrinsic techniques has its origin in this work.

Here, we restrict ourselves to the case of the first–order variational bicomplex on a fibred manifold whose base is one–dimensional. We give isomorphisms of the quotient sheaves of the bicomplex with subsheaves of the sheaves of forms on a jet space of suitable order. This order is always found as the minimal among all possible candidates; this aspect is not present in the infinite jet formalism. As a by–product, we find that the first–order variational bicomplex allows to deal with second–order affine Lagrangians and the related Euler–Lagrange morphism, rather than first–order ones. And it is worth to point out that most of second– order horizontal Lagrangians known in physics are affine. It is also seen that the sheaves that we find contain the sheaves of the standard objects of the variational calculus as proper subsheaves. Finally, we give a characterisation of the Helmholtz morphism, showing that the well–known local coordinate expression (see [GiMa90], [Kru90]) is intrinsic.

1991 Mathematics Subject Classification. 58A12, 58A20, 58E30, 58G05.

Key words and phrases. Fibred manifold, jet space, variational sequence, Euler–Lagrange map, Helmholtz map.

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

This work is partially supported by MURST (by national and local funds) and GNFM of Consiglio Nazionale delle Ricerche.

Throughout the paper, we will use as a fundamental tool the structure form on jet spaces, developed in [MaMo83]. Moreover, we make use of intrinsic techniques that are developed by means of the language of [Cos94], and which were first introduced in [Kol83].

We end the introduction with some mathematical conventions. In this paper, manifolds are connected and C^{∞}, and maps between manifolds are C^{∞}. All morphisms of fibred manifolds (and hence bundles) will be morphisms over the identity of the base manifold, unless otherwise specified. As for sheaves, we will use the definitions and the main results given in [Wel80]. In particular, we will be concerned only with sheaves of IR–vector spaces.

Acknowledgements. I am deeply indebted with Prof. Demeter Krupka for having kindly invited me at the University of Opava. Special thanks are also due to Prof. Ivan Kolář for several helpful suggestions. Finally, I would like to thank Prof. Marco Modugno, Dr. Raffaella Paoletti and Prof. Jaroslav Štefanek for stimulating discussions.

1. Jet spaces

The variational sequence in mechanics is built in a natural way (i.e., by functorial techniques) starting from the following assumption.

Assumption We assume that a fibred manifold π : Y → X is given, with dim X = 1 and dim Y = m + 1.

In coordinate expressions indices i, j, … will run from 1 to m and will label fibre coordinates, and the index 0 will label the coordinate on X. A coordinate system on Y adapted to the fibring will be denoted by (x^{0}, y^{i}). We will denote by (∂_{0}, ∂_{i}) and (d^{0}, d^{i}) the local bases of vector fields and one–forms on Y induced by an adapted chart.

The fibred manifold Y → X yields in a natural way the jet manifolds J_{r}Y . A detailed account of the theory of jets can be found in [MaMo83], [Sau89]; here we just fix necessary notation. Let us set J_{0}Y := Y , and r, s ∈ ℕ, r ≥ s. We have the fibrings π_{s}^{r} : J_{r}Y → J_{s}Y and π^{r} : J_{r}Y → X.

One–dimensional multi–indexes will be denoted by underlined latin letters \underline{p} ∈ ℕ, \underline{q} ∈ ℕ, . . .. We have the adapted charts (x^{0}, yi ; \underline{p}) on J_{r}Y , with \underline{p} ≤ r (we have set

yi ; \underline{0} := y^{i}). Accordingly, we have the local bases (∂_{0}, ∂\underline{p} ; i) and (d^{0}, di ; \underline{p}) of vector fields and one–forms on J_{r}Y . When r = 0, 1, 2, 3 we will write y^{i}, y_{0}^{i}, y_{00}^{i}, y_{000}^{i}.

We have the natural complementary fibred morphisms (see [MaMo83])

�_{r} : J_{r}Y → T^{∗}X ⊗_{J_{r−1}Y} T J_{r−1}Y ,

ϑ_{r} : J_{r}Y → T^{∗}J_{r−1}Y ⊗_{J_{r−1}Y} V J_{r−1}Y ;

whose coordinate expressions are

�_{r} = d^{0} ⊗ �_{r} _{0} = d^{0} ⊗(∂_{0} + y_{\underline{p}+1}^{j}∂\underline{p} ; j) , 0 ≤ |\underline{p}| ≤ r − 1 ,

ϑ_{r} = ϑj ; \underline{p} ⊗ ∂\underline{p} ; j = (dj ; \underline{p} − y_{\underline{p}+1}^{j}d^{0}) ⊗ ∂\underline{p} ; j , 0 ≤ |\underline{p}| ≤ r − 1 .

The fibred morphism ϑ_{r}^{∗} : J_{r}Y ×_{J_{r−1}Y} V^{∗}J_{r−1}Y → J_{r}Y ×_{J_{r−1}Y} T^{∗}J_{r−1}Y yields

the splitting

J_{r}Y ×_{J_{r−1}Y} T^{∗}J_{r−1}Y = (J_{r}Y ×_{J_{r−1}Y} T^{∗}X) ⊕ imϑ_{r}^{∗} . (1)

Let us denote by Λ^{k}_{r} the sheaf of k–forms on J_{r}Y . The fibring π and the form

ϑ_{s} yield natural subsheaves of Λ^{k}_{r}. Let k > 0, and s ≤ r. We denote by C^{k}_{(r,s)} the

sheaf of fibred morphisms J_{r}Y → ⋀^{k}imϑ_{s}^{∗} over J_{r}Y → J_{s}Y , which are interpreted

as k–forms on J_{r}Y . We set also C^{k}_{r} := C^{k}_{(r,r)} and C^{0}_{(r,s)} := Λ^{0}_{r}. We denote by H^{1}_{r} the sheaf of fibred morphisms J_{r}Y → T^{∗}X over J_{r−1}Y → X, which are interpreted as 1–forms on J_{r}Y .

We denote by C^{1}_{(r,s)}^{A} ⊂ C^{1}_{(r,s)} the sheaf of fibred morphisms α : J_{r}Y → imϑ_{s}^{∗} over π_{s}^{r} which factorise as α = ϑ_{s}^{∗}∘α̃, where α̃ : J_{r}Y → J_{s}Y ×_{J_{s−1}Y} V^{∗}J_{s−1}Y is an affine

fibred morphism over π_{s}^{r}. In a similar way, we can define the subsheaf C^{k}_{(r,s)}^{A} ⊂ C^{k}_{(r,s)}.

Also, we denote by H^{1}_{r}^{A} ⊂ H^{1}_{r} the sheaf of fibred morphisms α : J_{r}Y → T^{∗}X that are affine fibred morphisms over π^{r−1} : J_{r−1}Y → X. The subsheaves with the superscript A are characterised by the fact that the components of their coordinate expressions are affine functions.

The splitting (1) provides the sheaf morphisms

h : Λ^{k}_{r−1} → C^{k−1}_{r}^{A} ∧ H^{1}_{r−1} v : Λ^{k}_{r−1} → C^{k}_{r}^{A}

where h + v = π_{r−1}^{r}^{∗}, and C^{k−1}_{r}^{A} ∧ H^{1}_{r−1} is done by means of the natural inclusion

H^{1}_{r−1} ⊂ H^{1}_{r}. We have h = i_{v}^{k−1}i_{h} and v = 1/k! i_{v}^{k}; moreover, h is surjective.

The morphisms �_{r} and ϑ_{r} induce two derivations along π_{r−1}^{r} of degree 1 (see [Sau89], [Cos94]), the horizontal and vertical differential d_{h} and d_{v}

d_{h} := i_{h}∘d − d∘i_{h} : Λ_{r−1} → Λ_{r} ,
d_{v} := i_{v} ∘d − d∘i_{v} : Λ_{r−1} → Λ_{r} ,

where i_{h} and i_{v} are defined to be, respectively, contractions with �_{r} and ϑ_{r}. It can be proved (see [Sau89]) that d_{h} and d_{v} fulfill the property d_{h} + d_{v} = (π_{r−1}^{r})^{∗} ∘d. One can find coordinate expressions of d_{h} and d_{v} in [Cos94], [Sau89]. Also, one can easily check that

d_{h} (C^{k}_{r−1}) ⊂ C^{k}_{r} ∧ H^{1}_{r} , d_{v} (C^{k}_{r−1}) ⊂ C^{k+1}_{r} .

2. The first–order variational sequence

We consider a short form of the first–order variational sequence of Krupka

[Kru90]. Let us set Θ^{0}_{r} := {0}, Θ^{1}_{r} := C^{1}_{r}, Θ^{k}_{r} := C^{k}_{r} + d C^{k−1}_{r} for all k ≥ 2. The following diagram is an exact bicomplex [Kru90]

0u  0u  0u  0u  0u

0  w0  w u  0  w u  Θ^{1}_{1}  w^{d} u  Θ^{2}_{1}  w^{d} u  dΘ^{2}_{1}  w^{d} u  0

0  wIR  w u  Λ^{0}_{1}  w^{d} u  Λ^{1}_{1}  w^{d} u  Λ^{2}_{1}  w^{d} u  dΛ^{2}_{1}  w^{d} u  0

0  wIR  w u  Λ^{0}_{1}  w^{E_{0}} u  Λ^{1}_{1}/Θ^{1}_{1}  w^{E_{1}} u  Λ^{2}_{1}/Θ^{2}_{1}  w^{E_{2}} u  E_{2} (Λ^{2}_{1}/Θ^{2}_{1})  w^{E_{3}} u  0

0  0  0  0  0

We say the bottom row of the above bicomplex to be the short variational sequence. Our main task is to find sheaves of fibred morphisms which are isomorphic to the quotient sheaves of the short variational sequence. It is possible to introduce a first simplification of the quotient sheaves; namely, the projection h induces the sheaf isomorphism

(Λ^{k}_{r}/Θ^{k}_{r}) → ( C^{k−1}_{r+1}^{A} ∧ H^{1}_{r}) /d_{h} C^{k−1}_{r} : [α] ↦ [h(α)] .

We notice that the natural inclusion Λ^{k}_{1}/Θ^{k}_{1} ⊂ Λ^{k}_{r−1}/Θ^{k}_{r−1} induces the natural inclusion

( C^{k−1}_{2}^{A} ∧ H^{1}_{1}) /d_{h} C^{k−1}_{1} → (C^{1}_{r}^{A} ∧ H^{1}_{r−1}) /d_{h} C^{k−1}_{r−1} , r ≥ 3 (2)

Proposition 2.1. We have the natural sheaf isomorphism

I_{1} : (Λ^{1}_{1}/Θ^{1}_{1}) → H^{1}_{2}^{A} : [α] ↦ h(α) .

If α ∈ Λ^{1}_{1} has the coordinate expression α = α_{0}d^{0} + α_{i}d^{i} + α_{i}^{0}d_{0}^{i}, then we have the coordinate expression

h(α) = (α_{0} + α_{i}y_{0}^{i} + α_{i}^{0}y_{00}^{i})d^{0} .

Let us set V^{1}_{1} := H^{1}_{2}^{A}. We say a section L ∈ V^{1}_{1} to be a first–order generalised Lagrangian. It is worth to note that the sheaf of the first–order Lagrangians of the

standard literature is H^{1}_{1}, and that H^{1}_{1} ⊂ H^{1}_{2}^{A}.

The following Lemma generalises the standard momentum (or Legendre transform) of a Lagrangian (see [Cos94]).

Lemma 2.1. There exists a natural surjective morphism

p : C^{1}_{2} ∧ H^{1}_{2} → C^{1}_{(2,1)} .

If, in coordinates, α = α_{i}ϑ^{i} ∧ d^{0}+α_{i}^{0}ϑ_{0}^{i} ∧ d^{0} (with α_{i}, α_{i}^{0} ∈ Λ^{0}_{2}), then p(α) = α_{i}^{0}ϑ^{i}.

Theorem 2.1. We have the injective sheaf morphism

I_{2} : (C^{1}_{2}^{A} ∧ H^{1}_{1}) /d_{h}C^{1}_{1} → Λ^{2}_{3} : [α] ↦ π_{2}^{3}^{∗}α + d_{h}p(α) ,

whose image is the sheaf

V^{2}_{1} := ((C^{1}_{2}^{A} ∧ H^{1}_{1}) + d_{h}C^{1}_{2}) ∩ (C^{1}_{(3,1)} ∧ H^{1}_{3}) .

Proof. One can test in a simple way that the morphism is well–defined via the

property p(d_{h}q) = −q fulfilled for any q ∈ C^{1}_{1}. The injectivity follows from (2).

Let α ∈ C^{1}_{2} ∧ H^{1}_{2}. Then one can prove that the form p(α) is the unique section of

the sheaf C^{1}_{2} such that α + d_{h}p(α) ∈ (C^{1}_{(3,1)} ∧ H^{1}_{3}) (see [Kol83] for the proof of a

similar statement). This yields the desired characterisation of the image of I_{2}.

Let α ∈ V^{1}_{1}, and let the coordinate expression of α be α = (α_{0} + α_{i}y_{00}^{i})d^{0}. Then, the identity E_{1}((α_{0} + α_{i}y_{00}^{i})d^{0}) = I_{2}([h ∘ dβ]), where β = α_{0}d^{0} + α_{i}d_{0}^{i}, yields E_{1}(α) = E_{i}ϑ^{i} ∧ d^{0}, where

E_{i} = ∂_{i}(α_{0} + y_{00}^{j}α_{j}) + �_{3} _{0}(∂_{0}α_{i} + y_{0}^{j}∂_{j}α_{i} + y_{00}^{j}∂_{j}^{0}α_{i} − ∂_{i}^{0}α_{0} − y_{00}^{j}∂_{i}^{0}α_{j})

We say E_{1} to be the first–order generalised Euler–Lagrange operator. Also, we

say a section E ∈ V^{2}_{1} to be a first–order generalised Euler–Lagrange morphism.

Remark 2.1. It is of fundamental importance to note that the standard higher– order Euler–Lagrange operator (see [Kol83], [Sau89]), when applied to a second–

order Lagrangian α ∈ H^{1}_{2}^{A}, produces exactly the same result as E_{1}(α).

Also, it is easy to verify that, when restricting E_{1} to the sheaf H^{1}_{1} of the Lagrangians of the standard literature, we obtain the standard Euler–Lagrange operator.

Now, we devote ourselves to a description of the sheaf E_{2} (V^{2}_{1}) ≃ (dΛ^{2}_{1}/dΘ^{2}_{1}).

Using (2) we find the natural injection

(dΛ^{2}_{1}/dΘ^{2}_{1}) → (C^{2}_{4}^{A} ∧ H^{1}_{3}) /d_{h}C^{2}_{3} : [dα] ↦ [dE_{1}(α)] .

We recall that to each α ∈ Λ^{1}_{r} ∧ H^{1}_{r} there is a unique E_{α} ∈ C^{1}_{(2r,1)} ∧ H^{1}_{2r} and a

unique p_{α} ∈ C^{1}_{(2r−1,r)} such that π_{r}^{2r}^{∗}α = E_{α} − d_{h}p_{α} (higher–order Euler–Lagrange morphism [Kol83]).

From now on, in this subsection we will suppose that \underline{p} is a multiindex such that |\underline{p}| ≥ 1. We introduce a notation in order to deal with repeated derivatives of functions by means of the fields �_{r} _{0}, �_{r+1} _{0}, … in a coordinate open subset.

Namely, if U ⊂ Y is a coordinate open subset and f ∈ (Λ^{0}_{r})_{U}, then we set

J\underline{p}f := �_{r+|\underline{p}|} _{0} … �_{r+1} _{0} .

Let u : Y → V Y be a vertical vector field with coordinate expression u = u^{i}∂_{i}. Then, the coordinate expression of the prolongation u_{r} : J_{r}Y → V J_{r}Y (see

[MaMo83]) is u_{r} = J\underline{p}u^{i}∂\underline{p} ; i.

The following Lemma is due to Krupka [Kru90].

Lemma 2.2. Let U ⊂ Y be a coordinate open subset, and β ∈

(Λ^{1}_{3} ∧ C^{1}_{(3,1)} ∧ H^{1}_{3})_{U} with coordinate expression

β = β_{ij}ϑ^{i} ∧ ϑ^{j} ∧ d^{0} + 2β\underline{p} ; ijϑi ; \underline{p} ∧ ϑ^{j} ∧ d^{0} , \underline{p} = 1, 2, 3.

Then there exist

H̃_{β}[U] ∈ ((C^{1}_{(6,4)} ∧ C^{1}_{(6,1)})^{A} ∧ H^{1}_{5})_{U} , q_{β}[U] ∈ (C^{1}_{(5,3)} ∧ C^{1}_{(5,2)})_{U}^{A} ,

such that

π_{3}^{6}^{∗}β = H̃_{β}[U] − d_{h}q_{β}[U]

Proof. We proceed by splitting the coefficients of π_{3}^{6}^{∗}β into symmetric and antisymmetric parts, as in [Kru90]. We obtain

H̃_{β}[U] = (β_{ij} − J_{0}β_{ij}^{0} + J_{00}β_{ij}^{00} − J_{000}β_{ij}^{000})ϑ^{i} ∧ ϑ^{j} ∧ d^{0}
+(β_{ij}^{0} + β_{ji}^{0} − 2J_{0}β_{ji}^{00} + 3J_{00}β_{ji}^{000})ϑ_{0}^{i} ∧ ϑ^{j} ∧ d^{0}
+(β_{ij}^{00} − β_{ji}^{00} + 3J_{0}β_{ji}^{000})ϑ_{00}^{i} ∧ ϑ^{j} ∧ d^{0} + (β_{ij}^{000} + β_{ji}^{000})ϑ_{000}^{i} ∧ ϑ^{j} ∧ d^{0}
q_{β}[U] = (−β_{ij}^{0} + J_{0}β_{ij}^{00} − J_{00}β_{ij}^{000})ϑ^{i} ∧ ϑ^{j}
−(β_{ij}^{00} + β_{ji}^{00} − J_{0}β_{ij}^{000} − 2J_{0}β_{ji}^{000})ϑ_{0}^{i} ∧ ϑ^{j}
+β_{ij}^{000}ϑ_{0}^{i} ∧ ϑ_{0}^{j} − (β_{ij}^{000} − β_{ji}^{000})ϑ_{00}^{i} ∧ ϑ^{j} .

Theorem 2.2. For every β ∈ Λ^{1}_{3} ∧ C^{1}_{(3,1)} ∧ H^{1}_{3} there is a unique

H_{β} ∈ (C^{1}_{(6,4)} ∧ C^{1}_{(6,1)})^{A} ∧ H^{1}_{5}

such that

(1) with respect to (2) we have [β] = [H_{β}];

(2) ∀ u : Y → V Y , E_{i_{u_{3}} β} = 0  ⇔  H_{β} = 0.

Proof. On each coordinate domain U, the form H̃_{β}[U] fulfills the second requirement. In fact, due to the arbitrariness of u, the condition E_{i_{u_{3}} β} = 0 gives rise to a set of local conditions whose symmetric and antisymmetric parts are precisely the same local conditions that arise from H̃_{β}[U] = 0 (see [GiMa90] for a local expression of E_{i_{u_{3}} β}).

Let U and U′ be two coordinate domains with non–empty intersection. Then, being E_{i_{u_{3}} β} = 0 an intrinsic condition, we obtain

H̃_{β}[U]|_{U∩U′} = 0  ⇔  H̃_{β}[U′]|_{U∩U′} = 0 .

But the above formula is clearly equivalent to H̃_{β}[U]|_{U∩U′} = H̃_{β}[U′]|_{U∩U′}. The result comes by setting H_{β}|_{U} := H̃_{β}[U] on each coordinate domain U ⊂ Y .

As a consequence, we obtain an intrinsic 3–form d_{h}q_{β}. Hence, due to the fact that dim X = 1, we obtain an intrinsic 2–form q_{β} (see [Kol83]). Moreover, the following corollary holds.

Corollary 2.1. E_{2} (V^{2}_{1}) is isomorphic to the image of the injective morphism

I_{3} : (dΛ^{2}_{1}/dΘ^{2}_{1}) → (C^{1}_{5} ∧ C^{1}_{(5,1)})^{A} ∧ H^{1}_{4} : [dα] ↦ H_{d(E_{1}([α]))} .

We can express E_{2} by the above morphism: if E ∈ V^{2}_{1}, then we have E_{2}(E) = H_{dE}. So, it is easy to find the coordinate expression of E_{2}. We say E_{2} to be the generalised Helmholtz operator.

We can summarise the results of this section in the following theorem.

Theorem 2.3. The first–order short variational sequence is isomorphic to the exact sequence

0 wIR wΛ^{0}_{1} ^{E_{0}} wV^{1}_{1} ^{E_{1}} wV^{2}_{1} ^{E_{2}} wE_{2} (V^{2}_{1})  w^{E_{3}} 0 .

Corollary 2.2. (Global inverse problem; see [Kru90]). Let E ∈ V^{2}_{1} be a global section such that E_{2}(E) = 0. Suppose, moreover, that

H_{de Rham}^{2}Y = 0 .

Then, there exists a global section L ∈ (V^{1}_{1})_{Y} such that E_{1}(L) = E.

Remark 2.2. We have obtained an intrinsic Helmholtz fibred morphism that is associated to each first–order generalised Euler–Lagrange morphism via the sheaf morphism E_{2}. The vanishing of the Helmholtz morphism is completely equivalent to the standard local Helmholtz conditions (see, for example, [GiMa90]).

As a by–product, to each first–order generalised Euler–Lagrange morphism E ∈

V^{2}_{1} we find a unique intrinsic two–form q_{dE} ∈ (C^{1}_{(4,3)} ∧ C^{1}_{(4,2)})^{A} ⊂ C^{2}_{4}^{A}.

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