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of variations

D. Krupka and M. Krupka

Dedicated to the 70-th anniversary of Professor Constantin Udriste

Abstract. Geometric structure of global integral variational functionals on higher order tangent bundles and Grassmann fibrations are investi-gated. The theory of Lepage forms is extended to these structures. The concept of a Lepage form allows us to introduce the Euler-Lagrange dis-tribution for variational functionals, depending on velocities, in a similar way as in the calculus of variations on fibred manifolds. Integral curves of this distribution include all extremal curves of the underlying variational functional. The generators of the Euler-Lagrange distribution, defined by the Lepage forms of the first order, are found explicitly.

M.S.C. 2000: 49Q99, 58A20, 58A30, 58E99.

Key words: Variational theory; velocity bundle; Grassmann bundle; Lepage form; Euler-Lagrange distribution.

1

Introduction

Our main objective in this paper is the geometric structure of the variational theory on higher order velocity spaces and Grassmann fibrations. We consider global integral variational functionals forcurvesand 1-dimensional submanifoldsof a given manifold. A specific feature of this theory is the absence of the concept of aLagrangian, a basic element of the classical calculus of variations and the variational theory on fibred manifolds; in this paper the role of a Lagrangian is played by a differential 1-form on a velocity manifold, called aLepage form.

We introduce Lepage forms, and derive a geometric (coordinate-free) first varia-tion formula on thehigher order velocity spaces. Our definition extends properties of the Cartan and Lepage forms, used in classical mechanics and the global variational theory on fibred manifolds. Main tools we use are properties of forms on the manifolds of velocities, and independence of the variational integral on parametrization. These notions are naturally characterized via the theory of jets and higher order contact elements. We also describe the Euler-Lagrange distribution, related with a Lepage form, whose integral curves include all extremals of the variational integral. In par-ticular, we derive chart expressions for a Lepage form as well as the generators of

∗

Balkan Journal of Geometry and Its Applications, Vol.15, No.1, 2010, pp. 68-79. c

the Euler-Lagrange destribution on the first order Grassmann fibration in terms of

adapted coordinates. The reader can easily understand that the presented theory can be extended to variational functionals for submanifolds of dimension greater than 1. Throughout this paper, R is the field of real numbers. We denote by Y a fixed smooth manifold of dimension m+ 1. Tr_{Y is the manifold of velocities of order r}

over Y (r-jets Jr

0ζ with source 0∈ R and target ζ(0) ∈ TrY, ) andτr,s : TrY →

Ts_{Y are the canonical jet projections. ImmT}r_{Y denotes an open submanifold of}

regular velocities inTr_{Y (r-jets of immersions), andL}r _{is the differential group of}

order r of R , consisting of regular r-jets Jr

0α∈ ImmTrR suchα(0) = 0, with the group multiplication the composition of jets. Gr_{Y is the Grassmann fibration of}

order r over Y (the quotient manifold ImmTr_{Y /L}r _{); ρ}r,s _{: G}r_{Y → G}s_{Y denotes}

the canonical projection of Grassmann fibrations. For simplicity, we restrict our coordinate considerations to lower order casesr= 1,2.

For the general theory of jets and contact elements the reader is referred to the papers [3, 5, 6]. The theorems, presented in this paper, namely the structure theory of Lepage forms and a new description of extremals in terms of a distribution, are an extension of the variational calculus on fibred manifolds as explained in [1, 2, 4, 7].

2

Velocities and Grassmann fibrations

Given a chart (V, ψ), ψ = (yK_{), on a manifold Y of dimension m+ 1, the}

associ-ated charts on the manifolds of regular velocities ImmT1_{Y and ImmT}2_{Y are denoted} by (V1_{, ψ}1_{), ψ}1 _{= (y}K_,y˙K_{), and (V}2_{, ψ}2_{), ψ}2 _{= (y}K_,y˙K_,y¨K_{). Each index L}

de-fines a partition of the index set, {L}, {1,2,· · ·, L−1, L+ 1,· · ·, m, m+ 1}, and thesubordinate charts on ImmT1_{Y and ImmT}2_{Y, denoted by (V}1,L_{, ψ}1,L_{), ψ}1,L ₌

(yL_,y˙L_{, y}σ_,y˙σ_{), and (V}2,L_{, ψ}2,L_),ψ2,L_{= (y}L_,y˙L_,y¨L_{, y}σ_,y˙σ_,y¨σ_{) ; recall that the sets}

V1,L_andV2,L _{are defined by}

˙

wL_{6= 0.}

(2.1)

The corresponding second subordinate charts are denoted by (V1,L_{, χ}1,L_{), χ}1,L ₌

(wL_,w˙L_{, w}σ_{, w}σ

1), and (V2,L, χ2,L), χ2,L = (wL,w˙L,w¨L, wσ, wσ1, w2σ). The second subordinate charts can be introduced by the transformation equations.

Lemma 1. The transformation equations between the charts (V2,L_{, ψ}2,L_{) and}

(V2,L_{, χ}2,L_)are

yL_=wL_{, y˙}L_{= ˙w}L_{, y¨}L _{= ¨w}L

(2.2)

yσ_=wσ_{, y˙}σ_=wσ

1w˙L, y¨σ=wσ2( ˙wL)2+w1σw¨L.

LetLr _{be the differential groupof order r of the real line R; in the context of this}

work,Lr_{describes the change of parameter in variational functionals for curves inY.}

Lr_{acts canonically onT}r_{Y to the right by composition of jets,}

Tr_{Y ×L}r_∋(Jr

0ζ, J0rα)→J0rζ◦J0rα=J0r(ζ◦α)∈TrY. (2.3)

Clearly, this group action restricts to the submanifold ofregularvelocities ImmTr_Y.

Recall that the canonical coordinates a1, a2,· · ·, ar on Lr are the functions on Lr

Lemma 2. (a)The group multiplication(J2

0α, J02β)→J02(α◦β)in the groupL2

is expressed by the equations

a1(Jr

0α◦J0rβ) =a1(J0rα)·a1(J0rβ), (2.4)

a2(J0rα◦J0rβ) =a2(J0rα)·(a1(J0rβ))2+a1(J0rα)·a2(J0rβ).

(b)The group action (2.3), restricted to the set ImmT2_{Y, is expressed in a}

sub-ordinate chart(V2,L_{, χ}2,L_),χ2,L_{= (w}L_,w˙L_,w¨L_{, w}σ_{, w}σ

1, w2σ), by the equations

wL_(J2

0ζ◦J02α) =wL(J02ζ), (2.5)

˙

wL_(J2

0ζ◦J02α) = ˙wL(J02ζ)a1(J02α), ¨

wL_(J2

0ζ◦J02α) = ˙wL(J02ζ)a2(J02α) + ¨wL(J02ζ)a1(J02α)2,

wσ_(J2

0ζ◦J02α) =wσ(J02ζ),

wσ1(J02ζ◦J02α) =wσ1(J02ζ),

wσ

2(J02ζ◦J02α) =wσ2(J02ζ).

Lemma 3. Suppose we have two charts on Y,(V, ψ), ψ= (yK_{), and(V,ψ¯),ψ¯=}

(¯yK_{)such thatV ∩V¯ 6=¡❢, and the transformation equations}

¯

yM _=fM_(yL_{, y}σ_{), y¯}ν _=fν_(yL_{, y}σ_).

(2.6)

Then the transformation equations between the subordinate charts(V1,L_{, χ}1,L_),χ1,L₌

(wL_,w˙L_{, w}σ_{, w}σ

1)and( ¯V1,M,χ¯1,M),χ¯1,M = ( ¯wM,w˙¯M,w¯σ,w¯1σ), are ¯

wM _=fM_(wL_{, w}σ_{), w¯}ν _=fν_(wL_{, w}σ_{), w˙¯}M ₌ ∂fM

∂wLw˙

L₊∂fM

∂wσw˙ L_wσ

1, (2.7)

¯

wν

1 =

1

∂fL

∂wL+

∂fL

∂wτw τ

1 µ

∂fν

∂wL +

∂fν

∂wσw σ

1 ¶

.

Let Gr_{Y be the Grassmann fibration of order r over Y. We denote by [J}r

0ζ] the Lr_{-orbit of a regular velocity J}r

0ζ, and by ρr,s : GrY → GsY the canonical projection. For every indexLthe pair ( ˜V2,L_,χ˜2,L_{), where ˜V}2,L_{= (ρ}2,0₎−1_(V),χ˜2,L₌ ( ˜wL_,w˜σ_,w˜σ

1,w˜2σ), and for allJ02ζ∈V2,L ˜

wL_([J2

0ζ]) =wL(J02ζ), (2.8)

˜

wσ_([J2

0ζ]) =wσ(J02ζ), w˜σ1([J02ζ]) =wσ1(J02ζ), w˜σ2([J02ζ]) =w2σ(J02ζ), is a chart onG3_Y.

LetIbe an open interval, containing 0∈R, and letγ:I→Y be a curve. Denote by trt0 the translationt→t−t0 of R. γdefines ther-jet prolongation

I∋t→(Trγ)(t) =J0r(γ◦tr−t)∈TrY.

(2.9)

Further on, we suppose thatγis animmersionsuch that for a chart onY,γ(I)⊂V

Lemma 4. The 2-jet prolongationT2_{γ is expressed by}

wL_◦T2_γ=wL_{γ, w˙}L_◦T2_γ=D(wL_{γ), w¨}L_◦T2_γ=D2_(wL_γ),

(2.10)

wσ_◦T2_γ=wσ_{γ, w}σ

1 ◦T2γ=

D(wσ_γ)

D(wL_γ),

wσ2◦T2γ= 1 (D(wL_γ))2

µ

D2(wσγ)−D

2_(wL_γ)

D(wL_γ)D(w σ_γ)

¶

.

Ther-jet prolongation of an immersionγ:I→Y defines a curve inGr_Y,

I∋t→Grγ(t) = [Trγ(t)]∈GrY,

(2.11)

called theGrassmann prolongationofγ of orderr.

We examine the behavior of the mappingT2_{γunderreparametrizations. LetJ be} an open interval, containing the origin 0, and letµ:J→Ibe a diffeomorphism. µis defined by an equation

t=µ(s).

(2.12)

Setting for everys∈J

µs(t) = trµ(s)◦µ◦tr−s(t) =−µ(s) +µ(s+t), (2.13)

we get another diffeomorphism µs : Js →Is of open intervals, containing 0. Since

Dµs(t) =Dµ(s+t) andD2µs(t) =D2µ(s+t),µssatisfies

µs(0) = 0, Dµs(0) =Dµ(s), D2µs(0) =D2µ(s).

(2.14)

In particular, the 2-jetJ2

0µsis an element of the differential groupL2for alls, whose

canonical coordinates area1(s) = Dµ(s), a2(s) =D2_{µ(s). µinduces a differentiable} mappings→J2

0µsof the domainJ ofµintoL2;µalso induces a diffeomorphism

ImmT2Y ∋J02ζ→J02ζ◦J02µs∈ImmT2Y,

(2.15)

defined by the canonical action of L2 _{on ImmT}2

0Y. The mappings s → J02µs and

J2

0ζ→J02ζ◦J02µs are said to beassociatedwithµ.

A diffeomorphismµ:J →I assigns to the immersionγan immersionγ◦µ:J →

Y, and its 2-jet prolongationT2_(γ◦µ).

Lemma 5. (a)The mappings→T2_{(γ◦µ)(s) satisfies}

T2(γ◦µ)(s) =T2γ(µ(s))◦J02µs.

(2.16)

(b)The mapping s→T2_{(γ◦µ)(s)is expressed in the chart (V}2,L_{, χ}2,L_),χ2,L ₌

(wL_{, w}L

1, w2L, wσ, wσ1, wσ2)by

wL(T2(γ◦µ)(s)) =wL(T2γ(µ(s))),

(2.17)

wL

1(T2(γ◦µ)(s)) =w1L(T2γ(µ(s)))a1(J02µs),

wL

2(T2(γ◦µ)(s)) =w1L(T2γ(µ(s)))a2(J02µs) +wL2(T2γ(µ(s)))a1(J02µs)2,

wσ_(T2_{(γ◦µ)(s)) =w}σ_(T2_γ(µ(s))),

wσ

1(T2(γ◦µ)(s)) =w1σ(T2γ(µ(s))),

There exists a bijection between the set of diffeomorphismsα: (−a, a)→R such

equal to the identity elementJ2

0idR of the groupL2,

immediately be modified for vector fields by means of flows; we denote by Tr_{ξ the}

r-jet prolongationof a vector fieldξonY.

Lemma 6. Letξ be a vector field onY, and let

3

The calculus of variations on velocity manifolds

Choose a velocityP ∈T3_{Y and a representativeζ ofP; then P =J}3

0ζ. ζdefines a mappingT2_{ζof a neighbourhood of 0∈R intoT}2_{Y and the tangent mapping at 0,}

T0T2ζ :T0R→TJ2 0ζT

2_{Y, sending a vector ξ∈T0R to a vector of T}2_{Y at the point}

T2_{ζ(0) =J}2

0ζ =τ3,2(J03ζ). Express ξis in the canonical basis of the 1-dimensional vector spaceT0R as ξ =ξ0·(d/dt)0 and define a vector field δalong the projection

τ3,2 _by

δ(J03ζ) =T0T2ζ· µ

d dt

¶

0

.

(3.1)

The vector fieldδinduces a mappingη→hη, defined on differential 1-forms onT2_Y, with values in the module of functions onT3_{Y by}

hη(J03ζ) =η(J02ζ) ¡

δ(J03ζ) ¢

.

(3.2)

In particular, iff is a function on an open setW ⊂T2_{Y, then the formula δ(f) =}

h(df) defines a functionδ(f) on the set (τ3,1₎−1_(W)⊂T3_Y.

Lemma 7. Let η be a 1-form, let (V, ψ), ψ = (yK_{), be a chart onY, and let γ} be a curve with values inV.

(a)δ has a chart expression

δ= ˙yK ∂

∂yK + ¨y K ∂

∂y˙K.

(3.3)

(b)If η is expressed asη=AkdyK+BKdy˙K, then

hη=AKy˙K+BKy¨K.

(3.4)

We callδ theformal derivative morphism; the functionδ(f) is called theformal derivativeoff. The mappinghis called thehorizontalization.

Remark 1. From the definitions (3.1) and (3.2) we easily derive the formulas

hdwL= ˙wL, hdw˙L = ¨wL, hdwσ = ˙wLw1σ, hdw1σ= ˙wLw2σ. (3.5)

LetW be an open set inY, and suppose we have a 1-formη, defined on the set (τr,0₎−1_(W)⊂ImmTr_{Y. We say thatηiscontact, ifT}r_ζ∗_{η= 0 for all immersionsζ,} defined on an open interval in R, with values inW. The ideal of the exterior algebra of differential forms on the set (τr,0₎−1_{(W), locally generated by contact 1-forms, is} called thecontact ideal, and is denoted by Ω1cW. By a contactk-form we mean any

k-form, belonging to the contact ideal.

Lemma 8. LetW be an open set inY, letη be a1-form on(τ2,0₎−1_{(W), and let} (V, ψ), ψ= (yK_{), be an chart onY such thatV ⊂W. Then the following conditions} are equivalent:

(a)η is a contact form.

(b)For every subordinate chart(V2,L_{, ψ}2,L_{), ψ}2,L_{= (y}L_,y˙L_,y¨L_{, y}σ_,y˙σ_,y¨σ_),

η= ˙ALη˙L+Aσησ+ ˙Aση˙σ,

where

˙

ηL_=dy˙L₋y¨L

˙

yLdy

L_{, η}σ _=dyσ₋y˙σ

˙

yLdy

L_{, η˙}σ_=dy˙σ₋y¨σ

˙

yLdy L_.

(3.7)

(c)For every subordinate chart(V2,L_{, χ}2,L_),χ2,L_{= (w}L_{, w}L

1, w2L, wσ, wσ1, wσ2),

η=B_L1ωL

1 +Bσ0ωσ0 +Bσ1ω1σ, (3.8)

where

ω1L=dw1L−

wL

2

wL

1

dwL, ω0σ=dwσ−wσ1dwL, ωσ1 =dwσ1 −w2σdwL. (3.9)

(d)η belongs to the kernel of the horizontalization h, i.e.,hη= 0.

Clearly, the forms (3.7), and the forms (3.8) are linearly independent.

Suppose we have a 1-form η, defined on ImmT1_{Y. Let I be an open interval,} and let γ : I → Y be an immersion. Any compact subintervalK of I defines the

variational integral, associated withη,

ηK(γ) =

Z

K

(T1γ)∗_η. (3.10)

The mappingηK is theintegral variational functional, associated with theη.

The function hηis the Lagrange function, associated withη. The following gives us a description of variational functionals in terms of Lagrange functions.

Lemma 9. Let η be a 1-form on T1_{Y, and let γ : I → Y be an immersion,}

defined on an open intervalI⊂R. Then

Trγ∗_{η= (hη◦T}r+1_γ)·dt. (3.11)

Lemma 9 says that the Lagrange functionLη, associated withη, is given by

Lη(J0r+1ζ) =hη◦Tr+1(ζ◦trt)(t).

(3.12)

LetC2

KY denote the set of curves inY of classC2, defined on a compact interval

K⊂R. We have for every isomorphismαofY and every curveγ∈C2

KY

ηK(αγ) =

Z

K

(T1(αγ))∗_η. (3.13)

But by definitions,T1_{(αγ) =T}1_α◦T1_{γ, so (3.13) reduces to}

ηK(αγ) =

Z

K

(T1α◦T1γ)∗_η=

Z

K

T1γ∗_T1_α∗_η. (3.14)

Consequently, the variational functionalC2

KY ∋γ→ηK(αγ)∈R (3.13) satisfies

ηK(αγ) = (T1α∗η)K(γ),

(3.15)

This property of variational functionals can be transfered to vector fields. Letξ

be a vector field on Y and αξ

s its flow. Then for all sufficiently smalls, ηK(αξsγ) =

((T1_αξ

s)∗η)K(γ). Differentiating, we have

µ_dη

K(αξsγ)

ds

¶

0 =

Z

K

T1γ∗_∂

T1 ξη,

(3.16)

where ∂T1

ξη is the Lie derivative of the form η by the vector field T1ξ, the 1-jet

prolongation ofξ. The mapping

C_K2Y ∋γ→(∂T1

ξη)K(γ) =

Z

K

(T1γ)∗_∂

T1 ξη∈R

(3.17)

is called thefirst variationof the variational functional ηK by the vector fieldξ.

Note that the Lie derivative∂T1

ξηunder the integral sign in (3.17) can be

decom-posed as ∂T1

ξη = iT1

ξdη+diT1

ξη. The form η is said to be a Lepage form, if the

2-formdη belongs to the contact ideal Ω1

cY.

Theorem 1. (The structure of Lepage forms)The following conditions are equivalent:

(a)η is a Lepage form on ImmT1_Y.

(b)η has in any subordinate chart(V1,L_{, ψ}1,L_{)an expression}

η=PLdyL+

∂PL

∂y˙σy˙

L_ησ_+dF L,

(3.18)

whereFL andPL are function onV1,L, andPL satisfies

∂PL

∂y˙Ly˙

L₊∂PL

∂y˙σy˙ σ_{= 0.}

(3.19)

(c)η has in a subordinate chart(V1,L_{, χ}1,L_{)an expression}

η=PLdwL+

∂PL

∂wσ

1

ωσ_+dF,

(3.20)

wherePL andF are functions onV1,L such that

∂PL

∂wL

1 = 0.

(3.21)

To demonstrate basic ideas of the proof, we show that (b) follows from (a). Con-sider the chart expression

η=ALdyL+Aσησ+BLdy˙L+Bσdy˙σ,

(3.22)

where

ησ _=dyσ₋ y˙σ

˙

yLdy L

(Lemma 8). Now

The conditions thatdη be generated by the contact formsησ _imply

∂BL

Integrating these conditions we get

BL=

for a functionF. Then, however,

dF = ∂F

From this expression we have

dη= ∂A˜L

Suppose we have a Lepage form η on the manifold of velocities ImmT1_{Y. We} wish to describe a distribution ∆η on ImmT1Y, defined by differential 1-formsiΞdη, where Ξ runs through vector fields on ImmT1_{Y; ∆}

ηis theEuler-Lagrange distribution

associated withη.

We give an explicit characterization of the Euler-Lagrange distribution of a Lepage formη in terms of the second subordinate charts. We know that

η=LLdwL+

Theorem 2. (The Euler-Lagrange distribution) In a second subordinate chart, the Euler-Lagrange distribution∆η is generated by the 1-forms

To prove the theorem, we contract the formdη with a vector field

Ξ = ΞL ∂

∂wL + Ξ ν ∂

∂wν + Ξ L

1

∂ ∂wL

1 + Ξν

1

∂ ∂wν

1

,

(3.33)

and obtain the generators of the distribution by calculating the coefficients in the chart expression for the formiΞdη at the components ΞL, Ξσ, and Ξν1.

Theorem 2 describes important special cases. In particular, if the matrix

∂2_L

L

∂wν

1∂w1σ (3.34)

is non-singular, then each integral curve of the Euler-Lagrange distribution is holo-nomic, i.e., is the prolongation of a curve in the manifoldY. In this case the generators of the Euler-Lagrange distribution reduce to

ωσ_,

µ

−∂LL

∂wσ +

∂2_L

L

∂wL_∂wσ

1

+ ∂

2_L

L

∂wν_∂wσ

1

wν

1 ¶

dwL₊ ∂2LL

∂wν

1∂wσ1

dwν

1. (3.35)

4

Parameter invariance

Suppose we have a 1-formηon ImmT1_{Y. Consider the variational integral (3.10). We} are interested in the case when the numberηK(γ) is independent of parametrization

of the setγ(I). This is characterized by the following theorems.

Theorem 3. Let η be a 1-form on T1_{Y, let γ : I → Y be an immersion, J}

an open interval, and µ : J → I a diffeomorphism. The following conditions are equivalent:

(a)For any two compact intervalsL⊂J andK⊂I such that µ(L) =K,

ηK(γ) =ηL(γ◦µ).

(4.1)

(b)η satisfies

(T1γ)∗_{η= (µ}−1₎∗_T1_(γ◦µ)∗_η. (4.2)

Condition (4.2) is called the invariance condition; we say that η and γ satisfy the invariance condition, if (4.2) holds for all diffeomorphisms µ. We say thatη is

parameter-invariant, if (4.2) holds for allγ andµ.

Consider the variational functional (3.10). The form (T1_γ)∗_{η has at every point}

t0∈I an expression

(T1γ)∗_{η(t0) =L ◦T}2_{γ(t0)·dt(t0).} (4.3)

We can now give a version of the invariance condition in terms of the Lagrange functionL.

Lemma 10. Let the immersionγ and the diffeomorphism µbe given. Then the following two conditions are equivalent:

(2)For alls∈K,

L¡

J02(γ◦tr−µ(s))

¢

·Dµs(0) =L¡J02(γ◦tr−µ(s))◦J02µs¢.

(4.4)

The following is a criterion of invariance of the formηunder changes of parametriza-tion; the criterion says that the Lagrange functionL, associated with η, should be

L2_{-equivariant.}

Theorem 4. η satisfies the invariance condition if and only if

L¡

J02γ ¢

·Dα(0) =L¡

J02γ◦J02α ¢ (4.5)

for allJ2

0α∈L2 .

Remark 2. According to Lemma 2, the group action ofL2 _{on ImmT}2_{Y is in a} chart (V, ψ), ψ= (yK_{), given by the equations ¯y}K_=yK_,y˙¯K _=ay˙K_,y¨¯K _=ay¨K_+by˙K_,

where a and b are the canonical coordinates on L2_{. Hence condition (3.32) can be} expressed asL(yK_{, ay˙}K_{, ay¨}K_+by˙K_{) =aL(y}K_,y˙K_,y¨K_).

Remark 3. Condition (4.5) can also be expressed in a subordinate chart (V2,L_{, χ}2,L_{), χ}2,L _{= (w}L_,w˙L_,w¨L_{, w}σ_{, w}σ

1, w2σ). Since the group action of L2 in this chart is given by the equations ¯wL _{= w}L_,w˙¯L _{= aw˙}L_,w¯¨L _{= bw˙}L _+a2_w¨L_,w¯σ ₌

wσ_,w¯σ

1 =wσ1,w¯σ2 =w2σ, whereaandb are the canonical coordinates onL2, we have

L(wL_{, aw˙}L_{, bw˙}L_+a2_w¨L_{, w}σ_{, w}σ

1, w2σ) =aL(wL,w˙L,w¨L, wσ, wσ1, wσ2).

We are now in a position to give a complete description of Lepage forms on ImmT1_{Y that satisfy the invariance condition.}

Theorem 5. Let η be a 1-form on ImmT1_{Y. The following two conditions are}

equivalent:

(a)η is a Lepage form, and satisfies the invariance condition.

(b) In every subordinate chart (V1,L_{, χ}1,L_{), χ}1,L _{= (w}L_,w˙L_{, w}σ_{, w}σ

1) has an

ex-pression

η=PLdwL+

∂PL

∂wσ

1

ωσ_+dF L,

(4.6)

wherePL andFL are functions on the setV1,L such that

∂PL

∂w˙L = 0,

∂FL

∂w˙L = 0.

(4.7)

Express the Lepage formη as in Theorem 1, by

η=PLdwL+

∂PL

∂wσ

1

ωσ_+dF,

(4.8)

and compute the corresponding Lagrange function L = hη. We have, using the formulashdwL_{= ˙w}L_{, hdw˙}L_{= ¨w}L_{, hdw}σ_{= ˙w}L_wσ

1, andhdw1σ= ˙wLw2σ (2.10),

L= µ

PL+

∂F ∂wL

¶ ˙

wL₊ ∂F

∂w˙Lw¨

L₊ ∂F

∂wσw˙ L_wσ

1 +

∂F ∂wσ

1 ˙

wL_wσ

ButL(wL_{, aw˙}L_{, bw˙}L_+a2_w¨L_{, w}σ_{, w}σ

1, w2σ) =aL(wL,w˙L,w¨L, wσ, wσ1, wσ2) for alla, b∈ R,a6= 0, sinceηsatisfies the invariance condition (Theorem 3), andPL=PL(wL, wσ, wσ1) sinceη is Lepage; then (a) implies∂F/∂w˙L_{= 0.}

Remark 4. From Theorem 5 we conclude that Theorem 2 remains valid for the Euler-Lagrange distribution of parameter-invariant variational problems.

Acknowledgments. The research of the first author was supported by grants 201/09/0981 (Czech Science Foundation) and MEB 080808 (Kontakt).

References

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Authors’ addresses:

D. Krupka

Institute of Theoretical Physics and Astrophysics, Faculty of Science, Masaryk University,

Kotlarska 2, 638 00 Brno, Czech Republic;

Department of Mathematics, La Trobe University, Melbourne, Bundoora, Victoria 3086, Australia.

M. Krupka

Department of Computer Science, Faculty of Science, Palacky University,