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on frame bundles

J. Brajerˇ

c´ık

Abstract. Variational principles on frame bundles, given by the first and the second order Lagrangians invariant with respect to the struc-ture group, are considered. Noether’s currents, associated with the corre-sponding Lepage equivalents, are obtained. It is shown that for the first and the second order invariant variational problems, the system of the Euler-Lagrange equations for a frame field are equivalent with the lower order system of equations.

M.S.C. 2000: 49Q99, 49S05, 58A20, 58E30, 58J99.

Key words: Frame, variational principle, Lagrangian, Euler-Lagrange equations, invariant transformation, Noether’s current.

1

Introduction

LetF X be the frame bundle over ann-dimensional manifoldX, and letJr_{F Xbe the} r-jet prolongation ofF X. We shall consider Jr_{F X with the canonical prolongation}

of the right action of the general linear groupGln(R) onF X. For foundations of the

variational theory in fibered space we refer to [5], [7], [8], [10], [14], and the notions related to the frame bundles and invariance can be found in [6], [9], [11], [12], [15]. In this paper we study the consequences ofGln(R)-invariance for variational problems

onJ1_{F X andJ}2_{F X. In particular, we discuss the corresponding Noether’s currents.} The generators of invariant transformations are the fundamental vector fields of the

Gln(R)-action. Then the Noether’s theorem gives us a conservation law for each one

ofn2_{linearly independent fundamental vector fields. Our main object is to study how} the Noether’s currents can be used to simplify the Euler-Lagrange equations for a frame field. We show that in case of first order invariant Lagrangian, the system of

n2_{second order Euler-Lagrange equations is equivalent with the system of the same} number of first order equations. Analogously, for the second order Lagrangian, the system of fourth order Euler-Lagrange equations is equivalent to the system of third order equations coming from the corresponding Noether’s currents.

For variational problems on principal fiber bundles there are several different con-cepts of invariance. Castrill´on, Garc´ıa, Ratiu and Shkoller [3], [4] consider invariance of

∗

Balkan Journal of Geometry and Its Applications, Vol.13, No.1, 2008, pp. 11-19. c

the first order Lagrangians on principal fiber bundleP, which determine constrained variational problems on the bundleC(P) of connections ofP. Mu˜noz and Rosado [13] study first order variational problems, invariant under diffeomorphisms of the base manifold (first ordernaturalvariational problems in the sense of Krupka [7]).

2

Invariant Lagrange structures

In this section we recall basic notions of the theory of invariant Lagrangians, and introduce our notation. For a more complete discussion we refer to [2].

IfY is a fibered manifold over ann-dimensional manifoldX, of dimensionn+m, we denote byJr_{Y ther-jet prolongationofY, andπ}r,s_:Jr_{Y →J}s_Y,πr_:Jr_{Y →X}

are the canonical jet projections. Ther-jetof a section γ of Y at a point x∈X, is denotedJr

xγ; andx→Jrγ(x) =Jxrγ is ther-jet prolongationofγ. Anyfibered chart

(V, ψ),ψ= (xi_{, y}σ_{), onY, where 1≤i≤n, 1≤σ≤m, induces theassociated charts}

onXand onJr_{Y, (U, ϕ),ϕ= (x}i_{), and (V}r_{, ψ}r_),ψr_{= (x}i_{, y}σ_{, y}σ j1, y

σ

j1j2, . . . , y

σ j1j2...jr), respectively; hereVr_{= (π}r,0₎−1_{(V), andU =π(V). Recall that theformal derivative} operatoris defined by

di= ∂ ∂xi +y

σ i

∂ ∂yσ +y

σ i1i

∂ ∂yσ

i1

+. . .+yσi1i2...iri

∂ ∂yσ

i1i2...ir

.

For any open set W ⊂ Y, Ωr

0W denotes the ring of smooth functions on Wr. The Ωr

0W-module of differentialq-forms onWr is denoted by ΩrqW, and the exterior

algebra of forms on Wr _{is denoted by Ω}r_{W. The module of π}r,0_{-horizontal (π}r

-horizontal)q-forms is denoted by Ωr

q,YW (Ωrq,XW, respectively).

Thehorizontalization is the exterior algebra morphismh: Ωr_{W → Ω}r+1_W, de-fined, in any fibered chart (V, ψ),ψ= (xi_{, y}σ_{), by}

hf=f◦πr+1,r, hdxi=dxi, hdyjσ1j2...jp =y

σ

j1j2...jpkdx

k_,

where f : Wr _{→ R is a function, and 0 ≤ p ≤ r. A form η ∈ Ω}r

kW is contact, if hη= 0. For any fibered chart (V, ψ),ψ= (xi_{, y}σ_{), the 1-forms}

ωσj1j2...jp=dy

σ

j1j2...jp−y

σ

j1j2...jpkdx

k_,

where 0≤p≤r−1, are examples of contact 1-forms. η isπr_{-horizontalif and only}

if (πr+1,r₎∗_η=hη.

A Lagrangian (of order r) for Y is any πr_{-horizontal n-form on some W}r_{. A}

differential form ρ ∈ Ωs

nW, where n = dimX, is called a Lepage form, if p1dρ is

πs+1,0_{-horizontal, i.e.p}

1dρ∈Ωsn+1+1,YW. A Lepage formρis aLepage equivalent of a

Lagrangianλ∈Ωr

n,XW, ifhρ=λ(possibly up to a jet projection).

In a fibered chart (V, ψ),ψ= (xi_{, y}σ_{), denote}

ω0=dx1∧dx2∧. . .∧dxn, ωk=i∂/∂xkω0.

In this fibered chart, a Lagrangian, defined onVr_{= (π}r,0₎−1_{(V), has an expression}

where L : Vr _{→ R is the Lagrange function associated with λ and (V, ψ). A pair}

(Y, λ), consisting of a fibered manifoldY and a Lagrangianλof orderrforY is called aLagrange structure(of orderr).

For our purpose we give the following examples of Lepage equivalents. (1) Every first order Lagrangian λ ∈ Ω1

n,XW has a unique Lepage equivalent

Θλ∈Ω1n,YW whose order of contactness is≤1. Ifλis expressed by (2.1), then

(2.2) Θλ=Lω0+

∂L

∂yσ i

ωσ_∧ω i.

Θλ is thePoincar´e-Cartan equivalentofλ, or thePoincar´e-Cartan form.

(2) Formula

(2.3) Θλ=Lω0+

Ã

∂L

∂yσ i

−dp ∂L

∂yσ pi

!

ωσ_∧ω i+

∂L

∂yσ ji

ωσ j ∧ωi

generalizes the Poincar´e-Cartan form tosecond orderLagrangiansλ∈Ω2

n,XW.

Ifρis a Lepage equivalent of a Lagrangianλ∈Ωr

n,XW,λ=Lω0, then by a direct

calculationp1dρ=Eσ(L)ωσ∧ω0, where

Eσ(L) = r

X

k=0

(−1)kdi1di2. . . dik

∂L

∂yσ i1i2...ik are theEuler-Lagrange expressions. The (n+ 1)-form

Eλ=p1dρ

is theEuler-Lagrange formassociated with λ.

By anautomorphismofY we mean a diffeomorphismα:W →Y, whereW ⊂Y

is an open set, such that there exists a diffeomorphism α0 : π(W) → X such that

πα = α0π. If α0 exists, it is unique, and is called the π-projection of α. The r-jet

prolongationofαis an automorphismJr_α:Wr_→Jr_{Y ofJ}r_{Y, defined by}

Jr_α(Jr

xγ) =Jαr0(x)(αγα

−1

0 ).

Ifξ is aπ-projectable vector field onY, and αt is the local one-parameter group

ofξwith projection α(0)t, we define ther-jet prolongationof ξto be the vector field Jr_ξonJr_{Y whose local one-parameter group isJ}r

αt. Thus,

Jrξ(Jxrγ) =

½_d

dtJ r

α(0)t(x)(αtγα

−1

(0)t)

¾

0

.

The chart expression forJr_{ξcan be found in [8] or [9].}

We now compute the Lie derivative∂Jr_ξλ. Choose to this purpose a Lepage equiv-alent ρof λ, and denote by s the order of ρ. Since λ = hρ, or, which is the same,

Jr_γ∗_λ=Js_γ∗_{ρfor all sections γ, we obtain}

Jrγ∗_∂

Jr_ξλ=Jsγ∗∂_Js_ξρ=Jsγ∗(i_Js_ξdρ+di_Js_ξρ).

∂Jr_ξλ=hi_Js+1_ξE_λ+hdi_Js_ξρ.

This is thedifferential first variation formula; the first term on the right is the Euler-Lagrange term, and the second one is theboundary term.

An automorphismα:W →Y ofY is said to be aninvariant transformationof a formη∈Ωs

pW, if

Jsα∗_η=η.

We say that aπ-projectable vector fieldξis thegeneratorof invariant transformations ofη, if

∂Js_ξη= 0.

The following simple consequence of the first variation formula is known as the

Noether’s theorem. Let λ ∈ Ωrn,XW be a Lagrangian, let ρ ∈ ΩsnW be a Lepage

equivalent of λ, and let γ be an extremal. Then for any generator ξ of invariant transformations ofλ,

dJs_γ∗_i

Js_ξρ= 0.

An (n−1)-formiJs_ξρis called theNoether’s currentassociated with a Lepage form

ρand a vector fieldξ.

3

Frame bundle and its second jet prolongation

Let X be an n-dimensional smooth manifold, and let µ : F X → X be the frame bundleoverX.F X has the structure of a right principalGln(R)-bundle. Recall that

for every chart (U, ϕ),ϕ= (xi_{), onX, theassociated chartonF X, (V, ψ),ψ= (x}i_{, x}i j),

is defined byV =µ−1_{(U), and}

xi(Ξ) =xi(µ(Ξ)), Ξj =xij(Ξ)

µ _∂

∂xi

¶

x ,

where Ξ∈V,x=µ(Ξ), and Ξ = (x,Ξj). We denote by ykj theinverse matrixofxij.

The right action F X ×Gln(R) ∋ (Ξ, A) → RA(Ξ) = Ξ·A ∈ F X is given by the

equations

¯

xi=xi◦RA=xi, x¯ji =xij◦RA=xikakj,

whereA=ai

j is an element of the groupGln(R).

For the formulation of variational principles on the frame bundles in this paper we need the r-jet prolongations of F X, the manifolds Jr_{F X, where r = 1,2,3,4.}

These manifolds are constructed from sections of the frame bundleF X in a standard way. We introduce basic concepts for J2_{F X, more general description of J}r_{F X is}

available in [1]. To the charts (U, ϕ), and (V, ψ), introduced above, we associate a chart (V2_{, ψ}2_),ψ2_{= (x}i_{, x}i

j, xij,k, xij,kl), as follows. We denote byV2 the set of 2-jets

of smooth frame fieldsU ∋x→γ(x)∈V ⊂F X. IfJ2

xγ∈V2, we set

xi_(J2

xγ) =xi(x), xij(Jx2γ) =xij(γ(x)),

As usual, the 2-jets are equivalence classes of frame fields, which have the contact up to the second order, and have the canonicaljet prolongation x →J2_{γ(x) =J}2

xγ of

any frame fieldγ. The general linear group acts onJ2_{F X on the right by the formula}

J2

xγ·A=Jx2(γ·A); the action is expressed by the equations

(3.1) x¯i=xi, x¯ji =xikakj, x¯ij,k=xim,kamj , x¯ij,kl=xim,klamj .

It is easy to determine the orbits of the action (J2

xγ, A)→Jx2(γ·A). Denoting

We have the following result.

Lemma 1.Every Gln(R)-invariant function onJ2F X depends onxi,Γikp,Γiklp.

In other words Lemma 1 says that Gln(R)-invariant functions coincide with the

functions on thebundle of second order connections C2_{X =J}2_{F X/Gl}

n(R) over X.

From equations (3.1) we can obtain an extension of Lemma 1 to differential forms.

Lemma 2. A k-form η on J2_{F X is Gl}

be a vector belonging to the Lie algebragln(R). Then the corresponding fundamental

vector field onJ2_{F X is given by}

4

Reduction of the Euler-Lagrange equations

Using the results of Section 3, we determine in this section Gln(R)-invariant

Lagrangians for the Euler-Lagrange equations. Our main tool is the first variation formula (Section 2).

Let us denote by Ψλ,ξ the Noether’s current associated with the Lepage form Θλ

(2.2), (2.3) and a vector fieldξ, and byωi

j the contact forms defined by

ωij=dxij−xij,mdxm=dxij+x p

jΓimpdxm.

Lemma 3.Let λ∈Ω1

n,XF X be a Lagrangian expressed byλ=Lω0. (a)λisGln(R)-invariant if and only if Ldepends on xi,Γikj only.

(b)The Euler-Lagrange form of aGln(R)-invariant Lagrangian has an expression

Eλ=yj_l

à −Γpqi

∂L

∂Γp_ql+ Γ

l pq

∂L

∂Γi pq

+ ∂ 2_L

∂xp_∂Γi pl

+(Γk

mpr+ ΓkmqΓqpr) ∂2_L

∂Γk mr∂Γipl

!

ωi j∧ω0.

(c) If λ is Gln(R)-invariant, then the Noether’s current associated with the Poincar´e-Cartan form ofλand any fundamental vector field ξis given by

(4.1) Ψλ,ξ =−ξjmy j lxim

∂L

∂Γi kl

ωk.

LetX be ann-dimensional manifold, letF X be the bundle of frames overX, and letµbe the bundle projection. Suppose that we have a Lagrangianλ∈Ω1

n,XF X and

aµ-verticalvector fieldξ onF X. Then in our standard notation

(4.2) ∂J1_ξλ=i_J2_ξE_λ+hdi_J1_ξΘ_λ,

where Θλ is the Poincar´e-Cartan equivalent ofλ.

Theorem 1. Let λ∈Ω1

n,XF X be aGln(R)-invariant Lagrangian, letn≥2, and letγbe a section ofF X. The following conditions are equivalent.

(a)γ satisfies the Euler-Lagrange equations,

Eλ◦J2γ= 0.

(b)For any chart(U, ϕ),ϕ= (xi_{), on X, and allj,k, there exist (n−2)-forms} ηj_k such that

J1γ∗

µ

yljxik ∂L

∂Γi ml

ωm−dηkj

¶

= 0.

Proof. By hypothesis, for any fundamental vector field ξ on F X, ∂J1_ξλ = 0.

Consequently, sinceξis alwaysµ-vertical, the first variation formula (4.2) reduces to

We can write this identity in a chart (U, ϕ), ϕ = (xi_{), on X. Using (3.2) we have,}

and the Noether’s current Ψλ,ξ is given by (4.1). It is convenient to denote

ψmj =y

Suppose now that a sectionγsatisfies the Euler-Lagrange equations. Then the form

Eji(L)xikω0 vanishes alongJ2γ, so we haveJ2γ∗dψ_kj =dJ2γ∗ψj_k = 0. Integrating we

Conversely, if a sectionγ satisfies condition (4.5), then by (4.4),γ is necessarily an extremal. ✷

For second order Lagrangians onF X we have the following results.

Lemma 4.Let λ∈Ω2

n,XF X be a Lagrangian expressed byλ=Lω0.

(a)λisGln(R)-invariant if and only if Ldepends on xi,Γikj,Γiklj only.

(b)The Euler-Lagrange form of aGln(R)-invariant Lagrangian has an expression

Eλ=yjl

(c)IfλisGln(R)-invariant, then the Noether’s current associated with the Lepage form(2.3)and any fundamental vector fieldξ is given by

(a)γ satisfies the Euler-Lagrange equations,

Eλ◦J4γ= 0.

(b)For any chart(U, ϕ),ϕ= (xi_{), on X, and all j,k, there exist (n−2)-forms} ηj_k such that

J3γ∗

Ã

yj_lxik

Ã

∂L

∂Γi ml

−Γq_pi ∂L

∂Γq_pml −Γ

l pq

∂L

∂Γi pmq

−dp ∂L

∂Γi pml

!

ωm−dηkj

!

= 0.

Proof.The first variation formula for a second order Lagrangian λhas the form

(4.6) ∂J2_ξλ=i_J4_ξE_λ+hdi_J3_ξΘ_λ,

where Θλ is the Lepage equivalent ofλgiven by (2.3). Again, left hand side vanishes

and formula (4.6) reduces to

iJ4_ξE_λ+hdΨ_λ,ξ= 0,

where the formsEλand Ψλ,ξ are given by Lemma 4. The rest of the proof is analogous

to the Proof of Theorem 1. ✷

Acknowledgments.The author is grateful to the Ministry of Education of the Slovak Republic (Grant VEGA No. 1/3009/06) and to the Czech Grant Agency (Grant No. 201/06/0922).

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Author’s address:

J´an Brajerˇc´ık