on frame bundles

on frame bundles

J. Brajerˇc´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 letJ^rF Xbe the r-jet prolongation of F X. We shall consider J^rF 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 onJ¹F X andJ²F 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 ofn²linearly 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 n²second 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

°Balkan Society of Geometers, Geometry Balkan Press 2008.

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 byJ^rY ther-jet prolongationofY, andπ^r,s:J^rY →J^sY,π^r:J^rY →X are the canonical jet projections. Ther-jetof a section γ of Y at a point x∈X, is denotedJ_x^rγ; andx→J^rγ(x) =J_x^rγ is ther-jet prolongationofγ. Anyfibered chart (V, ψ),ψ= (xⁱ, y^σ), onY, where 1≤i≤n, 1≤σ≤m, induces theassociated charts onXand onJ^rY, (U, ϕ),ϕ= (xⁱ), and (V^r, ψ^r),ψ^r= (xⁱ, y^σ, y_j^σ₁, y^σ_j1j2, . . . , y_j^σ_1j2...jr), respectively; hereV^r= (π^r,0)⁻¹(V), andU =π(V). Recall that theformal derivative operatoris defined by

di= ∂

∂xⁱ +y^σ_i ∂

∂y^σ +y_i^σ_1i ∂

∂y^σ_i1 +. . .+y^σ_i1i2...iri ∂

∂y^σ_i1i2...ir.

For any open set W ⊂ Y, Ω^r₀W denotes the ring of smooth functions on W^r. The Ω^r₀W-module of differentialq-forms onW^r is denoted by Ω^r_qW, and the exterior algebra of forms on W^r is denoted by Ω^rW. The module of π^r,0-horizontal (π^r- horizontal)q-forms is denoted by Ω^r_q,YW (Ω^r_q,XW, respectively).

Thehorizontalization is the exterior algebra morphismh: Ω^rW → Ω^r+1W, de- fined, in any fibered chart (V, ψ),ψ= (xⁱ, y^σ), by

hf=f◦π^r+1,r, hdxⁱ=dxⁱ, hdy_j^σ_1j2...jp =y^σ_j1j2...jpkdx^k,

where f : W^r → 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

ω^σ_j1j2...jp=dy_j^σ_1j2...jp−y^σ_j1j2...jpkdx^k,

where 0≤p≤r−1, are examples of contact 1-forms. η isπ^r-horizontalif and only if (π^r+1,r)^∗η=hη.

A Lagrangian (of order r) for Y is any π^r-horizontal n-form on some W^r. A differential form ρ ∈ Ω^s_nW, where n = dimX, is called a Lepage form, if p1dρ is π^s+1,0-horizontal, i.e.p1dρ∈Ω^s+1_n+1,YW. A Lepage formρis aLepage equivalent of a Lagrangianλ∈Ω^r_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_∂/∂x^kω0.

In this fibered chart, a Lagrangian, defined onV^r= (π^r,0)⁻¹(V), has an expression

(2.1) λ=Lω0,

where L : V^r → 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∈Ω¹_n,YW whose order of contactness is≤1. Ifλis expressed by (2.1), then

(2.2) Θλ=Lω0+ ∂L

∂y_i^σω^σ∧ωi.

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

(2) Formula

(2.3) Θλ=Lω0+ Ã

∂L

∂y^σ_i −dp ∂L

∂y^σ_pi

!

ω^σ∧ωi+ ∂L

∂y^σ_jiω^σ_j ∧ωi

generalizes the Poincar´e-Cartan form tosecond orderLagrangiansλ∈Ω²_n,XW. Ifρis a Lepage equivalent of a Lagrangianλ∈Ω^r_n,XW,λ=Lω0, then by a direct calculationp1dρ=Eσ(L)ω^σ∧ω0, where

Eσ(L) = Xr k=0

(−1)^kdi1di2. . . di_k ∂L

∂y^σ_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 automorphismJ^rα:W^r→J^rY ofJ^rY, defined by

J^rα(J_x^rγ) =J_α^r_0(x)(αγα⁻¹₀ ).

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 J^rξonJ^rY whose local one-parameter group isJ_α^r_t. Thus,

J^rξ(J_x^rγ) =

½d dtJ_α^r

(0)t(x)(α_tγα⁻¹_(0)t)

¾

0

.

The chart expression forJ^rξcan be found in [8] or [9].

We now compute the Lie derivative∂J^rξλ. 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, J^rγ^∗λ=J^sγ^∗ρfor all sections γ, we obtain

J^rγ^∗∂J^rξλ=J^sγ^∗∂J^sξρ=J^sγ^∗(iJ^sξdρ+diJ^sξρ).

Omittingγ and using the Euler-Lagrange form we get

∂J^rξλ=hi_J^s+1_ξEλ+hdiJ^sξρ.

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

An automorphismα:W →Y ofY is said to be aninvariant transformationof a formη∈Ω^s_pW, if

J^sα^∗η=η.

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

∂J^sξη= 0.

The following simple consequence of the first variation formula is known as the Noether’s theorem. Let λ ∈ Ω^r_n,XW be a Lagrangian, let ρ ∈ Ω^s_nW be a Lepage equivalent of λ, and let γ be an extremal. Then for any generator ξ of invariant transformations ofλ,

dJ^sγ^∗iJ^sξρ= 0.

An (n−1)-formiJ^sξρis 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= (xⁱ, xⁱ_j), is defined byV =µ⁻¹(U), and

xⁱ(Ξ) =xⁱ(µ(Ξ)), Ξj =xⁱ_j(Ξ) µ ∂

∂xⁱ

¶

x

,

where Ξ∈V,x=µ(Ξ), and Ξ = (x,Ξj). We denote by y_k^j theinverse matrixofxⁱ_j. The right action F X ×Gln(R) 3 (Ξ, A) → RA(Ξ) = Ξ·A ∈ F X is given by the equations

¯

xⁱ=xⁱ◦RA=xⁱ, x¯ⁱ_j=xⁱ_j◦RA=xⁱ_ka^k_j, 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 J^rF 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 J²F X, more general description of J^rF X is available in [1]. To the charts (U, ϕ), and (V, ψ), introduced above, we associate a chart (V², ψ²),ψ²= (xⁱ, xⁱ_j, xⁱ_j,k, xⁱ_j,kl), as follows. We denote byV² the set of 2-jets of smooth frame fieldsU 3x→γ(x)∈V ⊂F X. IfJ_x²γ∈V², we set

xⁱ(J_x²γ) =xⁱ(x), xⁱ_j(J_x²γ) =xⁱ_j(γ(x)),

xⁱ_j,k(J_x²γ) =Dk(xⁱ_jγϕ⁻¹)(ϕ(x)), xⁱ_j,kl(J_x²γ) =DkDl(xⁱ_jγϕ⁻¹)(ϕ(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 →J²γ(x) =J_x²γ of any frame fieldγ. The general linear group acts onJ²F X on the right by the formula J_x²γ·A=J_x²(γ·A); the action is expressed by the equations

(3.1) x¯ⁱ=xⁱ, x¯ⁱ_j=xⁱ_ka^k_j, x¯ⁱ_j,k=xⁱ_m,ka^m_j , x¯ⁱ_j,kl=xⁱ_m,kla^m_j .

It is easy to determine the orbits of the action (J_x²γ, A)→J_x²(γ·A). Denoting Γⁱ_kp=−y_p^mxⁱ_m,k, Γⁱ_klp=−y_p^mxⁱ_m,kl,

we obtainGln(R)-invariant functions onJ²F X, andequations of Gln(R)-orbits Γⁱ_kp=cⁱ_kp, Γⁱ_klp=cⁱ_klp,

wherecⁱ_kp, cⁱ_klp∈R are arbitrary numbers. The functions Γⁱ_klp are symmetric ink, l.

We have the following result.

Lemma 1.Every Gln(R)-invariant function onJ²F X depends onxⁱ,Γⁱ_kp,Γⁱ_klp. In other words Lemma 1 says that Gln(R)-invariant functions coincide with the functions on thebundle of second order connections C²X =J²F X/Gln(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 J²F X is Gln(R)-invariant if and only if it has an expression

η= ∆0+y^q_r1¹dx^p_q1¹∧∆^r_p¹₁+y^q_r1¹y^q_r2²dx^p_q1¹∧dx^p_q²₂∧∆^r_p¹₁^r_p²₂ +. . .+y_r^q1₁y^q_r2². . . y_r^qk_kdx^p_q1¹∧dx^p_q2². . .∧dx^p_q^k_k∧∆^r_p¹₁^r_p²₂^...r_...p^k_k, where∆0, ∆^r_p¹₁, ∆^r_p¹₁^r_p²₂, . . . , ∆^r_p¹₁^r_p²₂^...r_...p^k_k are arbitrary forms defined onC²X.

Let

ξ0=ξ_jⁱ Ã ∂

∂aⁱ_j

!

e

be a vector belonging to the Lie algebragln(R). Then the corresponding fundamental vector field onJ²F X is given by

(3.2) J²ξ=ξⁱ_s Ã

x^t_i ∂

∂x^t_s +x^t_i,k ∂

∂x^t_s,k +x^t_i,kl ∂

∂x^t_s,kl

! .

4 Reduction of the Euler-Lagrange equations

Using the results of Section 3, we determine in this section Gln(R)-invariant La- grangians on J¹F X and J²F X. Then we give explicit expressions of the Euler- Lagrange forms, and the Noether’s currents associated with the Lepage equivalents Θλof these Lagrangians. Then we discuss consequences ofGln(R)-invariance of these

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 byω_jⁱ the contact forms defined by ωⁱ_j=dxⁱ_j−xⁱ_j,mdx^m=dxⁱ_j+x^p_jΓⁱ_mpdx^m.

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,Γⁱ_kj only.

(b)The Euler-Lagrange form of aGln(R)-invariant Lagrangian has an expression Eλ=y^j_l

Ã

−Γ^p_qi ∂L

∂Γ^p_ql+ Γ^l_pq ∂L

∂Γⁱ_pq + ∂²L

∂x^p∂Γⁱ_pl

+(Γ^k_mpr+ Γ^k_mqΓ^q_pr) ∂²L

∂Γ^k_mr∂Γⁱ_pl

!

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 =−ξ_j^my^j_lxⁱ_m ∂L

∂Γⁱ_klωk.

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_ξEλ+hdi_J¹_ξΘλ, 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 all j,k, there exist (n−2)-forms η^j_k such that

J¹γ^∗ µ

y_l^jxⁱ_k ∂L

∂Γⁱ_mlωm−dη_k^j

¶

= 0.

Proof. By hypothesis, for any fundamental vector field ξ on F X, ∂J¹ξλ = 0.

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

(4.3) i_J²_ξEλ+hdΨλ,ξ= 0.

We can write this identity in a chart (U, ϕ), ϕ = (xⁱ), on X. Using (3.2) we have, according to Lemma 3,

iJ²ξEλ=E^j_i(L)ξ_j^kxⁱ_kω0, where

E^j_i(L) = Ã

−Γ^p_qi ∂L

∂Γ^p_ql + Γ^l_pq ∂L

∂Γⁱ_pq+ ∂²L

∂x^p∂Γⁱ_pl+ (Γ^k_mps+ Γ^k_mqΓ^q_ps) ∂²L

∂Γ^k_ms∂Γⁱ_pl

! y_l^j,

and the Noether’s current Ψλ,ξ is given by (4.1). It is convenient to denote ψ_m^j =y^j_lxⁱ_m ∂L

∂Γⁱ_klωk.

Then Ψλ,ξ=−ξ_j^mψ_m^j, and the first variation formula (4.3) can equivalently be written in the formE_i^j(L)ξ_j^kxⁱ_kω0−ξ_j^khdψ_k^j = 0. But the numbersξ^k_j ∈Rare arbitrary, so we have

(4.4) E_i^j(L)xⁱ_kω₀−hdψ^j_k= 0.

Suppose now that a sectionγsatisfies the Euler-Lagrange equations. Then the form E^j_i(L)xⁱ_kω0 vanishes alongJ²γ, so we haveJ²γ^∗dψ_k^j =dJ²γ^∗ψ^j_k = 0. Integrating we can find an (n−2)-formη^j_k onU such that

(4.5) J¹γ^∗ψ^j_k=dη_k^j.

Conversely, if a sectionγ satisfies condition (4.5), then by (4.4), γ is necessarily an extremal. 2

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)λisGl_n(R)-invariant if and only if Ldepends on xⁱ,Γⁱ_kj,Γⁱ_klj only.

(b)The Euler-Lagrange form of aGln(R)-invariant Lagrangian has an expression Eλ=y^j_l

Ã

−Γ^p_qi ∂L

∂Γ^p_ql −Γ^p_qmi ∂L

∂Γ^p_qml + Γ^l_pq ∂L

∂Γⁱ_pq +dp ∂L

∂Γⁱ_pl −Γ^l_pqm ∂L

∂Γⁱ_pqm

−2Γ^l_pt µ

Γ^t_qm ∂L

∂Γⁱ_pqm+d_q ∂L

∂Γⁱ_pqt

¶

−d_pd_q ∂L

∂Γⁱ_pql

!

dxⁱ_j∧ω₀.

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

Ψλ,ξ =ξ_j^my_l^jxⁱ_m Ã

− ∂L

∂Γⁱ_kl + Γ^q_pi ∂L

∂Γ^q_pkl + Γ^l_pq ∂L

∂Γⁱ_pkq +dp ∂L

∂Γⁱ_pkl

! ωk.

Theorem 2. Let λ∈Ω²_n,XF X be aGl_n(R)-invariant Lagrangian, letn≥2, and letγbe a section ofF X. The following conditions are equivalent.

(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 η^j_k such that

J³γ^∗ Ã

y^j_lxⁱ_k Ã

∂L

∂Γⁱ_ml −Γ^q_pi ∂L

∂Γ^q_pml −Γ^l_pq ∂L

∂Γⁱ_pmq −dp ∂L

∂Γⁱ_pml

!

ωm−dη_k^j

!

= 0.

Proof.The first variation formula for a second order Lagrangian λhas the form (4.6) ∂J²ξλ=iJ⁴ξEλ+hdiJ³ξΘλ,

where Θλ is the Lepage equivalent ofλgiven by (2.3). Again, left hand side vanishes and formula (4.6) reduces to

i_J⁴_ξEλ+hdΨλ,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. 2

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

Department of Mathematics, University of Preˇsov, Ul. 17. novembra 1, 081 16 Preˇsov, Slovakia.

E-mail: [email protected]