We define a class of almost S(M)-multilinear maps. The Euler-Lagrange operator is given by means of the trace of an almost S(M)-bilinear map

Milewski, J.

Zurnal matematiceskoj fiziki analiza geometrii

Наукова електронна бібліотека періодичних видань НАН України (Vernadsky National Library of Ukraine)

Journal of Mathematical Physics, Analysis, Geometry2008, vol. 4, No. 2, pp. 267277An Invariant Form of the EulerLagrange OperatorJan MilewskiInstitute of Mathematics, Poznan University of Technologyul. Piotrowo 3A, 60-965 Poznan, PolandE-mail:jsmilew@wp.plReceived March 14, 2007We dene a class of almost S(M)-multilinear maps. The EulerLagrangeoperator is given by means of the trace of an almost S(M)-bilinear map.Key words: calculus of variations on manifolds, jets, dierential opera-tors, integration on manifolds.Mathematics Subject Classication 2000: 49Q99, 58A20, 58C35.1. IntroductionThe aim of the paper is to give a geometric interpretation of the EulerLagrange operator. The problem considered here will be described in terms ofbre bundle theory. First, let us consider a functional of the formI[y] =ZF (x; y(x); : : : ; y()(x); : : :)dkx; (1)where  is a domain in Rk, y :   ! Rmis a smooth function,  = (i1; : : : ; ik)denotes a multiindex, and y()=DjjyDx. We assume here that the functional (1)has a nite rank (say N), which means that F depends only on partial derivativesy()of the rank less or equal n. It is well known [1, 2] that the EulerLagrangeequationsX( 1)jjDjjDx@F@y()i= 0 for i = 1; 2; : : : ;m; (2)are necessary conditions for the extremum of the functional (1) in a suitableclass of functions satisfying the well-known boundary conditions. These equationsThe work has been supported by the projects: grant KBN No. 1 P03A 039 29 and PB-43-047/06 BW of Poznan University of Technology.c Jan Milewski, 2008Jan Milewskiimply that for any vector eld h :   ! Rmthe formÆhF =XihiX( 1)jjDjjDx@F@y()i(3)is identically equal to zero on the extreme of the functional. The operator Æhiscalled the EulerLagrange operator. This operator can be expressed in the formÆhF =Xi;( 1)jj!8<:DjjDx24X DjjDx((x  x0)hi(x))!x0=x@F@y()i359=;: (4)To show the equality of the formulas (3) and (4), it is enough to use the identityPk( 1)k nk kl= ( 1)nÆn;land the Leibnitz rule. Using the following dieren-tial operatorsVhF =Xi;h()i@F@y()i(5)and (VfghF )x= (Vhx0F )x0=x, where hx0(x) = (x  x0)h(x), we haveÆhF =X( 1)jj!DjjDx(VfghF ): (6)W.M. Tulczyjew [2] considered the analogous to (6) expressions for jet bundlesin a local system of coordinates. The aim of the paper is to give a geometricalformulation of the problem. For the functionals of rank one, the invariant formulafor the EulerLagrange equation is known [3, 4]. In [5] the authors consideredthe EulerLagrange equation invariant with respect to a Lie group. In this paperwe will generalize (6) to the functionals dened on a bre bundle, and the resultwill be obtained without using the coordinate system. Here we deal with thefunctionals of arbitrary but nite rank. The paper is organized as follows. InSection 2 we introduce some notation concerning bre bundles, jets and dierentialoperators. In Section 3 we dene what an almost S(M)multilinear map betweenthe sections of vector bundles is (S(M) denotes here the ring of smooth realfunctions on M). By a smooth function on a Crmanifold we understand anyCrfunction. The trace dened here can be considered as a generalization of thetrace of matrices or linear operators. In Section 4 we deal with the adjoint ofdierential operators. In Section 5 we consider the jet vertical derivative on brebundles. Further, using the adjoint of dierential operators, we give an invariantexpression for the EulerLagrange operator by means of the trace of a suitablealmost bilinear map (61) which is the main result of the paper.268 Journal of Mathematical Physics, Analysis, Geometry, 2008, vol. 4, No. 2An Invariant Form of the EulerLagrange Operator2. NotationAll the manifolds considered in this paper have nite dimensions and we dealonly with dierential bundles. Let us consider the following notation for thehomomorphism of vector bundles. Let K =  : K  ! X, E = p : E  ! Y begiven vector bundles, and f : X  ! Y be a map. A map  : K  ! E is calleda homomorphism of the vector bundles K and E over f , if p Æ  = f Æ , and therestriction jEf(x)Kx: Kx ! Ef(x)is a linear homomorphism for any x 2 X. Wedenote the set of all homomorphisms of these bundles over f by HOMf(K; E).In the case X = Y , f = idXwe write HOM(K; E) instead of HOMidX(K; E).For a bundle E =  : E  !M by V E we denote the vertical bundle of E andby JnE  the nth jet extension of E [6, 7]. We denote the total space of JnEalso by JnE and so on. The mapsn: JnE  !M ; nk: JnE  ! JkE for k  n (7)denote suitable projections. For a local section s of the bundle E , [s]ndenote thelifting of s to JnE . If the bundle E is a vector bundle, then JnE is also a vectorbundle.A linear dierential operator D of the rank less or equal n from a vector bundleE1= 1: E1 ! M to a vector bundle E2= 2: E2 ! M can be treated as ahomomorphism of the vector bundles JnE1and E2over idM. The bundle of theseoperators is denoted by Ldin(E1; E2). In particular, Ldi0(E1; E2) is isomorphicto HOM(E1; E2). In the case E1= E2we write Ldin(E1) instead of Ldin(E1; E1).For any f 2 S(M) by mf2  (Ldi0(E)) we understand the dierential operatorof the rank 0 of multiplication by the function f .So, we regard the vector bundle Ldin(M;R) of the linear dierential opera-tors acting on S(M) of the rank less or equal n as a dual one to the nth jet bundleJn(M;R). There exist two kinds of multiplication of functions and sections ofthe nth jet bundle Jn(M;R):(f)x= f(x)xand (f  )x= [fg]nx; (8)where x 2M , f 2 S(M),  2  (Jn(M;R)), and g 2 S(M) is a representative ofx. For any D 2  (Ldin(M;R)) the following equalitiesD(f) = fD() and (D(f  ))(x) = (D(fg))(x) (9)are fullled. The consequence of the second formula in (9) is(Dmf)() = D(f  ): (10)We use the following notation. A bundle  : P  ! B is denoted as PBif theprojection is obvious.Journal of Mathematical Physics, Analysis, Geometry, 2008, vol. 4, No. 2 269Jan MilewskiNow, let the map E =  : E  ! M be a bundle, not necessarily a vectorbundle. Using this notation, we can write the equality(Jn((V E)M))JnE= (V (JnE))JnE; (11)which is a very well-known canonical isomorphism [3, 6] of the bundles over JnE .The following equalityJn((V E)M) = (V (JnE))M(12)is an analogue of (11) for the bundles over M .Further, letE =  : E  !M ; KE= p : K  ! E ; KM=  Æ p : K  !M (13)be given bundles. Then we put(JnKM)JnE= Jn(p) : JnKM ! JnE ; (14)where the projection is given by the formulaJn(p)([]nx) = [p Æ ]nx; (15)and  is a local section of KMin a neighborhood of x 2M .If the bundle KEin (13) is a vector bundle, then there exists a pointwisemultiplication of functions on M and sections of KM. For any f 2 S(M) and Xbeing a local section of KMin a neighborhood x 2M , we can put(fX)x= f(x)Xx; (16)because Xxis an element of the total space of the bundle KE. Moreover, eitherthe composition p ÆX or p Æ (fX) is the same section of E . In a similar way onecan dene the addition of local sections X;Y of KMfor the case p ÆX = p Æ Y .One can also dene two kinds of multiplication of functions and local sectionsof Jn(KM) in the following way:(fh)x= f(x)hxand (f  h)x= [fX]nx; (17)where x 2M , and h is a local section of Jn(KM) such that hx= [X]nx.Moreover, we put(h)x= (f  h)xfor  2  (Jn(M;R)); x= [f ]nx: (18)270 Journal of Mathematical Physics, Analysis, Geometry, 2008, vol. 4, No. 2An Invariant Form of the EulerLagrange Operator3. Almost Multilinearity and TraceLet Ei= pi: Ei ! M; i = 1; : : : ; n, and P =  : P  ! M be given vectorbundles. An Rmultilinear mapA :  (E1)  (E2) : : :  (En)  !  (P) (19)will be called almost S(M)multilinear if the equalitiesA(f1; 2; : : : ; n) = A(1; f2; : : : ; n) = : : : = A(1; 2; : : : ; fn) (20)are satised for any sections i2  (Ei) and a smooth function f 2 S(M).Any almost S(M)multilinear map A can be treated as an Rlinear mapA :  (E1 E2     En)  !  (P) (21)given byA(1 2 : : : n) = A(1; 2; : : : ; n): (22)Example I.Any S(M)-multilinear map is an almost S(M)-multilinear map.Example II.A Lie bracket [; ] :  (TM)   (TM)  !  (TM) of vector elds is not analmost S(M)bilinear map.Example III.The map A :  (TM) (VjM jM)  !  (VjM jM) given by formula A(X;!) =LX! is an almost S(M)bilinear map, where LXis the Lie derivation and jM jdenotes a dimension of M . Certainly,A(fX; !) = LfX! = (@Xf)! + fLX! = LX(f!) = A(X; f!):Let us notice that for ! 2  (VkM) the expression LX! is not an almost S(M)-bilinear one for k < jM j.For the case when n = 2, E1= E and E2= E, where E : E  !M is a vectorbundle, we can dene the trace of map A by the following formula:TrA = A(id (E)); (23)where we treat the bundle E  Eas End(E) using the identication(u v)(w) = v(w)u: (24)For two sections T1; T22  (End(E)) we putlT1T2= T1T2; rT1T2= T2T1: (25)Journal of Mathematical Physics, Analysis, Geometry, 2008, vol. 4, No. 2 271Jan MilewskiThen, from (23) we getTr(A Æ lT) = Tr(A Æ rT) = A(T ) (26)for any T 2  (End(E)).Example IV.The trace of a linear endomorphism.Let V be a real k-dimensional space. Let A be an endomorphism of V and [Aij]be a representation of A in a basis (ei)i=1;:::; kin V ,Aij= ei(Aej); i; j = 1; : : : ; k; (27)where (ei)i=1;:::; kis the dual basis to ei, it means thatei(ej) = Æij: (28)Further, using the Eq.(24), we can write the decomposition of idVin the way:idV=Xiei ei: (29)Certainly, for any basis vector eiwe have(Xiei ei)ej=Xiei(ej)ei=XiÆijei= ej: (30)So, the right side of the Eq.(29) is the idV. Moreover,A(idV) = A(Xiei ei) =XiAii= TrA: (31)This example shows that our denition of the trace is a generalization of the traceof a linear endomorphism.Example V. (The Taylor expansion in terms of jets.)Lemma. Let f 2 S(M) be a smooth function and the S(M)-bilinear mapAf:  (Jn(M;R))   (Ldin(M;R))  !  (Jn(M;R)) (32)be given by the formulaAf(;D) = (Df): (33)Then for any point x 2M the equality(TrAf)x= [f ]nx(34)is fullled.272 Journal of Mathematical Physics, Analysis, Geometry, 2008, vol. 4, No. 2An Invariant Form of the EulerLagrange OperatorP r o o f. The proof follows immediately from the denition of the trace (23).R e m a r k. One can treat the equality (34) as the Taylor formula in the Peanoform. The basis of Jnx0(M;R) given by e= [f]nx0, where f(x) = (ex   ex0), isa dual one to the basis e=1!DjjDexof dierential operators, for any choice of achart ' at a neighborhood of the point x0, where ex = '(x) denotes coordinatesof x. Certainly,ee=1!DjjDex(ex  ex0)jx=x0= Æ; (35)andPeeis the identity operator in the space of jets Jnx0(M;R).4. Adjoint of Dierential OperatorsLet us dene the adjoint of dierential operators:  (Ldi(M;R))  !  (Ldi(^jM jM)): (36)For any function f 2 S(M) and any vector eld X on M we put(mf)= mf; (@X)=  LX: (37)We also put(D1D2)= D2D1(38)for D1;D22  (Ldi(M;R)).By a domain in M we understand here an jM j-dimensional oriented compactconnected submanifold  of M with boundary @. The operation of taking theadjoint satises the following integral formula:Z(Df)! =Zf(D!) (39)for any dierential operator D 2  (Ldin(M;R), with  being a domain in M ,f 2 S(M) and ! 2  (VjM jM) such that [f!]n 1j@ 0. The Equation (39) isa consequence of integration by parts the formulaZ(@Xf)! =Z@fiX!  ZfLX!; (40)where X 2  (TM).Now we can formulate the following theorem.Theorem 1. LetE =  : E  !M ; KE= p : K  ! E; KM=  Æ p : K  !M (41)Journal of Mathematical Physics, Analysis, Geometry, 2008, vol. 4, No. 2 273Jan Milewskibe given bundles, where KEis a vector bundle. Let  be a domain in M , and! 2 HOMn(JnKM)JnEje;^jM j); h 2  (Jn(KM)j); (42)wheree = (n) 1(). Let the mapA!h:  (Jn(;R))   (Ldin(;R))  !  (^jM j) (43)be given by the formulaA!h(;D) = D(!(h)): (44)Then A!his an almost S(M)-bilinear map, and the equalityTrA!fh= fTrA!h(45)holds for any smooth function f .First, let us prove the following:Lemma. The map A!hsatises the equalities:A!fh(;D) = A!h(f  ;D) (46)andA!h(;D Æmf) = fA!h(;D): (47)T h e p r o o f o f t h e l e m m a.We can writeA!fh(;D) = D(!((f  h))) = D(!((f  )h)) = A!h(f  ;D) ; (48)because (f  h) = (f  )h. The formula (47) is a consequence of the fact(Dmf)= fD.T h e p r o o f o f T h e o r e m 1. Using the identication (22), we canwrite the equality (46) asA!fh= A!hÆ lf: (49)Further, from (10, 47) we getA!hÆ rf= fA!h: (50)So, from Eq.(26) we obtain Eq.(45).274 Journal of Mathematical Physics, Analysis, Geometry, 2008, vol. 4, No. 2An Invariant Form of the EulerLagrange Operator5. EulerLagrange OperatorLet E = p : E  !M be a bre bundle (not necessarily a vector bundle), andF : JnE  !^jM jM (51)be a bre map (over idM). The map (51) induces the functional of the formI[s] =ZF ([s]n) ; (52)where s denotes a section of E over the domain  M .Let X be a local section of (V E)Mover a neighborhood of x 2M and let t,t 2 ( ; ) be a one-parameter family of local sections of E such thatX =ddttjt=0: (53)The expression(VXF )x=ddtF ([t]nx)jt=0(54)is a vertical jet derivative of the map F at point x in the direction of eld X. Itis well known that (VXF )xdepends on the n-th jet of X at x. So, we can putVhF = (VXF )x(55)for h = [X]nx.R e m a r k. The derivative operator (5) is an analogue of the operator (54).The eld X in (54) corresponds to the eld h in (5).Now let h be a local section of Jn((V E)M) = (V (JnE))M. The vertical jetderivative satises the following equalities:VfhF = fVhF; (VfXF )x= (VfhF )x; (56)where x 2M and X is a local section of (V E)Mjsuch that [X]nx= hx.The EulerLagrange operator ÆXfor the functional (52) is uniquely determinedby the following two conditions:ÆfXF = fÆXF (57)(it means that the operator Æ is S(M)-linear with respect to the eld X), andZÆXF =ZVXF (58)Journal of Mathematical Physics, Analysis, Geometry, 2008, vol. 4, No. 2 275Jan Milewskifor any X over  such that [X]n 1j@ 0.The following theorem expresses the EulerLagrange operator by means of thetrace of an S(M)-bilinear map.Theorem 2. Let F : JnE  !VjM jM be a bre map over idM,  be a domainin M , andAFh:  (Jn(;R))   (Ldin(;R))  !  (^jM j) (59)be given by the formulaAFh(;D) = D(VhF ) (60)for h 2  (Jn((V E)M)j): Then AFhis an almost S(M)bilinear map, and thetrace TrAF[X]nis the EulerLagrange operatorÆXF = TrAF[X]n(61)for any X 2  ((V E)M)j).P r o o f. Let hF := TrAFh. Theorem 1 shows that the equality fhF =fhF is fullled for any smooth function f 2 S(M). Now let us consider anoperator of the form D Æ @Yfor D 2  (Ldik 1(M;R)), where k  n and Y 2 (TM). Let  2  (Jn(M;R)) be a section of the class of innitesimal jets ofrank k. It means that D0()  0 for any operator D0of the rank lower than k. SoZ(D Æ @Y)(V[X]nF ) =Z@iY(D(V[X]nF )) = 0 (62)for [X](n 1)j@ 0. The bundle Ldin(M;R) can be decomposed into the directsumLdin(M;R) = Ldi0(M;R)  Ldin;1(M;R); (63)where Ldi0(M;R) is the bundle of vanishing on constant functions operators ofrank 0 (multiplication by functions) and Ldin;1(M;R) is the bundle of operatorsof the rank not higher than n.The decomposition of Jn(M;R), dual to (63), isJn(M;R) = Jn;0(M;R) Jn;1(M;R); (64)where Jn;0(M;R) is the bundle of jets represented by locally constant functionsand Jn;1(M;R) is the bundle of innitesimal jets. We can decompose hFhF = Trn;0AFh+ Trn;1AFh(65)in accordance with (63, 64). SoTrn;0AFh= AFh([1];m1) = VhF;276 Journal of Mathematical Physics, Analysis, Geometry, 2008, vol. 4, No. 2An Invariant Form of the EulerLagrange Operatorand from (62) we obtainZTrn;1AF[X]n= 0for [X]n 1j@ 0.R em a r k. Our formalism gives a geometric approach to the theory withoutusing local coordinates. W.M. Tulczyjew considered the bundles of jets of inniterank J1(E). In the paper [2] there was introduced a family of operators actingon dierential forms on J1(E). Using these operators, W.M. Tulczyjew derivedan expression analogous to the formula (6). We have given here a denition ofthe EulerLagrange operator by means of the trace of an almost S(M)bilinearmap.In a local coordinate the Taylor formula and the EulerLagrange operator aregiven by means of dierential operators of the formDjjDx, which commute eachother, that is a consequence of the Schwarz theorem. Our approach is global andit can be applied to a noncommutative basis of dierential operators. The wayis natural for to study functionals on the noncommutative Lie groups. The rstorder operators can be globally dened as the left-invariant vector elds and theyform Lie algebra of the group. Any left-invariant dierential operator is a linearcombination of compositions of the rst order operators. Using a suitable basisin the space of the left-invariant dierential operators we can study globally theEulerLagrange operators on Lie groups.References[1] B.S. DeWitt, Dynamical Theory of Groups and Fields. Gordon and Breach, NewYork, 1965.[2] W.M. Tulczyjew, The EulerLagrange Resolution. In: Di. Geom. Meth. Math.Phys. (P.L. Garcia et all, Ed.).  Lect. Notes Math. Salamanca 836 (1979), 2248.[3] H. Goldschmidt and S. Sternberg, The HamiltonCartan Formalism of Variations. Ann. Inst. Fourier Grenoble 23 (1973), No. 1, 203267.[4] M.J. Munoz and M.E. Rosado Maria, Invariant Variational Problem on LinearFrame Bundles.  J. Phys. A 35 (2002), No. 8, 2013.[5] I.A. Kogan and P.J. Olver, Invariant EulerLagrange Equations and InvariantVariational Bicomplex.  Acta Appl. Math. 76 (2003), No. 2, 137.[6] J.F. Pommaret, Systems of Partial Dierential Equations and Lie Pseudogroups.Gordon and Breach, New York, 1978.[7] A.M. Vinogradov, I.S. Krasilschik, and V.V. Lytchagin, Introducion to Geometryof Nonlinear Dierential Equations. Nauka, Moscow, 1986. (Russian)Journal of Mathematical Physics, Analysis, Geometry, 2008, vol. 4, No. 2 277

An Invariant Form of the Euler-Lagrange Operator

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An Invariant Form of the Euler-Lagrange Operator

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