Di erential Geometry and Its Applications Proc. Conf. Opava (Czechoslovakia), August 24{28, 1992 Silesian University, Opava, 1993, 473{495
473
Topics in the calculus of variations: Finite order variational sequences D. Krupka
Abstract : It is known that there exists a mapping assigning to a rst order lagrangian its Lepagean equivalent () in such a way that d() = 0 if and only if the Euler-Lagrange form E vanishes identically, i.e., E = 0. In this paper we discuss within the theory of nite order variational sequences an analogue of for higher order lagrangians. It turns out that () is a class of forms rather than a di erential form de ned on the same domain as . The main use of is to de ne the order of a lagrangian in a more adequate way than the usual one. We show how the order of a lagrangian can be determined. We nd the order of a variationally trivial lagrangian. Applying these results to the classi cation problem of symmetry transformations we obtain a higher order analogue of the Noether-Bessel-Hagen equation. Keywords : Fibered manifold, r-jet, lagrangian, contact form, variational sequence, order of a lagrangian, Euler-Lagrange mapping, Helmholtz-Sonin mapping, variationally trivial lagrangians. MS classi cation : 58E30, 58A10, 58G05, 49F99.
1. Introduction Let X be an n-dimensional manifold, Y a bered manifold of dimension n + m with projection : Y ! X . The r-jet prolongation of Y will be written as J r Y , and r : J r Y ! X; r;s : J r Y ! J s Y , where 0 s r, will denote the canonical jet projections. If W Y is an open set, we write Wr = r;?01 (W ), and denote by 0r W the ring of smooth functions de ned on Wr . For k 1 we denote by kr W the 0r W -module of smooth k-forms on Wr . The submodule of kr W consisting of r -horizontal (resp. r;0horizontal, resp. r;s -horizontal) forms will be written as kr;X W (resp. kr;Y W resp.
kr;Z W , where Z = J s Y ). The standard horizontalization mapping is denoted by h; h maps kr W to kr+1;X W , where 1 k n. Notice that restricting h to kr;Z W kr W where Z = J r Y , we obtain a mapping h : kr;Z W ! kr W kr+1;X W . We also denote h by p0 and write pi for This paper is in nal form and no version of it will be submitted for publication elsewhere.
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D. Krupka
the i-contact component of a form (the de nitions of these concepts will be recalled in Sec. 2). Let E denote the Euler-Lagrange form of a lagrangian 2 n1;X Y . The following result concerns the rst order variational calculus, and is taken from [8]. There exists an R-linear mapping : n1;X Y ! n1;Y Y satisfying the following three conditions : (1) For every lagrangian 2 n1;X Y ,
h(()) = : (2) If 2 n1 Y is a 1;0-projectable form, then (h)) = :
(1:1) (1:2)
(3) For every lagrangian 2 n1;X W the form E = p1 d() 2 n2 +1 Y is 2;0horizontal, and E = 0 if and only if
d() = 0:
(1:3)
The rst part of (3) means that () is a Lepagean form; condition (1) then means that () is a Lepagean equivalent of . The second part of (3) shows that EulerLagrange form E vanishes if and only if d() = 0; in this sense () has a remarkable property to stand \closer" to the exterior derivative operator d as the well-known Poincare-Cartan equivalent (). The mapping was discovered by Krupka in connection with the study of lagrangians 2 n1;X Y of the form h(), where 2 n0 Y (see [12]). Recall that a -projectable vector eld on Y is said to generate generalized invariant transformations of a lagrangian 2 n1;X Y , if
@J 2 E = 0;
(1:4)
i.e., if the Lie derivative of the Euler-Lagrange form E by the 2-jet prolongation J 2 of vanishes. The following additional properties of have been proved (see [11] and [10]): (4) If 2 n1;X Y is such that E = 0, then the form () is a unique form such that h(()) = and d() = 0. This form is 1;0-projectable. (5) A -projectable vector eld on Y generates generalized invariant transformations of a lagrangian 2 n1;X Y if and only if there exists an n-form on Y such that @J 2 = h(); d = 0: (1:5) Clearly, applying (1.2) to (1.5) we get = (@J 2 ), and (1.5) reduces to the condition d(@J 2 ) = 0: (1:6) Condition (1.6) which may be used to determine the generators of the generalized invariant transformation, is a generalization of the classical Noether-Bessel-Hagen equation. Practically the same theory as above was developed by Betounes [5] (see also [6]) who obtained, moreover, the following result [4]:
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Topics in the calculus of variations
(6) For every isomorphism : Y ! Y of the bered manifold Y , (J 1 ) () = ((J 1 ) ); (1:7) where J 1 : J 1 Y ! J 1 Y is the rst jet prolongation of . Combining this result with (1.6) we obtain @J 1 d() = 0 which is another version of the Noether-Bessel-Hagen equation. A comparison of the papers [8] and [4] from the point of view of the Lepagean forms can be found in [9]. Denote
!0 = dx1 ^ dx2 ^ . . . ^ dxn; ! = dy ? yi dxi :
(1:8)
= L!0;
(1:9)
If is expressed by then
n X
@qL 1 j1 ?1 ^ ! 1 ^ dxj1 +1 1 @y 2 . . . @y dx ^ . . . ^ dx @y j q=1 j1 j2 ^ . . . ^ dxj ?1 ^ ! ^ dxj +1 ^ . . . ^ dxn: (1:10)
() = L!0 +
q
q
q
q
q
The aim of this paper is to discuss a possible extension of the mapping to the higher order variational calculus. Our discussion is going on within the framework of the theory of nite order variational sequences [13]. Since the calculations in the general case of r-th order jet spaces are extremely dicult and long instead of presenting the details of the proofs we shall explain the main ideas of how the mapping is introduced for the rst order case; a complete exposition including proofs will be given elsewhere. Let W Y be an open set, and 2 n0 W . Write in a bered chart (V; ); = i (x ; y ), where V W , h = n1! Bi1 i2 ...i dxi1 ^ . . . ^ dxi ; X pj = j !(n1? j )! B1 ... i +1 ...i !1 ^ . . . ^ ! ^ dxi +1 ^ . . . ^ dxi ; 1 j n ? 1; 1 (1:11) pn = n! B1 2 ... !1 ^ ! 2 ^ . . . ^ ! : n
n
j j
n
j
n
j
n
n
Using the decomposition 1;0 = h + p1 + . . . + pn
and the formula d1;0 = 1;0d, we obtain hd + p1 d + . . . + pn+1 d = dh + dp1 + . . . + dpn . Hence 2;1hd = hdh; 2;1pj d = pj dpj?1 + pj dpj ; 1 j n; 2;1pn+1 d = pn+1 dpn (1:12)
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(here p0 = h). Express both sides in terms of the basis of linear forms dxi ; ! ; !j ; dyij on V2 J 2Y . Then the forms on the left contain the exterior factors dxi; ! only, while the right-hand side expressions depend also on !j . The sum of such terms should therefore vanish identically. We shall nd the corresponding identities. The rst or the conditions (1.12) reduces to the identity 0 = 0. Consider the case 1 j n. We get @B1 ... ?1 i ...i 1 1 !p ^ ! ^ . . . ^ ! ?1 ^ dxi ^ . . . ^ dxi (j ? 1)!(n ? j + 1)! @yp + j !(n 1? j )! j (?1)j B1 ... i +1 ...i ! 1 ^ . . . ^ ! ?1 ^ !i ^ dxi ^ dxi +1 ^ . . .. . . ^ dxi @B ?1 i ...i 1 pB = (j ? 1)!(n1 ? j + 1)! 1 ...@y ? ... i ... i 1 +1 1 (j ? 1)!(n ? j )! i j
n
j
j
j
j
j
n
j j
n
j
j
n
j
j
n
j
p
j
j
!p ^ !1 ^ . . . ^ ! ?1 ^ dxi ^ . . . ^ dxi : j
Therefore
@B1... ?1 i ...i ? B1 ... ?1 i ...i ip = 0: n?j+1 @yp Contracting the left-hand side in p; ij we nally obtain 1 @B1 ... ?1 qi +1 ...i ? B 1 ... ?1 i +1 ...i = 0; 1 j n: j @yq If j = n + 1, we get the term 1 @B1 2 ... 1 2 n! @yp !p ^ ! ^ ! . . . ^ ! ; 1
j
j
j
j
(1:14)
n
j
j
n
j
(1:15)
n
j
n
j
n
n
(1:13)
n
j
j
(1:16)
n
which gives
@B1 2... = 0: @yp
(1:17)
n
Conditions (1.15), (1.16) and (1.17) determine pj with the help of h. In particular, writing Bi1 i2 ...I = L"i1 i2 ...i ; (1:18) we get j pj = j !(n 1? j )! j1! 1 @2L "q1 q2...q i +1 ...i !1 ^ . . . ^ ! @yq1 @yq2 . . . @yq (1:19) ^ dxi +1 ^ . . . ^ dxi : Denote by (p1; . . . ; pj ) the unique sequence of indices between 1 and n, complementary to (ij +1; . . . ; in), such that p1 < . . . < pj . Then ! 1 ^ . . . ^ ! ^ dxi +1 ^ . . . ^ dxi = "p1 ...p i +1 ...i dx1 ^ . . . ^ dxp1 ?1 ^ ! 1 ^ dxp1+1 ^ . . . ^ dxp ?1 ^ ! ^ dxp +1 ^ . . . ^ dxn: (1:20) n
n
n
j
j
n
j j
j j
n
j
j
j j
j
j
n
j
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Since
"q1 ...g i +1 ...i "p1 ...p i +1 ...i = j !(n ? j )!qp11 . . . qp j j
n
j j
n
j j
antisym(p1 . . . pj ); (1:21)
we easily obtain the above expression (1.10). We note that this paper is not intended to compare the techniques and result of the theory of nite order variational sequences and the variational bicomplexes which are based on nite jet constructions. For the theory of variational bicomplexes the reader is referred to [1{3], [7], and [14{16], where further references can be found.
2. Finite order variational sequences In this section we recall basic notions of the theory of di erential forms on jet prolongations of a bered manifold which are necessary for next sections. (see [9], [12], [13]). We use the same notation as in Introduction. Let r 1 be an integer. To every tangent vector 2 TJ r+1 Y at a point Jxr+1 2 J r+1 Y one can assign a tangent vector h 2 TJ r Y at the point r+1;r (Jxr+1 ) = Jxr , and a tangent vector p 2 TJ r Y at Jxr by h = Tx J r Tr+1 ; p = Tr+1;r ? h: (2:1) Clearly, p is always a r -vertical vector. is r+1 -vertical if and only if h = 0; is r+1;r -vertical if and only if h = 0; p = 0. The mappings ! h; ! p are vector bundle homomorphisms of TJ r+1 Y to r TJ Y over the jet projection r+1;r . We call h the horizontalization. The horizontalization h : TJ r+1 Y ! TJ r Y induces a decomposition of the modules of di erential forms on J r Y in the following sense. Let W Y be an open set, and let 2 kr W; k 1. Let Jxr+1 2 Wr+1 be a point, 1 ; 2; . . . ; k 2 TJ r+1 Y tangent vectors at this point. Write for each i = 1; 2; . . . ; k Tr+1;r i = hi + pi ; (2:2) and denote p0 (Jxr+1 )(1; 2; . . . ; k ) = (Jxr )(h1; h2; . . . ; hk ); p1(Jxr+1 )(1; 2; . . . ; k ) = (Jxr )(p1; h2; . . . ; hk ) + (Jxr )(h1; p2; h3; . . . ; hk ) + . . . + (Jxr )(h1; . . . ; hk?1 ; pk ); ... pk (Jxr )(1; 2; . . . ; k ) = (Jxr )(p1; p2; . . . ; pk ): (2:3) pi is a k-form on Wr+1 , called the i-contact component of . Sometimes we denote k X
pi
(2:4)
and call h (resp. p) the horizontal (resp. contact) component of . Since by the de nition of the pull-back, r+1;r (Jxr+1 )(1; 2; . . . ; k ) = (Jxr )(Tr+1;r 1 ; Tr+1;r 2 ; . . . ; Tr+1;r k );
(2:5)
h = p0; p =
i=1
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we easily obtain a decomposition formula
r+1;r =
k X i=0
pi = h + p
(2:6)
is called 0-contact, or horizontal, if p = 0; is called contact, if h = 0. We say that is q-contact, if r+1;r = pq , or, which is the same, if pj = 0 for j 6= q. We say that is of order of contactness q, if h = 0; p1 = 0; . . ., pq?1 = 0. In what follows we often use multiindices I = (i1i2 . . . ip ), where 1 i1 ; i2; . . . ; ip n. For such a multiindex I we denote p = jI j, and call this number the length of I . It will be convenient to denote by Ii the multiindex (i1i2 . . . ip i). Let (V; ); = (xi ; y ), be a bered chart on Y . Denote for every multiindex I such that 0 jI j r, and every ; 1 m, !I = dyI ? yIi dxi : Then on Vr+1, with the obvious notational convention, hdxi = dxi; hdyI = yIi dxi; pdxi = 0; pdyI = !I :
(2:7) (2:8)
Obviously, the forms
dxi ; !I ; dyJ ; 0 jI j r; jJ j = r + 1; de ne a basis of linear forms on Vr+1 . We have d!I = ?!Ii ^ dxi; 0 jI j r ? 1; d!I = ?dyIi ^ dxi; jI j = r: If Jxr+1 2 Vr+1 is a point and 2 TJ r+1 Y a tangent vector at Jxr+1 , X = i @x@ i + I @y@ ; I 0jI jr+1
(2:9)
(2:10) (2:11)
then
dxi (Jxr+1 )() = i; !I (Jxr+1 )() = I ? yI i ; 0 jI j r; dyJ (Jxr+1 )() = J ; jJ j = r + 1: If f : Vr ! R is a smooth function, we have r+1;r df = hdf + pdf; i
where
@f + X @f y ; hdf = dif dxi ; dif = @x Ii i jI jr @yI pdf =
@f ! : I jI jr @yI X
(2:12) (2:13) (2:14) (2:15)
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Topics in the calculus of variations
In particular,
diyI = yIi :
(2:16)
We also use the notation
df dxi + d0f = dx i
df dy ; I dy jI jr?1 I X
(2:17)
and de ne a function d0if : Vr ! R by Then
hd0 f = d0i f dxi:
(2:18)
df dy ; I jI j=r dyI X df dif = d0if + yIi : dy jI j=r I
(2:19)
df = d0f +
X
(2:20)
Let r 1; k 0 be xed. We denote by kr the direct image of the sheaf of k-forms over J r Y by the projection r;0 : J r Y ! Y . If pi : kr ! kr+1 is considered as a sheaf morphism, we denote ker h; 1 k n; k (2:21)
r(c) = ker pk?n ; k n + 1: Then we set (2:22) d kr(c) = im dk ; where dk is the sheaf morphism de ned by the exterior derivative d of k-forms, and kr = d rk(?c)1 + kr(c) : (2:23) We get a diagram
0 0
0
-
?
1r
0
?
d- 2 r
0
?
d- 3 r
- ...
? ? ? ? - R - 0r d- 1r d- 2r d- 3r - . . . @@ ? R 1=1 E1- 2?=2 E2- 3?=3 - . . . @ r r r r r r ? ? ? 0
0
0
(2:24)
in which the sequence 0 ! R ! 0r ! 1r =1r ! 2r =2r ! 3r =3r ! . . .
(2:25)
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D. Krupka
is an acyclic resolution of the constant sheaf R over Y . This sequence is called the variational sequence of order r over Y . The sheaf morphism En (resp. En+1 ) is the Euler-Lagrange mapping (resp. the Helmholtz-Sonin mapping). If 2 nr is a germ and [] = = f0 !0 its class in nr =nr , then
En() = " ! ^ !0 ;
(2:26)
where
" =
r+1 X
0 : (?1)k dj1 . . . dj @y@f j1 ...j k=0
(2:27)
k
k
If 2 nr +1 is a germ and [] = " = " ! ^ !0 its class in nr +1 =nr +1 , where " are de ned on Vs J s Y , then s X @" ? (?1)i @" 1 En+1 (") = 2 @y @yj1 ...j j1 ...j i=1
?
i
s X
k=i+1
(?1)k
i
k d . . . d @" ! ^ ! ^ ! : 0 i j +1 j @yj1 ...j j1 ...j
i
k
i
(2:28)
k
A well-known conclusion of these results for the calculus of variations consists in a cohomology description of the di erence between the local and global triviality of global sections of nr (or nr =nr ), and between the local and global variationality of global sections of nr +1 (or nr +1 =nr +1 ) (see [1], [2], [13{16]).
3. Projectable and -decomposable forms We begin by presenting three auxiliary algebraic assertions which can be used in the proofs of our main theorems. If in the formulas below a symbol like antisym(i1i2 . . . is ) (resp. sym(j1j2 . . . jr )) is used, then the expression preceding this symbol should be considered as antisymmetrized in i1 ; i2; . . . ; is (resp. symmetrized in j1; j2; . . . ; jr ).
Lemma 1. Let r; s, and n be positive integers, and let Aj1 ...j i1...i , 1 j1; . . . ; jr, i1; . . . ; is n, be a system of real numbers symmetric in the superscripts and antisymmetric in the subscripts. Put for every p; q , 1 p; q n, r
s
Qj1 ...j p i1 ...i q = Aj1...j i1 ...i qp antisym(i1 . . . isq) sym(j1 . . . jrp); r
Bj2 ...j
r
r
s
=
i2 ...is
s
rs Apj2 ...j h+r?s
r
pi2 ...is :
(3:2)
(a) The system Aj1 ...j i1 ...i obeys the identity Qj1 ...j p i1 ...i p = (r n++1)(r s?+s 1) (Aj1 ...j i1 ...i ? B j2 ...j i2 ...i ij11 ) antisym(i1i2 . . . is ) sym(j1j2 . . . jr ): r
r
s
(3:1)
s
r
s
r
s
(3:3)
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Topics in the calculus of variations
(b) The system C j1 ...j i1 ...i de ned by the decomposition Aj1 ...j i1 ...i = B j2 ...j i2 ...i ij11 + C j1 ...j i1 ...i antisym(i1i2 . . . is ) sym(j1j2 . . . jr ) is traceless, i.e., for all j2; . . . jr ; i2 ; . . . ; is r
r
s
r
s
r
s
s
(3:4)
C pj2 ...j pi2 ...i = 0: r
(3:5)
s
(c) Equation
Qj1 ...j p i1 ...i q = 0 r
(3:6)
s
is satis ed if and only if there exists a system B j2 ...j i2 ...i symmetric in the superscripts and antisymmetric in the subscripts such that r
Aj1 ...j
r
i2 ...is
s
= B j2 ...j i2 ...i ij11 antisym(i1i2 . . . is ) sym(j1j2 . . . jr ): r
(3:7)
s
The system B j2 ...j i2 ...i is unique, and is traceless, i.e. Bpj3 ...j pi3...i = 0: r
s
r
(3:8)
s
Lemma 2. Let n; k; j be integers such that 2 k n, 1 j k ? 1. Let r1; r2; . . . ; rj be positive integers, and let I1 (resp. I2 ; . . . ; Ij ) be a multiindex of length r1 (resp. r2; . . . ; rj ) whose entries are arbitrary indices belonging to the set f1; 2; . . . ; ng. Consider the system of linear, homogeneous equations
AI1 I2 ...I i +1 i +2 ...i ip11 ip22 . . . ip = 0 antisym(i1i2 . . . ik ) sym(I1 p1) sym(I2 p2) . . . sym(Ij pj ); (3:9) where 1 p1 ; p2; . . . ; pj ; i1; i2; . . . ; ik n, and the unknown system AI1 I2 ...I i +1 i +2 ...i j
j
j
j
k
j
j
j
j
k
is antisymmetric in the subscripts, and symmetric in the superscripts entering every of the multiindices I1; I2; . . . ; Ij . Then every solution of this system is of the form
AI1 I2 ...I i +1 i +2 ...i = B Q1 I2 ...I i +2 ...i iq1+1 + B I1 Q2 I3 ...I i +2 ...i iq2+1 + B I1 ...I ?1 Q i +2 ...i iq +1 antisym(ij +1 ij +2 . . . ik ) sym(Q1 q1 ) . . . sym(Qj qj ); (3:10) where I1 = Q1 q1 ; . . ., Ij = Qj qj , and B Q1 I2 ...I i +2 ...i ; . . . ; B I1 ...I ?1 Q i +2 ...i are arj
j
j
k
j
j
k
j
j
s
j
j
k
k
j
j
j
j
j
j
j
j
k
j
k
bitrary constants antisymmetric in the subscripts, and symmetric in the superscripts entering every of the multiindices Q1; . . . ; Qj ; I1; . . . ; Ij .
Lemma 3. Let n; k; j be integers such that 1 j n ? 1; k n + 1. Let r be a positive
integer, and let I1 ; I2; . . . ; In?k+j be multiindices of length r whose entries are arbitrary indices from the set f1; 2; . . . ; ng. Consider the system of linear, homogeneous equations pk?n+1 . . . pk?n+j = 0 I ?n+1 ...Ik?n+j ij+1 ...in i1 ij antisym(i1i2 . . . in ) sym(Ik?n+1 pk?n+1 ) . . . sym(Ik?n+j pk?n+j )
AI1 ...I ? k
n k
(3:11)
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D. Krupka
for the unknowns AI1 ...I ? I ? +1 ...I ? + i +1 ...i , antisymmetric in the subscripts and in the multiindices I1; . . . ; Ik?n+j , and symmetric in the superscripts entering every of the multiindices I1 ; . . . ; Ik?n+j , where 1 p1; . . . ; pj ; i1; . . . ; in n. Every solution of this system has the form k
n k
n
k
n j
n
j
AI1 ...I ?
I ?n+1 ...Ik?n+j ij+1 ...in Q ... Q Q = C 1 k?n k?n+1 Ik?n+2 ...Ik?n+j
k
n k
pk?n pk?n+1 p ij+k?n+2 ...in ij1+1 . . . ij+k?n ij+k?n+1 + . . . + C I1 ...Ij?1 Qj Qj+1 ...Qk?n+j ij+k?n+2 ...in ipjn+1 . . . ipjk+?kn?+nj?1 ipjk+?kn?+nj+1 antisym(ij +1 . . . in ) sym(Q1p1 ) . . . sym(Qk?n+j pk?n+j );
j 2n ? k ? 1;
AI1 ...Ik?n Ik?n+1 ...Ik?n+j i
= 0; j > 2n ? k ? 1; (3:12) where I1 = Q1p1 ; . . . ; Ik?n+j = Qk?n+j pk?n+j , and C Q1 ...Q ? +1 Q ? +2 ...I ? + i + ? +2 ...i ; . . ., C I1...I ?1 Q ...Q ? + i + ? +2 ...i are arbitrary constants antisymmetric in the subscripts and symmetric in the superscripts entering every of the multiindices Q1 ; . . . ; Qk?n+j , I1 ; . . . ; Ik?n+j . k
n
k
n
k
n j
j k
j
+1 ...in
n
n
n
n
k
n j
j k
n
n
Let 2 Kr W be a form, and consider the pull-back r+1;r . We have an invariant decomposition of r+1;r into the summands pi (2.6). In the following theorem we derive some identities for the chart expressions of the coecients in pi .
Theorem 1. Let k 1, and let 2 kr W be a form. Let for every j; 0 j k; pj be expressed by
pj =
j X
1 1 ^ . . . ^ ! J1 J I +1 I ! B ... ... i ... i J J 1 +1 +1 1 s=0 s!(j ? s)!(k ? j )! ^ !I +1+1 ^ . . . ^ !I ^ dxi +1 ^ . . . ^ dxi ; j
s s s s
s
j
s
j
j j
s s
k
j
k
(3:13)
where J1 ; . . . ; Js (resp. Is+1 ; . . . ; Ij ) are multiindices of length < r (resp. = r). Denote for j < k
CJ11... J I +1+1 ...I ??11p2...p i +2 ...i I ?1 pp2 ...p = n +r(rk??kj )+ j BJ11 ...J I +1 pi +2 ...i : +1 ... ?1 I Then the coecients BJ11 ...J I +1 +1 ... +1 ...i obey the following conditions: s s s s
r
j j
j
j
k
s s s s
r
j j
s s s s
j
j
k
j
j j
k
@BJ11 ...J I +1+1 ... I = 0; 0 s k; jLj = r + 1; @yL s s s s
k k
BJ11 ... J I +1+1 ...I ??11I i +1 ...i ? CJ11...J I +1+1 ...I ??11p2 ...p i +2...i ip1+1 @BJ11... J I +1+1 ...I ??11 qi +1 ...i r + 1 antisym(ij +1 . . . ik ) = n+r?k+j @yI q sym(p1 . . . pr ); Ij = (p1 . . . pr ); 1 j k; 0 s j ? 1; s s s s
j
j
j
j j
j j
s s s s
k
r
j
j
j
s s s s
j
j
j
j
k
k
j
(3:14)
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Topics in the calculus of variations
@BJ11 ...J I +1+1 ... I ??11 i ...i (r + 1)(k ? j + 1) @BJ11... J I +1+1 ...I ??11 qi +1 ...i p ? n+r?k+j i = 0 @yI q @yI q antisym(ij . . . ik ) sym(Ij p); 1 j k; 0 s j ? 1: (3:15) j
s s s s
j
j
j
s s s s
k
j
j
j
j
j
j
k
j
The proof of Theorem 1 is based on analogous identities as (1.12), and on Lemma 1,(a). Let be a k-form on an open set W J r Y . Let 1 k n. We say that is -decomposable if its horizontal (i.e. 0-contact) component h is r+1;r -projectable. If 1 k dim J r Y we say that is -decomposable if its (k ? n)-contact component pk?n is r+1;r -projectable. If 1 k n and is -decomposable, then
= 1 + 2 ; (3:16) where 1 is a r -horizontal form, and 2 is a contact form; in fact, 1 is the r+1;r projection of h. Analogous decompositions arise for k-forms with k n + 1; in this case 1 is the (k ? n)-contact component of , and 2 is a form whose order of contactness is k ? n + 1. Our second result represents a criterion for a form to be -decomposable.
Theorem 2. Let W Y be an open set, 2 kr W a form, and let (V; ), = (xi; y ), be any bered chart such that V W . (a) Let k = 1. Then is -decomposable if and only if
= i dxi + J !J (3:17) for some functions i ; J : Vr ! R, where jJ j r ? 1. (b) Let 2 k n. Then is -decomposable if and only if (3:18) = i1 ...i dxi1 ^ . . . ^ dxi + !J ^ J + d!I ^ I ; where i1 ...i (resp. J , resp. I ) are some functions (resp. (k ? 1)-forms, resp. (k ? 2)forms) on Vr , and jJ j r ? 1; jI j = r ? 1. (c) Let n + 1 k dim J r Y . Then is -decomposable if and only if = i1 ...i ^ dxi1 ^ . . . ^ dxi + !J11 ^ . . . ^ !J ^ d!I1 ^ . . . (3:19) ^ d!I ^ J11...J I11 ...I ; p + q = k ? n + 1; k
k
k
p
n
k
p
q
p p
q
i
q q
for some (k ? n)-forms i1 ...i and (k ? p ? 2q )-forms J11 ...J I11 ... I on Vr where jJ1j; . . . ; jJpj r ? 1; jI1j; . . . ; jIqj = r ? 1. p p
n
q q
The proof of (a) is straightforward. To prove (b), suppose that 2 k n. Then has an expression
= 0 + Ai1 i2...i dxi1 ^ dxi2 ^ . . . ^ dxi + AI11 i2 ...i dyI11 ^ dxi2 ^ . . . ^ dxi + . . . + AI11 ... I ??11 i dyI11 ^ . . . ^ dyI ??11 ^ dxi + AI11 ... I dyI11 ^ . . . ^ dyI ; (3:20) k
k
k
k
k
k
k
k
k
k
k k
k k
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D. Krupka
where 0 belongs to the ideal of forms generated by the 1-forms !J , where jJ j r ? 1. Since ? h = Ai1 i2 ...i + AI11 2...i yI11i1 + . . . + AI11 ...I ??11 i yI11i1 . . . yI ??11i ?1 (3:21) + AI11 ...I yI11i1 . . . yI i dxi1 ^ . . . ^ dxi ; the form h is r+1;r -projectable if and only if k
k
k
k
k
k
k k
k
k
k
k
k k
AI11 ...I i +1 ...i ip11 . . . ip = 0 antisym(i1 . . . ij ij+1 . . . ik ) sym(I1 p1) . . .sym(Ij pj ); 1 j k: j
j
j j
j
k
(3:22)
If j = k, this implies (3:23) AI11 ...I = 0: If 1 j k ? 1, we apply Lemma 2, (3.10). To prove (c), rst consider the case k = n + 1, and suppose that we have an (n + 1)form such that p1 is r+1;r -projectable. has a unique decomposition in the form k k
= 0 + 0 + 00; (3:24) where 0 is a multiple of !0 , 0 contains at most n ? 1 factors dxi and is generated by the forms !J , jJ j r ? 1, and 00 contains at most n ? 1 factors dxi and none of the factors !J , jJ j r ? 1. We wish to show that the r+1;r -projectability of p1 implies that both 0 and 00 have the desired structure (3.19). Notice that 0 and 00 are of the form 0 = !J ^ J ; 0 jJ j r ? 1;
(3:25)
1 i2 i 1 00 = AJI 1 i2 ...i dyJ ^ dyI1 ^ dx ^ . . . ^ dx 2 1 i3 i 1 I2 + AJI 1 2 i3 ...i dyJ ^ dyI1 ^ dyI2 ^ dx ^ . . . ^ dx + . . . ?1 I ?1 1 i + AJI 11 ... ?1 i dyJ ^ dyI1 ^ . . . ^ dyI ?1 ^ dx 1 I (3:26) + AJI 11 ... dyJ ^ dyI1 ^ . . . ^ dyI ; jJ j; jI1j; . . . ; jIn j = r: We apply (b) to the form h J , and Lemma 2(a) to p100, and get the required result n
n
n
n
n
n
n
n
n
n
n n
n n
after several steps. If k n + 1, we proceed in the same way.
Remark 1. Taking i = 0; i1...i = 0, and i1...i = 0 in Theorem 2 we obtain necessary k
n
and sucient conditions for a k-form to be contact, resp. strongly contact.
4. Order of a lagrangian Usually, a lagrangian 2 nr;X W is said to be of order r, if it is not r;r?1 -projectable. In this paper we adopt a little di erent terminology. We shall say that 2 nr;X W is of order r, if there is no form 2 nr?1 W for which h = or, which is the same, if h 6= for every 2 nr?1 W . If is not of order r we call the order of the smallest integer s such that there exists a form 2 ns W for which = h.
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Topics in the calculus of variations
Clearly, if 2 nr;X W is r;s -projectable, and is not r;s?1 -projectable, then the order of is s or s ? 1; the latter case takes place if and only if = h with 2 ns?1 W ). (up to r;s For example, the scalar curvature lagrangian (the Hilbert lagrangian for the Einstein equations in vacuo), is of the 1st order (see [9]). Our problem to be considered in this section is to nd the order of a lagrangian, that is, to solve the equation
= h
(4:1)
with respect to . Let us consider the module of r-th order lagrangians nr W . Our aim will be to describe the image of nr W by the horizontalization h, i.e., the subspace h nr W
nr+1;X W . We have the canonical decomposition of h,
h
nr W
-
?n
nr+1;X W
6
r W=nr W g- h nr W
(4:2) in which the right vertical arrow represents the canonical inclusion, and g is a bijection. We wish to construct the inverse mapping = g ?1
(4:3)
in terms of coordinates. To this purpose we specify the result of Theorem 1 to the case k = n. Let us consider a form 2 nr W , and a bered chart (V; ); = (xi ; y ), where V W . has a unique decomposition
= 0 + 0 ;
(4:4)
where 0 is a contact form, i.e., is generated by !J ; jJ j r ? 1 and d!I ; jI j = r ? 1 (see Sec. 3, Remark 1), and
0 =
n X
1 AI1 I 1 i i 1 ... i +1 ...i dyI1 ^ . . . ^ dyI ^ dx +1 ^ . . . ^ dx ; (4:5) p !( n ? p )! p=0 p
p
p p
n
n
p
p
where jI1j; . . . ; jIpj = r and all the coecients AI11 ;i2 ... i ; . . . ; AI11 ... I ??11 i are completely traceless; in particular, 0 does not contain any non-zero term of the form d!J ^ J with jJ j = r ? 1. This decomposition may be referred to as the canonical decomposition of relative to the bered chart (V; ). Clearly, 0 may be viewed as a representative of the class [] 2 nr W=nr W in the chart (V; ). We have n
n
h0 =
n X
1
n
n
AI11 ... I i +1 ...i yI11i1 . . . yI i dxi1 ^ . . . ^ dxi : p !( n ? p )! p=0 p
p p
p
n
p p
n
(4:6)
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D. Krupka
If we write h0 in the form h0 = Bdx1 ^ . . . ^ dxn , or (4:7) h0 = n1! Bi1 ...i dxr1 ^ . . . ^ dxi ; it is clear that one can reconstruct 0 from B , or Bi1 ...i , by a di erentiation process (the same method has been used in [8], [12], [11]). Consider the form (4.5) and write n
n
n
r+1;r 0 =
n X
1
1 ^ . . . ^ ! ^ dxi +1 ^ . . . ^ dxi : (4:8) I1 I B ! ... i ... i I I 1 +1 1 p=0 p!(n ? p)! p
p
p p
n
p
n
p
This form has a unique decomposition
r+1;r 0 = 00 + 00;
(4:9)
where 00 is of the form d!J ^ J ; jJ j = r ? 1,
00 =
n X
1 C I1 I 1 i +1 ^ . . . ^ dxi ; (4:10) 0 1 ... i +1 ...i !I1 ^ . . . ^ !I ^ dx p !( n ? p ) p=0 p
p
p p
n
p
n
p
and the coecients CI11 i2 ...i ; CI11... I ??11 i are completely traceless. If : Vr ! Vr+1 is the section of J r+1 Y described by the equations n
n
n
n
yI = 0; jI j = r + 1;
(4:11)
then Theorem 1, (3.15) implies, since the coecients in (4.6) are supposed to be traceless,
00 = 0
(4:12)
0 = 00:
(4:13)
hence On the other hand, we have the following result.
Lemma 4. If 0 is expressed by (4.5) and 00 by (4.10), we have CI11 i2 ...i = @C@yqi2...1 i ; I1 q + 1 @CI11 qi3 ...i ; CI11I22 i3 ...i = rr + 2 @yI22q n
n
n
n
...
C I11...I ??22 q CI11 ... I ??11 i = r +r +n ?1 1 @y ?1 ; I ?1 q CI11 ... I ??11 qi r + 1 I I 1 C1 ... = r + n @y ; I q n n
n
n
in
n
n
n
n n
n n
n n
n
(4:14)
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Topics in the calculus of variations
and
CI11 ... I = 0; jLj = r + 1; @yL CI11 ...I ??11 +1 ... CI11 ... I ??11 ... ( r + 1)( n ? j + 1) ? I q = 0; r+1 @yI p @yI q 1 j n; antisym(i1i2 . . . in ) sym(Ij p): n n
j j
ij
j j
in
qij
j
j
j
j
in
j
j
Formulas (4.14) and (4.15) follow from (3.15). In particular, 00 is completely determined by h00 = h0 = h, where h = n1! Ci1 ...i dxi1 ^ . . . ^ dxi : Using (4.14) we get for every p; 1 p n, n
n
p @ pC CI11 ... I i +1 ...i = (r + p) . (.r.(+r +1) 2)(r + 1) 1q1...q i +1...i : @yI1 q1 . . . @yI q p p
p
p p
n
p
n
p
p
(4:15)
(4:16)
(4:17)
Now we shall extend this formula to arbitrary lagrangians belonging to nr+1;X W . Let 2 nr+1;X W be a lagrangian. Write in a bered chart (V; ); = (xi; y ),
= Ldx1 ^ . . . ^ dxn = n1! L"i1 ...i dxi1 ^ . . . ^ dxi ; Li1 ...i = L"i1 ...i ; n
n
n
(4:18)
n
and put (V; )() = where
n X
1
I1 Ip
C1 ... p=0 p!(n ? p)!
p i i i +1 ...in !I11 ^ . . . ^ !Ip ^ dx p+1 ^ . . . ^ dx n ;
p p
p @ pL CI11 ... I i +1 ...i = (r + p) . (.r.(+r +1) 2)(r + 1) 1q1...q i +1...i ; @yI1 q1 . . . @yI q jI1j; . . . ; jIpj = r: p p p
p p
n
(4:19)
n
p
p p
(4:20)
Theorem 3. Let 2 nr+1;X W . The following two conditions are equivalent: (1) There exists a form 2 nr W such that = h. (2) For every bered chart (V; ); projectable.
= (xi ; y ), the form (V; )() is r+1;r -
Suppose that (1) is satis ed. Then we have a decomposition
= 0 + (V; )(h) = 0 + (V; )() in which 0 is a contact form, and both 0, (V; )() are de ned on Vr W .
(4:21)
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D. Krupka
Conversely, suppose that (2) holds. Is (V ; ); 2 I , is a system of bered charts such that V W for every 2 I and [V = W , and if ( ) is a partition of unity subordinate to the covering (V ) of W , we put
=
X
(V ; ) ():
(4:22)
Then h = . Recall that a lagrangian on a bered manifold : Y ! X is called variationally trivial, if its Euler-Lagrange form E = En () vanishes, i.e., E = 0. Our goal now is to discuss the local and global structure of variationally trivial lagrangians.
Theorem 4. Let 2 nr;X Y be a variationally trivial lagrangian on a bered manifold : Y ! X , n = dim X . Then to each point Jxr 2 J r Y there exist a neighborhood V of (x) 2 Y and an n-form V de ned on Vr?1 J r?1 Y such that (1) = hV on Vr ,
and (2) dV = 0.
To prove Theorem 4 we may suppose that V = U W , where U Rn is an open set and W Rm is an open ball with centre 0. Let xi ; y ; 1 i n; 1 m, be the canonical coordinate on V . We de ne a mapping : R Vr ! Vr by the formula
(s; (xi; y ; yj1 ; . . . ; yj1 ...j )) = (xi; y ; yj1 ; . . . ; yj1 ...j ?1 ; syj1 ...j ): r
r
(4:23)
r
Let be any (n ? 1)-form on Vr such that hd is r+1;r -projectable. Write
= 0 + 1 ;
(4:24)
where 0 is generated by the forms ! ; !j1 ; . . . ; !j1 ...j ?1 and r
1 = Bi2 ...i dxi2 ^ . . . ^ dxi + BI22 i3 ...i dyI22 ^ dxi3 ^ . . . ^ dxi + BI22I33 i4 ...i dyI22 ^ dyI33 ^ dxi4 ^ . . . ^ dxi + . . . + BI22 ...I ??11 i dyI22 ^ . . . ^ dyI ??11 ^ dxi + BI22 ... I dyI22 ^ . . . ^ dyI : (4:25) n
n
n
n
n
n
n
n
n
n
n n
n n
n
n
Clearly,
hd = hd1: In particular, hd1 is r+1;r -projectable. By a direct computation hd1 = d0i1 Bi2 ...i + @B@yi2...1i ? d0i1 BI11 i3 ...i I1 @B I2 2 i3 ...i + d0 B I1 I2 1 2 + @y 1 i3 1 2 i4 ...i yI1 i1 yI2 i2 + . . . I1 @B I2 I ?1 2 ... ?1 i n?1 d0 B I1 I ?1 y 1 . . . y ?1 + ( ? 1) + i 1 ... ?1 I ?1 i ?1 I1 i1 @yI11 @BI22 ...I 1 + @y 1 yI1 i1 . . . yI i dxi1 ^ . . . ^ dxi : I1
(4:26)
n
n
n
n
n
n
n
n
n
n n
n n n
n
n n
n
n
n
(4:27)
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Topics in the calculus of variations
This formula enables us to express the conditions of r+1;r -projectability of hd1 explicitly. Put
1 = CI33 ...I dyI33 ^ . . . ^ dyI ; where
CI3j ...Inn
= yJ
(4:28)
n n
n n
Z
JI3 I sn?2 ds B 3 ...
(4:29)
n n
(integration from 0 to 1). Write d 1 in the form
d 1 = d0CI33 ...I ^ dyI33 ^ . . . ^ dyI @C I3 I + @y3...2 dyI22 ^ dyI33 ^ . . . ^ dyI ; I2 n n
n n
n n
where
(4:30)
n n
Z @CI33 ...I 2 = dy ^ . . . ^ dy BI22 ...I sn?2 ds I @yI22 I2 Z JI3 I @B 3 ... sn?1 ds dy 2 ^ . . . ^ dy + yJ I I2 @yI22 Z Z JI3 I JI2 I4 I @B 1 I I n ? 2 3 ... ? @B2 4 ... ? . . . 2 = B2 ... s ds + n?1 @yI22 @yI33 JI3 I ?1 I2 3 ... ?1 2 y sn?2 ds dy 2 ^ . . . ^ dy : (4:31) ? @B@y J I I2 I n n
n n
n n
n n
n n
n n
n n
n n
n n
n n
n n
By (4.27) JI3 I JI2 I4 I JI3 I ?1 I2 @BI22I33... I @B 3 ... ? @B2 4 ... ? ? @B3 ... ?1 2 ; = @yJ @yI22 @yI33 @yI n n
n n
n n
n n
n n
(4:32)
from which it follows that
@CI33 ...I dy2 ^ . . . ^ dy = Z BI2 I sn?2 ds 2 ... I @yI22 I2 Z @BI22 ...I y sn?2 dsdy2 ^ . . . ^ dy 1 + n?1 I I2 @yI22 J Z d . . .I2 I 1 sn?1 ds dy 2 ^ . . . ^ dy = ds 2 I I2 n?1 = BI22 ... I dyI22 ^ . . . ^ dyI : n n
n n
n n
n n
n n
n n
n n
n n
n n
(4:33)
Consequently ? 1 = 2 + d 1 ? p d0CI33 ... I ^ dyI33 ^ . . . ^ dyI ; n n
n n
(4:34)
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D. Krupka
where
2 = Bi2 ...i dxi2 ^ . . . ^ dxi + BI22 i3 ...i dyI22 ^ dxi3 ^ . . . ^ dxi + BI22I33 i4 ...i dyI22 ^ dyI33 ^ dxi4 ^ . . . ^ dxi + . . . ? + BI22 ... I ??11 i ? (?1)n?2 d0i CI22 ... I ??11 i dyI22 ^ . . . ^ dyI ??11 ^ dxi : (4:35) n
n
n
n
n
n
n
n
n
n
n
n
n
n
n
n
Clearly
hd1 = hd2;
(4:36)
Now we can use (4.27) again replacing 1 by 2. After several steps we conclude that
is expressible in the form = + d' + ; where is r -horizontal,
(4:37)
= Bi2 ...i dxi2 ^ . . . ^ dxi ; (4:38) and is contact; the form hd = hd is r+1;r -projectable which means that @ Bi2 ...i = r(n ? 1) j1 @ Bqi3 ...i @yI11 r + 1 i2 @yqj 2 ...j antisym(i2i3 . . . in ) sym(j1j2 . . . jr ); I1 = (j1j2 . . . jr ): (4:39) Let U Rn be an open set, W Rm an open ball with centre 0, and denote by (xi ; y ) the canonical coordinates on U W . Put for every (s; (xi; y )) 2 R (U W ) n
n
n
n
r
(s; (xi; y )) = (xi ; sy ): (4:40) is a mapping from R (U W ) into U W . Let k 1, and let be a k-form on U W . There exists a unique decomposition. = ds ^ 0(s) + 0 (s); where the form 0(s); 0(s) do not contain ds. We set Z
I = 0(s);
(4:41) (4:42)
where the right-hand side means that we integrate the coecients in the form 0(s) over s from 0 to 1. Let : U W ! U be the canonical projection, and let : U ! U W be the zero section. Then it is standart to show that
= Id + dI + :
(4:43)
In what follows we apply this formula to the case of the jet projection r;r?1 = Vr ! Vr?1 , where V = U W . Let be a variationally trivial lagrangian on Vr . Using the variational sequence (2.25) we get r+1;r = En?1 h = hd , where is an (n ? 1)-form on Vr . Since hd is
Topics in the calculus of variations
491
r+1;r -projectable we may suppose, by the rst part of the proof, that is r -horizontal. Put
= d; = Id;
(4:44) where I is de ned by (4.42) (with respect by the projection r;r?1 : Vr ! Vr?1 . Is : Vr?1 ! Vr ). By (4.43) : = d + r;r (4:45) ?1 Clearly,
d = d = 0;
(4:46)
and it is easily veri ed with the help of (4.41) and (4.42) that is r -horizontal. Let U be any compact, n-dimensional submanifold with boundary @ . Let us consider a section w : Vr?1 ! Vr satisfying
supp w Vr?1 \ r??11 ( ): If : U ! V is a section and x 2 @ , then (w J r?1 )(x) = (w J r?1 )(x):
(4:47) (4:48)
Now we integrate both sides of the equality
w = dw +
(4:49)
arising from (4.45). We obtain by (4.48) and (4.46) Z
so that
Z
Z
Z
(w J r1 ) @
@
Z Z Z r ? 1 r 1 = ( J ) = J i = J r?1 di = 0
@
@
J r?1 dw =
J r?1 w =
J r?1 (w ? ) = 0:
(4:50) (4:51)
Since this condition holds for every we get on
hw = hi
(4:52)
and (4.49) gives
hdw = 0:
(4:53)
Writing
= Bi2 ...i dxi2 ^ . . . ^ dxi
(4:54)
n
n
we get
hdw = di1 (Bi2 ...i w)dxi1 ^ . . . ^ dxi = 0; n
n
(4:55)
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D. Krupka
i.e.,
@Bi2 ...i w + @Bi2 ...i w y + . . . + @Bi2 ...i w y i1 @xi1 @y @yq1 ...q ?1 i1 q1 ...q ?1 i1 i2 ...i w @yj1 ...j w = @
[email protected] i w + @B @yj1 ...j @xi1 i2 ...i w @yj1 ...j w y + . . . + @B@yi2 ... i w + @B i1 @yj1...j @y @B i2 ...i w + @Bi2 ...i w @yj1 ...j w y + @y @yj1 ...j @yq1...q 1 q1 ...q ?1 i1 q1 ...q ?1 = @
[email protected] i w + @B@yi2 ... i w yi1 + . . . + @B i2 ...i w yq1 ...q ?1 i1 + @Bi2 ...i w di1 (yj1 ...j w) @y @y n
n
n
r
r
n
n
r
r
r
n
n
r
r
n
r
n
n
r
r
r
n
n
n
q1 ...qr?1
r
j1 ...jr
r
= 0: Since the functions yj1 ...j w are arbitrary, we have
(4:56)
r
@Bi2 ...i w = 0; @yj1 ...j n
(4:57)
r
and
@Bi2 ...i = 0: (4:58) @yj1 ...j In particular, is de ned on Vr?1 , and so must be (4.43) as required by Theorem 4. n
r
Using the variational sequence it is easy to obtain the following slightly di erent version of Theorem 4 re ning the results known so far on the kernel of the higher order Euler-Lagrange mapping (see [1]).
Theorem 5. Let 2 nr;X Y be a lagrangian on a bered manifold : Y ! X .
is variationally trivial if and only if there exists an n-form on J r?1 Y such that (1) = h, (2) d 2 nr?+11 Y .
Only necessity of conditions (1), (2) needs proof. We have an exact sequence 0 ! nr?1 ! nr?1 ! nr?1 =nr?1 ! 0
(4:59)
in which nr?1 ; nr?1 are soft sheaves. Hence nr?1 =nr?1 is also soft, and the induced sequence of global sections must be exact. But by Theorem 4 a variationally trivial lagrangian 2 nr;X Y belongs to nr?1 =nr?1 . This ensures the existence of a form on J r?1 Y such that h = . Then obviously d 2 nr?+11 .
Corollary 1. A lagrangian 2 n1;X Y is variationally trivial if and only if there exists an n-form on Y such that (1) = h, (2) d = 0.
By Theorem 4, if En () = 0, then = h for an n-form on J 0 Y = Y such that d 2 n0 +1 . But n0 +1 = f0g.
Topics in the calculus of variations
493
Remark 2. Condition (1) of Theorem 5 completely describes the polynomial structure
of variationally trivial lagrangians in the variables yj1 ...j ; the polynomial structure is therefore de ned by the horizontalization h. Condition (2) of Theorem 4 then determines relations among the coecients of the polynomials. r
In the following theorem we summarize our result on the structure of variationally trivial lagrangians; in its second part we use a global hypothesis and derive global results, based on the variational sequence (2.25). We denote by H n Y the n-th cohomology group of Y .
Theorem 6. Let 2 nr;X Y be a lagrangian, n = dim X . is variationally trivial if and only if to each point Jxr 2 J r Y there exists a neighborhood V of (x) 2 Y and an (n ? 1)-form V de ned on Vr?1 J r?1 Y such that on Vr , = hdV :
(4:60)
(b) Suppose that is variationally trivial and moreover, H n Y = f0g. Then there exists an (n ? 1)-form de ned on J r?1 Y such that
= hd:
(4:61)
To prove (a) we use the identity En?1 hV = hdV , where = En?1 hV . Suppose that is variationally trivial and H n Y = f0g. Then = En?1 , where is a global section of nr??11 =nr??11 . Since the induced sequence of global section associated to the exact sequence 0 ! nr?1 ! nr?1 ! nr?1 =nr?1 ! 0
(4:62)
is exact, = h for some global section of nr?1 . Then = En?1 h = hd . If a lagrangian is variationally trivial and there exists a form such that = hd , we say that is globally variationally trivial. The condition H n Y = f0g guarantees that every variationally trivial lagrangian 2 nr;X Y is globally variational, and is of the form = hd , where 2 nr?1Y . As mentioned in Introduction, a complete characterization of the so called generalized invariant transformations of a lagrangian requires the knowledge of the kernel of the Euler-Lagrange mapping. The de nition of a generalized invariant transformation of , saying that the transformed lagrangian should belong to the kernel of the Euler-Lagrange mapping, can then e ectively be applied. Having now a proper theorem on the structure of the variationally trivial lagrangians (Theorem 6), we are in a position to give a precise description of generators of the generalized invariants transformations, quite in the fashion of the rst order case [10]. Recall that a -projectable vector eld on Y is said to generate generalized invariant transformations of a lagrangian 2 nr;X Y , if its one-parameter group preserves the Euler-Lagrange form E = En () i.e.,
@J 2 E = 0; r
(4:63)
where J 2r is the 2r-jet prolongation of . The following result gives a complete description of the generators of generalized invariants transformations.
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D. Krupka
Theorem 7. Let 2 nr;X Y be a lagrangian, and let be a -projectable vector eld.
The following three conditions are equivalent: (1) generates generalized invariants transformations of . (2) There exists an n-form on J r?1 Y such that
@J = h; d 2 nr?+11 Y: (4:64) (3) For every bered chart (V; ) on Y; d(V; )@J is r;r?1 -projectable, and r
r
d(V; )@J 2 nr?+11 V:
(4:65)
r
Suppose that generates generalized invariants transformations of . Then @J 2 E = E@ 2 = 0; (4:66) r
J r
and we apply Theorem 6. Suppose that (2) is satis ed, and choose a bered chart (V; ) on Y . Write
= 0 + (V; )h = 0 + (V; )@J ;
(4:67)
r
where 0 is contact. Since by hypothesis, d 2 nr?+11 Y , and obviously d0 2 nr?+11 V , we have d(V; )@J 2 nr?+11 V (4:68) as required. Suppose that (3) holds, and consider the lagrangian r
h(V; )@J = @J : r
(4:69)
r
The Euler-Lagrange form En (@J ) is the class of the form d(V; )@J . By hypothesis this class is equal to 0. Equation (4.65) replaces the generalized Noether-Bessel-Hagen equation (1.6) on a higher order jet space J r Y . If r = 1 it reduces to (1.6). r
r
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Topics in the calculus of variations
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[9] D. Krupka, Geometry of Lagrangean structures 3, Suppl. ai Rend. del Circ. Mat. di Palermo, Serie II, No. 14 (1987) 187{224. [10] D. Krupka, On generalized invariant transformations, Rep. Math. Phys. 5 (1974) 355{360. [11] D. Krupka, On the structure of the Euler mapping, Arch. Math. (Brno) 10 (1974) 55{62. [12] D. Krupka, Some geometric aspects of the calculus variations in bered manifolds, Folia Fac. Sci. Nat. UJEP Brunensis, Brno University (Czechoslovakia), 14, 1973. [13] D. Krupka, Variational sequences on nite order jet spaces, Di . Geom. and Appl., Proc. Conf., Aug. 27{Sept. 2, 1989, Brno, Czechoslovakia, World Scienti c, Singapore, 1990, 236{254. [14] F. Takens, A global version of the inverse problem of the calculus of variations, J. Di erential Geometry 14 (1979) 543{562. [15] W. M. Tulczyjew, The Euler-Lagrange resolution, Internat. Coll. on Di . Geom. Methods in Math. Physics, Aix-en-Provence, Sept. 1979; Lecture Notes in Math. 836, Springer, Berlin, 1980, 22{48. [16] A. M. Vinogradov, The C -spectral sequence, Lagrangian formalism, and conservation laws. I. The linear theory, II. The nonlinear theory, Math. Anal. Appl. 100 (1984) 1{40, 41{129. D. Krupka Department of Mathematics Silesian University Opava Bezrucovo nam. 17 746 01 Opava Czech Republic Received 28 October 1992
