T pT r’2 M , . . . , T · · · T pM (r ’ 1 times) over T r’1 M .

De¬nition. A sector r-form on M is a map σ : T r M ’ R linear with respect

to all r vector bundle structures on T r M over T r’1 M .

A sector r-form on M at a point x is the restriction of a sector r-form an

M to the ¬ber (T r M )x . Denote by T— M ’ M the ¬ber bundle of all sector

r

r-forms at the individual points on M , so that a sector r-form on M is a section

r r

of T— M . Obviously, T— M ’ M has a vector bundle structure induced by the

linear combinations of R-valued maps. If f : M ’ N is a local di¬eomorphism

and A : (T r M )x ’ R is an element of (T— M )x , we de¬ne (T— f )(A) = A —¦

r r

(T r f ’1 )f (x) : (T r N )f (x) ’ R, where f ’1 is constructed locally. Since T r f is a

r

linear morphism for all r vector bundle structures, (T— f )(A) is an element of

r r

(T— N )f (x) . Hence T— is a natural bundle. In particular, for every vector ¬eld X

on M and every sector r-form σ on M we have de¬ned the Lie derivative

r

LX σ = LT—r X σ : M ’ T— M.

For every sector r-form σ : T r ’ R we can construct its tangent map T σ : T T r M

’ T R = R — R, which is of the form (σ, δσ). Since the tangent functor preserves

vector bundle structures,

δσ : T r+1 M ’ R

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49. Lie derivatives of morphisms of ¬bered manifolds 387

is linear with respect to all r + 1 vector bundle structures on T r+1 M over T r M ,

so that this is a sector (r + 1)-form on M .

48.12. De¬nition. The operator δ : C ∞ T— M ’ C ∞ T— M will be called the

r r+1

di¬erential of sector forms.

By de¬nition, δ is a natural operator. Obviously, δ is a linear operator as

well. Applying proposition 48.1, we obtain

48.13. Corollary. δ commutes with the Lie di¬erentiation, i.e.

δ(LX σ) = LX (δσ)

for every sector r-form σ and every vector ¬eld X.

49. Lie derivatives of morphisms of ¬bered manifolds

We are going to show a deeper application of the geometrical approach to

Lie di¬erentiation in the higher order variational calculus in ¬bered manifolds.

For the sake of simplicity we restrict ourselves to the geometrical aspects of the

problem.

49.1. By an r-th order Lagrangian on a ¬bered manifold p : Y ’ M we mean

a base-preserving morphism

» : J r Y ’ Λm T — M, m = dim M.

For every section s : M ’ Y , we obtain the induced m-form » —¦ j r s on M .

We underline that from the geometrical point of view the Lagrangian is not a

function on J r Y , since m-forms (and not functions) are the proper geometric

objects for integration on X. If xi , y p are local ¬ber coordinates on Y , the in-

duced coordinates on J r Y are xi , y± for all multi indices |±| ¤ r. The coordinate

p

expression of » is

L(xi , y± )dxi § · · · § dxm

p

but such a decomposition of » into a function on J r Y and a volume element on

M has no geometric meaning.

If · is a projectable vector ¬eld on Y over ξ on M , we can construct, similarly

to 47.8.(2), the Lie derivative L· » of » with respect to ·

L· » := L(J r ·,Λm T — ξ) » : J r Y ’ Λm T — M

which coincides with the classical variation of » with respect to ·.

49.2. The geometrical form of the Euler equations for the extremals of » is

the so-called Euler morphism E(») : J 2r Y ’ V — Y — Λm T — M . Its geometric

de¬nition is based on a suitable decomposition of L· ». Here it is useful to

introduce an appropriate geometric operation.

Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993

388 Chapter XI. General theory of Lie derivatives

De¬nition. Given a base-preserving morphism • : J q Y ’ Λk T — M , its formal

exterior di¬erential D• : J q+1 Y ’ Λk+1 T — M is de¬ned by

D•(jx s) = d(• —¦ j q s)(x)

q+1

for every local section s of Y , where d means the exterior di¬erential at x ∈ M

of the local exterior k-form • —¦ j q s on M .

If f : J q Y ’ R is a function, we have a coordinate decomposition

Df = (Di f )dxi

‚f p

‚f

where Di f = ‚xi + |±|¤q ‚y± y±+i : J q+1 Y ’ R is the so called formal (or total)

p

derivative of f , provided ±+i means the multi index arising from ± by increasing

its i-th component by 1. If the coordinate expression of • is ai1 ...ik dxi1 §· · ·§dxik ,

then

D• = Di ai1 ...ik dxi § dxi1 § · · · § dxik .

To determine the Euler morphism, it su¬ces to discuss the variation L· » with

‚

respect to the vertical vector ¬elds. If · p (x, y) ‚yp is the coordinate expression

of such a vector ¬eld, then the coordinate expression of J r · is

‚

(D± · p ) p

‚y±

|±|¤r

where D± means the iterated formal derivative with respect to the multi index

±. In the following assertion we do not indicate explicitly the pullback of L· »

to J 2r Y .

49.3. Proposition. For every r-th order Lagrangian » : J r Y ’ Λm T — M ,

there exists a morphism K(») : J 2r’1 Y ’ V — J r’1 Y — Λm’1 T — M and a unique

morphism E(») : J 2r Y ’ V — Y — Λm T — M satisfying

J r’1 ·, K(»)

L· » = D

(1) + ·, E(»)

for every vertical vector ¬eld · on Y .

Proof. Write ω = dx1 § · · · § dxm , ωi = i ±i p

kp dy± — ωi ,

ω, K(») =

‚ |±|¤r’1

‚xi

E(») = Ep dy p — ω. Since L· » = T » —¦ J r ·, the coordinate expression of L· » is

‚L p

(2) p D± · .

‚y±

|±|¤r

Comparing the coe¬cients of the individual expressions D± · p in (1), we ¬nd the

following relations

Lj1 ...jr = Kp 1 ...jr )

(j

p

.

.

.

Lj1 ...jq = Di Kp1 ...jq i + Kp 1 ...jq )

j (j

p

(3)

.

.

.

Lj = Di Kp + Kp

ji i

p

i

Lp = Di Kp + Ep

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49. Lie derivatives of morphisms of ¬bered manifolds 389

j ...j j ...j i

where Lp1 q = ±! ‚y± and Kp1 q = ±! kp , provided ± is the multi index

‚L ±i

p

q! q!

corresponding to j1 . . . jq , |±| = q. Evaluating Ep by a backward procedure, we

¬nd

‚L

(’1)|±| D±

(4) Ep = p

‚y±

|±|¤r

for any K™s, so that the Euler morphism is uniquely determined. The quantities

j ...j i

Kp1 q , which are not symmetric in the last two superscripts, are not uniquely

determined by virtue of the symmetrizations in (3). Nevertheless, the global

existence of a K(») can be deduced by a recurrent construction of some sections

of certain a¬ne bundles. This procedure is straightforward, but rather technical.

The reader is referred to [Kol´ˇ, 84b]

ar

We remark that one can prove easily by proposition 49.3 that a section s of

Y is an extremal of » if and only if E(») —¦ j 2r s = 0.

49.4. The construction of the Euler morphism can be viewed as an operator

transforming every base-preserving morphism » : J r Y ’ Λm T — M into a base-

preserving morphism E(») : J 2r Y ’ V — Y — Λm T — M . Analogously to L· », the

Lie derivative of E(») with respect to a projectable vector ¬eld · on Y over ξ

on M is de¬ned by

L· E(») := L(J 2r ·,V — ·—Λm T — ξ) E(»).

An important question is whether the Euler operator commutes with Lie

di¬erentiation. From the uniqueness assertion in proposition 49.3 it follows that

E is a natural operator and from 49.3.(4) we see that E is a linear operator.

49.5. We ¬rst deduce a general result of such a type. Consider two natural

bundles over m-manifolds F and H, a natural surjective submersion q : H ’ F

and two natural vector bundles over m-manifolds G and K.

Proposition. Every linear natural operator A : (F, G) (H, K) satis¬es

Lξ (Af ) = A(Lξ f )

for every base-preserving morphism f : F M ’ GM and every vector ¬eld ξ on

M.

Proof. By 47.8.(2) and an analogy of 47.4.(1), we have

G(Flξ ) —¦ f —¦ F (Flξ ) ’ f .

1

Lξ f = lim ’t t

t’0 t

Since A commutes with limits by 19.9, we obtain by linearity and naturality

K(Flξ ) —¦ Af —¦ H(Flξ ) ’ Af = Lξ (Af )

1

ALξ (f ) = lim ’t t

t’0 t

Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993

390 Chapter XI. General theory of Lie derivatives

49.6. Our original problem on the Euler morphism can be discussed in the same

way as in the proof of proposition 49.5, but the functors in question are de¬ned

on the local isomorphisms of ¬bered manifolds. Hence the answer to our problem

is a¬rmative.

Proposition. It holds

L· E(») = E(L· »)

for every r-th order Lagrangian » and every projectable vector ¬eld · on Y .

49.7. A projectable vector ¬eld · on Y is said to be a generalized in¬nitesimal

automorphism of an r-th order Lagrangian », if L· E(») = 0. By proposition

49.6 we obtain immediately the following interesting assertion.

Corollary. Higher order Noether-Bessel-Hagen theorem. A projectable

vector ¬eld · is a generalized in¬nitesimal automorphism of an r-th order La-

grangian » if and only if E(L· ») = 0.

49.8. An in¬nitesimal automorphism of » means a projectable vector ¬eld ·

satisfying L· » = 0. In particular, corollary 49.7 and 49.3.(4) imply that every

in¬nitesimal automorphism is a generalized in¬nitesimal automorphism.

50. The general bracket formula

50.1. The generalized Lie derivative of a section s of an arbitrary ¬bered man-

ifold Y ’ M with respect to a projectable vector ¬eld · on Y over ξ on M is

¯

˜

a section L· s : M ’ V Y . If · is another projectable vector ¬eld on Y over ξ

¯

on M , a general problem is whether there exists a reasonable formula for the

˜·

generalized Lie derivative L[·,¯] s of s with respect to the bracket [·, · ]. Since

¯

L· s is not a section of Y , we cannot construct the generalized Lie derivative of

˜

L· s with respect to · . However, in the case of a vector bundle E ’ M we have

¯

de¬ned L· L· s : M ’ E.

¯

Proposition. If · and · are two linear vector ¬elds on a vector bundle E ’ M ,

¯

then

L[·,¯] s = L· L· s ’ L· L· s

(1) ¯ ¯

·

for every section s of E.

At this moment, the reader can prove this by direct evaluation using 47.10.(2).

But we shall give a conceptual proof resulting from more general considerations

in 50.5. By direct evaluation, the reader can also verify that the above proposi-

tion does not hold for arbitrary projectable vector ¬elds · and · on E. However,

¯

if F M is a natural vector bundle, then Fξ is a linear vector ¬eld on F M for

every vector ¬eld ξ on M , so that we have

Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993

50. The general bracket formula 391

Corollary. If F M is a natural vector bundle, then

L[ξ,ξ] s = Lξ Lξ s ’ Lξ Lξ s

¯ ¯ ¯

¯

for every section s of F M and every vector ¬elds ξ, ξ on M .

This result covers the classical cases of Lie di¬erentiation.

50.2. We are going to discuss the most general situation. Let M , N be two

¯

manifolds, f : M ’ N be a map, ξ, ξ be two vector ¬elds on M and ·, · be two

¯

vector ¬elds on N . Our problem is to ¬nd a reasonable expression for

˜¯·

L([ξ,ξ],[·,¯]) f : M ’ T N.

(1)

˜

Since L(ξ,·) f is a map of M into T N , we cannot construct its Lie derivative

¯¯

with respect to the pair (ξ, · ), since · is a vector ¬eld on N and not on T N .

¯

However, if we replace · by its ¬‚ow prolongation T · , we have de¬ned

¯ ¯

˜¯ ¯ ˜

L(ξ,T ·) L(ξ,·) f : M ’ T T N.

(2)

On the other hand, we can construct

˜ ˜ ¯·

L(ξ,T ·) L(ξ,¯) f : M ’ T T N.

(3)

Now we need an operation transforming certain special pairs of the elements

of the second tangent bundle T T Q of any manifold Q into the elements of T Q.

Consider A,B ∈ T Tz Q satisfying

(4) πT Q (A) = T πQ (B) and T πQ (A) = πT Q (B).

Since the canonical involution κ : T T Q ’ T T Q exchanges both projections, we

have πT Q (A) = πT Q (κB), T πQ (A) = T πQ (κB). Hence A and κB are in the

same ¬ber of T T Q with respect to projection πT Q and their di¬erence A ’ κB

satis¬es T πQ (A ’ κB) = 0. This implies that A ’ κB is a tangent vector to the

¬ber Tz Q of T Q and such a vector can be identi¬ed with an element of Tz Q,

which will be denoted by A · B.

50.3. De¬nition. A · B ∈ T Q is called the strong di¬erence of A, B ∈ T T Q

satisfying 50.2.(4).

In the case Q = Rm we have T T Rm = Rm — Rm — Rm — Rm . If A =

(x, a, b, c) ∈ T T Rm , then B satisfying 50.2.(4) is of the form B = (x, b, a, d) and

one ¬nds easily

A · B = (x, c ’ d)

(1)

From the geometrical de¬nition of the strong di¬erence it follows directly

Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993

392 Chapter XI. General theory of Lie derivatives

Lemma. If A, B ∈ T T Q satisfy 50.2.(4) and f : Q ’ P is any map, then

T T f (A), T T f (B) ∈ T T P satisfy the condition of the same type and it holds

T T f (A) · T T f (B) = T f (A · B) ∈ T P.

50.4. We are going to deduce the bracket formula for generalized Lie derivatives.

First we recall that lemma 6.13 reads

¯ ¯ ¯

[ζ, ζ] = T ζ —¦ ζ · T ζ —¦ ζ

(1)

for every two vector ¬elds on the same manifold.

The maps 50.2.(2) and 50.2.(3) satisfy the condition for the existence of

˜¯ ¯ ˜ ˜

the strong di¬erence. Indeed, we have πT N —¦ L(ξ,T ·) L(ξ,·) f = L(ξ,·) f since

˜ ˜

any generalized Lie derivative of L(ξ,·) f is a vector ¬eld along L(ξ,·) f . On

¯

the other hand, T πN —¦ (L(ξ,T ·) L(ξ,·) f ) = T πN ‚ T (Fl· ) —¦ L(ξ,·) f —¦ Flξ =

¯

˜¯ ¯ ˜ ˜

’t

t

‚t 0

¯

(Fl·

¯

Flξ ) ˜¯¯

‚

—¦f —¦ = L(ξ, · )f .

’t t

‚t 0

Proposition. It holds

˜¯· ˜ ˜ ¯· ˜¯ ¯ ˜

L([ξ,ξ],[·,¯]) f = L(ξ,T ·) L(ξ,¯) f · L(ξ,T ·) L(ξ,·) f

(2)

Proof. We ¬rst recall that the ¬‚ow prolongation of · satis¬es T · = κ —¦ T ·. By

¯

˜¯ ¯ ˜

47.1.(1) we obtain L(ξ,T ·) L(ξ,·) f = T (T f —¦ ξ ’ · —¦ f ) —¦ ξ ’ T · —¦ (T f —¦ ξ ’ · —¦ f ) =

¯

¯ ¯

T T f —¦ T ξ —¦ ξ ’ T · —¦ T f —¦ ξ ’ κ —¦ T · —¦ T f —¦ ξ + κ —¦ T · —¦ · —¦ f as well as a similar

¯ ¯

˜(ξ,T ·) L(ξ,¯) f . Using (1) we deduce that the right hand side of

˜ ¯·

expression for L

¯ ¯ ¯

(2) is equal to T f —¦(T ξ —¦ ξ ·T ξ —¦ξ)’(T · —¦ · ·T · —¦·)f = T f —¦[ξ, ξ]’[·, · ]—¦f .

¯ ¯ ¯

50.5. In the special case of a section s : M ’ Y of a ¬bered manifold Y ’ M