ńņš. 41 |

algebraic relations.

Of most interest to us is the inļ¬nite prolongation (M (ā) , E (ā) ) of an EDS

(M, E). As a space, M (ā) is deļ¬ned as the inverse limit of

Ļk+1 Ļk Ļ

Ā· Ā· Ā· ā’ M (k) ā’ Ā· Ā· Ā· ā’ M (0) = M ;

1

that is,

M (ā) = {(p0, p1, . . . ) ā M (0) Ć— M (1) Ć— Ā· Ā· Ā· : Ļk (pk ) = pkā’1 for each k ā„ 1}.

An element of M (ā) may be thought of as a Taylor series expansion for a possi-

ble integral manifold of (M, E). M (ā) is generally of inļ¬nite dimension, but its

presentation as an inverse limit will prevent us from facing analytic diļ¬culties.

In particular, smooth functions and diļ¬erential forms are by deļ¬nition the cor-

responding objects on some ļ¬nite M (k), pulled up to M (ā) by the projections.

It therefore makes sense to deļ¬ne

E (ā) = E (k),

k>0

which gives an EDS on M (ā) whose transverse integral manifolds are the

inļ¬nite-jet graphs of integral manifolds of (M (0), E (0)). E (ā) is a Pfaļ¬an sys-

tem, diļ¬erentially generated by its degree-1 part I (ā) = I (k) , where each

I (k) is the degree-1 part of E (k). In fact, we can see from (4.38) that E (ā) is

algebraically generated by I (ā) ; that is, I (ā) is a formally integrable Pfaļ¬an

system, although this is not true of any ļ¬nite I (k) . However, there is no ana-

log of the Frobenius theorem for the inļ¬nite-dimensional M (ā) , so we must be

cautious about how we use this fact.

Vector ļ¬elds on M (ā) are more subtle. By deļ¬nition, V(M (ā) ) is the Lie

algebra of derivations of the ring R(M (ā) ) of smooth functions on M (ā) . In

case M (ā) = J ā (Rn, Rs), a vector ļ¬eld is of the form

v = vi ā‚xi + v0 ā‚zĪ± + Ā· Ā· Ā· + vI ā‚pĪ± + Ā· Ā· Ā· .

Ī±ā‚ Ī±ā‚

ā‚

I

Each coeļ¬cient vI is a function on some J k (Rn, Rs), possibly with k > |I|.

Ī±

Although v may have inļ¬nitely many terms, only ļ¬nitely many appear in its

application to any particular f ā R(M (ā) ), so there are no issues of conver-

gence.

4.3.2 Noetherā™s Theorem

To give the desired generalization of Noetherā™s theorem, we must ļ¬rst discuss a

generalization of the inļ¬nitesimal symmetries used in the classical version. For

convenience, we change notation and let (M, E) denote the inļ¬nite prolongation

of an exterior diļ¬erential system (M (0) , E (0)).

4.3. HIGHER-ORDER CONSERVATION LAWS 173

Deļ¬nition 4.10 A generalized symmetry of (M (0), E (0)) is a vector ļ¬eld v ā

V(M ) such that Lv E ā E. A trivial generalized symmetry is a vector ļ¬eld

v ā V(M ) such that v E ā E. The space g of proper generalized symmetries

is the quotient of the space of generalized symmetries by the subspace of trivial

generalized symmetries.

Several remarks are in order.

ā¢ The Lie derivative in the deļ¬nition of generalized symmetry is deļ¬ned by

the Cartan formula

Lv Ļ• = v dĻ• + d(v Ļ•).

The usual deļ¬nition involves a ļ¬‚ow along v, which may not exist in this

setting.

ā¢ A trivial generalized symmetry is in fact a generalized symmetry; this is

an immediate consequence of the fact that E is diļ¬erentially closed.

ā¢ The space of generalized symmetries has the obvious structure of a Lie

algebra.

ā¢ Using the fact that E is a formally integrable Pfaļ¬an system, it is easy

to show that the condition v E ā E for v to be a trivial generalized

symmetry is equivalent to the condition v I = 0, where I = E ā© ā„¦1(M )

is the degree-1 part of E.

ā¢ The vector subspace of trivial generalized symmetries is an ideal in the

Lie algebra of generalized symmetries, so g is a Lie algebra as well. The

following proof of this fact uses the preceding characterization v I = 0

for trivial generalized symmetries: if Lv E ā‚ E, w I = 0, and Īø ā Ī“(I),

then

ā’w (v dĪø) + v(w Īø) ā’ w(v Īø)

[v, w] Īø =

ā’w (Lv Īø ā’ d(v Īø)) + 0 ā’ w d(v

= Īø)

ā’w Lv Īø

=

= 0.

The motivation for designating certain generalized symmetries as trivial comes

from a formal calculation which shows that a trivial generalized symmetry is

tangent to any integral manifold of the formally integrable Pfaļ¬an system E.

Thus, the āļ¬‚owā of a trivial generalized symmetry does not permute the integral

manifolds of E, but instead acts by diļ¬eomorphisms of each āleafā.

The following example is relevant to what follows. Let M (0) = J 1(Rn , R)

be the standard contact manifold of 1-jets of functions, with global coordinates

(xi , z, pi) and contact ideal E (0) = {dz ā’ pi dxi, dpi ā§ dxi}. The inļ¬nite prolon-

gation of (M (0) , E (0)) is

M = J ā (Rn , R), E = {ĪøI : |I| ā„ 0},

174 CHAPTER 4. ADDITIONAL TOPICS

where M has coordinates (xi , z, pi, pij , . . . ), and ĪøI = dpI ā’ pIj dxj . (For the

empty index I = ā…, we let p = z, so Īø = dp ā’ pi dxi is the original contact form.)

Then the trivial generalized symmetries of (M, E) are the total derivative vector

ļ¬elds

ā‚ ā‚ ā‚

Di = ā‚xi + pi ā‚z + Ā· Ā· Ā· + pIi ā‚pI + Ā· Ā· Ā· .

We will determine the proper generalized symmetries of (M, E) shortly.

There is another important feature of a vector ļ¬eld on the inļ¬nite prolon-

gation (M, E) of (M (0) , E (0)), which is its order. To introduce this, ļ¬rst note

that any vector ļ¬eld v0 ā V(M (0) ) on the original, ļ¬nite-dimensional manifold

induces a vector ļ¬eld and a ļ¬‚ow on each ļ¬nite prolongation M (k), and therefore

induces on M itself a vector ļ¬eld v ā V(M ) having a ļ¬‚ow. A further special

property of v ā V(M ) induced by v0 ā V(M (0) ) is that Lv (Ik ) ā Ik for each

k ā„ 1. Though it is tempting to try to characterize those v ā V(M ) induced

by such v0 using this last criterion, we ought not to do so, because this is not

a criterion that can be inherited by proper generalized symmetries of (M, E).

Speciļ¬cally, an arbitrary trivial generalized symmetry v ā V(M ) only satisļ¬es

Lv (Ik ) ā Ik+1,

so a generalized symmetry v can be equivalent (modulo trivials) to one induced

by a v0 ā V(M (0) ), without satisfying Lv (Ik ) ā Ik . Instead, we have the

following.

Deļ¬nition 4.11 For a vector ļ¬eld v ā V(M ), the order of v, written o(v), is

the minimal k ā„ 0 such that Lv (I0 ) ā Ik+1.

With the restriction o(V ) ā„ 0, the orders of equivalent generalized symmetries

of E are equal. A vector ļ¬eld induced by v0 ā V(M (0) ) has order 0. Further

properties are:

ā¢ o(v) = k if and only if for each l ā„ 0, Lv (Il ) ā Il+k+1 ;

ā¢ letting gk = {v : o(v) ā¤ k}, we have [gk, gl ] ā gk+l .

We now investigate the generalized symmetries of the prolonged contact

system on M = J ā (Rn, R). The conclusion will be that the proper gener-

alized symmetries correspond to smooth functions on M ; this is analogous to

the ļ¬nite-dimensional contact case, in which we could locally associate to each

contact symmetry its generating function, and conversely. Recall that we have

a coframing (dxi, ĪøI ) for M , satisfying dĪøI = ā’ĪøIj ā§ dxj for all multi-indices I.

To describe vector ļ¬elds on M , we will work with the dual framing (Di , ā‚/ā‚ĪøI ),

which in terms of the usual framing (ā‚/ā‚xi , ā‚/ā‚pI ) is given by

ā‚ ā‚

Di = + pIi ,

ā‚xi ā‚pI

|I|ā„0

ā‚ ā‚

= .

ā‚ĪøI ā‚pI

4.3. HIGHER-ORDER CONSERVATION LAWS 175

The vector ļ¬elds Di may be thought of as ātotal derivativeā operators, and

applied to a function g(xi , z, pi, . . . , pI ) on some J k (Rn , R) give

ā‚g ā‚g ā‚g ā‚g

(Di g)(xj , z, pj , . . . , pI , pIj ) = + pi ā‚z + pij ā‚pj + Ā· Ā· Ā· + pIi ā‚pI ,

ā‚xi

which will generally be deļ¬ned only on J k+1(Rn , R) rather than J k (Rn, R).

These operators can be composed, and we set

DI = D i 1 ā—¦ Ā· Ā· Ā· ā—¦ D i k , I = (i1 , . . . , ik ).

We do this because the proper generalized symmetries of I = {ĪøI : |I| ā„ 0} are

uniquely represented by vector ļ¬elds

ā‚ ā‚ ā‚

+ gi i + Ā· Ā· Ā· + g I +Ā·Ā·Ā· ,

v=g (4.40)

ā‚Īø ā‚Īø ā‚ĪøI

where g = v Īø and

|I| > 0.

gI = DI g, (4.41)

To see this, ļ¬rst note that any vector ļ¬eld is congruent modulo trivial gener-

alized symmetries to a unique one of the form (4.40). It then follows from a

straightforward calculation that a vector ļ¬eld of the form (4.40) is a generalized

symmetry of I if and only if it satisļ¬es (4.41). If one deļ¬nes Rk ā‚ R(M ) to

consist of functions pulled back from J k (Rn, R), then one can verify that for

any proper generalized symmetry v ā g,

o(v) ā¤ k āā’ Īø ā Rk+1 .

g=v

The general version of Noetherā™s theorem will involve proper generalized

symmetries. However, recall that our ļ¬rst-order version requires us to distin-

guish among the symmetries of an Euler-Lagrange system the symmetries of the

original variational problem; only the latter give rise to conservation laws. We

therefore have to give the appropriate corresponding notion for proper general-

ized symmetries.

For this purpose, we introduce the following algebraic apparatus. We ļ¬lter

the diļ¬erential forms ā„¦ā— (M ) on the inļ¬nite prolongation (M, E) of an Euler-

Lagrange system (M (0), E (0)) by letting

I p ā„¦p+q (M ) = Image(E ā— Ā· Ā· Ā· ā— E ā—ā„¦ā— (M ) ā’ ā„¦ā— (M )) ā© ā„¦p+q (M ). (4.42)

p

We deļ¬ne the associated graded objects

ā„¦p,q (M ) = I p ā„¦p+q (M )/I p+1 ā„¦p+q (M ).

Because E is formally integrable, the exterior derivative d preserves this ļ¬ltration

and its associated graded objects:

d : ā„¦p,q (M ) ā’ ā„¦p,q+1 (M ).

176 CHAPTER 4. ADDITIONAL TOPICS

We deļ¬ne the cohomology

Ker(d : ā„¦p,q (M ) ā’ ā„¦p,q+1 (M ))

p,q

HĪ (M ) = .

Im(d : ā„¦p,qā’1(M ) ā’ ā„¦p,q (M ))

A simple diagram-chase shows that the exterior derivative operator d induces

p,q p+1,q

(M ).8 Now, the PoincarĀ“-Cartan form Ī ā

a map d1 : HĪ (M ) ā’ HĪ e

n+1 (0) 2 n+1

(M ) pulls back to an element Ī ā I ā„¦

ā„¦ (M ) which is closed and

2,nā’1

therefore deļ¬nes a class [Ī ] ā HĪ (M ).

It follows from the deļ¬nition that a generalized symmetry of E preserves the

2,nā’1

ļ¬ltration I p ā„¦p+q (M ), and therefore acts on the cohomology group HĪ (M ).

The generalized symmetries appropriate for Noetherā™s theorem are exactly those

generalized symmetries v of E satisfying the additional condition

2,nā’1

Lv [Ī ] = 0 ā HĪ (M ).

In other words, v is required to preserve Ī modulo (a) forms in I 3 ā„¦n+1 (M ),

and (b) derivatives of forms in I 2 ā„¦n (M ). We also need to verify that a trivial

generalized symmetry v preserves the class [Ī ]; this follows from the fact that

v I = 0, for then v Ī ā I 2 ā„¦n (M ), so that

Lv [Ī ] = [d(I 2 ā„¦n(M ))] = 0.

We now have the Lie subalgebra g[Ī ] ā g of proper generalized symmetries of

the variational problem. It is worth noting that this requires only that we have

Ī deļ¬ned modulo I 3 ā„¦n+1(M ). A consequence is that even in the most general

higher-order, multi-contact case where a canonical PoincarĀ“-Cartan form is not

e

known to exist, there should be a version of Noetherā™s theorem that includes

both the ļ¬rst-order multi-contact version discussed in the previous section, and

the higher-order scalar version discussed below. However, we will not pursue

this.

The other ingredient in Noetherā™s theorem is a space of conservation laws,

deļ¬ned by analogy with previous cases as

ĀÆ 0,nā’1

C(E) = H nā’1(ā„¦ā— /E, d) = HĪ (M ),

where the last notation refers to the cohomology just introduced. It is a sub-

stantial result (see [BG95a]) that over contractible subsets of M (0) we can use

the exterior derivative to identify C(E) with

def

ĀÆ 1,nā’1 2,nā’1

C(E) = Ker(d1 : HĪ (M ) ā’ HĪ (M )).

Now we can identify conservation laws as classes of n-forms, as in the previous

case of Noetherā™s theorem, and we will do so without comment in the following.

ā—,ā—

8 Of course, HĪ (M ) is the E1 -term of a spectral sequence. Because we will not be using

any of the higher terms, however, there is no reason to invoke this machinery. Most of this

theory was introduced in [Vin84].

4.3. HIGHER-ORDER CONSERVATION LAWS 177

ĀÆ

We deļ¬ne a Noether map g[Ī ] ā’ C(E) as v ā’ v Ī . To see that this is

ńņš. 41 |