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(4.37), satisfying structure equations (4.38), typically with additional linear-
algebraic relations.
Of most interest to us is the in¬nite prolongation (M (∞) , E (∞) ) of an EDS
(M, E). As a space, M (∞) is de¬ned as the inverse limit of
πk+1 πk π
· · · ’ M (k) ’ · · · ’ M (0) = M ;
1




that is,

M (∞) = {(p0, p1, . . . ) ∈ M (0) — M (1) — · · · : πk (pk ) = pk’1 for each k ≥ 1}.

An element of M (∞) may be thought of as a Taylor series expansion for a possi-
ble integral manifold of (M, E). M (∞) is generally of in¬nite dimension, but its
presentation as an inverse limit will prevent us from facing analytic di¬culties.
In particular, smooth functions and di¬erential forms are by de¬nition the cor-
responding objects on some ¬nite M (k), pulled up to M (∞) by the projections.
It therefore makes sense to de¬ne

E (∞) = E (k),
k>0


which gives an EDS on M (∞) whose transverse integral manifolds are the
in¬nite-jet graphs of integral manifolds of (M (0), E (0)). E (∞) is a Pfa¬an sys-
tem, di¬erentially generated by its degree-1 part I (∞) = I (k) , where each
I (k) is the degree-1 part of E (k). In fact, we can see from (4.38) that E (∞) is
algebraically generated by I (∞) ; that is, I (∞) is a formally integrable Pfa¬an
system, although this is not true of any ¬nite I (k) . However, there is no ana-
log of the Frobenius theorem for the in¬nite-dimensional M (∞) , so we must be
cautious about how we use this fact.
Vector ¬elds on M (∞) are more subtle. By de¬nition, V(M (∞) ) is the Lie
algebra of derivations of the ring R(M (∞) ) of smooth functions on M (∞) . In
case M (∞) = J ∞ (Rn, Rs), a vector ¬eld is of the form

v = vi ‚xi + v0 ‚z± + · · · + vI ‚p± + · · · .
±‚ ±‚

I


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

Although v may have in¬nitely many terms, only ¬nitely many appear in its
application to any particular f ∈ R(M (∞) ), so there are no issues of conver-
gence.

4.3.2 Noether™s Theorem
To give the desired generalization of Noether™s theorem, we must ¬rst discuss a
generalization of the in¬nitesimal symmetries used in the classical version. For
convenience, we change notation and let (M, E) denote the in¬nite prolongation
of an exterior di¬erential system (M (0) , E (0)).
4.3. HIGHER-ORDER CONSERVATION LAWS 173

De¬nition 4.10 A generalized symmetry of (M (0), E (0)) is a vector ¬eld v ∈
V(M ) such that Lv E ⊆ E. A trivial generalized symmetry is a vector ¬eld
v ∈ V(M ) such that v E ⊆ 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.
• The Lie derivative in the de¬nition of generalized symmetry is de¬ned by
the Cartan formula
Lv • = v d• + d(v •).
The usual de¬nition involves a ¬‚ow along v, which may not exist in this
setting.
• A trivial generalized symmetry is in fact a generalized symmetry; this is
an immediate consequence of the fact that E is di¬erentially closed.
• The space of generalized symmetries has the obvious structure of a Lie
algebra.
• Using the fact that E is a formally integrable Pfa¬an system, it is easy
to show that the condition v E ⊆ E for v to be a trivial generalized
symmetry is equivalent to the condition v I = 0, where I = E © „¦1(M )
is the degree-1 part of E.
• 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 ‚ E, w I = 0, and θ ∈ “(I),
then

’w (v dθ) + v(w θ) ’ w(v θ)
[v, w] θ =
’w (Lv θ ’ d(v θ)) + 0 ’ w d(v
= θ)
’w Lv θ
=
= 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 Pfa¬an system E.
Thus, the “¬‚ow” of a trivial generalized symmetry does not permute the integral
manifolds of E, but instead acts by di¬eomorphisms of each “leaf”.
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 ’ pi dxi, dpi § dxi}. The in¬nite prolon-
gation of (M (0) , E (0)) is

M = J ∞ (Rn , R), E = {θI : |I| ≥ 0},
174 CHAPTER 4. ADDITIONAL TOPICS

where M has coordinates (xi , z, pi, pij , . . . ), and θI = dpI ’ pIj dxj . (For the
empty index I = …, we let p = z, so θ = dp ’ pi dxi is the original contact form.)
Then the trivial generalized symmetries of (M, E) are the total derivative vector
¬elds
‚ ‚ ‚
Di = ‚xi + pi ‚z + · · · + pIi ‚pI + · · · .
We will determine the proper generalized symmetries of (M, E) shortly.
There is another important feature of a vector ¬eld on the in¬nite prolon-
gation (M, E) of (M (0) , E (0)), which is its order. To introduce this, ¬rst note
that any vector ¬eld v0 ∈ V(M (0) ) on the original, ¬nite-dimensional manifold
induces a vector ¬eld and a ¬‚ow on each ¬nite prolongation M (k), and therefore
induces on M itself a vector ¬eld v ∈ V(M ) having a ¬‚ow. A further special
property of v ∈ V(M ) induced by v0 ∈ V(M (0) ) is that Lv (Ik ) ⊆ Ik for each
k ≥ 1. Though it is tempting to try to characterize those v ∈ 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).
Speci¬cally, an arbitrary trivial generalized symmetry v ∈ V(M ) only satis¬es

Lv (Ik ) ⊆ Ik+1,

so a generalized symmetry v can be equivalent (modulo trivials) to one induced
by a v0 ∈ V(M (0) ), without satisfying Lv (Ik ) ⊆ Ik . Instead, we have the
following.
De¬nition 4.11 For a vector ¬eld v ∈ V(M ), the order of v, written o(v), is
the minimal k ≥ 0 such that Lv (I0 ) ⊆ Ik+1.
With the restriction o(V ) ≥ 0, the orders of equivalent generalized symmetries
of E are equal. A vector ¬eld induced by v0 ∈ V(M (0) ) has order 0. Further
properties are:
• o(v) = k if and only if for each l ≥ 0, Lv (Il ) ⊆ Il+k+1 ;

• letting gk = {v : o(v) ¤ k}, we have [gk, gl ] ⊆ gk+l .
We now investigate the generalized symmetries of the prolonged contact
system on M = J ∞ (Rn, R). The conclusion will be that the proper gener-
alized symmetries correspond to smooth functions on M ; this is analogous to
the ¬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, θI ) for M , satisfying dθI = ’θIj § dxj for all multi-indices I.
To describe vector ¬elds on M , we will work with the dual framing (Di , ‚/‚θI ),
which in terms of the usual framing (‚/‚xi , ‚/‚pI ) is given by

‚ ‚
Di = + pIi ,
‚xi ‚pI
|I|≥0
‚ ‚
= .
‚θI ‚pI
4.3. HIGHER-ORDER CONSERVATION LAWS 175

The vector ¬elds Di may be thought of as “total derivative” operators, and
applied to a function g(xi , z, pi, . . . , pI ) on some J k (Rn , R) give
‚g ‚g ‚g ‚g
(Di g)(xj , z, pj , . . . , pI , pIj ) = + pi ‚z + pij ‚pj + · · · + pIi ‚pI ,
‚xi

which will generally be de¬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 —¦ · · · —¦ D i k , I = (i1 , . . . , ik ).

We do this because the proper generalized symmetries of I = {θI : |I| ≥ 0} are
uniquely represented by vector ¬elds
‚ ‚ ‚
+ gi i + · · · + g I +··· ,
v=g (4.40)
‚θ ‚θ ‚θI
where g = v θ and

|I| > 0.
gI = DI g, (4.41)

To see this, ¬rst note that any vector ¬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 ¬eld of the form (4.40) is a generalized
symmetry of I if and only if it satis¬es (4.41). If one de¬nes Rk ‚ R(M ) to
consist of functions pulled back from J k (Rn, R), then one can verify that for
any proper generalized symmetry v ∈ g,

o(v) ¤ k ⇐’ θ ∈ Rk+1 .
g=v

The general version of Noether™s theorem will involve proper generalized
symmetries. However, recall that our ¬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 ¬lter
the di¬erential forms „¦— (M ) on the in¬nite prolongation (M, E) of an Euler-
Lagrange system (M (0), E (0)) by letting

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

We de¬ne the associated graded objects

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

Because E is formally integrable, the exterior derivative d preserves this ¬ltration
and its associated graded objects:

d : „¦p,q (M ) ’ „¦p,q+1 (M ).
176 CHAPTER 4. ADDITIONAL TOPICS

We de¬ne the cohomology

Ker(d : „¦p,q (M ) ’ „¦p,q+1 (M ))
p,q
HΛ (M ) = .
Im(d : „¦p,q’1(M ) ’ „¦p,q (M ))

A simple diagram-chase shows that the exterior derivative operator d induces
p,q p+1,q
(M ).8 Now, the Poincar´-Cartan form Π ∈
a map d1 : HΛ (M ) ’ HΛ e
n+1 (0) 2 n+1
(M ) pulls back to an element Π ∈ I „¦
„¦ (M ) which is closed and
2,n’1
therefore de¬nes a class [Π] ∈ HΛ (M ).
It follows from the de¬nition that a generalized symmetry of E preserves the
2,n’1
¬ltration I p „¦p+q (M ), and therefore acts on the cohomology group HΛ (M ).
The generalized symmetries appropriate for Noether™s theorem are exactly those
generalized symmetries v of E satisfying the additional condition
2,n’1
Lv [Π] = 0 ∈ HΛ (M ).

In other words, v is required to preserve Π modulo (a) forms in I 3 „¦n+1 (M ),
and (b) derivatives of forms in I 2 „¦n (M ). We also need to verify that a trivial
generalized symmetry v preserves the class [Π]; this follows from the fact that
v I = 0, for then v Π ∈ I 2 „¦n (M ), so that

Lv [Π] = [d(I 2 „¦n(M ))] = 0.

We now have the Lie subalgebra g[Π] ⊆ g of proper generalized symmetries of
the variational problem. It is worth noting that this requires only that we have
Π de¬ned modulo I 3 „¦n+1(M ). A consequence is that even in the most general
higher-order, multi-contact case where a canonical Poincar´-Cartan form is not
e
known to exist, there should be a version of Noether™s theorem that includes
both the ¬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 Noether™s theorem is a space of conservation laws,
de¬ned by analogy with previous cases as
¯ 0,n’1
C(E) = H n’1(„¦— /E, d) = HΛ (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
¯ 1,n’1 2,n’1
C(E) = Ker(d1 : HΛ (M ) ’ HΛ (M )).

Now we can identify conservation laws as classes of n-forms, as in the previous
case of Noether™s theorem, and we will do so without comment in the following.
—,—
8 Of course, HΛ (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

¯
We de¬ne a Noether map g[Π] ’ C(E) as v ’ v Π. To see that this is

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