ńņš. 40 |

to regard dsQ (df(Ā·), w) as a 1-form on P , and then

Ļ•w = ā—P (ds2 (df(Ā·), w)).

Q

Because this expression depends linearly on w, we can simplify further by letting

a denote the Lie algebra of inļ¬nitesimal symmetries of Q, and then the map

w ā’ w Ī is an element of aā— ā— ā„¦nā’1(P ). If we deļ¬ne an aā— -valued 1-form on

P by

Ī±(v) = ds2 (df(v), Ā·), v ā Tp P,

Q

then our conservation laws read

d(ā—P Ī±) = 0 ā aā— ā— ā„¦n (P ). (4.35)

The aā— -valued (n ā’ 1)-form ā—P Ī± may be formed for any map f : P ā’ Q, and

it is closed if f is harmonic. In fact, if Q is locally homogeneous, meaning that

6 Thedivergence of a symmetric 2-form S is the 1-form div S = ei S(ei , Ā·), where is the

Levi-Civita covariant derivative and (ei ) is any orthonormal frame. Equation (4.34) is true of

any symmetric 2-form S and inļ¬nitesimal isometry v.

168 CHAPTER 4. ADDITIONAL TOPICS

inļ¬nitesimal isometries span each tangent space Tq Q, then (4.35) is equivalent

to the harmonicity of f.

An important special case of this last phenomenon is when Q itself is a Lie

group G with bi-invariant metric ds2 . Examples are compact semisimple Lie

G

groups, such as O(N ) or SU (N ), with metric induced by the Killing form on

the Lie algebra g. Now a map f : P ā’ G is uniquely determined up to left-

translation by the pullback f ā— Ļ• of the left-invariant g-valued Maurer-Cartan

1-form Ļ•. Using the metric to identify g ā¼ gā— , the conservation laws state that

=

ā—

if f is harmonic, then d(ā—P (f Ļ•)) = 0. Conversely, if P is simply connected,

then given a g-valued 1-form Ī± on P satisfying

dĪ± + 1 [Ī±, Ī±] = 0,

2

d(ā—P Ī±) = 0,

there is a harmonic map f : P ā’ G with f ā— Ļ• = Ī±, uniquely determined up

to left-translation. This is the idea behind the gauge-theoretic reformulation

of certain harmonic map systems, for which remarkable results have been ob-

tained in the past decade.7 Quite generally, PDE systems that can be written

as systems of conservation laws have special properties; one typically exploits

such expressions to deļ¬ne weak solutions, derive integral identities, and prove

regularity theorems.

4.3 Higher-Order Conservation Laws

One sometimes encounters a conservation law for a PDE that involves higher-

order derivatives of the unknown function, but that cannot be expressed in terms

of derivatives of ļ¬rst-order conservation laws considered up to this point. An

2

example is the (1+1)-dimensional wave equation ā’ztt +zxx = 0, for which (ztt +

2

ztx )dt+2ztt ztx dx is closed on solutions, but cannot be obtained by diļ¬erentiating

any conservation law on J 1(R2 , R). In this section, we introduce the geometric

framework in which such conservation laws may be found, and we propose a

version of Noetherā™s theorem appropriate to this setting. While other general

forms of Noetherā™s theorem have been stated and proved (e.g., see [Vin84] or

[Olv93]), it is not clear how they relate to that conjectured here.

We also discuss (independently from the preceding) the higher-order rela-

tionship between surfaces in Euclidean space with Gauss curvature K = ā’1

1

and the sine-Gordon equation ztx = 2 sin(2z), in terms of exterior diļ¬erential

systems.

4.3.1 The Inļ¬nite Prolongation

We begin by deļ¬ning the prolongation of an exterior diļ¬erential system (EDS).

When this is applied to the EDS associated to a PDE system, it gives the EDS

associated to the PDE system augmented by the ļ¬rst derivatives of the original

7 The literature on this subject is vast, but a good starting point is [Woo94].

4.3. HIGHER-ORDER CONSERVATION LAWS 169

equations. This construction then extends to that of the inļ¬nite prolongation,

an EDS on an inļ¬nite-dimensional manifold which includes information about

derivatives of all orders.

The general deļ¬nition of prolongation uses a construction introduced in Ā§4.2,

in the discussion of multi-contact manifolds. Let X n+s be a manifold, and

Ļ

Gn (T X) ā’ X the bundle of tangent n-planes of X; points of Gn(T X) are of

the form (p, E), where p ā X and E ā‚ Tp X is a vector subspace of dimension

n. As discussed previously, there is a canonical Pfaļ¬an system I ā‚ T ā— Gn(T X)

of rank s, deļ¬ned at (p, E) by

def

I(p,E) = Ļā— (E ā„ ).

Given local coordinates (xi, z Ī± ) on X, there are induced coordinates (xi , z Ī±, pĪ±)

i

on Gn(T X), in terms of which I is generated by the 1-forms

ĪøĪ± = dz Ī± ā’ pĪ± dxi. (4.36)

i

We let I ā‚ ā„¦ā— (Gn(T X)) be the diļ¬erential ideal generated by I.

Now let (M, E) be an exterior diļ¬erential system; that is, M is a manifold of

dimension m+s and E ā‚ ā„¦ā— (M ) is a diļ¬erential ideal for which we are interested

in m-dimensional integral manifolds. We then deļ¬ne the locus M (1) ā‚ Gm (T M )

to consist of the integral elements of E ā‚ ā„¦ā— (M ); that is, (p, E) ā M (1) if and

only if

ā—

(E ā— ) for all Ļ• ā E.

Ļ•E = 0 ā

Ī¹

We will assume from now on that M (1) ā’ Gm (T M ) is a smooth submanifold.

Then we deļ¬ne

def

E (1) = Ī¹ā— I ā‚ ā„¦ā— (M (1) )

as the restriction to M (1) of the multi-contact diļ¬erential ideal. This is the

same as the diļ¬erential ideal generated by the Pfaļ¬an system Ī¹ā— I ā‚ T ā— M (1),

and the ļ¬rst prolongation of (M, E) is deļ¬ned to be the exterior diļ¬erential

system (M (1), E (1)). Note that the ļ¬rst prolongation is always a Pfaļ¬an system.

Furthermore, if Ļ : M (1) ā’ M is the obvious projection map, and assuming

that E is a Pfaļ¬an system, then one can show that Ļ ā— E ā E (1) . However,

the projection Ļ could be quite complicated, and need not even be surjective.

Finally, note that any integral manifold f : N ā’ M of E lifts to an integral

manifold f (1) : N ā’ M (1) of E (1), and that the transverse integral manifold of

E (1) is locally of this form.

Inductively, the kth prolongation (M (k) , E (k)) of (M, E) is the ļ¬rst prolonga-

tion of the (k ā’ 1)st prolongation of (M, E). This gives rise to the prolongation

tower

Ā· Ā· Ā· ā’ M (k) ā’ M (kā’1) ā’ Ā· Ā· Ā· ā’ M (1) ā’ M.

An integral manifold of (M, E) lifts to an integral manifold of each (M (k), E (k))

in this tower.

Two examples will help to clarify the construction. The ļ¬rst is the pro-

longation tower of the multi-contact system (Gn(T X), I) itself, and this will

170 CHAPTER 4. ADDITIONAL TOPICS

give us more detailed information about the structure of the ideals E (k) for gen-

eral (M, E). The second is the prolongation tower of the EDS associated to a

ļ¬rst-order PDE system, most of which we leave as an exercise.

Example 1. Consider the multi-contact ideal I on Gn(T X), over a manifold

X of dimension n + s with local coordinates (xi , z Ī±). We can see from the

coordinate expression (4.36) that its integral elements over the dense open subset

where i dxi = 0 are exactly the n-planes of the form

EpĪ± = {dz Ī± ā’ pĪ±dxi , dpĪ± ā’ pĪ± dxj }ā„ ā‚ T (Gn(T X)),

i i ij

ij

for some constants pĪ± = pĪ± . These pĪ± are local ļ¬ber coordinates for the pro-

ij ji ij

(1) (1)

longation (Gn (T X) , I ). Furthermore, with respect to the full coordinates

(xi , z Ī±, pĪ±, pĪ± ) for Gn(T X)(1) ā‚ Gn(T Gn (T X)), the 1-jet graphs of integral

i ij

manifolds of I ā‚ ā„¦ā— (Gn(T X)) satisfy

dz Ī± ā’ pĪ± dxi = 0, dpĪ± ā’ pĪ± dxj = 0.

i i ij

It is these s + ns 1-forms that diļ¬erentially generate the prolonged Pfaļ¬an

system I (1). It is not diļ¬cult to verify that we have globally Gn(T X)(1) =

G2,n(X), the bundle of 2-jets of n-dimensional submanifolds of X, and that

I (1) ā‚ ā„¦ā— (G2,n(X)) is the Pfaļ¬an system whose transverse integral manifolds

are 2-jet graphs of submanifolds of X.

More generally, let Gk = Gk,n(X) ā’ X be the bundle of k-jets of n-

dimensional submanifolds of X. Because a 1-jet of a submanifold is the same

as a tangent plane, G1 = Gn (T X) is the original space whose prolongation

tower we are describing. Gk carries a canonical Pfaļ¬an system Ik ā‚ ā„¦ā— (Gk ),

whose transverse integral manifolds are k-jet graphs f (k) : N ā’ Gk of n-

dimensional submanifolds f : N ā’ X. This is perhaps clearest in coordi-

nates. Letting (xi , z Ī±) be coordinates on X, Gk has induced local coordinates

(xi , z Ī±, pĪ±, . . . , pĪ±), |I| ā¤ k, corresponding to the jet at (xi , z Ī±) of the subman-

i I

ifold

{(ĀÆi, z Ī±) ā X : z Ī± = z Ī± + pĪ± (ĀÆi ā’ xi ) + Ā· Ā· Ā· + 1Ī±

ā’ x)I }.

xĀÆ ĀÆ ix I! pI (ĀÆ

x

In terms of these coordinates, the degree-1 part Ik ā‚ T ā— (Gk ) of the Pfaļ¬an

system Ik is generated by

ĪøĪ± = dz Ī± ā’ pĪ± dxi,

i

Īøi = dpĪ± ā’ pĪ± dxj ,

Ī±

i ij

(4.37)

.

.

.

ĪøI = dpĪ± ā’ pĪ± dxj ,

Ī±

|I| = k ā’ 1.

I Ij

It is not hard to see that the transverse integral manifolds of this Ik are as

described above. The point here is that (Gk , Ik) is the ļ¬rst prolongation of

(Gkā’1, Ikā’1) for each k > 1, and is therefore the (k ā’ 1)st prolongation of the

original (G1 , I1) = (Gn(T X), I).

4.3. HIGHER-ORDER CONSERVATION LAWS 171

For future reference, we note the structure equations

dĪøĪ± = ā’Īøi ā§ dxi,

Ī±

dĪøi = ā’Īøij ā§ dxj ,

Ī± Ī±

.

. (4.38)

.

dĪøI = ā’ĪøIj ā§ dxj ,

Ī± Ī±

|I| = k ā’ 2,

Ī± Ī± j

dĪøI = ā’dpIj ā§ dx , |I| = k ā’ 1.

There is a tower

Ā· Ā· Ā· ā’ Gk ā’ Gkā’1 ā’ Ā· Ā· Ā· ā’ G1 , (4.39)

and one can pull back to Gk any functions or diļ¬erential forms on Gk , with

k < k. Under these maps, our diļ¬erent uses of the coordinates pĪ± and forms

I

Ī±

ĪøI are consistent, and we can also write

Ik ā‚ I k ā‚ T ā— G k , for k < k.

None of the Ik is an integrable Pfaļ¬an system. In fact, the ļ¬ltration on Gk

Ik ā Ikā’1 ā Ā· Ā· Ā· ā I1 ā 0

coincides with the derived ļ¬‚ag of Ik ā‚ T ā— Gk,n (cf. Ch. II, Ā§4 of [B+ 91]).

Example 2. Our second example of prolongation relates to a ļ¬rst-order PDE

system F a(xi , z Ī±(x), zxi (x)) = 0 for some unknown functions z Ī± (x). The equa-

Ī±

tions F a(xi , z Ī±, pĪ±) deļ¬ne a locus MF in the space J 1 (Rn , Rs) of 1-jets of maps

i

z : Rn ā’ Rs , and we will assume that this locus is a smooth submanifold which

submersively surjects onto Rn. The restriction to MF ā‚ J 1 (Rn, Rs) of the

multi-contact Pfaļ¬an system I1 = {dz Ī± ā’ pĪ± dxi} generates an EDS (MF , IF ).

i

Now, the set of integral elements for (MF , IF ) is a subset of the set of integral

elements for I1 in J 1(Rn , Rs); it consists of those integral elements of I1 which

are tangent to MF ā‚ J 1(Rn , Rs). Just as in the preceding example, the inte-

gral elements of I1 may be identiļ¬ed with elements of the space J 2(Rn , Rs) of

2-jets of maps. The collection of 2-jets which correspond to integral elements of

(MF , IF ) are exactly the 2-jets satisfying the augmented PDE system

F a (xi, z Ī± (x), zxi (x)),

Ī±

0=

ā‚F a ā‚F a Ī± ā‚F a Ī±

0= + z i+ z i j.

ā‚z Ī± x ā‚pĪ± x x

ā‚xi j

Therefore, integral manifolds of the prolongation of the EDS associated to a

PDE system correspond to solutions of this augmented system. For this reason,

prolongation may generally be thought of as adjoining the derivatives of the

original equations.

It is important to note that as the ļ¬rst prolongation of arbitrary (M, E) is

embedded in the canonical multi-contact system (Gn(T M ), I), so can all higher

prolongations (M (k), E (k)) be embedded in the prolongations (Gk,n(M ), Ik ).

172 CHAPTER 4. ADDITIONAL TOPICS

Among other things, this implies that E (k) is locally generated by forms like

ńņš. 40 |