Π ∈ I 1 „¦n(M ),

v

so v Π represents an element of „¦1,n’1(M ), which we shall also denote as

v Π. Furthermore, its exterior derivative is

Π) = Lv Π,

d(v (4.43)

and this lies in I 2 „¦n+1, simply because v preserves E and therefore also the

¬ltration (4.42). Consequently,

Π ∈ Ker(d : „¦1,n’1(M ) ’ „¦1,n(M )),

v

and we therefore have an element

1,n’1

Π] ∈ HΛ

[v (M ).

Finally, we need to verify that

1,n’1 2,n’1

Π] ∈ Ker(d1 : HΛ (M ) ’ HΛ

[v (M )).

This follows from the hypothesis that v preserves not only the Euler-Lagrange

system E and associated ¬ltration (4.42), but also the class [Π]. Speci¬cally, the

image

2,n’1

d1 ([v Π]) ∈ HΛ (M )

is represented by the class (see (4.43))

Π)] = [Lv Π] = Lv [Π] = 0.

[d(v

This proves that

v ’ [v Π]

de¬nes a map between the appropriate spaces.

We can now make the following proposal for a general form of Noether™s

theorem.

Conjecture 4.1 Let (M, E) be the in¬nite prolongation of an Euler-Lagrange

q

system, and assume that the system is non-degenerate and that HdR (M ) = 0

for all q > 0. Then the map v ’ [v Π] induces an isomorphism

∼¯

g[Π] ’’ C(E).

It is quite possible that this is already essentially proved in [Vin84] or [Olv93],

but we have not been able to determine the relationship between their state-

ments and ours. In any case, it would be illuminating to have a proof of the

present statement in a spirit similar to that of our Theorem 1.3.

178 CHAPTER 4. ADDITIONAL TOPICS

To clarify this, we will describe how it appears in coordinates. First, note

that for the classical Lagrangian

L(xi , z, pi)dx,

the Euler-Lagrange equation

E(xi , z, pi, pij ) = ( Dj Lpj ’ Lz )(xi , z, pi, pij ) = 0

de¬nes a locus M (1) ‚ J 2 (Rn, R), and the ¬rst prolongation of the Euler-

Lagrange system (J 1(Rn , R), EL) discussed previously is given by the restriction

of the second-order contact Pfa¬an system on J 2 (Rn, R) to this locus. Higher

prolongations are de¬ned by setting

def

M (k) = {EI = DI E = 0, |I| ¤ k ’ 1} ‚ J k+1 (Rn, R)

and restricting the (k + 1)st-order contact system I (k+1). We will consider

generalized symmetries of (M (∞) , E (∞) ) which arise as restrictions of those

generalized symmetries of (J ∞ (Rn , R), I) which are also tangent to M (∞) ‚

J ∞ (Rn, R). This simpli¬es matters insofar as we can understand generalized

symmetries of I by their generating functions. The tangency condition is

Lv (EI )|M (∞) = 0, |I| ≥ 0. (4.44)

This Lie derivative is just the action of a vector ¬eld as a derivation on functions.

Now, for a generalized symmetry v of the in¬nite-order contact system, all of

the conditions (4.44) follow from just the ¬rst one,

Lv (E)|M (∞) = 0.

If we let v have generating function g = v θ ∈ R(J ∞ (Rn, R)), then we can

see from (4.40, 4.41) that this condition on g is

‚E

= 0 on M (∞) .

‚pI DI g (4.45)

|I|≥0

pii ’ f(z)

We are again using p = z for convenience. For instance, E =

de¬nes the Poisson equation ∆z = f(z), and the preceding condition is

Di g ’ f (z)g = 0 on M (∞) .

2

(4.46)

i

We now consider the Noether map for M (∞) ‚ J ∞ (Rn, R). We write the

Poincar´-Cartan form pulled back to M (∞) using coframes adapted to this in-

e

¬nite prolongation, starting with

dLpi = Dj (Lpi )dxj + Lpi z θ + Lpi pj θj ,

4.3. HIGHER-ORDER CONSERVATION LAWS 179

and then the Poincar´-Cartan form on J ∞ (Rn , R) is

e

= d(L dx + θ § Lpi dx(i))

Π

= θ § (’Di (Lpi ) + Lz )dx ’ θj § Lpi pj dx(i) .

Restriction to M (∞) ‚ J ∞ (Rn, R) kills the ¬rst term, and we have

Π = ’Lpi pj θ § θj § dx(i).

We then apply a vector ¬eld vg with generating function g, and obtain

Π = ’gLpi pj θj § dx(i) + (Dj g)Lpi pj θ § dx(i).

vg

This will be the “di¬erentiated form” of a conservation law precisely if Lvg [Π] =

0, that is, if

Π) ≡ 0 (mod I 3 „¦n+1(M ) + dI 2 „¦n(M )).

d(vg

Concerning generalized symmetries of a PDE, note that in the condition

(4.45) for g = g(xi, p, pi, . . . , pI ) ∈ Rk = C ∞ (J k (Rn , R)), the variables pI with

|I| > k appear only polynomially upon taking the total derivatives DJ g. In

other words, the condition on g is polynomial in the variables pI for |I| > k.

Equating coe¬cients of these polynomials gives a PDE system to be satis¬ed

by a generalized symmetry of an Euler-Lagrange equation. With some e¬ort,

one can analyze the situation for our Poisson equation ∆z = f(z) and ¬nd the

following.

Proposition 4.6 If n ≥ 3, then a solution g = g(xi , p, pi, . . . , pI ) of order k

to (4.46) is equal on M (∞) to a function that is linear in the variables pJ with

|J| ≥ k ’ 2. If in addition f (z) = 0, so that the Poisson equation is non-

linear, then every solution™s restriction to M (∞) is the pullback of a function on

M (0) ‚ J 2 (Rn, R), which generates a classical symmetry of the equation.

In other words, a non-linear Poisson equation in n ≥ 3 independent variables

has no non-classical generalized symmetries, and consequently no higher-order

conservation laws.

Proof. Because the notation involved here becomes rather tedious, we will

sketch the proof and leave it to the reader to verify the calculations. We pre-

viously hinted at the main idea: the condition (4.46) on a generating function

g = g(xi , p, . . . , pI ), |I| = k, is polynomial in the highest order variables with

coe¬cients depending on partial derivatives of g. To isolate these terms, we

¬lter the functions on M (∞) by letting Rl be the image of Rl under restriction

to M (∞) ; in other words, Rl consists of functions which can be expressed as

functions of xi, pJ , |J| ¤ l, after substituting the de¬ning relations of M (∞) ,

d|J | f

pJ ii = .

dxJ

180 CHAPTER 4. ADDITIONAL TOPICS

To calculate in Rl we will need to de¬ne variables qJ to be the harmonic parts

of pJ ; that is,

qi = pi ,

1

pij ’ n δij pll

qij =

1

pij ’ n δij f(p),

=

1

pijk ’ n+2 (δij pkll + δjk pill + δki pjll )

qijk =

1

pijk ’

= (δ p + δjk pi + δki pj )f (p), &c.

n+2 ij k

These, along with xi and p, give coordinates on M (∞) . In addition to working

modulo various Rl to isolate terms with higher-order derivatives, we will also at

times work modulo functions that are linear in the qI . In what follows, we use

the following index conventions: p(l) = (p, pj , . . . , pJ ) (with |J| = l) denotes the

derivative variables up to order l, and the multi-indices I, K, A, satisfy |I| = k,

|K| = k ’ 1, |A| = k ’ 2.

Now, starting with g = g(xj , p(k)) ∈ Rk , we note that

2

Di g ’ f g ∈ Rk+1;

0=

i

that is, the possible order-(k + 2) term resulting from two di¬erentiations of g

already drops to order k when restricted to the equation manifold. We consider

this expression modulo Rk 9 , and obtain a quadratic polynomial in qIj with

coe¬cients in Rk . We consider only the quadratic terms of this polynomial,

which are

‚2g

0≡ qIiqI i . (4.47)

‚pI ‚pI

|I|,|I |=k

1¤i¤n

2

To draw conclusions about ‚p‚ ‚p from this, we need the following fundamental

g

I I

lemma, in which the di¬erence between the cases n = 2 and n ≥ 3 appears:

If Hl ‚ Sym l (Rn )— denotes the space of degree-l homogeneous har-

monic polynomials on Rn , n ≥ 3, then the O(n)-equivariant con-

traction map Hl+1 — Hm+1 ’ Hl — Hm given by

XC z C — Y Q z Q ’ XBi z B — YP i z P

i

is surjective; here, |C| ’ 1 = |B| = l, |Q| ’ 1 = |P | = m.

We will apply this in situations where a given g BP ∈ Sym l (Rn) — Sym m (Rn) is

known to annihilate all XBi YP i with XC , YQ harmonic, for then we have g BP

9 Inthis context, “modulo” refers to quotients of vector spaces by subspaces, not of rings

by ideals, as in exterior algebra.

4.3. HIGHER-ORDER CONSERVATION LAWS 181

orthogonal to Hl — Hm ‚ Sym l (Rn)— — Sym m (Rn )— . In particular, from (4.47)

we have

‚2g

qI qJ ≡ 0.

‚pI ‚pJ

d|A| f

This means that the restriction of the function g to the hyperplanes pAii = dxA

is linear in the highest pI ; in other words, we can write

g(xi , p(k)) = h(x, p(k’1)) + hI (x, p(k’1))pI

for some hI ∈ Rk’1. We can further assume that all hAii = 0, where |A| = k’2.

This completes the ¬rst step.

The second step is to simplify the functions hI (x, p(k’1)), substituting our

new form of g into the condition (4.46). Again, the “highest” terms appear

modulo Rk , and are

‚hI ‚hI

0≡2 qKiqIi + 2 qAiqIi ;

‚pK ‚pA

here we recall our index convention |A| = k ’2, |K| = k ’1, |I| = k. Both terms

must vanish separately, and for the ¬rst, our lemma on harmonic polynomials

def ‚hI

gives that [h]IK = ‚pK is orthogonal to harmonics; but then our normalization

hypothesis hAii = 0 gives that [h]IK = 0. We conclude that

g(xi , p(k)) = h(x, p(k’1)) + hI (x, p(k’2))pI , |I| = k.

‚hI

For the second term, our lemma gives similarly that = 0, so have

‚pA

g(xi , p(k)) = h(x, p(k’1)) + hI (x, p(k’3))pI .

This completes the second step.

For the third step, we again substitute the latest form of g into the condition

(4.46), and now work modulo Rk’1. The only term non-linear in the pI is

‚2h

0≡ pKi pK i ,

‚pK ‚pK

and as before, the lemma implies that h is linear in pK . Now we have