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( 48 .)


well-de¬ned, ¬rst note that

Π ∈ I 1 „¦n(M ),

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 )),

and we therefore have an element
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
d1 ([v Π]) ∈ HΛ (M )
is represented by the class (see (4.43))

Π)] = [Lv Π] = Lv [Π] = 0.

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

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

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
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

= 0 on M (∞) .
‚pI DI g (4.45)

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 (∞) .

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-
¬nite prolongation, starting with

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

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

= d(L dx + θ § Lpi dx(i))
= 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).

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 )).

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
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 = .

To calculate in Rl we will need to de¬ne variables qJ to be the harmonic parts
of pJ ; that is,

qi = pi ,
pij ’ n δij pll
qij =
pij ’ n δij f(p),
pijk ’ n+2 (δij pkll + δjk pill + δki pjll )
qijk =
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
Di g ’ f g ∈ Rk+1;

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
0≡ qIiqI i . (4.47)
‚pI ‚pI
|I|,|I |=k

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

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.

orthogonal to Hl — Hm ‚ Sym l (Rn)— — Sym m (Rn )— . In particular, from (4.47)
we have
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.
For the second term, our lemma gives similarly that = 0, so have

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

0≡ pKi pK i ,
‚pK ‚pK
and as before, the lemma implies that h is linear in pK . Now we have


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( 48 .)