<<

. 5
( 48 .)



>>


dΞ = ’θ § dΨ = θ § df § dx.

Therefore, we consider • satisfying

θ § df § dx = • § Ξ,

or equivalently
df § dx ≡ ’• § Ψ (mod {I}),
and ¬nd that they are exactly those 1-forms of the form

fpi dxi + c θ
•=

for an arbitrary function c. The problem is reduced to describing those f(x, z, p)
fpi dxi + c θ is closed. We
for which there exists some c(x, z, p) so that • =
can determine all such forms explicitly, as follows. The condition that • be
closed expands to

cpi dpi § dz
0=
j
+(fpi pj ’ cδi ’ cpj pi )dpj § dxi
+ 2 (fpi xj ’ fpj xi ’ cxj pi + cxi pj )dxj § dxi
1

+(fpi z ’ cxi ’ cz pi)dz § dxi .

These four terms must vanish separately. The vanishing of the ¬rst term implies
that c = c(xi, z) does not depend on any pi . Given this, the vanishing of the
second term implies that f(xi , z, pi) is quadratic in the pi , with diagonal leading
term:
f(xi , z, pi) = 1 c(xi, z) p2 + ej (xi, z)pj + a(xi , z)
j
2

for some functions ej (xi , z) and a(xi , z). Now the vanishing of the third term
reduces to
0 = e i j ’ ej i ,
x x
implying that for some function b(xj , z),

‚b(xi , z)
j i
e (x , z) = ;
‚xj
this b(xj , z) is uniquely determined only up to addition of a function of z.
Finally, the vanishing of the fourth term reduces to

(bz ’ c)xi = 0,

so that c(xi , z) di¬ers from bz (xi, z) by a function of z alone. By adding an
antiderivative of this di¬erence to b(xi, z) and relabelling the result as b(xi, z),
1.2. THE EULER-LAGRANGE SYSTEM 13

we see that our criterion for the Monge-Ampere system to be Euler-Lagrange is
that f(xi , z, pi) be of the form
f(xi , z, pi) = 1 bz (x, z) p2 + bxi (x, z)pi + a(x, z)
i
2

for some functions b(x, z), a(x, z). These describe exactly those Poisson equa-
tions that are locally contact-equivalent to Euler-Lagrange equations.

Example 2. An example that is not quasi-linear is given by
2
z) ’ g(x, z,
det( z) = 0.
The n-form Ψ = dp ’ g(x, z, p)dx and the standard contact system generate a
Monge-Ampere system whose transverse integral manifolds correspond to solu-
tions of this equation. A calculation similar to that in the preceding example
shows that this Monge-Ampere system is Euler-Lagrange if and only if g(x, z, p)
is of the form
g(x, z, p) = g0 (x, z) g1(p, z ’ pi xi).

Example 3. The linear Weingarten equation aK + bH + c = 0 for a surface
in Euclidean space having Gauss curvature K and mean curvature H is Euler-
Lagrange for all choices of constants a, b, c, as we shall see in §1.4.2. In this
case, the appropriate contact manifold for the problem is M = G2(T E3 ), the
Grassmannian of oriented tangent planes of Euclidean space.

Example 4. Here is an example of a Monge-Ampere system which is locally,
but not globally, Euler-Lagrange, suitable for those readers familiar with some
complex algebraic geometry. Let X be a K3 surface; that is, X is a simply con-
nected, compact, complex manifold of complex dimension 2 with trivial canon-
ical bundle, necessarily of K¨hler type. Suppose also that there is a positive
a
holomorphic line bundle L ’ X with a Hermitian metric having positive ¬rst
Chern form ω ∈ „¦1,1(X). Our contact manifold M is the unit circle subbundle
of L ’ X, a smooth manifold of real dimension 5; the contact form is
i
θ= 2π ±, dθ = ω,
where ± is the u(1)-valued Hermitian connection form on M . Note θ§(dθ)2 = 0,
because the 4-form (dθ)2 = ω2 is actually a volume form on M (by positivity)
and θ is non-vanishing on ¬bers of M ’ X, unlike (dθ)2 .
Now we trivialize the canonical bundle of X with a holomorphic 2-form
¦ = Ψ + iΣ, and take for our Monge-Ampere system
E = {θ, dθ = ω, Ψ = Re(¦)}.
We can see that E is locally Euler-Lagrange as follows. First, by reasons of type,
ω § ¦ = 0; and ω is real, so 0 = Re(ω § ¦) = ω § Ψ. In particular, Ψ is primitive.
With Ξ = θ § Ψ, we compute
dΞ = ω § Ψ ’ θ § dΨ = ’θ § dΨ,
´
14 CHAPTER 1. LAGRANGIANS AND POINCARE-CARTAN FORMS

but dΨ = Re(d¦) = 0, because ¦ is holomorphic and therefore closed.
On the other hand, (M, E) cannot be globally Euler-Lagrange; that is, Ξ =
θ § Ψ cannot be exact, for if Ξ = dξ, then

Ξ§Σ= d(ξ § Σ) = 0,
M M

but also
¯
Ξ§Σ = θ§Ψ§Σ= c ¦ § ¦,
M M X
for some number c = 0.


1.3 Noether™s Theorem
The classical theorem of Noether describes an isomorphism between a Lie alge-
bra of in¬nitesimal symmetries associated to a variational problem, and a space
of conservation laws for its Euler-Lagrange equations. We will often assume
without comment that our Lagrangian is non-degenerate in the sense discussed
earlier.
There are four reasonable Lie algebras of symmetries that we might consider
in our setup. Letting V(M ) denote the Lie algebra of all vector ¬elds on M ,
they are the following.
• Symmetries of (M, I, Λ):

gΛ = {v ∈ V(M ) : Lv I ⊆ I, Lv Λ = 0}.

• Symmetries of (M, I, [Λ]):

g[Λ] = {v ∈ V(M ) : Lv I ⊆ I, Lv [Λ] = 0}.

• Symmetries of (M, Π):

gΠ = {v ∈ V(M ) : Lv Π = 0}.

(Note that Lv Π = 0 implies Lv I ⊆ I for non-degenerate Λ.)
• Symmetries of (M, EΛ):

gEΛ = {v ∈ V(M ) : Lv EΛ ⊆ EΛ}.

(Note that Lv EΛ ⊆ EΛ implies Lv I ⊆ I.)
We comment on the relationship between these spaces. Clearly, there are
inclusions
gΛ ⊆ g[Λ] ⊆ gΠ ⊆ gEΛ .
Any of the three inclusions may be strict. For example, we locally have g [Λ] = gΠ
because Π is the image of [Λ] under the coboundary δ : H n („¦— /I) ’ H n+1(I),
1.3. NOETHER™S THEOREM 15

which is invariant under di¬eomorphisms of (M, I) and is an isomorphism on
contractible open sets. However, we shall see later that globally there is an
inclusion
n
gΠ/g[Λ] ’ HdR (M ),
and this discrepancy between the two symmetry algebras introduces some sub-
tlety into Noether™s theorem.
Also, there is a bound

dim (gEΛ /gΠ ) ¤ 1. (1.7)

This follows from noting that if a vector ¬eld v preserves EΛ, then it preserves
Π up to multiplication by a function; that is, Lv Π = fΠ. Because Π is a closed
form, we ¬nd that df § Π = 0; in the non-degenerate case, this implies df = uθ
for some function u. The de¬nition of a contact form prohibits any uθ from
being closed unless u = 0, meaning that f is a constant. This constant gives a
linear functional on gEΛ whose kernel is gΠ , proving (1.7). The area functional
and minimal surface equation for Euclidean hypersurfaces provide an example
where the two spaces are di¬erent. In that case, the induced Monge-Ampere
system is invariant not only under Euclidean motions, but under dilations of
Euclidean space as well; this is not true of the Poincar´-Cartan form.
e
The next step in introducing Noether™s theorem is to describe the relevant
spaces of conservation laws. In general, suppose that (M, J ) is an exterior dif-
ferential system with integral manifolds of dimension n. A conservation law for
(M, J ) is an (n ’ 1)-form • ∈ „¦n’1(M ) such that d(f — •) = 0 for every integral
manifold f : N n ’ M of J . Actually, we will only consider as conservation
laws those • on M such that d• ∈ J , which may be a strictly smaller set. This
will not present any liability, as one can always “saturate” J to remove this
discrepancy. The two apparent ways in which a conservation law may be trivial
are when either • ∈ J n’1 already or • is exact on M . Factoring out these cases
leads us to the following.
De¬nition 1.5 The space of conservation laws for (M, J ) is

C = H n’1(„¦— (M )/J ).

It also makes sense to factor out those conservation laws represented by
• ∈ „¦n’1(M ) which are already closed on M , and not merely on integral
manifolds of J . This can be understood using the long exact sequence:
π
· · · ’ HdR (M ) ’ C ’ H n(J ) ’ HdR (M ) ’ · · · .
n’1 n




¯ n’1
De¬nition 1.6 The space of proper conservation laws is C = C/π(HdR (M )).
¯
Note that there is an inclusion C ’ H n(J ). In case J = EΛ is the Euler-
Lagrange system of a non-degenerate functional Λ on a contact manifold (M, I),
we have the following.
´
16 CHAPTER 1. LAGRANGIANS AND POINCARE-CARTAN FORMS

Theorem 1.3 (Noether) Let (M, EΛ) be the Euler-Lagrange system of a non-
degenerate functional Λ. There is a linear isomorphism

· : gΠ ’ H n (EΛ ),
¯
taking the subalgebra g[Λ] ‚ gΠ to the subspace ·(g[Λ] ) = C ‚ H n(EΛ ).
Before proceeding to the proof, which will furnish an explicit formula for ·,
we need to make a digression on the algebra of in¬nitesimal contact transfor-
mations
gI = {v ∈ V(M ) : Lv I ⊆ I}.
The key facts are that on any neighborhood where I has a non-zero generator θ,
a contact symmetry v is uniquely determined by its so-called generating function
g = v θ, and that given such θ, any function g is the generating function of
some v ∈ gI . This can be seen on a possibly smaller neighborhood by taking
pidxi. Working in a basis ‚θ , ‚ i, ‚i dual to
Pfa¬ coordinates with θ = dz ’
the basis θ, dpi, dxi of T — M , we write

v i ‚i + vi ‚ i .
v = g ‚θ +

Now the condition
Lv θ ≡ 0 (mod {I})
can be made explicit, and it turns out to be

‚ ‚ ‚g
vi = ’‚ i g = ’
vi = ‚ i g = + pi g, .
i
‚x ‚z ‚pi

This establishes our claim, because the correspondence between v and g is now
given by

(gxi + pigz )‚ i .
v = g‚θ ’ g pi ‚ i + (1.8)

As we have presented it, the correspondence between in¬nitesimal contact
symmetries and their generating functions is local. But a simple patching ar-
gument shows that globally, as one moves between di¬erent local generators θ
for I, the di¬erent generating functions g glue together to give a global section
g ∈ “(M, I — ) of the dual line bundle. In fact, the formula (1.8) describes a
canonical splitting of the surjection

“(T M ) ’ “(I — ) ’ 0.

Note that this splitting is not a bundle map, but a di¬erential operator.
Returning to Noether™s theorem, the proof that we present is slightly in-
complete in that we assume given a global non-zero contact form θ ∈ “(I),
or equivalently, that the contact line bundle is trivial. This allows us to treat
generating functions of contact symmetries as functions rather than as sections
of I — . It is an enlightening exercise to develop the patching arguments needed
1.3. NOETHER™S THEOREM 17

to overcome this using sheaf cohomology. Alternatively, one can simply pull
everything up to a double cover of M on which I has a global generator, and
little will be lost.

Proof of Theorem 1.3. Step 1: De¬nition of the map ·. The map in question
is given by
for v ∈ gΠ ‚ V(M ).

<<

. 5
( 48 .)



>>