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 ).