Note that locally v Π = (v θ)Ψ ’ θ § (v Π lies in EΛ .

Ψ), so that v

Furthermore, the condition Lv Π = 0 gives

0=v dΠ + d(v Π) = d(v Π),

Π is closed, and gives a well-de¬ned class ·(v) ∈ H n(EΛ ).

so that v

Step 2: · is injective. Write

θ)Ψ ’ θ § (v

·(v) = (v Ψ).

Suppose that this n-form is cohomologous to zero in H n(EΛ ); that is,

θ)Ψ ’ θ § (v d(θ § ± + dθ § β)

(v Ψ) =

’θ § d± + dθ § (± + dβ).

=

Regarding this equation modulo {I} and using the primitivity of Ψ, we conclude

that

v θ = 0.

An in¬nitesimal symmetry v ∈ gI of the contact system is locally determined

by its generating function v θ as in (1.8), so we conclude that v = 0, proving

injectivity.

Step 3: · is locally surjective. We start by representing a class in H n (EΛ ) by a

closed n-form

¦ = gΨ + θ § ±. (1.9)

We can choose the unique contact vector ¬eld v such that v θ = g, and our

goals are to show that v ∈ gΠ and that ·(v) = [¦] ∈ H n(EΛ ).

For this, we need a special choice of Ψ, which so far is determined only

modulo {I}; this is reasonable because the presentation (1.9) is not unique. In

fact, we can further normalize Ψ by the condition

dθ § Ψ = 0.

To see why this is so, ¬rst note that by symplectic linear algebra (Proposi-

tion 1.1),

dΨ ≡ dθ § “ (mod {I}), (1.10)

´

18 CHAPTER 1. LAGRANGIANS AND POINCARE-CARTAN FORMS

for some “, because dΨ is of degree n + 1. Now suppose we replace Ψ by

¯ ¯

Ψ = Ψ ’ θ § “, which certainly preserves the essential condition Π = θ § Ψ.

Then we have

¯

dθ § Ψ = dθ § (Ψ ’ θ § “)

= (dΠ + θ § dΨ) ’ dθ § θ § “

= θ § (dΨ ’ dθ § “)

= 0, by (1.10),

and we have obtained our re¬ned normalization.

Now we combine the following three equations modulo {I}:

• 0 ≡ Lv θ ≡ dg + v dθ, when multiplied by Ψ, gives

dg § Ψ + (v dθ) § Ψ ≡ 0 (mod {I});

• 0 = d¦ = d(gΨ + θ § ±), so using our normalization condition dθ § Ψ = 0

(which implies dΨ ≡ 0 (mod {I})),

dg § Ψ + dθ § ± ≡ 0 (mod {I});

• Ordinary primitivity gives dθ § Ψ ≡ 0 (mod {I}), and contracting with v,

dθ) § Ψ + dθ § (v Ψ) ≡ 0 (mod {I}).

(v

These three equations combine to give

dθ § (± + v Ψ) ≡ 0 (mod {I}),

and from symplectic linear algebra, we have

Ψ ≡ 0 (mod {I}).

±+v

This allows us to conclude

Π = gΨ + θ § ±

v (1.11)

which would complete the proof of surjectivity, except that we have not yet

shown that Lv Π = 0. However, by hypothesis gΨ + θ § ± is closed; with (1.11),

this is enough to compute Lv Π = 0.

The global isomorphism asserted in the theorem follows easily from these

local conclusions, so long as we maintain the assumption that there exists a

global contact form.

Step 4: · maps symmetries of [Λ] to proper conservation laws. For this, ¬rst

note that there is an exact sequence

i

¯

0 ’ C ’ H n(EΛ ) ’ HdR (M ) ’ · · ·

n

1.3. NOETHER™S THEOREM 19

so it su¬ces to show that for v ∈ gΠ ,

Lv [Λ] = 0 if and only if ·(v) ∈ Ker i. (1.12)

Recall that Π = d(Λ ’ θ § β) for some β, and we can therefore calculate

·(v) = vΠ

Lv (Λ ’ θ § β) ’ d(v (Λ ’ θ § β))

=

Lv Λ (mod d„¦n’1(M ) + I n).

≡

This proves that j —¦ i(·(v)) = Lv [Λ] in the composition

j

i

H n(EΛ ) ’ HdR (M ) ’ H n („¦— (M )/I).

n

The conclusion (1.12) will follow if we can prove that j is injective.

To see that j is injective, note that it occurs in the long exact cohomology

sequence of

0 ’ I ’ „¦— (M ) ’ „¦— (M )/I ’ 0;

namely, we have

j

· · · ’ H n (I) ’ HdR (M ) ’ H n(„¦— (M )/I) ’ · · · .

n

So it su¬ces to show that H n(I) = 0, which we will do under the standing

assumption that there is a global contact form θ. Suppose that the n-form

• = θ § ± + dθ § β = θ § (± + dβ) + d(θ § β)

is closed. Then regarding 0 = d• modulo {I}, we have by symplectic linear

algebra that

± + dβ ≡ 0 (mod {I})

so that actually

• = d(θ § β).

This says that • ∼ 0 in H n(I), and our proof is complete.

It is important in practice to have a local formula for a representative in

n’1

(M ), closed modulo EΛ , for the proper conservation law ·(v). This is

„¦

obtained by ¬rst writing as usual

Π = d(Λ ’ θ § β), (1.13)

and also, for a given v ∈ g[Λ] ,

Lv Λ ≡ dγ (mod I). (1.14)

We will show that the (n ’ 1)-form

• = ’v Λ + (v θ)β + γ (1.15)

´

20 CHAPTER 1. LAGRANGIANS AND POINCARE-CARTAN FORMS

is satisfactory. First, compute

d• = (’Lv Λ + v dΛ) + d((v θ)β) + dγ

≡ v (Π + d(θ § β)) + d((v θ)β) (mod I)

≡ ·(v) + Lv (θ § β) (mod I)

≡ ·(v) (mod I).

Now we have d• = ·(v) + Ξ for some closed Ξ ∈ I n. We proved in the last

part of the proof of Noether™s theorem that H n(I) = 0, which implies that

Ξ = dξ for some ξ ∈ I n’1. Now we have d(• ’ ξ) = ·(v), and • ∼ • ’ ξ in

C = H n’1(„¦— (M )/EΛ ). This justi¬es our prescription (1.15).

Note that the prescription is especially simple when v ∈ gΛ ⊆ g[Λ] , for then

we can take γ = 0.

Example. Let Ln+1 = {(t, y1 , . . . , yn )} ∼ Rn+1 be Minkowski space, and

=

2n+3 1 n+1

let M = J (L , R) be the standard contact manifold, with coordinates

(t, y , z, pa) (where 0 ¤ a ¤ n), θ = dz ’ p0 dt ’ pi dyi . For a Lagrangian, take

i

1 2

2 ||p|| + F (z) dt § dy

Λ=

for some “potential” function F (z), where dy = dy 1 § · · · § dyn and ||p||2 =

’p2 + p2 is the Lorentz-signature norm. The local symmetry group of this

0 i

functional is generated by two subgroups, the translations in Ln+1 and the linear

isometries SOo (1, n); as we shall see in Chapter 3, for certain F (z) the symmetry

group of the associated Poincar´-Cartan form is strictly larger. For now, we

e

calculate the conservation law corresponding to translation in t, and begin by

¬nding the Poincar´-Cartan form Π. Letting f(z) = F (z), we di¬erentiate

e

(’p0 dp0 + pi dpi + f(z)θ) § dt § dy

dΛ =

θ § (f(z)dt § dy) + dθ § (p0 dy + pj dt § dy(j));

=

with the usual recipe Π = θ § (± + dβ) whenever dΛ = θ § ± + dθ § β, we obtain

Π = θ § f(z)dt § dy + dp0 § dy + dpj § dt § dy(j) .

We see the Euler-Lagrange equation using

Ψ = f(z)dt § dy + dp0 § dy + dpj § dt § dy(j);

an integral manifold of EΛ = {θ, dθ, Ψ} of the form

{(t, y, z(t, y), zt (t, y), zyi (t, y))}

must satisfy

‚2z ‚2z

’ + f(z) dt § dy.

0 = Ψ|N =

‚t2 ‚yi2

With the independence condition dt§dy = 0, we have the familiar wave equation

z(t, y) = f(z).

1.4. HYPERSURFACES IN EUCLIDEAN SPACE 21

‚

∈ gΛ, the Noether

Now considering the time-translation symmetry v = ‚t

prescription (1.15) gives

’v

• = Λ + (v θ)β

||p||2

’ + F (z) dy ’ p0 p0dy + pj dt § dy(j)

=

2

1

p2 + F (z) dy ’ p0dt § (

’

= pj dy(j) ).

a

2

One can verify that • is closed when restricted to a solution of z = f(z). The

question of how one might use this conservation law will be taken up later.

1.4 Hypersurfaces in Euclidean Space

We will apply the the theory developed so far to the study of hypersurfaces in

Euclidean space

N n ’ En+1 .

We are particularly interested in the study of those functionals on such hyper-

surfaces which are invariant under the group E(n + 1) of orientation-preserving

Euclidean motions.

The Contact Manifold over En+1

1.4.1

Points of En+1 will be denoted x = (x0 , . . . , xn), and each tangent space Tx En+1

will be canonically identi¬ed with En+1 itself via translation. A frame for En+1

is a pair

f = (x, e)

consisting of a point x ∈ En+1 and a positively-oriented orthonormal basis

e = (e0 , . . . , en ) for Tx En+1 . The set F of all such frames is a manifold, and

the right SO(n + 1, R)-action

a a a

(x, (e0, . . . , en)) · (gb ) = (x, ( ea g0 , . . . , ea gn))

gives the basepoint map

x : F ’ En+1

the structure of a principal bundle.5 There is also an obvious left-action of E(n+

1, R) on F, and a choice of reference frame gives a left-equivariant identi¬cation

F ∼ E(n + 1) of the bundle of frames with the group of Euclidean motions.

=

The relevant contact manifold for studying hypersurfaces in En+1 is the

manifold of contact elements

M 2n+1 = {(x, H) : x ∈ En+1 , H n ‚ Tx En+1 an oriented hyperplane}.

5 Throughout this section, we use index ranges 1 ¤ i, j ¤ n and 0 ¤ a, b ¤ n.

´

22 CHAPTER 1. LAGRANGIANS AND POINCARE-CARTAN FORMS