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О·(v) = v О
Note that locally v О  = (v Оё)ОЁ в€’ Оё в€§ (v О  lies in EО› .
ОЁ), so that v
Furthermore, the condition Lv О  = 0 gives

0=v dО  + d(v О ) = d(v О ),

О  is closed, and gives a well-deп¬Ѓned class О·(v) в€€ H n(EО› ).
so that v

Step 2: О· is injective. Write

Оё)ОЁ в€’ Оё в€§ (v
О·(v) = (v ОЁ).

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

Оё)ОЁ в€’ Оё в€§ (v d(Оё в€§ О± + dОё в€§ ОІ)
(v ОЁ) =
в€’Оё в€§ dО± + dОё в€§ (О± + dОІ).
=

Regarding this equation modulo {I} and using the primitivity of ОЁ, we conclude
that
v Оё = 0.
An inп¬Ѓnitesimal symmetry v в€€ gI of the contact system is locally determined
by its generating function v Оё as in (1.8), so we conclude that v = 0, proving
injectivity.

Step 3: О· is locally surjective. We start by representing a class in H n (EО› ) by a
closed n-form

О¦ = gОЁ + Оё в€§ О±. (1.9)

We can choose the unique contact vector п¬Ѓeld v such that v Оё = g, and our
goals are to show that v в€€ gО  and that О·(v) = [О¦] в€€ H n(EО› ).
For this, we need a special choice of ОЁ, 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 ОЁ by the condition

dОё в€§ ОЁ = 0.

To see why this is so, п¬Ѓrst note that by symplectic linear algebra (Proposi-
tion 1.1),

dОЁ в‰Ў dОё в€§ О“ (mod {I}), (1.10)
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18 CHAPTER 1. LAGRANGIANS AND POINCARE-CARTAN FORMS

for some О“, because dОЁ is of degree n + 1. Now suppose we replace ОЁ by
ВЇ ВЇ
ОЁ = ОЁ в€’ Оё в€§ О“, which certainly preserves the essential condition О  = Оё в€§ ОЁ.
Then we have
ВЇ
dОё в€§ ОЁ = dОё в€§ (ОЁ в€’ Оё в€§ О“)
= (dО  + Оё в€§ dОЁ) в€’ dОё в€§ Оё в€§ О“
= Оё в€§ (dОЁ в€’ dОё в€§ О“)
= 0, by (1.10),

and we have obtained our reп¬Ѓned normalization.
Now we combine the following three equations modulo {I}:
вЂў 0 в‰Ў Lv Оё в‰Ў dg + v dОё, when multiplied by ОЁ, gives

dg в€§ ОЁ + (v dОё) в€§ ОЁ в‰Ў 0 (mod {I});

вЂў 0 = dО¦ = d(gОЁ + Оё в€§ О±), so using our normalization condition dОё в€§ ОЁ = 0
(which implies dОЁ в‰Ў 0 (mod {I})),

dg в€§ ОЁ + dОё в€§ О± в‰Ў 0 (mod {I});

вЂў Ordinary primitivity gives dОё в€§ ОЁ в‰Ў 0 (mod {I}), and contracting with v,

dОё) в€§ ОЁ + dОё в€§ (v ОЁ) в‰Ў 0 (mod {I}).
(v

These three equations combine to give

dОё в€§ (О± + v ОЁ) в‰Ў 0 (mod {I}),

and from symplectic linear algebra, we have

ОЁ в‰Ў 0 (mod {I}).
О±+v

This allows us to conclude

О  = gОЁ + Оё в€§ О±
v (1.11)

which would complete the proof of surjectivity, except that we have not yet
shown that Lv О  = 0. However, by hypothesis gОЁ + Оё в€§ О± is closed; with (1.11),
this is enough to compute Lv О  = 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: О· maps symmetries of [О›] to proper conservation laws. For this, п¬Ѓrst
note that there is an exact sequence
i
ВЇ
0 в†’ C в†’ H n(EО› ) в†’ HdR (M ) в†’ В· В· В·
n
1.3. NOETHERвЂ™S THEOREM 19

so it suп¬ѓces to show that for v в€€ gО  ,

Lv [О›] = 0 if and only if О·(v) в€€ Ker i. (1.12)

Recall that О  = d(О› в€’ Оё в€§ ОІ) for some ОІ, and we can therefore calculate

О·(v) = vО
Lv (О› в€’ Оё в€§ ОІ) в€’ d(v (О› в€’ Оё в€§ ОІ))
=
Lv О› (mod dв„¦nв€’1(M ) + I n).
в‰Ў

This proves that j в—¦ i(О·(v)) = Lv [О›] in the composition
j
i
H n(EО› ) в†’ HdR (M ) в†’ H n (в„¦в€— (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 в†’ I в†’ в„¦в€— (M ) в†’ в„¦в€— (M )/I в†’ 0;
namely, we have
j
В· В· В· в†’ H n (I) в†’ HdR (M ) в†’ H n(в„¦в€— (M )/I) в†’ В· В· В· .
n

So it suп¬ѓces to show that H n(I) = 0, which we will do under the standing
assumption that there is a global contact form Оё. Suppose that the n-form

П• = Оё в€§ О± + dОё в€§ ОІ = Оё в€§ (О± + dОІ) + d(Оё в€§ ОІ)

is closed. Then regarding 0 = dП• modulo {I}, we have by symplectic linear
algebra that
О± + dОІ в‰Ў 0 (mod {I})
so that actually
П• = d(Оё в€§ ОІ).
This says that П• в€ј 0 in H n(I), and our proof is complete.

It is important in practice to have a local formula for a representative in
nв€’1
(M ), closed modulo EО› , for the proper conservation law О·(v). This is
в„¦
obtained by п¬Ѓrst writing as usual

О  = d(О› в€’ Оё в€§ ОІ), (1.13)

and also, for a given v в€€ g[О›] ,

Lv О› в‰Ў dОі (mod I). (1.14)

We will show that the (n в€’ 1)-form

П• = в€’v О› + (v Оё)ОІ + Оі (1.15)
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20 CHAPTER 1. LAGRANGIANS AND POINCARE-CARTAN FORMS

is satisfactory. First, compute
dП• = (в€’Lv О› + v dО›) + d((v Оё)ОІ) + dОі
в‰Ў v (О  + d(Оё в€§ ОІ)) + d((v Оё)ОІ) (mod I)
в‰Ў О·(v) + Lv (Оё в€§ ОІ) (mod I)
в‰Ў О·(v) (mod I).
Now we have dП• = О·(v) + Оћ for some closed Оћ в€€ I n. We proved in the last
part of the proof of NoetherвЂ™s theorem that H n(I) = 0, which implies that
Оћ = dОѕ for some Оѕ в€€ I nв€’1. Now we have d(П• в€’ Оѕ) = О·(v), and П• в€ј П• в€’ Оѕ in
C = H nв€’1(в„¦в€— (M )/EО› ). This justiп¬Ѓes our prescription (1.15).
Note that the prescription is especially simple when v в€€ gО› вЉ† g[О›] , for then
we can take Оі = 0.

Example. Let Ln+1 = {(t, y1 , . . . , yn )} в€ј 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 в‰¤ a в‰¤ n), Оё = dz в€’ p0 dt в€’ pi dyi . For a Lagrangian, take
i

1 2
2 ||p|| + F (z) dt в€§ dy
О›=
for some вЂњpotentialвЂќ function F (z), where dy = dy 1 в€§ В· В· В· в€§ dyn and ||p||2 =
в€’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 PoincarВґ-Cartan form is strictly larger. For now, we
e
calculate the conservation law corresponding to translation in t, and begin by
п¬Ѓnding the PoincarВґ-Cartan form О . Letting f(z) = F (z), we diп¬Ђerentiate
e
(в€’p0 dp0 + pi dpi + f(z)Оё) в€§ dt в€§ dy
dО› =
Оё в€§ (f(z)dt в€§ dy) + dОё в€§ (p0 dy + pj dt в€§ dy(j));
=
with the usual recipe О  = Оё в€§ (О± + dОІ) whenever dО› = Оё в€§ О± + dОё в€§ ОІ, we obtain
О  = Оё в€§ f(z)dt в€§ dy + dp0 в€§ dy + dpj в€§ dt в€§ dy(j) .
We see the Euler-Lagrange equation using
ОЁ = f(z)dt в€§ dy + dp0 в€§ dy + dpj в€§ dt в€§ dy(j);
an integral manifold of EО› = {Оё, dОё, ОЁ} of the form
{(t, y, z(t, y), zt (t, y), zyi (t, y))}
must satisfy
в€‚2z в€‚2z
в€’ + f(z) dt в€§ dy.
0 = ОЁ|N =
в€‚t2 в€‚yi2
With the independence condition dtв€§dy = 0, we have the familiar wave equation
z(t, y) = f(z).
1.4. HYPERSURFACES IN EUCLIDEAN SPACE 21

в€‚
в€€ gО›, the Noether
Now considering the time-translation symmetry v = в€‚t
prescription (1.15) gives

в€’v
П• = О› + (v Оё)ОІ
||p||2
в€’ + F (z) dy в€’ p0 p0dy + pj dt в€§ dy(j)
=
2
1
p2 + F (z) dy в€’ p0dt в€§ (
в€’
= pj dy(j) ).
a
2

One can verify that П• 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 в†’ 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 identiп¬Ѓed with En+1 itself via translation. A frame for En+1
is a pair
f = (x, e)
consisting of a point x в€€ 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)) В· (gb ) = (x, ( ea g0 , . . . , ea gn))

gives the basepoint map
x : F в†’ 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 identiп¬Ѓcation
F в€ј 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 в€€ En+1 , H n вЉ‚ Tx En+1 an oriented hyperplane}.
5 Throughout this section, we use index ranges 1 в‰¤ i, j в‰¤ n and 0 в‰¤ a, b в‰¤ n.
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22 CHAPTER 1. LAGRANGIANS AND POINCARE-CARTAN FORMS
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