C 1 -close to N0 satis¬es the independence condition

dx1 § · · · § dxn = 0,

and can therefore be described locally as a graph

N = {(xi , z(x), pi(x))}.

1 Recall

our convention that braces {·} denote the algebraic ideal generated by an object;

for instance, {θ} consists of exterior multiples of any contact form θ, and is smaller than I.

We sometimes use {I} as alternate notation for {θ}.

1.1. LAGRANGIANS AND CONTACT GEOMETRY 3

In this case, we have

‚z

θ|N = 0 if and only if pi (x) = (x).

‚xi

Therefore, N is determined by the function z(x), and conversely, every function

z(x) determines such an N ; we informally say that “the generic Legendre sub-

manifold depends locally on one arbitrary function of n variables.” Legendre

submanifolds of this form, with dx|N = 0, will often be described as transverse.

Motivated by (1) in the Introduction, we are primarily interested in func-

tionals given by triples (M, I, Λ), where (M, I) is a (2n + 1)-dimensional contact

manifold, and Λ ∈ „¦n (M ) is a di¬erential form of degree n on M ; such a Λ will

be referred to as a Lagrangian on (M, I).2 We then de¬ne a functional on the

set of smooth, compact Legendre submanifolds N ‚ M , possibly with boundary

‚N , by

FΛ (N ) = Λ.

N

The classical variational problems described above may be recovered from this

notion by taking M = J 1(Rn , R) ∼ R2n+1 with coordinates (xi , z, pi), I gener-

=

ated by θ = dz ’ pidx , and Λ = L(xi , z, pi)dx. This formulation also admits

i

certain functionals depending on second derivatives of z(x), because there may

be dpi -terms in Λ. Later, we will restrict attention to a class of functionals

which, possibly after a contact transformation, can be expressed without sec-

ond derivatives.

There are two standard notions of equivalence for Lagrangians Λ. First,

note that if the di¬erence Λ ’ Λ of two Lagrangians lies in the contact ideal I

then the functionals FΛ and FΛ are equal, because they are de¬ned only for

Legendre submanifolds, on which all forms in I vanish. Second, suppose that

the di¬erence of two Lagrangians is an exact n-form, Λ ’ Λ = d• for some

• ∈ „¦n’1(M ). Then we ¬nd

FΛ(N ) = FΛ (N ) + •

‚N

for all Legendre submanifolds N . One typically studies the variation of FΛ along

1-parameter families Nt with ¬xed boundary, and the preceding equation shows

that FΛ and FΛ di¬er only by a constant on such a family. Such Λ and Λ are

sometimes said to be divergence-equivalent.

These two notions of equivalence suggest that we consider the class

[Λ] ∈ „¦n (M )/(I n + d„¦n’1(M )),

where I n = I © „¦n (M ). The natural setting for this space is the quotient

¯¯ ¯

¯

(„¦— , d) of the de Rham complex („¦— (M ), d), where „¦n = „¦n (M )/I n, and d

is induced by the usual exterior derivative d on this quotient. We then have

2 In the Introduction, we used the term Lagrangian for a function, rather than for a di¬er-

ential form, but we will not do so again.

´

4 CHAPTER 1. LAGRANGIANS AND POINCARE-CARTAN FORMS

¯ ¯¯

characteristic cohomology groups H n = H n („¦— , d). We will show in a moment

that (recalling dim(M ) = 2n + 1):

for k > n, I k = „¦k (M ). (1.1)

In other words, all forms on M of degree greater than n lie in the contact

ideal; one consequence is that I can have no integral manifolds of dimension

greater than n. The importance of (1.1) is that it implies that dΛ ∈ I n+1, and

we can therefore regard our equivalence class of functionals as a characteristic

cohomology class

¯

[Λ] ∈ H n.

This class is almost, but not quite, our fundamental object of study.

To prove both (1.1) and several later results, we need to describe some of the

pointwise linear algebra associated with the contact ideal I = {θ, dθ} ‚ „¦— (M ).

Consider the tangent distribution of rank 2n

I⊥ ‚ T M

given by the annihilator of the contact line bundle. Then the non-degeneracy

condition on I implies that the 2-form

def

˜ = dθ

restricts ¬berwise to I ⊥ as a non-degenerate, alternating bilinear form, deter-

mined by I up to scaling. This allows one to use tools from symplectic linear

algebra; the main fact is the following.

2

V—

Proposition 1.1 Let (V 2n, ˜) be a symplectic vector space, where ˜ ∈

is a non-degenerate alternating bilinear form. Then

(a) for 0 ¤ k ¤ n, the map

n’k n+k

V— ’ V—

(˜§)k : (1.2)

is an isomorphism, and

(b) if we de¬ne the space of primitive forms to be

n’k n+k+2

P n’k (V — ) = Ker ((˜§)k+1 : V— ’ V — ),

then we have a decomposition of Sp(n, R)-modules

n’k n’k’2

V — = P n’k(V — ) • ˜ § V — .3

3 Of

course, we can extend this decomposition inductively to obtain the Hodge-Lepage

decomposition

P 1 (V — ) = V — for n ’ k odd,

V — ∼ P n’k (V — ) • P n’k’2 (V — ) • · · · •

n’k

=

P 0 (V — ) = R for n ’ k even,

1.1. LAGRANGIANS AND CONTACT GEOMETRY 5

Proposition 1.1 implies in particular (1.1), for it says that modulo {θ} (equiv-

alently, restricted to I ⊥ ), every form • of degree greater than n is a multiple of

dθ, which is exactly to say that • is in the algebraic ideal generated by θ and

dθ.

Proof. (a) Because n’k V — and n+k V — have the same dimension, it su¬ces

to show that the map (1.2) is injective. We proceed by induction on k, downward

from k = n to k = 0. In case k = n, the (1.2) is just multiplication

2n

V —,

(˜n )· : R ’

which is obviously injective, because ˜ is non-degenerate.

Now suppose that the statement is proved for some k, and suppose that

n’(k’1)

ξ∈ satis¬es

˜k’1 § ξ = 0.

This implies that

˜k § ξ = 0,

so that for every vector X ∈ V , we have

(˜k § ξ) = k(X ˜) § ˜k’1 § ξ + ˜k § (X

0=X ξ).

Now, the ¬rst term on the right-hand side vanishes by our assumption on ξ (our

second use of this assumption), so we must have

0 = ˜k § (X ξ),

and the induction hypothesis then gives

X ξ = 0.

This is true for every X ∈ V , so we conclude that ξ = 0.

n’k

V — has a unique decomposition as the sum of

(b) We will show that any ξ ∈

a primitive form and a multiple of ˜. For the existence of such a decomposition,

we apply the surjectivity in part (a) to the element ˜k+1 § ξ ∈ n+k+2 V — , and

n’k’2 —

¬nd · ∈ V for which

˜k+2 § · = ˜k+1 § ξ.

n’k

V — can be written uniquely as

under which any element ξ ∈

n’k

ξ = ξ0 + (˜ § ξ1 ) + (˜2 § ξ2 ) + · · · + §ξ

˜ ,

2 n’k

2

with each ξi ∈ P n’k’2i (V — ). What we will not prove here is that the representation

of Sp(n, R) on P n’k (V — ) is irreducible for each k, so this gives the complete irreducible

decomposition of n’k (V — ).

´

6 CHAPTER 1. LAGRANGIANS AND POINCARE-CARTAN FORMS

Then we can decompose

ξ = (ξ ’ ˜ § ·) + (˜ § ·),

where the ¬rst summand is primitive by construction.

To prove uniqueness, we need to show that if ˜ § · is primitive for some

n’k’2

(V — ), then ˜ § · = 0. In fact, primitivity means

·∈

0 = ˜k § ˜ § ·,

which implies that · = 0 by the injectivity in part (a).

Returning to our discussion of Lagrangian functionals, observe that there is

a short exact sequence of complexes

¯

0 ’ I — ’ „¦— (M ) ’ „¦— ’ 0

giving a long exact cohomology sequence

¯δ

· · · ’ HdR (M ) ’ H n ’ H n+1 (I) ’ HdR (M ) ’ · · · ,

n n+1

where δ is essentially exterior di¬erentiation. Although an equivalence class

¯

[Λ] ∈ H n generally has no canonical representative di¬erential form, we can

now show that its image δ([Λ]) ∈ H n+1 (I) does.

Theorem 1.1 Any class [Π] ∈ H n+1(I) has a unique global representative

closed form Π ∈ I n+1 satisfying θ § Π = 0 for any contact form θ ∈ “(I),

or equivalently, Π ≡ 0 (mod {I}).

Proof. Any Π ∈ I n+1 may be written locally as

Π = θ § ± + dθ § β

for some ± ∈ „¦n (M ), β ∈ „¦n’1(M ). But this is the same as

Π = θ § (± + dβ) + d(θ § β),

so replacing Π with the equivalent (in H n+1(I)) form Π ’ d(θ § β), we have the

local existence of a representative as claimed.

For uniqueness, suppose that Π1 ’ Π2 = d(θ § γ) for some (n ’ 1)-form γ

(this is exactly equivalence in H n+1(I)), and that θ § Π1 = θ § Π2 = 0. Then

θ § dθ § γ = 0, so dθ § γ ≡ 0 (mod {I}). By symplectic linear algebra, this

implies that γ ≡ 0 (mod {I}), so Π1 ’ Π2 = 0.

Finally, global existence follows from local existence and uniqueness.

We can now de¬ne our main object of study.

De¬nition 1.2 For a contact manifold (M, I) with Lagrangian Λ, the unique

representative Π ∈ I n+1 of δ([Λ]) satisfying Π ≡ 0 (mod {I}) is called the

Poincar´-Cartan form of Λ.

e

1.2. THE EULER-LAGRANGE SYSTEM 7

Poincar´-Cartan forms of Lagrangians will be the main object of study in

e

these lectures, and there are two computationally useful ways to think of them.

The ¬rst is as above: given a representative Lagrangian Λ, express dΛ locally

as θ § (± + dβ) + d(θ § β), and then

Π = θ § (± + dβ).

The second, which will be important for computing the ¬rst variation and the

Euler-Lagrange system of [Λ], is as an exact form:

Π = d(Λ ’ θ § β).

In fact, β is the unique (n ’ 1)-form modulo {I} such that

d(Λ ’ θ § β) ≡ 0 (mod {I}).

This observation will be used later, in the proof of Noether™s theorem.

1.2 The Euler-Lagrange System

In the preceding section, we showed how one can associate to an equivalence

class [Λ] of Lagrangians on a contact manifold (M, I) a canonical (n + 1)-form

Π. In this section, we use this Poincar´-Cartan form to ¬nd an exterior dif-

e

ferential system whose integral manifolds are precisely the stationary Legendre

submanifolds for the functional FΛ . This requires us to calculate the ¬rst vari-

ation of FΛ , which gives the derivative of FΛ (Nt ) for any 1-parameter family

Nt of Legendre submanifolds of (M, I). The Poincar´-Cartan form enables us

e

to carry out the usual integration by parts for this calculation in an invariant

manner.