variations, which asks whether a given PDE of the appropriate type is equivalent

to the Euler-Lagrange equation for some functional. We answer this by giving

a necessary and su¬cient condition for an EDS of the appropriate type to be

locally equivalent to the Euler-Lagrange system of some [Λ]. We ¬nd these

conditions by reducing the problem to a search for a Poincar´-Cartan form.

e

1.2.1 Variation of a Legendre Submanifold

Suppose that we have a 1-parameter family {Nt } of Legendre submanifolds of

a contact manifold (M, I); more precisely, this is given by a compact manifold

with boundary (N, ‚N ) and a smooth map

F : N — [0, 1] ’ M

which is a Legendre submanifold Ft for each ¬xed t ∈ [0, 1] and is independent

of t ∈ [0, 1] on ‚N — [0, 1]. Because Ft— θ = 0 for any contact form θ ∈ “(I), we

must have locally

F —θ = G dt (1.3)

´

8 CHAPTER 1. LAGRANGIANS AND POINCARE-CARTAN FORMS

for some function G on N — [0, 1]. We let g = G|N —{0} be the restriction to the

initial submanifold.

It will be useful to know that given a Legendre submanifold f : N ’ M ,

every function g may be realized as in (1.3) for some ¬xed-boundary variation

and some contact form θ, locally in the interior N o . This may be seen in Pfa¬

coordinates (xi , z, pi) on M , for which θ = dz ’ pi dxi generates I and such

that our given N is a 1-jet graph {(xi, z(x), pi(x) = zxi (x))}. Then (xi) give

coordinates on N , and a variation of N is of the form

F (x, t) = (xi , z(x, t), zxi (x, t)).

Now F —(dz ’ pidxi ) = zt dt; and given z(x, 0), we can always extend to z(x, t)

with g(x) = zt (x, 0) prescribed arbitrarily, which is what we claimed.

1.2.2 Calculation of the Euler-Lagrange System

We can now carry out a calculation that is fundamental for the whole theory.

Suppose given a Lagrangian Λ ∈ „¦n (M ) on a contact manifold (M, I), and a

¬xed-boundary variation of Legendre submanifold F : N — [0, 1] ’ M ; we wish

d

to compute dt ( Nt Λ).

To do this, ¬rst recall the calculation of the Poincar´-Cartan form for the

e

¯ n . Because I n+1 = „¦n+1 (M ), we can always write

equivalence class [Λ] ∈ H

θ § ± + dθ § β

dΛ =

θ § (± + dβ) + d(θ § β),

=

and then

Π = θ § (± + dβ) = d(Λ ’ θ § β). (1.4)

We are looking for conditions on a Legendre submanifold f : N ’ M to

be stationary for [Λ] under all ¬xed-boundary variations, in the sense that

d

( Nt Λ) = 0 whenever F |t=0 = f. We compute (without writing the

dt t=0

—

F s)

d d

(Λ ’ θ § β)

Λ=

dt dt

Nt Nt

L ‚ (Λ ’ θ § β)

=

‚t

Nt

‚ ‚

d(Λ ’ θ § β) + (Λ ’ θ § β)

= d

‚t ‚t

Nt Nt

‚

= Π (because ‚N is ¬xed).

‚t

Nt

One might express this result as

f — Π,

δ(FΛ )N (v) = v

N

1.2. THE EULER-LAGRANGE SYSTEM 9

where the variational vector ¬eld v, lying in the space “0(f — T M ) of sections of

‚

f — T M vanishing along ‚N , plays the role of ‚t . The condition Π ≡ 0 (mod {I})

allows us to write Π = θ § Ψ for some n-form Ψ, not uniquely determined, and

we have

d

g f — Ψ,

Λ=

dt t=0 Nt N

‚

where g = ( ‚t F —θ)|t=0 . It was shown previously that this g could locally be

chosen arbitrarily in the interior N o , so the necessary and su¬cient condition

for a Legendre submanifold f : N ’ M to be stationary for FΛ is that f — Ψ = 0.

De¬nition 1.3 The Euler-Lagrange system of the Lagrangian Λ is the di¬er-

ential ideal generated algebraically as

EΛ = {θ, dθ, Ψ} ‚ „¦— (M ).

A stationary Legendre submanifold of Λ is an integral manifold of EΛ . The

functional is said to be non-degenerate if its Poincar´-Cartan form Π = θ § Ψ

e

has no degree-1 divisors (in the exterior algebra of T — M ) other than multiples

of θ.

Note ¬rst that EΛ is uniquely determined by Π, even though θ and Ψ may

not be.4 Note also that the ideal in „¦— (M ) algebraically generated by {θ, dθ, Ψ}

is already di¬erentially closed, because dΨ ∈ „¦n+1 (M ) = I n+1.

We can examine this for the classical situation where M = {(xi, z, pi)},

θ = dz ’ pi dxi, and Λ = L(x, z, p)dx. We ¬nd

Lz θ § dx + Lpi dpi § dx

dΛ =

θ § Lz dx ’ dθ § Lpi dx(i),

=

so referring to (1.4),

Π = θ § (Lz dx ’ d(Lpi dx(i))) = θ § Ψ.

Now, for a transverse Legendre submanifold N = {(xi , z(x), zxi (x))}, we have

Ψ|N = 0 if and only if along N

‚L d ‚L

’ = 0,

dxi

‚z ‚pi

4 Actually, given Π we have not only a well-de¬ned EΛ , but a well-de¬ned Ψ modulo {I}

¯

which is primitive on I ⊥ . There is a canonical map E : H n („¦— ) ’ P n (T — M/I) to the space

of primitive forms, taking a Lagrangian class [Λ] to the corresponding Ψ in its Euler-Lagrange

system; and this map ¬ts into a full resolution of the constant sheaf

E

0 ’ R ’ „¦0 ’ · · · ’ „¦n’1 ’ H n („¦— ) ’ P n (T — M/I) ’ · · · ’ P 0 (T — M/I) ’ 0.

¯ ¯ ¯

This has been developed and applied in the context of CR geometry in [Rum90].

´

10 CHAPTER 1. LAGRANGIANS AND POINCARE-CARTAN FORMS

where

d ‚ ‚ ‚

= + z xi + zxi xj

dxi ‚xi ‚z ‚pj

j

is the total derivative. This is the usual Euler-Lagrange equation, a second-

order, quasi-linear PDE for z(x1, . . . , xn) having symbol Lpi pj . It is an exercise

to show that this symbol matrix is invertible at (xi, z, pi) if and only if Λ is

non-degenerate in the sense of De¬nition 1.3.

1.2.3 The Inverse Problem

There is a reasonable model for exterior di¬erential systems of “Euler-Lagrange

type”.

De¬nition 1.4 A Monge-Ampere di¬erential system (M, E) consists of a con-

tact manifold (M, I) of dimension 2n + 1, together with a di¬erential ideal

E ‚ „¦— (M ), generated locally by the contact ideal I and an n-form Ψ ∈ „¦ n (M ).

Note that in this de¬nition, the contact line bundle I can be recovered from E

as its degree-1 part. We can now pose a famous question.

Inverse Problem: When is a given Monge-Ampere system E on M equal to

the Euler-Lagrange system EΛ of some Lagrangian Λ ∈ „¦n (M )?

Note that if a given E does equal EΛ for some Λ, then for some local genera-

tors θ, Ψ of E we must have θ § Ψ = Π, the Poincar´-Cartan form of Λ. Indeed,

e

we can say that (M, E) is Euler-Lagrange if and only if there is an exact form

Π ∈ „¦n+1 (M ), locally of the form θ § Ψ for some generators θ, Ψ of E. How-

ever, we face the di¬culty that (M, E) does not determine either Ψ ∈ „¦n(M )

or θ ∈ “(I) uniquely.

This can be partially overcome by normalizing Ψ as follows. Given only

¯

(M, E = {θ, dθ, Ψ}), Ψ is determined as an element of „¦n = „¦n(M )/I n. We

can obtain a representative Ψ that is unique modulo {I} by adding the unique

multiple of dθ that yields a primitive form on I ⊥ , referring to the symplectic

n

(T — M/I) (see Proposition 1.1). With this choice, we have

decomposition of

a form θ § Ψ which is uniquely determined up to scaling; the various multiples

fθ § Ψ, where f is a locally de¬ned function on M , are the candidates to be

Poincar´-Cartan form. Note that using a primitive normalization is reasonable,

e

because our actual Poincar´-Cartan forms Π = θ § Ψ satisfy dΠ = 0, which in

e

particular implies that Ψ is primitive on I ⊥ . The proof of Noether™s theorem in

the next section will use a more re¬ned normalization of Ψ.

The condition for a Monge-Ampere system to be Euler-Lagrange is therefore

that there should be a globally de¬ned exact n-form Π, locally of the form fθ§Ψ

with Ψ normalized as above. This suggests the more accessible local inverse

problem, which asks whether there is a closed n-form that is locally expressible

as fθ § Ψ. It is for this local version that we give a criterion.

1.2. THE EULER-LAGRANGE SYSTEM 11

We start with any candidate Poincar´-Cartan form Ξ = θ § Ψ, and consider

e

the following criterion on Ξ:

dΞ = • § Ξ for some • with d• ≡ 0 (mod I). (1.5)

We ¬rst note that if this holds for some choice of Ξ = θ § Ψ, then it holds

for all other choices fΞ; this is easily veri¬ed.

Second, we claim that if (1.5) holds, then we can ¬nd • also satisfying

˜

dΞ = • § Ξ, and in addition, d• = 0. To see this, write

˜ ˜

d• = θ § ± + β dθ

(here ± is a 1-form and β is a function), and di¬erentiate using d2 = 0, modulo

the algebraic ideal {I}, to obtain

0 ≡ dθ § (± + dβ) (mod {I}).

But with the standing assumption n ≥ 2, symplectic linear algebra implies that

the 1-form ± + dβ must vanish modulo {I}. As a result,

d(• ’ β θ) = θ § (± + dβ) = 0,

so we can take • = • ’ β θ, verifying the claim.

˜

Third, once we know that dΞ = •§Ξ with d• = 0, then on a possibly smaller

neighborhood, we use the Poincar´ lemma to write • = du for a function u, and

e

then

d(e’uΞ) = e’u (• § Ξ ’ du § Ξ) = 0.

This proves the following.

Theorem 1.2 A Monge-Ampere system (M, E = {θ, dθ, Ψ}) on a (2n + 1)-

dimensional contact manifold M with n ≥ 2, where Ψ is assumed to be primitive

modulo {I}, is locally equal to an Euler-Lagrange system E Λ if and only if it

satis¬es (1.5).

Example 1. Consider a scalar PDE of the form

∆z = f(x, z, z), (1.6)

2

‚

; we ask which functions f : R2n+1 ’ R are such that (1.6)

where ∆ = ‚xi2

is contact-equivalent to an Euler-Lagrange equation. To apply our framework,

we let M = J 1(Rn , R), θ = dz ’ pidxi so dθ = ’ dpi § dxi, and set

dpi § dx(i) ’ f(x, z, p)dx.

Ψ=

‚z

Restricted to a Legendre submanifold of the form N = {(xi, z(x), ‚xi (x)}, we

¬nd

Ψ|N = (∆z ’ f(x, z, z))dx.

´

12 CHAPTER 1. LAGRANGIANS AND POINCARE-CARTAN FORMS

Evidently Ψ is primitive modulo {I}, and E = {θ, dθ, Ψ} is a Monge-Ampere

system whose transverse integral manifolds (i.e., those on which dx1 §· · ·§dxn =

0) correspond to solutions of the equation (1.6). To apply our test, we start with

the candidate Ξ = θ § Ψ, for which