An equivalence class [Λ] of Lagrangians corresponds to a section of the vector

n

bundle (J/I). It also de¬nes a functional on the space of compact integral

manifolds (possibly with boundary) of the Pfa¬an system I by

FΛ (N ) = Λ,

N

where Λ is any representative of the class. The notion of divergence equivalence

of Lagrangians will appear later. In the discussion in Chapter 1 of the scalar case

s = 1, we combined these two types of equivalence by emphasizing a character-

istic cohomology class in H n(„¦— (M )/I), and used facts about symplectic linear

algebra to investigate these classes. However, the analogous “multi-symplectic”

linear algebra that is appropriate for the study of multi-contact geometry is still

poorly understood.3

n

Our goal is to associate to any functional [Λ] ∈ “(M, (J/I)) a Lagrangian

n n

Λ ∈ “(M, J) ‚ „¦ (M ), not necessarily uniquely determined, whose exterior

derivative Π = dΛ has certain favorable properties and is uniquely determined

by [Λ]. Among these properties are:

3 The recent work [Gra00] of M. Grassi may illuminate this issue, along with some others

that will come up in the following discussion.

156 CHAPTER 4. ADDITIONAL TOPICS

• Π ≡ 0 (mod {I});

• Π is preserved under any di¬eomorphism of M preserving I, J, and [Λ];

• Π depends only on the divergence-equivalence class of [Λ];

• Π = 0 if and only if the Euler-Lagrange equations for [Λ] are trivial.

Triviality of the Euler-Lagrange equations means that every compact integral

manifold of I ‚ „¦— (M ) is stationary for FΛ under ¬xed-boundary variations.

Some less obvious ways in which such Π could be useful are the following, based

on our experience in the scalar case s = 1:

• in Noether™s theorem, where one would hope for v ’ v Π to give an

isomorphism from a Lie algebra of symmetries to a space of conservation

laws;

• in the inverse problem, where one can try to detect equations that are

locally of Euler-Lagrange type not by ¬nding a Lagrangian, but by ¬nding

a Poincar´-Cartan form inducing the equations;

e

• in the study of local minimization, where it could help one obtain a cali-

bration in terms of a ¬eld of stationary submanifolds.

Recall that in the case of a contact manifold, we replaced any Lagrangian

Λ ∈ „¦n (M ) by

Λ ’ θ § β,

the unique form congruent to Λ (mod {I}) with the property that

dΛ ≡ 0 (mod {I}).

n

What happens in the multi-contact case? Any Lagrangian Λ0 ∈ “(M, J) is

congruent modulo {I} to a form (in local coordinates)

L(xi , z ±, p±)dx,

i

and motivated by the scalar case, we consider the equivalent form

Λ = L dx + θ± § ‚L

‚p± dx(i) , (4.26)

i

which has exterior derivative

dΛ = θ± § ‚L ‚L

‚z ± dx ’ § dx(i) .

d (4.27)

‚p±

i

This suggests the following de¬nition.

De¬nition 4.5 An admissible lifting of a functional [Λ] ∈ “(M, n(J/I)) is a

n

Lagrangian Λ ∈ “(M, J) representing the class [Λ] and satisfying dΛ ∈ {I}.

4.2. EULER-LAGRANGE PDE SYSTEMS 157

The preceding calculation shows that locally, every functional [Λ] has an ad-

missible lifting. Unfortunately, the admissible lifting is generally not unique.

This will be addressed below, but ¬rst we show that any admissible lifting is

adequate for calculating the ¬rst variation and the Euler-Lagrange system of

the functional FΛ.

We mimic the derivation in Chapter 1 of the Euler-Lagrange di¬erential

system in the scalar case s = 1. Suppose that we have a 1-parameter family

{Nt } of integral manifolds of the multi-contact Pfa¬an system I, given as a

smooth map

F : N — [0, 1] ’ M,

for which each Ft = F |N —{t} : N ’ M is an integral manifold of I and such that

F |‚N —[0,1] is independent of t. Then choosing generators θ ± ∈ “(I), 1 ¤ ± ¤ s,

we have

F — θ± = G±dt

for some functions G± on N — [0, 1]. As in the contact case, it is not di¬cult to

show that any collection of functions g ± supported in the interior of N can be

realized as G±|t=0 for some 1-parameter family Nt .

The hypothesis that Λ is an admissible lifting means that we can write

θ ± § Ψ±

dΛ =

for some Ψ± ∈ „¦n(M ). Then we can proceed as in §1.2 to calculate

d

Ft— Λ L ‚ (F — Λ)

=

dt ‚t

Nt N0

t=0

(θ± § Ψ± ) +

‚ ‚

= d( ‚t Λ)

‚t

N0 N0

g ± Ψ± ,

=

N0

where in the last step we used the ¬xed-boundary condition, the vanishing of

—

F0 θ± , and the de¬nition g± = G±|N0 . Now the same reasoning as in §1.2 shows

that F0 : N ’ M is stationary for FΛ under all ¬xed-boundary variations if

and only if Ψ± |N0 = 0 for all ± = 1, . . . , s.

We now have a di¬erential system {θ ± , dθ±, Ψ±} whose integral manifolds are

exactly the integral manifolds of I that are stationary for [Λ], but it is not clear

that this system is uniquely determined by [Λ] alone; we might get di¬erent

systems for di¬erent admissible liftings. To rule out this possibility, observe

¬rst that if Λ, Λ are any two admissible liftings of [Λ], then the condition

Λ ’ Λ ∈ {I} allows us to write Λ ’ Λ = θ± § γ± , and then the fact that

dΛ ≡ dΛ ≡ 0 (mod {I}) along with the structure equations (4.24) allows us to

write

0 ≡ dθ± § γ± ≡ ’πi § ωi § γ± (mod {I}).

±

When n ≥ 2, this implies that γ± ≡ 0 (mod {I}), so while two general repre-

sentatives of [Λ] need be congruent only modulo {I}, for admissible liftings we

have the following.

158 CHAPTER 4. ADDITIONAL TOPICS

n

Proposition 4.5 Two admissible liftings of the same [Λ] ∈ “( (J/I)) are

2

congruent modulo { I}.

2

Of course, when s = 1, I = 0 and we have a unique lifting, whose derivative is

the familiar Poincar´-Cartan form. This explains how the Poincar´-Cartan form

e e

occurs in the context of point transformation as well as contact transformations.

We use the proposition as follows. If we take two admissible liftings Λ, Λ

of the same functional [Λ], and write

Λ ’ Λ = 1 θ± § θβ § γ±β ,

2

with γ±β + γβ± = 0, then

2

θ± § (Ψ± ’ Ψ± ) = d(Λ ’ Λ ) ≡ ’θ± § dθβ § γ±β (mod { I}).

A consequence of this is that Ψ± ’ Ψ± ∈ I for each ±, and we can therefore give

the following.

n

De¬nition 4.6 The Euler-Lagrange system EΛ of [Λ] ∈ “(M, (J/I)) is the

di¬erential ideal on M generated by I and the n-forms {Ψ1 , . . . , Ψs} ‚ „¦n (M ),

θ± § Ψ± . A stationary

where Λ is any admissible lifting of [Λ] and dΛ =

Legendre submanifold of [Λ] is an integral manifold of EΛ .

4.2.3 The Betounes and Poincar´-Cartan Forms

e

For scalar variational problems, the Poincar´-Cartan form Π ∈ „¦n+1(M ) on

e

the contact manifold (M, I) is an object of central importance. Some of its

key features were outlined above. Underlying its usefulness is the fact that we

are associating to a Lagrangian functional”a certain equivalence class of di¬er-

ential forms”an object that is not merely an equivalence class, but an actual

di¬erential form with which we can carry out certain explicit computations. We

would like to construct an analogous object in the multi-contact case.

def

We will do this by imposing pointwise algebraic conditions on Π = dΛ. Fix

an admissible coframing (θ ± , ωi, πi ) on a multi-contact manifold as in De¬nition

±

n

4.3. Then any admissible lifting Λ ∈ “( J) of a functional [Λ] has the form

«

min (n,s)

(k!)’2 FAθA § ω(I) ,

I

Λ=

k=0 |A|=|I|=k

A

for some functions FI , which are skew-symmetric with respect to each set of

n

indices. Because J is integrable, Π = dΛ lies in „¦n+1 (M )©{ J}; the “highest

weight” part can be written as

«

min(n,s)

¬ · n+1

¬(k!)’2 H±A πi § θA § ω(I) ·

iI ±

(mod {

Π= J}).

±,i

k=1

|A|=|I|=k

(4.28)

4.2. EULER-LAGRANGE PDE SYSTEMS 159

iI

The functions H±A are skew-symmetric in the multi-indices I and A. Notice

n+1

that the equation dΠ ≡ 0 (mod { J}) gives for the k = 1 term

ij ji

H±β = Hβ± .

To understand the relevant linear algebra, suppose that V n is a vector space

with basis {vi }, and that W s is a vector space with basis {w±} and dual basis

{w± }. Then we have for k ≥ 2 the GL(W ) — GL(V )-equivariant exact sequence

„

k k k+1 k+1

0 ’ Uk ’ W — — V — ( W— — W— —

k

V) ’ V ’ 0.

Here the surjection is the obvious skew-symmetrization map, and Uk is by def-

inition its kernel. The term k = 1 will be exceptional, and we instead de¬ne

„ 2 2

0 ’ U1 ’ Sym2 (W — — V ) ’ W— — V ’ 0,

so that U1 = Sym2 W — — Sym2 V .

iI

Now we can regard our coe¬cients H±A, with |I| = |A| = k, at each point

of M as coe¬cients of an element

k k

Hk = H±Aw± — vi — wA — vI ∈ W — — V — (

iI

W— — V ).

n

De¬nition 4.7 The form Π ∈ „¦n+1 (M )©{ J} is symmetric if its expansion

(4.28) has Hk ∈ Uk for all k ≥ 1.

For k = 1 the condition is

ij ji ij

H±β = H±β = Hβ± .

We ¬rst need to show that the condition that a given Π be symmetric is

independent of the choice of admissible coframe. Equivalently, we can show

that the symmetry condition is preserved under the group of coframe changes

of the form (4.23), and we will show this under three subgroups generating the

group. First, it is obvious that a change

¯ πi = a± πj (b’1)j ,

β

θ ± = a± θ β , ω i = bi ω j , ¯±

¯

β j β i

preserves the symmetry condition, because of the equivariance of the preceding

exact sequences under (a± ) — (bi ) ∈ GL(W ) — GL(V ). Second, symmetry is

j

β