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congruent modulo {I}.

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

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
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.

• 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

• 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;

• 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}).
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,

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

Λ = L dx + θ± § ‚L
‚p± dx(i) , (4.26)

which has exterior derivative

dΛ = θ± § ‚L ‚L
‚z ± dx ’ § dx(i) .
d (4.27)

This suggests the following de¬nition.

De¬nition 4.5 An admissible lifting of a functional [Λ] ∈ “(M, n(J/I)) is a
Lagrangian Λ ∈ “(M, J) representing the class [Λ] and satisfying dΛ ∈ {I}.

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
Ft— Λ L ‚ (F — Λ)
dt ‚t
Nt N0

(θ± § Ψ± ) +
‚ ‚
= d( ‚t Λ)
N0 N0

g ± Ψ± ,

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
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.

Proposition 4.5 Two admissible liftings of the same [Λ] ∈ “( (J/I)) are
congruent modulo { I}.
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 ,

with γ±β + γβ± = 0, then
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.
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
For scalar variational problems, the Poincar´-Cartan form Π ∈ „¦n+1(M ) on
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.
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
4.3. Then any admissible lifting Λ ∈ “( J) of a functional [Λ] has the form
« 
min (n,s)
(k!)’2 FAθA § ω(I)  ,
k=0 |A|=|I|=k

for some functions FI , which are skew-symmetric with respect to each set of
indices. Because J is integrable, Π = dΛ lies in „¦n+1 (M )©{ J}; the “highest
weight” part can be written as
« 
¬ · n+1
¬(k!)’2 H±A πi § θA § ω(I) ·
iI ±
(mod {
Π= J}).
 


The functions H±A are skew-symmetric in the multi-indices I and A. Notice
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— —
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 .
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 — (
W— — V ).

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 ,
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


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