’ = 0, ± = 1, . . . , s. (4.20)

‚p±

‚z ± dxi i

i

In the scalar case s = 1, we have examined the geometry of the equivalence

class of FL under contact transformations and found the canonically de¬ned

Poincar´-Cartan form to be of considerable use. In this section, we describe

e

a generalization of the Poincar´-Cartan form for s ≥ 1. Geometrically, we

e

study functionals on the space of compact submanifolds of codimension s, in an

(n + s)-dimensional manifold with local coordinates (xi , z ±).

An immediate di¬erence between the cases s = 1 and s ≥ 2 is that in the

latter case, there are no proper contact transformations of Rn+s+ns; that is, the

only smooth maps x = x (x, z, p), z = z (x, z, p), p = p (x, z, p) for which

{dz ± ’ p±dxi } = {dz ± ’ p± dxi }

i i

are point transformations x = x (x, z), z = z (x, z), with p = p (x, z, p) de-

termined by the chain rule. We will explain why this is so, and later, we will

see that in case s = 1 our original contact-invariant Poincar´-Cartan form still

e

appears naturally in the more limited context of point transformations. Our

¬rst task, however, is to introduce the geometric setting for studying function-

als (4.19) subject to point transformations, analogous to our use of contact

manifolds for (4.18).

Throughout this section, we have as always n ≥ 2 and we use the index

ranges 1 ¤ i, j ¤ n, 1 ¤ ±, β ¤ s.

4.2.1 Multi-contact Geometry

Having decided to apply point transformations to the functional (4.19), we in-

terpret z(x) = (z ± (xi )) as corresponding to an n-dimensional submanifold of

±

Rn+s . The ¬rst derivatives p± = ‚z i specify the tangent n-planes of this sub-

i ‚x

manifold. This suggests our ¬rst level of geometric generalization.

π

Let X be a manifold of dimension n + s, and let Gn (T X) ’ X be the

Grassmannian bundle of n-dimensional subspaces of tangent spaces of X; that

is, a point of Gn(T X) is of the form

p ∈ X, E n ‚ Tp X.

m = (p, E),

152 CHAPTER 4. ADDITIONAL TOPICS

Any di¬eomorphism of X induces a di¬eomorphism of Gn(T X), and either of

these di¬eomorphisms will be called a point transformation.

We can de¬ne on Gn(T X) two Pfa¬an systems I ‚ J ‚ T — (Gn(T X)),

of ranks s and n + s, respectively, which are canonical in the sense that they

are preserved by any point transformation. First, J = π — (T — X) consists of

all forms that are semibasic over X; J is integrable, and its maximal integral

submanifolds are the ¬bers of Gn (T X) ’ X. Second, we de¬ne I at a point

(p, E) ∈ Gn(T X) to be

I(p,E) = πp (E ⊥ ),

—

where E ⊥ ‚ Tp X is the s-dimensional annihilator of the subspace E ‚ Tp X.

—

I is not integrable, and to understand its integral submanifolds, note that any

n-dimensional immersion ι : N ’ X has a 1-jet lift ι(1) : N ’ Gn (T X). In

fact, such lifts are the transverse integral submanifolds of the Pfa¬an system

I ‚ T — (Gn (T X)).

To see this explicitly, choose local coordinates (xi , z ±) on U ‚ X n+s . These

induce local coordinates (xi , z ±, p±) corresponding to the n-plane E ‚ T(xi ,z± ) U

i

de¬ned as

E = {dz 1 ’ p1dxi, . . . , dz s ’ psdxi}⊥ .

i i

These coordinates are de¬ned on a dense open subset of π ’1 (U ) ‚ Gn(T X),

consisting of n-planes E ‚ T X for which dx1 § · · · § dxn|E = 0. In terms of

these local coordinates on Gn(T X), our Pfa¬an systems are

{dxi, dz ±},

J =

{dz ± ’ p±dxi}.

I = i

An immersed submanifold N n ’ U for which dx1 § · · · § dxn|N = 0 may be

regarded as a graph

N = {(xi , z ±) : z ± = f ± (x1 , . . . , xn)}.

Its lift to N ’ Gn(T X) lies in the domain of the coordinates (xi , z ±, p±), and

i

equals the 1-jet graph

‚f ±

N (1) = {(xi, z ± , p±) : z ± = f ± (x1 , . . . , xn), p± = 1

, xn)}.

‚xi (x , . . . (4.21)

i i

Clearly this lift is an integral submanifold of I. Conversely, a submanifold

N (1) ’ π’1 (U ) ‚ Gn (T X) on which dz ± ’ p±dxi = 0 and dx1 §· · ·§dxn = 0

i

is necessarily given locally by a graph of the form (4.21). The manifold M =

Gn (T X) with its Pfa¬an systems I ‚ J is our standard example of a multi-

contact manifold. This notion will be de¬ned shortly, in terms of the following

structural properties of the Pfa¬an systems.

Consider on Gn (T X) the di¬erential ideal I = {I, dI} ‚ „¦— (Gn(T X)) gen-

erated by I ‚ T — (Gn(T X)). If we set

¯

θ± = dz ± ’ p± dxi, ωi = dxi, πi = dp±,

¯±

¯

i i

4.2. EULER-LAGRANGE PDE SYSTEMS 153

then we have the structure equations

¯

dθ± ≡ ’¯i § ωi

π± ¯ (mod {I}), 1 ¤ ± ¤ s. (4.22)

It is not di¬cult to verify that the set of all coframings (θ ± , ωi , πi ) on Gn(T X)

±

for which

• θ1 , . . . , θs generate I,

• θ1 , . . . , θs , ω1, . . . , ωn generate J, and

• dθ± ≡ ’πi § ωi (mod {I})

±

are the local sections of a G-structure on Gn(T X). Here G ‚ GL(n + s + ns, R)

may be represented as acting on (θ ± , ωi , πi ) by

±

± ¯± ±β

θ = aβ θ ,

ω i = ci θ β + bi ω j ,

¯ (4.23)

j

β

± ± β ’1 j

’1 k j

±β ±

πi = diβ θ + ekj (b )i ω + aβ πj (b )i ,

¯

where (a± ) ∈ GL(s, R), (bi ) ∈ GL(n, R), and e± = e± . From these properties

j ij ji

β

we make our de¬nition.

De¬nition 4.3 A multi-contact manifold is a manifold M n+s+ns, with a G-

structure as in (4.23), whose sections (θ ± , ωi, πi ) satisfy

±

dθ± ≡ ’πi § ωi (mod {θ1 , . . . , θs }),

±

(4.24)

dωi ≡ 0 (mod {θ1 , . . . , θs , ω1, . . . , ωn}). (4.25)

Note that the G-structure determines Pfa¬an systems I = {θ 1 , . . . , θs} and

J = {θ1 , . . . , θs , ω1, . . . , ωn }, and we may often refer to (M, I, J) as a multi-

contact manifold, implicitly assuming that J is integrable and that there are

coframings for which the structure equations (4.22) hold. The integrability of

J implies that locally in M one can de¬ne a smooth leaf space X n+s and a

surjective submersion M ’ X whose ¬bers are integral manifolds of J. When

working locally in a multi-contact manifold, we will often make reference to this

quotient X.

It is not di¬cult to show that any multi-contact structure (M, I, J) is locally

equivalent to that of Gn(T X) for a manifold X n+s . The integrability of J

implies that there are local coordinates (xi, z ± , qi ) for which dxi, dz ± generate

±

J. We can relabel the xi , z ± to assume that dz ± ’ p±dxi generate I for some

i

functions p±(x, z, q). The structure equations then imply that dxi, dz ±, dp±

i i

are linearly independent, so on a possibly smaller neighborhood in M , we can

replace the coordinates qi by p±, and this exhibits our structure as equivalent

±

i

to that of Gn(T X).

We will see below that if s ≥ 2, then the Pfa¬an system I of a multi-contact

manifold uniquely determines the larger system J. Also, if s ≥ 3, the hypothesis

154 CHAPTER 4. ADDITIONAL TOPICS

(4.25) that J = {θ ± , ωi} is integrable is not necessary; it is easily seen to be a

consequence of the structure equation (4.24). However, in the case s = 2, J is

determined by I but is not necessarily integrable; our study of Euler-Lagrange

systems will not involve this exceptional situation, so we have ruled it out in

our de¬nition.

It is not at all obvious how one can determine, given a Pfa¬an system I of

rank s on a manifold M of dimension n + s + ns, whether I comes from a multi-

contact structure; deciding whether structure equations (4.24) can be satis¬ed

for some generators of I is a di¬cult problem. Bryant has given easily evaluated

intrinsic criteria characterizing such I, generalizing the Pfa¬ theorem™s normal

form for contact manifolds, but we shall not need this here (see Ch. II, §4 of

[B+ 91]).

Aside from those of the form Gn(T X), there are two other kinds of multi-

contact manifolds in common use. One is J 1 (Y n , Z s), the space of 1-jets of

maps from an n-manifold Y to an s-manifold Z. The other is J“ (E n+s , Y n ),

1

the space of 1-jets of sections of a ¬ber bundle E ’ Y with base of dimension n

and ¬ber Z of dimension s. These are distinguished by the kinds of coordinate

changes considered admissible in each case; to the space J 1 (Y n , Z s), one would

apply prolonged classical transformations x (x), z (z), while to J“ (E n+s , Y n ),

1

one would apply prolonged gauge transformations x (x), z (x, z). These are

both smaller classes than the point transformations x (x, z), z (x, z) that we

apply to Gn (T X), the space of 1-jets of n-submanifolds in X n+s .

Recall our claim that in the multi-contact case s ≥ 2, every contact transfor-

mation is a prolonged point transformation. This is the same as saying that any

local di¬eomorphism of M which preserves the Pfa¬an system I also preserves

J; for a local di¬eomorphism of M preserving J must induce a di¬eomorphism

of the local quotient space X, which in turn uniquely determines the original

local di¬eomorphism of M . To see why a local di¬eomorphism preserving I

must preserve J, we will give an intrinsic construction of J in terms of I alone,

2 —

for the local model Gn(T X). First, de¬ne for any 2-form Ψ ∈ (Tm M ) the

space of 1-forms

C(Ψ) = {V Ψ : V ∈ Tm M }.

This is a pointwise construction. We apply it to each element of the vector

space

{»±dθ± : (»± ) ∈ Rs},

intrinsically given as the quotient of I2, the degree-2 part of the multi-contact

di¬erential ideal, by the subspace {I}2, the degree-2 part of the algebraic ideal

{I}. For example,

C(dθ± ) ≡ Span{πi , ωi } (mod {I}).

±

The intersection

C(˜)

˜∈I2 /{I}2

is a well-de¬ned subbundle of T — M/I. If s ≥ 2, then its preimage in T — M is

¯¯

J = {θ± , ωi}, as is easily seen using the structure equations (4.22). Any local

4.2. EULER-LAGRANGE PDE SYSTEMS 155

di¬eomorphism of M preserving I therefore preserves I2 , I2/{I}2, C(˜), and

¬nally J, which is what we wanted to prove. Note that in the contact case

s = 1, C(˜) ≡ C(dθ) ≡ {πi, ωi} modulo {I}, so instead of this construction

giving J, it gives all of T — M . In this case, introducing J in the de¬nition of a

multi-contact manifold reduces our pseudogroup from contact transformations

to point transformations.

We have given a generalization of the notion of a contact manifold to acco-

modate the study of submanifolds of codimension greater than one. There is

a further generalization to higher-order contact geometry which is the correct

setting for studying higher-order Lagrangian functionals, and we will consider

it brie¬‚y in the next section.

In what follows, we will carry out the discussion of functionals modelled

on (4.19) on a general multi-contact manifold (M, I, J), but the reader can

concentrate on the case M = Gn(T X).

4.2.2 Functionals on Submanifolds of Higher Codimension

Returning to our functional (4.19), we think of the integrand L(xi , z ±, pi )dx as

±

an n-form on a dense open subset of the multi-contact manifold Gn(T Rn+s ).

Note that this n-form is semibasic for the projection Gn(T Rn+s ) ’ Rn+s , and

that any n-form congruent to L(xi, z ± , pi )dx modulo {dz ± ’ p±dxi} gives the

± i

same classical functional. This suggests the following.

De¬nition 4.4 A Lagrangian on a multi-contact manifold (M, I, J) is a smooth

n

J) ‚ „¦n(M ). Two Lagrangians are equivalent if they are

section Λ ∈ “(M,