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. 36
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‚L d ‚L
’ = 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,

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. 36
( 48 .)



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