having prescribed mean curvature.

A particularly interesting branch of the equivalence problem for neo-classical

Poincar´-Cartan forms includes some highly symmetric Poincar´-Cartan forms

e e

corresponding to Poisson equations, discussed in Chapter 3. Some of these

equations have good invariance properties under the group of conformal trans-

formations of the n-sphere, and we ¬nd that the corresponding branch of the

equivalence problem reproduces a construction that is familiar in conformal ge-

ometry. We will discuss the relevant aspects of conformal geometry in some

detail; these include another application of the equivalence method, in which

the important conceptual step of prolongation of G-structures appears for the

¬rst time. This point of view allows us to apply Noether™s theorem in a partic-

ularly simple way to the most symmetric of non-linear Poisson equations, the

one with the critical exponent:

n+2

∆u = Cu n’2 .

Having calculated the conservation laws for this equation, we also consider the

case of wave equations, and in particular the very symmetric example:

n+3

z = Cz n’1 .

Here, conformal geometry with Lorentz signature is the appropriate background,

and we present the conservation laws corresponding to the associated symmetry

group, along with a few elementary applications.

The ¬nal chapter addresses certain matters which are thus far not so well-

developed. First, we consider the second variation of a functional, with the

goal of understanding which integral manifolds of an Euler-Lagrange system are

local minima. We give an interesting geometric formula for the second variation,

in which conformal geometry makes another appearance (unrelated to that in

the preceding chapter). Speci¬ally, we ¬nd that the critical submanifolds for

certain variational problems inherit a canonical conformal structure, and the

second variation can be expressed in terms of this structure and an additional

scalar curvature invariant. This interpretation does not seem to appear in the

classical literature. Circumstances under which one can carry out in an invariant

manner the usual “integration by parts” in the second-variation formula, which

is crucial for the study of local minimization, turn out to be somewhat limited.

xi

We discuss the reason for this, and illustrate the optimal situation by revisiting

the example of prescribed mean curvature systems.

We also consider the problem of ¬nding an analog of the Poincar´-Cartan

e

form in the case of functionals on vector-valued functions and their Euler-

Lagrange PDE systems. Although there is no analog of proper contact trans-

formations in this case, we will present and describe the merits of D. Betounes™

construction of such an analog, based on some rather involved multi-linear alge-

bra. An illuminating special case is that of harmonic maps between Riemannian

manifolds, for which we ¬nd the associated forms and conservation laws.

Finally, we consider the appearance of higher-order conservation laws for

¬rst-order variational problems. The geometric setting for these is the in¬nite

prolongation of an Euler-Lagrange system, which has come to play a major

role in classifying conservation laws. We will propose a generalized version of

Noether™s theorem appropriate to our setting, but we do not have a proof of

our statement. In any case, there are other ways to illustrate two of the most

well-known but intriguing examples: the system describing Euclidean surfaces

of Gauss curvature K = ’1, and that corresponding to the sine-Gordon equa-

tion, z = sin z. We will generate examples of higher-order conservation laws

by relating these two systems, ¬rst in the classical manner, and then more sys-

tematically using the notions of prolongation and integrable extension, which

come from the subject of exterior di¬erential systems. Finally, having explored

these systems this far, it is convenient to exhibit and relate the B¨cklund trans-

a

formations that act on each.

One particularly appealing aspect of this study is that one sees in action so

many aspects of the subject of exterior di¬erential systems. There are particu-

larly beautiful instances of the method of equivalence, a good illustration of the

method of moving frames (for a¬ne hypersurfaces), essential use of prolonga-

tion both of G-structures and of di¬erential systems, and a use of the notion of

integrable extension to clarify a confusing issue.

Of course, the study of Euler-Lagrange equations by means of exterior di¬er-

ential forms and the method of equivalence is not new. In fact, much of the 19th

century material in this area is so naturally formulated in terms of di¬erential

forms (cf. the Hilbert form in the one-variable calculus of variations) that it is

di¬cult to say exactly when this approach was initiated.

´

However, there is no doubt that Elie Cartan™s 1922 work Le¸ons sur les

c

invariants int´graux [Car71] serves both as an elegant summary of the known

e

material at the time and as a remarkably forward-looking formulation of the use

of di¬erential forms in the calculus of variations. At that time, Cartan did not

bring his method of equivalence (which he had developed beginning around 1904

as a tool to study the geometry of pseudo-groups) to bear on the subject. It was

not until his 1933 work Les espaces m´triques fond´s sur la notion d™aire [Car33]

e e

and his 1934 monograph Les espaces de Finsler [Car34] that Cartan began to

explore the geometries that one could attach to a Lagrangian for surfaces or

for curves. Even in these works, any explicit discussion of the full method of

equivalence is supressed and Cartan contents himself with deriving the needed

xii INTRODUCTION

geometric structures by seemingly ad hoc methods.

After the modern formulation of jet spaces and their contact systems was put

into place, Cartan™s approach was extended and further developed by several

people. One might particularly note the 1935 work of Th. de Donder [Don35]

and its development. Beginning in the early 1940s, Th. Lepage [Lep46, Lep54]

undertook a study of ¬rst order Lagrangians that made extensive use of the al-

gebra of di¬erential forms on a contact manifold. Beginning in the early 1950s,

this point of view was developed further by P. Dedecker [Ded77], who undertook

a serious study of the calculus of variations via tools of homological algebra.

All of these authors are concerned in one way or another with the canonical

construction of di¬erential geometric (and other) structures associated to a La-

grangian, but the method of equivalence is not utilized in any extensive way.

Consequently, they deal primarily with ¬rst-order linear-algebraic invariants of

variational problems. Only with the method of equivalence can one uncover the

full set of higher-order geometric invariants. This is one of the central themes of

the present work; without the equivalence method, for example, one could not

give our unique characterizations of certain classical, “natural” systems (cf. §2.1,

§2.5, and §3.3).

In more modern times, numerous works of I. Anderson, D. Betounes, R.

Hermann, N. Kamran, V. Lychagin, P. Olver, H. Rund, A. Vinogradov, and

their coworkers, just to name a few, all concern themselves with geometric

aspects and invariance properties of the calculus of variations. Many of the

results expounded in this monograph can be found in one form or another in

works by these or earlier authors. We certainly make no pretext of giving a

complete historical account of the work in this area in the 20th century. Our

bibliography lists those works of which we were aware that seemed most relevant

to our approach, if not necessarily to the results themselves, and it identi¬es

only a small portion of the work done in these areas. The most substantially

developed alternative theory in this area is that of the variational bicomplex

associated to the algebra of di¬erential forms on a ¬ber bundle. The reader can

learn this material from Anderson™s works [And92] and [And], and references

therein, which contain results heavily overlapping those of our Chapter 4.

Some terminology and notation that we will use follows, with more intro-

duced in the text. An exterior di¬erential system (EDS) is a pair (M, E) con-

sisting of a smooth manifold M and a homogeneous, di¬erentially closed ideal

E ⊆ „¦— (M ) in the algebra of smooth di¬erential forms on M . Some of the EDSs

that we study are di¬erentially generated by the sections of a smooth subbun-

dle I ⊆ T — M of the cotangent bundle of M ; this subbundle, and sometimes

its space of sections, is called a Pfa¬an system on M . It will be useful to use

the notation {±, β, . . .} for the (two-sided) algebraic ideal generated by forms

±, β, . . . , and to use the notation {I} for the algebraic ideal generated by the

sections of a Pfa¬an system I ⊆ T — M . An integral manifold of an EDS (M, E)

def

is a submanifold immersion ι : N ’ M for which •N = ι— • = 0 for all • ∈ E.

Integral manifolds of Pfa¬an systems are de¬ned similarly.

xiii

A di¬erential form • on the total space of a ¬ber bundle π : E ’ B is said

to be semibasic if its contraction with any vector ¬eld tangent to the ¬bers of π

—

vanishes, or equivalently, if its value at each point e ∈ E is the pullback via πe

of some form at π(e) ∈ B. Some authors call such a form horizontal. A stronger

condition is that • be basic, meaning that it is locally (in open subsets of E)

the pullback via π — of a form on the base B.

Our computations will frequently require the following multi-index notation.

If (ω1 , . . . , ωn ) is an ordered basis for a vector space V , then corresponding to

a multi-index I = (i1 , . . . , ik ) is the k-vector

k

ω I = ω i1 § · · · § ω ik ∈ (V ),

and for the complete multi-index we simply de¬ne

ω = ω1 § · · · § ωn .

Letting (e1 , . . . , en) be a dual basis for V — , we also de¬ne the (n ’ k)-vector

· · · (ei1 ω) · · · ).

ω(I) = eI ω = e ik (eik’1

This ω(I) is, up to sign, just ωIc , where Ic is a multi-index complementary to

I. For the most frequently occurring cases k = 1, 2 we have the formulae (with

“hats” ˆ indicating omission of a factor)

= (’1)i’1 ω1 § · · · § ωi § · · · § ωn ,

ω(i) ˆ

= (’1)i+j’1 ω1 § · · · § ωi § · · · § ωj § · · · § ωn

ω(ij) ˆ ˆ

= ’ω(ji) , for i < j,

and the identities

ωi § ω(j) i

= δj ω,

ωi § ω(jk) i i

δk ω(j) ’ δj ω(k).

=

We will often, but not always, use without comment the convention of sum-

ming over repeated indices. Always, n ≥ 2.2

2 For the case n = 1, an analogous geometric approach to the calculus of variations for

curves may be found in [Gri83].

xiv INTRODUCTION

Chapter 1

Lagrangians and

Poincar´-Cartan Forms

e

In this chapter, we will construct and illustrate our basic objects of study. The

geometric setting that one uses for studying Lagrangian functionals subject to

contact transformations is a contact manifold, and we will begin with its def-

inition and relevant cohomological properties. These properties allow us to

formalize an intuitive notion of equivalence for functionals, and more impor-

tantly, to replace such an equivalence class by a more concrete di¬erential form,

the Poincar´-Cartan form, on which all of our later calculations depend. In

e

particular, we will ¬rst use it to derive the Euler-Lagrange di¬erential system,

whose integral manifolds correspond to stationary points of a given functional.

We then use it to give an elegant version of the solution to the inverse problem,

which asks when a di¬erential system of the appropriate algebraic type is the

Euler-Lagrange system of some functional. Next, we use it to de¬ne the isomor-

phism between a certain Lie algebra of in¬nitesimal symmetries of a variational

problem and the space conservation laws for the Euler-Lagrange system, as de-

scribed in Noether™s theorem. All of this will be illustrated at an elementary

level using examples from Euclidean hypersurface geometry.

1.1 Lagrangians and Contact Geometry

We begin by introducing the geometric setting in which we will study Lagrangian

functionals and their Euler-Lagrange systems.

De¬nition 1.1 A contact manifold (M, I) is a smooth manifold M of dimen-

sion 2n + 1 (n ∈ Z+ ), with a distinguished line sub-bundle I ‚ T — M of the

cotangent bundle which is non-degenerate in the sense that for any local 1-form

θ generating I,

θ § (dθ)n = 0.

1

´

2 CHAPTER 1. LAGRANGIANS AND POINCARE-CARTAN FORMS

Note that the non-degeneracy criterion is independent of the choice of θ; this is

¯

because if θ = fθ for some function f = 0, then we ¬nd

¯ ¯

θ § (dθ)n = f n+1 θ § (dθ)n .

For example, on the space J 1 (Rn, R) of 1-jets of functions, we can take

coordinates (xi , z, pi) corresponding to the jet at (xi ) ∈ Rn of the linear function

f(¯) = z + pi (¯i ’ xi ). Then we de¬ne the contact form

x x

pi dxi,

θ = dz ’

for which

dpi § dxi,

dθ = ’

so the non-degeneracy condition θ § (dθ)n = 0 is apparent. In fact, the Pfa¬

theorem (cf. Ch. I, §3 of [B+ 91]) implies that every contact manifold is locally

isomorphic to this example; that is, every contact manifold (M, I) has local

coordinates (xi , z, pi) for which the form θ = dz ’ pidxi generates I.

More relevant for di¬erential geometry is the example Gn(T X n+1 ), the

Grassmannian bundle parameterizing n-dimensional oriented subspaces of the

tangent spaces of an (n + 1)-dimensional manifold X. It is naturally a contact

manifold, and will be considered in more detail later.

Let (M, I) be a contact manifold of dimension 2n + 1, and assume that I

is generated by a global, non-vanishing section θ ∈ “(I); this assumption only

simpli¬es our notation, and would in any case hold on a double-cover of M .

Sections of I generate the contact di¬erential ideal

I = {θ, dθ} ‚ „¦— (M )

in the exterior algebra of di¬erential forms on M .1 A Legendre submanifold of

M is an immersion ι : N ’ M of an n-dimensional submanifold N such that

ι— θ = 0 for any contact form θ ∈ “(I); in this case ι—dθ = 0 as well, so a Legendre

submanifold is the same thing as an integral manifold of the di¬erential ideal I.

In Pfa¬ coordinates with θ = dz ’ pidxi, one such integral manifold is given

by

N0 = {z = pi = 0}.