Exterior Di¬erential Systems and

Euler-Lagrange Partial Di¬erential Equations

Robert Bryant Phillip Gri¬ths Daniel Grossman

July 3, 2002

ii

Contents

Preface v

Introduction vii

1 Lagrangians and Poincar´-Cartan Forms

e 1

1.1 Lagrangians and Contact Geometry . . . . . . . . . . . . . . . . 1

1.2 The Euler-Lagrange System . . . . . . . . . . . . . . . . . . . . . 7

1.2.1 Variation of a Legendre Submanifold . . . . . . . . . . . . 7

1.2.2 Calculation of the Euler-Lagrange System . . . . . . . . . 8

1.2.3 The Inverse Problem . . . . . . . . . . . . . . . . . . . . . 10

1.3 Noether™s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 14

1.4 Hypersurfaces in Euclidean Space . . . . . . . . . . . . . . . . . . 21

1.4.1 The Contact Manifold over En+1 . . . . . . . . . . . . . . 21

1.4.2 Euclidean-invariant Euler-Lagrange Systems . . . . . . . . 24

1.4.3 Conservation Laws for Minimal Hypersurfaces . . . . . . . 27

2 The Geometry of Poincar´-Cartan Forms

e 37

2.1 The Equivalence Problem for n = 2 . . . . . . . . . . . . . . . . . 39

2.2 Neo-Classical Poincar´-Cartan Forms . . . . . .

e . . . . . . . . . . 52

2.3 Digression on A¬ne Geometry of Hypersurfaces . . . . . . . . . . 58

The Equivalence Problem for n ≥ 3 . . . . . . .

2.4 . . . . . . . . . . 65

2.5 The Prescribed Mean Curvature System . . . . . . . . . . . . . . 74

3 Conformally Invariant Systems 79

3.1 Background Material on Conformal Geometry . . . . . . . . . . . 80

3.1.1 Flat Conformal Space . . . . . . . . . . . . . . . . . . . . 80

3.1.2 The Conformal Equivalence Problem . . . . . . . . . . . . 85

3.1.3 The Conformal Laplacian . . . . . . . . . . . . . . . . . . 93

3.2 Conformally Invariant Poincar´-Cartan

e

Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

3.3 The Conformal Branch of the Equivalence Problem . . . . . . . . 102

n+2

3.4 Conservation Laws for ∆u = Cu n’2 . . . . . . . . . . . . . . . . 110

3.4.1 The Lie Algebra of In¬nitesimal Symmetries . . . . . . . 111

3.4.2 Calculation of Conservation Laws . . . . . . . . . . . . . . 114

iii

iv CONTENTS

3.5 Conservation Laws for Wave Equations . . . . . . . . . . . . . . . 118

3.5.1 Energy Density . . . . . . . . . . . . . . . . . . . . . . . . 122

3.5.2 The Conformally Invariant Wave Equation . . . . . . . . 123

3.5.3 Energy in Three Space Dimensions . . . . . . . . . . . . . 127

4 Additional Topics 133

4.1 The Second Variation . . . . . . . . . . . . . . . . . . . . . . . . 133

4.1.1 A Formula for the Second Variation . . . . . . . . . . . . 133

4.1.2 Relative Conformal Geometry . . . . . . . . . . . . . . . . 136

4.1.3 Intrinsic Integration by Parts . . . . . . . . . . . . . . . . 139

4.1.4 Prescribed Mean Curvature, Revisited . . . . . . . . . . . 141

4.1.5 Conditions for a Local Minimum . . . . . . . . . . . . . . 145

4.2 Euler-Lagrange PDE Systems . . . . . . . . . . . . . . . . . . . . 150

4.2.1 Multi-contact Geometry . . . . . . . . . . . . . . . . . . . 151

4.2.2 Functionals on Submanifolds of Higher Codimension . . . 155

4.2.3 The Betounes and Poincar´-Cartan Forms . . . . . .

e . . . 158

4.2.4 Harmonic Maps of Riemannian Manifolds . . . . . . . . . 164

4.3 Higher-Order Conservation Laws . . . . . . . . . . . . . . . . . . 168

4.3.1 The In¬nite Prolongation . . . . . . . . . . . . . . . . . . 168

4.3.2 Noether™s Theorem . . . . . . . . . . . . . . . . . . . . . . 172

4.3.3 The K = ’1 Surface System . . . . . . . . . . . . . . . . 182

4.3.4 Two B¨cklund Transformations . . . . . . . . . . . .

a . . . 191

Preface

During the 1996-97 academic year, Phillip Gri¬ths and Robert Bryant con-

ducted a seminar at the Institute for Advanced Study in Princeton, NJ, outlin-

ing their recent work (with Lucas Hsu) on a geometric approach to the calculus

of variations in several variables. The present work is an outgrowth of that

project; it includes all of the material presented in the seminar, with numerous

additional details and a few extra topics of interest.

The material can be viewed as a chapter in the ongoing development of a

theory of the geometry of di¬erential equations. The relative importance among

PDEs of second-order Euler-Lagrange equations suggests that their geometry

should be particularly rich, as does the geometric character of their conservation

laws, which we discuss at length.

A second purpose for the present work is to give an exposition of certain

aspects of the theory of exterior di¬erential systems, which provides the lan-

guage and the techniques for the entire study. Special emphasis is placed on

the method of equivalence, which plays a central role in uncovering geometric

properties of di¬erential equations. The Euler-Lagrange PDEs of the calculus

of variations have turned out to provide excellent illustrations of the general

theory.

v

vi PREFACE

Introduction

In the classical calculus of variations, one studies functionals of the form

„¦ ‚ Rn ,

FL(z) = L(x, z, z) dx, (1)

„¦

¯

where x = (x1, . . . , xn), dx = dx1 § · · · § dxn, z = z(x) ∈ C 1(„¦) (for ex-

ample), and the Lagrangian L = L(x, z, p) is a smooth function of x, z, and

p = (p1, . . . , pn). Examples frequently encountered in physical ¬eld theories are

Lagrangians of the form

L = 2 ||p||2 + F (z),

1

usually interpreted as a kind of energy. The Euler-Lagrange equation describ-

ing functions z(x) that are stationary for such a functional is the second-order

partial di¬erential equation

∆z(x) = F (z(x)).

For another example, we may identify a function z(x) with its graph N ‚ Rn+1 ,

and take the Lagrangian

L = 1 + ||p||2,

whose associated functional FL (z) equals the area of the graph, regarded as

a hypersurface in Euclidean space. The Euler-Lagrange equation describing

functions z(x) stationary for this functional is H = 0, where H is the mean

curvature of the graph N .

To study these Lagrangians and Euler-Lagrange equations geometrically, one

has to choose a class of admissible coordinate changes, and there are four natural

candidates. In increasing order of generality, they are:

• Classical transformations, of the form x = x (x), z = z (z); in this

situation, we think of (x, z, p) as coordinates on the space J 1 (Rn, R) of

1-jets of maps Rn ’ R.1

• Gauge transformations, of the form x = x (x), z = z (x, z); here, we

think of (x, z, p) as coordinates on the space of 1-jets of sections of a

bundle Rn+1 ’ Rn, where x = (x1, . . . , xn) are coordinates on the base

Rn and z ∈ R is a ¬ber coordinate.

1A 1-jet is an equivalence class of functions having the same value and the same ¬rst

derivatives at some designated point of the domain.

vii

viii INTRODUCTION

• Point transformations, of the form x = x (x, z), z = z (x, z); here, we

think of (x, z, p) as coordinates on the space of tangent hyperplanes

{dz ’ pidxi}⊥ ‚ T(xi ,z) (Rn+1)

of the manifold Rn+1 with coordinates (x1 , . . . , xn, z).

• Contact transformations, of the form x = x (x, z, p), z = z (x, z, p),

p = p (x, z, p), satisfying the equation of di¬erential 1-forms

pi dxi = f · (dz ’ pi dxi)

dz ’

for some function f(x, z, p) = 0.

We will be studying the geometry of functionals FL (z) subject to the class of

contact transformations, which is strictly larger than the other three classes.

The e¬ects of this choice will become clear as we proceed. Although contact

transformations were recognized classically, appearing most notably in studies

of surface geometry, they do not seem to have been extensively utilized in the

calculus of variations.

Classical calculus of variations primarily concerns the following features of

a functional FL .

The ¬rst variation δFL (z) is analogous to the derivative of a function, where

z = z(x) is thought of as an independent variable in an in¬nite-dimensional

space of functions. The analog of the condition that a point be critical is the

condition that z(x) be stationary for all ¬xed-boundary variations. Formally,

one writes

δFL(z) = 0,

and as we shall explain, this gives a second-order scalar partial di¬erential equa-

tion for the unknown function z(x) of the form

‚L d ‚L

’ = 0.

dxi

‚z ‚pi

This is the Euler-Lagrange equation of the Lagrangian L(x, z, p), and we will

study it in an invariant, geometric setting. This seems especially promising in

light of the fact that, although it is not obvious, the process by which we asso-

ciate an Euler-Lagrange equation to a Lagrangian is invariant under the large

class of contact transformations. Also, note that the Lagrangian L determines

the functional FL , but not vice versa. To see this, observe that if we add to

L(x, z, p) a “divergence term” and consider

‚K i (x, z) ‚K i (x, z) i

L (x, z, p) = L(x, z, p) + + p

‚xi ‚z

for functions K i (x, z), then by Green™s theorem, the functionals FL and FL

di¬er by a constant depending only on values of z on ‚„¦. For many purposes,

ix

such functionals should be considered equivalent; in particular, L and L have

the same Euler-Lagrange equations.

Second, there is a relationship between symmetries of a Lagrangian L and

conservation laws for the corresponding Euler-Lagrange equations, described by

a classical theorem of Noether. A subtlety here is that the group of symmetries

of an equivalence class of Lagrangians may be strictly larger than the group

of symmetries of any particular representative. We will investigate how this

discrepancy is re¬‚ected in the space of conservation laws, in a manner that

involves global topological issues.

Third, one considers the second variation δ 2 FL , analogous to the Hessian

of a smooth function, usually with the goal of identifying local minima of the

functional. There has been a great deal of analytic work done in this area

for classical variational problems, reducing the problem of local minimization to

understanding the behavior of certain Jacobi operators, but the geometric theory

is not as well-developed as that of the ¬rst variation and the Euler-Lagrange

equations.

We will consider these issues and several others in a geometric setting as

suggested above, using various methods from the subject of exterior di¬erential

systems, to be explained along the way. Chapter 1 begins with an introduc-

tion to contact manifolds, which provide the geometric setting for the study

of ¬rst-order functionals (1) subject to contact transformations. We then con-

struct an object that is central to the entire theory: the Poincar´-Cartan form,

e

an explicitly computable di¬erential form that is associated to the equivalence

class of any Lagrangian, where the notion of equivalence includes that alluded

to above for classical Lagrangians. We then carry out a calculation using the

Poincar´-Cartan form to associate to any Lagrangian on a contact manifold an

e

exterior di¬erential system”the Euler-Lagrange system”whose integral man-

ifolds are stationary for the associated functional; in the classical case, these

correspond to solutions of the Euler-Lagrange equation. The Poincar´-Cartan

e

form also makes it quite easy to state and prove Noether™s theorem, which gives

an isomorphism between a space of symmetries of a Lagrangian and a space of

conservation laws for the Euler-Lagrange equation; exterior di¬erential systems

provides a particularly natural setting for studying the latter objects. We illus-

trate all of this theory in the case of minimal hypersurfaces in Euclidean space

En , and in the case of more general linear Weingarten surfaces in E3 , providing

intuitive and computationally simple proofs of known results.

In Chapter 2, we consider the geometry of Poincar´-Cartan forms more

e

´ Cartan™s method of equivalence, by which

closely. The main tool for this is E.

one develops an algorithm for associating to certain geometric structures their

di¬erential invariants under a speci¬ed class of equivalences. We explain the

various steps of this method while illustrating them in several major cases.

First, we apply the method to hyperbolic Monge-Ampere systems in two inde-

pendent variables; these exterior di¬erential systems include many important

Euler-Lagrange systems that arise from classical problems, and among other

results, we ¬nd a characterization of those PDEs that are contact-equivalent

x INTRODUCTION

to the homogeneous linear wave equation. We then turn to the case of n ≥ 3

independent variables, and carry out several steps of the equivalence method for

Poincar´-Cartan forms, after isolating those of the algebraic type arising from

e

classical problems. Associated to such a neo-classical form is a ¬eld of hypersur-

faces in the ¬bers of a vector bundle, well-de¬ned up to a¬ne transformations.

This motivates a digression on the a¬ne geometry of hypersurfaces, conducted

using Cartan™s method of moving frames, which we will illustrate but not dis-

cuss in any generality. After identifying a number of di¬erential invariants for

Poincar´-Cartan forms in this manner, we show that they are su¬cient for char-

e

acterizing those Poincar´-Cartan forms associated to the PDE for hypersurfaces