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arXiv:math.DG/0207039 v1 3 Jul 2002




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

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