Exterior differential systems and Euler-Lagrange PDEs

219 Pages·2002·1.331 MB·English

by  Bryant R., Griffiths P., Grossman D.

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Exterior Di(cid:11)erential Systems and

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3 Euler-Lagrange Partial Di(cid:11)erential Equations
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0 Robert Bryant Phillip Gri(cid:14)ths Daniel Grossman
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Contents
Preface v
Introduction vii
1 Lagrangians and Poincar(cid:19)e-Cartan Forms 1
1.1 Lagrangians and Contact Geometry . . . . . . . . . . . . . . . . 1
1.2 The Euler-Lagrange System . . . . . . . . . . . . . . . . . . . . . 7
1.2.1 Variationof a Legendre Submanifold . . . . . . . . . . . . 7
1.2.2 Calculationof 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 Manifoldover En+1 . . . . . . . . . . . . . . 21
1.4.2 Euclidean-invariantEuler-Lagrange Systems . . . . . . . . 24
1.4.3 Conservation Laws for MinimalHypersurfaces . . . . . . . 27
2 The Geometry of Poincar(cid:19)e-Cartan Forms 37
2.1 The Equivalence Problem for n=2 . . . . . . . . . . . . . . . . . 39
2.2 Neo-Classical Poincar(cid:19)e-Cartan Forms . . . . . . . . . . . . . . . . 52
2.3 Digression on A(cid:14)ne Geometry of Hypersurfaces . . . . . . . . . . 58
2.4 The Equivalence Problem for n 3 . . . . . . . . . . . . . . . . . 65
(cid:21)
2.5 The Prescribed Mean Curvature System . . . . . . . . . . . . . . 74
3 Conformally Invariant Systems 79
3.1 Background Material on ConformalGeometry . . . . . . . . . . . 80
3.1.1 Flat ConformalSpace . . . . . . . . . . . . . . . . . . . . 80
3.1.2 The ConformalEquivalence Problem . . . . . . . . . . . . 85
3.1.3 The ConformalLaplacian . . . . . . . . . . . . . . . . . . 93
3.2 ConformallyInvariantPoincar(cid:19)e-Cartan
Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
3.3 The Conformal Branch of the Equivalence Problem . . . . . . . . 102
n+2
3.4 Conservation Laws for (cid:1)u=Cun 2 . . . . . . . . . . . . . . . . 110
(cid:0)
3.4.1 The Lie Algebra of In(cid:12)nitesimal Symmetries . . . . . . . 111
3.4.2 Calculationof Conservation Laws . . . . . . . . . . . . . . 114
iii
iv CONTENTS
3.5 Conservation Laws for Wave Equations. . . . . . . . . . . . . . . 118
3.5.1 Energy Density . . . . . . . . . . . . . . . . . . . . . . . . 122
3.5.2 The ConformallyInvariant 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 Formulafor the Second Variation . . . . . . . . . . . . 133
4.1.2 Relative ConformalGeometry . . . . . . . . . . . . . . . . 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 Submanifoldsof Higher Codimension . . . 155
4.2.3 The Betounes and Poincar(cid:19)e-Cartan Forms . . . . . . . . . 158
4.2.4 Harmonic Maps of RiemannianManifolds . . . . . . . . . 164
4.3 Higher-Order Conservation Laws . . . . . . . . . . . . . . . . . . 168
4.3.1 The In(cid:12)nite Prolongation . . . . . . . . . . . . . . . . . . 168
4.3.2 Noether’s Theorem . . . . . . . . . . . . . . . . . . . . . . 172
4.3.3 The K = 1 Surface System . . . . . . . . . . . . . . . . 182
(cid:0)
4.3.4 Two Ba(cid:127)cklund Transformations . . . . . . . . . . . . . . . 191
Preface
During the 1996-97 academic year, Phillip Gri(cid:14)ths and Robert Bryant con-
ducted aseminar at the Institute for Advanced Study in Princeton, NJ, outlin-
ingtheir recent work (withLucas Hsu) on ageometric approach tothe calculus
of variations in several variables. The present work is an outgrowth of that
project; it includes allof the material presented in the seminar, with numerous
additionaldetails and a few extra topics of interest.
The material can be viewed as a chapter in the ongoing development of a
theoryofthegeometryofdi(cid:11)erentialequations. Therelativeimportanceamong
PDEs of second-order Euler-Lagrange equations suggests that their geometry
shouldbeparticularlyrich,asdoesthegeometriccharacteroftheirconservation
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(cid:11)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(cid:11)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
(z)= L(x;z; z)dx; (cid:10) Rn; (1)
L
F r (cid:26)
Z(cid:10)
where x = (x1;:::;xn); dx = dx1 dxn; z = z(x) C1((cid:10)(cid:22)) (for ex-
^ (cid:1)(cid:1)(cid:1)^ 2
ample), and the Lagrangian L = L(x;z;p) is a smooth function of x, z, and
p=(p ;:::;p ). Examplesfrequently encountered inphysical(cid:12)eldtheories are
1 n
Lagrangians of the form
L= 1 p 2+F(z);
2jj jj
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(cid:11)erential equation
(cid:1)z(x)=F (z(x)):
0
Foranotherexample,wemayidentifyafunctionz(x)withitsgraphN Rn+1,
(cid:26)
and take the Lagrangian
L= 1+ p 2;
jj jj
whose associated functional L(z) pequals the area of the graph, regarded as
F
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.
TostudytheseLagrangiansandEuler-Lagrangeequationsgeometrically,one
hastochooseaclassofadmissiblecoordinatechanges,andtherearefournatural
candidates. In increasing order of generality,they are:
Classical transformations, of the form x = x(x), z = z (z); in this
0 0 0 0
(cid:15)
situation, we think of (x;z;p) as coordinates on the space J1(Rn;R) of
1-jets of maps Rn R.1
!
Gauge transformations, of the form x = x(x), z = z (x;z); here, we
0 0 0 0
(cid:15)
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 (cid:12)ber coordinate.
2
1A 1-jet is an equivalence class of functions having the same value and the same (cid:12)rst
derivativesatsomedesignatedpointofthedomain.
vii
viii INTRODUCTION
Point transformations, of the form x = x(x;z), z = z (x;z); here, we
0 0 0 0
(cid:15)
think of (x;z;p) as coordinates on the space of tangent hyperplanes
dz p dxi T (Rn+1)
f (cid:0) i g? (cid:26) (xi;z)
of the manifoldRn+1 with coordinates (x1;:::;xn;z).
Contact transformations, of the form x = x(x;z;p), z = z (x;z;p),
0 0 0 0
(cid:15)
p =p(x;z;p),satisfying the equation of di(cid:11)erential 1-forms
0 0
dz p dxi =f (dz p dxi)
0(cid:0) 0i 0 (cid:1) (cid:0) i
for some function f(x;z;Pp)=0. P
6
We will be studying the geometry of functionals (z) subject to the class of
L
F
contact transformations, which is strictly larger than the other three classes.
The e(cid:11)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 .
L
F
The(cid:12)rstvariation(cid:14) (z)isanalogoustothederivativeofafunction,where
L
F
z = z(x) is thought of as an independent variable in an in(cid:12)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 (cid:12)xed-boundary variations. Formally,
one writes
(cid:14) (z)=0;
L
F
andasweshallexplain,thisgivesasecond-order scalarpartialdi(cid:11)erentialequa-
tion for the unknown function z(x) of the form
@L d @L
=0:
@z (cid:0) dxi @p
(cid:18) i(cid:19)
X
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 , but not vice versa. To see this, observe that if we add to
L
F
L(x;z;p) a \divergence term" and consider
@Ki(x;z) @Ki(x;z)
L(x;z;p)=L(x;z;p)+ + pi
0
@xi @z
(cid:18) (cid:19)
X
for functions Ki(x;z), then by Green’s theorem, the functionals and
L L
F F 0
di(cid:11)er by a constant depending only on values of z on @(cid:10). For many purposes,
ix
such functionals should be considered equivalent; in particular, L and L have
0
the same Euler-Lagrange equations.
Second, there is a relationship between symmetries of a Lagrangian L and
conservation lawsforthecorrespondingEuler-Lagrangeequations,described by
a classicaltheorem ofNoether. A subtlety here is that the group ofsymmetries
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(cid:13)ected in the space of conservation laws, in a manner that
involves globaltopologicalissues.
Third, one considers the second variation (cid:14)2 , analogous to the Hessian
L
F
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
forclassicalvariationalproblems,reducingtheproblemoflocalminimizationto
understandingthebehaviorofcertainJacobioperators,butthegeometrictheory
is not as well-developed as that of the (cid:12)rst variation and the Euler-Lagrange
equations.
We will consider these issues and several others in a geometric setting as
suggested above, usingvariousmethods fromthe subject ofexterior di(cid:11)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 (cid:12)rst-order functionals (1) subject to contact transformations. We then con-
struct anobject that is central to the entire theory: the Poincar(cid:19)e-Cartan form,
an explicitly computable di(cid:11)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(cid:19)e-Cartan form to associate to any Lagrangianon a contact manifoldan
exterior di(cid:11)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(cid:19)e-Cartan
formalsomakes itquite easy tostate andprove Noether’s theorem,which gives
an isomorphism between a space of symmetries of a Lagrangianand a space of
conservation laws forthe Euler-Lagrange equation;exterior di(cid:11)erentialsystems
provides aparticularly naturalsetting forstudying the latter objects. We illus-
trate all of this theory in the case of minimalhypersurfaces in Euclidean space
En,and inthe case ofmore generallinearWeingarten surfaces inE3, providing
intuitive and computationallysimple proofs of known results.
In Chapter 2, we consider the geometry of Poincar(cid:19)e-Cartan forms more
closely. The main tool for this is E(cid:19). Cartan’s method of equivalence, by which
one develops an algorithm for associating to certain geometric structures their
di(cid:11)erential invariants under a speci(cid:12)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(cid:11)erential systems include many important
Euler-Lagrange systems that arise from classical problems, and among other
results, we (cid:12)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
(cid:21)
independent variables,andcarryoutseveralstepsoftheequivalencemethodfor
Poincar(cid:19)e-Cartan forms, after isolating those of the algebraic type arising from
classicalproblems. Associated tosuchaneo-classicalformisa(cid:12)eldofhypersur-
faces in the (cid:12)bers of a vector bundle, well-de(cid:12)ned up to a(cid:14)ne transformations.
This motivates a digression on the a(cid:14)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(cid:11)erential invariants for
Poincar(cid:19)e-Cartanformsinthismanner,weshowthattheyaresu(cid:14)cientforchar-
acterizingthosePoincar(cid:19)e-CartanformsassociatedtothePDEforhypersurfaces
having prescribed mean curvature.
Aparticularlyinterestingbranchoftheequivalenceproblemforneo-classical
Poincar(cid:19)e-Cartan forms includes some highly symmetric Poincar(cid:19)e-Cartan forms
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 (cid:12)nd that the corresponding branch of the
equivalence problemreproduces a construction that is familiarin conformalge-
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
(cid:12)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
(cid:1)u=Cun 2:
(cid:0)
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
(cid:3)z =Czn 1:
(cid:0)
Here,conformalgeometrywithLorentzsignatureistheappropriatebackground,
andwepresent theconservationlawscorrespondingtotheassociatedsymmetry
group, alongwith a few elementary applications.
The (cid:12)nal chapter addresses certain matters which are thus far not so well-
developed. First, we consider the second variation of a functional, with the
goalofunderstandingwhichintegralmanifoldsofanEuler-Lagrangesystemare
localminima. Wegiveaninterestinggeometricformulaforthesecondvariation,
in which conformal geometry makes another appearance (unrelated to that in
the preceding chapter). Speci(cid:12)ally, we (cid:12)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
classicalliterature. Circumstancesunderwhichonecancarryoutinaninvariant
manner the usual\integrationbyparts" inthe second-variationformula,which
is crucial for the study of localminimization,turn out to be somewhat limited.