Calculus of the Variations

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C A M B R I D G E STUDIES I NA D V A N C ED M A TH EM A TI C S 6 4E D I T O R I A L B O A R DD.J.H. G A R L I N G , W . F U L T O N , K . R I B E T , T . T O M D I E C K ,P. WALTERS

CALCULUS OF VARIATIONS


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Calculus of Variations

Jiirgen Jost and Xianqing Li-JostMax-Planck-Institute for Mathematics in the Sciences,Leipzig

C A M B R I D G EU N I V E R S I T Y PRESS


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P U B L I S H E D B Y T H E P R E S S S Y N D I C A T E O F T H E U N I V E R S I T Y O F C A M B R I D G ET h e Pitt Building, Trumpington Street, Cambridge CB2 1RP, United KingdomC A M B R I D G E U N I V E R S I T Y P R E S S

T h e Edinburgh Building, Cambridge C B2 2RU, U K http://www.cup.ac.uk40 West 20th Street, New Y o r k , N Y 10011-4211, USA http://www.cup.org

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F i r s t published 1998Typeset in Computer Modern by the authors using I A l ^ X 2e

A catalogue record of this book is available from the British LibraryLibrary of Congress Cataloguing in Publication data

Jost, Jurgen, 1956-Calculus of variations / Jurgen Jost and Xianqing L i - J o s t .

p. cm.Includes index.

I S B N 0 521 64203 5 (he.)1. Calculus of variations. I. L i - J o s t , Xianqing, 1956-

I I . Title.QA315.J67 1999

515'.64-dc21 98-38618 C IPI S B N 0 521 64203 5 hardback

Transferred to digital printing 2003
http://www.cup.ac.uk/http://www.cup.org/http://www.cup.org/http://www.cup.ac.uk/


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Dedicated to Stefan Hildebrandt


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Contents

Preface and summary page xRemarks on notation xv

Part one: One-dimensional variational problems 11 The classical theory 31.1 The Euler-Lagrange equations. Examples 31.2 The idea of the direct methods and some regularityresults 101.3 The second variation. Jacobi fields 181.4 Free boundary conditions 241.5 Symmetries and the theorem of E. Noether 262 A geometric example: geodesic curves 322.1 The length and energy of curves 322.2 Fields of geodesic curves 432.3 The existence of geodesies 513 Saddle point constructions 623.1 A finite dimensional example 623.2 The construction of Lyusternik-Schnirelman 674 The theory of Hamilton and Jacobi 794.1 The canonical equations 794.2 The Hamilton-Jacobi equation 814.3 Geodesies 874.4 Fields of extremals 894.5 Hilbert's invariant integral and Jacobi's theorem 924.6 Canonical transformations 95

v i i


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v i i i Contents5 Dynamic optimization 1045.1 Discrete control problems 1045.2 Continuous control problems 1065.3 The Pontryagin maximum principle 109

Part two: Multiple integrals in the calculus ofvariations 115

1 Lebesgue measure and integration theory 1171.1 The Lebesgue measure and the Lebesgue integral 1171.2 Convergence theorems 1222 Banach spaces 1252.1 D e f i n i t i o n and basic properties of Banach and Hilbert

spaces 1252.2 Dual spaces and weak convergence 1322.3 Linear operators between Banach spaces 1442.4 Calculus in Banach spaces 1503 L p and Sobolev spaces 1593.1 L p spaces 1593.2 Approximation of LP functions by smooth functions

(mollification) 1663.3 Sobolev spaces 1713.4 Rellich's theorem and the Poincare and Sobolev

inequalities 1754 The direct methods in the calculus of variations 1834.1 Description of the problem and its solution 1834.2 Lower semicontinuity 1844.3 The existence of minimizers for convex variational

problems 1874.4 Convex functional on Hilbert spaces and Moreau-

Yosida approximation 1904.5 The Euler-Lagrange equations and regularity questions 1955 Nonconvex functionals. Relaxation 2055.1 Nonlower semicontinuous functionals and relaxation 2055.2 Representation of relaxed functionals via convex

envelopes 2136 T-convergence 2256.1 The definition of T-convergence 225


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Contents i x6.2 Homogenization 2316.3 T h i n insulating layers 2357 BV-functionals and T-convergence: the example of

Mod ica and Mortola 2417.1 The space BV{Q) 2417.2 The example of Modica-Mortola 248Appendix A The coarea formula 257Appendix B The distance function from smooth hypersurfaces 2628 Bifurcation theory 2668.1 Bifurcation problems in the calculus of variations 2668.2 The functional analytic approach to bifurcation theory 2708.3 The existence of catenoids as an example of a bifurca

t i o n process 2829 The PalaisSmale condition and unstable critical

points of variational problems 2919.1 The Palais-Smale condition 2919.2 The mountain pass theorem 3019.3 Topological indices and critical points 306Index 319


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Preface and summary

The calculus of variations is concerned w i t h the construction of optimalshapes, states, or processes where the optimality criterion is given inthe form of an integral involving an unknown function. The task of thecalculus of variations then is to demonstrate the existence and to deducethe properties of some function that realizes the optimal value for thisintegral. Such variational problems occur in many-fold applications, inparticular in physics, engineering, and economics, and the variationalintegral may represent some action, energy, or cost functional. The calculus of variations also has deep and important connections w i t h otherfields of mathematics. For instance, in geometrically defined classes ofobjects, a variational principle often permits the selection of a uniqueoptimal representative, and the properties of this representative can frequently be used to much advantage to deduce additional informationabout its class. For these reasons, the calculus of variations is a r i c h andample mathematical subject, and a good impression of this diversitycan be obtained by reading the beautiful book by S. Hildebrandt andA . Tromba, The Parsimonious Universe, Springer, 1996.

I n this textbook, we have attempted to present some of the many facesof the calculus of variations, and a br i e f summary may be useful beforeputting the contents into a broader perspective. At the same time, weshall also describe the logical connections between the various chapters,i n order to facilitate reading for readers w i t h a specific aim. The bookis divided into two parts. The first part treats variational problems forfunctions of one independent variable; the second, problems for functionsof several variables. The distinction between these two parts, however, isalso that the first treats the more elementary and more classical aspectsof the subject, while the second is concerned w i t h some more dif f icul ttopics and uses somewhat more abstract reasoning. In this second part,

x


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Preface and summary x ialso some examples are presented in detail that occurred in recent applications of the calculus of variations. This second part leads the readerto some topics and questions of current research in the calculus of v a r i ations.

The f i rs t chapter of Part I is of a somewhat introductory nature andattempts to develop some i n t u i t i o n for the properties of solutions of v a r i ational problems. In the basic Section 1.1, we derive the Euler-Lagrangeequations that any smooth solution of a variational problem has to satisfy. The topics of the other sections of that chapter contain some regularity questions and an outline of the so-called direct methods of thecalculus of variations (a subject that w i l l be taken up in much more det a i l in Chapter 4 of Part I I ) , Jacobi's theory of the second variation andstability of solutions, and Noether's theorem that deduces conservationlaws f rom invariance properties of variational integrals. A l l those resultsw i l l not be directly applied in subsequent chapters, but should ratherserve as a motivation. In any case, basically all the chapters of Part I canbe read independently, after the reader has gone through Section 1.1.

In Chapter 2, we treat one of the most important variational problems, namely that of geodesies, i.e. of finding (locally) shortest curvesunder smooth geometric constraints. Geodesies are of fundamental importance in Riemannian geometry and several physical applications. Weshall make use of the geometric nature of this problem and develop someelementary geometric constructions, to deduce the existence not only oflength-minimizing curves, but also of curves that furnish unstable c r i t i cal points of the length functional. In Chapter 3, we present some moreabstract aspects of such so-called saddle point constructions. At thispoint, however, we can only treat problems that allow the reduction toa finite dimensional situation. A deeper treatment needs additional toolsand therefore has to wait u n t i l Chapter 9 of Part I I . Geodesies w i l l onlyoccur once more in the remainder, namely as an example in Section 4.3.

Chapter 4 is concerned w i t h one of the classical highlights of the calculus of variations, the theory of Hamilton and Jacobi. This theory isof particular importance in mechanics. Presently, its global aspects areresurging in connection w i t h symplectic geometry, one of the most activefields of present mathematical research.

Chapter 5 is a br i e f introduction to dynamic optimization and controltheory The canonical equations of Hamilton and Jacobi of Section 4.1br ief ly reoccur as an example of the Pontryagin maximum principle atthe end of Section 5.3.

As mentioned, Part I I is of a less elementary nature. We therefore need


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x i i Preface and summaryto develop some general theory f i rs t . In Chapter 1 of that part, Lebesgueintegration theory is summarized (without proofs) for the convenienceof the reader. While in Part I , the Riemann integral entirely suffices( w i t h the exception of some places in Section 1.2), the function spacesthat are basic for Part I I , namely the LP and Sobolev spaces, are essentially based on Lebesgue's notion of the integral. In Chapter 2, wedevelop some results from functional analysis about Banach and Hilbertspaces that w i l l be applied in Chapter 3 for deriving the fundamental properties of the L p and Sobolev spaces. (In fact, as the tools fromfunctional analysis needed in subsequent chapters are of a quite variednature, Chapter 2 can also serve as a br i e f introduction into the field offunctional analysis itself.) These chapters serve the purpose of makingthe book self-contained, and for most readers the best strategy mightbe to start w i t h Chapter 4, or at most w i t h Chapter 3, and look up theresults of the previous chapters only when they are applied. Chapter 4is fundamental. I t is concerned w i t h the existence of minimizers of v a r i ational integrals under appropriate convexity and lower semicontinuityassumptions. We treat both the standard method based on weak compactness and a more abstract method for minimizing convex functionalsthat does not need the concept of weak convergence. Chapters 5-7 essent i a l l y discuss situations where those assumptions are no longer satisfied.Chapter 5 deals w i t h the method of relaxation, while Chapters 6 and7 present the important concept of T-convergence for minimizing functionals that can be represented only in an indirect manner as l i m i t s ofother functionals. Such problems occur in many applications, includinghomogenization and phase transitions, and several such examples aretreated in detail. Chapter 8 discusses bifurcation theory. We f i rs t discuss the variational aspects (Jacobi fields), taking up the constructionsof Sections 1.1 and 1.3 of Part I , then develop a general functional analytic framework for analyzing bifurcation phenomena and then treatthe example of minimal surfaces of revolution (catenoids) in the l i g h tof that framework. Chapter 8 is independent of Chapters 4-7, and of amore elementary nature than those. The key tool is the i m p l i c i t functiontheorem in Banach spaces, proved in Section 2.4. The last Chapter 9 returns to the topic of the existence of non-miminizing, unstable criticalpoints of variational integrals. While such solutions usually cannot beobserved in physical applications because of their unstable nature, theyare of considerable mathematical interest, for example in the context ofRiemannian geometry. Chapter 9 is independent of Chapters 4-8.


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Preface and summary x i i iThe present book is self-contained, w i t h very few exceptions. Prere

quisites are only the calculus of one and several variables.Although, as indicated, there are important connections between the

calculus of variations and geometry, the present book is of an analyticnature and does not explore those connections. One such connection concerns the global aspects of the space of solutions of one-dimensional v a r i ational problems and their trajectories that started w i t h the qualitativeinvestigations of Poincare and is for example represented in V . I . A r n o l d ,Mathematical Methods of Classical Mechanics, GT M 60, Springer, NewY o r k , 2nd edition, 1987. Here, geometric methods are used to studyvariational problems. In the opposite direction, variational methods canoften be used to solve geometric problems. This is the topic of geometricanalysis; we refer the interested reader to J. Jost, Riemannian Geometry and Geometric Analysis, Springer, B e r l i n , 2nd edition, 1998, and thereferences contained therein.

There is one important omission in this textbook. Namely, the regularity theory for solutions of variational problems is not treated, w i t hthe exception of the one-dimensional case in Section 1.2 of Part I , andthe simplest example of the multi-dimensional theory, namely harmonicfunctions (plus an easy generalization) in Section 4.5 of Part I I . Therefore, the solutions of the variational problems that are discussed usuallyonly are obtained in some Sobolev space. We think that a detailed treatment of regularity theory more properly belongs to the realm of partialdifferential equations, and therefore we have to refer the reader to textbooks and monographs on partial differential equations, for exampleD . Gilbarg and N. Trudinger, Elliptic Partial Differential Equations ofSecond Order, Springer, B e r l i n , 2nd edition, 1983, or J. Jost, PartielleDifferentialgleichungen, Springer, B e r l i n , 1998.

I n any case, the present textbook cannot cover all the many diverseaspects of the calculus of variations. For readers who are interested in amore extensive treatment, we strongly recommend M . Giaquinta and St.Hildebrandt, Calculus of Variations, several volumes, Springer, B e r l i n ,1996 ff., as w e l l as E. Zeidler, Nonlinear Functional Analysis and itsApplications, Vols. I l l and I V, Springer, New Y o r k , 1984 ff. (a secondedition of V o l . I V appeared in 1995). Additional references are giveni n the course of the text. Since the present book, however, is neither aresearch monograph nor an account of the historical development of thecalculus of variations, references to individual contributions are usuallynot given. We just l i s t our sources, and refer the interested readers aswel l as the contributing mathematicans to those for references to theoriginal contributions.


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x i v Preface and summaryThe authors thank Felicia Bernatzki, R a l f Muno, Xiao-Wei Peng, Mar-

ianna Rolf, and Wilderich Tuschmann for their help in proofreading andchecking the contents and various corrections, and Michael Knebel andMicaela Krieger for their competent typing.The present authors owe much of their education in the calculus ofvariations to their teacher, Stefan Hildebrandt. In particular, the presentation of the material of Chapters 1 and 4 in Part I is influencedby his lectures that the authors attended as students. For example, theregularity arguments in Section 1.2 are taken directly f rom his lectures.For these reasons, and for his generous support of the authors over manyyears, and for his profound contributions to the subject, in particular togeometric variational problems, the authors dedicate this book to him.


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Remarks on notation

A dot always denotes the Euclidean scalar product in R d , i.e. ifx = (x\...,x d) ,y = (y\...,yd)eRd,

thend

x - y x%y% x%y% (Einstein summation convention) ,2=1

and| x | 2 = X X.

For a function u(t), we write

I n Part I , the independent variable is usually called t, because in manyphysical applications, it is interpreted as the time parameter. Here, thedependent variables are mostly called u(t) or x(t). In Part I I , the independent variables are denoted by x = ( x 1 , . . . , x d ) , conforming to established conventions.We use the standard notation

ck(n)for the space of A;-times continuously differentiable functions on someopen set Q C M d , for k = 0 (continuous functions), 1,2,.. ., oo ( i n f i n i t e lyoften differentiable functions). For vector valued functions, w i t h valuesi n M d , we write

C k(fl,R d)x v


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x v i Remarks on notationfo r the corresponding spaces.

Co(ft)denotes the space of functions of class C on ft that vanish identicallyoutside some compact subset K C ft (where K may depend on thefunction, of course). Occasionally, we also use the notation


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P a r t o n eOne-dimensional variational problems


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1

The classical theory

1.1 The Eu ler- Lagrange equations. ExamplesThe classical calculus of variations consists in minimizing expressions ofthe fo rm

where F : [a, 6] x Rd x Rd > E is given. One seeks a function u : [a, 6] R d minimizing J. More generally, one is also interested in other criticalpoints of J. Usually, u has to satisfy some constraints, the most commonone being a Dirichlet boundary condition

Also, one needs to specify a class of admissible functions among whichone seeks a minimizing u. For example, one might want to take theclass of continuously differentiable or piecewise continuously differen-tiable functions. Let us consider some examples of such variational problems:

(1) We want to minimize the arc-length of the graph of a function u :[a, 6] K, i.e. the length of the curve (t,u(t)) C K 2 among allgraphs w i t h prescribed boundary values u(a),u(b). This leads tothe variational problem

O f course, one knows and easily proves that the solution is thestraight line between u(a) and u(b), i.e. satisfies u(t) = 0.

u(a) = u\u(b) = 1*2-

mm.

3


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4 The classical theory(2) Historically, the calculus of variations started w i t h the so-called

brachystochrone problem that was posed by Johann Bernoulli.Here, one wants to connect two points (to,yo) and (t\,y\) in R 2by such a curve that a particle obeying Newton's law of gravitation and moving without f r i c t i on travels the distance betweenthose points in the fastest possible way. After fa l l ing the height y,the particle has speed {2gy)^ where g is the gravitational acceleration. The time the particle needs to traverse the path y = u(t)then is

I [ u ) = L i - ^ w d t(3) A generalization of (1) and (2) is

fb Jl 4- u(t)2 ,I(u) = / ~rdt,V ; Ja 7 ( t , ( t ) )where 7 : [a,6] x R R is a given positive function. This v a r i ational problem also arises from Fermat's principle..That p r i n c i ple says that a l i g h t ray chooses the path that needs the shortesttime to be traversed among all possible paths. I f the speed ofl i g h t in a given medium is y(t,u(t)), we obtain the precedingvariational problem.

I f one seeks a minimum of a smooth function/ : fi R ( fi open in R d ) ,

one knows that at a minimizing point Zo fi, one necessarily hasDf(z0) - 0,

where Df is the derivative of / . The first variation of / actually hasto vanish at any stationary point, not only at minimizers. In order todistinguish a minimizer from other critical points, one has the additionalnecessary condition that the Hessian D 2f(z0) is positive semidefinite and(at least for a local minimizer) the sufficient condition that it is positivedefinite.

I n the present case, however, we do not have a function / of finitelymany independent real variables, but a functional Z o n a class of functions. Nevertheless, we expect that a first derivative of J somethings t i l l to be defined needs to vanish at a minimizer, and moreover thata suitably defined second derivative is positive (semi)definite.


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1.1 The Euler-Lagrange equations. Examples 5I n order to investigate this more closely, we assume that F is of class

C 1 and that we have a minimizer or, more generally, a critical point of /that also is C1. We also assume prescribed Dirichlet boundary conditionsu(a) = u i , u(b) = U 2 . In other words, we assume that u minimizes / inthe class of all functions of class C 1 satisfying the prescribed boundarycondition. We then have for any 77 G CQ ([a, 6], M d ) f and any s G R

I(u + sri) > I(u).Now

I(u + sri)= I F(t,u(t) + sri(t),u(t) + sf}(t))dt.J aSince F, u, and 77 are assumed to be of class C 1 , we may differentiatethe preceding expression w.r.t. s and obtain at s = 0

I(u + sr,)Uo (1.1.1)

= J {F u(t,u{t),u{t))-r)(t) + F p(t,u{t),u{t))-T](t)}dt,J awhere Fu is the vector of partial derivatives of F w.r.t. the componentsof u, and Fp the one w.r.t. the components of u(t).

We now keep 77 fixed and let s vary. We are thus just in the situation ofa real valued f(s), s G R, (f(s) = I(u + srj)), and the condition / ' ( 0 ) = 0translates into

0 = / (1.1.2)/ a

and this actually then has to hold for all rj CQ. We now assume that Fand u are even of class C 2 . Equation (1.1.2) may then be integrated byparts. Noting that we do not get a boundary term since 77(a) = 0 = 77(6),we thus obtain

0 = ^ b | ( F ( t , u ( t ) , u ( < ) ) - | ( F p ( i ) U ( < ) ) ^ ) ) ) ) !,() J d (1.1.3)

for all 7] Co ([a, 6],Rd) . In order to proceed, we need the so-calledfundamental lemma of the calculus of variations:f This means that r) is continuously differentiable as a function on [a, b) with values

in Rd

and that there exist a < a\ < b\ < b with rj(x) = 0 if x is not contained in[aiM]-


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6 The classical theoryLemma 1.1.1. If h e C ((a,6),Rd) satisfies

b

h(t)(p(t)dt = 0 for all


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1.1 The Euler-Lagrange equations. Examples 7Written out, the Euler-Lagrange equations are

F pp(t, u(t),u(t))u(t) + F pu(t, u(t), u(t))u(t)+ F pt(t,u(t),u(t)) - F u(t,u(t),u(t)) = 0, (1.1.5)

i. e. a system of d ordinary differential equations of second order that arelinear in the second derivatives of the unknown function u.

Let us compute the Euler-Lagrange equations for our preceding threeexamples:

(1 ) Here Fu = 0, Fp = / t ^ a > and we geti / i+u(t)d u(0 = d u(t) u(t) u(t)

2il(t)

3 '

i.e.u(t) = 0

meaning that u has to be a straight line, a fact that we know ofcourse.

(3 ) For the general example (3), we obtain as Euler-Lagrange equations

o = | + 2 ^ + ^d*7(t,(*))>/l + (*) 2 T 8

u{t) ii{t) 2u(t) 7 t

hence0 = u(t) - ^ u ( t ) (1 + 7 i ( t ) 2 ) + ^ (1 + < i ( t ) 2 ) . (1.1.6)

(2 ) We just need to insert 7 = yj2gu(t) into (1.1.6) to obtain0 = fi(t) + ( l + ( * ) )


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The classical theoryActually, (2) is an example of an integrand F(t,u,ii) that does not

depend explicitly on t, i.e. Ft = 0. In this casej t( F - uFp) = u(Fu - jFp) = 0 by (1.1.4),

and hence every solution of the Euler-Lagrange equation (1.1.4) satisfiesF(t,u{t),u{t)) -u(t)Fp(t,u(t),u(t)) = constant. (1.1.7)

Conversely, every solution of (1.1.7), with the exception of ii = 0, i.e.u = constant, also satisfies (1.1.4).

I n the case of example (2), we have F = and (1.1.7) becomes= ^(1 + u2), if we denote the constant in (1.1.7) by A.

I n all examples ( l ) - (3) , we actually had d = 1. If one modifies e.g. (1)and seeks a curve g(t) = ( # i ( ) , . . . , 9d(t)) C Rd connecting two givenpoints g(a) and ^(6), our variational problem becomes

The Euler-Lagrange equations in this case ared d

, . 9i J2(9j)2-9i E 9j9jQ = d ^ ) _ _ ^ = _ J = i i - i

d t ( d \ * ( d \

f j g f t W 3 ) L s { ^ ' ) 2 )for i = 1 , . . . , d.

We now recall that any smooth curve g(t) C Rd may be parameterizedby arc-length, i.e.

= 1. (1.1.9)We also know that a reparameterization of a curve g(t) does not changeits arc-length 1(g). Consequently, we may assume (1.1.9) in (1.1.8). Thelatter then becomes

0 for i = l d

so that we see again that a length minimizing curve in E d is a straightline.


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1.1 The Euler-Lagrange equations. ExamplesOften, one also meets the task of minimizing

I(u) = / F(t,u{t),u(t))dtJ a

subject to some constraint, for example

S(u)= G(t,u(t),u(t))dt = c0 (a given constant). (1.1.10)J aAs in the case of finite dimensional minimization problems, one thenfinds a Lagrange multiplier A w i t h

0 = A (I(u + srj) + \S{u + sr])) | 5 = o (1.1.11)asfor all rj G Cg([a,6],E d) . This leads to the Euler-Lagrange equations

j t (F p(t,u(t),ii(t)) + \G p(t,u(t),ii(t)))- (F u(t,u(t),u(t)) + \Gu(t,u(t),u(t))) = 0. (1.1.12)

Example. We wish to miminize

J(M) = / min,w i t h

I(u) = J F(t,u(t),ii(t))dtJ a


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10 The classical theoryfor given F and unknown u : [a,6] Rd . I f F and u are differentiate,one may consider some k i n d of partial derivative, namely

I f F and u are of class C 2 , this leads to the Euler-Lagrange equations

consists in solving the Euler-Lagrange equations and then investigatingwhether a solution of the equations is a minimum of / or not.

1.2 The idea of the direct methods and some regularityresults

So far, our formulation of the variational problem

has been rather vague, because we did not specify in which class offunctions u we are trying to minimize /. The only things we did prescribewere boundary conditions of Dirichlet type, i.e. we prescribed the valuesu(a) and u(b) for our functions u : [a,6] * Rd .

Because of our derivation of the Euler-Lagrange equations in the preceding section, it would be desirable to have a solution u of class C2.So one might want to specify in advance that one minimizes / onlyamong functions of class C2. This, however, directly leads to the quest i o n whether / achieves its i n f i m u m among functions of class C2 ( w i t hprescribed Dirichlet boundary conditions, as always) or not, and if itdoes, whether the i n f i m u m of / in some larger class of functions, say C 1 ,could be strictly smaller than the one in C2. In the l i g h t of this question,i t might be preferable to minimize / in the class of all functions u forwhich

61{u,rj) := I{u + srj)yfor rj E Co([a,6],Rd) . For a minimizer u then

61(u, rj) = 0 for all such rj.

-F p(t,u(t),ii(t)) - F*(t,u(t),u(t)) = 0.The classical strategy for solving the problem

I(u) > min

I(u) > min


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1.2 Direct methods, regularity results 11is meaningful. Here, we assume that F(t,u,p) is continuous in u and pand measurable in t. For this purpose, one needs the class of functionsfor which the derivative u(t) exists almost everywhere and is finite. Thisis the class

AC([a,b})of absolutely continuous functions. A function u G AC([a,b]) satisfies forti,t2e [a,6]

u{t2)-u(h) = / ii(t)dt.JtiNote that F(t, u(t),u(t)) is a measurable function of t for u AC by ourassumptions on F and the fact that the composition of a measurable anda continuous function is measurablef. The idea of the direct methods inthe calculus of variations, as opposed to the classical methods describedi n the preceding section then consists in minimizing / in a class of functions l ike AC([a,b]) and then trying to show that a solution u becauseof its minimizing character actually enjoys better regularity properties,for example to be of class C 2 , provided F satisfies suitable assumptions.

This minimizing procedure w i l l be treated later J, since we want toreturn to the classical theory for a while. Nevertheless, even for theclassical theory, one occasionally needs certain regularity results, andtherefore we now br ief ly address the regularity theory. To s i m p l i f y ournotation, we put / := [a,6]. A class of functions intermediate betweenC 1 and AC is

D 1 ( / , E d ) := {u : / M d , u continuous and piecewisecontinuously differentiable, i.e. there exista = to < t\ < ... < tm = b w i t h u GC H M j + i ] , M d ) for j = 0 , . . . , m - l } .

u G D 1 then has left and right derivatives u~(tj) and u+(tj) even at thepoints where the derivative is discontinuous, and

f Lebesgue integration theory is summarized in Chapter 1 of Part II. The requiredcomposition property is stated there as Theorem 1.1.2. Here, this point will notbe pursued or used any further.

t See Chapter 4 of Part II . We shall use the same letter J to denote the functional to be minimized and the

domain of definition of the functions, inserted into this functional. This conformsto standard notations. The reader should be aware of this and not be confused.


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12 The classical theoryExamples

Example 1.2.1. [a, 6] = [-1,1], d = 1I(u) = ( l - ( f k ( t ) ) 2 ) 2 di ,

t i ( - l ) = 1 = u( l ) .A minimizer is

t i ( 0 = | t | D 1 ( / , R )which is not of class C

1

. The minimizer of / is not unique (exercise:determine all minimizers), but none of them is of class C 1 .Example 1.2.2. [a,6] = [-1,1],d = 1

I(u) = J (l-u(t))2u(t)2dt

u ( - l ) = 0 , t i ( l ) = l .Here, the unique minimizer is

, x . f 0 for - 1 < t < 0u { t ) = \ t f o r O < * < !

which again is of class D 1 , but not C 1 .Example 1.2.3. [a, 6] = [-1,1], d = 1

J(u) = ^ (2t~u(t))2u(t)2dt,

u ( - l ) = 0 , t i ( l ) = l .The unique minimizer is

, . f 0 for - 1 < t < 0= f o r O < * < !

which is of class C 1 , but not of class C2.Theorem 1.2.1. Let F(t,u,p) be of class C1 w.r.t. u and p and continuous w.r.t t ( F : J x Rd x Rd -+ R), and let u G AC(I,Rd) be asolution of

6I(u,rj) = J {F u(t,u,u)-rj + Fp(t,u,u)-f)}dt = 0 (1.2.1)J a


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1.2 Direct methods, regularity results 13for all rj G AC0(I,R d) (i.e. rj G AC(I,R d)) and we require that if I =[a, 6], there exist a < a\


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14 The classical theoryand


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1.2 Direct methods, regularity results 15F(to,u0,po) for any to G I . Thus, there exists a neighbourhood U of(to,uo,qo) such that for each (t,u,q) G /, 0 has a unique solutionp =


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16 The classical theoryfor almost all f i n a neighbourhood of to. Since u(t) and F p(t,u(t),u(t))are absolutely continuous w.r.t. t (the latter by Theorem 1.2.1), u(t)coincides for almost all t near to w i t h an absolutely continuous functionv(t). We put

w then is of class C 1 . Since u is absolutely continuous, by a theorem ofLebesgue

Since v = u almost everywhere, we conclude u = w, hence u G C1 nearto, which was arbitrary in / . Theorem 1.2.2 then gives u G C2.

Corollary 1.2.1. Under the assumptions of Theorem 1.2.3, any AC-solution of 6I(u,rj) = 0 for all rj G ACo(I,R

d

) is a solution of theEuler-Lagrange equations

or equivalently ofF pp(t, u(t),u(t))ii(t) + F pu(t, u(t), u(t))u(t)

+ F pt(t,u(t),ii(t)) - F u(t,u(t),u(t)) = 0. (1.2.8)The same holds under the assumptions of Theorem 1.2.2 for a Cl- solution of6I(u,rj) = 0 for all rj G C


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1.2 Direct methods, regularity results 17solves (1.2.8). Because of (1.2.9), F pp(t, u(t),u(t)) is an invertible matrix,hence

il(t) = F pp 1(t,u(t),u(t)){-F pu(t,u(t),u(t)) - F pt(t,u(t),u(t)) + F u(t,u(t),ii(t))}

(1.2.10)Let now j < k, and suppose inductively u E C3. The right hand side of(1.2.10) then is of class C3~x. Therefore, u is of class C- 7" 1, hence u isof class Cj+1.

q.e.d.The preceding proof most clearly shows the importance of the as

sumption det(Fptpj(t, u(t),ii(t))) ^ 0 that already occurred in the proofof Theorem 1.2.2. Namely, it implies that the Euler-Lagrange equations(1.2.8) can be solved for u in terms of u and u.Corollary 1.2.2. If under the assumption of Theorem 1.2.3, F and Fpare of class Ck, then a solution u of 6I(u,rj) = 0 for all rj ACo is ofclass C k + 1 .

q.e.d.

Summary. I f one wants to solveI(u) > min

by a direct minimization procedure, it is preferable to admit a class ofcomparison functions u that is as large as possible. AC (I, E d ) seems tobe a good choice, because this is the largest class for which

7(u)= J F(t,u(t),u(t))is wel l defined, assuming F(t,u,p) to be continuous in u and p andmeasurable in t. However, if one then finds a minimizer u, it might notbe a solution of the Euler-Lagrange equations, because it is not regularenough. If the i n v e r t i b i l i t y condition de t F p p ^ 0 is satisfied, however,one may show that a minimizer u is as regular as F allows. Namely, ifF and Fp are of class C f c, k G { 1 , 2 , . . . , o o } , then u is of class C k + 1 .Examples show that without such an i n v e r t i b i l i t y condition, regularityneed not hold. This i n v e r t i b i l i t y condition det Fpp ^ 0 implies that theEuler-Lagrange equations allow the expression of u(t) in terms of u(t)and u(t).


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18 The classical theory1.3 The second variation. Jacobi fields

We assume that u G D 1 ( / , E d ) is a critical point ofI ( u ) = / F(t,u(t),u(t))dt,

J ai.e.

6I(u,T]) = 0 for all 77 G >o(/,Md). (1.3.1)We recall that

:= ^ / ( u + s77) u = 0 ,and 8I(u,rj) = 0 is equivalent to s = 0 being a crit ical point of thefunction

f(s)=I(u + sri).I f we want to decide if a given solution u minimizes J instead of justbeing a critical point, we immediately see that a necessary conditionwould be

/ " ( 0 ) > 0 (1.3.2)f o r the above function / and all 77 G Do(J,Rd ) . Namely, by Taylor'stheorem, since /'(0) = 0

m-f(0) = \s2f"(0)+o(s2) f o r s ^ O .More precisely, (1.3.2) is needed for u to minimize / when compared w i t hu 4 - srj for sufficiently small s. In other words, we want u to minimize i"i n a D1-neighbourhood of itself, i.e. among functions

w i t hu ( a ) = v(a), u(b) = v(b) and (1.3.3)

sup (\u(t) - v(t)\ + \ii-(t) - v-(t)\ 4- \ii+(t) - < e (1.3.4)

f o r some e > 0. (Note: It is not clear that e may be chosen independentlyo f v.) We define the second variation of / at u in the direction rj e DQas

d262I(u,rj) : = _ / ( w - f s 7 ? ) u = 0 .


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1.3 The second variation. Jacobi fields 19I n order that this variation exists, we require for the rest of the section

that F is of class C 2 . We then compute

62I(u, 77) = ^ J" F(t, u(t) + sr](t), ii(t) + 7 ( < ) ) d * | . rbI

J a+ 2F piUJ(t,u(t),u(t))r)i{t)r )j{t)+ F^j^ui^^uit^rji^rjjit)} dt. (1.3.5)

Here, and in the sequel, we employ the standard summation conventions,e.g.

dFpipjrjirjj = ] T Fpipjfiifjj.

We abbreviate (1.3.5) asfb6 2I{u,r])= {F ppr)r) + 2F pur)rj + Fuurjrj} dt. (1.3.6)

J aOur preceding considerations i m p l y :Theorem 1.3.1. SupposeF e < 7 2 ( J x R d xR d xR) andletu G D l(I,R d)satisfy I(u) < I(v) for all v with {1.3.3), (1.3.4). Then

62I(u,rj)>0 forallrjeDl(I,R d). (1.3.7)We now put, for given u,

min among all 77 G Z^( J ,R d ) .

(1.3.8)I f u satisfies the assumptions of Theorem 1.3.1, thenQ(rj) > 0 for all 77 G >J, (1.3.9)

and hence 77 = 0 is a t r i v i a l solution of (1.3.8). We are interested in thequestion whether there are others. The Euler-Lagrange equations for(1.3.8) are

= ^ ( * , r ? W , i ) W ) , (1.3.10)


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20 The classical theoryi.e.

~ (F pp(t, u(t), u(t))f)(t) + Fpu{t, u(t), u(t))ri(t))= F pu(t, u(t), u(t))fj(t) + F uu(t, u(t), u(t))rj(t). (1.3.11)

Since u is considered as given, our f i rs t observation is that (1.3.11) is alinear homogeneous system of second order equations for the unknown77. These equations are called Jacobi equations.Definition 1.3.1. A solution 77 G C 2(I,R d) of the Jacobi equations(1.3.11) is called a Jacobi field along u(t).Lemma 1.3.1. Let F G C3(I x Rd x R d ,R) , det Fpp{t, u{t), u{t)) ^ 0for all t e I , u e C 2 ( J , R d ) . Then any solution of rj e AC0{I,Rd),6Q (rj,(p) = 0 for all

V(t)) = Fpp(t, u{t), u(t)) for all t and 77and so the assumption det F pp(t, u(t), u(t)) ^ 0, that is seemingly weakerthan the one of Theorem 1.2.3, indeed suffices to apply that Theorem.

q.e.d.We now derive the so-called necessary Legendre condition:Theorem 1.3.2. Under the assumption of Theorem 1.3.1, i.e. u GD

1

( / , Rd

) minimizes I in the sense described there, we have thatF pp(t,u(t),u(t)) is positive semidefinite for all t G / ,

i.e.Fpipj (t, u(t), u{t))?? > 0 for all = (\ ..., d) e Rd.

(At points where ii(t) is discontinuous, this holds for the left and rightderivatives.)Proof. We may assume that t0 e I and ii is continuous at t0. The resultat the points where u jumps then follows by taking appropriate l i m i t s ,and likewise at to a, 6. We then consider 0 < e < min(to a, b to)and define 77 G Z^ ( J ,R d ) by

{ 0 for a < t < 10 e and to 4- e < t < be for t - tolinear for to e < t < 10 and for to < t < to + e


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1.3 The second variation. Jacobi fields 21for given Rd. Then

{ 0 for a < t < t0 or t0 + e < t < 6 for t 0 - e < t < t0- for t 0 < t < t 0 + c.We apply Theorem 1.3.1 to obtain0 < 62I(u, rj ) = r + C F p V (t, i x ( t ) , u(t))CZJdt + 0(e 2 ) for c 0,

Jto-esince all other terms contain a factor e, and we integrate over an intervalof length 2e. Hence

F pipJ(t 0,u{t0),u(to))Cl; j - lim - / F pipj{t,u{t),u(t))C^ jdt > 0.0 Jt0-e

q.e.d.The Jacobi equations and the notion of Jacobi fields are meaningful

for arbitrary solutions of the Euler-Lagrange equations, not only forminimizing ones. In fact, Jacobi fields are solutions of the linearizedEuler-Lagrange equations. Namely:Theorem 1.3.3. Let F e C3{I x Rd x R d ,R) , and let us(t) be a familyo f C2-solutions of the Euler-Lagrange equations

j tF p(t,u s(t),iis(t)) - F u(t,u s(t),u s(t)) = 0, (1.3.12)with us depending differentiably on a parameter s 6 (e,e). Then

dsrj(t) := - ^ s ( % = 0is a Jacobi field along u = uo.Proof. We differentiate (1.3.12) w . r . t . s at s = 0 to obtain

~ (F pp(t, u(t), u(t))r)(t) + F pu(t, u(t), u(t))ri(t))-F pu(t,u(t),u(t))r)(t) - F uu{t,u(t),u(t))r](t) = 0.

i.e. the Jacobi equation (1.3.11). q.e.d.Lemma 1.3.2. Let a < a\ < a2


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2 2 The classical theoryProof. Since is homogeneous of second order in (77,7r), we have

2(t, 77, 7T) = (f)v(t, 77, 7r)77 -h(f>n(t, 77, 7T)7T.Therefore

2 / (t,r),r,)dt = / {^,(,?, 1 7 ) + ) ) ' ) } * (1-3-14)/ai /aiComparing (1.3.10) and (1.3.11), we see that (f>n is of class C 1 as afunction of t. We may hence integrate the last term in (1.3.14) by parts.Since 77(01) = 0 = 7 7 ( 0 2 ) , we obtain

2 j (t,T 7 ,r))dt = j (j>ri{tiT7,rj) - jfA^V, ^ = 0 ,since 77 is a Jacobi field. q.e.d.

As before, let F be of class C 3 , and let u(t) be a solution of class C2on [a, 6] of the Euler-Lagrange equations

j tFp{t, u(t), ii(t)) - Fu(t, u(t), ii(t)) = 0.Definition 1.3.2. Let a < a\ < a2 < b. We call the parameter valuea2 conjugate to a\ and the point (a2,u(a2)) conjugate to (a\,u(a\)) ifthere exists a not identically vanishing Jacobi field 77 on [a\,a2] with77(01) = 0 = 77(02) .

We may derive the important result of Jacobi:Theorem 1.3.4. LetF e < 7 3 ( JxR d xR d , R) and suppose u e C 2(I,R d).Suppose that Fpp(t,u(t),u(t)) is positive definite on I. If there exists a*with a < a* < b that is conjugate to a, then u cannot be a local minimum of I. More precisely, for any e > 0, there exists v Dl(I, Rd) withv(a) = u(a), v(b) = u(b),

sup (\u{t) - v(t)\ 4 \u{t) - v(t)\) < etlan d

I(v) < I(u).Proof. Let rj(t) be a nontrivial Jacobi field on [a, a*]. We put

rj(t) for a


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1.3 The second variation. Jacobi fields 23Then 77* G ^(J , ]Rd ) , and by Lemma 1.3.2

Q(V*)= f*


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24 The classical theoryIf, for fixed u, we consider the variational problem Q(rj) * 0, we are ledto the Jacobi equations

~ (F pp(t, u(t), u(t))rj(t) + F pu(t, u(t),u(t))rj(t))= F up(t, u(t),u(t))r)(t) + F uu(t, u(t),u(t))ri(t)

for 77.Solutions rj w i t h 77(a) = 77(6) = 0 are called Jacobi fields, a* G (a, 6) for

which there exists a nontrivial Jacobi field on [a, a*] is called conjugateto a, and i f there exists such a*, u cannot be locally minimizing on [a, 6].I n other words, a solution of the Euler-Lagrange equations cannot beminimizing beyond the first conjugate point.

1.4 Free boundary conditionsWe recall the definition of an n-dimensional embedded differentiable sub-manifold M of R d : For every p G M , there have to exist a neighbourhoodV = V(p) C M d , an open set U cRn and an injective differentiable map/ : U * V of everywhere maximal rank n (i.e. for every z U, thederivative Df(z), a linear map from E n to E d , has rank n) w i t h

M nv = f(U).A n example is the sphere Sn described in detail in Section 2.1 (Example 2.1.1). The tangent space TPM of M at p then is the vector spaceD / ( z ) ( E n ) . I t can be considered as a subspace of the vector space T p E d ,the tangent space of E d at p.

As in 1.1, we now consider the variational problem

I(u)= / F(t,u(t),u(t))dt > minJa

w i t h F of class C2. This time, however, we do not impose the Dirichletboundary condition that the values of u(a) and u(b) were prescribed,but the more general condition that for given submanifolds M i , M2(differentiable, embedded) of E d , we require that

u(a) GM i , i i ( 6 ) G M 2 .(Dirichlet boundary conditions constitute the special case where M\ andM 2 are points.)

I n this section, we do not consider regularity questions. As an exercise,


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1.4 Free boundary conditions 25the reader should supply the necessary regularity assumptions on F , w,etc. at each step.

Let u be a solution. Then, as before, u has to satisfy the Euler-Lagrange equations, because if u(a) G M i , 77(a) = 0, then also u(a) +577(a) G M i for any s, and likewise at 6, and so we may again considervariations of the form 72 + 577, 77 G DQ. This time, however, also moregeneral variations are admissible. Namely, let us(t) be a family of mapsfrom / into M.d depending differentiably on s G (e, c), w i t h u(t) = Uo(t)and

u s(a) G M i , us(b) G M 2 for all s.Let

Then again

0 = ^ / K ) | . _ 0 = F(t,u(t),u(t))dt^0= f {Fp{t,u(t),u(t))-f,{t) + Fu{t,u(t),u{t))-T}(t)}dt

J a

= fa \ - j F P + ^ } - v + F P ( * . ( * ) , ( * ) ) m l z i= F p(t ,u (t)M *))-V (t)\iZa>

since u solves the Euler-Lagrange equations.We now observe that 77(a) G T u ( a ) M i (and likewise at 6), since we may

find a 'local chart' / as above w i t h MiDV(u(a)) = f(U) for a neighbourhood V of u(a) and some open set U C M n i (ni = dim M i ) . By choosinge smaller if necessary, we may assume us(a) G M i f l V = f(U) for 5 G(~, e). Since / is injective, there then has to exist a curve 7 ( 5 ) C U w i t hu s(a) = fo>y(s) for all s. Hence 77(a) = u s(a )u= 0 = D / ( / - 1 7 i ( a ) ) 7 , ( 0 )is indeed tangent to M i at u(a). Moreover, any tangent vector to M i atu(a) can be realized in this manner. Therefore, since we may choose thevalues of 77 at a and 6 independently of each other, we conclude

Fp(a,u(a),u(a)) V = 0 for al l V G Tu{a)Muand likewise

F p ( 6 , u(6), u(b)) -W = 0 for all W G r u ( 6 ) M 2 .


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26 The classical theoryWe have thus shown:Theorem 1.4.1. Let u be a critical point of I among curves withu(a) GM i, u(b) G M2 {Mi, M2 given differentiable embedded submanifoldso f Rd), i.e. ^ ^ ( ^ s ) | s = 0 = 0 for all variations us(t) differentiable ins with us(a) G M b us(b) G M2 for all s G (-e,e) (e > 0). Thenu is a solution of the Euler-Lagrange equations for I , and in addition, F p(a,u(a),u(a)) and F p(b,u(b),u(b)) are orthogonal to Tu^Miand T U ( 5 ) M 2 , respectively. In particular, if for example Mi = Rd, thenF p(a,u(a),u(a)) = 0.Summary. I f instead of a Dirichlet boundary condition, we more generally impose a free boundary condition that u(a) and u(b) are onlyrequired to be contained in given differentiable submanifolds Mi andM 2 , respectively, of E d , then F p(a,u(a),u(a)) and F p(b,u(b),u(b)) areorthogonal to these submanifolds for a critical point of / under thoseboundary conditions.

1.5 Symmetries and the theorem of E . NoetherI n the variational problems of classical mechanics, one often encountersconserved quantities, l ike energy, momentum, or angular momentum. Itwas realized by E. Noether that all those conservation laws result froma general theorem stating that invariance properties of the variationalintegral / lead to corresponding conserved quantities. We f i rs t treat aspecial case.Theorem 1.5.1. We consider the variational integral

I(u) = / F(t,u(t),u(t))dt,J a

with F G C2([a,6] x l d x E d , E ) . We suppose that there exists a smoothone-parameter family of differentiable maps

hs : Rd -> Rd(the precise smoothness requirement is that

h(s,z) := hs(z)is of class C 2 ( ( - e 0 ,e 0 ) x E d ,E ) for some e0 > 0),with

h0(z) = z for all zeRd


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1.5 Symmetries and the theorem of E. Noether 27and satisfying

j\(t,h s(u(t)), f th s(u(t)^ J dt = j\(t,u(t), j tu(t^J dt (1.5.1)fo r all s G (~e,e) and all u G C2 ([a, 6], Rd).Then, for any solution u(t) of the Euler-Lagrange equations (1.1.4)

fori,Fp (t, u(t),ii(t)) ~h s(u(t))\ s=0 (1.5.2)

is constant in t G [ a , 6].Definition 1.5.1. A quantity C(t,u(t),u(t)) that is constant in t foreach solution of the Euler-Lagrange equations of a variational integralI(u) is called a (first) integral of motion.Proo f of Theorem 1.5.1: Equation (1.5.1) yields for any t0 G [a, 6], usingh0(z) = z,

= ^ s J a F { t ' h s ^ ^ J t k s { u { t ) ) ) d t ^ s = 0= jT {Fu (t,u(t),ii(t)) ^hs(u(t)) (1.5.3)

+ FP (t, u(t),u(t)) J tf shsW))}dt\s=o-We recall the Euler-Lagrange equations (1.1.4) for u:

0 = j tF p (t, u(t), ii(t)) - Fu (t, u(t),u(t)) . (1.5.4)Using (1.5.4) in (1.5.3) to replace Fu, we obtain

0 = f [jF p{t,u(t),u(t)) fh a{u{t))+ FP (t,u{t),u(t)) J t-ff shs( u(t))}dt\ s=o (1-5.5)

= f j t (F p(t,u(t),u(t))^- sh 3(u(t))\ s=0) dt.Therefore

F p(t 0,u(to),u(t0))~h s(u(t0)) \s=0 = F p(a,u(a),u(a))~h s(u(a))\s=0(1.5.6)

for any to G [a, 6]. This means that (1.5.2) is constant on [a, 6].q.e.d.


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28 The classical theoryExamples

Example 1.5.1. We consider for u : E > E 3 n , u = (ui,...,un) w i t h

F(t,u(t),u(t)) = pmj-^f-- ^ I I ^ H 2 = E ^ i j ,

i.e. a mechanical system in E 3 w i t h point masses m*, and a potentialV(u) that is independent of the third coordinates of the Ui. Then

ha(z) = z + se 3 ,where e 3 is the third unit vector in M 3 , leaves F invariant in the senseof Theorem 1.5.1. Since

d-hs\s=o = e 3 ,we conclude that

1=1i.e. the third component of the momentum vector of the system is conserved.Example 1.5.2. Similarly, i f a system as in Example 1.5.1 is invariantunder rotations about the e3-axis, and if h8 now denotes such rotations,then (up to a constant factor)

d L , ns\s=oUi = es A Ui.asHence, the conserved quantity is the angular momentum w.r.t. the e 3-axis,

n^ Fvez A U i = ] P (

m

iu

i) ' (e

3 A Ui) = ] P (ui A ra^) e 3 .i= l i iWe now come to the general form of E. Noether's theoremTheorem 1.5.2 (Theorem of E . Noether). We consider the variational integral

rbI(u) = I F(t,u(t),u(t))dtJ a


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1.5 Symmetries and the theorem of E. Noether 29with F C2([a,6] x Rd x E d , E ) . We suppose that there exists a smoothone-parameter family of differentiate maps

hs = (h s,h s) : [a,b] x E d > E x E d(s G (-eo, e 0 ) as before) with

h0(t, z) = (t, z) for all (t, z) G [a, b] x E dand satisfying

rh 3(b ) ( d \ rb/ F(t s,h s(u(ts)),h s(u(ts)))dts= / F(t,u(t),u(t))dt(1.5.7)

forts = h s(t), all s G ( - e 0 , e 0 ) and a// it G C 2 ( [a,6] ,E d ) . Then, for anysolution u(t) of the Euler-Lagrange equations (1.1.4) for I ,

'ds'F p{t,u(t),u(t)) h s{u{t))\ s=0+ ( F ( t , t i ( t ) , t i ( t ) ) - F p ( * , t i ( * ) , A ( * ) ) w W ) ^ f c 2WI - =o (1-5.8)

is constant in t G [a, 6].Proof. We reduce the statement to the one of Theorem 1.5.1 by a r t i f i c ia l ly considering t as a dependent variable on the same footing w i t h u.Thus, we consider the integrand

F(t(T),u(t(T)), , u(t(r))

\ dr dr=F t M t ) , * ^ ) Z (1.5-9)

ThenI(t,u) := j H F ( i ( r ) ,u( i ( r ) ) , | :(*(r ) ) ) dr

= J F(t,u(t),u{t))dt, if i ( r 0 ) = a, f ( n) = 6 (1.5.10)= / ( )

B y our assumption, F remains invariant under replacing (t,u) byhs(t,u). Consequently, Theorem 1.5.1 applied to I yields that

F p{t,u(t)M t))~h s(u(t))\s=Q ^


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30 The classical theoryw i t h p standing for the place of the argument ^ of F (while p standsas before for the arguments ii), is invariant. Since, by (1.5.9),

F Fp piF po = F - Fpuat s = 0 (note ^ = 1 for s = 0 since h^t) = t), this implies theinvariance of (1.5.8).

q.e.d.Example 1.5.3. Suppose F = F(u,u), i.e. F does not depend e x p l i c i t l yon t. Then

hs(t,z) = (t + s,z)leaves / invariant as required in Theorem 1.5.2. Therefore, the 'energy'

F(t, u(t),u{t)) - Fp(t, u(t), u(t))u{t)is conserved. We shall see another proof of this fact in Section 4.1.Summary. The theorem of E. Noether identifies a quantity that is preserved along any solution u(t) of the Euler-Lagrange equations of avariational integral, a so-called f i rs t integral of motion, w i t h any differentiable symmetry of the integrand. For example, in classical mechanics, conservation of momentum and angular momentum correspond totranslational and rotational invariance of the integral, respectively, whiletime invariance leads to the conservation of energy.

Exercises1.1 For mappings u : [a, 6] > E d , consider

E ( u ) : = i f\u{t)\ 2dt(| | is the Euclidean norm of E d , i.e. for z (zl,..., zd),| z | 2 _ J2i=i(z1)2). Compute the Euler-Lagrange equations andthe second variation. Also, let

L(u) := I \ii{t)\dt.J a

Show thatL(u) < yj2(b-a)E(u),


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Exercises 3 1

1.2

1.3

1.4

w i t h equality if \u(t)\ = constant almost everywhere. (Whatis an appropriate regularity class for the mappings u that areconsidered here?)Determine all minimizers of the variational integral

w i t h u(~l) = 0 = u{l).Develop a theory of Jacobi fields for variational problems w i t hfree boundary conditions. In particular, you should obtain ananalogue of Jacobi's theorem.For mappings u : [a, 6] E d , consider

Compute the f irst and second variation of / and the Jacobiequation. Can you f ind Jacobi fields?


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2A geometric example: geodesic curves

2.1 The length and energy of curvesWe let M be an n-dirnensional embedded submanifold of R d . In thissection, we assume that / is of class C 3 , i.e. that all local charts arethrice differentiable. We let c G AC([0,T],M ) be a curve on M . Thismeans that c is an absolutely continuous map from the interval [0, T] intoRd w i t h the property that c(t) G M for every t G [ 0 , T ] . The derivativeof c w.r.t. t w i l l be denoted by a dot ' ,

c(t) := Jt{t).The length of c is given by

L(c):=\c(t)\dt = ^ ( n a y j dt, (2.1.1)

where ( c 1 , . . . , cd) are the coordinates of c = c(t) . We also define theenergy of c as

E(c) := ^ \c(t)\ 2 d t = \ Y . (V , f(U) = Mf)V

be a local chart for M as defined in Section 1.4. We assume for a momentthat c([0, T]) is contained in /(J7). Since / maps U bijectively onto f(U),there exists a curve

7 ( t ) C C/32


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2.1 The length and energy of curves 33w i t h

c(t) = f( 1(t)). (2.1.3)Since the derivative Df(z) has maximal rank everywhere (by definitionof a chart, cf. 1.4), 7 is absolutely continuous, since c is, and we havethe chain rule

c(t) = (Df) ( 7 ( ) ) o 7 ( t ) ,or

where the index i is summed f rom 1 to n. ThusL(c) = (^(7W)7


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34 Geodesic curvesRemark 2.1.1. The use of local charts for M seerns to have the obviousdisadvantage that the expressions for length and energy of curves become more complicated. The advantage of this approach, namely not toconsider curves on M as curves in Rd satisfying a constraint, is that thisconstraint now is automatically fu l f i l l ed . Al l curves represented in localcharts lie on M . This more than compensates for the complication inthe formulae for L and E.

O u r aim w i l l be to find curves of shortest length or of smallest energyon M , i.e. to minimize the functionals L and E among curves on M . Forthis purpose it w i l l be useful to observe certain invariance properties ofL and E. First of a l l , whenever i : H &d > H &d is a Euclidean isometry, i.e.i(y) = Ay + b w i t h A G 0(d) , the orthogonal group, and b G E d , then

L(i(c)) = L(c) (2.1.7)

E(i(c)) = E(c) (2.1.8)for any curve c : [0, T] -> Rd.

Secondly, L is parameterization invariant in the sense that wheneverT : [0 ,S] ->[0 ,r]

is a diffeornorphisrn (i.e. r is bijective, and both r and its inverse r~lare everywhere differentiable), then

L(c) = L ( c o r ) ,Namely

L(cor)


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2.1 The length and energy of curves 35E , however, is not parameterization invariant. By the Schwarz inequality,we have instead

L(c) = J l


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36 Geodesic curves(i.e. we keep the interval of definition fixed, namely [0, L(c)]), the parameterization by arc-length leads to the smallest energy. Namely, ifc : [0, L(c)] E d is parameterized by arc-length

L(c) = 2JE7(c), (2.1.12)whereas for any other parameterization of c on the same interval,

L(c) < 2E(c). (2.1.13)We now return to those curves c that are confined to lie on M , in order

to discover a t h i r d invariance. Namely,we compare the two expressions(2.1.1) and (2.1.5) for the length of c, and similarly (2.1.2) and (2.1.6)for its energy. (2.1.1) is obviously independent of the chart / : U Vand its metric tensor, and therefore (2.1.5) has to be independent ofthem, too. In order to study this more closely, let

f:U-+Vbe another chart w i t h

C ( [ 0 , T ] ) c / ( / ) .Then there exists a curve 7 in U w i t h c(t) = /(7()) for all t. Putting

QfOt Q fOt

we then also have

L{c) = (s(7(*))7*7 (*)) * dt. (2.1.14)I n order to study this invariance property more closely, we define

f == r 1 of-.r1 (/([/) n /( #) ) ^ r 1 (/([/) n /"(#))(see Figure 2.1).

(p is called a coordinate transformation, (p is a diffeomorphism, i.e. abijective map between open subsets of E n whose derivative Dp(z) hasmaximal rank ( = n) at every z. Then from

/ o 7 ( t ) = c(t) = /o 7 ( t ) ,

7(0 = ^ ( 7 ( 0 ) , hence ?(t) = ^ ( 7 ( t ) ) V ( t ) (2.1.15)and from

fob)) = f(z)


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2.1 The length and energy of curves 37

Figure 2.1.

we get9ij(z) = ~9ki(


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38 Geodesic curvesw . r . t . the chart / . Then

m

m lf x= E / ( ^ ( 7 , W ) 7 i W 7 i W ) ' *

t i y 1

where c(t) = fvO~fu(t) for t G By the preceding considerations,this does not depend on the choice of charts / . For this reason, oneusually just says that for a curve c on M

L(c) = fT (9ij^(t))f(t)jHt)) h dt, (2.1.18)Jowhere 7 is the representation for c w . r . t . a local chart, and (9ij)ij=i,..., nis the metric tensor of M w . r . t . this chart. Similarly

E(c) = \ j T ^ ( 7 ( * ) ) f ( * ) y (*)* (2-1.19)We now assume that the charts for M are twice differentiable and returnto the question of finding shortest curves on M , for example between twogiven points. By Corollary. 2.1.1, it is preferable to minimize E insteadof L, because a minimizer for E contains more information than onefor L; namely, minimizers for E are precisely those minimizers for Lthat are parameterized proportionally to arc-length. Thus, minimizingE not only selects shortest curves but also convenient parameterizationsof such curves.

We now compute the Euler-Lagrange equations for E as given by(2.1.19):

d0 = E i for i = 1 , . . . ,m^ 0 = j t (29ij(7(WJ(t)) - ( ^ ) (7(*))7*(t)7>(*)(the factor 2 in the f i rs t term results from the symmetry = gji)& 0 = 2 9irf + 2 ^ y 7 f c 7 > - ^ i f l y y V - (2.1.20)

We now introduce some further notation:(g ij) ,W / t,j = l,... ,n


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2.1 The length and energy of curves 39is the matrix inverse to (gij)ij i.e.

9*9jk = 6k := 1 for i = k0 for i ^ k for all i,

and f ina l ly the Christoffel symbols

Equation (2.1.20) then becomes0 = f 4- \gil (2gij,kih - 9kj trikV)

= f + ^ +Abu - ffiM) 7* yby using symmetries. Thus:Lemma 2.1.2. The Euler-Lagrange equations for the energy E forcurves on M are

0 = f(t) + rijk(1(t)W(t)jk(t) fori = l,...,n. (2.1.21)The theorem of Picard-Lindelof about solutions of ordinary differentialequations implies:Lemma 2.1.3. For any z E U, v e K n , the system (2.1.21) has aunique solution y(t) with

7(0) = z , 7(0) = v for t E [e, e] and some e > 0.Moreover, 7(2) depends differentiably on the initial values z, v.Definition 2.1.2. The solutions of (2.1.21) are called geodesies on M.

is a differentiable manifold of dimension n. In order to construct localcharts, we put

ExamplesExample 2.1.1. The sphere

in+l

:= Sn \ { ( 0 , 0 , . . . , 0 , l ) } , f i 2 :=S n\ { ( 0 , 0 , . . . , 0 , - 1 ) }


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and

40 Geodesic curvesand define

gi : fix - E n , g2 : fi2 - E n

as

*"+,> - ( I ^ j t nSsr)( # 1 and g2 are the stereographic projections from the south and northpole, respectively). We then obtain charts

/ 1 = j r 1 : r - ^ \ { ( o 0,1)}/ 2 = 9 2 - 1 : R " - 5 n \ { ( 0 , . . . , 0 , - l ) } .

More e x p l i c i t l y , / i can be computed as fol lows:W i t h

/ 1 n , _ f X1 Xn \

1 = X


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2.1 The length and energy of curves 41Hence

9ij(z) = = ^-6ij. (2.1.22)

A c t u a l l y , the metric tensor w . r . t . the chart f2 is given by the sameformula. In order to compute the expression for geodesies, we also needto compute the Christoffel symbols. It turns out that adding a l i t t l egenerality w i l l actually facilitate the computations. We consider a metricof the form

9ij = (2.1-23)where (\>: Rn E + is positive and differentiable. Then

g ij = 28 ij. (2.1.24)We also put

Then


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42 Geodesic curvesTherefore, the equations for geodesies become

n no = f + 2 r*, . (7) 7 V' - r j i ( 7 ) 7 i 7 i + W W

(using the symmetry F\j = T^)

= V - 2 r r r V y V + S TTTPW - ( 2 - L 2 7 )1 + |7l j x 1 + 171

We now claim that the geodesic 7 ( t ) through the o r i g i n , i.e. 7( 0) = 0,w i t h 7(0) = a R

n

is given by7() = aa(t), (2.1.28)

where a : E E then satisfies a (0) = 0, d ( 0 ) = 1. Making the ansatz(2.1.28) in (2.1.27) leads to

i 2a3 a . . o 2a* a . 2fr[l + a2\a\2 fril + a*\a\2

if.. 2\a\2a . 2 \ . 1= a1 a L - L c r i = 1 , . . . , n.V l + \a\2a* y

Since we may assume a ^ 0 (otherwise the solution w i t h 7( 0) = a isa point curve, hence uninteresting), this equation holds, if a(t) satisfiesthe ordinary differential equation (ODE)

0 = a ! - 4 a . (2.1.29)1 4- |a| a 2

The theorem of Picard-Lindelof implies that (2.1.29) has a unique solut i o n in a neighbourhood of t = 0. We then have found a solution j(t) of(2.1.27) of the desired form (2.1.28). The image of 7 ( f ) is a straight linethrough 0. By Lemma 2.1.3, we have thus found all solutions through 0.The images of the straight lines under the chart / 1 are the great circleson Sn through the south pole. We can now use a symmetry argumentto conclude that all the geodesic lines on Sn are given by the great circles on Sn. Namely, the south pole does not play any distinguished role,and we could have constructed a local chart by stereographic projectionfrom any other point on Sn as wel l , and the metric tensor would haveassumed the same form (2.1.22). More generally, one may also argue asfollows: We want to find the geodesic arc j(t) on Sn w i t h 7( 0) = po,7 ( 0 ) = V0 for some p0 e Sn,V0 e TPoSn. Let c0(t) be the great circle on


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2.2 Fields of geodesic curves 43Sn parameterized such that Co(0) = po> co(0) = V0. Co is contained in aunique two-dimensional plane through the o r i g i n in E n + 1 . Let i denotethe reflection across this plane. This is an isometry of R n + 1 mapping Snonto itself. It therefore maps geodesies on Sn onto geodesies, becausewe have observed that the length and energy functionals are invariantunder isornetries, and so isometries have to map critical points to c r i t ical points. Now i maps po and Vo to themselves. I f 7 were not invariantunder i , i o 7 would be another geodesic w i t h i n i t i a l values po, Vo, contradicting the uniqueness result of Lemma 2.1.3. Therefore, 2 0 7 = 7 ,and therefore 7 = c0.We draw some conclusions:

The geodesic arc through two given points need not be unique. Namely,let p, q be antipodal points on 5 n , e.g. north and south pole. Then thereexist i n f i n i t e ly many great circles that pass through both p and q.

We shall later on see that the f i rs t conjugate point of a point p Snalong a great circle is the antipodal point q ofp. One also sees by explicitcomparison that a geodesic arc on Sn ceases to be minimizing beyondthe f i rs t conjugate point, in accordance w i t h Theorem 1.3.4.

2.2 Fields of geodesic curvesLet M be an embedded, differentiate submanifold of E d , or, more generally, a Riemannian manifold of dimension nf, again of class C 3 . LetM q be a submanifold of M ; this means that Mo i t se l f is a differentiablesubmanifold of E d , respectively a Riemannian manifold, and that theinclusion i : M 0 ^ M is a differentiable embedding. We assume thatM q has dimension n 1, and that it is also of class C 3 .Theorem 2.2.1. For any x 0 M 0 , there exist a neighbourhood V ofx0 in M, and a chart f : U * V with the following properties:

( i ) U contains the origin o/ E n , / (0) = #o-( i i ) M0nV = f{UD{x n = 0})(Hi) The curves xl = C\, C \ constant, i = l , . . . , n 1, are geodesies

parameterized by arc-length. The arcs 1 < xn < 2 on any suchf We do not introduce the concept of an abstract Riemannian manifold here, but

some readers may know that concept already, and in fact it provides the naturalsetting for the theory of geodesies. On the other hand, the embedding theoremof J.Nash says that any Riemannian manifold can be isometrically embedded intosome Euclidean space E d , hence considered as a submanifold of Md. Therefore, fromthat point of view, no generality is gained by considering Riemannian manifoldsinstead of submanifolds of Rd.


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44 Geodesic curvescurve between the hypersurfaces xn = 1 and xn = 2 are M ofthe same length 1 2

( i v ) The metric tensor on U satisfies9nn = 1, gin = 0 for all i = 1 , . . . , n - 1 (2.2.1)

(T/ie second relation means that the curves xl = C\, i = 1 , . . . , n 1, intersect the hypersurfaces xn = constant orthogonally.)

Proof. Since Mo is a hypersurface, for every p G Mo, there exist twounit normal vectors n(p) to Mo at p, i.e.

n(p)eTpM,

| | n ( p ) | | = l( n ( p ) , v ) = 0 for all v G r p M 0 C T P M .

I n a sufficiently small neighbourhood V of #o, we may assume that sucha normal vector n(p) may be chosen so that it depends smoothly onp G Mo f l V =: Vo. We assume that there is a local chart (p0 : Uo * VQfor M q (Uo C M 7 1 " " 1 ) , possibly choosing V smaller, i f necessary. For everyp G Mo f l V, we then consider the geodesic arc 7 P ( ) w i t h

7P(0) = P,7 P (0 ) =n(p) . (2.2.2)

This geodesic exists for |f| < e = e(p) by Lemma 2.1.3. By choosingV smaller if necessary, we may assume that e > 0 is independent of p.Instead of 7P(), we write ~/(p,t). Since the solution of (2.2.2) dependsdifferentiably on its i n i t i a l values (see Lemma 2.1.3), hence on p, themap

/ : 0 b * ( - e , ) - M(x, t) - > 7 ( ( ^ ( x ) , f )

is likewise differentiable, where (p : C/o Vo is a local chart for Mo- Wemay assume

x0 = y?(0),by composing


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2.2 Fields of geodesic curves 45are orthogonal to all the vectors J- r X ^ ^ M o , j = 1 , . . . , n 1). Therefore, by the inverse function theorem, / yields a chart in some neighbourhood U of (0,0) e Uo x (,). / obviously satisfies (i), (ii) (afterredefining V) . (ii i) also holds by construction (putt ing x n = t). Next,g nn = 1, since the curves x% = c^, namely / ( c i , . . . , c n _ i , ), t 6 (e, e),are geodesies parameterized by arc-length, hence gnn = = 1.Finally, the system of equations for these curves to be geodesic is

- + r * ^ S : (* n = *) * * * = ! , . . . , n .(d x n)2 t3dxndxnHence in particular

r*n = 0 for fc = l , . . . , n .

1Now

1r n n = ^9M(l9nl,n ~ 9nn,l) = ^ ' f f n l . n ,

since # n n = 1. Therefore

g nkjn = 0 for all k = 1,... , n .Since furthermore ^ ( x 1 , . . . , x n _ 1 , 0 ) = 0, because the geodesic arcxn ~ t, xl Ci ~ constant, is orthogonal to the surface ^ ( x 1 , . . . , x n _ _ 1 )= / ( x 1 , . . . , x n ~ 1 , 0 ) , we obtain

9nk = 0.

DejRnition 2.2.1. T/ie coordinates whose existence is affirmed by Theorem 2.2.1 are called geodesic parallel coordinates based on the hyper-surface M0.Theorem 2.2.2. Let f : U V be a chart with the properties describedin Theorem 2.2.1. In particular, the curves x% = Ci, Ci constant,, fori = l , . . . , n 1 are geodesic arcs. Then any such curve is the shortest connection of its endpoints when compared with all curves containedentirely in U and having the same endpoints.Proo f We consider the geodesic

7(t) = { x * = ^ , x n = t, -e < t < c},where U = / 7 0 x (-e, e). Let 7 ( f ) , t\ < t < t2 be another curve in U w i t h7 ( ^ 1 ) = 7( -e ) , 7 ( t 2 ) = 7(e). We have to prove

H i) > i (7) , (2-2.3)


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46 Geodesic curvesw i t h strict inequality, unless 7 is a reparameterization of 7 . Now

( 7 ) - ( 9ij (7( 0)7 W^W + ( T ^ ) ) j (2.2.4) / T L V , J = I /

since # n n = 1,gin = 0 fori = 1,..., n - 1 byTheorem 2.2.1(iv),

r i 7 n (o>= 1(7) .

* > 7 n ( < 2 ) - 7 n ( < i ) = 7 n ( e ) - 7 n ( - e )

The first inequality is strict, unless 7* is constant for i = 1,... , n 1,and thesecond one is strict, unless 7 n ( ) ismonotonic.

q.e.d.Following Weierstrafi, we say that thegeodesies

7( ) = {x* =C i , x n = t,-e


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2.2 Fields of geodesic curves 47We now define a map

e = eZ0 : {w G E n : \w\ < e0} -+ Uw H - 7tu( l ) .

Then e(0) = z 0 . We compute the derivative of e at 0 asDe(0)(v) = | 7 t ( l ) |

= ^ 7 ( % . o by (2.2.5)= 7(0)

Hence, the derivative of e at 0 G E n is the identity, and the inversemapping theorem implies:Theorem 2.2.3. e maps a neighbourhood of 0 G E n diffeomorphically(i.e. e is bijective, and both e and e~l are differentiable) onto a neighbourhood of Z Q G U. q.e.d.

We want to normalize our chart / : / V for M. First of all, we mayassume

z0 = 0 (2.2.6)for the point zo G U under consideration. Secondly, the transformationformula (2.1.16) implies that we may perform a linear change of coordinates (i.e. replace / by / o A, where A G GL(n,R)) in order to achieve

ffy(0) = fy. (2.2.7)We assume that / : U V satisfies these normalizations. We thenreplace / by / o e defined on {w G E n : \w\ < e 0 }.Theorem 2.2.4. In this new chart, the metric tensor satisfies

fti(0) =


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48 Geodesic curvesi n i t i a l direction v. We thus insert 7(f) = tv into the geodesic equation(2.1.21). Then 7 = 0, hence

Tijk(tv)vjvk = 0 for i = l , . . . , n .I n particular, inserting t = 0, we get

T j f c ( 0 y = 0 for all v e E n , i = 1 , . . . , n.We use t; = e*, where {ei) l=1 n is an orthonormal basis of E n . Then

17 , (0 ) = 0 for al i i and/.We next insert v = \(ei 4- e m ) , / ^ m. The symmetry TJ.fc = Tlh - (whichdirectly follows from the definition of Tl-k and the symmetry = gkj)then yields

r j m ( 0 ) = 0 for ali i , / , m.The vanishing of gij^ for all i , j , then is an easy exercise in linearalgebra. q.e.d.Definition 2.2.2. The local coordinates xl,...,xn constructed beforeTheorem 2.2.4 a r e called Riemannian normal coordinates.

We let x 1 , . . . , x n be Riemannian normal coordinates. We transformthem into polar coordinates r, y? 1 , . . . , (pn~l in the standard manner (e.g.i f n = 2, x1 = rcos^ 1 , x 2 = rsiny? 1) . This coordinate transformationis of course singular at 0. We now express the metric tensor w.r.t. thesepolar coordinates. We write grr instead of gn, and we write gr(p insteadof gu, I = 2, . . . , n , and g w instead of (9ki)k,i= 2,...,d' * n Particular, byTheorem 2.2.4 and the transformation rule (2.1.16)

ffrr(0) = l , f f r V ( 0 ) = 0 . (2.2.10)The lines through the origin are geodesies by the construction of Riemannian normal coordinates, and in polar coordinates, they now becomethe curves (p (y? 1 , . . . ,


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52 Geodesic curvesWe now proceed to establish a global result:Theorem 2.3.2. Let M be a compact connected differentiable subman-ifold ofW*, or, more generally, a compact connected Riemannian manifold. Then any two points p, q G M can be connected by a shortestgeodesic arc (i.e. of length d(p,q)).Proof. Let ( c n ) n N be a minimizing sequence. We may assume w. l .o .g .that all cn are parameterized on the interval [0,1] and proportionally toarc-length. Thus

C n (0) = p, c n ( l ) = q,L(c n) > d(p, q) for n oo.

For each n, we may f ind^0,n = 0 < i ? n < . . . < m , n = 1

w i t h

w i t h e0 given by Theorem 2.3.1. By Theorem 2.3.1, there exists a uniqueshortest geodesic arc between cn (tj-i,n) =: P j - i , n and cn (t j jTl) =: Pj, n .We replace cn\[t ^ t j by this shortest geodesic arc and obtain anew minimizing sequence, again denoted by c n, that now is piecewisegeodesic. Since the length of the cn are bounded because of the m i n i mizing property, we may actually assume that m is independent of n.Since M is compact, after selecting a subsequence of c n , the points pjyTlconverge to l i m i t points p^, ( j = 0,. . . , m) as n oo. c n j [ t i t ^ theunique shortest geodesic arc between P j _ i , n and Pj, n , then converges tothe unique shortest geodesic arc between Pj-\ and pj (for this point, oneverifies that l i m i t s of geodesic arcs are again geodesic arcs, that l i m i t sof shortest arcs are again shortest arcs, that d(pj-\,pj) < Q , and oneuses Theorem 2.3.1). We thus obtain a piecewise geodesic l i m i t curve c,w i t h c(0) = p, c( l) = g, and

L(c) = lim L ( c n ) ,noosince we have for the geodesic pieces

L ( c i i i - . . , . ) = B ^ L ( c i ^ - . - ^ - i )for all j (tj = lim ^ , n ) . Since the cn constitute a minimizing sequence,noo '

L ( c ) = d(p,g),


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2.3 The existence of geodesies 53and c thus is of shortest possible length. This implies that c is geodesic.Namely, otherwise we could f ind 0 < s\ < s2 < 1 w i t h L ( c | [ s i ) S 2 ] ) < e0,but w i t h C | ( S I not being geodesic. Replacing c j { a i 8 2 ) by the shortestgeodesic arc between c(s\) and c(s2) would y i e l d a shorter curve (cf.Theorem 2.2.6.), contradicting the minimizing property of c.

q.e.d.Thus, any two points on a compact M may be connected by a shortest

geodesic. We now pose the question whether they can be connected bymore than one geodesic, not necessarily the shortest. On 5 n , for example,this is clearly the case. Actually, the answer is that it is the case on anycompact M. That result needs a topological result that is not availableto us here, however. Therefore, we w i l l restrict ourselves to a specialcase which, however, already displays the crucial geometric idea of theconstruction for the general case, too.Theorem 2.3.3. Let M be a differentiable submanifold of Euclideanspace W*, (or more generally^, a Riemannian manifold), diffeomorphicto the sphere S2. The latter condition means that there exists a bijectivemap

h:S2 ~*Mthat is differentiable in both directions. Then any two points p, q G Mcan be connected by at least two geodesies.Proof. M is compact and connected since diffeomorphic to S2 which iscompact and connected. Let us assume p ^ q. We leave it to the readerto modify our constructions in order that they also apply to the casep = q. (In that case, Thm 2.3.3 asserts the existence of a nonconstantgeodesic c : [0,1] > M w i t h c(0) = p = c(l).) One may then constructa diffeomorphism

ho : S2 - Mw i t h the fo l lowing properties:Let S2 = { ( \ 2 , 3 ) G M 3 : |x| = l } . Then

p = M0,0,l), g = MO, 0,-1)and a shortest geodesic arc c : [0,1] M w i t h c(0) = p, c(l) = q isgiven by

c(t) = M0 , s i n 7 r , c o s 7 r ) .f See footnote on p. 43.


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54 Geodesic curvesLet us point out that these normalizations are not at all essential, butonly convenient for our constructions. We look at the family of curves

7(,s) = /io(sin27rssin7r, cos27rssin7r,cos7r), 0 < s,t < 1. (2.3.1)Then

7 (*,0) = 7 ( M ) = c(t) for all tand

7(0, s) = c(0), 7 (1 , s) = c( l) for all s.We find some number K w i t h

L{n(-,8)) 0 such that the shortest geodesic between any p, q G M, w i t hd(p, q) < o is unique. Let

0 = t0 < * i < .. . < tm = 1be a partition of [0,1] w i t h

h ~ tj-i < for j = 1 , . . . , m. (2.3.3)Let another partition ( T I , . . . , r m ) satisfy

To = t0 < r l < h < T2 < < T M < tm = T m + iand

T j ~ T j ~ 1 < : ? forj = l , . . . , m + l . (2.3.4)I f 7 : [0,1] M is any curve parameterized proportionally to arc-lengthw i t h

L(l) < K,we then have for j = 1 , . . . , m

d (7 (^ - 1) ,7(*j )) < I ( T | [ t j _ l ! ( j ) ) < * f =Therefore, by Theorem 2.3.1, the shortest geodesic from y(tj-i) to y(tj)is unique. We then define 7*1(7) to be that piecewise geodesic curve forwhich r i ( 7 ) j [ t i t ] coincides w i t h the shortest geodesic from 7(j_i)to 7 ( ^ j ) , j = 1 , . . . , m. Likewise, we let ^ ( 7 ) by that piecewise geodesiccurve for which r2 (7 ) |[r. r j coincides w i t h the again unique shortest geodesic from 7 ( T J _ I ) to 7 ( T J ) , j = 1 , . . . , m + 1. We now observe:


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2.3 The existence of geodesies 55Lemma 2.3.1. Suppose d(7(fy)7(fy-i)) < e 0 and d(y(Tj),7(r7_i)) inf F(x).xR dThen

{x G E d : F(x) < so}is compact and nonempty, and since F is continuous, it has to assumeits i n f i m u m on that set. We now assume that F even has two relativeminima, x i , #2 in E d , and that they are strict in the fo l lowing sense: Forx = # i , # 2 , we have

3 F(x). (3.1.3)Theorem 3.1.1. Under the above assumptions, F has a third criticalpoint 3 (i.e. VF(xs) = 0) with

F(x3) > max(F(x i ) ,F(x2 ) ) = : oProof. We consider curves 7 : [0,1] Rd w i t h

7(0) = x i ,7(1 )=*2. (3.1.4)62


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6 4 Saddle point constructions# 3 w i l l then be necessarily be different from X\ and #2- As a step towardsthe existence of such a point 3, we claim

Ve > 0 3 0 V curves 7 w i t h 7(0) = #1,7(1) = x2w i t h

sup F(7(0) i - 6 (3.1.10)| (VF) ( 7 ( * o ) ) | < c . (3.1.11)

Suppose this is not the case. Then3to > 0 V n N 3 curve 7 between Xi and x2 w i t h

supF (7 ( t ) ) < i + - (3.1.12)t nV < 0 w i t h F(7 ( t 0 ) ) > K I - 0 (3.1.13)

| ( V F ) ( 7 ( o ) ) | > 0 . (3.1.14)For s > 0, we define a new curve 7 i S by

7 , . ( < ) : = 7n ( ) - ( V F ) ( 7 ( * ) ) -

Since x\ and #2 are minima, VF{x\) 0 = VF(#2)> and so7n,*(0) = x i , 7 n , 5 ( l ) = x 2 ,

so that the curves 7 n > s are valid comparison curves. By our propernessassumption (3.1.2) and (3.1.12), 7 n () stays in a bounded subset of E d ,and VF w i l l then be bounded on that bounded set, and hence for anySo > 0 and all 0 < 5 < so, the curves 7n,s(0 stay in some boundedset, too. This set is independent of n (as long as 0 < 5 < 5 0 , for fixedS Q > 0). By Taylor's formula

F ( 7 n , , ( 0 ) = F( ln(t)) - sVF( ln(t)) V F ( 7 ( ) ) + o(s).Since F is continuously differentiable and 7 n , s ( ) is contained in a bounded set, 0(5) can be estimated independently of n and t (as long as 0 0 smaller,

F ( 7 n , . W ) < ^ ( 7 ( * ) ) - I | V F ( 7 n ( < ) ) | 2 (3.1.15)


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66 Saddle point constructions(cf. (3.1.15), (3.1.16), (3.1.14)), and finally for al l t w i t h F(y n(t)) >K l ~ 2

F( 7 ( t ) ) = F ( 7 n , . 0 ( * ) ) < ! (cf- (3.1.19)).Thus, (3.1.20) holds indeed. This, however, contradicts the definition of i . Therefore, the assumption that our claim was not correct led to acontradiction, and the claim holds. I t is now simple to prove the theorem.Namely, we let e n > 0 for n oo, and for e = e n, we find max(F(xi ) ,F(x 2 ) ) = 0 ,or it has infinitely many critical points.Proof. For the argument of the proof of Theorem 3.1.1, we only need

inf sup F(7(*)) > 0 , (3.1.26)7 t[0,l]

where the i n f i m u m again is taken over curves 7 : [0,1] E d w i t h 7(0) =# i , 7(1) = # 2 . So, suppose that (3.1.26) does not hold. We then want to


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3.2 The construction of Lyusternik-Schnirelman 67show the existence of i n f i n i t e ly many critical points. As in the proof ofTheorem 3.1.1, we may assume

F(x x) < F(x2).The argument at the beginning of the proof of Theorem 3.1.1 then showsthat (3.1.26) holds if x2 is a strict relative minimum. If x2 is a relativeminimum, which is not strict, for all sufficiently small 6 > 0, say


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68 Saddle point constructions

Figure 3.1.

Theorem 3.2.1. Let 7 be a closed convex Jordan curved of class C1 inthe plane E 2 . (7 then divides the plane into a bounded region A, and anunbounded one, by the Jordan curve Theorem. That 7 is convex meansthat the straight line between any two points of 7 is contained in theclosure A of A.) Then there exist at least two such straight lines betweenpoints on 7 meeting 7 orthogonally at both end points (see Figure 3.1).

Proof. We start by finding one such line. Let C be the set of all straightlines / in A w i t h dl C 7. We say that a sequence (l n)neN C convergesto / E , i f the end points of the ln converge to those of /. In order tohave a closed space, we allow lines to be t r i v i a l i.e. to consist of a singlepoint on 7 only. We denote the space of these point curves on 7 by 0 -We let / := [0,1] be the unit interval. We consider continuous maps

v:I-*Cw i t h the following two properties:

( i ) v(0) = v(l).( i i ) To any such family, we may assign two subregions A\(t) and A2(t)

of A in a certain manner. Namely, we let A\(t) and A2(t) be thetwo regions into which v(t) divides A. Having chosen A\(0) andA 2(0), A\(t) and A2(t) then are determined by the continuity

t A closed Jordan curve is a curve 7 : [0, T] Rd with 7( 0) = 7(T") that is injectiveon [0, T ) . Cf. the definition of a Jordan curve on p. 35.


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3.2 The construction of Lyusternik-Schnirelman 69

Figure 3.2.

requirement. We then requireA 1(1) = A 2(0).

We let Vi be the class of all such families v.The construction is visualized in Figure 3.2. (0 corresponds to 0 J,

/to\J/ to I / / / to | , 1 to 1)Actually, in order to s impl i fy the visualization, if v(0) is a point curve(on 7), i) may be relaxed to just requiring that v(l) also is a point curve

(on 7), not necessarily coinciding w i t h v(0) (see Figure 3.3). Namely, anypoint curves can be connected through point curves, i.e. w i t h vanishinglength.

We denote by L(l) the length oil C and defineK\ := inf supL(v(t)).

v^ yi tei1

Figure 3.3.


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70 Saddle point constructionsW e want to show that

i > 0.For this purpose, let p > 0 be the inner radius of 7, i.e. the largest p forwhich there exists a disc

B(x 0lp) C Afor some XQ G A (B(xo,p) := {x G E 2 : \x - x 0 | < p } ) - Then

i > i := inf sup L(v(t) P\ B ( X Q , p)).W e let Aft) := ^ ( t ) n B(xo,p), i = 1,2. Because of (ii) and the

continuous dependence of J4*() and hence also of A f{(t) on t, there existssome to I w i t h

Area ( t 0 )) = Area (A '2(t 0)) .Thus v(to) divides B(xo,p) into two subregions of equal area. v(to) thenhas to be a diameter of B(xo,p), i.e.

L(v(to)nB(x 0,p))=2p.Therefore

i > i = 2p > 0and K i is positive indeed. We are now going to show by a line of reasoningalready familiar from Sections 2.3 and 3.1 that K\ is realized by a c r i t i c a lpoint / of L among all lines w i t h end points in 7, i.e. by / meeting 7orthogonally (see Theorem 1.4.1). For that purpose we shall assume forthe moment that 7 is of class C 3 . Later on, we shall reduce the casewhere 7 is only C 1 to the present one by an approximation argument.W e now claim

V e > 0 3(5 > 0 : Vv Vi w i t hsupL (v(t)) < K i+ 6te i

3 to G I w i t h L (v(to)) > i - cand |cos(ai (v (t 0)))\ , |cos ( a 2 {v (t 0)))\ < c,

where a\(l) and a2(l) are the angles of / at its endpoints w i t h 7.


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3.2 The construction of Lyusternik-Schnirelman 71Otherwise

3e 0 > 0 : V n G N 3vn G V\ w i t hsupL(v n ( t ) ) < i 4- t

V t 0 w i t h L (vn(*o)) > K I _ o|cosai (vn(*o))| > eo

or |cosa2 (vn(*o))| > o-The idea to reach a contradiction from that assumption is simple, oncethe following Lemma is proved:Lemma 3.2.1. For every planar closed Jordan curve 7 of class C3,there exists (3 > 0 with the following property: Whenever x G E 2 satisfies

dist(x,7) := inf \x y\ < (3ye-ythere exists a unique y G 7 with dist(#,7) = \x y\.Proof. We consider 7 as an embedded submanifold of the Euclideanplane E 2 . 7 is then covered by the images of charts / : U V of thetype constructed in Theorem 2.2.1. Here, U and V are open in E 2 , and

7 n v = f (u n {x2 = 0 } ) .Furthermore, the curves x

1

= constant in U correspond to geodesies, i.e.straight lines in V perpendicular to 7, and they form shortest connections to 7 f l V. By shrinking U, if necessary, we may assume that it is ofthe form ( - , ) x ( -77 , rj), w i t h > 0, rj > 0. Since 7 is compact, it canbe covered by finitely many such charts

fi : ( - 6 , 6 ) x (-WiVi) ~*yi , i = l , . . . , m .I f we then restrict fi to ( - 6 , 6 ) x ( " f 1 ' ^ J , lines x1 = constant,~ k < x2 < ^ , then correspond to shortest geodesies to 7, since the partof 7 not contained in V{ is not contained in the image of fi, and hence hasdistance at least ^ from the image of the smaller set ( & ) x , ) .This is indicated in Figure 3.4 where the broken lines correspond tox2 = ^ and this is depicted for two different indices i.

Therefore, (3 := min ( ^ ) satisfies the claim.i=l,...,n q.e.d.


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Saddle point constructions

Figure 3.5.

We now return to the proof of Theorem 3.2.1:Without loss of generality eo < 0 < Assume e.g.

cosai (vn (to)) > e0.The following construction is depicted in Figure 3.5. Choose si(to)


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3.2 The construction of Lyusternik-Schnirelman 73vn(to) with

|ai(*o)-Pi(to)|=0,where p\(to) is the endpoint of vn(to) where it forms the angle a\(to)with 7. We replace the subarc v^(to) of vn(to) between p i ( t 0 ) and si(o)by the shortest line segment vfn(to) from s\(to) to 7. By the theorem ofPythagoras and the convexity of 7

L (v'n (t0)) < L (yln (t0)) s inai (vn (t0))

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