B. van Brunt - The Calculus of Variations-Springer (2004)

P refaceThe calculus of variations has a long history of interaction with other branches of m athem atics such as geom etry and differential equations, and with physics,particularly m ec

The (alculus

Bruce van Brunt

U ni versi text

Editorial Board

lNo4h Ameri cal:

B ruce van B runt

lnstitute of Fundamental Sciences

Palm erston Nort.h Cam pus

Private Bag 11222

M assey University

Palmerston Nort.h 5301

New Zealand

b.vanbrunt@ m assey.ac.nz

Editorial Board

tA%rf/ Americal:

S Axler

M athem atics Departgnent

San Francisco State University

San Francisco, CA 94132

USA

axler@ sfsu.edu

K A Ribet

M athem atics Departgnent

University of California, Berkeley

Berkeley, CA 94720-3840

USA

rilx t@ m attl.lxrkeley.edu

M athematics Subject Classification (2000)2 34Bxx, 49-0 1, 70Hxx

Library of Congress Cataloging-in-publication Data

van Brunt, B (Bmce)

The calculus of variations / Bmce van Bm nt.

ISBN 0- 387-40247- 0 Pr int ed on acid-fr ee paper.

( C) 2004 Spr i nger- ver l ag New Yor k lnc.

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P reface

The calculus of variations has a long history of interaction with other branches

of m athem atics such as geom etry and differential equations, and with physics,particularly m echanics M ore recently, the calculus of variations has foundapplications in other fields such as econom ics and electrical engineering M uch

of the m athem atics underlying control theory, for instance, can be regarded

as part of the calculus of variations

T his book is an introduction to the calculus of variations for m athem ati

-cians and scientists The reader interested prim arily in m athem atics w ill findresults of interest in geom etry and differential equations 1 have paused attim es to develop the proofs of som e of these results, and discuss briefly var-

ious topi cs not normall y found in an introductory book on thi s subject such

as the existence and uniqueness of solutions to boundary-value problem s, theinverse problem , and M orse theory 1 have m ade ttpassive use'' of f-unctional

analysis (in particular normed vector spaces) to place certain results in

con-text and reassure the m athem atician that a suitable fram ework is availablefor a m ore rigorous study For the reader interested m ainly in techniques andapplications of the calculus of variations, 1 leavened the book w ith num er-ous exam ples m ostly from physics ln addition, topics such as Ham ilton'sPrinciple, eigenvalue approxim ations, conservation law s, and nonholonom icconstraints in m echanics are discussed M ore im portantly, the book is w ritten

on two levels The technical details for m any of the results can be skipped

on the initial reading The student can thus learn the m ain results in eachchapter and return as needed to the proofs for a deeper understanding Sev-

eral key results in this subject have tractable analogues in finite-dimensional

optim ization W here possible, the theory is m otivated by first reviewing thetheory for finite-dim ensional problem s

T he book can be used for a one-sem ester course, a shorter course, or idependent study The final chapter on the second variation has been w rittenwith these options in m ind, so that the student can proceed directly fromChapter 3 to this topic Throughout the book, asterisks have been used toflag m aterial that is not central to a first course

n-graduate students in m athem atics, physics, or engineering The student is sum ed to have som e fam iliarity with linear ordinary differential equations,

as-m ultivariable calculus, and elem entary real analysis Som e of the m ore retical m aterial from these topics that is used throughout the book such asthe im plicit f-unction theorem and Picard's theorem for differential equationshas been collected in A ppendix A for the convenience of the reader

theo-Like m any textbooks in m athem atics, this book can trace its origins back

to a set of lecture notes The transform ation from lecture notes to textbook,how ever, is nontrivial, and one is faced with m yriad choices that, in part, re-

flect one's own interests and experiences teaching the subject W hi le writing

this book 1 kept in m ind three quotes spanning a few generations of m

athe-m aticians The first is from the introduction to a volum e of Spivak's m ulti

-volume treatise on di fferential geometry ( 641

1 feel som ewhat like a m an w ho has tried to cleanse the A ugean stableswith a Johnnp lklop

lt is tem pting, when w riting a textbook, to give som e m odicum of com

plete-ness W hen faced with the enorm ity of literature on this subject, however, the task proves daunting, and it soon becomes clear that there is just too

m uch m aterial for a single volum e ln the end, 1 could not face picking upthe Johnny-lklop, and m y solution to this dilem m a was to be savage w ith

m y choice of topics K eeping in m ind that the goal is to produce a bookthat should serve as a text for a one-sem ester introductory course, there were

m any painful om issions Firstly, 1 have tried to steer a reasonably consistentpath by keeping the focus on the sim plest type problem s that illustrate aparticular aspect of the theory Secondly, 1 have opted in m ost cases for the

tno frills'' version of results if the ttull feature'' version w ould take us toofar afield, or require a substantially m ore sophisticated m athem atical back-ground Topics such as piecew ise s11100th extrem als, fields of extrem als, andnum erical m ethods arguably belong in any introductory account N onethe-less, 1 have om itted these topics in favour of other topics, such as a solution

m ethod for the H am ilton-lacobi equation and N oether's theorem , that areaccessible to the general m athem atically literate undergraduate student but

often postponed to a second course in the subject.

T he second quote com es from the introduction to Titchm arsh's book on

ei genf - unction expansions ( 701

1 believe in the future of tm athem atics for physicists', but it seem s

desirabl e that a writer on this subject shoul d understand 1 30th physics

as w ell as m athem atics

The w ords of Titchm arsh rem ind m e that, although 1 am a m athem aticianinterested in the applications of m athem atics, 1 am not a physicist, and it

is best to leave detailed accounts of physical m odels in the hands of experts.This is not to say that the m aterial presented here lies in som e vacuum of pure

m athem atics, w here we m erely acknowledge that the m aterial has found som eapplications lndeed, the book is written with a definite slant towards ttapplied

m athem atics,' but it focuses on no particular field of applied m athem atics inany depth O ften it is the application not the m athem atics that perplexesthe student, and a study in depth of any particular field would require eitherthe student to have the necessary prerequisites or the author to develop the

subject The former case restricts the potential audience' , the latter case shifts

away from the m ain topic ln any event, 1 have not tried to w rite a book onthe calculus of variations with a particular em phasis on one of its m any fields

of applications There are m any splendid books that m erge the calculus ofvariations w ith particular applications such as classical m echanics or control

theory Such texts can be read wi th profit in conjunction with this book.

T he third quote com es from G H H ardy, who m ade the follow ing com m ent

about A R Forsyth' s 656-page treatise (271 on the calculus of variati ons : 1

ln this enorm ous volum e, the author never succeeds in proving thatthe shortest distance betw een two points is a straight line

H ardy did not m ince words w hen it cam e to m athem atics The prospectiveauthor of any text on the calculus of variations should bear in m ind that,although there are m any m athem atical avenues to explore and endless m inu-tia? to discuss, certain basic questions that can be answ ered by the calculus

of variations in an elem entary text should be answered There are certainproblem s such as geodesics in the plane and the catenary that can be solvedwithin our self-im posed regim e of elem entary theory 1 do not hesitate to usethese sim ple problem s as exam ples At the sam e tim e, 1 also hope to give the

reader a glimpse of the power and elegance of a subject that has fascinated

m athem aticians for centuries

1 wish to acknow lege the help of m y form er students, whose input shapedthe final form of this book 1 w ish also to thank Fiona D avies for helping m ewith the figures Finally, 1 would like to acknowledge the help of m y colleagues

at the lnstitute of Fundam ental Sciences, M assey U niversity

T he earlier drafts of m any chapters were w ritten w hile travelling on various m ountaineering expeditions throughout the South lsland of N ew Zealand.The hospitality of Clive M arsh and H eather North is gratef-ully acknowledgedalong w ith that of A ndy Backhouse and Zoe Hart 1 should also like to ac-know ledge the New Zealand A lpine C lub, in whose huts 1 w rote m any early

-(and later) drafts during periods of bad weather ln particular, 1 would like

to thank G raham and Eileen Jackson of Unw in H ut for providing a second

home conduci ve to writing (and climbing).

Fox G lacier, N ew Zealand Bruce van BruntFebruary 2003

1 F.Sm ithies reported this com m ent in an unpublished talk, ttlardy as 1 Knew

Him, ' gi ven to the Bvit isl t Society Jt pr f/ zc Hi stovy 6, / Matl tematics 19 December

1990

C ontents

1 lntroduction 1

1.1 lntroduction 1

1.2 The Catenary and Brachystochrone Problem s 3

1.2.1 The Catenary 3

1.2.2 Brachystochrones 7

Ham ilton's Principle 10

Som e Variational Problem s from G eom etry 14

1.4.1 D ido's Problem 14

1.4.2 G eodesics 16

1.4.3 M inim al Surfaces 20

Optim al H arvest Strategy 21

2 T he First Variation 23

2.1 The Finite-D im ensional Case 23

2.1.1 Functions of O ne Variable 23

2.1.2 Functions of Several Variables 26

The Euler-lsagrange Equation 28

Som e Special Cases 36

2.3.1 Case 1: No Explicit y D ependence 36

2.3.2 Case 11: N o Explicit z Dependence 38

2.4 A Degenerate C ase 42

2.5 lnvariance of the Euler-lsagrange Equation 44

2.6 Existence of Solutions to the Boundarp value Problem * 49

3 Som e G eneralizations 55

3.1 Functionals Containing H igher-order D erivatives 55

3.2 Several D ependent Variables 60

3.3 Two lndependent Variables* 65

3.4 The lnverse Problem * 70

lsoperim etric P roblem s 73

4.1 The Finite-D im ensional Case and Lagrange M ultipliers 73

4.1.1 Single Constraint 73

4.1.2 M ultiple C onstraints 77

4.1.3 A bnorm al Problem s 79

The lsoperim etric Problem 83

Som e G eneralizations on the lsoperim etric Problem 94

4.3.1 Problem s Containing H igher-order Derivatives 95

4.3.2 M ultiple lsoperim etric Constraints 96

4.3.3 Several Dependent Variables 99

5 A pplications to Eigenvalue P roblem s* 103

5.1 The Sturm -lsiouville Problem 103

5.2 The First Eigenvalue 109

5.3 Higher Eigenvalues 115

6 H olonom ic and N onholonom ic C onstraints 119

6.1 Holonom ic C onstraints 119

6.2 Nonholonom ic Constraints 125

6.3 Nonholonom ic Constraints in M echanics* 131

P roblem s w ith V ariable Endpoints 135

7.1 Natural B oundary Conditions 135

7.2 The G eneral C ase 144

7.3 Transversality Conditions 150

8 T he H am iltonian Form ulation 159

8.1 The Legendre Transform ation 160

8.2 Ham ilton's Equations 164

8.3 Sym plectic M aps 171

8.4 The H am ilton-lacobi Equation 175

8.4.1 The General Problem 175

8.4.2 Conservative System s 181

Separation of Variables 184

8.5.1 The M ethod of A dditive Separation 185

8.5.2 Conditions for Separable Solutions* 190

9 N oetherhs T heorem 201

9.1 Conservation Law s 201

9.2 Variational Sym m etries 202

9.3 Noether's Theorem 207

9.4 Finding Variational Sym m etries 213

Contents X111

10 T he Second Variation 221

10.1 The Finite-D im ensional Case 221

10.2 The Second Variation 224

10.3 The Legendre Condition 227

10.4 The Jacobi N ecessary Condition 232

10.4.1 A Reformulation of the Second Variation 232

10.4.2 The Jacobi Accessory Equation 234

10.4.3 The Jacobi Necessary Condition 237

10.5 A Sufllcient Condition 241

10 6 M ore on Conjugate Points 244

10 6.1 Findi ng Conjugate Points 245

10.6.2 A G eom etrical lnterpretation 249

10.6.3 Saddle Points* 254

10.7 Convex lntegrands 257

A A nalysis and D ifferential E quations 261

A 1 Taylor's Theorem 261

A 2 The lm plicit Function Theorem 265

A 3 Theory of Ordinary Differential Equations 268

B Function Spaces 273

B 1 Norm ed Spaces 273

B 2 Banach and Hilbert Spaces 278

R eferences 283

lndex 287

1 1 lntroduction

The calculus of variations is concerned with finding extrem a and, in this sense,

it can be considered a branch of optim ization The problem s and techniques

in this branch, however, differ m arkedly from those involving the extrem a

of f-unctions of several variables owing to the nature of the dom ain on thequantity to be optim ized A functional is a m apping from a set of functions

to the real num bers T he calculus of variations deals w ith finding extrem afor functiona.ls as opposed to f-unctions The candidates in the com petitionfor an extremum are thus functions as opposed to vectors in R'', and this

gives the subject a disti nct character The f - unctionals are generally defined

by definite integrals', the sets of f-unctions are often defined by boundary ditions and sm oothness requirem ents, which arise in the form ulation of theproblem m odel

con-T he calculus of variations is nearly as old as the calculus, and the two

subjects were developed somewhat i n parallel ln 1927 Forsyth ( 271 noted that

the subject t t attracted a rather fickle attention at m ore or l ess isol ated interval s

in its grow th.' ln the eighteenth century, the Bernoulli brothers, Newton,

Leibniz, Euler, Lagrange, and Legendre contributed to the subject, and thei r

work was extended significantly in the next century by Jacobi and W eierstraB

Hil bert ( 381 i n hi s renowned 1900 l ecture to the lnternational Congress of

M athemati ci ans, outlined 23 (now famous) problems for mathematicians Hi s

231.d problem is entitled Fnrtlter developvtent o.f tl te mc// ztl s o f tlte calcnln, s

o.f variations lmm ediatel y before describing the problem , he rem arks:

1 should like to close with a general problem , nam ely with theindication of a branch of m athem atics repeatedly m entioned in thislecture- w hich, in spite of the considerable advancem ent lately given

it by W eierstraB, does not receive the general appreciation w hich in

m y opinion it is due- l m ean the calculus of variations

H ilbert's lecture perhaps struck a chord with m athem aticians.l In the earlytwentieth century Hilbert, Noether, Tonelli, Lebesgue, and Hadam ard am ongothers m ade significant contributions to the field Although by Forsyth's tim e

the subject m ay have t tattracted rather fi ckle attention,' ' m any of those who

did pay attention are num bered am ong the leading m athem aticians of the

last three centuries The reader is directed to Gol dstine ( 361 for an i n-depth

account of the history of the subject up to the late nineteenth century.

T he enduring interest in the calculus of variations is in part due to its

ap-pli cations Of particul ar note is the relationship of the subject with classical

m echanics, w here it crosses the boundary from being m erely a m athem atical tool to encom passing a general philosophy Variational principles abound

-in physics and particularly -in m echanics T he application of these principles

usually entails fi nding functions that mi nimize definite integrals (e.g., energy integral s) and hence the calcul us of variations comes natural ly to the fore.

H am ilton's Principle in classical m echanics is a prom inent exam ple A n earlierexam ple is Ferm at's Principle of M inim um Tim e in geom etrical optics Thedevelopm ent of the calculus of variations in the eighteenth and nineteenthcenturies w as m otivated largely by problem s in m echanics M ost textbooks on

cl assical mechanics (old and new) discuss the calculus of variations in some

depth Conversely, m any books on the calculus of variations discuss applitions to classical m echanics in detail ln the introduction of C arath4odory's

ca-book ( 211 he states:

1 have never lost sight of the fact that the calculus of variations, as it

is presented in Part 11, should above all be a servant of m echanics

C ertainly there is an intim ate relationship between m echanics and the calculus of variations, but this should not com pletely overshadow other fieldswhere the calculus of variations also has applications Aside from applications

-in traditional fields of continuum m echanics and electrom agnetism , the lus of variations has found applications in econom ics, urban planning, and ahost of other ttnontraditional fields.' lndeed, the theory of optim al control iscentred largely around the calculus of variations

calcu-Finally it should be noted the calculus of variations does not exist in a

m athem atical vacuum or as a closed chapter of classical analysis H istorically,this field has always intersected with geom etry and differential equations,

and continues to do so ln 1974, Stampacchia (171, writing on Hilbert' s 231

problem , sum m ed up the situation:

O ne m ight infer that the interest in this branch of Analysis is ening and that the Calculus of Variations is a C hapter of C lassical

weak-A nalysis ln fact this inference would be quite w rong since new lem s like those in control theory are closely related to the problem s of

prob-1 His nineteenth and twentieth problem s were also devoted to the calculus of variations

-the C alculus of Variations while classical theories, like that of ary value problem s for partial differential equations, have been deeplyaffected by the developm ent of the C alculus of Variations M oreover,the natural developm ent of the Calculus of Variations has producednew branches of m athem atics w hich have assum ed different aspectsand appear quite different from the Calculus of Variations.

bound-The field is far from dead and it continues to attract new researchers

ln the rem ainder of this chapter we discuss som e typical problem s in the

calcul us of variati ons that are easy to model (although perhaps not so easy

to sol ve) These problems i llustrate the above comments and gi ve the reader

a taste of the subject W e return to m ost of these exampl es later in the book

as the m athem atics to solve them develops

1.2 T he C atenary and B rachystochrone P roblem s

1.2.1 T he C atenary

Consider a thin heavy uniform flexible cable suspended from the top of two

poles of height yç j and yï spaced a distance d apart (figure 1.1) At the base of

each pole the cable is assum ed to be coiled The cable follow s up the pole tothe top, runs through a pulley, and then spans the distance d to the next pole.The problem is to determ ine the shape of the cable between the tw o poles

T he cable w ill assum e the shape that m akes the potential energy m inimum The potential energy associated w ith the vertical parts of the cable will bethe sam e for any configuration of the cable and hence we m ay ignore thiscom ponent lf ?rz denotes the m ass per unit length of the cable and g thegravitational constant, the potential energy of the cable between the poles is

U nfortunately, we do not know L in this form ulation W e can, however, cast the above expression for W 's in term s of Cartesian coördinates since we

re-do know the coördinates of the pole tops The differential arclength elem ent

in Cartesian coördinates is given by ds = 1 + y?2, and this leads to thefollow ing expression for W 'p ,

N ote that unlike our first expression for W 's, the above one involves the tive of y W e have im plicitly assum ed here that the solution curve can be

deriva-represented by a f - unction y : g zt ), z11 + R and that this f - unction is continuous

and at least piecew ise differentiable G iven the nature of the problem theseSeem reasonable assum ptions

T he cable w ill assum e the shape that m inim izes W 's The constant factor

m,g in the expression for W 's can be ignored for the purposes of optim izing thepotential energy The essence of the problem is thus to determ ine a f-unction

y such that the quantity

is minimum The model requires that any candidate : ' è for an extremum

sat-isfies the boundary conditions

: h(al o) = #0, 5(z1) = #1.

ln addition, the candidates m ust also be continuous and at least piecew ise

dif ferentiabl e in the interval g zt ), z11

W e find the extrem a for J in Chapter 2, where w e show that the shape ofthe cable can be described by a hyperbolic cosine f-unction The curve itself iscalled a catenary.z

T he sam e functional J arises in a problem in geom etry concerning a m i

n-im al surface of revolution, i.e., a surface of revolution having m inim al surfacearea Suppose that the z-axis corresponds to the axis of rotation Any surface

of revol ution can be generated by a curve in the zp-plane (figure 1 2) The

2 The name ttcatenary'' is particularly descriptive.The nam e com es from the Latinword catena meaning chain Catenary refers to the curve form ed by a uniformchain hanging freely between two poles Leibniz is credited with coining the term

(ca 1691).

problem thus translates to finding the curve y that generates the surface ofrevolution having the m inim al surface area A s with the catenary problem , we

make the assumption that y can be described by a function y : g zt ), z11 + R that is continuous and piecewise differentiable in the interval g zt ), z11 Under

these assum ptions we have that the surface area of the corresponding surface

of revolution is

Here we need also make the assumption that ylz) > 0 for all z CE g zt ), z11.3 The

problem of finding the m inim al surface thus reduces to finding the f-unction ysuch that the quantity

zj

u yy J( t - J ïJ 20 l 1 + y

is m inim um The two problem s thus produce the sam e f-unctional to be m ini

-m ized The generating curve that produces the m inim al surface of revolution

is thus a catenary T he surface itself is called a catenoid

3 lf p = 0 at some point k ( E (zo , z 1 we can sti ll generate a rotati onally symmetri c

t object, ' but technical ly it would not be a surface Near (k, 0, 0) the t t object' ' would resemble (i e., be homeomorphic to) a double cone The doubl e cone fail s

the requirements to be a surface because any neighbourhood containing the com

-m on vertex is not hom eomorphic to the plane

Let us return to the original problem A m odification of the problem would

be to first specify the length of the cable Evidently, if L is the length of thecable w e m ust require that

L k (zl - zo)2 + (#1 - y( jlz

in order that the cable span the two poles M oreover, it is intuitively clearthat in the case of equality there is only one configuration possible viz., the

line segment from (zo, y( t l to (z1, 3 /1) ln this case, there is no optimization to

be done as there is only one candidate W e m ay thus restrict our attention tothe case

L > (z1 - zo)2 + (3/1 - yv)2.

G iven a cable of length L, the problem is to determ ine the shape the cableassum es w hen supported between the poles The problem was posed by JacobBernoulli in 1690 By the end of 1691 the problem was solved by Leibniz,

H uygens, and Jacob's younger brother Johann Bernoulli lt should be notedthat G alileo had earlier considered the problem , but he thought the catenarywas essentially a parabola.'

Since the arcl ength L of the cable is given, we can use expression (1.1)

to look for a m inim um potential energy configuration lnstead, we start

with expression (1.2) The modified probl em is now to find the f - unction

y : g zt ), z11 + R such that W' s is mini mized subject to the arclength

con-straint

1 + y?2 dzand the boundary conditions

!/(zo) = yç j, ! /(z1) = yk.

This problem is thus an exam ple of a constrained variational problem The

constraint (1.6) can be regarded as an i ntegral equation (with, it is hoped, nonunique solutions) Constraints such as (1 6) are call ed isoperimetric W e

discuss problem s having isoperim etric constraints in Chapter 4

Suppose that we use expressi on (1 1), which prim, a Jacie seems simpler than expression (1.2) We know L, so that the l imi ts of the i ntegral are known,

but the param eter s is special and corresponds to arclength W e m ust som how build in the requirem ent that s is arclength if w e are to use expression

e-(1.1) ln order to do this we must use a parametric representation of the curve (z(s), ! /(s)), s CE ( 0, fvl The arclength parameter for such a curve is character-

ized by the differential equation

z/2(s) + : ( // 2(s) = 1.

z(0) = z( ) z(L) = zl : v(0) = yç j, y(L) = yk.

ln general, a constraint of this kind is m ore difllcult to deal w ith than anisoperim etric constraint

1.2.2 B rachystochrones

The history of the calculus of variations essentially begins with a problem

posed by Johann Bernoull i (1696) as a challenge to the mathematical munity and in particular to his brother Jacob (There was significant si bling rival ry between the two brothers.) The problem is important i n the history of

com-the calculus of variations because the m ethod developed by Johann's pupil,Euler, to solve this problem provided a sufllciently general fram ework to solveother variational problem s

T he problem that Johann posed w as to find the shape of a wire alongwhich a bead initially at rest slides under gravity from one end to the other

in m inim al tim e The endpoints of the w ire are specified and the m otion ofthe bead is assum ed frictionless The curve corresponding to the shape of thewire is called a brachystochrones or a curve of fastest descent

T he problem attracted the attention of a num ber of m athem atical lum iies including Huygens, L'H ôpital, Leibniz, and Newton, in addition of course

nar-to the Bernoulli brothers, and later Euler and Lagrange T his problem w as atthe cutting edge of m athem atics at the turn of the eighteenth century

Jacob was up to the chall enge and sol ved the probl em M eanwhile tand independentl y) Johann and Lei bniz also arri ved at correct solutions Newton

was late to the party because he learned about the problem som e six m onthslater than the others Nonetheless, he solved the problem that sam e eveningand sent his solution anonym ously the next day to Johann N ew ton's coverwas blow n instantly Upon looking at the solution, Johann exclaim ed (tA h! 1recognize the paw of the lion.''

To m odel Bernoulli's problem w e use Cartesian coördinates w ith the

pos-iti ve l /-axis ori ented in the direction of the gravi tational force (figure 1.3) Let (zo, y( tl and (z1, 3/1) denote the coördi nates of the i niti al and fi nal posi -

tions of the bead, respectively Here, we require that ztl < zl and yçj < yï.The Bernoulli problem consists of determ ining, am ong the curves that have

(zo, y( t l and (z1, 3/1) as endpoints, the curve on whi ch the bead sl ides down from (zo, y( tl to (z1, 3/1) in mi nimum time The problem makes sense only for continuous curves We make the additional simpl ifjri ng ( but reasonable) as- sumptions that the curve can be represented by a functi on y : g zt ), z11 + R

5 The word com es from the Greek words bvakltistos m eaning ddshortest'' and klwonos

m eaning tim e

t

-L Js

T( y = l s( s ,

where L denotes the arclength of the curve, s is the arclength param eter, and

z? is the velocity of the bead s units down the curve from (zo, y( j) As with

the catenary problem , we do not know the value of L, so we m ust seek analternative form ulation

Our first job is to get an expression for the veloci ty i n terms of the f unction

y W e use the law of conservation of energy to achieve this At any position

(z, !/(z)) on the curve, the sum of the potential and ki netic energies of the

bead is a constant Hence

Christiaan discovered that a bead sliding down a cycloid generated by a circle

of radius p under gravity reaches the bottom of the cycloid arch after theperiod r p g wlterever on the arch the bead starts from rest This notableproperty of the cycloid earned it the appellation isochrone T he cycloid thussports the nam es isochrone and brachystochrone.6 Christiaan used the curve

to good effect and designed what w as then considered a rem arkably accuratependulum clock based on the laudable properties of the cycloid, which wasused to govern the m otion of the pendulum The reader m ay find a diagram

of the pendulum and further details on this interesting curve in an article by

Tee ( 671 wherein several ori ginal references may be f ound.

Finally, we note that brachystochrone problem s have proliferated in the

three centuries fol lowing Bernoulli' s chall enge Some m odels subjected the

bead to a resisting m edium w hilst others changed the force field from a sim pleuniform gravitational field to m ore com plicated scenarios R esearch is stillprogressing on brachystochrones The reader is directed to the w ork of Tee

( 671, ( 681, ( 691 f or more references.

1.3 H am ilton's Principle

There are many fine books on classical (analytical ) mechanics (e g., (11 , (61 ( 351, ( 481 ( 491 ( 591 and ( 731) and we make no attempt here to gi ve even a basic

account of this seemingly vast subject Nonetheless, it would be demeaning

to the calculus of variations to ignore its rich heritage and fruitful interactionwith classical m echanics M oreover, m any of our exam ples com e from classical

m echanics, so a few words from our sponsor seem in order

C lassical m echanics is teem ing with variational principles of w hich Ham ilton's Principle is perhaps the m ost im portant I ln this section we give a brief

-tno frills'' statem ent of Ham ilton's Principle as it applies to the m otion ofparticles The serious student of m echanics should consult one of the m any

special ized texts on this subject.

Let us first consider the m otion of a single particle in R 3.Let r(f) = (z(f), y(t), z(f)) denote the positi on of the particle at time f The kinetic

energy of this particle is given by

1 2 2 2T'

= - w 2 ( ) ( f+ # ( f) + ( f)) ,

where ?rz is the m ass of the particle and ' denotes d dt W e assum e that theforces on the particle can be derived from a single scalar function Specifically,

we assum e there is a function U' such that:

6 It is also called a tautochrone,but we do not count this since the word is derivedfrom the Greek word tauto m eaning ttsam e.' The prefzx iso comes from the Greekword isos which also means ttsame.'

V One need only scan through Lanczos' book ( 481 to fi nd the t t princi pl e of

Vir-tual W ork '' ttzphtlembert's Principle '' ttG auss' Principle of Least Constraint ''

tlacobi's Principle '' and of course ttlamilton's Principle'' am ong others

1.3 Hamilton's Principle

1 U' depends only on time and posi tion; i e., U' = U(f, z, y, z);

2 the force f = (/1, A, /a) acti ng on the particl e has the components

The function L is called the Lagrangian Suppose that the initial position of

the particle r(fo) and final positi on r(f1) are specif ied Ham iltonh s Principle states that the path of the parti cle r(f) i n the ti me i nterval (), f11 is such that

the f-unctional

is stationary, i e., a local extremum or a t t saddl e point.' ' (W e define t t ary' ' more precisely i n Secti on 2 2.) ln the l ingo of mechanics J is called the

station-action integral or sim ply the station-action

Problems in mechanics often invol ve several particles (or spatial coördi natesl;

m oreover, Cartesian coördinates are not always the best choice V ariationalprinciples are thus usually given in term s of generalized coördinates.The letter q has been universally adopted to denote generalized positioncoördinates The configuration of a system at tim e t is thus denoted by

q(f) = (t ?1(f), , ?, z(f)), where the qk are position vari ables 1f, for

exam-ple, the system consists of three free particles in 1:.3 then n, = 9

T he kinetic energy T of a system is given by a quadratic form in the

generali zed velociti es t ik,

L(t, q, ù) - T(q, ù) - t-tf, q).

ln this fram ew ork H am ilton's Principle takes the follow ing form

Theorem 1.3.1 (Ham iltonhs Principle) Tlte mt a/ït ?zz o f a sps/cm o f cles q(f) /' rt ? m a g/wcz z initial conji guration q(fo) to a given ji nal conjignration q(f1) in t/ Jc tim, e ïz z/crrt zl (f() , f11 is suclt tl tat t/ Jc Junctional

parti-is stationary

(x(r ), y(r))

T he dynam ics of a system of particles is thus com pletely contained in thesingle scalar function L W e can derive the fam iliar equations of m otion from

Hamilton's Pri nci pl e (cf Secti on 3.2) The reader mi ght rightf - ul ly question

whether the m otion predicted by Ham ilton's Principle depends on the choice

of coördinates The variational approach w ould surely be of lim ited value were

it sensitive to the observer's choice of coördinates W e show in Section 2.5 that

H am ilton's Principle produces equations that are necessarily invariant w ithrespect to coördinate choices

Exam ple 1.3.1: Sim ple Pendulum

Consider a sim ple pendulum of m ass ?rz and length f in the plane Let

(z(f), y(t)) denote the posi tion of the mass at time f Since z2 + y2 = :2

we need in fact only one position variable Rather than use z or y it is natural

to use polar coördinates and characterize the position of the m ass at tim e t

by the angl e 4(t) between the vertical and the string to which the mass i s attached (fi gure 1.5) Now, the kinetic energy is

and H am ilton's Principle im plies that the m otion from a given initial angle

4(t( j ) to a fl xed angl e 4(f1) is such that the f - uncti onal

J(/)

-is stationary

1.3 Hamilton's Principle

Exam ple 1.3.2 : K epler problem

The Kepler problem m odels planetary m otion lt is one of the m ost heavilystudied problem s in classical m echanics Keeping with our no frills approach,

we consider the sim plest problem of a single planet orbiting around the sun,and ignore the rest of the solar system A ssum ing the sun is fixed at the origin,the kinetic energy of the planet is

l

wt : ) + 2 ) = j z , / ) + , ) J2( j j ? T=è

where r and 0 denote polar coördinates and ?rz is the m ass of the planet

W e can deduce the potential energy f-unction U' from the gravitational law ofattraction

Gnzi v / = -

z , r

where / is the force (acting i n the radial direction), M is the mass of the sun,

and G is the universal gravitation constant G iven that

:U /

= -

,ôr

C?vza vf U(r) = -

H am ilton's Principle im plies that the m otion of the planet from an initial

observation (r(fo), p(f( ) )) to a f inal observation (r(f1), p(f1)) i s such that

is stationary

T he reader m ay be wondering about the fate of the constant of integration

in the last exam ple Any potential energy of the form G'm,M r + const w ill

produce the requi site force / ln the pendulum problem we tacitl y assumed

that the potential energy was proportional to the height of the m ass above the

m inim um possible height ln fact, for the purposes of describing the dynam ics

it does not matter; i e., U(f, q) and U(f, q) + cl produce the same results for

any constant c1 W e are optim izing J and the addition of a constant in the

Lagrangian simply alters the f - unctional J(q) to /(q) = J(q) + const lf one

functional is stationary at q the other m ust also be stationary at q

ln the lore of classical m echanics there is another variational principlethat is som etim es called the ttprinciple of Least Action'' or ttklaupertuis'

Principle,' w hich predates H am ilton's Principle This principle is som etim esconf-used w ith H am ilton's and the situation is not m itigated by the fact that

H am ilton's Principle is som etim es called the Principle of Least Action S M pertuis' Principle concerns system s that are conservative ln a conservativesystem w e have that the total energy of the system at any tim e t along thepath of m otion is constant ln other words, L + U' = k, where k is a con-stant For this special case L = 2T - k, and H am ilton's Principle leads to

au-M aupertuis' Principle that the f-unctional

tlA-tql - T(q, il) dt

te

is stationary along a path of m otion Hence, M aupertuis' Principle is a specialcase of H am ilton's Principle M ost books on classical m echanics discuss these

princi ples (along with others) Lanczos ( 481 gi ves a particularly complete and

readable account that, in addition to m echanics, deals with the history and

philosophy of these principles The eminent scientist E Mach ( 511 also writes

at length about the history, significance, and philosophy underlying theseprinciples His perspective and sym pathies are som ew hat different from those

A frica w ith Pygm alion in pursuit U pon landing in North A frica, legend has itthat she struck a deal with a local chief to procure as m uch land as an oxhidecould contain She then selected an ox and cut its hide into very narrow strips,

which she joined together to form a thread of oxhi de m ore than two and a half

m iles long Dido then used the oxhide thread and the North African sea coast

to define the perim eter of her property lt is not clear what the im m ediatereaction of the chief was to this particular interpretation of the deal, but it is

8 The translators of Landau and Li lhitz ( 491, p 131, go so far as to draf t a tabl e

to elucidate the different usages

9 M ach is not so generous with M aupertuis.ln connexion with M aupertuis' ciple he writes ddlt appears that M aupertuis reached this obscure expression by

Prin-an unclear mingling of his ideas of vis wïwtz and the principle of virtual velocities''

(p 365) ln defense of Mach, we must note that Maupertuis suf fered no lack of

critics even in his own day Voltaire wrote the satire Histoive d,u docteuv 412a/:/a cf

d, u ' r zt z / de Saint Mal o about M aupertuis The si tuation at Frederi ck the Great' s court regarding Maupertui s, Köni g, and Voltaire i s the stuf f of soap operas (see Pars ( 591 p 634).

1.4 Som e Variational Problem s from G eom etry 15

clear that D ido sought to enclose the m axim um area within her ox and thesea The city of Carthage was then built within the perim eter defined by thethread and the sea coast D ido called the place Byrsa m eaning hide of bull.10

The problem that Dido faced on the shores of North Africa (aside from fami ly dif llculties) was to determine the optimal path along whi ch to place

the oxhide thread so as to provide Byrsa with the m axim um am ount of land

D ido did not have the luxury of waiting som e 2500 years for the calculus ofvariations to develop and thus settled for an ttintuitive solution.'

Dido' s problem entai led determining the curve y of fl xed length (the thread) such that the area encl osed by y and a gi ven curve o' (the North African shorel ine) is maximum Although this is perhaps the original version

of D ido's problem , the term has been used to cover the m ore basic problem :

am ong all closed curves in the plane of perim eter L determ ine the curve thatencloses the m axim um area The problem did not escape the attention of an-

cient m athem aticians, and as early as perhaps 200 B.C the m athem aticianZenodorusll is credited with a proof that the solution is a circle. Unfortu-

nately, there were some technical l oopholes i n Zenodorus' proof (he compared the area of a circle with that of polygons having the same peri meter) The

first com plete proof of this result was given som e 2000 years later by K arl

W eierstraB in his Berlin lectures

Prior to Wei erstraB, Stei ner (ca 1841) proved that z l / there exi sts a t t

fig-ure'' y w hose area is never less than that of any other ttfigure'' of the sam eperim eter, then y is a circle Not content w ith one proof, Steiner gave fiveproofs of this result The proofs are based on sim ple geom etric considerations

(no calculus of variations) The operati ve word in the statement of his result,

how ever, is ttif.'' Steiner's contem porary, D irichlet, pointed out that his proofs

do not actually establish the existence of such a figure W eierstraB and his follow ers resolved these subtle aspects of the problem A lively account of Dido's

-problem and the fi rst of Stei ner's proofs can be found in Körner ( 451

Som e sim ple geom etrical argum ents can be used to show that if y is a

simple closed curve solution to Dido' s probl em then y is convex (cf Körner,

op cï /.) This means that a chord joi ning any two points on y lies within y

10 The reader will find various bits and pieces of Dido's history scattered in Latinworks by authors such as Justin and Virgil One account of the hide story comesfrom the Aeneid Bk 1 vs 367 The story gets even better once Aeneas arrives onthe scene Finally good ideas never die lt is said that the Anglo-saxon chieftains

Hengi st and Horsa (ca 449 A D.) acqui red thei r l and by ci rcl ing it with oxhide stri ps ( 371 Beware of real estate transacti ons that i nvolve an ox.

the author of the proof that appears in the comm entary of Theon to Ptolemy's

Al magest Zenodorus quotes Archi medes (who di ed in 2 12 B C ) and i s quoted

by Pappus (ca 340 A D ) Asi de f rom these rough dates we do not know exactl y

when Zenodorus lived At any rate the solution was of little com fort to Dido's

heirs as the Romans obliterated Carthage Byrsa in the third Punic war just after

200 B.C and sowed salt on the scorched ground so that nothing would grow

and the area enclosed by y The convexity of y is then used to show that

D ido's problem can be distilled down to the problem of finding a f-unction

!/ : ( al o, zll - -+ R such that

21

yitt vl - t vtzl dz

20

is maxi mum subject to the constraint that the arclength of the curve y' j

described by y is L 2 lf we assum e that y is at least piecewise differentiablethen this am ounts to the condition

r, z 1

= 1 + y ?2 d z

2 zoThe problem w ith this form ulation is that we do not know the lim its of theintegral The geom etrical character of the problem indicates that we do not

need to know 1 30th zt l and zl (we could always normalize the constructi on so that zt l = 0 < z1), but we do need to know zl - zo This problem is ef fecti vely

the opposite of the problem we had w ith the first form ulation of the catenary.Since we know arclength, a natural form ulation to use w ould be one in term s

of arclength

Suppose that y' j- i s described parametrical ly by (z(s), ! /(s)), s CE ( 0, L 21

where s is arclength Suppose further that z and y are at least piecew isedifferentiable G reen's theorem in the plane can then be used to show thatthe area of the set enclosed by y'j and the z-axis is

z-/21

.4(t v) - j' yls) 1 - y'2(s) ds,

0

where we have used the relation z?2(s) + y? 2 (s) = 1 The basic Dido probl em

is thus to determine a posi tive function y : ( 0, L 21 + R such that - 4 i s

m axim um

1.4.2 G eodesics

Let E be a surface, and let yo, pl be tw o distinct points on E The geodesic

problem concerns f inding the curvets) on E wi th endpoi nts yb, pl for whi ch

the arclength is m inim um A curve having this property is called a geodesic

The theory of geodesics is one of the most developed subjects i n dif ferenti al

geom etry The general theory is com plicated analytically by the situation thatsim ple, com m on surfaces such as the sphere require m ore than one vectorfunction to describe them com pletely ln the language of geom etry, the sphere

is a m anifold that requires at least two charts W e have encountered and stepped the analogous problem for curves, and we do so here in the interest ofsim plicity W e focus on the local problem and refer the reader to any general

side-text on dif ferenti al geometry such as Stoker ( 661 or W ill more ( 751 for a more

precise and in-depth treatm ent of geodesics.lz

Suppose that E is described by the position vector function r : tz + 1:.3where o' is a nonem pty connected open subset of R2,and for (u, z?) CE c,

rtz', z, ) - (z(z ', z, ), ylu, z, ztz', z, ))

W e assum e that r is a s11100th f-unction on (z;

functi ons of (u, z? ), and that

t ?r t ' ?r I

A

-I / 0 ,

ç)u t'?r

so that r is a one-to-one m apping of o' onto E lf y is a curve on E , thenthere is a curve ym in o' that m aps to y under r A ny curve on E m ay thus

be regarded as a curve in o' Suppose that the points yb and pl correspond

to rt l = rtztt ), zo) and rl = rtztl, z? 1), respecti vely Any curve y from rt l to rl maps to a curve y m from t' ? to, z? ) to tul, z ?1).

For the geodesic problem we restrict our attention to s11100th sim ple curves

(no self-intersections) on E f rom rt l to r1 Let F denote the set of all such

curves Thus, if y CE F , then there exists a param etrization of y of the form

f = El zt/ 2 + 2F' t /r/ + (7r/ 2

is positive definite

T he arclength of y is given by

utfol - z' o, rtfol - ' t ?o utfzl - z'z, rtfzl - ' t ?z.

Exam ple 1.4.1: G eodesics on a Sphere

Let E be an octant of the unit sphere The surface E can be described

para-m etrically by

= I (cos t z cos z? , cos z t sin z?, - sin zt ) 1 2

The arclength integral is thus

A feature of the basic geodesic problem described above is that it doesnot involve the f-unction r directly The arclength of a curve depends only onthe three scalar f-unctions E ,F , and G G eodesics are part of the intrinsicgeom etry of the surface, i.e., the geom etry defined by the m etric tensor The

m etric tensor does not define a surface uniquely even m odulo translations and

rotations There are any num ber of distinct surfaces in 1:.3 that have the sam e

m etric tensor For exam ple, a plane, a cone, and a cylinder all have the sam e

m etric tensor lf a cylinder is ttunrolled'' and ttflattened'' to form a portion ofthe plane, then a geodesic on the cylinder w ould becom e a geodesic on theplane

O ne direction for a generalization of the above problem is to focus on the

space a f ; 12 and dehne the components of the metric tensor For notational

sim plicity, let zt = ztl,z? = zt2

, and u = @, z? ) W e can choose scal ar functions

gjk : t z + R j, k = 1, 2 and define the arclength element ds by

(/.$:2 =

.g11(( /tt1)2 + gyzt yttlt yuz + w yt yuzgztl + ggg Ld t g2)2

= g kduiidukwhere the last expression uses the Einstein sum m ation convention: sum m ation

of repeated indices when one is a superscript and the other is a subscript O fcourse we m ust place som e restrictions on the gjk in order to ensure that ourarclength elem ent is positive and that the length of a curve does not depend

on the choice of coördinates u W e can take care of these concerns by requiringthat the gjk produce a quadratic form that is positive definite and that thegjk form a second order covariant tensor To m im ic the earlier case we also

im pose the sym m etry condition

so that

(/.52 =

.g11(( /tt1)2 + z qyz( /ttlt y t tz + gggLdu2)2 (J.J6)

ln term s of the form er notation, E = .t.711, F = .ty12 = t721, and G = gnLt Forthis case, the positive definite requirem ent am ounts to the condition

2 > ()

t.711 t 722 - t?12

with .ty11 > 0 The condition that the gjk form a second-order covariant tensor

means that under a s1 11 00th coördi nate transformation from u = (ul, zt2) to

fl = (fl 1, i )2), the components pklu) transform to ji s? ztl 4l according to the

relation

çluii t'?'tzk

.

l il m = gjk gt?fl ogjvn .

The set o' equipped with such a tensor can be view ed as defining a geom etrical

object i n itsel f ( as the surface E was) lt i s a special case of what is called a

R iem annian m anifold Let ' tt denote this geom etrical object A curve y m in

o' generates a curve y in Azt , and the arclength is given by

tlZ(' 7) = qj i ktt i ltt k' t/ f

tewhere (zt1(f), zt2(f)), t (E g ft f1j is a parametrizati on of q The condition that the gjk form a second-order covariant tensor ensures that L(y) is i nvariant

with respect to changes in the curvilinear coördinates u used to represent

'' tt Note also that L(y) is invariant with respect to orientation-preserving

aram etrizations of ym

T he advantage of the above abstraction is that it can be readily m odified to

accommodate hi gher dimensions Suppose that o' f ;l R? z and u = (ul, u? zl.

W e can defi ne an zz-di mensi onal (Riemannian) manifold Azt by introducing a

m etric tensor w ith com ponents gjk such that:

1 the quadratic form gjkduiiduk is positive definite;

2 gjk = gkj fOr j, k = 1, 2, , z z;

3 under any s1 11 00th transformation u = uttll the gjk transform to ql v

according to the relation

çlut t'?ztk9lm = 9jk jôh dûsz

A curve y on Azt is generated by a curve J f in o' (; R' ' Suppose that u(f) = (zt1(f), , ? t? z(f)), t ( E g fl ), f1j is a parametrization of J f The arclength of y i s

then defined as

tlZ(' 7) = qj i ktt i ltt k' dt.

te

A generalizati on of the geodesic problem i s thus to find the curvet s) y m in o' with specified endpoints ut l = u(fo), ul = u(f1) such that L(y) is a minimum.

G eodesics are of interest not only in differential geom etry, but also in

m athemati cal physics and other subjects lt turns out that many probl ems

can be interpreted as geodesic problem s on a suitably defined m anifold.l3 Inthis regard, the geodesic problem is even m ore im portant because it provides

a unifjring fram ework for m any problem s

1.4.3 M inim al Surfaces

W e have already encountered a special m inim al surface problem in our cussion of the catenary The rotational sym m etry of the problem reduced theproblem to that of finding a f-unction y of a single variable z, the graph ofwhich generates the surface of revolution having m inim al surface area Locally,any surface can be represented in ttgraphical'' form ,

dis-rlz, y) = (z, y, zlz, y)),

where r is the position function in R 3.U nless som e sym m etry condition is

im posed, a surface param etrization requires two independent variables Thusthe problem of finding a surface w ith m inim al surface area involves tw o inde-pendent variables in contrast to the problem s discussed earlier

13 ln the theory of relativity,where differential geometry is widely used, the condi

-tion that the m etric tensor be positive definite is relaxed to positive semidefinite

G iven a sim ple closed space curve y, the basic m inim al surface problementails finding, am ong all s11100th sim ply connected surfaces with y as a bound-ary, the surface having m inim al surface area Suppose that the curve y can be

represented parametrically by (z(f), y(t), z(f)) for t CE )( ), f11 and for simpli

c-ity suppose that the projection of y on the zp-pl ane is al so a simple closed

curve; i e , the curve J f described by (z(f), y(t)) for t CE )() , f11 i s a simple closed

curve in the zp-plane Let J2 denote the region in the zp-plane enclosed by Jf.Suppose further that w e restrict the class of surfaces under consideration to

those that can be represented in the form (1 17), where z i s a sm00th f - unction for (z, y) CE Q The dif ferenti al area el ement is gi ven by

m inim um There is a substantial body of inform ation about m inim al surfaces

The reader can fi nd an overvi ew of the subject i n Osserman ( 581

1.5 O ptim al H arvest Strategy

O ur final exam ple in this chapter concerns a problem in econom ics dealingwith finding a harvest strategy that m axim izes profit Here, we follow the

example gi ven by W an (711 p 6 and use a fi shery to ill ustrate the model Let ylt) denote the total tonnage of fi sh at time t in a region J2 of the

ocean, and let yc denote the carrying capacity of the region J2 for the fish.The grow th of the fish population w ithout any harvesting is typically m odelled

by a first-order differential equation

lf y i s sm al l compared to yc, then / i s often approxi m ated by a linear f - unction

in y; i e., /(f, y) = ky + g(t), where k is a constant More complicated model s are avai lable for a wi der range of ylt) such as l ogistic growth

f y - k yl t (1 - Vt f 1

The ordi nary dif ferential equati on (1 18) i s accompanied by an i ni tial condi

-tion

3/(0) = : %

that reflects the initial fish population

Suppose now that the fish are harvested at a rate ' t t;(f) Equati on (1.18)

for the population grow th can then be m odified to the relation

y'lt) - /(f, y) - zc(f) (1 20)

Given the f - unction /, the problem i s to determine the f - uncti on w so that the

profit in a given tim e interval T is m axim um

lt is reasonable to expect that the cost of harvesting the fish depends on

the season, the fish population, and the harvest rate Let c(f, y, t; ) denote

the cost to harvest a unit of fish biom ass Suppose that the fish com m ands aprice p per unit fish biom ass and that the price is perhaps season dependent,but not dependent on the volum e of fish on the m arket The profit gained by

harvesting the fi sh in a small ti me i ncrement is (p(f) - c(f, y, ' t t; ))' t t;(f) dt Given

a fixed period T w ith which to plan the strategy, the total profit is thus

T

Ply, z&) - (r(f) - ctf, y, zc))zc(f) dt.

0

The problem is to identifjr the f-unction w so that P is m axim um

T he above problem is an exam ple of a constrained variational problem The

functi onal P is optimized subject to the constraint defi ned by the dif ferenti al

equation (1.20) (a nonholonomi c constraint) and i nitial condition (1.19) W e

can convert the problem into an unconstrained one by sim ply elim inating

w from the integrand defining P usi ng equation (1 20) The problem then

becom es the determ ination of a function y that m axim izes the total profit.This approach is not necessarily desirable because we want to keep track of't;, the only physical quantity we can regulate

A feature of this problem that distinguishes it from earlier problem s is theabsence of a boundary condition for the fish population at tim e T A lthough

we are given the initial fish population, it is not necessarily desirable to specify

the final fi sh population after ti me T As W an poi nts out, the conditi on y(T) =

0, for exam ple, is not always the best strategy: ttgreen issues'' aside, it m ay costfar m ore to harvest the last few fish than they are worth This sim ple m odelthus provides an exam ple of a variational problem with only one endpointfixed in contrast to the catenary and brachystochrone

ln passing we note that econom ic m odels such as this one are generallyfram ed in term s of ttpresent value.' A pound sterling invested earns interest,and this should be incorporated into the overall profit lf the interest is com-pounded continuously at a rate r, then a pound invested yields crt poundsafter tim e f Another way of looking at this is to view a pound of incom e attim e t as worth c-rt pounds now C onsiderations of this sort lead to profitfunctiona.ls of the form

T he F irst V ariation

ln this chapter we develop a necessary condition for a f-unction to yield anextrem um for a f-unctional The centrepiece of the chapter is a second-orderdifferential equation, the Euler-lsagrange equation, w hich plays a rôle analo-gous to the gradient of a f-unction W e first m otivate the analysis by review ingnecessary conditions for functions to have local extrem a The Euler-lsagrangeequations are derived in Section 2.2 and som e special cases where the differ-ential equation can be sim plified are discussed in Section 2.3 T he rem ainingthree sections are devoted to m ore qualitative topics concerning degeneratecases, invariance, and existence of solutions W e postpone a discussion of suf-ficient conditions until Chapter 10

2.1 T he F inite-llim ensional C ase

The theory underlying the necessary conditions for extrem a in the calculus

of variations is m otivated by that for f-unctions of n, independent variables.Problem s in the calculus of variations are inherently infinite-dim ensional Thecharacter of the analytical tools needed to solve infinite-dim ensional problem sdiffers from that required for finite-dim ensional problem s, but m any of theunderlying ideas have tractable analogues in finite dim ensions ln this section

we review a necessary condition for a function of n, independent variables tohave a local extrem um

2.1.1 Functions of O ne V ariable

Let / be a real-valued functi on defi ned on the interval f f ; R The f - unction / : f + R is said to have a local m axim um at z CE f i f there exists a number

t î > 0 such that for any k C E (z - 6, z + 6) ( : z f , Jlk) < /(z) The f - unction

/ : f + R is said to have a local m inim um at z CE f if -/ has a local

m axim um at z A f-unction m ay have several local extrem a in a given interval

lt m ay be that a function attains a m axim um or m inim um value for the

entire interval The f - unction / : f + R has a global m axim um on f at

z CE f if Jlk) < /(z) for all k CE f The function / is said to have a global

m inim um on f at z CE f i f -/ has a global maxi m um at z Note that if f has boundary points then / may have a global maximum on the boundary lf / is di fferentiable on f then the presence of local maxima or mi nima on f i s

characterized by the first derivative

Theorem 2.1.1 Let / be a real -valned J' unction dt jferentiable on tlte

interval I f/ / It as a local cz/rcm' t zm at z CE f tl ten /? (z) = 0.

P roof: The proof of this result is essentially the sam e for a local m axim um

or m inim um Suppose that z is a local m axim um Then there is a num ber

t î > 0 such that for any k CE (z - 6, z + 6) ( : z f the i nequality /(z) k Jlk) i s

satisfi ed Now the deri vati ve of / at z is gi ven by

/'(z) - l l ina (/(:) - /(z)) (: - z).->zThe numerator of this l imi t is never posi tive since /(z) i s a maxi mum, but

the denom inator is positive w hen k > z and negative w hen k < z Since the

functi on / is dif ferentiable at z the right- and l eft-sided li m its exist and are

equal The only way this can be true is if /?(z) = 0 E

lt is illum inating to exam ine the situation for s11100th f-unctions W e usethe generic term ttsm ooth'' to indicate that the f-unction has as m any continu-ous derivatives as are necessary to perform whatever operations are required

Suppose that / is sm00th i n the interval (z - 6, z + 6), where t î > 0 Let

k - z = 6p Taylor's theorem i ndi cates that, for t î suf l lci ently sm al l, / can be

sign for all p B ut it is clear that p can be positive or negative and hence

p/?(z) can be positive or negati ve We must therefore have that /?(z) = 0 lf /?(z) = 0, then the above expansion indicates that the sign of the dif ference i s that of the quadratic term, i e the si gn of /?? z) lf thi s derivati ve i s negati ve then /(z) is a local maximum; if it is positi ve then /(z) is a local minimum.

lt may be that /? ?(z) = 0 ln thi s case the sign of the difference depends on

the cubic term , which contains a factor p3.Like the linear term , however, thisfactor can be either positive or negative depending on the choice of p Thus, if

/?? ?(z) # 0, /(z) cannot be a local extremum We can conti nue in this manner

as l ong as / has the requisi te deri vati ves i n (z - 6, z + 6).

For a differentiable function it is easy to see graphically why the condition

/?(z) = 0 is necessary for a l ocal extremum The Taylor expansion for a

s11100th f-unction indicates that at any point z at w hich the first derivative

vani shes an 0/) change in the independent variable produces an 0(62) change

in the f-unction value as tî + 0 For this reason points such as z are called

stationary points The f - unctions /z z(z) = z' ', where n, CE N, z CE R provide

sim ple paradigm s for the various possibilities

Example 2.1.1: Let /(z) = 3z2 - z3 The function / i s sm00th for z CE

R and therefore if any local extrem a exist they must satisfjr the equation6z- 3z2 = 0.This equation is satisfied if z = 0 or z = 2 The second derivative

is 6 - 6z, so that // /(0) = 6 and consequently /(0) i s a local mini mum On the other hand, // /(2) = -6 and thus /(2) i s a local maximum.

Exam ple 2.1.2 :

,2 sin2 (1 z), if z , # 0 /(

z - t ( i z = (

This f - unction is dif ferentiable for all z CE R Now //(0) = 0, and thus z = 0

is a stationary point but the deri vative is not continuous there and so ///(0)

does not exist W e can deduce that / has a local mini mum at z = 0 because

/(z) k 0 for a11 z CE R.

Example 2.1 3: Let /(z) = I zI This function is dif ferenti abl e for all z CE

R - (0/ The deri vative is given by //(z) = -1 for z < 0, and //(z) = 1 for

z > 0 Thus / cannot have a l ocal extremum in R (0/ Nonetheless i t is clear that /(0) = 0 is a l ocal (and global ) minimum for / i n R.

Example 2.1 4: Let /(z) = c2 This f - uncti on is sm00th for all z CE R and

its deri vati ve never vanishes; consequently, / does not have any local extrema.

T he relationship between local and global extrem a is lim ited C ertainly if

/ has a global extremum at some interior point z of an interval then /(z)

is also a local extremum 1f, in addi tion, / is dif ferentiabl e in f , then it must

also satisfjr the condition /?(z) = 0 But it may be (as often is the case) that

a global extrem um is attained at one of the boundary points of f , in w hich

case even i f / is di fferentiable nothing regarding the value of the deri vati ve

can be asserted