The Calculus Of Variations

The (alculus ofVariations

Bruce van Brunt

U niversitext EditorialBoard

lNo4hAmerical: S.Axler F.W .Gehring K.A.Ribet

SPrin:er Ncw Fork Berlin H eidelberg Hong Kong London M ilan Paris Tokyo

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B ruce van B runt

T he C alculus of V a riatio ns Wi th 24 Fi gures

.T#4* ' SPringer

Bruce van Brunt lnstitute ofFundamentalSciences Palmerston Nort . h Campus Private Bag 11222 M assey University Palmerston Nort . h 5301 New Zealand b. vanbrunt@m assey. ac. nz

EditorialBoard

tA%rf/zAmerical: S.Axler M athem aticsDepartg nent San Francisco State University San Francisco,CA 94132 USA axler@ sfsu. edu

F.W .Gehring M athem aticsDepartment EastHall University ofMichigan Ann Arbor,M 148109-1109 USA fgehring@m ath. lsa.urnich. edu

K. A .Ribet M athem aticsDepartg nent University ofCalifornia,Berkeley Berkeley,CA 94720-3840 USA rilxt@mattl. lxrkeley.edu

M athematicsSubjectClassification(2000)234Bxx,49-01,70Hxx Library ofCongress Cataloging-in-publication Data

van Brunt,B.(Bmce)

The calculus of variations/Bmce van Bm nt.

p.cm.- (Universitext) lncludesbibliographicalreferencesandindex.

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1.Calculus of variations. 1.Title.

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P reface

Thecalculusofvariationshasa long history ofinteraction with otherbranches ofm athem aticssuch asgeom etry and differentialequations,and with physics, particularly m echanics.M ore recently,the calculus ofvariations has found applicationsin otherfieldssuch aseconom icsand electricalengineering.M uch ofthe m athem atics underlying controltheory,for instance,can be regarded as partofthe calculusofvariations. T hisbook isan introduction to the calculus ofvariationsfor m athem aticiansand scientists.Thereader interested prim arily in m athem aticsw illfind results of interest in geom etry and differentialequations.1 have paused at tim esto develop the proofsofsom e ofthese results,and discussbriefly var-

ioustopicsnotnormally found in an introductory book on this subject such as theexistence and uniquenessofsolutions toboundary-value problem s,the inverse problem ,and M orse theory.1have m ade ttpassive use''offunctional

analysis(in particularnormed vectorspaces)to place certain resultsin context and reassure the m athem atician that a suitable fram ework is available fora m ore rigorousstudy.Forthereaderinterested m ainly in techniquesand applications ofthe calculus ofvariations,1 leavened the book w ith num erous exam ples m ostly from physics.ln addition,topics such as Ham ilton's Principle,eigenvalue approxim ations,conservation law s,and nonholonom ic constraintsin m echanicsarediscussed.M oreim portantly,the book isw ritten on two levels.The technicaldetails for m any of the results can be skipped on the initialreading.The student can thuslearn the m ain results in each chapterand return asneeded to the proofsfor a deeperunderstanding.Sev-

eralkey resultsin thissubject have tractable analoguesin finite-dimensional optim ization.W here possible,the theory is m otivated by firstreviewing the theory forfinite-dim ensionalproblem s. T he book can be used for a one-sem estercourse,a shortercourse,or independent study.The finalchapter on the second variation has been w ritten with these options in m ind,so that the student can proceed directly from Chapter 3 to this topic.Throughout the book,asterisks have been used to flag m aterialthatisnotcentralto a first course.

T he target audience forthisbook isadvanced undergraduate beginning graduate studentsin m athem atics,physics,or engineering.The studentisassum ed to have som e fam iliarity with linear ordinary differentialequations, m ultivariable calculus,and elem entary realanalysis.Som e ofthe m ore theoreticalm aterialfrom these topics that is used throughout the book such as the im plicitf-unction theorem and Picard's theorem fordifferentialequations hasbeen collected in A ppendix A forthe convenienceofthe reader. Like m any textbooksin m athem atics,thisbook can trace itsoriginsback to a set oflecture notes.The transform ation from lecture notesto textbook, how ever,isnontrivial,and one isfaced with m yriad choicesthat,in part,re-

flectone'sown interestsand experiencesteaching thesubject.W hile writing thisbook 1keptin m ind three quotes spanning a few generations ofm athem aticians.The first isfrom the introduction to a volum e of Spivak's m ulti-

volumetreatiseon differentialgeometry (641: 1feelsom ewhatlikeam an w ho hastriedto cleansetheA ugean stables with a Johnnp lklop. ltistem pting,when w riting a textbook,to give som e m odicum ofcom plete-

ness.W hen faced with the enorm ity ofliterature on this subject,however, the task proves daunting,and it soon becomes clear that there isjust too m uch m aterialfor a single volum e.ln the end,1 could not face picking up the Johnny-lklop,and m y solution to this dilem m a was to be savage w ith m y choice oftopics.K eeping in m ind that the goal is to produce a book thatshould serve asa textfora one-sem esterintroductory course,therewere m any painfulom issions.Firstly,1have tried to steer a reasonably consistent path by keeping the focus on the sim plest type problem s that illustrate a particular aspect ofthe theory.Secondly,1have opted in m ostcases forthe ttno frills''version of results if the ttfullfeature''version w ould take us too farafield,or require a substantially m ore sophisticated m athem aticalbackground.Topics such as piecew ise s1100th extrem als,fields ofextrem als,and num ericalm ethods arguably belong in any introductory account.N onetheless,1have om itted these topics in favourofothertopics,such asa solution m ethod for the H am ilton-lacobi equation and N oether's theorem ,that are accessible to the generalm 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'sbook on

eigenf -unctionexpansions(701: 1 believe in the future oftm athem atics for physicists',but it seem s

desirablethata writeron thissubjectshould understand 130th physics asw ellasm athem atics. The w ords ofTitchm arsh rem ind m e that,although 1 am a m athem atician interested in the applications ofm athem atics,1 am not a physicist,and it is bestto leave detailed accounts ofphysicalm odelsin the handsofexperts. Thisisnotto say thatthem aterialpresented hereliesin som evacuum ofpure

m athem atics,w herewe m erely acknowledgethatthem aterialhasfound som e applications.lndeed,thebook iswritten with adefiniteslanttowards ttapplied m athem atics,''butit focuseson no particularfield ofapplied m athem aticsin any depth.O ften it is the application not the m athem atics that perplexes the student,and a study in depth ofany particularfield would requireeither the student to have the necessary prerequisitesorthe authorto develop the

subject.Theformercaserestrictsthe potentialaudience' ,thelattercaseshifts away from the m ain topic.ln any event,1have nottried to w rite a book on thecalculusofvariationswith a particularem phasison oneofitsm any fields of applications.There are m any splendid books that m erge the calculus of variationsw ith particularapplicationssuch asclassicalm echanics orcontrol

theory.Such textscan be read with profitin conjunction with thisbook. T hethird quotecom esfrom G .H .H ardy,whom adethefollow ingcom m ent

aboutA.R.Forsyth's656-pagetreatise(271on thecalculusofvariations:1 ln this enorm ousvolum e,the authornever succeedsin proving that theshortestdistance betw een two pointsisa straightline. H ardy did not m ince wordsw hen it cam e to m athem atics.The prospective author of any text on the calculus of variations should bear in m ind that, although thereare m any m athem aticalavenuesto explore and endlessm inutia?to discuss,certain basic questions thatcan be answ ered by the calculus of variations in an elem entary text should be answered.There are certain problem ssuch asgeodesicsin the plane and the catenary thatcan be solved within ourself-im posed regim e ofelem entary theory.1do nothesitate to use these sim ple problem sas exam ples.Atthesam e tim e,1also hope to givethe

readera glimpse ofthe power and elegance ofa subjectthathasfascinated m athem aticiansforcenturies. 1wish to acknow lege the help ofm y form erstudents,whose inputshaped the finalform ofthisbook.1w ish also to thank Fiona D aviesfor helping m e with thefigures.Finally,1would liketoacknowledgethehelp ofm y colleagues at the lnstituteofFundam entalSciences,M assey U niversity. T he earlier draftsofm any chapterswerew ritten w hiletravelling on variousm ountaineering expeditionsthroughouttheSouth lsland ofN ew Zealand. Thehospitality ofClive M arsh and H eatherNorth isgratef-ully acknowledged along w ith that ofA ndy Backhouse and Zoe Hart.1should also like to acknow ledge the New Zealand A lpine C lub,in whosehuts 1w rote m any early

(and later)draftsduring periodsofbad weather.ln particular,1would like

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

homeconducivetowriting (and climbing). Fox G lacier,N ew Zealand February 2003

Bruce van Brunt

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

Him,''given to theBvitisltSocietyJt pr f/zcHistovy 6, / Matl tematics 19 December 1990.

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C ontents

1

lntroduction ............................................... 1 1.1 lntroduction ............................................ 1 1.2 The Catenary and BrachystochroneProblem s ............... 3 1.2.1 The Catenary ..................................... 3 1.2.2 Brachystochrones.................................. 7 Ham ilton's Principle ..................................... 10 Som e VariationalProblem sfrom G eom etry ................. 14 1.4.1 D ido'sProblem ................................... 14 1.4.2 G eodesics ........................................ 16 1.4.3 M inim alSurfaces.................................. 20 Optim alH arvestStrategy ................................ 21

2

T he First Variation ........................................ 23 2.1 The Finite-D im ensionalCase ............................. 23 2.1.1 FunctionsofO ne Variable .......................... 23 2.1.2 FunctionsofSeveralVariables....................... 26 The Euler-lsagrangeEquation ............................. 28 Som e SpecialCases...................................... 36 2.3.1 Case 1:No Explicity D ependence ................... 36 2.3.2 Case 11:N o Explicit z Dependence .................. 38 2.4 A DegenerateC ase ...................................... 42 2.5 lnvariance oftheEuler-lsagrange Equation .................. 44 2.6 ExistenceofSolutionsto theBoundarp value Problem * ...... 49

3

Som 3.1 3.2 3.3 3.4

e G eneralizations ...................................... 55 FunctionalsContaining H igher-orderD erivatives ............ 55 SeveralD ependentVariables .............................. 60 Two lndependentVariables* .............................. 65 The lnverse Problem *.................................... 70

X11

lsoperim etric P roblem s .................................... 73 4.1 The Finite-D im ensionalCase and Lagrange M ultipliers....... 73 4.1.1 Single Constraint.................................. 73 4.1.2 M ultiple C onstraints............................... 77 4.1.3 A bnorm alProblem s ............................... 79 The lsoperim etric Problem ............................... 83 Som e G eneralizations on the lsoperim etric Problem .......... 94 4.3.1 Problem sContaining H igher-order Derivatives........ 95 4.3.2 M ultiple lsoperim etric Constraints................... 96 4.3.3 SeveralDependentVariables........................ 99 5

A pplications to Eigenvalue P roblem s* .....................103 5.1 The Sturm -lsiouville Problem .............................103 5.2 The FirstEigenvalue.....................................109 5.3 HigherEigenvalues ......................................115

6

H olonom ic and N onholonom ic C onstraints .................119 6.1 Holonom icC onstraints ...................................119 6.2 Nonholonom icConstraints................................125 6.3 Nonholonom icConstraintsin M echanics*...................131 P roblem s w ith V ariable Endpoints .........................135 7.1 NaturalB oundary Conditions.............................135 7.2 The G eneralC 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-lacobiEquation ............................175 8.4.1 The GeneralProblem ..............................175 8.4.2 Conservative System s ..............................181 Separation ofVariables...................................184 8.5.1 The M ethod ofA dditive Separation..................185 8.5.2 ConditionsforSeparable Solutions*..................190

9

N oetherhs T heorem ........................................201 9.1 Conservation Law s ......................................201 9.2 VariationalSym m etries ..................................202 9.3 Noether'sTheorem ......................................207 9.4 FindingVariationalSym m etries ...........................213

Contents

X111

10 T he Second Variation ......................................221 10.1 The Finite-D im ensionalCase .............................221 10.2 The Second Variation ....................................224 10.3 The Legendre Condition..................................227 10.4 The JacobiN ecessary Condition ...........................232 10.4.1 A Reformulation ofthe Second Variation .............232 10.4.2 The JacobiAccessory Equation .....................234 10.4.3 The JacobiNecessary Condition .....................237 10.5 A SufllcientCondition ...................................241

10.6 M ore on Conjugate Points................................244 10.6.1 Finding Conjugate Points ..........................245 10.6.2 A G eom etricallnterpretation .......................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 plicitFunction Theorem ...........................265 A .3 Theory ofOrdinary DifferentialEquations ..................268

B

Function Spaces ...........................................273 B .1 Norm ed Spaces..........................................273 B .2 Banach and HilbertSpaces ...............................278

R eferences .....................................................283 lndex ..........................................................287

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Introduction

1.1 lntroduction Thecalculusofvariationsisconcerned with finding extrem aand,inthissense, it can be considered a branch ofoptim ization.The problem sand techniques in this branch,however,differ m arkedly from those involving the extrem a of functions ofseveralvariables owing to the nature of the dom ain on the quantity to be optim ized.A functionalisa m apping from a setoffunctions to the realnum bers.T he calculus ofvariations dealsw ith finding extrem a forfunctiona. ls as opposed to f-unctions.The candidates in the com petition for an extremum are thus functions as opposed to vectors in R'',and this

givesthe subject a distinctcharacter.The f-unctionals are generally defined by definiteintegrals' ,thesetsoff-unctionsare often defined by boundary conditions and sm oothness requirem ents,which arise in the form ulation ofthe problem m odel. T he calculus ofvariations is nearly as old as the calculus,and the two

subjectsweredevelopedsomewhatinparallel.ln1927Forsyth (271noted that thesubjectttattractedaratherfickleattention atm oreorlessisolatedintervals in its grow th.'' ln the eighteenth century,the Bernoullibrothers,Newton,

Leibniz,Euler,Lagrange,and Legendre contributed to the subject,and their work wasextended significantly in thenextcenturyby Jacobiand W eierstraB.

Hilbert ( 381,in his renowned 1900 lectureto the lnternationalCongressof M athematicians,outlined23 (now famous)problemsformathematicians.His 231. d problem isentitled Fnrtlter developvtent o.f tlte mc//ztacls o. ftlte calcnln, s o.fvari ations.lmm ediately before describing the problem ,he rem arks: .. .1 s hould like to close with a general problem ,nam ely with the indication of a branch of m athem atics repeatedly m entioned in this lecture- w hich,in spite ofthe considerableadvancem entlately given it by W eierstraB,does not receive the generalappreciation w hich in m y opinion itisdue- lm ean thecalculusofvariations.

2

1 lntroduction

H ilbert's lecture perhaps struck a chord with m athem aticians.l In the early twentieth century Hilbert,Noether,Tonelli,Lebesgue,and Hadam ard am ong othersm adesignificantcontributionsto thefield.Although by Forsyth'stim e

the subjectm ay have tattracted rather fickleattention,''m any ofthose who did pay attention are num bered am ong the leading m athem aticians ofthe

lastthree centuries.The readerisdirected to Goldstine (361foran in-depth accountofthe history ofthe subjectup to the late nineteenth century. T he enduring interestin the calculusofvariationsisin partdue to itsap-

plications.Ofparticular note isthe relationship ofthe subjectwith classical m echanics,w here it crosses the boundary from being m erely a m athem aticaltoolto encom passing a generalphilosophy.Variationalprinciples abound in physicsand particularly in m echanics.T he application ofthese principles

usuallyentailsfindingfunctionsthatminimizedefiniteintegrals(e.g.,energy integrals) and hence the calculusofvariationscomes naturally to the fore. H am ilton'sPrinciplein classicalm echanicsisa prom inentexam ple.A n earlier exam ple is Ferm at'sPrinciple ofM inim um Tim e in geom etricaloptics.The developm ent of the calculus ofvariations in the eighteenth and nineteenth centuriesw asm otivated largely by problem sin m echanics.M osttextbookson

classicalmechanics (old and new)discussthe calculusofvariationsin some depth.Conversely,m any bookson the calculus ofvariationsdiscuss applicationsto classicalm echanics in detail.ln the introduction ofC arath4odory's

book (211hestates: 1have neverlostsightofthefactthatthecalculusofvariations,as it ispresented in Part11,should above allbe a servantofm echanics. C ertainly thereisan intim ate relationship between m echanicsand the calculus ofvariations,but this should not com pletely overshadow other fields wherethecalculusofvariationsalso hasapplications.Asidefrom applications in traditionalfieldsofcontinuum m echanicsand electrom agnetism ,thecalculusofvariations has found applications in econom ics,urban planning,and a hostofother ttnontraditionalfields.''lndeed,the theory ofoptim alcontrolis centred largely around the calculusofvariations. Finally it should be noted the calculus ofvariations does not exist in a m athem aticalvacuum orasaclosed chapterofclassicalanalysis.H istorically, this field has always intersected with geom etry and differential equations,

and continuesto do so.ln 1974,Stampacchia (171,writingon Hilbert's231.d problem ,sum m ed up the situation: O ne m ight infer that the interestin this branch ofAnalysis isweakening and that the Calculus ofVariations is a C hapter ofC lassical A nalysis.ln factthisinference would bequitew rong since new problem slike thosein controltheory areclosely related to theproblem sof 1Hisnineteenth and twentieth problem swere also devoted to the calculus ofvariations.

theC alculusofVariationswhile classicaltheories,like thatofboundary value problem sforpartialdifferentialequations,havebeen deeply affected by the developm ent ofthe C alculus ofVariations.M oreover, the naturaldevelopm ent ofthe Calculus ofVariations has produced new branches ofm athem aticsw hich have assum ed different aspects and appearquite differentfrom the CalculusofVariations. The field isfar from dead and itcontinuesto attract new researchers. ln the rem ainder ofthischapterwe discusssom e typicalproblem sin the

calculusofvariationsthatareeasy tomodel(although perhapsnotso easy to solve).Theseproblemsillustratetheabovecommentsand givethereader a taste ofthesubject.W e return to m ostoftheseexampleslaterin the book as the m athem aticsto solve them develops.

1.2 T he C atenary and B rachystochrone P roblem s 1.2.1 T he C atenary

Considera thin heavy uniform flexible cable suspended from the top oftwo

polesofheightyç jand yïspaced adistancedapart(figure1.1).Atthebaseof each polethecable isassum ed to be coiled.The cable follow sup the pole to thetop,runsthrough apulley,and then spansthedistance dto the nextpole. The problem isto determ inetheshapeofthe cablebetween the tw o poles. T hecablew illassum etheshapethatm akesthepotentialenergym inimum . The potentialenergy associated w ith the verticalparts ofthe cable willbe the sam e for any configuration of the cable and hence we m ay ignore this com ponent.lf?rz denotes the m ass per unit length ofthe cable and g the gravitationalconstant,thepotentialenergy ofthe cable between the polesis

1 lntroduction Z'

u,s(? ,)-jv,, z,: vtslds, wheresdenotesarclength,andyls)denotestheheightofthecableabovethe ground sunitsin length alongthecablefrom thetop ofthepoleat(zo,y(j). The numberL denotesthearclength ofthecable from (zo,y(tl to (z1,3/1). U nfortunately,we do notknow L in this form ulation.W e can,however,recastthe above expression for W ' s in term s ofCartesian coördinates since we do know the coördinatesofthe pole tops.The differentialarclength elem ent in Cartesian coördinates is given by ds = 1+ y?2,and this leads to the follow ing expression forW p ',

N otethatunlikeourfirstexpression forW ' s,theaboveoneinvolvesthe derivative ofy.W e have im plicitly assum ed here that the solution curve can be

representedbyaf-unction y:gzt ),z11--+R and thatthisfunctioniscontinuous and at least piecew ise differentiable.G iven the nature ofthe problem these Seem reasonable assum ptions. T he cablew illassum e the shapethatm inim izes W ' s.The constantfactor m,g in theexpression forW ' s can be ignored forthepurposesofoptim izing the potentialenergy.The essence ofthe problem is thusto determ ine a f-unction y such thatthequantity

is minimum .The modelrequires thatany candidate :' èfor an extremum satisfiestheboundary conditions

: ih(alo)= #0, 5(z1)= #1. ln addition,the candidates m ust also be continuous and at least piecew ise

differentiablein theintervalgzt),z11.

W efind the extrem a for J in Chapter2,wherew eshow thatthe shapeof thecablecan be described by a hyperboliccosine f -unction.Thecurveitselfis called a catenary.z T he sam efunctionalJ arisesin a problem in geom etry concerning a m inim alsurface ofrevolution,i.e.,a surfaceofrevolution having m inim alsurface area.Supposethatthez-axiscorrespondsto the axisofrotation.Any surface

ofrevolution can be generated by a curvein the zp-plane (figure 1.2).The 2 The name ttcatenary''isparticularly descriptive. The nam e com es from the Latin word catena meaning chain.Catenary refers to the curve form ed by a uniform chain hanging freely between two poles.Leibniziscredited with coining the term

(ca.1691).

problem thus translates to finding the curve y that generatesthe surface of revolution havingthem inim alsurfacearea.A swith thecatenary problem ,we

maketheassumption thaty can bedescribed by afunction y :gzt ),z11--+R thatiscontinuousand piecewisedifferentiablein theintervalgzt ),z11.Under these assum ptionswe havethatthe surfacearea ofthecorresponding surface ofrevolution is

Hereweneed alsomaketheassumptionthatylz)> 0forallz CEgzt),z11.3The problem offinding the m inim alsurface thusreducesto finding thef-unction y such that thequantity zj

J(t v)-J ïl1+yuyy J 20

ism inim um .The two problem sthusproduce the sam ef-unctionalto bem inim ized.Thegenerating curvethatproduces the m inim alsurface ofrevolution isthusa catenary.T he surface itselfis called a catenoid.

3lfp= 0atsomepointk(E(zo,z1)wecanstillgeneratearotationallysymmetric ttobject,''buttechnicallyitwould notbeasurface.Near(k,0,0)the ttobject'' wouldresemble(i.e.,behomeomorphicto)adoublecone.Thedoubleconefails the requirementsto be a surfacebecause any neighbourhood containing the com m on vertex is nothom eomorphic to the plane.

6

1 lntroduction

Letusreturn to theoriginalproblem .A m odification oftheproblem would be to firstspecify thelength ofthe cable.Evidently,ifL isthe length ofthe cablew e m ustrequire that

L k (zl- zo)2+ (#1- y(jlz in order that the cable span the two poles.M oreover,it is intuitively clear thatin the case ofequality there is only one configuration possible viz.,the

linesegmentfrom (zo,y(tlto (z1,3/1).ln thiscase,thereisnooptimizationto be doneasthereis only onecandidate.W e m ay thusrestrictourattention to the case

L > (z1- zo)2+ (3/1- yv)2. G iven acable oflength L,theproblem isto determ inetheshapethecable assum esw hen supported between thepoles.Theproblem wasposed by Jacob Bernoulli in 1690.By the end of 1691 the problem was solved by Leibniz, H uygens,and Jacob's younger brother Johann Bernoulli.ltshould be noted thatG alileo had earlierconsidered the problem ,buthe thought the catenary was essentially a parabola.' l

Sincethearclength L ofthe cableisgiven,wecan use expression (1.1) to look for a m inim um potential energy configuration. lnstead,we start

with expression (1.2).The modified problem is now to find the f-unction y :g zt),z11--+ R such that W' s isminimized subjectto the arclength constraint

1+ y?2dz and the boundary conditions

!/(zo)= yçj, !/(z1)= yk. This problem is thus an exam ple ofa constrained variationalproblem .The

constraint (1.6)can be regarded asan integralequation (with,itishoped, nonuniquesolutions).Constraintssuchas(1.6)arecalledisoperimetric.W e discussproblem s having isoperim etric constraintsin Chapter4.

Supposethatwe use expression (1.1),which prim,aJacie seemssimpler thanexpression (1.2).Weknow L,sothatthelimitsoftheintegralareknown, but the param eters isspecialand correspondsto arclength.W e m ustsom ehow build in the requirem entthat s is arclength ifw e are to use expression

(1.1).lnordertodothiswemustuseaparametricrepresentation ofthecurve (z(s),!/(s)),sCE(0,fvl.Thearclength parameterforsuchacurveischaracter-

ized by the differentialequation

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

z(0)= z(), z(L)= zl : v(0)= yç j, y(L)= yk. ln general,a constraint ofthis kind is m ore difllcult to dealw ith than an isoperim etric constraint. 1.2.2 B rachystochrones

The history ofthe calculus ofvariations essentially begins with a problem

posed by Johann Bernoulli(1696)asa challengeto themathematicalcommunityand in particulartohisbrotherJacob.(Therewassignificantsibling rivalry between thetwobrothers.)Theproblem isimportantin thehistoryof the calculusofvariations because the m ethod developed by Johann's pupil, Euler,to solvethisproblem provided a sufllciently generalfram ework to solve othervariationalproblem s. T he problem that Johann posed w as to find the shape of a wire along which a bead initially at restslidesundergravity from one end to the other in m inim altim e.The endpoints ofthew ire are specified and the m otion of thebead isassum ed frictionless.Thecurve corresponding to theshapeofthe wire iscalled a brachystochrones ora curveoffastestdescent. T heproblem attracted theattention ofa num berofm athem aticallum inaries including Huygens,L'H ôpital,Leibniz,and Newton,in addition ofcourse to theBernoullibrothers,and laterEulerand Lagrange.T hisproblem w asat the cutting edgeofm athem atics atthe turn ofthe eighteenth century.

Jacob wasup to thechallenge and solved theproblem.M eanwhile tand independently)Johann and Leibnizalsoarrived atcorrectsolutions.Newton waslate to the party because helearned about the problem som e six m onths later than the others.Nonetheless,he solved the problem thatsam e evening and sent his solution anonym ously the next day to Johann.N ew ton's cover was blow n instantly.Upon looking at the solution,Johann exclaim ed (tA h!1 recognizethe paw ofthe lion.'' To m odelBernoulli'sproblem w e use Cartesian coördinatesw ith the pos-

itive l/-axis oriented in the direction ofthe gravitationalforce (figure 1.3). Let(zo,y(tland (z1,3/1)denotethe coördinatesofthe initialand finalpositionsofthe bead,respectively.Here,we require thatzt l < zl and yç j < yï. The Bernoulliproblem consistsofdeterm ining,am ong the curvesthat have

(zo,y(tland (z1,3/1)asendpoints,thecurve on which thebead slidesdown from (zo,y(tlto (z1,3/1)inminimum time.Theproblem makessenseonlyfor continuouscurves.We maketheadditionalsimplifjring (butreasonable)assumptionsthatthecurve can be represented by afunction y :gzt ),z11--+R 5 Theword com esfrom theGreek wordsbvakltistos m eaning ddshortest''and klwonos m eaning tim e.

8

1 lntroduction

tL Js

T(y)=ljs(s), whereL denotesthearclength ofthecurve,s isthe arclength param eter,and

z?isthevelocity ofthe bead s unitsdown the curvefrom (zo,y(j).Aswith the catenary problem ,we do not know the value of L,so we m ust seek an alternative form ulation.

Ourfirstjob isto getan expression forthevelocity in termsofthef-unction y.W e use the law ofconservation ofenergy to achieve this.Atany position

(z,!/(z))on the curve,the sum ofthe potentialand kinetic energiesofthe

bead isa constant.Hence

1

a

c= j.zrz. ? p(z()+ vrtgylzv). Solving equation (1.9)forz?gives 2c

r(z)= --- 2gy(z). Equation (1.8)thusimpliesthat

1 lntroduction

Christiaan discovered thata bead sliding down a cycloid generated bya circle of radius p under gravity reaches the bottom of the cycloid arch after the period r p g wlterever on the arch the bead starts from rest.This notable property ofthecycloid earned ittheappellation isochrone.T he cycloid thus sportsthenam esisochrone and brachystochrone.6 Christiaan used the curve to good effectand designed whatw asthen considered a rem arkably accurate pendulum clock based on the laudable properties ofthe cycloid,which was used to govern the m otion ofthe pendulum .The readerm ay find a diagram ofthe pendulum and furtherdetailson thisinteresting curve in an article by

Tee (671wherein severaloriginalreferencesmay befound.

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

three centuries following Bernoulli's challenge.Some m odels subjected the bead to aresisting m edium w hilstotherschanged theforcefield from asim ple uniform gravitationalfield to m ore com plicated scenarios.R esearch is still progressing on brachystochrones.The reader is directed to the w ork ofTee

(671,(681,(691formorereferences. 1.3 H am ilton's Principle

Therearemany finebookson classical(analytical)mechanics (e.g.,(11,(61, (351,(481,(491,( 591,and (731)andwemakenoattemptheretogiveevenabasic account ofthis seemingly vast subject.Nonetheless,itwould be demeaning to thecalculusofvariationsto ignore itsrich heritage and fruitfulinteraction with classicalm echanics.M oreover,m any ofourexam plescom efrom classical m echanics,so a few words from oursponsorseem in order. C lassicalm echanicsisteem ing with variationalprinciplesofw hich Ham ilton'sPrincipleisperhapsthem ostim portant.Iln thissection we give a brief ttno frills'' statem ent ofHam ilton's Principle as it applies to the m otion of particles.The serious student ofm echanics should consult one ofthe m any

specialized textson thissubject.

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 position ofthe particle at time f.The kinetic

energy ofthisparticle isgiven by T' 1 2

2

2

2 (: i)(f)+# (f)+. , 2(f)),

= -w

where?rzis the m ass ofthe particle and 'denotes d dt.W e assum e that the forceson the particlecan bederived from a single scalarfunction.Specifically, we assum e there isa function U'such that: 6 Itisalso called a tautochrone, butwe do notcountthissincetheword isderived from the Greek word tauto m eaning tsam e.''Theprefzx iso comes from the Greek word isos which also means tt same.''

VOne need only scan through Lanczos'book (481tofind the ttprincipleofVirtual W ork '' ttl zpha ïtlembert's Principle'' tt G auss' Principle of Least Constraint'' ttlacobi's Principle''and ofcourse ttllamilton'sPrinciple''am ong others.

1.3 Hamilton'sPrinciple

1.U'dependsonlyon timeand position;i.e.,U'= U(f,z,y,z); 2.theforcef= (/1,A,/a)actingon theparticlehasthecomponents / :U :U :U 1= - t'?z , /2= - ç')y , /a= - t'?z . The function U'iscalled the potentialenergy.Let L = T - U. Thefunction L iscalled theLagrangian.Supposethattheinitialposition of

theparticler(fo)andfinalpositionr(f1)arespecified.Ham iltonhsPrinciple statesthatthepathoftheparticler(f)in thetimeinterval(f(),f11issuch that the f -unctional

isstationary,i.e.,a localextremum ora ttsaddlepoint.''(W edefine ttstationary''moreprecisely in Section 2.2.)ln thelingoofmechanicsJ iscalled the action integralorsim ply the action.

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

q(f) = (t ?1(f),...,t ?,z(f)),where the qk are position variables.1f,for example,thesystem consistsofthree free particlesin 1:.3then n, = 9. T he kinetic energy T of a system is given by a quadratic form in the

generalized velocitiest ik,

L(t,q,ù)- T(q,ù)- t-tf,q). ln thisfram ew ork H am ilton'sPrincipletakes the follow ing form .

Theorem 1.3.1 (Ham iltonhsPrinciple) Tltemta/ït?zzo. fasps/cm o. fparticlesq(f)/' rt?m ag/wczzinitialconjiguration q(fo)toagivenjinalconjignration q(f1)in t/Jctim,eïzz/crrtzl(f(),f11issuclttltatt/JcJunctional

is stationary.

1 lntroduction

(x(r),y(r))

T he dynam icsofa system ofparticlesisthuscom pletely contained in the single scalarfunction L.W e can derive the fam iliar equationsofm otion from

Hamilton'sPrinciple (cf.Section 3.2).The readermightrightf-ully question whetherthe m otion predicted by Ham ilton's Principle dependson the choice ofcoördinates.Thevariationalapproachw ould surely beoflim ited valuewere itsensitivetotheobserver'schoiceofcoördinates.W eshow in Section 2.5 that H am ilton's Principle produces equations that are necessarily invariantw ith respectto coördinatechoices. 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 position ofthe mass at time f.Since z2+ y2 = :2 we need in factonly oneposition variable.Ratherthan usez ory itisnatural to use polar coördinates and characterize the position ofthe m ass attim e t

by the angle 4(t) between theverticaland thestring to which the mass is attached (figure1.5).Now,thekineticenergyis 1 1 z T= j . w t: ilz(f)+ #z(f))= j . zrzé rp (j;,

and the potentialenergy is

t-= vtzgît= zrz# tl- cos4(f)), whereg is a gravitation constant.Thus,

L(4,./' 1zr? )- . j . za$a-?wz#tl-cos4), and H am ilton's Principle im plies that the m otion from a given initialangle

4(t(j)to aflxed angle4(f1)issuch thatthef-unctional J(/)is stationary.

1.3 Hamilton'sPrinciple

Exam ple 1.3.2: K epler problem The Kepler problem m odels planetary m otion.lt is one ofthe m ostheavily studied problem sin classicalm echanics.Keeping with our no frillsapproach, we considerthe sim plest problem ofa single planetorbiting around the sun, and ignoretherestofthesolarsystem .A ssum ingthesun isfixed attheorigin, the kinetic energy oftheplanet is l

T=èwt: i lz(f)+2 )2(f))=1 j. zrz, / . 2(f)+,. 2(j)J2(j)j? where r and 0 denote polar coördinates and ?rzis the m ass of the planet. W e can deduce the potentialenergy function U'from thegravitationallaw of attraction

/=

-

Gnziv rz

,

where/ istheforce(acting in theradialdirection),M isthemassofthesun, and G isthe universalgravitation constant.G iven that

/=

U(r)=

-

1

-

:U ôr ,

jJlr)dr=-C?vzpavf; g2 .

Llr,0)=Vzrz yw2j2j. yGmr, M. H am ilton's Principle im plies that the m otion of the planet from an initial

observation (r(fo),p(f()))toa finalobservation (r(f1),p(f1))issuch that

is stationary.

T hereaderm ay bewonderingaboutthefateoftheconstantofintegration in the lastexam ple.Any potentialenergy ofthe form --G' m, M r+ const.w ill

producetherequisiteforce/.ln thependulum problem we tacitly assumed thatthepotentialenergywasproportionalto theheightofthem assabovethe m inim um possible height.ln fact,forthepurposesofdescribing thedynam ics

itdoesnotmatter;i.e.,U(f,q)and U(f,q)+ clproducethesameresultsfor any constantc1.W e are optim izing J and the addition ofa constantin the

Lagrangian simply altersthefunctionalJ(q)to ./(q)= J(q)+ const.lfone functionalisstationary atq theotherm ustalso bestationary atq. ln the lore of classicalm echanics there is another variational principle that is som etim es called the ttprinciple of Least Action'' or tlklaupertuis'

1 lntroduction

Principle,''w hich predatesH am ilton's Principle.This principle issom etim es conf-used w ith H am ilton'sand thesituation isnotm itigated by the fact that H am ilton'sPrincipleissom etim escalled thePrincipleofLeastAction.SM aupertuis'Principle concernssystem sthatareconservative.ln a conservative system w e have that the totalenergy ofthe system atany tim e talong the path ofm otion is constant.ln other words,L + U'= k,where k is a constant.For this specialcase L = 2T - k,and H am ilton's Principle leads to M aupertuis'Principle thatthe f-unctional tl

A-tql-

te

T(q,il)dt

isstationary along a path ofm otion.Hence,M aupertuis'Principleisa special caseofH am ilton'sPrinciple.M ostbookson classicalm echanicsdiscussthese

principles(alongwith others).Lanczos(481givesaparticularlycompleteand readable account that,in addition to m echanics,deals with the history and

philosophyoftheseprinciples.TheeminentscientistE.Mach (511alsowrites at length about the history,significance,and philosophy underlying these principles.Hisperspective and sym pathiesare som ew hatdifferentfrom those ofLanczos.9

1.4 Som e V ariational P roblem s from G eom etry 1.4.1 D idots P roblem

DidowasaCarthaginianqueen (ca.850B.C.?)whocamefrom adysfunctional family.Her brother,Pygmalion,murdered herhusband (who was also her uncle)and Dido,with thehelp ofvariousgods,fled to the shoresofNorth A fricaw ith Pygm alion in pursuit.U pon landingin North A frica,legend hasit thatshestruck a dealwith a localchiefto procureasm uch land asan oxhide could contain.She then selected an ox and cutitshideintovery narrow strips,

which shejoined togethertoform a thread ofoxhidem ore than twoand a half m ileslong.Dido then used the oxhidethread and theNorth African sea coast to define the perim eter ofher property.lt is not clear what the im m ediate reaction ofthe chiefwas to thisparticular interpretation ofthedeal,butit is

8ThetranslatorsofLandauand Lilhitz(491,p.131,gosofarastodraftatable to elucidate the different usages. 9 M ach is not so generouswith M aupertuis. ln connexion with M aupertuis'Principle he writes dd lt appears that M aupertuis reached this obscure expression by an unclear mingling ofhisideasofviswïwtzand the principle ofvirtualvelocities''

(p.365).lndefenseofMach,wemustnotethatMaupertuissuffered nolack of criticseven in hisown day.Voltaire wrote the satire Histoive d, u docteuv . 412 a/ :/a cf

d, u' rztzz/ deSaintMalo aboutM aupertuis.Thesituation atFrederick theGreat's

courtregardingMaupertuis,König,andVoltaireisthestuffofsoapoperas(see Pars(591p.634).

1.4 Som e VariationalProblem sfrom G eom etry

15

clear that D ido sought to enclose the m axim um area within her ox and the sea.Thecity ofCarthage wasthen builtwithin the perim eterdefined by the thread and the sea coast.D ido called the place Byrsa m eaning hideofbull.10

The problem thatDido faced on theshoresofNorth Africa (aside from family difllculties)wasto determine the optimalpath alongwhich to place

the oxhide thread so as to provide Byrsa with the m axim um am ountofland. D ido did not have theluxury ofwaiting som e 2500 years forthe calculusof variationsto develop and thus settled foran ttintuitivesolution.''

Dido's problem entailed determining the curve y of flxed length (the thread)such that the area enclosed by y and a given curve o'(the North African shoreline)ismaximum.Although thisisperhapstheoriginalversion

ofD ido'sproblem ,the term hasbeen used to coverthem ore basic problem : am ong allclosed curvesin theplane ofperim eterL determ inethecurve that enclosesthe m axim um area.The problem did notescapetheattention ofancient m athem aticians,and asearly as perhaps 200 B.C.the m athem atician Zenodorusll is credited with a proofthat the solution is a circle. Unfortu-

nately,thereweresometechnicalloopholesin Zenodorus'proof(hecompared thearea ofa circlewith thatofpolygonshaving the same perimeter).The

first com plete proofofthis result was given som e 2000 years later by K arl W eierstraB in his Berlin lectures.

Priorto WeierstraB,Steiner(ca.1841)proved thatz l / thereexistsa ttfigure''y w hose area is never less than thatof any other ttfigure''ofthe sam e perim eter,then y is a circle.Not content w ith one proof,Steiner gave five proofsofthisresult.The proofsare based on sim plegeom etricconsiderations

(nocalculusofvariations).Theoperativeword inthestatementofhisresult,

how ever,is ttif.''Steiner'scontem porary,D irichlet,pointed outthathisproofs do notactually establish the existenceofsuch a figure.W eierstraB and hisfollow ersresolved these subtle aspectsoftheproblem .A lively accountofDido's

problem and thefirstofSteiner'sproofscan befound inKörner(451.

Som e sim ple geom etricalargum ents can be used to show that ify is a

simpleclosed curvesolution toDido'sproblem then y isconvex (cf.Körner, op.cï/.).Thismeansthata chord joining any twopointson y lieswithin y 10 The reader willfind various bits and pieces of Dido's history scattered in Latin worksby authorssuch as Justin and Virgil.One account ofthe hide story comes from the Aeneid Bk.1 vs.367.Thestory gets even betteronce Aeneasarriveson the scene.Finally good ideasneverdie.ltis said thattheAnglo-saxon chieftains

Hengistand Horsa (ca.449A.D.)acquiredtheirlandbycirclingitwith oxhide strips(371.Bewareofrealestatetransactionsthatinvolveanox.

11The proof may have been know n even earlier, but Zenodorus in any event is the author ofthe proofthat appears in the comm entary ofTheon to Ptolemy's

Almagest.ZenodorusquotesArchimedes(who died in 212 B.C.)and isquoted byPappus(ca.340A.D.).Asidefrom theseroughdateswedonotknow exactly when Zenodorus lived.At any rate the solution was of little com fort to Dido's

heirsastheRomansobliteratedCarthage Byrsa in thethird Punicwarjustafter 200 B.C.and sowed salt on the scorched ground so that nothing would grow.

1 lntroduction

and the area enclosed by y.The convexity ofy is then used to show that D ido's problem can be distilled down to the problem of finding a f-unction

!/:(alo,zll--+R such that 21

yitt vl-

t vtzldz 20

is maximum subject to the constraint that the arclength of the curve y' jdescribed by y isL 2.lfwe assum e thaty is atleastpiecewise differentiable then thisam ountsto the condition r,

z1

-

=

-

2

zo

1+ y?2dz.

The problem w ith this form ulation is that we do notknow the lim itsofthe integral.The geom etricalcharacterofthe problem indicatesthatwe do not

needto know 130th ztlandzl(wecouldalwaysnormalizetheconstruction so thatzt l= 0< z1),butwedoneedtoknow zl- zo.Thisproblem iseffectively

theoppositeoftheproblem we had w ith the firstform ulation ofthecatenary. Sincewe know arclength,a naturalform ulation to usew ould be one in term s ofarclength.

Supposethaty' j-isdescribed parametrically by (z(s),!/(s)),s CE(0,L 21, where s is arclength.Suppose further that z and y are at least piecew ise differentiable.G reen's theorem in the plane can then be used to show that the area ofthe setenclosed by y' j-and the z-axis is 1

z-/2

4(t v)- j' 0

.

yls) 1- y'2(s)ds,

wherewehaveused therelation z?2(s)+ y?2(s)= 1.ThebasicDido problem is thus to determine a positive function y :(0,L 21 --+ R such that -4 is m axim um .

1.4.2 G eodesics

Let E be a surface,and letyo,plbe tw o distinctpointson E .The geodesic

problem concernsfinding thecurvets)on E with endpointsyb,plforwhich thearclength ism inim um .A curvehaving thisproperty iscalled a geodesic.

The theory ofgeodesics isone ofthe most developed subjects in differential geom etry.Thegeneraltheory iscom plicated analytically bythesituation that sim ple, com m on surfaces such as the sphere require m ore than one vector function to describe them com pletely.ln thelanguageofgeom etry,thesphere isa m anifold thatrequiresatleasttwo charts.W ehaveencountered and sidestepped the analogousproblem forcurves,and we do so herein theinterestof sim plicity.W e focuson the localproblem and refer the readerto any general

texton differentialgeometry such asStoker(661orW illmore(751foramore precise and in-depth treatm entofgeodesics.lz Suppose that E isdescribed by the position vector function r :tz --+ 1 :.3 whereo'is a nonem pty connected open subsetofR2, and for(u,z?)CEc,

rtz',z,)- (z(z',z,),ylu,z,),ztz',z,)). W e assum e that r is a s11100th function on (z;

functionsof(u,z?),and that '

It' ?r A t '?rI/ 0, -

ç ' )u t ' ?r so that r is a one-to-one m apping of o'onto E .lfy is a curve on E ,then there is a curve y m 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 thatthe pointsyb and pl correspond

to rt l= rtztt ),zo)and rl= rtztl,z?1),respectively.Anycurvey from rtlto rl mapstoacurvey mfrom t' ? . to,z?())to tul,z?1). Forthegeodesicproblem werestrictourattention tos11100th sim plecurves

(no self-intersections)on E from rtlto r1.LetF denote the setofallsuch curves.Thus,ify CEF ,then thereexistsa param etrization ofy ofthe form

lt, (f)= r(zt(f),z?(f)), te g' àt ),flj,

where

t' ?r 2 E= ô

u

F=

t ' ?r t' ?r t' ?r 2 . , G=

du 37

37

The f-unctionsE ,F,and G are called com ponentsofthe first fundam ental form or m etric tensor.N ote that these com ponentsdepend only on ztand v.Note also thatthe identity t' ?r

A

t' ?r

= EG - . /7'2

ôu 37 12A morespecializeddiscussioncanbefound in Postnikov ( 621.

1 lntroduction

f = .Elzt/2+ 2F' t/r/+ (7r/2 ispositivedefinite. T he arclength ofy isgiven 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 oftheunit sphere.The surface E can bedescribed param etrically by

=

I(cos. tzcosz?,cosztsinz?,- sinzt)12

The arclength integralisthus

A feature ofthe basic geodesic problem described above is that it does notinvolvethe f -unction r directly.Thearclength ofa curvedepends only on the three scalar functions E ,F ,and G .G eodesics are partofthe intrinsic geom etry ofthe surface,i.e.,the geom etry defined by them etrictensor.The m etrictensordoesnotdefinea surfaceuniquely even m odulotranslationsand

rotations.Thereare any num berofdistinctsurfacesin 1 :.3thathave thesam e m etrictensor.Forexam ple,a plane,a cone,and a cylinderallhave thesam e m etrictensor.lfa cylinderis tunrolled''and ttflattened''to form a portion of the plane,then a geodesic on the cylinderw ould becom e a geodesic on the plane. O nedirection fora generalization ofthe aboveproblem isto focus on the

spacea.f;l1:.2and dehnethecomponentsofthemetrictensor.Fornotational sim plicity,letzt= ztl, z? =

zt2

,

and u = @,z?).W ecan choosescalarfunctions

gjk :tz--+R j,k = 1,2 and definethearclength elementds by (/. $ :2= .g11(( /tt1)2+ gyztyttltyuz+ w ytyuzgztl+ gggLd. tg2)2 = g kduiiduk wherethelastexpression usestheEinstein sum m ation convention:sum m ation ofrepeated indiceswhen one isa superscriptand the otheris a subscript.O f course wem ustplacesom erestrictions on thegjk in orderto ensure thatour arclength elem entispositive and thatthelength ofa curvedoesnotdepend on thechoiceofcoördinatesu.W ecan takecareoftheseconcernsby requiring that the gjk produce a quadratic form that is positive definite and that the gjk form a second order covariant tensor.To m im ic the earliercase we also im pose the sym m etry condition

so that (/. 52 = .g11(( /tt1)2+ z. qyz(/ttlty. ttz+ gggLdu2)2.

(J.J6)

ln term softhe form er notation,E = .t711,F = .t y12 = . t721,and G = gnLt.For thiscase,the positive definiterequirem ent am ounts to the condition .

t711. t.722 -

..

. ?2 t 12 > ()

with .t y11> 0.Thecondition thatthegjk form a second-ordercovarianttensor

meansthatundera s11100th coördinatetransformation from u = (ul,zt2)to fl= (fl1,i )2),thecomponentspklu)transform to jis? ztl 4laccording to the relation

ç' lui it ' ?' t zk

lilm = gjkt'?flgogjvn. .

Theseto'equipped with sucha tensorcan beview ed asdefining ageom etrical

objectin itself(asthesurfaceE was).ltisaspecialcaseofwhatiscalled a R iem annian m anifold.Let. 'ttdenotethisgeom etricalobject.A curvey m in o'generates a curvey in . Azt,and the arclength isgiven by tl

Z(' 7)=

te

qjikttiilttk't/f .

where(zt1(f),zt2(f)),t(Egft ;,f1jisa parametrization ofq.Thecondition that thegjk form a second-ordercovarianttensorensuresthatL(y)isinvariant

20

1 lntroduction

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

' tt.Note also thatL(y)isinvariantwith respectto orientation-preserving

.'

param etrizationsofy m. T headvantageoftheaboveabstraction isthatitcan bereadilym odified to

accommodatehigherdimensions.Supposethato'f;lR?zand u = (ul,...,u?zl. W ecan definean zz-dimensional(Riemannian)manifold . Aztby introducinga m etric tensorw ith com ponentsgjk such that:

1.thequadratic form gjkdui iduk ispositive definite;

2.gjk = gkj fOrj,k = 1,2,...,zz;

3.under any s11100th transformation u = uttll the gjk transform to . qlv. n according to therelation

ç 'lut t' ?ztk 9lm = 9jkôhjdûs? z.

A curvey on .Aztisgenerated by a curveJfin o'(;R''.Supposethatu(f)= (zt1(f),...,. ? . t? z(f)),t(Egfl),f1jisa parametrization ofJf.Thearclength ofy is then defined as

tl

Z(' 7)=

te

qjikttiilttk'dt. .

A generalization ofthe geodesicproblem isthusto find thecurvets)y m in o' with specifiedendpointsutl= u(fo),ul= u(f1)suchthatL(y)isaminimum. G eodesics are of interest not only in differential geom etry,but also in

m athematicalphysics and other subjects.lt turns out that many problems can be interpreted asgeodesic problem son a suitably defined m anifold.l3 In thisregard,thegeodesicproblem iseven m oreim portant because itprovides a unifjring fram ework form any problem s. 1.4.3 M inim al Surfaces

W e have already encountered a specialm inim alsurface problem in our discussion ofthe catenary.The rotationalsym m etry ofthe problem reduced the problem to that offinding a f -unction y ofa single variable z,the graph of which generatesthesurfaceofrevolution having m inim alsurfacearea.Locally, any surface can be represented in ttgraphical''form ,

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 requirestwo independentvariables.Thus the problem offinding a surfacew ith m inim alsurface area involvestw oindependentvariablesin contrastto theproblem sdiscussed earlier. 13 ln the theory ofrelativity, where differentialgeometry is widely used,the condition that the m etric tensor be positive definite isrelaxed to positive semidefinite.

G iven a sim ple closed space curve y,the basic m inim alsurface problem entailsfinding,am ongalls11100th sim plyconnected surfaceswith y asaboundary,thesurfacehaving m inim alsurfacearea.Suppose thatthecurve y can be

represented parametricallyby (z(f),y(t),z(f))fortCE)(),f11,and forsimplicity suppose that the projection ofy on the zp-plane is also a simple closed

curve;i.e.,thecurveJfdescribedby (z(f),y(t))fortCE)(),f11isasimpleclosed

curve in the zp-plane.LetJ2denote the region in the zp-planeenclosed by Jf. Suppose further thatw e restrictthe class ofsurfacesunder consideration to

thosethatcan berepresentedintheform (1.17),wherezisasm00thf-unction for(z,y)CE. Q.Thedifferentialareaelementisgiven by CI-,b-

1+

(, '?z)2+ts '?zl2csdy,

and the surfacearea isthus

,tzl-jjJ21+(, 'z)2+(, ??.)2dzo. ,

.

The(simplified)minimalsurfaceproblem thusconcernsdeterminingas11100th functionz:J2--+R such thatz(z(f),y(t))= z(f)fortCE(f(),f11,and dtzlisa

m inim um .There isa substantialbody ofinform ation aboutm inim alsurfaces.

Thereadercan find anoverview ofthesubjectin Osserman (581. 1.5 O ptim al H arvest Strategy O ur finalexam ple in this chapter concerns a problem in econom ics dealing with finding a harvest strategy that m axim izes profit.Here,we follow the

examplegiven byW an (711,p.6anduseafishery toillustratethemodel. Letylt)denote the totaltonnage offish attime tin a region J2ofthe ocean,and let yc denote the carrying capacity ofthe region J2 for the fish. Thegrow thofthefish populationw ithoutany harvestingistypicallym odelled by a first-orderdifferentialequation

y'lt)- /(f,y).

(1.18)

lfyissmallcomparedtoyc,then / isoften approximatedbyalinearf-unction

in y;i.e.,/(f,y)= ky+ g(t),wherekisaconstant.Morecomplicated models areavailableforawiderrangeofylt)such aslogisticgrowth

/(f,y)-kylt)(1-VVtc f)1. Theordinarydifferentialequation (1.18)isaccompanied by an initialcondition

1 lntroduction

3/(0)= : % thatreflectsthe initialfish population.

Supposenow thatthefish are harvested ata rate ' tt;(f).Equation (1.18)

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

y'lt)- /(f,y)- zc(f).

(1.20)

Given thef-unction /,theproblem istodeterminethef-unction w so thatthe profit in a given tim e intervalT is m axim um . ltis reasonable to expect that the cost ofharvesting the fish depends on

the season,the fish population,and the harvestrate.Letc(f,y,' tt;)denote

the costto harvesta unitoffish biom ass.Suppose thatthe fish com m andsa pricep perunitfish biom assand thatthe price isperhaps season dependent, but notdependenton the volum e offish on them arket.The profitgained by

harvestingthefishin asmalltimeincrementis(p(f)- c(f,y,' t;))' t;(f)dt.Given a fixed period T w ith which to plan the strategy,the totalprofitisthus T

Ply,z&)-

0

(r(f)- ctf,y,zc))zc(f)dt.

The problem isto identifjrthe f-unction w so thatP ism axim um . T heaboveproblem isanexam pleofa constrainedvariationalproblem .The

functionalP isoptimized subjectto the constraintdefined by the differential

equation (1.20)(anonholonomicconstraint)and initialcondition (1.19).W e can convert the problem into an unconstrained one by sim ply elim inating

w from the integrand defining P using equation (1.20).The problem then becom es the determ ination ofa function y that m axim izes the totalprofit. This approach is not necessarily desirable because we wantto keep track of ' tt;,the only physicalquantity we can regulate. A feature ofthisproblem thatdistinguishesitfrom earlierproblem sisthe absence ofa boundary condition forthe fish population attim e T .A lthough wearegiven theinitialfish population,itisnotnecessarily desirableto specify

thefinalfishpopulationaftertimeT.AsW anpointsout,theconditiony(T)=

0,forexam ple,isnotalwaysthebeststrategy:ttgreen issues''aside,itm ay cost farm ore to harvestthe lastfew fish than they areworth.This sim ple m odel thus provides an exam ple ofa variational problem with only one endpoint fixed in contrastto the catenary and brachystochrone. ln passing we note that econom ic m odels such as this one are generally fram ed in term sof ttpresentvalue.''A pound sterling invested earnsinterest, and thisshould beincorporated into the overallprofit.lftheinterestiscompounded continuously at a rate r,then a pound invested yields crt pounds aftertim e f.Anotherway oflooking at thisis to view a pound ofincom e at tim e tas worth c-rt pounds now .C onsiderations ofthis sort lead to profit functiona. lsoftheform

T he F irst V ariation

ln this chapterwe develop a necessary condition fora function to yield an extrem um for a f -unctional.The centrepiece ofthe chapter isa second-order differentialequation,the Euler-lsagrange equation,w hich playsa rôle analogoustothegradientofa f-unction.W efirstm otivate the analysisby review ing necessary conditionsforfunctionsto have localextrem a.TheEuler-lsagrange equationsare derived in Section 2.2 and som e specialcases where thedifferentialequation can be sim plified are discussed in Section 2.3.T he rem aining three sections are devoted to m ore qualitative topics concerning degenerate cases,invariance,and existence ofsolutions.W epostpone a discussion ofsufficientconditions untilChapter10.

2.1 T he F inite-llim ensional C ase The theory underlying the necessary conditions for extrem a in the calculus ofvariations is m otivated by that for functions ofn,independent variables. Problem sin the calculusofvariationsare inherently infinite-dim ensional.The characteroftheanalyticaltoolsneeded to solve infinite-dim ensionalproblem s differs from that required for finite-dim ensional problem s,but m any ofthe underlying ideashavetractable analoguesin finite dim ensions.ln thissection we review a necessary condition fora function ofn,independentvariablesto havea localextrem um . 2.1.1 Functions of O ne V ariable

Let/ bea real-valued function defined on theintervalf f;lR.Thef-unction / :f--+R issaidtohavealocalmaximum atzCEfifthereexistsanumber

t î> 0 such thatforany k CE (z - 6,z + 6)( :z f,Jlk)< /(z).Thef-unction / :f --+ R issaid to have a localm inim um atz CEf if-/ has a local m axim um atz.A function m ay haveseverallocalextrem a in a given interval.

2 The FirstVariation

lt m ay be that a function attainsa m axim um or m inim um value forthe

entireinterval.The f-unction / :f --+R has a globalmaximum on f at

z CEf ifJlk)< /(z)for allk CEf.The function / issaid to have a global m inimum on f atz CEf if-/ hasaglobalmaximum atz.Notethatiff hasboundary pointsthen / mayhaveaglobalmaximum on theboundary.lf / isdifferentiableon f then thepresenceoflocalmaxima orminima on f is characterized by the first derivative.

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

intervalI.f// Itasa localcz/rcm' tzm atzCEf tlten /?(z)= 0. P roof: The proofofthis result is essentially the sam e for a localm axim um or m inim um .Suppose that z is a localm axim um .Then there is a num ber

t î> 0 such thatforany k CE(z- 6,z+ 6)( :zf theinequality /(z)k Jlk)is satisfied.Now thederivativeof/atz isgiven by

/'(z)- llin>za(/(:)- /(z)) (:- z). -

The numeratorofthislimitisneverpositivesince /(z)isa maximum,but the denom inatorispositive w hen k > z and negativew hen k < z.Since the

function / isdifferentiableatz theright-and left-sided limitsexistand are

equal.Theonly way thiscanbetrueisif/?(z)= 0.

En

lt is illum inating to exam ine the situation for s1100th f-unctions.W e use thegenericterm ttsm ooth''to indicate thatthef-unction hasasm any continuousderivatives asare necessary to perform whateveroperationsare required.

Suppose that / issm00th in the interval(z - 6,z + 6),where t î> 0.Let k - z = 6p.Taylor'stheorem indicatesthat,fort îsufllcientlysmall,/ can be represented by 2

/(g)- /(z)+ ' î07/t(z)+ u. t îpz/??(z)+ otya;. lf/?(z)# 0 and t îissmall,thesign ofJlk)- /(z)isdetermined by p/?(z). Suppose that /?(z) # 0.lf/ hasa localextremum at z then the sign of Jlk)- /(z)cannotchangein (z- 6,z+ 6),sothatp/?(z)musthavethesame sign for allp.B ut it is clear that p can be positive or negative and hence

p/?(z)canbepositiveornegative.Wemustthereforehavethat/?(z)= 0.lf /?(z)= 0,then theaboveexpansionindicatesthatthesignofthedifferenceis thatofthequadraticterm,i.e.,thesign of/??(z).lfthisderivativeisnegative then /(z)isalocalmaximum;ifitispositivethen /(z)isalocalminimum. ltmay bethat/??(z)= 0.ln thiscasethesign ofthedifferencedependson thecubicterm ,which containsa factorp3. Likethelinearterm ,however,this factorcan beeitherpositiveornegativedepending on thechoiceofp.Thus,if

/???(z)# 0,/(z)cannotbealocalextremum.Wecan continuein thismanner aslongas/ hastherequisitederivativesin (z- 6,z+ 6). Fora differentiablefunction itiseasy to see graphically why thecondition

/?(z) = 0 isnecessary fora localextremum.The Taylor expansion for a

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

vanishesan 0/)changeintheindependentvariableproducesan0(62)change in the function value as tî--+ 0.For this reason points such as z are called

stationary points.Thef-unctions/zz(z)= z'',wheren,CEN,z CER provide sim ple paradigm s forthe variouspossibilities

Example 2.1.1: Let/(z)= 3z2- z3.Thefunction / issm00th forz CE R and therefore if any local extrem a exist they must satisfjrthe equation 6z- 3z2 = 0. Thisequation issatisfied ifz = 0orz = 2.The second derivative

is6- 6z,so that///(0)= 6 and consequently /(0)isa localminimum.On theotherhand,///(2)= -6and thus/(2)isalocalmaximum. Exam ple 2.1.2:

/(

,2sin2(1 z),ifz , #0

z)-t(),

ifz=().

Thisf -unction isdifferentiableforallz CER.Now //(0)= 0,and thusz = 0 isastationary pointbutthederivativeisnotcontinuousthereand so ///(0) doesnotexist.W ecan deducethat/ hasalocalminimum atz = 0 because

/(z)k 0fora11z CER. Example 2.1.3: Let/(z)= IzI.Thisfunction isdifferentiable forallz CE R - (0/..Thederivativeisgiven by //(z)= -1 forz < 0,and //(z)= 1for z > 0.Thus/cannothavealocalextremum in R (0/..Nonethelessitisclear that/(0)= 0isa local(and global)minimum for/ in R. Example 2.1.4: Let/(z)= c2.Thisf-unction issm00th forallz CER and itsderivativenevervanishes;consequently,/doesnothaveanylocalextrema. T he relationship between localand globalextrem a islim ited.C ertainly if

/ hasa globalextremum atsome interior pointz ofan intervalthen /(z) isalso alocalextremum.1f,in addition,/ isdifferentiablein f,then itmust

alsosatisfjrthecondition /?(z)= 0.Butitmay be(asoften isthecase)that a globalextrem um isattained at one ofthe boundary points off,in w hich

caseeven if/ isdifferentiablenothing regarding thevalue ofthe derivative can be asserted.

26

2 The FirstVariation

2.1.2 Functions of Several V ariables

The definitionsforlocaland globalextrem a in n,dim ensions areform ally the

sam easfor the one-variable case.LetJ2 f;lR?zbe a region and suppose that

/ :J2--+R.Fort î> 0andx = (z1,z2,...,z,zl,let The function / :J2 --+R hasa globalm mximum (globalm inim um )on J2 atx CEJ2if/(k) < /(x) (/(k) k /(x))fora11#)CE fî.The function / hasalocalm mxim um (localm inim um)atx CEJ2ifthereexistsanumber t î> 0 such that forany E k CEB (x;6)( : z. 62,Jlk) < /(x) (/(2) k /(x)).As with the one-variable case ifJ2hasboundary points / may have a global m axim um m inim um on the boundary. N ecessary conditions for a s11100th function oftwo independentvariables to havelocalextrem a can be derived from considerationssim ilarto thoseused

in thesingle-variablecase.Suppose that/ :J2--+R isa s1100th f-unction on

theregion J2f;lR2,and that/ hasa localextremum atx = (z1,z2)CE. Q. Thenthereexistsan t î> 0such that/(k)- /(x)doesnotchangesignforall E k CE.B(x;6).Letl k = x + 6p,wherep = (p1,m )CER2.Fort îsmall,Taylor's theorem im plies

andthesignofJlk)-/(x)isgivenbythelinearterm in theTaylorexpansion, unlessthisterm iszero.But,ifx + tîr/CEB (x;6),then x - t îr/CEB (x;6)and these pointsyield differentsigns forthe linear term unlessitiszero.lfx isa localextrem um w e m usttherefore have that

('?z,'?a)'(::/ ,:/ )- 0, z1 :z2

forallp CER2. In particular,equation (2. 2)musthold forthespecialchoices

el= (1,0)and ez= (0,1).Theformerchoiceimpliesthatt ' ?/ t ' ?zl= 0 and thelatterchoiceimpliesthatt ' ?/ t' ?zz= 0.Wethushavethatif/ hasalocal extrem um atx then

V/(X)= 0. Geometrically,equation (2.2)impliesthatthetangentplaneto the graph of / ishorizontalata localextremum.Pointsx atwhich V/(x)= 0arecalled stationary points.lfx isastationarypointandl k= x-h6p,then J(k)- /(x) is0(62)ast î--+0,in contrasttothegenericcasewherean 0(6)changeinthe independentvariablesproducesan 0/)changeinthedifference.

Example 2.1.5: Let/(z1,z2)= z2 1- z2 z+ z3 y.Thestationary pointsfor / are given by V/(z1,z2)= (2z1+ 3z2 1,-2zz)= 0.Thisequation hastwo solutions(0,0)and (-2 3,0).ltcan beshown that(0,0)producesneithera localminimum nora localmaximum for/ (itisasaddlepoint).ln contrast, at(-2 3,0)itcan beshown that/ hasa localmaximum. Exampl e 2.1.6: Themonkeysaddlelisasurfacedescribedby/(z1,z2)= zz 3 -

3z2 yzz. Ifx i sastationarypointfor/ then theequations 6z1za = 0, 3z2 2 - 3z2 1 = O, -

mustbesatisfied and thismeansthatzl= z2= 0.Thef-unction / doesnot havea localextrem um atthispoint.Note thateven the second derivativesat thispointare zero. T he extension ofthe above argum entsto f-unctionsofn,independentvari-

ablesisstraightforward.Let/ :J2--+R bea s11100th f-unction on theregion J2( :zR'',and supposethat/ hasalocalextremum atx CE. Q.Then,fort î> 0

sufllcientlysmall,thesign of/(k)- /(x)doesnotchangeforallk CEB (x;6). Letk = x+ 6p,wherep = (p1,w ,...,p,z).Fort îissufllciently small,Taylor's theorem im plies

and thesign of/(2)- /(x)isdetermined by thelinearterm in theTaylor expansion,provided thisterm is not zero.Butthe linear term m ustbe zero

sincex + t îr/and x - t îr/are1 30th in . B(x;6);hence, p'V/(x)= 0 fora11p CER''.Thespecialchoicesel= (1,0,...,0),e2= (0,1,...,0),...,ezz= (0,0,...,1)forp yield then,conditionst ' ?/ t ' ?zk= 0 atx fork= 1,2,...,zz. ln sum m ary we have thefollow ing result.

Theorem 2.1.2 Let/ :J2--+R bea smtat a//zJ' unction on tlteregion J2f;lR''. f// Itasalocalcz/rcm' tzm atapointx CEJ2tlten

V/(X)= 0.

28

2 The FirstVariation

2.2 T he E uler-tzagrange Equation Localextrem a fora f -unctionalcan be defined in a m anneranalogous to that used forfunctionsofn,variables.T hetransition from finiteto infinitedim ensionaldom ains,how ever,carries with it som e com plications.For instance, there m ay beseveralvectorspacesforw hich the problem iswelldefined,and once a f-unction space is chosen,there m ay be severalsuitable norm s avail-

able.Thevectorspace C'zgzt),zll,forexample,can beequipped with any of the 11.I lk,x norms,k - 1,2,...,n oreven a. ny LPnorm.2 Unlikethefinitedim ensionalcase,differentnorm sneed notbeequivalentand thusm ay lead to differentextrem a.Functions tclose''in onenorm need notbeclose in another norm .ln applications,thechoiceofa vectorspace and norm form an integral partofthe m athem aticalm odel.

LetJ :X --+R bea f -unctionaldefined on the function space (. X,11.II) and let S f;lX .The functionalJ is said to have a localm axim um in S at

y CES ifthereexistsan t î> 0 such thatJ(L)- J(y)< 0forall:' èCES such that115- : v11< E.ThefunctionalJ issaidtohavealocalm inim um in S at y CES ify isa localm axim um in S for- J.ln thischapter,the set S is a set offunctions satisfying certain boundary conditions.

Functions:' èCES in an 6-neighbourhood ofa f-unction y CES can berepresented in a convenientway as a perturbation ofy.Specifically,if: ' èCES and

115- : v11< 6,then thereissomepCEX such that b= #+ 6r/. A llthefunctionsin an 6-neighbourhood ofy can begenerated from a suitable

setHLoffunctionsp.Certainly any such p mustbean elementofX,butp m ustalso be such that y+ t îr/CES.The setIL isthusdefined by

Since the inequalitiesdefining the extrem a m ustbe valid w hen t îis replaced by any num beré such that 0 < ? < 6,itis clear that t îcan always be m ade

arbitrarily sm allwhen convenient.The auxiliary setIL can thus be replaced by theset

bI= (p CEX :y+ t îr/CESj,

.

forthe purposesofanalysis. A tthisstagewespecialize to a particularclassofproblem called thefixed

endpointvariationalproblem ,3andworkwiththevectorspaceCzgzl),z1j thatconsistsoffunctionson gzt ),z11thathavecontinuoussecond derivatives. LetJ :Czgzt ),z1j--+R beafunctionaloftheform 2 See Appendix B .1. 3 M ore accurately, it is called the nonparam etric fixed endpoint problem in the plane.

2.2 The Euler-luagrange Equation

where/isafunctionassumed tohaveatleastsecond-ordercontinuouspartial derivatives w ith respect to z,y,and y?. Given two values yç j,yï CE R,the fixed endpointvariationalproblem consistsofdeterm ining the functionsy CE

(72g zt),z1jsuchthat!/(zo)= yç j,!/(z1)= 3/1,and J hasalocalextremum in S at y CES.Here,

S = (!/CE(72gzt ),z1j:ylzç j)= yçj and !/(z1)= 3/1/., bI = (pCE(72gzl),z1j:p(z())= p(z1)= 0j

.

(cf.figure2.1). Suppose that J hasa localextremum in S aty.For definiteness,let us assum e that J has a localm aximum at y.Then there isan t î> 0 such that

J(L)- J(y)< 0forallL (tCES such that115- : v11< E.Fora. ny?)e S thereisan p CE. bI such that:' è= y+ 6p,and fortîsmallTaylor'stheorem im pliesthat

/(z,b,? )/)- /(z,y+ t î07,y'+ 6p/)

=/(z,y,y?)+,t,?ç ')y /+,?t ' ?: t . /,)+0/2). Here,weregard / asafunction ofthethreeindependentvariablesz,y,and y?, and thepartialderivativesin the aboveexpression are allevaluated atthe

point(z,y,!/?).Now,

30

2 The FirstVariation

The quantity

é'.

l(r/,#)= 20zl r/oj 1-07/çoj t' ?!/' ly? dr

is called the first variation of J.Evidently,if p CE. bI then -p CE . bI,and

JJ(p,y)= -JJ(-p,y).Fort îsmall,thesign ofJIL)- J(y)isdetermined by thesignofthefirstvariation,unlessJJ(p,y)= 0forallp CE. bI.Thecondition thatJ(y)bealocalmaximum in S,however,requiresthatJ(L)- J(y)does notchangesign forany :' èCES such that 15- : v11< 6;consequently,ifJ(y)is a localm aximum then

é'.

t oj 7'(r/,#)= 2z /toj 0 r/ç ' )y + / ' 93/? dr = 0,

forallp CE. bI.A sim ilarchain ofargum entscan beused to show thatequation

(2.6)mustbesatisfied forallpCE. bI ifJ hasa localminimum in S aty.

Sofarwehaveshow nthatifJ hasalocalextrem um in S aty then equation

(2.6)mustbe satisfied fora11p CE. bI.Asin the finite-dimensionalcase,the converseisnottrue:satisfaction ofequation (2.6)doesnotnecessarily mean thaty producesa localextremum forJ.lfy satisfiesequation (2.6) forall p CEf. J,w esay thatJ isstationary aty,and follow ing com m on convention,y iscalled an extrem alforJ even though itm ay notproducealocalextrem um forJ.

Equation (2.6)istheinfinite-dimensionalanalogueoftheequation (2.5). Recallthatthecondition V/ = 0 isderived from thefactthatp.V/ = 0 m usthold forallp CER''.By a suitable choiceofvectors in R?zit wasshow n

thateach componentofV/ mustvanish separately.A similarstrategy can

beused to divorcethenecessary condition (2.6)from thearbitrary f-unction p.lt is not yet clear,however,w hich specialchoices off-unctions in . bI w ill

accomplish this.M oreovertheintegrand in equation (2.6)containsnotonly p butalsoz/ tocomplicatematters. Thep?term inequation (2.6)canbeeliminated usingintegrationbyparts. ln detail,

zlr//ooj dz = p oj zj- ztp (j oj dz zo #t o#tzo zo ' jt , / oyt zl (j oj = - 2 v:IV ôy? dz, 0

2.2 The Euler-luagrange Equation

j20 2'ptô t ?y ' /-yd(o ôy /,j).( s-() . N ow ,

*/ d */ ôy t)' 7 ç2y? '

*/ *2/ d2/ t d2/ tt (' ?y ozoy? (' ???(' ????V ç ' lvoy'ïl'

and given that/ hasatleasttwocontinuousderivatives,weseethatforany

Jzccly CE(72gzt),z1jthef -unction E :gzt),z11--+R defined by E(z)= t '?/ d t '?/ oy - gg

(oy,(

iscontinuouson theintervalgzt),z11.Here,foragiven f-unction y thepartial derivativesdefiningE areevaluatedatthepoint(z,ylz),y?(z)).lnfact,E can beregardedasanelementintheHilbertspacefvzgzll,z1j4andsinceanyp CEf' f isalso in L2gzt ),z1jwecandraw acloseranalogy with thefinite-dimensional casebynotingthatequation (2.7)isequivalenttotheinnerproductcondition

for a11p CE . bI.As with the finite-dim ensionalcase,w e can show that the abovecondition leadsto E = 0 by considering a specialsubsetof. bI.Firstwe establish two technicalresults. Lem m a 2.2.1 Letctand, d be two realzz' t zm scrs suclttltatct< , 3. Tlten tltere

ezistsaJunctionv CEC2(R)suclttltatzztzl> 0JorallzCE(ct,, d)andzztzl= 0 Jorallz CER - (ct,, J). P roof: Let

( z- 0,)3(p - z)3,i (cste ,, J) zztzl- ( ' ) oftz he( rEwi .

The function v clearly has allthe properties claim ed in the lem m a except perhapscontinuousderivativesat z = ctand z = , 3.Now ,

li ,(z)- ,(a) = Iina (z- G)'(/ - z)'- 0

m z--ycr'h

z - ct

z--ycr'h

=

z - ct

lina (z- a)2(p - z)3= 0,

z--ycr'h

and 4 Hilbert spaces are discussed in Appendix B.2.Any function continuous on the

intervalgzo,zljisinthisspace.Therearealottrougher''functionsinthisspace aswell.

32

2 The FirstVariation

li M?(z)- M?(a)== jkru 3(z- a)2(4 - z)2(p :-a - 2z)- 0

rn z--ycr'h

I - ct

z--ycr'h

=

I - ct

lina 3(z- a)(p - z)2(p+ a - 2z)= 0,

z--ycr'h

and

ifz CE(ct,, t ?) otherwise and itisclear that

lim zz??(z)= zz??(ct)= 0 and

lim zz??(z)= zz??(, J)= 04

z->:

hence,v CE(72(R).

Proof:Supposethatg , # 0forsomecCEgzt ),z11.W ithoutlossofgenerality it can beassumed thatglc)> 0,and bycontinuity thatcCE(zo,z1).Sinceg is continuouson gzt ),z11therearenumbersct,, d such thatztl< ct< c< , d < zl andglz)> 0 forz CE(ct,, d).Lemma2.2.1impliesthatthereexistsaf-unction v CECzgzlj,z1jsuch thatzztzl > 0 for allz CE (ct,/7) and zztzl = 0 for all z CEgzt ),z11- (ct,, d).Therefore,vCE. bIand

2.2 The Euler-luagrange Equation

wltere/ Itascontinuon, spartialderivativeso. fsecondorder' ttl ï//zrespecttoz,y, and : (/? and ztl< z1.Let

2d 7(o t ?y ' /?)-o t ' ?y /-0 Jorallz CEgzo,z11. Equation (2.9) isa second-order (generally nonlinear)ordinary differential equationthatany(smooth)extremalymustsatisfjr.Thisdifferentialequation is called the Euler-Lagrange equation.The boundary values associated with thisequation for thefixed endpointproblem are

!/(zo)= #0, !/(z1)= #1. T he Euler-lsagrange equation is the infinite-dim ensionalanalogue ofthe

equation (2.5).lnthetransitionfrom finitetoinfinitedimensions,analgebraic condition forthe determ ination ofpointsx CER?zw hich m ight lead to local extrem a is replaced by a boundary-value problem involving a second-order differentialequation. Exam ple 2.2.1: G eodesics in the P lane

Let(zo,y(tl= (0,0)and (z1,3/1)= (1,1).Thearclength ofa curvedescribed by!/(z),z CE(0,11isgiven by

The geodesic problem in the plane entails determ ining the f -unction y such that the arclength is m inim um .W e lim it our investigation to functions in

(72g 0,ljsuch that v(0)- 0, v(1)- 1. lfy isan extrem alforJ then the Euler-lsagrangeequation m ustbesatisfied; hence,

d t' ?/ 77 ôy?

-

t' ?/ = d ta?/ Tf

: (/ 1+ y?2

-

()= ()4

34

2 The FirstVariation

#(z)= clz+ c2, whereczisanotherconstantofintegration.Since3/(0)= 0,weseethatcz= 0, and since3/(1)= 1,weseethatcl= 1.Thus,theonly extremaly isgiven by ylz)= z,which describesthe linesegmentfrom (0,0)to (1,1)in theplane (asexpected).Wehavenotshown thatthisextremalisin facta minimum. (Thisisshown in Example10.7.1.) Example 2.2.2: Let (zo,y(tl functionaldefined by

J(t v)The Euler-lsagrange equation forthisfunctionalis

#//y.# uuuz.

Thehomogeneoussolutionisyulz)= clcostz)+ czsintz),whereclandczare constants,and the particularsolution isyplz)= z.Thegeneralsolution to the Euler-lsagrange equation isthusgiven by

intz) ylz)= z - s sintl).

Exam ple 2.2.3: Let k denote som e positive constant and let J be the functionaldefined by X

J(t v)-

(t v/2- kyz)dz, 0

with endpointconditions3/(0)= 0and!/(r)= 0.lfyisanextremalforJ then

2.2 The Euler-luagrange Equation

d (2y,)+ 2ky=

0;

ylz)= clcos(VIz)+ czsin(VIz). Now 3/(0)= 0 impliesthatcl= 0,and !/(r)= 0impliesthatczsintvffr)= 0. lfsfk isnotan integer, then cz = 0,and the only extremalisy = 0.lfsfk is

an integer,then sintx/lr)= 0and cz can beany number.ln thelattercase wehavean infinitenumberofextremalsoftheform ylz)= czsintxf/cz). Exercises 2.2:

1.Alternative ProofofCondition (2.6): LetyCES andp CE. bIbeflxed functions.Then thequantity J(y+ 6p)can beregarded asa function of thesinglerealvariableE.Show thattheequation d. l (/6= 0 att î= 0leads

tocondition (2.6)underthesamehypothesesfor/. 2.T he F irst V ariation: Let J :S --+ J2 and K :S --+J2,be functionals defined by

where/ andg ares11100th f -unctionsoftheindicated argumentsand J2(; R.

(a) Show thatforany realnumbers4 and B, (i.e.,tîisalinearoperator),and

é

(' ?J

(' ?J

(7(JlA-llp,y)= (' ?JéJ(p,y)+ ( -)IL-éA-lp,y) ,

(a ttchain rule''fortheJoperator). 3.Letn,beany positiveinteger.Extend Lem m a 2.2.1 by showing thatthere

existsav CEC''(R)such thatzztzl> 0forallz CE(ct,/7)and v = 0forall z CER (ct,, d).

36

2 The FirstVariation

4.Let J be the functionaldefined by

with boundary conditions3/(0)= 0and 3/(1)= 1.Find theextremalts)in (72g 0,ljforJ. 5.Considerthe f-unctionaldefined by 1

J(t v)-

1

z4t v'2dz

(a) Show thatnoextremalsin C2g-1,ljexistwhich satisfjrtheboundary conditions3/(-1)= -1,3/(1)= 1. (b) W ithoutresortingtotheEuler-lsagrangeequation,provethatJ cannothave a localm inim um in the set

2.3 Som e Special C ases TheEuler-lsagrangeequation isa second-ordernonlineardifferentialequation, and such equations are usually difllcult to sim plify let alone solve.T here are,however,certain casesw hen this differentialequation can be sim plified. W e exam ine tw o such casesin this section.W e suppose throughoutthatthe functionalsatisfiesthe conditions ofTheorem 2.2.3. 2.3.1 C ase 1: N o E xplicit v D ependence

Suppose thatthe f-unctionalisoftheform 21

J(: v)-

/(z,y')dz, 20

where the variable y does not appearexplicitly in the integrand.Evidently, the Euler-lsagrange equation reduces to

whereclisa constantofintegration.Now t ' ?/ ç 'ly?isa known function ofz

and y?,so thatequation (2.14)isa first-orderdifferentialequation fory.ln principle,equation (2.14)issolvablefory?,provided t' ?2/ ( ')y?2: /:O 5so that equation (2.14)couldberecastintheform 5Onecaninvokeavariantoftheimplicitfunctiontheorem (AppendixA.2).

2.3 Som e SpecialCases

: (/= t ?(al,c1), .

forsom efunction g and then integrated.ln practice,however,solving equation

(2.14)fory?can proveformidableifnotimpossible,and theremaybeseveral

solutionsavailable.N onetheless,theabsenceofy intheintegrand sim plifiesthe problem ofsolving a second-orderdifferentialequation to solving an im plicit equation and quadratures. Exam ple 2.3.1:

The Euler-lsagrange equation forthisfunctionalleadsto the equation

01 == C*V/

ç ' ly?

1+ #/2

== c1?

whereclis a constantofintegration.Note that 1 3// 1+ : 4/21< 1 so that Icll< czo.Equation (2.15)can besolved fory?toget

Exam ple 2.3.2: G eodesics on a Sphere

lnExample1.4.1,letzt= 0andr = 4.Supposethatwechooset= zt,sothat weregard4 asafunction of0.Thearclength f -unctionalforthesphereisthen

where 4/denotesd4 d0.The integrand does not contain 4 explicitly,and therefore theEuler-lsagrangeequation gives

4/sin2p == C1, 1+ 4?2sin2p whereclisaconstant.Now,4?2sin40< 4?2sin20< 1+4?2sin20 andtherefore -

1< cl< 1.Hence,wecan replaceclbytheconstantsinct.Equation (2.17)

im plies

38

2 The FirstVariation

4/=

sinct 2

2

.

,

sin0 sin 0- sin ct

thus,

tan ct

costy+ , J)= tan0, or in C artesian coördinates, z cos, d - ysin, d = ztanct.

Equation (2.19)istheequation ofa plane through thecentreofthesphere. The geodesic corresponds to the intersection ofthis plane with the sphere' , hence,itm ustbe an arcofgreatcircle.

2.3.2 C ase ll:N o Explicit tr D ependence

Theorem 2.3.1 LetJ beaJ' unctionalo. ftlte/' t a?wz

I

I(y,t v')- y'ô:/ y?- /.

'

P roof: Suppose thaty isan extrem alforJ.Now,

and since y isan extremal,the Euler-lsagrange equation (2.9) issatisfied; hence,

2.3 Som e SpecialCases

ddz I1'(y,t v,)- 0. Consequently,. bI m ustbe constantalong an extrem al. N otethatthe f-unction . bI dependsonly on y and y?,and thustheequation

V (#,!//)= const. isaJrs/-orderdifferentialequation fortheextremaly. Exam ple 2.3.3: C atenary

Thecatenary problem (Section 1.2)hasa f -unctionaloftheform z1

J(t v)-J ïl1+y4 ,jya;. F 20

The above integrand doesnotcontain z explicitly and therefore

V2

1 -q #/2

-

J1'

whereclisaconstant.lfcl= 0,then theonly solution to equation (2.21)is y = 0.Supposethatcl, # 04thenequation (2.21)can bereplaced by (2.22)

c1c(z-cal/ct ==#-r #2 --c2 1,

40

2 The FirstVariation

therefore, = #V

2

2 C1

2

# - C1V

#+

:2 - c2 1

The extrem alsare thus given by I - C2

ylz)= clcosht

C1

).

Exam ple 2.3.4: B rachystochrone

Thebrachystochroneproblem (Section 1.2)hasafunctionaloftheform

The integrand doesnotdepend on z explicitly;thus,

Hly,y')- y'ô :/ - / y?

is constantalong an extrem a. l.lfy is an extrem alfor J then itm ustsatisfy the first-orderdifferentialequation

y(1+ : v/2)= c1,

wheresl= cl 2.Now ,

dy= --z lslcos' ? lsin' ? ldbb

2.3 Som e SpecialCases

dz = cot#dy= -4s1cosz. 4dbb =

-

2s1(1+ cos(2#))d' ().

Therefore,

z = sz- s1(2' ? )+ sin(2#)), (2.25) where s2 is an integration constant.Equations(2.24) and (2.25)provide a param etric solution to the problem .The solution curve isa well-known class

ofplanecurvescalled cycloids(Section 1.2). Thesimplification when / doesnotdependon y explicitly ismoreorless obviousfrom theEuler-lsagrangeequation' ,thesim plification when z isabsent

in /islessobvious.ln particular,whatleadsonetoconsideraf-unction such

as . bI in the firstplace? Equation (2.20)isan example ofa conservation

law :along any extrem al,the quantity . bI isconserved.ln problem s concerning classical m echanics,. bI often represents the totalenergy ofthe system . O ne can thus be led to consider a function such as . bI from the physics of whateverthe f-unctionalism odelling ifa conservation law isknow n.M athem atically,thisapproach is notvery satisfactory.O ne im m ediately questions whetherotherconservation lawsexistand ifthereare any otherspecialcases forthe integra. nd leading to conservation laws.ln fact,there are waysto deduceconservation law sm athem atically.N oether'stheorem providesa general fram ework in w hich to derive conservation law s.W e discuss this theorem in Chapter9. Exercises 2.3:

1.Find the generalsolution to the Euler-lsagrange equation corresponding to the functional 21

J(t v)-

/(z) 1+ y'2dz, 20

whereztl> 0,and investigatethespecialcases:(i)/(z)= xfz,(ii)/(z)= I.

2.Find the extrem a. lsforthe functionaldefined by # dI XG

where zo> 0. 3.Let

z3

,

2 The FirstVariation

2.4 A D egenerate C ase ln the exam ples so far,theintegrand ofthef-unctionaldependson y?in som e nonlinearway.lfthe integrand islinearin y?,the problem becom esdegenerate in a sense that is explained in thissection. SupposethatJ is a functionalofthe form zl

J(#)=jvv(X(t r,#)#t1-yqy,y))cy, d

' . ) ;'yttz'v-(y'' o 9A y+' 9B o y(-() . But

d

tg-zt

?DA

27yttz,y)- dz + y ôy , so thatthe Euler-lsagrange equation reduces to dd d z

-

OB = 0. oy

(2.26)

Notethatequation (2.26)isnoteven a differentialequation foryLitis an im plicit equation for y that m ay or m ay not have solutions depending on

thegiven functions4 and . B.Moreover,equation (2.26)containsnoarbitrary constants so that arbitrary boundary conditions cannot be im posed on any solutions.

ltmaybethatequation (2.26)issatisfied forallz and y;i.e.,ytz = By is an identity.ln thiscaseequation (2.26)placesnorestriction on y,butitdoes imply the existence ofa f-unction 4tz,y)such that4y = -4 and 4z = B.ln thiscase the integra. nd can bew ritten as

J= t g4 + y?t g4 = X d; t gz

ôy

dz

sothatJdependsonlyon4 andtheendpoints(zo,!/(zo))and (z1,!/(z1)).The value ofJ isthusindependent ofy,so thatthe integralispath independent.

6Equation(2.26)isawell-knownintegrabilitycondition(cf.(441,p.529).

2.4 A Degenerate Case

isindependentofthechoice ofy.A function 4 can befound by integrating theequationsB = 4zand -4 = 4y.Forexample4z= B = 2z!/;hence,

4 = z2y+ (7(y), 4#- ,2+ c?(?y)= -4 = z2+ : jy2, 4 = z2y+ y3+ k

J(y)- 4(z,,?/(z,))- 4(zo,?/(zo)) - z

lt vz+ yl- (zpt vo+ : ( /( )). '

(Notethatthearbitraryconstantkvanishesfrom thefinalanswer.) ln summary,variationalproblemswith integrandsoftheform dtz,yly?+ B (z,y)aredegenerateinthateitheryisdeterminedimplicitlyandcansatisfy onlyvery specialsetsofboundary data,orthevalueofthecorrespondingfunctionaldoesnotdepend on thechoiceofy.ln thelattercase thedeterm ination oflocalextrem a isvacuous. A n im m ediateconcern isthattherem ay beotherform sofintegrandsthat lead to path independent functionals.Thesefunctiona. lsare characterized by the property that the Euler-lsagrange equation reduces to an identity valid forallz and y in thespace underconsideration.Thenexttheorem showsthat in fact the integrand mustbe linearin y?for such an identity to bevalid.

Theorem 2.4.1 SnpposetltattlteJunctionalJ satishestlteconditionso. fTlte-

t arcm 2.2.3 andtltattlteEnler-Lagrangeequation (2.9)rccluccsto an identity. Tlten, tlte integrand m ' tzs/ be linear ïzzy?) and tlte valn,e o.ftlte jhnctionalïs independento.fy. P roof: lfthe Euler-lsagrange equation isan identity,then

( ' V t' ?2/ t#2/ / (92/ // u:uu y y () t' ?!/ ç 'lzoy? t ' A tgl/? (' )y?2 .

forallz CEgzt ),z11and y CES.Now,: (/??appearsonly in the lastterm on the left-hand side ofthe equation,and since equation (2.27) must hold for all

2 The FirstVariation

/(z,y,y')- yttz,yly'+ B(z,y),

2.5 lnvariance of the E uler-tzagrange Equation The principlesin physicsthatlead tovariationalform ulationsdo notdepend on coördinate system s.G eom etricalproblem s such as the determ ination of geodesics are likewise ttcoördinate free''in character.The path ofa particle, forinstance,does not depend on the coördinate system the observer usesto describe it;a geodesic does not depend on a particular param etrization of the surface.These types ofproblem s can be fram ed in term sofm axim izing functiona. ls and ultim ately lead to solutions to an Euler-lsagrange equation.

On physical(and geometrical)groundsonethusexpectstheEuler-lsagrange equation to also be invariant with respect to coördinate transform ations.ln thissection we take an inform albut practicallook at the invariance ofthe Euler-lsagrangeequation. A coördinate transform ation

z = ztzt,z?), y= ylu,z?), iscalled sm ooth ifthe f -unctionsz and y have continuouspartialderivatives with respectto ztand v.A s1100th transform ation is called nonsingular if the Jacobian 8(1,#)= det z,zy. a

t ' ?tu,z?)

alv yv

satisfiesthe condition

t ' ?tz,y)# 0 t ' ?tu,z?) .

(2.29)

H ere we use the notation z,z= t ' ?z ç' )u etc.for succinctness.Note thatcondi-

tion (2.29)impliesthatthetransformation isinvertible:to every pair (z,y) therecorrespondsauniquepair(u,z?)satisfjringequation (2.28).7W eassume thatthecoördinatetransformation defined byequation (2.28)issm00th and nonsingular. LetJ bea functionalofthe form 21

J(t v)-

/(z,y,t v')dz, 20

(2.30)

S = (!/CE(72gzt ),z1j:ylzç j)= yçj and !/(z1)= 3/1/., whereyç jand yïaregiven num bers.Suppose now thatwewritethe functional

in termsofthe (u,z?)coördinatesand,fordefiniteness,letusregard z?asa function ofzt.Then,

% d dz

-

dy du - l/,z+ yvl' ) dz du

zu + zvè'

and

ztl= ztztt),zo1,zl= ztul,z?1), yç j= yluçj,r0),yk= : v(' tz1,r1). Forclarity,let

and letT bethe setdefined by

Given acurveinthezp-planedescribed byafunction y= y(z),thetransformation (2.28)definesthecurveintheztzsplanedescribedbysomef-unction z?= z?(zt).Theessenceoftheinvaria. ntquestion is:ifz?CET isan extremalfor K ,isy CES and extremalforJ (and viceversal?Thenexttheorem resolves thisquestion.

Theorem 2.5.1 Lety CES andz?CET betwoJunctionstltatsatisj' ytltesmt ata//z

nonsingnlartransjbrvtation (2.28).Tlten y ïsan cz/rcmtzlJorJ z/andonlyzl / z?isan eztrevtalJorK . P roof: Supposethatz?CET is an extrem alforK .Then,z?satisfiestheEulerLagrange equation d( ' ?F (' )F X (% ôv = (). (2.a2) -

46

2 The FirstVariation

F(' tz,' u+ yv? ))(z, t?,i))= /(z(' tz,r),ylu,r),y. y z+ zv' t' ?l, zu + zv ,

so that

(' )F ç ' )è

=

t' ?/

t ' ? yu+ yvè

)s. zu+ zvè t' ?!/t(z.+ zvï?

and

( ')F ç' )v

' ?/ t ' ?/ t ' ?/ t ' ? yu+ yvè = t t '?zzv+ ç ' )yyv+ t ' ??/?s . zu + zvv

(zu+ zv' ll

A straightforw ard but tediouscalculation show sthat

?F - ( /d-f('ç ' )è 'ov)F - tol'?tzu,,yr))(l.'zd;-oyt'?/?- toy'?/). '

.

dt ' ?/

.

(2. 33)

t' ?/

X t ' A/ ç ' )y = 0, -

lt is philosophically reassuring thatthe path ofa particle isindependent ofthe observer'schoice ofcoördinates.There is also a practicalim plication: coördinate transform ations can be m ade in the f -unctionalbefore the EulerLagrange equation is form ulated.A n exam plesufllces to illustrate the value ofthisobservation. Exam ple 2.5.1: LetJ be thefunctionaldefined by

d

W

: 712+ y2 ?

1+ y?2 V

1+ y?2

-

y z2 + y2 = ().

z = z(4,r)= rcos4, y= y(4,r)= rsin4. Thistransform ation isevidently sm 00th,and since

the transformation is nonsingular,provided r , # 0.Now,suppose thatr is regardedasafunction of4,then / 3//+ !/r/ = rcos4+ sin4/ , # = z/+ zr/ -rsin4+ cos4/ so that

1+ y?2dz = . :2+ /2d$. The functionalJ thusbecom es

Nlrl=

/1

(,Ftr'/)d$'

.,

Theintegra. nd doesnotdependon 4 explicitly,and thereforethecorresponding Euler-lsagrange equation hasa firstintegral

?'.= r

c2. 1,.4

-

1,

48

2 The FirstVariation 51

W = sin(-24 + sz) = - sintz/lcoss2+ costz/lsins2 =

-

2sin4cos4cossz+ (2cosz4-1)sinsz.

ln term s ofthe originalC artesian coördinate system ,the above expression is equivalentto

1 zsinsz- 2zycoss2- Usinsz.

s = z

Exercises 2.5:

1.Change ofVariable: Let' ?û:(f(),f11--+R beasm00th f-unction on the interval)(),f11such that' ?)?(f)> 0 fora11tCE(f(),f11and let' ?)(fo)= zo, ' ? )(f1)= z1.Usingthetransformation z = ' ?)(f),thefunctionalJ defined by

where,forF(f)= y(' ()(t)),' frdenotes(/F dtand J7(f,1$f')--/(,9(f),1$f')?9?(f). Proveby directcalculation that

2.6 Existence of Solutions to the B oundary-v alue P roblem * ln thissection w ediscuss briefly and inform ally thequestion ofexistenceand uniquenessofsolutionsto theboundary-valueproblem associated with finding extrem als.Generally questions ofthisnature are difllcultto answereven for specific casesowing to tw o features.Firstly,the Euler-lsagrange equation is usually a nonlineardifferentialequation and thusdifllcultifnotim possibleto solveanalytically.Secondly,boundary-valueproblem sareglobalin character:

the solutionsmustbedefined on the entire intervalgzt),z11.ln contrastto

initial-value problem s,which are localin character,S there are few general results analogous to Picard's theorem g available. O ur discussion is lim ited prim arily to exam ples that illustrate som e of the pathologies ofboundaryvalueproblem s.An exam pleofageneralexistenceresultforcertain boundaryvalue problem s isgiven atthe end ofthissection. U nderthe conditionsofTheorem 2.2.3,thedeterm ination ofextrem alsfor a functionalJ ofthe form

with ztl< zl and given boundary values yç j,3/1,entails finding solutions to the Euler-lsagrange equation

dt ' ?/ t' ?/ =

27 ôy?

-

t a?/

()

(2.a9)

subjectto theconditions

!/(zo)= yç j, !/(z1)= yk. ln thiscontexta solution to the boundary-valueproblem isa function y such that:

(a) y CEC2(zo,zll; (b) ysatisfiestheEuler-lsagrangeequation (2.39)forallzCEgzt ),z11;and (c) ysatisfiestheboundary conditions(2.40). The definition of a solution can certainly be relaxed to include ttrougher'' functionssuch as piecewise s11100th functions,butw edo not pursue thisgeneralization and lim itourdiscussion to s1100th solutions. M uch ofthe discussion in the earlier sections ofthis chapter focused on

determining the generalsolution ylz,c1,c2)to equation (2.39).Even ifthe 8Initial-value problemsentailsolving a differentialequation subjectto conditions

oftheform ptzo)= po,p/tzo)= po /.Theconditionsaredefinedatthesamepoint zo and the solution need exist only in a sm allneighbourhood ofzo. 9 See Appendix A .3.

50

2 The FirstVariation

two-param eterfam ily off-unctionsthat com prisesthe generalsolution can be found,how ever,there isno guarantee that constantscland cz can be found such that

#(J 7l0,C1,C2)= : % , #(J7l1,C1,C2)= #1, fora given choiceofpoints (zo,y(tland (z1,3/1).ln fact,thereisno apriori reasonwhy ylz,c1,c2)need evenbeinthespaceCzgzt ),z1jforany particular choice ofconstants.ltm ay be that no solution exists to the boundary-value problem even though a generalsolution can be found to the Euler-lsagrange

equation.Attheotherextreme,equations(2.41)mayhaveaninfinitenumber of solutions for cl and or c2,and in this case the boundary-value problem would have an infinite number ofsolutions.Fuxam ples 2.2.1,2.2.3,and Exercises2.2-5 illustrate som e ofthe possible scenarios. Exam ple 2.2.1: T he generalsolution forgeodesicsin the plane is

#(z,c1,c2)= c1z+ c2, and given any setofpoints(zo,y(j),(z1,3/1) (such thatztl, # z1)itisclear thatthe f-unction

j vlalo #(z)= zyk1 -- yç I+ tvoazll1 -- tzt ztl l

isthe unique solution to the boundary-value problem . Exam ple 2.2.3: T he generalsolution to this problem is

ylz,c1,c2)- clcostx/tz)+ casintx/tz),

ylz)- casintx/lz) isa solution to theboundary-value problem .ln the above expression cz is an arbitrary num ber and hence there are an infinite num berofsolutions to the boundary-value problem . Exercises 2.2-5: T hegeneralsolution to the Euler-lsagrange equation isof the form

and

t(-1,0,c2)- -1 v : v(1,0,c2)= 14

A m ore involved butillum inating exam ple isafforded by the catenary.10 Exam ple 2.6.1: C atenary RecallthatthefunctionalJ defined by

with boundary conditionsylzçj) = yç jk 0 and !/(z1)= yï k 0 modelsthe shapeofa uniform flexiblecablesuspendedfrom a pole ofheightyç jto another pole ofheight 3/1,where the polesare a distance ofzl- zt lapart.The cable isassum ed to becoiled atthebaseofeach pole so thatthereisno restriction regarding the arclength ofcablebetween the poles. T hereare three im portantparam etersin them odel:theheightsyç jand 3/1, and the separation distance zl- z().W e can always norm alize the problem by assum ing thatthe separation distanceisone unit,say ztl= 0 and zl= 1. W e can then work w ith the param etersyçjand yï. R ecallfrom Exam ple2.3.3 thatthegeneralsolution totheboundary-value problem is

yv= slcoshtsz), 1 yï= slcosh - + s2 . r;1

Atthisstageletusspecialize(andsimplify)theproblem byassumingthat yçj= 1.A lthough this does not capture allthe possibilities,it doesdisplay the basic pathologies.W e thus look atthe availability ofsolutionsforvarious valuesofthe rem aining param eteryï > 0.Undertheassum ption thatyçj= 1, the above equationsim ply that

garding the existence and uniquenessofsolutionswereresolved only because the general solution was available explicitly.Typically,the Euler-lsagrange equation cannotbesolved and w edo nothavetheluxury ofknow ing the generalsolution beforeweinvestigatethesequestions.Even ifwe cannotsolvethe Euler-lsagrangeequation analytically,qualitativepropertiessuch asexistence and uniqueness ofsolutions to the boundary-value problem are nonetheless im portant.These properties test the veracity ofthe m odel especially w hen experim ent shows that a solution m ustexist.M oreover,the investigation of solution existence and uniqueness highlights any special param eter regions whereno solution orm ultiplesolutionsm ay exist.G enerally this type ofinvestigation providesusefulinform ation in preparation fora m oreefllcientnum ericalapproach to the problem . T here is,unfortunately,apaucity ofgeneralresultsconcerning boundaryvalue problem s involving nonlinear second-order differential equations,and

theresultsthatareavailable areoften fettered with numerousspecialtand usually complicated)conditions.ltiswellbeyond thescopeofthisbook to giveeven a briefoverview ofthevariousresults techniquesused toaddressexistence uniquenessquestionsforboundary-value problem s.lnstead,we leave

thereaderwith an ttolder''butusefulresultdueto Bernstein (71,whichwedo not prove.

Theorem 2.6.1 (Bernstein) Considertlteboundary-valn,eproblevttltatconsists o.fsol ving t/Jc equation

#V= FV,#,#/),

(2.43)

subjectto t/Jc bonndary conditions

!/(zo)= : % , !/(z1)= #1, wlterel/tland!/1aregiven realnnm bersandzt l# z1.Supposetltaton tlteset

J2= gzt ),z11x R xR tlteJunction F ïscontinuon, sandItascontinuon, sy' tzràïtzl derivatives' tl ï//zrespecttoyand!/?.SupposeJnrtltertltattltereezistsapositive constantp,suclttltat

t' ?F(z,y,!/?)> p, ( ' Ly

IF(z,y,: v/)1< yttz,yly'z+ B(z,y). Tlten,tltere ezistsprecisely oneJunction y suclttltatequations (2.43) and (2.44)aresatished.

2 The FirstVariation

R em arks' .

(a) W eneed continuity in a setsuch asJ2sinceyisunknown and hence its range asw ellasthatofy?is unknown.

(b) Althoughthetheorem doesnotstateitexplicitly,asolution requiresyto be atleasttw ice differentiable.Thism eans that y and y?are continuous

functionson theintervalgzt ),z11.SinceF iscontinuous,equation (2.43) impliesthat: (/??mustalsobecontinuouson theintervalgzt ),z11;i.e.,y CE (72g zl),z1j. ln closing,westressthatexistenceand uniquenessresultsfortheboundaryvalue problem do notnecessarily transferto the originalvariationalproblem , which is generally concerned w ith finding localextrem a.The catenary is an exam ple ofthis situation.The basic question concernsthe existence ofa local extrem um for a given functionalnot m erely an extrem a. l.Som e results concerning this question can be found in C arath4odory loc.cit.and Ew ing

(261.

Som e G eneralizations

3.1 Functionals C ontaining H igher-o rder D erivatives The argum ents leading to the Euler-lsagrange equation in Section 2.2 can be extended to functiona. ls involving higher-order derivatives.Naturally,the function spaces m ust be further restricted to account for the higher-order derivatives.Considera functionalofthe form 21

J(y)-

/(z,y,y',y'')dz, 20

alongwith boundary conditionsoftheform ylzçj)= yç j,!/?(zo)= y6,!/(z1)= 3/1,andy?(z1)= 3// 1.Hereweassumethat/hascontinuouspartialderivatives ofthe third orderwith respectto z,y,!/?,and : (/?? and thaty CEC4(zo,z11. The set S isthus

and the setLI isdefined by

Suppose that J has a localextrem um in S at y CE S.Proceeding as in

Section 2.2,let: ' 2= y+ t îr/and considerthedifference J(L)- J(y).Taylor's theorem im pliesthat

and consequently,

J(? ))-. J(y)-zj20 -,(w,t ç ?y ' /+w,?t ' ?t v .'+,??ô t ' ? y. t ')c u+0/2).

3 Some Generalizations

The firstvariation for thisf-unctionalis therefore zt o.f ( ' ?/

f tî' . llr/,#)= 20 r/t'?# + 07/t'?!? + r///t'o. ?!?/ dI.

dz

using the boundary conditionsp(zo) = 0 and p(z1)= 0,and so condition (3.1)reducestotheequation 21r/ t ' ?/ - d t ' ?/ + d2 ( ' ?/ dr = 0, 20 ç ' )y 2V ç' ly? dz2 ( ' ?: (/?? whichmustholdforallpCE. bI.Theintegrand /byassumptionhascontinuous

third orderpartialderivativessothatforany yCEC4(zo,z11theterm E(z)= t ' ?/ - yd o t' ?/t + d2 ( '?/ t ' ?!/

y

ts z oytt

mustbecontinuousontheintervalgzt ),z11.A suitablemodificationofLemma 2.2.2 (cf.Exercises2.2-3)canbeusedtoshow thatymustsatisfythefourthorder Euler-lsagrangedifferentialequation

d2 ( ' ?/ z2 ( ' ?: (/??

d

-

d (' ?/ + ( ' ?/ = (). W ( 'ly? t ' ?: l/

The above equation is a necessary condition for a f -unction y CE S to be an extrem alforthe f-unctionalJ.

3.1 FunctionalsContaining Higher-order Derivatives

Exam ple 3.1.1:

#(ï' &)(z)= g, which hasthe generalsolution 1 4

a

2

ylz)= Y pz + c1z + c2z + caz+ c4, wherethecksareconstants.Theconditions3/(0)= 0and3//(0)= 0implythat c4= ca= 0.Theconditions3/(1)= 1and3//(1)= 1implythatp 41+ cl+ c2= 1 and p 31+ 3c1 + 2c2= 14hence,cl= - 1 - p 12 and c2 = 2 + p 24.The extrem alisthusgiven by

y(z)-2 X4- 1+X) 12 z3+ 2+2 X) 4 z2. R esults such as Theorem 2.3.1 have analogues for f-unctionals containing higher-orderderivatives.Forthe second-ordercase,iftheintegrand does not contain y explicitly then itisplain thata firstintegralfortheEuler-lsagrange equation can be obtained,viz.,

d t' ?/ IV ç' ly'?

-

t' ?/ t' A /= const.

?/tt- y? d t ' ?/ t ' ?/ bI(y,: (/,!///)= : (//ot' # (yz o#tt- o#t - /= covtst.

.

Exam ple 3.1.2:

J(: y.y ?,),dz. v)- zt(1. , , # zo

Theintegrand defining J doesnotcontain y explicitly' ,therefore,any extrem al y satisfies the differentialequation

t ' ? ' ?!/ 'kt 2d 7( ' ? !/ ///h )j- tt ' ? //'' UC1' wherecl issom e constant.The integrand also doesnotcontain z explicitly, and so forany extrem al

3 Some Generalizations

wherecz isanotherconstant.Theaboveexpression can be recastin the form ??kïy?+ k, = !/ 1

(1+ y?z)z

,

wherellland k,areconstants.Thetw osim plificationsthusenableustoreduce

thefourth-orderEuler-lsagrange equation (3.3)to asecond-orderdifferential equation.Wecan solveequation (3.5)parametrically:let #?= tan. 4;

(/clcos' ?ûsin' ?û+kzcosz#)#?=1. lntegrating 130th sidesofthe above equation yields lll k, z = ks+ --(1- cos(2#))+ -jzf

1

?? ' û+ jsintz#l ,

where ks is an integration constant.Sim plifjring the above expression and using ll = 4s1,k,= 4s2,sa= ks+ lll 4 gives

z = sa+ 2sz' ? )+ szsintz' t))- slcostzkl. Equations(3.6)and (3.7)implythat dy = tan' ?ûdz = =

(2s2(1+ cos(2#))+ 2s1sin(2#))tan' ?ûdbb (2s1+ zszsintz#l- 2s1cos(2#))dbb;

hence,

y= s4+ 2s1' ? )- szcostzkl+ slsintzkl,

(3.8)

wheres4 isanotherintegration constant.Thesolution isthusgiven param et-

rically by equations(3.7)and (3.8). T he m ethodsused for integrandscontaining second-orderderivativescan be extended to integrands containing derivatives ofthe sàth order.W e leave as an exercise the proofthatthe Euler-lsagrange equation for a f-unctionalof the form

3.1 FunctionalsContaining Higher-order Derivatives

Exercises 3.1:

1.Find the generalsolution forthe extrem alsto the f -unctionalJ defined by

2.Conservation Law: Supposetheintegrand / defining the functional

J doesnotdepend on z explicitly.Provethatequation (3.4)issatisfied

along any extrem al. 3.Forthef-unctionalJ defined by

1

J(t v)-

0

y' 1+ (: v//)2dz,

find an extremalsatisfying theconditions3/(0)= 0,3//(0)= 0,3/(1)= 1, and 3//(1)= 2.

4.D egenerate C ase: LetJ bea functionalofthe form

where 4 and B ares11100th f-unctionsofz,y,and y?.ProvethattheEulerLagrange equation forthis f -unctionalisa differentialequation ofatm ost second orderand thatconsequently any solutions can satisfjratm osttwo arbitrary boundary conditions. 5.Let J and K befunctiona. lsdefined by

F(z,y,y',y'')= /(z,y,y',y'')+ dz (7tz,y,y').

60

3 Some Generalizations

6.Let J be a functionaloftheform

where yln' s denotes thesàth derivative ofy.

(a) Formulatethe flxed endpointvariationalproblem forthisfunctional and provethatany s11100th extrem alm ustsatisfjrtheEuler-lsagrange

equation (3.9).Noteanyassumptionson thefunction /and thefunction space.

(b) If/ isoftheform yttz,y,!/?,...,!/(rL-1)): 4 /(?z)+ . B(z,y,: ( /?,...,: 4/(rL-1)) whatisthem axim um orderthe Euler-lsagrange equation can be?

3.2 Several D ependent V ariables Variationalproblem s typically involve functionalsthatdepend on severaldependent variables.ln classicalm echanics,forexam ple,even the m otion ofa

singleparticlein spacerequiresthreedependentvariables(z(f),y(t),z(f))to describe the position ofthe particle at tim e f.ln this section we derive the Euler-lsagrange equations for functionals that depend on severaldependent variables and oneindependentvariable.

LetC12gf( ;,f1jdenote the setoff-unctionsq :(f(),f11--+R?zsuch thatfor q = (t ?1,q2,...,qnlwehaveqkCEC2gf(),f1jfork = 1,2,...,zz.ThesetC12gflj,f1j iSa.Vector Spafleal' lcla.l' lorm SLICIAa. S

11q11- k- mz,aax,...,-zssu-p I' ?k(')I g z z-j

can be defined on thisspace.A swith the single dependentvariablecase,the choice ofnorm really dependson the application. C onsidera functionalofthe form tl

J(q)-

to

Llt,q,ù)dt,

where 'denotes differentiation w ith respect to f,and L isa f-unction having continuous partialderivatives ofsecond order with respect to f,qk,and q' k, fork = 1,2,...,zz.G iven tw o vectors qo,ql CER '',the fixed endpoint prob-

lem consistsofdetermining the localextrem a forJ subject to the conditions

q(fo)= qoand q(f1)= q1.Here, S = (q CEC12gfl ),f1j:q(f())= ql l and q(f1)= qlj. A gain we can representa ttnearby''f-unction q n asa perturbation, m

q = (j+ (;vk,

wherep = (p1,w ,...,p,z).Forthiscase, bI= (p CEC2gf(),f1j:z7(f( ))= z7(f1)= Oj. .

consequently,

The firstvariation for thisf-unctionalis thus

éJ('?,

tt ,z

ogy

t j?rs .

q)-j,,, j) ()tpkoc,+bkog)dt. .

,

DJITI,tR)= 0 forallp CE. bI.

Condition (3.11)ismorecomplicated than itsanalogue(2.6)owingtothe presenceofn,arbitrary f-unctionsand theirderivatives,butjudiciouschoicesof functionsp CE. bI can be m ade to m ake the problem m ore tractable.Consider

thesetoffunctions. r.J1definedby.r.J1= ((p1,0,...,0)CEff)..Condition(3.11) m usthold for allp CEff1,and forany p CE.r.J1 thiscondition reducesto

Jzo tlqgj gô ( ' yq . r s.yg k jg( % ' y. r s( z yy.( ;. W e know from Section 2.2 that this condition leads to the Euler-lsagrange equation

)' it ' ?til ç'kï= 0, -

as a necessary condition foran extrem al.Evidently w ecan m odify theabove approach by selecting appropriatesubsets of. bI to argue thatifJ has a local extrem um at q then

62

3 Some Generalizations = 7/t ' Al ç' kï 0, -

X't ' Az ç'k, = 0, -

The above condition is a system ofn,second-order differentialequations for the n,unknown functionst 71,...,qn.Note that ifq satisfies thissystem then

condition (3.11)issatisfied foranyp CE. bI.ln summary,wehavethefollowing result.

wltereq = (t71,q,,...,qnl,andL Itascontinuon, ssecond-ordery't zràïtzlderivatives ' ttlï//zrespectto f,qk,and q' k,k = 1,2,...,n,.Let

S = (q CE(72gft ),f1j:q(f())= qt l a. nd q(f1)= qlj,

/@ô4k :vk = 0 -

Jork= 1,2,...,n,. Exam ple 3.2.1: Let

with q(0)= qo,q(1)= q1.TheEuler-lsagrangeequationsforthisfunctional correspond to the system

2 (ï,p)

1

q, - 2(2 - 2 -t. ?2= 0.

The characteristic equation forthis lineardifferentialequation is

1

2/. / - 2/ .:2 - = 0, which hasroots

Thegeneralsolution toequation (3.16)istherefore wherethecksaredetermined by theboundaryconditionsq(0)= q(),q(1)= q1.Thefunction t 71(f)can bereadily deduced from qnlt)by useofequation (3.15). T he specialcases detailed in Section 2.3 can also be extended to several dependentvariables.ln particular,ifL does notdepend on texplicitly itcan be shown that

along any extrem al. Exam ple 3.2.2: T he fam iliar equations of m otion for a particle can be

derivedfrom Hamilton'sPrinciple(Section1.3).Letq(f)= (t 71(f),q,(t),t ?a(f)) denote the C artesian coördinatesofa free particle ofm ass?rzat tim e f.The kinetic energy ofthisparticleis

T(q, ù)- j 1. wtt it+ . 1/+ /). LetU(f,q)denotethepotentialenergy.TheLagrangian is

TheEuler-lsagrangeequations(3.13)giveimmediatelytheLagrangeequations ofm otion,

64

3 Some Generalizations

:U

' tvtkk= -w-, Uqk

fork= 1,2,3.Recallfrom Section 1.3thatthekthcomponentofforce,A on the particle is given by

X

=

-

( ' ?U

0qk. H ence,the Euler-lsagrange equations im ply N ew tonts equation

wherea = ( ' 2 iistheaccelerationandf= (/1,A,/a)istheforceontheparticle.

Forthisexam plenote thatifthe potentialenergy U'does not depend on tim e explicitly then neitherdoesL.ln thiscase,wehave theconservation law

(3.17),whichgives

Exercises 3.2:

Llt,q,ù)- 1 2(t ia ,+t ia a)-gq,, where g isa constant.

(a) Find theextremalsforthef -unctionalJ defined by

3.3 Two IndependentVariables' e

4.Let

s = stf,q) t '?s

'' t' ?s

&(f,q,ù)- t' ?f+k=1 S ôqk4k. (a) ProvethattheEuler-lsagrangeequations(3.13)forthefunctional

aresatisfied forany s1100th functiony.(Thisisthedegeneratecase.) (b) Let tl J(q)- te Llt,q,il)dt and

3.3 T w o lndependent V ariables* This book is concerned prim arily with f-unctionalsw hose integrands contain a single independentvariable.W e pause here,how ever,to discussbriefly the firstvariation forfunctiona. ls defined by m ultiple integrals.W e focus on the sim plestcase w hen the integrand containstw o independentvariables. LetJ2bea sim ply connected bounded region in 1 :.2with boundary (' ?. Q and

closureL = :. 62LJtî.Let(72(J-))denotethespaceofallf-unctionszt:L --+R such thatzthascontinuous derivativesofsecond order.Consider a functional

J :(72(. 0)-+R oftheform J(z')-

e

/(z,y,'. ',r,q)dzdy,

u

wherep = ztz,q = uy,and / isa s11100th function ofz,y,zt,p,and q. An analoyue ofthe flxed-endpoint variationalproblem is to find a f-unction

ztCEC2(J2)such thatJ isan extremum subjectto a boundary condition of the form

z'tz,y)- z'otz,y), (z,y)e ôtî,

(3.19)

66

3 Some Generalizations

W e can approach this problem in the sam e m anner as the single inde-

pendent variable case.Suppose that ztis an extrem al for J subject to the

boundarycondition (3.19),and let Z(z,y)= ' tzlz,y)+ 6r/lz,y). Here,t îisasmallparameterand p CEC2(. 0).In addition,itisrequired thatfl satisfy theboundary condition (3.19)andhence Tl(1,#)= 0

/(z,y,fl,:,1)- /(z,y,z'+ t îp,r+ 6pz,q+ Evly) /(z,y,u,r,q)+ t î '?:/ t' ?.?.t+ '?z:/ y:/ oq ç ' )p + vl + (?(62),

jj. ( )tn' ( t+wo ' gp î+n,: ,q 7)dzo-o, forallp CE(72(J'))satisfjring condition (3.20).Lemma 2.2.2 can be generalized to accom m odate m ultiple integrals.A sw ith thefixed-endpointproblem , how ever,w e need to elim inate the derivatives ofthe arbitrary f -unction from

condition (3.21).

G reen'stheorem states that

//J 2(7t+' l' y)d-'' vforanyfunctions4,' ?û:L --+R suchthat4,' ? ),4z,andbbyarecontinuous.Let 4

:/

= pôp,

3.3 Two IndependentVariables' e

Here,P -*z denotespartialdifferentiation holding (only) y flxed,and o -*y denotespartialdifferentiation holdingzflxed.Condition (3.20)impliesthatthe boundary integralis zero;therefore,

Condition (3.21)thusimplies

//,-n), 1(: . t)+' o ty(' o' q)-' t)d-',-o. Equation(3.22)mustbesatisfiedforarbitraryp,andthecoefllcientofpin theintegrandisa continuousf-unction sinceztCEC2(. 0)and / issm00th.W e can thusinvokea generalization ofLem m a 2.2.2 togetthe necessary condition

t g t g z(ô t 9p /( j.yj(t 9/( A j.ô t gu /.( ;. Equation (3.23)isasecond-orderpartialdifferentialequationfortheunknown functionzt,whichmustalsosatisfytheboundarycondition (3.19).Thisdifferentialequation isthe analogueofequation (2.9)4itisalsocalled theEulerLagrange equation.

Exam ple 3.3.1: LetJ2be the disc defined by : 712+ y2< 1, and 1et

(3.24) uölz,y)= 2z2- 1 t' ?zu ( ')2u + =0 t' ?z2 (' )y2

(3.26)

(Laplace's equation).lfJ hasan extremum atztCEC2(. 0),then ztmustbe a solution tothepartialdifferentialequation (3.26)andsatisfytheboundary condition (3.25).Thereadercan verifjrthatthefunction zttz,y)= : 712- y2is a solution to thissim ple problem .

68

3 Some Generalizations

Exam ple 3.3.2: Letr :J2--+1 :.3bea function ofthe form

rlz,y)= (z,y,' tzlz,!/)).

J(zt)=jj 1+p2+q2dzdy. Supposeweconsidertheminimalsurfaceproblem (Section 1.4),whichconsists offindingaminimum forJ subjecttoboundaryconditionsoftheform (3.19). G eom etrically,the problem entails finding a surface that can be described

parametrically in theform (3.27)such thatthesurfacecontainsthe (closed) space curve y described by rtl::. 62 --+1 :.3 where

rotz,y)- (z,y,z'otz,y)), and thesurface area ism inim um com pared to others11100th surfacescontaining the spacecurve y.TheEuler-lsagrange equation forthisproblem reduces to

(1+ pzlt- 2w s+ (1+ qzlr= 0, where r=

t' W tt

( '72tt

t' ?z2

( 'lzç' )y

, s=

, t=

(' 72tt

.

( 12

Them ean curvatureofasurfacedescribedparametricallyin theform (3.27) isgiven by

' FI= (1+ pzlt- zpqs+ (1+ qzlr 2(1+ 372+ :2)3/2 , so thatsolutionsto the m inim alsurface problem are characterized geom etrically by the condition

lfJ hasan extrem um atzt,then ztm ustsatisfjran equation ofthe form

ytr+ 2. Bs+ Ct+ D = 0,

(3.30)

where -4,. B,C,and D are f-unctions of the variables z,y,zt,p,q.The EulerLagrangeequation isthusa quasilinearsecond-orderpartialdifferentialequation fortheextrem alzt.Boundary-valueproblem sinvolvingsuch equationscan be exceedingly difllcultto solve and basic questionsconcerning the existence and uniqueness ofsolutionsfor specific problem s can be difllcult to answer. The boundary conditions for these problem s play a centralpart in the solution m ethod,and there are concerns here that do not m anifest them selves strongly in the one-variable case such asw hetherthe problem iswell-posed.

3.3 Two lndependentVariables' e

A w ell-posed boundary-valueproblem hasa unique solution,and the solution isstablewith respectto sm allperturbations ofthe boundary conditions. W e eschew a generaldiscussion on w ell-posed boundary-value problem s.

Sufllceittosaythatthematteriscomplicated especiallyforquasilineartand fullynonlinear)partialdifferentialequations.Thereaderisreferred tostandard workssuch asGarabedian (301and John (421fora fuller introductory account.

ln som e cases,itispossible to classify the Euler-lsagrange equation,and then generalresults concerning the class ofequation can be exploited.The

differentialequation (3.30)iscalled: (a) hyperbolic,ifAC - . B2< 0' , (b) parabolic,ifAC - . B2= 0' , (c) elliptic,ifAC - . B2> 0. The classification is based on the existence ofa specialclassofcurvescalled characteristics on the integralsurface.Roughly speaking,a characteristic is a curve on the integralsurface along w hich the differentialequation and the initial boundary data do not determ ine allthe second-order derivatives uniquely.H yperbolic equations have integralsurfaces with two realfam ilies of characteristics.Parabolic equations have integral surfaces w ith only one characteristic.Ellipticequationshaveintegralsurfacesw ith no realcharacteristics.The presenceofcharacteristicsinfluencesstrongly the typeofproblem forw hich thedifferentialequation iswell-posed.T he type ofboundary-value problem considered in thissection is called a D irichlet problem .lt iswell know n that Dirichletproblem sinvolving hyperbolic partialdifferentialequationsareill-posed.ln contrast,Dirichletproblem sare generally well-posed for elliptic partialdifferentialequations. ln general,thecoefllcients-4,B ,and C depend on thevariablesz,y,zt,p,q, so thatan Euler-lsagrangeequation need notfitintoany ofthecategoriesm entioned.The signs and m agnitudes ofthese coefllcientscan change,an equation m ay be hyperbolicatsom e pointsin J2and elliptic atotherpoints.M ore im portantly,the coefllcients depend on the solution itself.The classification really depends on the equation,the dom ain,and the solution.N onetheless, there are cases w here the equation can be classified w ithout know ing solutions.lfthe coefllcientsare a11constants,forexam ple,then the classification

dependspurelyontheseconstants.Laplace'sequation(3.26)isclearlyelliptic' , thewave equation, r - t = 0,

isclearlyhyperbolic.Thereadercan alsoverifythatequation(3.29)iselliptic. GilbargandTrudinger( 341discusstheDirichletproblem forquasilinearelliptic partialdifferentialequationsofthis type in som e depth.

3 Some Generalizations

3.4 T he lnverse P roblem * The variational form ulation of a boundary-value problem has som e advantages.For exam ple,in Chapter5 we show how one can exploit the isoperim etric problem to approxim ate eigenvalues for Sturm -lsiouville problem s.ln Chapter 8,w e show how variationalproblem s lead to Ham ilton's equations and theHam ilton-lacobiequation,w hich m ay besolvablethrough separation

ofvariables.lnaddition,Noether'stheorem (Chapter9)providesasystematic algorithm forfinding conservation law sforvariationalproblem s.1T hese and

otherfeatures(e.g.,Rayleigh-ltitznumericalmethods)makeitattractiveto identifjra given differentialequation asthe Euler-lsagrangeequation ofsom e functiona. l. G iven a differentialequation

#?? F(z,y,?y?)= 0, -

theinverse problem isto determinea f-unction /(z,y,!//)such thaty isa solution to(3.31)ifand onlyifyisasolutiontotheEuler-lsagrangeequation dt 9/ t9/ 2V % ? tg?/ = 0. -

ln this section we discuss briefly and inform ally som e qualitative aspects of the inverse problem . Letusfirst considerthe generalsecond-orderlineardifferentialequation

y??+ Py?+ Qy- G = 0,

(3.33)

whereP,Q,andG arefunctionsofz.W eknow from thetheoryofdifferential equationsthat such equations can beputin an equivalent self-adjoint form

(>'#/)/+ qïl- # = 0, where

2

p = exp

P31)t/$ , 20

A quickcomparisonwith equation (3.32)showsthatequation (3.34)isequivalentto theEuler-lsagrangeequation for

ln thismanner,weseethatthegenerallinearequation (3.33)can alwaysbe transform ed intoan Euler-lsagrangeequation.W e discussthisrelationship for Sturm -lsiouville problem sin m ore detailin Section 5.1.

1ln fact, there are versions ofN oether's theorem that do not require a variational

formulation.AncoandBluman g 2j,g3jdescribethealgorithm.

3.4 The lnverse Problem ' e

y''Jy'y'+ y'hy'+ hy'- Jy- 0, and ?/??can beeliminated from theaboveequationusing (3.31)togive Fl' y/y'+ VJyy'+ /zp/- h = 0. Equation (3.35)canberegardedasasecond-orderpartialdifferentialequation forthe function /.From a practical standpoint,the above equation is of lim ited value owing to the paucity of m ethods for solving such equations.

Fortunately,itispossibletotransform equation (3.35)intoafirst-orderpartial differentialequationforli= Jy/y'.Differentiating130thsidesofequation(3.35) with respectto y?gives

Fy'Jy/y'+ Fly/y/y'+ y'Jyy/y'+ Jzy/y'= 0' ,

(3.36) Thereisa generalm ethod forsolving first-orderpartialdifferentialequations, the m ethod ofcharacteristics,that entails solving a system offour ordinary differentialequations.W e do not go into thism ethod here,but sim ply note

thatitcan beusedtoshow thatsolutionstoequation (3.36)exist,zand hence the generalsecond-order nonlinearequation (3.31)does have a variational form ulation. T he inverse problem for system s of second-order differential equations poses a m ore form idable problem .Fortunately,there is a result that helps characterizesystem sthathave variationalform ulations.Let

denoteasystem ofn,second-orderdifferentialequationsforq = (t 71,...,qn), and let

Aj(f,q,il,ik)= Ej(L),

(3.37)

forj = 1,...,zz,isthatA satisfiesthefollowing integrability conditions,3

2Wecould alsoappealtoresultssuchastheCauchy-lfowalevskitheorem ((301)if F is analytic in z,p,and p/. 3 These conditions correspond to the requirem ent thatthe Fréchet derivative ofA

beself-adjoint(cf.(571,p.355).

3 Some Generalizations

forj,k = 1,...,zz.Relations (3.38)are called the Helm holtz conditions. lfthe Aj satisfjrthe Helm holtz conditions,then it can be shown that the function L defined by 1 n

Llt,q,ù)= 0 / )()qkzkklt,$q,. i' tl,. i'ik)dl k 1 =

satisfiesequation (3.37).TheHelmholtzconditionsarediscussed in moredetailin (571. N otethatfailure oftheH elm holtz conditionsdoesnotprecludethepossibility ofa system having a variationalform ulation.A lthough theseconditions

precludedirectrelationshipssuchas(3.37),itmaybethatthereisamultiplier m atrix B ,forexam ple,such that

Here,B isa nonsingular n,x n,matrix with entries bij = bij(f,q,($.For example,considerthesimplecasen,= 1,dtz,y,!/?,!/??)= : (/??- F(z,y,!/?).For thiscase,the Helm holtz conditions reduce to the condition Fy'= 0.But we

know thatallthesecond-orderequationsoftheform (3.31)haveavariational formulation.Supposenow thatweintroduceamultiplierB = B (z,y,!/?)and apply theHelmholzconditionsto B (!/??- F).Then,theHelmholtzcondition reducesto

F By'+ y'B y+ B.+ Fy'B = 0, .

which isthesameasdifferentialequation (3.36).

T he determ ination of a m atrix B such that B A satisfies the H elm holz condition iscalled the tm ultiplierproblem ''in the calculusofvariations.The

difllculties and conditions on the bij escalate substantially for n,k 2.The readercan find a sum m ary ofthe problem ,generalizations,and furtherresults

in themonograph by Anderson and Thompson (41.

Isop erim etric P rob lem s

Variationalproblem s are often accom panied by oneor m oreconstraints.The presence ofa constraint further lim its the space S in which w e search for extrem als.Constraintsm ay be prescribed in any num ber ofw ays.Forexample,one m ight require the functions q CES to satisfjran algebraic condition, a differentialequation,or an inequality.O ften there are different w ays to im pose the sam e constraint.ln this chapter w e discuss problem s that have isoperim etric constraints.Problem sthathavealgebraic equations ordifferentialequations asconstraints arediscussed in Chapter6.

4.1 T he F inite-llim ensional C ase and Lagrange M ultipliers ltisusefulto investigatea sim plefinite-dim ensionalexam pleofa constrained optim ization problem to gain som e insight into the infinite-dim ensionalcase. M oreover,the theory underlying the Lagrange m ultipliertechniqueforvariationalproblem s restson thatfor finite-dim ensionalproblem s.ln thissection we review Lagrange m ultipliersfor finite-dim ensionaloptim ization problem s. 4.1.1 Single C onstraint

Considertheproblem ofdetermininglocalextremaforaf-unction /:1:. 2-+R subjecttothecondition thatthevaluesof/aresampled on acurvey ( :zR2. ln otherwords,determinethepointson y atwhich / hasalocalextremum relativetovaluesof/ sampled atnearbypointson y.Thisproblem isinherently one-dim ensionalin character,butthe approach to locating theextrem a really depends on the constraint used to define y.W e assum e for sim plicity

that/ isa s1100th function and thaty isas11100th curve. T here are m any ways to define a curve.Suppose,for exam ple,that y is defined param etrically by som e f-unction r :f --+ R2, where f f;l R is an interva. l,and fortCEf,

4 lsoperim etric Problem s

r(f)= (z(f),y(t)). Then we can build the constraintdirectly into the problem by constructing

the function F :f --+ R defined by F(f) = /(z(f),!/(f)).Given that the param etrization issm 00th,anecessary condition fora localextrem um attis

d17(t)= :/z?(f)+ :/y?(f)= 0. Now,z(f)andylt)areknownandthust ' ?/ t' ?zandt ' ?/ t ' ?zareknownfunctions off.ln principlewecan thussolvetheaboveequation forthevaluesoft(if any)thatmakeF an extremum.Notethata specialcaseoftheparametric representation isthe ttgraphical''representationsr(z)= (z,!/(z))and r(!/)= (z(: v),y). A curvemay bedefined implicitlyby an equation oftheform glz,y)= 0. lfg isa s11100th f -unction and Tg , # 0,then in principlewecould solvethe

equation foroneofthevariablesand proceed asdescribed above.lIn practice, how ever,finding an explicit solution m ight not be possible or convenient. M oreover,even ifgiss11100th forallvaluesofz and y,theresultingsolution for

z orymaynotbe.Consider,forexample,theequationglz,y)= z2+: (/2-1= 0 thatdescribestheunitcirclecentred at (0,0).lfwesolvethisequation for, say y,wegetylz)= 1- z2,and y isnotsm00th atz = +I.Yetanother concern with thisapproach is thatitoften leadsto an artificialdistinction of dependentvariables.ln m any problem sin geom etry and physicsthevariables are allon the sam e footing and itisnotdesirable to m ake such a distinction forthe purposesofanalysis. A n eleganttechnique thatavoids the problem ofdirectly solving im plicit equationsinvolvestheintroduction ofaconstantcalled aLagrangem ultiplier.

Thetechniquehasasimplegeometricalinterpretation.Supposethat/ and g ares11100th functions.Wewish to find a necessary condition for/ to havea localextremum subjectto the constraint

1(1,#)= 0.

1

..

W e suppose furtherthat

% .qlz,y)# 0.

(4.2) The equation (4.1)defines a curve y implicitly,and since Tg , # 0 the

curve is sm ooth' ,i.e.,y has a w ell-defined unittangentvector ateach point thatvaries sm oothly along y.This m eans that locally y can be represented

parametricallybyas11100thvectorfunction r(f)= (z(f),!/(f)),tCEf,suchthat r?(f), # 0 foralltCEf.A necessary condition for/ to havea localextremum on y at(z(f),y(t))is 1lfoneofthe derivativesisnonzero, then wecan use the im plicitfunction theorem to assert the existence ofa solution to the equation.Unfortunately the theorem does not actually provide a m eans ofobtaining the solution.

4.1 The Finite-Dim ensionalCase and Lagrange M ultipliers

,f)-F:/ 2J7/(z(f),v(f))--: y/ :y v,(f)--0. z z( d

:# ?

ôg t

70v(z(f),: v(f))- oz z (f)+ oy y (f)- 0, .

foralltCEf.Equation (4.2)impliesthatatanypointon y atleastoneofthe derivativesç' )g t' ?z,ç' )g t' ?!/isnonzero.Supposefordefinitenessthatç' )g t ' ?!/, # 0.

Then equation (4.4)impliesthat

and consequently equation (4.3)can bereplaced by z?(t) t' ?/

V )t ' ? zô t ' ?y.-t ô ?y ' /t ç ' )z ? g)-0. ...

Now r?(f) = (z?(f),y?(t)) # 0 so thatz?(f)and y?(t)cannot 1 30th be zero; hence,equation (4.5)precludesthepossibility thatz?(f)= 0.Equation (4.3) thusreducesto the condition

t' ?/t ' ?j. t' ?/t ' ?g = 0, ....

-

t' ?zç' )y

t ' ?!/t' ?z

which isequivalentto thecondition

V/A Tg = 0,

where A denotes the exterior (cross) product. v w CE1 :.2

I v Aw l= IvIIw Isin4, where4 isthe anglebetween v and w.Equation (4.6)indicatesthatV/ is parallelto Tg atan extremum (i.e.,4 = 0).Since V/ and Tg areparallel, thereisaconstant,LsuchthatV/= hs'g.Thenecessarycondition (4.3)thus reducesto thecondition

V(/- hg)- 0. The constant,Liscalled a Lagrange m ultiplier.

ltisevidentgraphicallythatV/isparalleltoTgatan extremum.Figure 4.1depictslevelcurvesof/ and thecurvey.Theconditionsong ensurethat y doesnothave any discontinuities or ttcorners,''and since / is s11100th /

hass1100th levelcurves.Suppose that/ hasan extremum on y at(z,y).lf thelevelcurve of/ through thepoint (z,y)intersectsy transversally,then / isincreasing decreasingalongy at(z,y)and hence(z,y)willnotyield an

4 lsoperim etric Problem s

extremum for/ on y.Thelevelcurveof/ through (z,y)mustthereforebe tangentto yat(z,y)andconsequentlytheunitnormalto ymustbeparallel to theunitnormaltothelevelcurveat(z,y).ln otherwords,V/ isparallel to Tg at(z,y). Undertheabove conditions,if/ hasan extremum subject to condition

(4.1),then equation (4.7)mustbe satisfied.Thisvectorequation provides two scalarequationsfor the three unknow n quantitiesz,y,and à.Equation

(4.1)providesthethird equation. Exampl e 4.1.1: Findthelocalextremaforthefunctiondefinedby/(z,y)= z2 y2subjectto theconditionglz,y)= : 712+ y2- 1= 0. Equation (4.7)impliesthat -

V (z2-y2-à(z2+ y2-1)j= (), z(1+ 1)= 0, : v(-1+ 1)= 0. Thefirstequation indicates thateitherz = 0 or, L= - 1.Supposethatz = 0. Then the second equation im pliesthat eithery = 0 or,L = 1,butz = 0 and the condition z2+ y2- 1 = 0 im plies that y = + I. Thus,there are critical

pointsat(0,1)and (0,-1).Supposeinstead thatz , # 0 and , L = -1.Then the second equation im pliesthat- 23/= 0,i.e.,y = 0,so thattheconstraint

impliesthatz = +I.Hencetherearecriticalpointsat(1,0)and (-1,0).

4.1 The Finite-Dim ensionalCase and Lagrange M ultipliers

T he Lagrangem ultiplier technique can be adapted to problem sin higher dim ensions.For exam ple,to find the extrem a for a function of the form

/(z,y,z) subjectto a constraint ofthe form glz,y,z) = 0,we can form thefunction F = /- hg,where, Lisan unknown constant,and look forsolutions to the three equations given by VF = 0,where V is the operator

(t' ?t ' ?z,t ' ?0y,t ' ? 0z).Theconstraintprovidesthefourth equation fortheunknown quantitiesz,y,z,and à.Thisapproach isvalid provided Tg , # 0.ln sum m ary,wehavethe following result.

Theorem 4.1.1 (Lagrange M ultiplierRule) LetJ2( J R?zbearcg/t?zzand let/ :J2--+R andg :J2--+R besmtat a//zJunctions.IJ/ Itasa localcz/rcm' tzm atx CEJ2 subjectto tlte condition tltatt y(x)= 0 andz/V. t y(x)# 0,tlten tltere .

ïs a zz' t zm scr,L suclttltat

X7(/(x)- lp(x))- 0.

4.1.2 M ultiple C onstraints

Letx = (z1,zz,...,z,zland let/ :J2--+R beas11100th function defined on a region J2f;lR''.lfn,> 2 itispossible to impose more than one constraint. Supposethat?rz< n,and considertheproblem offinding the localextrem a of

/ in J2subjecttothe?rzconstraints

#k(X)= 0, wherek = 1,2,...,?rzand thefunctionsgk:J2--+R aresm 00th.Forthesim ple

casewheren,= 2 and ?rz= 1,wesaw that/ and g sharethe sametangent line atan extrem um .ln higher dim ensionsthe analogue ofthis condition is

thatthetangentspace(hyperplane)of/ ata criticalpointx iscontained in the tangentspace defined by the gk atx.G eom etrically,thism eans thatthe

normalvectorV/(x)liesin thenormalspaceNglx)spanned by thevectors Vt yk(x).lntermsoflinearalgebra,thevectorV/(x)islinearly dependenton thesetofvectorsLs' gk(x),k= 1,2,...,zrz)..Thus,if/hasa localextremum .

atx,then there existconstants àl,,Lz,...,à,,zsuch that

4 lsoperim etric Problem s

Theapproachisvalidprovided V/(x)islinearlydependentontheV. t yk(x), and thiscondition leadsto thegeneralization ofthecondition Tglz,y), # 0. LetM (x)bethen,x ?rzmatrix

and letM y(x)betheaugmented matrix

Mg( x)-(Mty x)j. ThelineardependenceofV/ isassuredif RankM

r(x)< RankM (x).

.

Condition (4.10)providesthe analogue ofthe gradient condition (4.2).ln sum m ary,wehavethe following extension ofTheorem 4.1.1.

Theorem 4.1.2 LetJ2(= R?zbearegion andlet/ :J2--+R andgk :J2--+R be szrztat a//zJunctionsJork = 1,...,zrz.f// Itasa localcz/rcm' tzm atx CEJ2

subjecttotlte?rzconstraintstltatgk(x)= 0,andz /ineqztality (4.10)issatished atx,tlten tltere ezistconstantsàl,, Lz,...,à,,zsuclttltat

Exam ple 4.1.2: Find the localextrem a forthe f-unction defined by

/(x)= z2 a 2- zlzz subjectto theconditions

v1(x)- z2 1+ za- 1= 0, v2(x)- zl+ za- 1= 0.

. .

Here,n,= 3and?rz= 2.Equation (4.9)producestheequations z2 + 2à1z1+ . :2= 0, I1+ . X1= 0, za - , à. 2 = 0, thatalong with the constraints provide five equations forthe five quantities z1,z2,za,àl,and , Lz.This system ofequations has the tw o solutions w =

4.1 The Finite-Dim ensionalCase and Lagrange M ultipliers

M(w)=(-120 11 0j, which hasrank 2.Theaugm ented m atrix is -

M /,(w)=

210

1 01 , 0 12

andsincethedeterminantofM y(x)iszero,wemusthavethatRankM y(w)< RankM (w).A similarcalculationindicatesthatcondition (4.10)isalsosatisfied forthesecond solution.Hence,if/hasanylocalextremaunderthegiven constraintsthen they m ustoccurateitherw orz.

4.1.3 A bnorm al P roblem s

TheLagrangemultipliertechniquebreaksdownifcondition (4.2)(orcondition (4.10))isnotsatisfied.Thetechnique,however,can beadapted tocopewith these cases. W e consider here only the optim ization problem offinding the localex-

tremafora f-unction / oftwo independentvariablessubjectto a singlecon-

straintg = 0.We assume (asalways)that / and g ares11100th f-unctions.lf (z,y)isa localextremum forthisproblem and Tglz,y), # 0,then we have theexistenceofa number,Lsuch thatV (/(z,y)- hglz,y))= 0.Wecalla problem ofthistypenormal.ln contrast,if(z,y)isalocalextremum forthe problem and Tglz,y)= 0,then theexistenceofa Lagrangemultiplierisnot assured.Thistype ofproblem iscalled abnorm al.

lfglz,y)= 0and Tglz,y)= 0 thentheimplicitf-unction theorem cannot be invoked to deduce thatthe equation g = 0 can besolved uniquely forz in term s ofy orvice versa.Geom etrically,this m eans thatthe set ofsolutions

to g = 0 need notform a s11100th curve in a neighbourhood of(z,y).This doesnotmeanthatthecurvemusthavesomesingularityat(z,y)sothatthe

tangent vector to the curve is not well-defined,only that it is a possibility. Various nasty thingscan happen to ttcurves''defined by an im plicit relation when the gradient vanishes.For exam ple,it m ay be that the curve has a ttcorner''or a cusp at thispoint.A nother possibility is thatthe curve has a

self-intersection,orthattwo distinct solution curvesintersectat (z,y).An even more degenerate possibility isthat (z,y) represents an isolated point

in the set of solutions to the equation.ln these cases it is clear that the geom etricalargum ents leading to the existence ofa Lagrange m ultiplier are notapplicable.Thefollow ing barrageofexam plesillustratesthesepathologies.

1

*

@ 1

K-4 1

1

82

4 lsoperim etric Problem s

(4.15) (4.16) But the equation z2 + y2 = O has only one solution, viz.,(z,y) = (0,0), and for this solution there is no , L such that equations (4.15) and (4.16) are satisfied.Theproblem is abnormalbecauseTglz,y)= (2z,2y)= 0 at the only candidate for optimization (z,y) = (0,0).Technically,the func-

1- 2àz = 0 1- 2hy = 0.

tion / hasan extremum atthispointundertheconstraintg = 0 because thereareno otherchoices.ln thisproblem / ispassiveand playsno rôle in the optim ization process:the constraintdictatesthe criticalpoint.Note that

V/(0,0)- (1,-1)# 0. W ecan adaptT heorem 4.1.1 to include the abnorm alcase by introducing

an additionalmultiplier,L().Suppose that/ has a localextremum at (z,y) subjectto theconstraintg = 0.Let

/J= hltj+ ,Ll. t y.

lfTglz,y), # 0 then theproblem isnormal.Hencewecan choosezïtl= 1and use Theorem 4.1.1to show thatthereisa z ïlsuch that

V/àtz,y)- V (/(z,y)+ l1. v(z,y))- 0. Supposenow thattheproblem isabnormalsothatglz,y)= 0andTglz,y)= 0.Then wecan salvagethecondition V/Jtz,y)= 0 byrequiring that àoV/(z,y)= 0. lfV/(z,y), # 0,then we mustchoosezïtl= 0.Theotherconstantzïlin this caseisnotdetermined.lfV/(z,y)= 0,then any choicesofzïtland z ïlwill sufllce. Exam ple4.1.5illustratesthecasewhere z ïtl= 0.lfwe m ustchoosez ïtl= 0,

then thefunction / doesnotparticipatein the optimization.Example4.1.4 illustratesthecasewhereV/ and Tg are 1 30th zero and weareatliberty to choose any values for z ïtland àl. T he abovediscussion show sthat,forany scenario,wecan alwaysfind two

numbers, L(),z ïlsuch thatatleastoneofthem isnonzeroand V/Jtz,y)= 0. W e sum m arize this form ally in the nexttheorem .A sim ilar extension can be m ade to Theorem 4.1.2.

Theorem 4.1.3 (Extended M ultiplier Rule) Let J2 ( J R?z be a region and let/ :J2 --+R andg :J2 --+R be smtat a//zJunctions.IJ/ Itas a local cz/rcm' t zm atx CE J2 subject to tlte condition tltat t y(x) = 0 tlten tltere are .

nnm bers, Llpand ,à. 1 not30th,zero suclttltat

4.2 The lsoperim etric Problem

4.2 T he lsoperim etric P roblem

where/ isa s11100th function ofz,y,and y?.Theisoperim etric problem consistsofdeterm ining the extrem als ofJ satisfying boundary conditionsof the form

!/(zo)= #0, !/(z1)= : v1

where g is a given function of z,y,and y?,and L is a specified constant.

Conditionsoftheform (4.19)arecalledisoperimetricconstraints.zInthis section wederiveanecessary condition forafunction to bea s1100th extrem al to the isoperim etricproblem .

Suppose that J has an extremum at y, subject to the boundary and isoperim etric conditions.W ecan proceed asw ew ould for the unconstrained

problem and consider neighbouring f-unctionsofthe form : ' 2= y + 6p,where

p CE(72gzt ),z1jand p(zo)= p(z1)= 0,buttheconstraint(4.19)complicates m attersbecause itplaces an additionalrestriction on the term tîr/and thereforeresultsthatarebased on thearbitrary characterofpsuch asLem m a 2.2.2 arenotvalidw ithoutfurtherm odifying thefunction space. bI.lfw eproceed in

thismannerwewillhaveto determinethe classoff-unctionsin . bI such that: ' è satisfiesthe isoperim etric condition.W e can avoid this problem by introducinganotherfunction and param eter.W ethusconsiderneighbouringfunctions ofthe form

b = #+ 61071+ 62072, (4.20) wherethe6ksaresmallparameters,pk(z)CECzgz()?zlj,andpklzt))= pklJ rll= 0 for k = 1,2.Roughly speaking,theintroduction ofthe additionalterm 62072 can be view ed as a ttcorrection term . ''The function pl can be regarded as

arbitrary,butthe term 62072mustbeselected so that: ' 2satisfiesthe condition

(4.19). 2 Literally, theword isopevim etvic meanssam eperim eter.The m ostfam ousisoperi-

metricproblem isDido'sproblem (Section 1.4),wheretheconstrainttook the form ofa specified arclength.lndeed many isoperim etricproblem shave arclength constraints.Theusage oftheterm tt isoperim etric constraint''in the literature has

simplycometomeanconditionsoftheform (4.19)ofwhicharclengthisaprominent example.

84

4 lsoperim etric Problem s

Even with the introduction ofthe extra term (ï nvtz,itis not im m ediately obvious that we can always choose an arbitrary pl and then find an appropriate term to m eet the isoperim etric problem .Consider,for exam ple,the constraint

alongwith theboundaryconditions3/(0)= 0and 3/(1)= 1.Thereisonlyone s11100th f-unction thatwillmeetthisconstraint,viz.,the f-unction ylz)= z, and thereforetherearenovariationsoftheform (4.20)available(apartfrom :' è= y).Thissituation arisesbecause thechoice L = , /2 -happensto bethe minimalvaluethef-unctionalfcantake.Notethatylz)= z mustalsobean extrem alfor the functionalf.Extrem als such asthe above one that cannot be varied owing to theconstraintare called rigid extrem als. A lthough rigid extrem als are a concern,it turnsoutthatfor the isoperim etricproblem they have a tractable characterization.C onsiderthe quantity

Fora fixed choice ofpk wecan regard f(L)asa f -unction oftheparameters 61,62,say f(L)= S(61,62).Sinceg isa s11100th function wehavethatE is also a s11100th function.M oreover,ifJ has an extremum aty subject to the

boundary and isoperimetric condition,we have thatS(0,0) = L.We can appealtotheimplicitf-unctiontheorem toassertthatfor11611- maxtl6ll,I621) sufllciently smallthere existsa curve62 = 62(61)(or61= t 51(t52))such that S(61,62(61))= L,Provided VS # 0

at(0,0).Thus,ifyisarigidextremal,thenVS = 0at(0,0).Wereturntoan

interpretation ofthis condition laterin thissection.Forthe present,we shall

supposethatcondition (4.21)issatisfied sothatweavoid rigidextremals.

R atherthan usetheTaylorseries approach ofChapter 4,itiseasier here to convertthe problem to a finite-dim ensionalconstrained optim ization problem as discussed in Section 4.1.Suppose that y is a sm 00th extrem alto the

isoperimetric problem and thatcondition (4.21)issatisfied.Then there are neighbouring functionsofthe form (4.20)which meet the boundary conditions(4.18)and theisoperimetriccondition (4.19),whereplisan arbitrary function.

Thequantity J(L)can beregarded asa function oftheparameters61,62. LetJ(L)= 8(61,62).SinceJ hasan extremum aty subjectto theconstraint f(y)= L,the f -unction @ must havean extremum at (0,0)subjectto the constraintS(61,62)- L = 0.Theresultsoftheprevioussectionindicatethat foranycriticalpoint(61,62)thereisaconstant,Lsuch that V (&(6z,6a)- l(, 8'(6z,6a)- L))- 0.

4.2 The lsoperim etric Problem

t % 1 (t?(6z,6a)- lEE(6z,6a)) E=c,--0, and t ' ?62 (t?(6z,6a)- lEF(6z,6a)) E=t )--0.

(4.23)

N ow ,

and integrating by partsw esee that

t' ? 8(

zl

t ' ?/

d t' ?/

t %1 61,62) j,ploy-z: yoy,dz. =

,.(,

Sim ilarly,w ehave that t ' ? ' )g d t ' ?. S(61,62) = zlpl çyq t/ l E-c, zo y - y;-oy,

dz.

Equation (4.22)can thusbewritten 21

Jzo (yd.( ' $ ?y /'.t ( ' ?: / v.z j(yd.% ( ' ?g'.. ( t . ' ? . y : v. (;4.;y.(;. yy ,

The function pl isarbitrary and Lem m a 2.2.2 im pliesthat d :F

DF

X t l ? ôy = 0, -

where

F = /- hg.

The extremaly musttherefore satisfjrthe Euler-lsagrange equation (4.24). Theconcernnow isthatequation (4.23)mightoverdeterminetheproblem.ln fact,thesameargumentsusedtoderiveequation (4.24)leadtotheexpression

Jzo zl62Wdt o Xy/-o 0y : 4 /V*U'0' which isalwayssatisfied foranyw provided equation (4.24)issatisfied.

86

4 lsoperim etric Problem s

and

Jzo z1V1u o t ' ?#-W(yt o ' ?# g/V*'' U0 Jzo z162u o ( ' Ly-W( y( o ' Ly o?V*'U0* d t ' ?t y d .

ç ' )g -

zt' A/ ç ' )y

=

0.

The latterequation isautomatically satisfied ifequation (4.25)issatisfied. Hence thecondition thatVS = 0 att î= 0 reducesto condition (4.25),and this m eans that y is an extrem al for the functionalf.Rigid extrem als for the isoperim etric problem are thus characterized as functions that are also extrem als forthe f-unctionaldefining the isoperim etric condition. ln sum m ary,we have the follow ing result.

Theorem 4.2.1 SnpposetltatJ Itasan cz/rcm' tzm atyCE(72gzt ),z1jsubjectto tlteboundary conditions (4.18)and tlte ïscwcrïzrzc/r/c constraint(4.19).SnpposeJnrtltertltaty ïsnotan cz/rczrztzlJortlteJunctionalI.Tlten tltereezists

a constant, L suclttltaty satishesequation (4.24). ln lightofthe above theorem ,the isoperim etric problem reduces to the

unconstrained fixed endpoint problem with / replaced by F.The general

solution to theEuler-lsagrangeequation (4.24)willcontain twoconstantsof integrationalongwiththeconstantà.Theboundaryconditions(4.18)andthe constraint(4.19)providethreeequationsfortheseconstants.lnthissense,the isoperim etric problem is m ore com plicated than the unconstrained problem of Section 2.2.A nother com plication with the isoperim etric problem is the possibility of rigid extrem als. To validate the m ethod we m ust verifjrthat

thesolution to theEuler-lsagrangeequation (4.24)isnotalso a solution to equation (4.25),i.e.,an extremalforf. lfy isan extrem alforJ subjecttotheisoperimetriccondition f = L,and y isnotan extrem alforf,then theproblem iscalled norm al.The ttnorm ality''

ofthisproblem isinherited from condition (4.21),which indicatesthatthe finite-dimensionalproblem ofdetermining the localextrema for @ subjectto

the condition E = 0 isnorm al.ln the sam e spirit,ify isan extrem al forf,

thenVS(0,0)= 0andtheproblem iscalledabnorm al.Becausewecanrelate

the isoperim etric problem back to a finite-dim ensionaloptim ization problem , we can readily extend T heorem 4.2.1 to cope w ith abnorm al problem s by introducing an additionalm ultiplier,L().

4.2 The lsoperim etric Problem

Theorem 4.2.2 SnpposetltatJ Itasan cz/rcm' tzm atyCE(72gzt ),z1jsubjectto tlteboundary conditions(4.18) andtlteïscwcr/zrzc/rïc constraint(4.19).Tlten tltere ezisttwo zz' tzm ôcrszït land z ïl not30th,zero suclttltat d :AX

27 ôy?

-

DK = t a?/ 0,

wltereK = hltj- , Ll. t y.f/y ïsnotan eztrevtalJorI tlten wemtz?/takezïtl= 1. f/y ïsan eztrevtalJorI tlten wetakezït l= 0,unlessy ïsalso an eztrevtalJor J.In tltelatter caseneitlterzït lnorzïl is determined. Exam ple 4.2.1: C atenary Considerthecatenary problem discussed in Section 1.2 and Exam ple2.3.3,but now suppose thelength ofthecableisspecified.Thisleadsto an isoperim etric problem .Forsim plicity,let zt l= 0,zl = 1,and letthe poles be ofthesam e height /J> 0.W e thus seek an extrem alto thefunctional

subjectto

1

L(#)=

o

1+ #/2dz = Z,

and theboundary conditions3/(0)= 3/(1)= h,.HereL > 1denotesthelength ofthe cable.

Theextremalsforfconsistoflinesegments(Fuxample2.2.1).SinceL > 1, nosolutiontotheEuler-lsagrangeequation (4.24)thatsatisfiestheboundary and isoperim etricconditionscan be an extrem alforf.T hus,ifJ has a local m inim um at y,then Theorem 4.2.1 im plies that y isa solution to equation

(4.24)with F = (y- ,L) 1+ y?2 .

N ow ,thef-unction F doesnotcontain z explicitly;hence,we havethefirst integral

(cf.Section 2.3).Letzt= y- à.Then ' tz/= y?,andtheaboveequation reduces to

88

4 lsoperim etric Problem s

utzl= clcosh

z - cz cl

,

wherec2 isa constant.The extrem alsto this problem arethusofthe form

(4.26)

#- l = slcoshtsa) and

therefore,

The isoperim etric condition im pliesthat

The isoperim etric condition thusreduces to L=

zslsinh

1 , 2s1

uponusing equation (4.29).Letï= 1 2s1.Equation (4.30)isequivalentto L1= sinhtï). Equation (4.31)is evidently satisfied for ï = 0,butthissolution corresponds to an infinite value for sl and thus produces the f -unction y =

à-hcoshto)= ctmsf.,whichcannotbeasolution totheisoperimetricproblem.

SinceL > 1,however,therearepreciselytwononzerosolutions1%and -jto equation (4.31)(seefigure 4.4).W ealwayshave two solutionsto the equationsgenerated by the boundary conditionsand theisoperim etric constraint.

Forthenonzerosolution 1,we havethatsl= 1 2ï,s2= -1,and therefore

equation (4.27)yields

ëw

90

4 lsoperim etric Problem s

curvedescribed by y isL > 2,and thearea enclosed by yand thelinesegm ent

(-1,11ofthez-axisisan extremum.(Thisisa simplified version ofDido's problem.) Thearea underacurvey:(-1,11--+R isgiven by

and the arclength ofthe curve described by y isgiven by

W ethusseek an extremum forJ subjectto the constraint f(y)= L.Note that the extrem als for f are line segm ents.Since L > 2,no solution to the

Euler-lsagrangeequation (4.24)satisfjringtheboundaryandtheisoperimetric conditionscan also be an extrem alforf.

lfJ hasan extremum aty underthe constraintf(y) = L,then y isa solution to equation (4.24)with F = y- ,L 1+ y?2.The Euler-lsagrange equation isthusequivalentto the equation

(4.33)

Equation (4.33)indicatesthatthecurvedescribed by y mustbeofconstant curvatures = 1 111' ,i.e.,thecurvemustbean arcofacircleofradius I ll. A notherway ofdeducing the shapeofthe extrem alcurve isto note that F doesnotcontain z explicitly and hence the quantity . bI = y?Fy/- F m ust be constantalong any extrem a. l.Therefore,

(y+ c1) 1+ y'2= , L.4 ?

V. W e thus have that

:2

(y+ c1)2. j..

4.2 The lsoperim etric Problem

l àay +(ycl:-c,)a(yV. z .ycz,

p

-

wherecz isa constant.Let

y+ cl= ,Lsin4.

Equations(4.36)and (4.37)aretheparametricequationsforacircleofradius Illcentred at(-ca,-c1). T he extrem a. lsarethusofthe form

(z+ ca)2+ (y+ c,)2- l2.

(-1+ c2)2+ ct= 12 and

L = 2IlII41, where 4 denotesthe angle between the p-axisand the line containing the

points(0,-c1),(1,0)(figure4.5).lntermsoftheconstantc1,theisoperimetric condition and equation (4.38)imply that L=

1

2 c2 1+ 1arctan(). cl

The condition thaty isa function (i.e.,single-valued)on (-1,11placesthe som ewhatartificialrestriction that clk 0,so that the centre ofthe circle is not above the z-axis.This in turn places a restriction on L for solutions of

thistype.ltiseasy to see geometrically thatwe musthave 2 < L < r under these circum stances.W ith these conditions it can be argued geom etrically

(and analytically)thatequation (4.41)hasauniquesolution forclin terms ofL,and thatequation (4.38)hasauniquepositivesolution forà.W erevisit thisproblem in Exam ple 4.3.3,w herewe liftthe restriction thaty be singlevalued. T he Lagrange m ultiplier,Lplaysa seem ingly form albutusefulrôle in the solution ofisoperim etricproblem s.Exam ple4.2.2showsthat,Lcan correspond

92

4 lsoperim etric Problem s

to a physically geom etrically significant param eter in the problem and this prom ptsus to look a bit deeper into the rôle ofà.The f -unctionalJ can be written in the form

Suppose that J has an extrem um at y.T he general solution to the corresponding Euler-lsagrange equation w illdepend on zo,z1,yçj,3/1,and L .The Lagrangem ultiplier, Lwillalsodepend on theseparam eters.Supposenow that theboundaryconditionsarefixed.T hen wem ay regard J asa function ofthe param eterL.Now

U'wlzo zt(( ' ?F ) y.Xdt ( , ' ?! F //yjO ( r )L y( yz.y( ' ?Z àqy.. yz -z otoqv,y,yt)t yai yj.ya jy where the firstintegralon the lastline was derived by integration by parts. Sincey is an extrem alforthe problem ,thefirstintegralon thelastline m ust vanish.The second term on the lastline vanishesbecause y m ustsatisfy the isoperim etric constraint.W e therefore havethat OJ OL = à.

4.2 The lsoperim etric Problem

The Lagrange m ultiplier therefore corresponds to the rate ofchange ofthe

extremum J(y)with respecttotheisoperimetricparameterL. W e note a certain duality that exists forthe isoperim etricproblem .Sup-

posethat, L, # 04then any extremaly totheproblym with F = / - hg must also be an extremalto a problem with G = g - hJ,where , $ = 1 à.More specifically,supposethaty minim izesJ subjectto theisoperimetricconstraint

f(y)= L.LetK denotetheminimum valueJ(y).Then K = J(y)- àtf(y)- L), and thus

L - I(y)- , $(Jy)- K' 4. W ehavethatJ - hl = -à(f- zsJ) and thisindicatesthatthe minimum forthef-unctional<20 (V3F dz correspondsto themaximum forthe functional ( V 3 G d z. A s i mi l a r s t a t e m e n t c a n b e ma d e i f y p r o du c e s a ma x i mum for J J20

subjecttof(y)= L.W ethushavethefollowing result. Theorem 4.2.3 Snppose tltaty produces amïzzïmw,m mtzzïmw,m valn,eJor

J subjecttotlte constraintI(y)= L and tltat, L, # 0.LetK = J(y).Tlten y produces a mtzzïm' t zm mïzzïm' t zm JorI subjectto tlte constraintJ(y) = ff, andI(y)= L. ln view oftheabove result,supposew erevisit,for exam ple,the catenary problem ofFuxam ple 4.2.1.W e saw thatthe catenary isthe curve alongw hich

the potentialenergy is an extremum subjectto the condition thatthe cable isoflength L.ln fact,itcan be show n thatthepotentialenergy ism inim um

alongacatenaryfortheappropriatechoiceofj.Theorem 4.2.3showsthat, fora fixed valueofpotentialenergy,thecatenary isthecurvealong which the arclength ism axim ized. T he duality relationship also helps to elucidate the condition that y not be an extrem alforf in Theorem 4.2.1.lfy is an extrem alforf,then in the

dualproblem , $= 0.Thismeansthatf(y)= L,independentoftheconstraint J(y)= K ,so thatK can beprescribed withoutchanging theextremum for f.Alternatively,if,L = 0,then J(y) = K,independentofthe constraint f(y)= L,so thatthe problem doesnotdepend on the constraint.At any rate,if, L isnotfinite orif, L = 0 the problem isdegenerate. Exercises 4.2:

'

z(? 7)-jvydz. Find theextremalsforJ subjecttotheconditions3/(0)= 0,3/(1)= 2,and f(: v)- L.

94

4 lsoperim etric Problem s

1

'T

J(r)= è*

''T

,.2d0, J(r)= . ()

,. . 2+ r?2d0, ()

wherer?= dr d0.FindanextremalforJ subjecttotheconditionsr(0)= 0,r(r)= 0,and f(r)= L > 0.

3.Let J and f bethe f -unctionalsdefined by

I(y)= intjyzdz. Supposethaty isan extremalforJ subjecttotheconditions3/(0)= 1, 3/(1)= 2,and f(y)= L. (a) FindafirstintegralfortheEuler-lsagrangeequationsforthisproblem and show thatforL = 3,

ylz)= 4- 3(z- 1)2. (b) ForL = 7 3 show thatthereexistsa linearf-unction thatisan extrem alfortheproblem .

(c) ForL = 5 2show thatthisproblem admitsthesolution ,L= 0.Find theextrem alcorresponding to thisvalue.

4.Letyt(!/)beas11100th f-unction and let

and

Form ulate the Euler-lsagrange equations for the isoperim etric problem

with 3/(0)= 0,3/(1)= 1,and f(y)= L > V2.Show that,L= 0,and that thereare an infinitenum berofsolutions to theproblem .Explain without

usingtheEuler-lsagrangeequations(oranyconservationlaws)why there

m ustbe an infinite num ber ofsolutionsto thisproblem . 5.Let y be the extrem alto the catenary problem ofExam ple 4.2.1.Show

thatforL sufllciently largethereisan z CE(0,1)such thatylz)< 0. 4.3 Som e G eneralizations on the lsoperim etric P roblem ln this section we present som e m odestgeneralizations on the isoperim etric problem discussed in Section 4.2.M ostofthedetailsare leftto the reader.

4.3.1 P roblem s C ontaining H igher-o rder D erivatives

Suppose thatJ and f aref-unctionalsofthe form

where/ and g aresm00th f-unctions.Thesameanalysisused intheprevious section can be used to show that any s1100th extremalto J subject to the

isoperimetricconstraintf(y)= L mustsatisfytheEuler-lsagrangeequation d2 :F dz2( A ??

-

d :F

ôF

X( :?/?+ ôy =

(),

(4.42)

where,forsom e constantà,

F = /- hg. The existence ofthe constant , L isassured provided y isnotalso an extrem al forthefunctionalf.lndeed,itisstraightforward to prove an analogousresult for functionals containing derivatives of order higher than two.A bnorm al problem scan betreated in am annersim ilartothatused forthebasicproblem in Section 4.2. Exam ple 4.3.1: Let

2V % ? tg?/ = 0, -

but ç ' )g t ' 93/? = 0 and ç' )g t' ??/ = 1,so that f has no extrem als.Any s11100th

extremaltotheproblem mustthereforesatisfyequation (4.42),whichreduces to

2: 4/1f' &)(z)- ,L= 0. The above differentialequation hasa generalsolution ofthe form

96

4 lsoperim etric Problem s

wherethecksareconstantsofintegration.Theboundary conditions3/(0)= 0 and 3//(0) = 0 imply thatctl= 0 and cl = 0,respectively.Atz = 1,the boundary conditionsyield the equations

à

u

c.

V + c4 + o3 = 1. .

The solution to this linear system ofequationsis , L = 6!,c2 = 30,and ca = 60.The extrem alisthus -

4.3.2 M ultiple lsoperim etric C onstraints

21

Ikly)-

gklz,y,y')dz, 20

fork= 1,2,...,111..Here,weassumethatthef -unctions/ and gkaresm00th,

and thatsome boundary conditions!/(zo)= yç j,!/(z1)= yïare prescribed. A generalization oftheisoperim etric problem consistsofdeterm ining theex-

tremalstoJ subjecttothe?rzisoperimetricconstraintsfk(y)= Zk,wherethe Lk are specified num bers.The Lagrangem ultipliertechnique can beadapted to this type of problem ,butthe analogue ofthe condition thaty is not an extrem alforthe isoperim etricf-unctionalislesstractable.W e discussthecase zrz= 2.

Supposethaty isan extremalforJ subjecttotheconstraintsf1(!/)= Lï and %(y)= L,.ln ordertomeet130th constraintsand stillhavean arbitrary term in ourvariation ofy we usea neighbouring function ofthe form

wheret î= (61,62,6a)and p = (p1,w,pa).ln addition,werequire thatpk CE (72g zt),z1jandp(zo)= p(z1)= 0.Thereisstilltheproblem ofrigidextremals. W edo notenterinto a generaldiscussion ofthisproblem .lnstead,weusethe conditionsdeveloped atthe end ofSection 4.1forfinite-dim ensionalproblem s with m ultiple constraints.

Asin the single constraintcase wecan regard JIL),f1(?)),and f2(?))as

functionsofE.Let

0(6)= and for k= 1,2 let

Sk(6)= lfy is an extrem alfor the problem ,then 0 is an extrem al for the f-unction

0 - subject to the constraints Ek = Zk.W e know from Section 4.1 thatthere existconstantszïl and zïz such thatthe criticalpointm ustsatisfy thevector equation

V (0 -- , L1=$ E% - ,L2ELt) 6=:0 = 0, oe

z.

o.f

(/ o.f

ç ' kjE=c, jzoo oy-s.oy,c fz -

()-

( ' )Ek = ()= 21vv 0gk - d t ' Vk dz, t' 90 E-c, zo ç ' )y W ç ' ly?

so thatequation (4.43)producesthethreeequations

Jzo zt* o t ' ? y #-W( yt o ' ?# y/V*U'0' where

F = / - ,Ll. t yl- hngLt. N ow we can regard theterm 6lplasan arbitrary f -unction with theterm st îzr/z and 6apaused as tcorrection''term sso thatthe constraintsare m et.W e can

thusapply Lemma 2.2.2 to equation (4.44)with j = 1,and thisgivesthe Euler-lsagrangeequation d( ' ?F d

(' )F -

zt ' A/ ç ' )y

=

0.

Aswith thesingleconstraintcase,theotherequationsforj = 2,3aresatisfied

automaticallyifequation (4.45)issatisfiedregardlessofwhatf -unctionsthepk mightbe.Thegeneralsolution to equation (4.45)willcontain twoconstants

98

4 lsoperim etric Problem s

ofintegration along with the constantszïl and ,Lz.The boundary conditions and the isoperim etricconstraintscan be used to determ ine these constants. T he condition that ensures the existence of constants àl, , L2 such that

equation (4.45)producesan extremaltotheproblem islesseasy tointerpret than thesingle constraintcase.W e know from Section 4.1 that thiscondition

translatestoinequality (4.10).Forthisproblem wehavethat M (0)= VS' 1(0) = c.llc.12ctla VS' 2(0) c.21ct22ctaa , where

M

M (0)

ctlltA12tA13

f(0)-(v( , y)(())- y oy , a)y oy , az ay oy , a, s,

.

zlTlj oj - (y oj? dI.

t73.j=

,

20

t' ?!/ dz t' ?!/

The Lagrange m ultipliertechnique w illbe valid provided there exists11100th functionspk such that:

(a) r/k(t ro)= Tlk(al1)= 04 (b) y+ (6,p)satisfiestheisoperimetricconstraintsfor11611small;and (c) RankM /,(0)< RankM (0). Exam ple 4.3.2:

#/2dz, and

f1(?/)L(y)Find theextrem als for J subjectto the constraintsfl= 2,f2 = 1 2 and the

boundaryconditions3/(0)= 3/(1)= 0. Let

y'= y?2- ,à1y - ïgzy.

.

The Euler-lsagrange equation forthischoice ofF is 23///+ , à1+ ,Lzz = 0, .

which hasthe generalsolution l2z3 à1z2 y = - 6 - 4 + c1z + c( ;?

wherect landclareconstantsofintegration.Theboundarycondition3/(0)= 0 impliesthatct l= 0,and theboundary condition3/(1)= 0yieldstheequation zïl . :2 cl = -4 + -6 .

The isoperim etric constraintsprovide the equations

2 - - -J.

+

-

i=

-

,

16 - 30 + 3.

is a solution to the Euler-lsagrange equation thatsatisfies the isoperim etric and boundary conditions.N ote that,forarbitrary p, 1 ct2.j =

1rljdI, o

and

4.3.3 Several D ependent V ariables

tl

J(q)f(q)-

te tl te

Llt,q,ù)dt, glt,q,ù)dt,

4 lsoperim etric Problem s

whereq = (t71,q,,...,qn),'denotesd dt,andL andg ares1100th functions. lfqisas11100th extremalforJ subjecttotheboundaryconditionsq(fo)= qo, q(f1)= qland theisoperimetriccondition f(q)= f,andq isnotan extremal forf,then there existsaconstant, Lsuch thatq satisfiesthen,Euler-lsagrange equations d (' ?F ( ' )F

7/A j t'kj = (),

(4.46)

-

wherej = 1,2,...,n,,and

Exam ple 4.3.3: Let us revisit the problem ofdeterm ining a curve y of

length f > 2 containing the points P-1 = (-1,0) and P1 = (1,0) such that the closed curve form ed by y and the line segm ent from P 1 to P 1 enclosesm axim um area.W eshow thatany s1100th extrem alforthisproblem m ustcorrespond to a circular arc,butwe liftthe restriction thaty m ust be described by a scalarf-unction y. -

Supposethatq(f)= (z(f),!/(f)),tCE(f(),f11isan extremalfortheproblem.

G reen's theorem im pliesthat the area under the curveis

and the isoperim etriccondition is

d

)-@ d

27 W e thereforehavethat

-1Jl

1

1

+1

1

:2+ :2 q'y - j'#= 0, -

-l#

:2+ :2 ' l'z + ' l'Jl= 0.

101

hence,theextremalmustbeacirculararcofradius, Lwithcentreat(c1,co1.lt follow sreadily from theboundary conditionsthatcl= 0.Theotherconstants require m ore effort,butcan be obtained essentially asdescribed in Exam ple 4.2.2.N ote that now the constant c2 m ay be negative orpositive depending on f.

Exercises 4.3 :

where z = z(f),y = y(t),and ?denotes d dt.Suppose that(z,y)isan extremalforJsubjecttotheconstraintf(z,y)= K,whereK isapositive constant.Provethatneitherz(f)norylt)can beidentically zero on the interval(f(),f11and thatthereisa constant. 4 such that , 2

2 3/2

z=.4(z +y) .

Find theextremalsforJ subjectto theconditionsf1(!/)= trl,%(y)= i' Lt and theboundary conditions3/(0)= 0,3/(1)= 1.

Thispage intentionally Jc.# blank

A pplications to E igenvalue P rob lem s*

Eigenvalue problem s infest applied m athem atics.These problem s consist of

finding nontrivialsolutionsto a lineardifferentialequation subject to boundary conditionsthatadm it the trivialsolution.The differentialequation containsan eigenvalue param eter,and nontrivialsolutions existonly forspecial values ofthis param eter,the eigenvalues.G enerally,finding the eigenvalues and the corresponding nontrivialsolutionsposesa form idable task. C ertain eigenvalue problem s can be recastas isoperim etric problem s.1ndeed,m any ofthe eigenvalue problem s have their origin in the calculus of variations.Although the Euler-lsagrange equation is essentially the original differentialequation and thusoflim ited valueforderiving solutions,thevariationalform ulation ishelpfulfor extracting resultsaboutthe distribution of eigenvalues.ln thischapterwediscussa few sim ple applicationsofthevariationalapproach to Sturm -lsiouville problem s.The standard reference on this

materialisCourantandHilbert(251.Moiseiwitsch (541alsodiscussesatlength

eigenvalue problem s in the fram ework ofthecalculusofvariations.O ur brief account is a blend of m aterialfrom C ourant and Hilbert op.cit.and W an

(711. 5.1 T he Sturm -tziouville Problem

The (regular)Sturm-lsiouvilleproblem entailsfinding nontrivialsolutionsto differentialequationsofthe form

ctot vtzol+ , t%: (/'(zo)- 0, c.1: v(z1)+ /71: v'(z1)- 0.

5 Applications to Eigenvalue Problem s' e

Here,q and r are functions continuous on the interval (z(),z11,and p CE Clg zt),z1j.ln addition,p(z)> 0 and r(z)> 0 forallz CEgzt ),zll.The ctk andpk intheboundary conditionsareconstantssuch thatct2 k+ / V2: /:0,and L is a param eter.

,

Generically,theonlysolutiontoequation (5.1)thatsatisfiestheboundary conditions(5.2)isthetrivialsolution,ylz)= 0forallz CEgzt ),z11.Thereare, how ever,certain valuesof,L that lead to nontrivialsolutions.T hese special valuesare called eigenvalues and the corresponding nontrivialsolutionsare called eigenfunctions.Thesetofalleigenvaluesfortheproblem iscalled the spectrum .

A n extensivetheory has been developed forthe Sturm -lsiouville problem . H ere,we lim itourselves to citing a few basic results and direct the readerto

standardworkssuch asBirkhoffandRota (91,CoddingtonandLevinson (241, and Titchmarsh (701forfurtherdetails. T he tnatural'' function space in which to study the Sturm -lsiouville

problem is the (real) Hilbert space L2g zt),z1j,which consists offunctions / :(alo,zll--+R such that

r(z)/2(z)dz,

21

II/IIa-

/2(z)dz, 20

because r iscontinuouson (z(),z11and positive' ,hence,r isbounded above and below by positive num bers.z Som e notableresultsfrom the theory are:

1Strictly spealdng, the integrals defining the Hilbert space are Lebesgue integrals and the elem entsofthe space are equivalence classes offunctions.W e dealhere with solutionsto theSturm-luiouville problem and these functionsare continuous

ongzo,zlj.ForsuchfunctionstheLebesgueandRiemannintegralsareequivalent. NotethatL2gzo,zjjalsoincludesmuch tt rougher''functionsthatarenotRiemann integrable. 2 See Appendix B .1.

5.1 The Sturm-luiouville Problem

(a) Thereexistan infinitenumberofeigenvalues.Alltheeigenvaluesarereal and isolated.The spectrum can be represented asa m onotonic increasing

sequencetà,zjwith limzz-sx àzz= xt.Theleastelementin thespectrum iscalled the first eigenvalue.

(b) The eigenvaluesaresimple.Thismeansthatthere existsprecisely one eigenf -unction (apartfrom multiplicative factors)corresponding to each eigenvalue.

(c) lfà,,zand àzzaredistincteigenvalueswith corresponding eigenfunctions 4,..and 4n.,respectively,theorthogonality relation

($m,Y,z)= 0 issatisfied.(Notethat(/z,z,4mj> 0,since4,..isanontrivialsolution.) (d) Thesetofalleigenfunctionst4zzjformsa basisforthespacefvzgzl l,z1j. lnotherwords,foranyfunction / CEL2gzt ),z1jthereexistconstants(tzszj such thattheseries

X

klimxrII/-n= j) l1laayalla-0. Theseriesrepresenting/iscalledan eigenfunction expansion orgeneralized Fourier seriesof/. T he Sturm -lsiouville problem can be recastas a variationalproblem .W e

do thisfor the case / % = /71 = 0.The formulation forthe generalboundary

conditions(5.2)canbefound in Wan,op.cï/.,p.285.LetJ bethefunctional defined by

v(zo)- v(z1)- 0, and the isoperim etricconstraint

The Euler-lsagrange equation forthe f-unctionalf is -

2r(z)p(z)= 0,

which issatisfied only forthetrivialsolution y = 0,because r ispositive.No

extremalsforf can thereforesatisfjrtheisoperimetriccondition (5.5).lfy is

5 Applications to Eigenvalue Problem s' e

d :F

DF

X t l ? ôy = 0, -

F = 37#/2+ qy2- hpyz. But the Euler-lsagrange equation for this choice of F is equivalent to the

differentialequation (5.1).Theisoperimetricproblem thuscorrespondstothe Sturm-lsiouvilleproblem augmented bythenormalizingcondition (5.5),which sim ply scalestheeigenf-unctions. H ere,theLagrangem ultiplierplaystherôle ofthe eigenvalue param eter.

Example 5.1.1: Letp(z)= 1,qlz) = 0,r(z) = 1,and gzt),z11= (0,rl. Then theEuler-lsagrange equation reduces to

!/V(z)+ . X!/(z)= 0, and the boundary conditionsare

t v(0)- : v(r)- 0. #(z)- Ae'' n--D + Be-wn-M , where-4 andB areconstants.Theboundaryconditionsim ply that-4 = B = 0, and thereforethere areonly trivialsolutionsif,L< 0.lf, L = 0,then equation

(5.7)hasthegeneralsolution #(z)= Xal+ 0. A gain the boundary conditionsim ply that4 = B = 0,and therefore preclude thepossibility ofnontrivialsolutions.H ence,anyeigenvaluesforthisproblem m ustbe positive.

lf,L> 0,then thegeneralsolutionto equation (5.7)is y(z)= 4cos(hfSz)+ B sin(hfSz). B sin(hfhr)= 0. Equation (5.9)issatisfied forB # 0 provided sfh isa positive integer,and thisleadsto thenontrivialsolution ylz)= B sintxfàz).Theeigenvaluesfor thisproblem are therefore àzz = n?,and the first eigenvalue is z ïl = 1.The eigenf -unctionscorresponding to àzzare oftheform

5.1 The Sturm-luiouville Problem

107

$n.V)= . Dsinlzzz), whereB is an arbitrary constant. ln term s of the isoperim etric problem ,there are an infinite num ber of Lagrangem ultipliersthatcan beused.Each Lagrangem ultipliercorresponds to an eigenvalue,and thelinearity oftheEuler-lsagrangeequation im pliesthat any function ofthe form

such that the Fourier series is convergent and twice term by term differen-

tiable,isan extremalfortheproblem,provided / satisfiestheisoperimetric

condition (5.5).Now,

wherewe have used the orthogonality relation

(sintzrzz),sintzzzl)= 0 ,if # zz, a f? wrz, 2:k . a. Hence,any eigenfunction expansion ofthe form (5.11)having therequisite convergenceproperties and satisfjring thecondition * >= 1

1

2 JF '

is an extrem alfor the problem .Any finitecom bination oftheeigenfunctions such as where forexam ple,is an extrem al. lfw earesearching am ong theeigenf-unction expansionsforextrem alsthat

makeJ aminimum,then thesituation changesconsiderably.Supposethat/ isan eigenfunction extrem alforthe problem .Then

5 Applications to Eigenvalue Problem s' e

H ere,w ehave used the orthogonality relation

2

v1(z)= : sintz), r and forthisextrem al

J'(3/1)= 1. ln fact,yïproducestheminimum valueforJ.Tosee this,let/ beanother extrem alfor the problem .Then the com pleteness property of the Fourier

seriesimpliesthat/ can be expressed asan eigenfunction expansion ofthe

form (5.11),wherethecoefllcientsan satisfy relation (5.12).lf/ isdistinct from yïthen thereisan integer?rzk 2suchthatavrv, # 0.Now,

and henceJ(/)> J(yï).

5.2 The First Eigenvalue

Exercises 5.1:

1.The C auchy-Euler equation is

(z7(z))?+ l: v(z)- 0. -

Show that

y(z)=c1cosh, $ . /-1l nz)+casinh, /-. $l nz), wherecland czareconstants,isa generalsolution to thisequation.G iven

theboundaryconditions3/(0)= yle' n' )= 0find theeigenvalues.

2.R eform ulate the differentialequation

ylivs(z)+ (à+ p(z))?/(z)= ()

5.2 T he F irst E igenvalue The first eigenvalue in Exam ple 5.1.1 hasthe notable property thatthecorresponding eigenfunction produced the m inim um value for J. lf fact,this relationship persists forthe generalSturm -lsiouvilleproblem .

Theorem 5.2.1 Letz ïl betltehrsteigenvaln, eJortlteS/w, rzq-. rzïch kzwïllcprob-

Jcm (5.1) nitltl't aqzzzclt zr? / conditions (5.4),and let3/1 be tlte corresponding eigenjunction nonvtalized tosatisj' y tlteïscwcr/mc/rïc constraint(5.5).Tlten, tzmtazzgJunctionsïzzCzgzt ),z1jtltatsatisj' ytlte boundary conditions(5.4)and tlteïscwcrïmc/r/ccondition (5.5),tlteJnnctionalJ dehnedby equation (5.3)is m ïzzïm ' tzm aty = yï.M oreover,

J'(3/1)= à1. P roof: SupposethatJ hasa m inim um aty.T hen y isan extrem aland thus

satisfiesequation (5.1)and conditions(5.4)and (5.5).Multiplying equation (5.1)by y and integrating from zt ltozlgives

The first term on the left-hand side of the above expression is zero since

!/(zo)= !/(z1)= 0' ,theintegralontheleft-handsideoftheequationisoneby the isoperim etric condition.Hence we have

5 Applications to Eigenvalue Problem s' e

J(t v)- 1. Any extremalto theproblem mustbeanontrivialsolution toequation (5.1) because oftheisoperim etriccondition' ,consequently,,Lm ustbean eigenvalue.

By property (a) there must be a least element in the spectrum,the first eigenvalue àl,and a corresponding eigenf-unction yïnorm alized to m eetthe

isoperimetriccondition.Hencetheminimum valueforJ iszïland J(yï)= àl. Eigenvalues for the Sturm -lsiouville problem signal a bifurcation: in a deleted neighbourhood ofan eigenvaluethereisonly thetrivialsolution avail-

able;atan eigenvaluethere arenontrivialsolutions(multiplesoftheeigenfunction)availablein addition tothetrivialsolution.ln applicationssuch as

those involving the stability ofelastic bodies,eigenvalues indicate potential abruptchanges.Often the m ostvitalpiece ofinform ation in a m odelis the location ofthe firsteigenvalue.Forexam ple,an engineerm ay w ish to design a colum n so that the firsteigenvalue in the problem m odelling the deflection of the colum n is sufllciently high that it willnot be attained under norm al loadings.S T heorem 5.2.1 suggests a characterization ofthe firsteigenvalue in term s ofthe f-unctionals J and f.LetR bethe f -unctionaldefined by

y). J?y)= J( gv)

.

The functionalR is called the R ayleigh quotient for the Sturm -lsiouville

problem.lff(y)= 1,then forany nontrivialsolution y wehave l - R(y). W ecan,how ever,drop thisnorm alization restriction on fbecause130th J and f arehom ogeneousquadraticfunctionsin y and y?so thatany norm alization

factorscanceloutin the quotient.Relation (5.14)isthusvalid forany nontrivialsolution,and we can m ake use ofthis observation to characterize the firsteigenvalue asthe m inim um ofthe Rayleigh quotient.

T heorem 5.2.2 LetS/denote tlteseto.fal lJunctionsïzzCzgzt),z1jthatsat-

isj' ytlteboundary conditions(5.4)ezcepttltetrivialsolution yEEE0.Tltezrzïzzïzrz' t zzrzoft lteRayleigltquotientR JortlteS/urzrz-fziouvilleproblevt(5.1),(5.4) .

overallJnnctionsïzzS?istltehrsteigenvalue;ï.c.,

mi àR(y)= ll. yqSr 3 The governing differentialequation for this m odelis in factoffourth order, but similarcom mentsapply.Thevariationalformulation ofthis modelis discussed in Courantand Hilbert op.cit. p.272.

5.2 The First Eigenvalue

b= !/+ 6r/,

wheretîissmalland p isas11100th f-unction such thatp(zo)= p(z1)= 0,to ensure that:' èCES?.Now,

1

1

f(5)-- I(y)q-()(,), wheref(y)# 0,and

'?((-vy')'+ qy- Ary)dz W e thus have

and since R ism inim um aty,the term s oforder t îm ustvanish in the above expression for arbitrary p.W e can apply Lem m a 2.2.2 to the num erator of

theordertîterm and deducethaty mustsatisfyequation (5.1).SinceyCES?, the constant . 4 m ustbe an eigenvalue.Any extrem alfor R m usttherefore be , a nontrivialsolution to the Sturm -lsiouvilleproblem . lfà,,zis an eigenvalue for the problem and yv. n is a corresponding eigenfunction,then the calculation in the proofofTheorem 5.2.1 can be used to show that

Xl#,? ym)= As? z)= Jl f(y m) z. SinceR ism inimum at y and . 4 isthe corresponding eigenvaluewe have that

l,, z= Rlymlk .4

5 Applications to Eigenvalue Problem s' e

Generally,theeigenvaluestandhenceeigenf -unctions)foraSturm-lsiouville

problem cannotbedeterm ined explicitly.Boundsforthefirsteigenvalue,however,can be obtained using the Rayleigh quotient.U pperbounds forzïl can

bereadily obtained sincez ïlisaminimum value:forany function 4 CES?we have

R(4)k l1,

(5.16)

so that an upperbound can be derived by using any function in S?.Lower boundsrequire a bitm orework. To geta lower bound,the strategy is to constructa com parison problem

that can be solved explicitly,the first eigenvalue ; $1 ofwhich is guaranteed to be no greater than àl.To construct a com parison problem ,we m ake the follow ing sim pleobservations.

(a) Let# CE Clg zt),z1jbeany function such thatp(z) k #(z) > 0 for all z CEgzt ),z11,and lett l /CEC0gzt ),z1jbeany function such thatqlz)k t )' (z) forallz CEgzt ),z11.Then,for

7(v)< J(y). (b) Let , izCEC0gzt),z11be any function such that' F(z) k r(z) > z CEgzt ),z11.Then,for

I(y)k Ily). J(y) J(y) pv), 2 -ky)- yy) s;go) -

.

and hence

à1< àl. (5.17) lnequality (5.17)isusefulonlyifwecandetermine; $1explicitly.W ehave ,

considerablefreedom,however,in ourchoicesfor#,t )' ,and , i%,and thesim plest choice iswhen these functions areconstants' ,i.e.,

#(z)= zemi n p(z)EEEpzzz, yo,zlj q(z)= zemi n qlz)EEEqvn, yo,zlj '

F(z)= zema x r(z)EEEru. yo,zlj

5.2 The First Eigenvalue

(-vmy')'+ qmy- ârpi.y- 0;

ya = 1

.

Pmn2r2 + qm .

ru (zo- z1)2

W e thus getthe low erbound

à1 =

.

1

p . ,.2

r' z

+ qm s;ll.

ru (zo- z1)2

Exam ple 5.2.1: M athieuhs Equation

Letp(z)= r(z)= 1,and qlz)= 20costzz),where0CER isa constant.Let zt l= 0and zl= r.Forthischoiceoffunctionsequation (5.1)isequivalentto yn+ (, à.- 20cos(2z))y = 0, and the boundary conditionsare

t v(0)- : v(r)- 0. Theexpression (5.20)iscalled M athieuhs equation,and itssolutionshave been investigated in depth (cf.Mclsachlan (521and W hittakerand W atson (741).lf0= 0,then theproblem reducestothatstudied in Example5.1.1.lf 0, # 0,then thenontrivialsolutionsto thisproblem cannotbeexpressed in closed form in term s ofelem entary functions.lndeed,this problem defines a

new classoffunctions(sczzjcalledM athieu functions?thatcorrespondto the eigenfunctions ofthe problem .T he determ ination ofthe eigenvalues for thisproblem is a m ore com plicated affaircom pared to thesim pleproblem of Exam ple 5.1.1.B riefly,it can be show n that the first eigenvalue zïl and the corresponding eigenf -unction sc1 are given asym ptotically by 1 z

1 a

1

4

11

:;

6

ïl= 1- 0- -0 8 +6 -40 - 15360 - 368640 + O(0 ),

z

(5.22)

5 Applications to Eigenvalue Problem s' e

forIpIsmall(cf.Mclsachlan,op.cit.p.10-14).

ln contrast,a rough low er bound for z ïl can be readily gleaned from in-

equality (5.19).Suppose that0 k 0,and letpv. n = ru = 1,qv . n = -2p < 20costzz).lnequality (5.19)thenimplies 1- 20< àl. (5.23) Given theasymptoticexpression (5.22),if0k 0issmallthen thelowerbound (5.23)can beverified directly.Butinequality (5.23)isalsovalid for0large, and thisisnotso obvious. N ote that if 0 < 0,w e cannot use qvrv= 20 in our com parison problem

since-2pk 20costzz)forz CEgzt ),z11.Forthiscasewecan useqv. n= 20and thusgetthe lowerbound

1+ 20 < zïl.

Exercises 5.2:

1.M athieu'sequation (5.20)can haveafirsteigenvaluez ïlthatisnegative depending on the constant0.W rite out the Rayleigh quotient forM ath-

ieu'sequation.Now,4 = sintz)isin thespaceS?.Usethisf-unction and inequality (5.16)to getan upperbound foràl,and show thatzïl < 0 whenever0 > 1.Compare thiswith expression (5.22).(Forthe choice 0= 5thevalueofzïlisgiven in table5.1attheendofSection 5.3.)

2. H alm hs equation is

(1+ z2)2?y??(z)+ àyytzl= 0. Undertheboundary conditions3/(0)= !/(r)= 0,find a lowerbound for

,L1. 3.The T itchm arsh equation is

!/??(z)+ (, à.- z2N.): (/(z)= 0, wheren,isanonnegativeinteger.Undertheboundaryconditions3/(0)= 3/(1)= 0 show thatthefirsteigenvaluezïl satisfiesr2 < àl< 11.(The function 4 = z(z - 1)can beused togettheupperbound.)

5.3 Higher Eigenvalues

5.3 H igher Eigenvalues The Rayleigh quotientcan beused to fram ea variationalcharacterization of higher eigenvalues. The eigenf-unctions for the Sturm-lsiouville problem are m utually orthogonal,and this relationship can be exploited to give such a characterization.Forexam ple,itcan be show n thatthe eigenvalue zïz correspondsto the m inim um ofR am ong f -unctions in y CES?thatalso satisfy the orthogonality condition

(#,#1)= 0, whereyïisan eigenf-unction corresponding to àl.M oregenerally,we havethe follow ing resultthe proofofw hich we om it.

Theorem 5.3.1 Let yk denote the eigenlhnction associated ' tl ï//zthe eigenvaln,eàk,and letSs /zbethe set o.fJnnctionsy CES/s' uch that

(#,Vk)= 0 Jork= 1,2,...,n,- 1.Tlten

la- pi mi )1t(y). E&r T he above theorem is oflim ited practicalvalue because,in general,the eigenvaluesàl,...,àzz-land corresponding eigenf-unction 3/1,...,l/,z-larenot

known explicitly.Constraintssuch as(5.24)requirepreciseknowledgeofthe eigenf -unctionsasopposed to approxim ations.Fortunately,wecan characterize higher eigenvalues w ith a ttm ax-m in''type principle involving the R ayleigh quotient,and circum vent the problem of finding eigenfunctions. The next resultswe state withoutproof.Som e detailscan be found in W an op.cï/.,p. 284,and in C ourantand Hilbert op.cï/.,p.406.

(y,zk)--0, ' îa skla. .

Lem m a 5.3.2 isa key result used to establish the follow ing ttm ax-m in''principle forhighereigenvalues.

àa - zef ma x (4a(z)), za-l

5 Applications to Eigenvalue Problem s' e

à,ztzl- r yi ))(.&(: (/):(y,zk)- 0, k- 1,...,n- 1). # ,

(-vy')'+ qy- hry- 0, : v(0)- : v(r)- 0, 4''(f)- /(f)4(f)+ l4(f)- 0, 4(0)- 4(t)- 0, by thetransform ation z

( j=, ( , / , VyyL=jj vp(jjj.jjc j)yf=ljx s wtz;gy. / = #//+ %, .

g

r

whereg = 4rp.We can thus restrictourattention to the problem (5.26), (5.27).5TheRayleigh quotientforthisproblem is R(4)= J( gii4) )l, .

where

and

k= tma x I/(f)I, egoyj

(5.28)

and 5 This formulation is called the Liouville norm al form of the problem . Details on thistransform ation and extensions to m ore generalintervals can be found in

BirkhoffandRotag9j,p.320.

5.3 Higher Eigenvalues

Then,

and,sinceJ-F(4)k J(4)k J-(44, 2+(4)k . 2(4)k R-(4); .

1.D(4)- . 2(4)1< k, tr ?z

1k(4)- /0f( 44)cjj.

(5.30)

The Rayleigh quotientdefined by equation (5.30)isassociated with the

Sturm -lsiouville problem

r#?+ . 14 = 0,

4(0)- 4(t)- 0, and the eigenvaluesfor thisproblem aregiven by

lnequality (5.29)indicatesthatR(4)can differfrom X(4)by no morethan +/ c.Bythe ttm ax-m in''principle forhighereigenvalueswe see thatàzzand àzz can differ by no m ore than + /cand thusdeducetheasym ptotic relation

(5.34) as n,--+ xt.The f-unction / influences only the 0(1)term (a term that is bounded asn,--+cx))4àzzisapproximately/, 2. ,r2 :2forlargevaluesofzz.lfwe return backtotheoriginalproblem,therelation (5.34)can berecastas

5 Applications to Eigenvalue Problem s' e > >2

4a

.

2

lim n? = 1 'T r(z)dz zz--.x Xzz ( /-' i' ( ; p(z)

.

N otethatq doesnotinfluencetheleading orderbehaviourfortheasym ptotic distribution ofeigenvalues.

Equation (5.35),for example,predicts that the higher eigenvalues for M athieu'sequation (Fuxample5.2.1)are lzz= /,2+ ()(1), as n,--+x t.ln fact,the approxim ation isnot ttoo bad''for0 sm alleven w ith

n,small(cf.Table5.1).

ln closing,we note that the results ofthis chapter can be extended for

thegeneralSturm-lsiouvilleboundary conditions(5.2).Someextensionscan also be m ade to copew ith singular Sturm -lsiouville problem s.The reader is directed to Courantand Hilbert,op.cit.Chapter5,fora fullerdiscussion and a w ealth ofexam plesfrom m athem aticalphysics. Exercises 5.3:

1.ForMathieu'sequation (5.20)show thatIlzz- n?I< 20fora11zz. 2.Determineaconstant. , 4 such thatforHalm'sequation (Exercise5.2-2) àzz= . , 4zz2+ 0(1).

H olonom ic and N onholonom ic C onstraints

6.1 H olonom ic C onstraints

glt,q)- 0,

#(f,Cl,tk)= 0. The analysis underlying variationalproblem s with holonom ic constraints is noticeably sim plerthan thatforproblem swith nonholonom ic constraints.ln thissection we focus on holonom ic constraints and postpone our discussion ofnonholonom ic constraintsuntilthenextsection.Forsim plicity we consider the sim plestcasew hen n,= 2. LetJ bea functionalofthe form

and supposethat J hasan extremum atq subjectto theboundary conditions

q(fo)= qo,q(f1)= q1,and the condition (6.1).Forconsistency werequire thatgltç j,qo) = 0 and . t y(f1,q1)= 0.We assumethatL and g aresm00th functions.W e also m ake theassum ption that 1The curious nam e for this type ofconstraint stems from the Greek word ltolos m eaning wholeorentire.ln thiscontextholonomic means ttintegrable.''Thistype ofconstraint isalso called dd finite'' 2 Som e authors use this term specifically to identify constraints that are not integrable i.e. cannot be reduced to a holonom ic condition.For our purposeswe simply callany constraintinvolving derivatives nonholonom ic.

6 Holonom ic and Nonholonomic Constraints

Tg =

(' Lg t '?.

8qk, t'?t?t?2 :# 0

fortheextremalqintheinterval(f(),f11.GiventhatTg, # 0,wecould(atleast in principle)solveequation (6.1)foroneoftheqk.3Wecould thusapplythe constraintim m ediately and reducetheproblem to an unconstrained problem involvinga singledependentvariable.T hisapproach,how ever,isfraughtw ith thesam e problem sasitsfinite-dim ensionalanalogue discussed in Section 4.1. Fortunately,the Lagrange m ultiplier technique can beadapted to copew ith these typesofproblem s.

W eseekanecessaryconditiononq forJ(q)tobeanextremum.Asbefore, we perturb q to get some nearby curve ( ' k= q+ t îr/and use the condition

J(4)- J(q)= 0(62)to geta necessary condition.We assume that the qk are in C2gf( ;,f1jfork = 1,2.A f-unction ( ' k= (41,t )' 2)iscalled an allowable variation forthe problem ifqm k CE (72gfl),flj,(' i j(f( ;) = q(;,( ' k(f1) = q1,and #(f,t * 1)= 0. Ourfirstconcerniswhetherthereareanyallowablevariations(apartfrom thetrivialone( ' k= q).Aswith theisoperimetricproblem,theremay berigid extremals.Letp= (p1,z?a).Theconditionson an allowablevariation require thatthepkbein theset(72gflj,f1jandthatr/(f())= p(f1)= 0.ln addition,we mustalso havethatglt,q+ 6p)= 0.Now,q isa flxed function and Tg , #0 at(f,q)fortCE(f(),f11.Supposefordefinitenessand simplicitythat (' )g / 0 (' Lqz

foralltCE(f(),f11.Thentheimplicitfunctiontheorem impliesthattheequation glt,q + 6p) = 0 can be solved for m in termsofpland 6,provided I6Iis sufllciently sm all.The sm oothness ofthe derivatives ofg also ensures that

w is in the set (72gfl),f1j.We can thus regard pl as an arbitrary f-unction in (72g fl),f1jsuch thatp1(f() = p1(f1) = 0,and w as the solution to the equation glt,q+ 6p)= 0.Theimplicitf -unctiontheorem guaranteesaunique solution to this equation.ln particular,w e know that at t = tç j we have

p1(fo) = 0 and that (f(),qo) isthe solution to gltç j,qt l+ 6p) = 04hence, w (f()) = 0.A similar argument can be framed to show that m (f1) = 0. Thuswealwayshavenontrivialallowablevariationsprovided condition (6.3) issatisfied.Thiscondition can berelaxed to condition (6.2),butwedo not pursue thisgeneralization.

Supposethat( ' i.isan allowablevariation.SinceJ isstationary atq,the

condition J(h)- J(q)= 0(62)leadstotheequation tl to

( ')L

-

d (?. r/

% k dzt' ?t il

pl+

(' )L

-

d (?. r/

(' k2 dz* 2

072 dt= 0.

W e cannotproceed asin Section 3.2 to deduce the Euler-lsagrange equations from the above expression because w cannotbe varied independently ofp1: 3 W e can use theimplicitfunction theorem to asserttheexistenceofsuch solutions.

6.1 Holonom ic Constraints

thesef-unctionsareconnectedbytheconstraint(6.1).Supposethatwechoose som es11100th butarbitrary function plthatsatisfiesthe boundary conditions.

Theimplicitfunction theorem indicatesthatforI6Ismallthereisasolution w totheequationglt,q n)= 0,thatdependsont îandpl.Foraflxed butarbitrary plwe can thus regard m as a f-unction ofE.M oreover,the im plicitf-unction

theorem impliesthat,forI6Isufllciently small,m isa s11100th function ofE. Now g(f,4)= 0,and thus d

t' ?.q

ç ' )g

27v(f,ù)E,-o= ôqïpl+ t'Azm = 0. .

A?

ç' )g

a(L)= l(f)ç'L kz.

W e can now apply Lem m a 2.2.2 and thus deduce that ( ' )L

d( -8. r/

ç' )g

%k Z t 'Ml+ l(f)ç'qk= 0. Equations(6.7)and (6.8)providetwodifferentialequationsforthethree unknownf-unctionst 71,q,,andà.Theconstraint(6.1)providesthethird equation.Thefunction , LisalsocalledaLagrangemultiplier.Equations(6.7)and (6.8)canbewritten in thecompactform -

6 Holonom ic and Nonholonomic Constraints

d :F

DF

= 7/ôq' k txk 0, -

wherek = 1,2,and F = L - hg.

Thederivation ofequations(6.9)hasthemeritofsimplicity,yetitseems disappointingly narrow .The Lagrange m ultiplier has a tractable geom etrical interpretation in finite-dim ensionalproblem s and even in the isoperim etric problem .H ere,the approach seem s som ewhatdestitute ofgeom etry.ln fact, thereisa satisfactory geom etricalinterpretation available,butitrequirescertain conceptsfrom differentialgeom etry such asfibrebundlesthatwould lead us astray from an introductory account.The reader isreferred to G iaquinta

and Hildebrandt(321fora geometry-based proofofthe Lagrange multiplier technique. ln sum m ary,we have the follow ing necessary condition.

Theorem 6.1.1 Snpposetltatq = (t 71,q,)ïsa szrztat a//zeztrevtalJortlteJ' unctionalJ subjecttotlteFztalt azzt azrzïcconstraintglt,q)= 0,andtltatTglt,q), #0 JortCE(f(),f11.Tlten tltere ezists aJnnction , L o. ft suclttltatq satishestlte Enler-Lagrangeequations(6.9). Exam ple 6.1.1: Let

and

q(f,q)= IqI 2- 1. Find theextremalsforJ subjecttotheconstraintglt,q)= 0and theboundary conditionsq(0)= (1,0)and q(r 2)= (0,1). .

Forthisproblem

F = 1412+1-l(f)(IqI2-1), and the Euler-lsagrange equationsare

t ?1(f)- cos4(f), q,(f)- sin4(f).

Now q'ï- -4'siny,4,- (cos4,and hence1ù12= 4'2.TheEuler-lsagrange equationsin termsof4 are

6.1 Holonom ic Constraints

and elim inating the function , L yieldsthe equation

4 = co

/(f)= cof1-cl, whereclisanotherconstantofintegration.The generalsolution oftheEulerLagrange equation for the extrem alq istherefore

t ?1(f)- costcl,f+ c1), q,(f)- sintct )f+ c1). Theboundaryconditionq(0)= (1,0)indicatesthatcl= 2zzr forsomeinteger zz.Theboundarycondition q(r 2)= (0,1)impliesthatct l= 4w + 1forsome integerzrz.The extrem alisthusgiven by

q(f)- (costf),sintfl),

Exam ple 6.1.2: Sim ple Pendulum Theparam etricequationsforthem otion ofasim plependulum ofm ass?rzand

length f can bederived using Lagrangemultipliers.Letq(f)= (t71(f),q,(f)) denotethe position ofthe pendulum attim e f.Here we associateq,w ith the verticalcom ponent ofposition.Them otion ofthe pendulum from tim e tçjto tim e f1is such thatthe functional

6 Holonom ic and Nonholonomic Constraints tl

J(q)- t s 'rzI ' k1 2+gq,)dt e

isan extremum subjectto the condition4

qk 2+ (q,- f)2- é2= 0. Here,theterm ?rz 21ù12isthekineticenergyandtheterm gqn,whereg isthe gravitation constant,isthe potentialenergy.The Euler-lsagrange equations forthe m otion ofa pendulum are thus

t i' l+ 2l(f)t ?1= 0, t i' 2- g+ 2l(f)(t ?2- f)= 0.

T he m ethod outlined in this section can be generalized in som e obvious ways as was done for the isoperim etric problem .For exam ple,w e could include functionals depending on higher-order derivatives,m ultiple holonom ic constraints,orfunctiona. lsdepending on n,dependentvariables,n,> 2.W e do

notpursuethesegeneralizations.Thereaderisreferred totheliterature(4121, (211,(271,(311,(321)fordetailson thesegeneralizations. W eclosethis section w ith a derivation oftheequationsforgeodesicson a

surfacedefined implicitly byglz,y,z)= 0. Exam ple 6.1.3: G eodesics Let g be a sm 00th function ofthe variables

z,y,z.lfTg , # 0,then anequation oftheform

q(1,#,. 2C)= 0 describesa surface implicitly.Forexample,ifglz,y,z)= z2+ y2+ . ,2- 1, then equation (6.11)describesasphereofradius1centred attheorigin. A generalspacecurvey offinite length isdescribed (atleastlocally)by .

param etric equationsofthe form

r(f)= (z(f),' !/(f)z(f)),

4 Fortheconnoisseuroftechnicalterm s,constraintsthatdo notinvolve time explicitly are called scleronom ic in mechanics.Constraintsthatinvolve tim eexplicitly

arecalled rheonomic.Condition (6.10)can thusbecalled ascleronomicholonom ic constraint.Need we say m ore?

LetE denotethesurfacedescribed by equation (6.11)and letPtland P1be two distinct points on E .A geodesic on E from P tlto P 1 is a curve on E with endpoints P t l and P 1 such that the arclength is stationary.Assum ing

thatsuchacurvecan berepresentedbya (single)parametricfunction ofthe form (6.11)with r(fo)= Pt land r(f1)= P1,a geodesicisthusa curvesuch thatthef-unctionalJ isstationarysubjecttotheconstraint(6.11).Let F = z?2+ y?2+ z?2- à(f). q(z?ytz). The(smooth)geodesicson E mustthereforesatisfjrtheEuler-lsagrangeequations

)d-@

z' + l(f)t'?.q = 0, z?2+ y?2+ a?2 t' ?z d y? ' ?.q = 0, /7 z?2+ y?2+ z?2 + l(f)t ' ç ' )y d z? ( ' ? X = 0. 7/ z?2+ y?2+ a?2 + l(f)ç ' )z ....

Exercises 6.1:

1.Geodesics on a Cylinder: Theequation glz,y,z)= : 712+ y2- 1 = 0 definesa rightcircularcylinder.Use them ultiplierrule to show thatthe geodesicson the cylinderare helices. 2. Catenary on a C ylinder: Let tl

J(q)-

te

qz t i2 ,+ t ia 2+ t i2 adt,

and

6.2 N onholonom ic C onstraints ln this section we discussvariationalproblem s that have nonholonom ic constraints.Theseproblem sarealso called Lagrange problem s.TheLagrange problem thusconsistsofdeterm ining the extrem a forfunctionalsofthe form

6 Holonom ic and Nonholonomic Constraints tl

J(q)-

te

Llt,q,ù)dt,

subjectto theboundary conditions

q(fo)= qo, q(f1)= q1, and a condition ofthe form

#(f,q,tk)= 0. lt is clear that Lagrange problem s include problem s with holonom ic con-

straints as a specialcase,butnotevery constraintofthe form (6.15)can be integrated to yield a holonom ic constraint.For exam ple,suppose n,= 3 and

glt,q,tk)- f'tqlt il+ Q(q)t i2+ .&tqlt ia- 0. Then itiswellknown thatthisequation is integrable only if p

(ç ' D kQz-t ' D ?t R2j+c(o ? ' gq Rk-t ' D kPaj+w(o ' 9q pz-t ' D kQlj-o

(cf.(611,p.140),and hence the constraint (6.16) can be converted into a holonomic one only for certain functions P,Q,R.For quasilinear nonholo-

nomicconstraintssuch as (6.16)thedimension n,iscrucial.lfn,= 2 and the constraintisofthe form

#(q)I1+ Q(q)I2- 0 then,in principle,this constraint can be reduced to a holonom ic condition.

Forexample,assuming Q(q), # 0,wecould recasttheaboveconstraintasan ordinary differentialequation

dq, f'(q) dqk Q(q) and appealto Picard'stheorem to asserttheexistenceofa solution q,(t71) to thisdifferentialequation.lfn,> 2,then condition (6.17)isnotgenerically satisfied forP,Q,R and hencethecondition isnotintegrable. lsoperim etricproblem scan also beconverted intoLagrangeproblem s.Suppose that theisoperim etric condition

t izz-hl- glt,q,ù)- 0,

t ?zz+1(f1)- t ?zz+1(fo)- f. ln thism annerwecan recasttheisoperim etricproblem asa Lagrangeproblem . Problem sthatcontain derivativesofordertwo or higherin the integra. nd can also be regarded as Lagrange problem s.For instance,consider a basic variationalproblem thatinvolves a f-unctionalofthe form

11- q2= 0.

Thisreformulation leads to a functionalofthe form (6.13) along with the above nonholonom icconstraint. T he theory behind the Lagrange problem iswelldeveloped for problem s involving one independentvariable,sbuttheproofofthe Lagrange m ultiplier rule fornonholonom ic constraintsis m ore com plicated than that forisoperim etricorholonom ic constraints.ln addition,theapplication oftheruleitself is awkw ard owing to the condition for an extrem alto be norm al.Som e of thedifllcultiesthatsurround the Lagrangeproblem concern the possibility of rigid extrem als. Consider,forinstance,the problem offinding extrem alsfor the f -unctionalJ defined by

subjectto theconstraint

g(f,q,tk)= t i2 1+ t J: 2= 0, and theboundaryconditionsq(fo)= qo,q(f1)= q1.Theonly (real)solution to the constraint equation is (1 = ( % = 0,so that qï and q, are constant

functions.lfq(fo)# q(f1)thentherearenosolutionsthatmeettheconstraint and the boundary conditions.lfq(f())= q(f1),then theonly solution tothe constraintequation thatsatisfiestheboundary conditionsisq = q(fo),and in thiscase J(q)= t 71(0)(f1- f()).ln short,thereareno arbitraryvariations available for this problem because only one function satisfies the constraint and the boundary conditions. ln the rem ainder ofthis section we present without proofthe Lagrange m ultiplier rule for nonholonom ic constraints and lim it our discussion to the

5TheLagrangeproblem forseveralindependentvariablesislesscomplete(cf.Giaquintaand Hildebrandt(321).

6 Holonom ic and Nonholonomic Constraints

sim plestcasesand exam ples.FulleraccountsoftheLagrangeproblem resplen-

dentwith gory detailscan befound in (101,(121,(211,and (631. W e begin firstw ith a generalm ultiplier rule thatincludes the abnorm al Case.

Theorem 6.2.1 Let J be tlte Jnnctionaldehned by (6.13), wltere q = (t 71,...,qnl and L ïs a smtata//zJunction o. ff,q, and (k.Suppose tltatJ Itas an cz/rcm' tzm atq CE(72gfl),f1jsubjectto thelouzzclt zr?/conditions (6.14) and tlteconstraint(6.15),wltereg ïs a smtat a//zJunction o. ff,q,and (ksuclttltat t' ?.t y ç' kj , # 0Jor stamcj,1 < j < n,.Tlten tltere ct c/s/s a constantzïtland a

jhnction , L1(f)not30th,zerosuclttltatJor Klt,q,ù)- hçjlult,q,ù)- l1(f). v(f,q,(%, q is a s/lu/ï/n to the systent = 7/A k t' kk 0, -

wltere k = 1,...,n,. T heaboveresultincludestheabnorm alcase,w hich correspondsto z ïtl= 0.

ln thiscasethef-unctionzïlisnotidentically zeroon theinterval(f(),f11,and equation (6.21)impliesthat

2d /tàyç ' l ' z t k . q k( )-a yyo %qk.() , fork = 1,...,zz.W e thus see that ifq is an extrem alfor the problem w ith

ït l= 0then z ïlmustbeanontrivialsolutionto(6.22).Theexistenceofanon-

z

trivialsolution zïlthus characterizes the abnorm alcase.A s1100th extrem al q is thus called abnorm al if there exists a nontrivial solution to system

(6.22)4otherwise,itiscallednorm al.W ehavethefollowingresultfornormal

extrem als.

d :F

DF

)è'A k txk = 0, -

F(f,q,ù)= L(t,q,tl)- l1(f). v(f,q,(%. N ote that,unlike the other constrained problem s,the differential equations

(6.23)willcontain theterm i1' ,moreover,thecondition (6.15)isadifferentialequation,so thatsolving problem sw ith nonholonom ic constraintsentails solving a system ofn,+ 1 differentialequations.

Exam ple 6.2.1: tl

J(q)-

te

(qI+vl)dt.

Find theextremalsforJ subjecttotheboundary conditions (6.14)and the constraint

v(f,q,tk)= 11+ qk+ q2= 0.

.

U sually,thepracticalapproach to constrained problem sisto firstidentify the candidates for extrem a. lsand then study w hether the problem is in fact norm al,i.e.,proceed under the assum ption that the problem is norm al.For thissim ple problem ,however,we can deduce readily that any extrem als to

theproblem mustbenormal.Specifically,equation (6.22)fork = 2gives

which has only the trivialsolution.W e thus know in advance that we have only norm alextrem als. Let

F(f,q,tk)= t ?2 1+ v2- . X1(f)(j1+ qï+ qLj). Theorem 6.2.2 showsthatifq isan extrem althen itm ustsatisfy the follow ing system ,

il- 11+ 2t ?1= 0 11- 2q, = 0.

Theseequationsandtheconstraint(6.24)imply 4' k- 2t ?1= 04

and

6 Holonom ic and Nonholonomic Constraints

Exam ple 6.2.2: C atenary Let us revisit the catenary problem ,butthis tim e as a Lagrange problem . Suppose that the length of the cable is f and the endpoints are given by

(alo,#0)and (t r1,: 4/1),Where lt r- t r012-1-(#- : 4/012< /2. The potentialenergy functionalisgiven by

wheres denotesarclength.ln order to ensure thats is arclength we need to add the constraint z/2+ y/2 - y = () .

(cf.Section 1.2).Usingthenotation ofthissection,letqï= z,q,= y,and

s = f.W e thusseek an extrem um forthe functional

(6.26) subjectto theconstraint

v(q)- 41+ t 2 -1- 0,

.

and the boundary conditions

q(0)- (zo,yç jl, q(t' )- (z1,: v1). W e can show directly thatany abnorm alextrem als to this problem m ust

belines.Supposethatthereisa nontrivialsolution toequation (6.22).Then there areconstants cl and c2 such that

1111- c1, 1112 - c2.

Theconstraint(6.27)impliesthat 42 1= c2 1+ c2 2,

.

and consequently qï and q, m ustbelinear functions off.The boundary con-

ditions(6.28)and theinequality (6.25),however,precludelinearsolutions. Let

F = q2- . 1.1(12 1+ t iz 2- 1). Equations(6.23)give 2. :111= /c1, 2. :14 % = t+ k,,

(6.29) (6.30)

6.3 Nonholonom ic Constraintsin M echanics' e

131

1

41= j. k;+ (f+ /ca)2;

.

hence,

qk=Si nh-,jt+/ c1p c at j. j .ks q,- kl+ (f+ /c2)2+ /c4, where ksand k4 are constants.The fam iliarparam etrization ofthe catenary in term softhehyperbolic cosinecan thusberecovered from theaboveexpressions.N ote that this problem is m erely a reform ulation ofthe isoperim etric problem so that the com m ents concerning the satisfaction of the boundary

conditions(Example4.2.1)stillapply. Exercises 6.2:

and

#(f,q,tk)= ft il+ 12+ qz- 1. Find the extremals for J subjectto the constraintglt,q,(k) = 0 and boundaryconditionsoftheform q(fo)= qo,q(f1)= q1. 2.Let J bea functionalofthe form (6.13),where n,= 3 and letg be of theform (6.16).Supposethatthereexistsaf -unctionylt,q)such thatthe nonconstantextrem alsfortheconstrained problem satisfjr

t gfv DL t% èdiô4 = = s ( f , c j ) k :vk ô4k, --

6.3 N onholonom ic C onstraints in M echanics*

W e digress briefly here to discuss the ticklish subjectof nonholonom ic constraints that occur in problem s from classicalm echanics.W e m ust first set the record straightconcerning our use oftheterm ttnonholonom ic.''The m echanics connoisseur is doubtless affronted by our slovenly use of this term

6 Holonom ic and Nonholonomic Constraints

as a labelfor any constraintgiven as a differentialequation.From our perspective,it is a handy catch-all term for such constraints, and it has the pleasing m erit thatwe need not continually distinguish nonintegrable from integrable constraints.M athem atically,Theorem 6.2.1 is valid forintegrable and nonintegrable constraintsalike,so thedistinction isnotim portant.From a m echanics perspective,however,the term isalways used in its pure sense:

a nonholonomiccondition isa differentialequation (orsystem ofdifferential equations)thatisnotintegrable.Onecannot,even in principle,convertsuch a constraintto a holonom ic onew ithoutessentially solving theproblem first. The distinction in m echanicsisim porta. ntnotso m uch form athem aticalreasons,butforphysicalreasons:a m ore generalvariationalprinciple isneeded to derivethe equationsofm otion forproblem sw ith nonholonom ic constraints in m echanics. Typically,nonholonom ic conditions in m echanics are ofthe form

i.e.,linearin thegeneralized velocities.Such constraintsarise,forexam ple,as

ttno slip''conditionsfor rolling objects.Forinstance,theproblem ofa penny rolling on the horizontalzp-plane such that the disc is always vertical has constraintsofthe form

: il- asinp4'- () #+ acosp4'- 0 where a is a constant,0 is the angle betw een the axis of the disc and the

z-axis,and4 istheangleofrotation aboutthediscaxis.A constraintofthis form cannotbereduced to a holonom icone. G iven the prom inence of nonholonom ic constraints in m echanics, the reader m ightw onder w hy we have studiously avoided them in the previous section.The directansw eris thatthe ttno frills''version ofHam ilton's Principle given in Section 1.3 is generally not applicable to these problem s.The appropriate principle forthese problem s is dhytlem berths P rinciple,w hich

statesthatthetotalvirtualworkoftheforcesiszero forall(reversible)variationsthatsatisfjrthe given kinem aticalconditions.Here,the forcesinclude

impressed forces along with inertialforces (forcesresulting from a mass in accelerated motion). Loosely speaking,wecan think ofd'A lem bert'sPrinciple asthe condition = 0.H am ilton's Principle com es from d'Alem bert's Principle by integration w ith respect to tim e.Forholonom ic problem swe have (q L

tl

tl (qL dt =

te

J

L dt, te

but,asPars(591(p.528)pointsout,fornonholonomicproblems

6.3 Nonholonom ic Constraintsin M echanics' e tl

tl

qLdt, #J

(

te

te

Ldt,

in general.

Fornonholonomicproblemswith?rzconstraintsoftheform (6.31)d'Alembert's

Principle yieldsequationsofthe form d (' ?z,

()L

'''

t) ' -@'ov oqj-k., ïllfzktqltzkptf,q), -

whereL isthe (unmodified)Lagrangian,i.e.,T - U,and thef-unctionsp. k are m ultipliers to be determ ined along with q using the n,Euler-lsagrange

equationsandthe?rzdifferentialequations(6.31).lngeneral,thesystem (6.33) isnotequivalentto the Euler-lsagrangeequationsofTheorem 6.2.1using the

modified LagrangianL - (, L1. t y1+ ...+ hvrvgvrv),becausethisapproach assumes thatHamilton'sPrincipleisvalid.Pars(Jt?c.cï/.)givesaninsightfuldiscussion of Ham ilton's Principle as it relates to nonholonom ic problem s and gives a

simpleconcreteexample to illustraterelation (6.32).Therolling penny and othernonholonomicproblemsaretreated indetailby Pars(op.cï/.),Webster (721,andW hittaker(731. T here appear to be divergent stream s of thought regarding the rôle of

Hamilton'sPrinciple in nonholonomic problems.Goldstein (351and others m aintain that H am ilton's Principle can be extended to cover nonholonom ic

problems' ,Rund (631statesthatsuch aprincipleisnotapplicableto nonholonom ic problem s.The confusion ofopinion on this m atterisin no sm allpart dueto differentinterpretationsofH am ilton'sPrincipleand theuseoftheterm nonholonom ic.W e end thissection with the follow ing quote from Goldstein

(op.cï/.,p.49)thatperhapsbringstherealissueinto perspective. ln view of the difllculties in form ulating a variationalprinciple for nonholonom ic system s,and the relative ease with which the equations ofm otion can be obtained directly,it is naturalto question the usef-ulness ofthevariationalapproach in this case.

Thispage intentionally Jc.# blank

P rob lem s w ith V ariab le E ndp oints

7.1 N atural B oundary C onditions The flxed endpoint variationalproblem entails finding the extrem als for a

functionalsubject to a given setofboundary conditions.For a functionalJ ofthe form

21

/(z,ïl,t v')dr 20

theseboundaryconditionstaketheform ylzçj)= yç j,!/(z1)= 3/1,whereyç jand yï are specified num bers.lfthe functionalcontains higherorderderivatives, then m ore boundary conditionsare required.Variationalproblem sarising in physicsand geom etry,however,arenotalwaysaccom panied by the appropriate num ber ofboundary conditions.For exam ple,the shape ofa cantilever beam issuch thatthe potentialenergy is m inim um .At the clam ped end of

thebeam we have boundary conditionsofthe form 3/(0)= 0 and 3//(0)= 0

reflecting the nature ofthe support.A t the free end,how ever,there are no conditions im posed on y.lndeed,it is part ofthe problem to determ ine y and y?atthis end.Now ,the differentialequation describing the shape ofthe beam isoffourth order,and four boundary conditions are thus required for uniqueness.W eexpecta unique solution totheproblem and hencetherem ust besom eboundary data im plicitin thevariationalform ulation oftheproblem . W e discussthisproblem further in Exam ple 7.1.3. O ne ofthe striking features ofcalculusofvariationsis that the m ethods always supply exactly the right num ber of boundary conditions.There are essentially tw o types ofboundary conditions.T here are boundary conditions thatareim posedon theproblem such asthoseattheclam pedendofthebeam , and there are boundary conditionsthatarise from the variationalprocess in lieu ofim posed conditions.The latter type ofboundary condition is called a naturalboundary condition.Even ifno boundary conditionsare im posed,the processtakescareofitselfand,asw eshow ,thecondition thatthe functional be stationary leads to precisely the correct number ofboundary conditions forthe problem .

7 Problem swith Variable Endpoints

Let J :(72g zt),z1j --+ R be a functionalofthe form (7.1),where / is a s11100th function.W e consider the problem of determ ining the functions

y CE(72gzt),z1jsuch thatJ hasan extremum.Noboundary conditionswillbe im posed on y.ln this section,we derive a necessary condition for J to have an extrem um at y.

Suppose that J has an extrem um aty.W e can proceed asin Section 2.2

byconsidering thevalueofJ ata ttnearby''f -unctionL.Let b= #+ 6r/,

wheret îisasmallparameterandpCE(72gzl),z1j.Sincenoboundaryconditions are imposed,we do notrequire p to vanish attheendpoints (Figure 7.1). Following the analysis ofSection 2.2,the condition thatJIL)- J(y) be of order 62as t î--+0 leadsto the condition

Jzo 21qyl ( r % )j.yytt ( ' ? )t v /'(gy.( ; (cf.equation (2.6)),and integratingtheterm containingp?by partsgivesthe condition

' ?/ zt+ ztr/ (' ?/ - d ( '?/ dr = 0. r/( ( ' Ly? zo zo (' Ly X t ' ?t v? Forthefixedendpointproblem theterm pt' ?/ ç 'ly?vanished attheendpoints

becausep(zo)= p(z1)= 0.Forthepresentproblem thisterm doesnotvanish

7.1 NaturalBoundary Conditions

137

forallp underconsideration.Nonetheless,equation (7.2)mustbesatisfiedfor allp CE Czgzt),zlj,and in particularthe subclass f. f offunctions that do vanish attheendpoints.Sinceequation (7.2)mustbesatisfied forallp CE. bI the argum entsofSection 2.2 apply and therefore

dt ' ?/ t' ?/ 2V % ? t'??/ = 0, -

forany y atwhich J hasan extremum.Equation (7.2)mustbe satisfied for allp CECzgzt ),zlj,however,and thisincludesfunctionsthatdo notvanish at theendpoints;consequently,equations(7.2)and (7.3)implythat r/:/? - r/:/? = 0, ç ' )y zt ç ' )y zo

t9/

=

0.

Sim ilarly,we can find f -unctions thatvanish at zl butnotat zo.Thisobservation leadsto the condition

t9/

=

0.

ln summary,ifJ hasan extremum aty CECzgzt),z1j,and there are no im posed boundary conditions,then y m ustsatisfjrthe Euler-lsagrange equa-

tion (7.3)alongwith equations(7.5)and (7.6).Equations(7.5)and (7.6)are

relationsinvolvingy and itsderivativesattheendpoints' ,i.e.,they areboundary conditions.Because these conditionsarise in thevariationalform ulation ofthe problem and not from considerationsoutside the f-unctional,they are called naturalboundary conditions. T he above processis com pletely ttm odular''in thesense thatifboundary conditionsareim posed ateachend,then thevariationalform ulation requiresp tovanish attheendpoints,and thustherearenonaturalboundary conditions. lfonly one boundary condition is im posed at say zo,then p is required to vanish atzt lbutnotatzl;hence,the problem issupplem ented by thenatural

boundary condition (7.5).lfno boundary conditionsare imposed,then we have130th naturalboundary conditions. Exam ple 7.1.1: D eterm ine a function y such that the functional 1+ y?2dz is an extrem um .

7 Problem swith Variable Endpoints

G eom etrically,the above problem correspondsto the problem offinding a curve y with one endpointon the linez = 0 and the other on the line z = 1 such that the arclength ofy is an extrem um . lntuitively,we see that any function ofthe form y = const.w illproduce a curve ofm inim um arclength. Letusseeifthenaturalboundary conditionslead us to thisconclusion. A ny extrem al to the problem m ust satisfjrthe Euler-lsagrange equation

(7.3).From Example2.2.1weknow thatsolutionsofthisequationareofthe form y = zrzz + 5,i.e.,linesegm ents.N o boundary conditionsare im posed on

theproblem and hencethenaturalboundary conditions(7.5)and (7.6)must be satisfied.N ow ,for y = zrzz + 5,

t 9/ = t 'V

y? = 1+ y?2

zrz , 1+ m2

so that the natural boundary conditions are satisfied only if?rz = 0.This m eansthaty = 5,w here thereis no restriction on the value ofthe constant 5. Exam ple 7.1.2: C atenary Suppose thatwe revisitthe catenary problem butim pose only one boundary condition.W e thusseek to find a function y such thatthe f -unctional

isan extremum subjecttotheboundarycondition 3/(0)= /J> 0. T he generalsolution to the Euler-lsagrange equation for this functional was determ ined in Exam ple2.3.3.H encew eknow thaty is ofthe form

/J= slcoshtsz). N o boundary condition has been im posed at z = 14consequently,y m ust

satisfy thenaturalboundarycondition (7.5).Therefore, t' ?/ = : v(1): 4?(1) = 0. (' ly?z=1 1+ y?2(1) Since/J> 0,sl# 0and consequently3/(1), # 0.Wemustthereforehavethat 3//(1)= 04i.e.,

7 Problem swith Variable Endpoints

along with the condition

' ?/ - )) d t '?/ - t' ?/ Tl?t

(. y#??

zt= (). j( ' /oyt? oy? za .

Equation (7.8)spawnsthefourboundaryconditions

forthefourth-order ordinary differentialequation (7.7).The proofofthese assertionsisleft as an exercise. Exam ple 7.1.3: W eapply theaboveresultsto thestudyofsm alldeflections of a beam of length f having uniform cross section under a 100 .1 Let y :

(0,f1--+R describetheshapeofthebeam and p :(0,f1--+R be theload per

unit length on the beam .A ssum ing sm alldeflections,the potential energy from elasticforcesis

The totalpotentialenergy isthus

Theshapeofthebeam issuch thatJ isa m inim um ;therefore,y m ustsatisfy

theEuler-lsagrangeequation (7.7),and thisproducestheequation (ï,J)(z)- P(: r). y S

Equation (7.13)hasthegeneralsolution 1ThisexampleisbasedononegiveninLanczos(481,p.70.

7.1 NaturalBoundary Conditions

'

z--jvp(-)/-, and thedifferentialequation (7.13)impliesthat s(: (/??(t)- : 4 ///(0))= F. Theterms3////(0)and!/???(tr )can beinterpreted asthereaction forcesatz = 0 andz = tr ,respectively,tokeepthebeam inequilibrium.Themoment(torque) produced by the im pressed force is

and hence

lfwe sum the momentsatthe z = 0 end ofthebeam,the term f!/???(f) is themomentproduced by thereaction forceatz = f,and thetermss!/??(0) and s!/??(f)can beinterpreted asthereaction momentsatz = 0and z = f, respectively. H aving m ade a physicalinterpretation of the higher derivatives of y at the endpoints,w e now exam ine the problem under a variety of boundary conditionscorresponding to supportsystem sforthe beam . C ase 1:D ouble C lam ped B eam

Supposethatthebeam isclampedateachend (figure7.3).Thebeam istflxed inthewall''ateachendsothatatz = 0wehave3/(0)= 3//(0)= 0,andatz = f wehavey(f)= y?(f)= 0.Here,therearefourimposed boundary conditions. Alltheallowablevariationsin thisproblem requirethatp(0)= p?(0)= 0and p(é)= p?@)= 0 so thatnonaturalboundary conditionsarise. C ase ll: C antilever B eam

Supposethatthebeam isclamped atz = 0 (figure 7.4).Then theboundary conditions3/(0) = 3//(0) = 0 are imposed.No boundary conditionsare imposed at the otherend ofthebeam and,consequently,the naturalboundary

conditions (7.10)and (7.12)must be satisfied.z Equation (7.10)yieldsthe relation

8J

??

ç ' ly'?z=z= ny (é)= 0, 2 The assum ption here is made thatthe unclamped endpointofthe beam stilllies

onthelinez= f(smalldeflections).

7 Problem swith Variable Endpoints

(p( x) )7/)= 0 )7:/)= (

y(0)= 0 y:(0)= 0

Fig. 7.3.

(p( x) y(0)= 0 y:(0)= 0

Fig. 7.4.

d t ' ?/ #

-

t' ?/ #

= s? y???(é)= ()

which statesthatthe reaction force atz = i'is zero.ln view ofthenature of thecantileversupport,the naturalboundary conditionsreflectthephysically evidentsituation atz = frequired forequilibrium . C ase 111: Sim ply Supported B eam Suppose that the beam is pinned at the ends,but no restrictions are m ade

concerning the derivatives ofy atthe endpoints (figure 7.5).The imposed boundary conditionsare3/(0)= 0 and y(f)= 0.Norestrictionsaremadeon the values ofp?at the endpoints,and hence we have the naturalboundary

7.1 NaturalBoundary Conditions

(p( x) y(0)= 0

conditions(7.9)and (7.10).Thenaturalboundary conditionsare3///(0)= 0 and !/??(t)= 04i.e.,thereaction momentsateach endpointarezero. C ase lV : U nsupported B eam Supposethatthebeam isunsupported.Then there areno im posed boundary conditions and we need all four naturalboundary conditions.The natural

boundary conditionsare3///(0) = 0,3////(0)= 0,!/??(é)= 0,and !/???(é) = 0. Theseconditionsstatethatthereaction forceand m om entateach end ofthe beam are zero.Note thatthe boundary conditionsalso im ply

byequation (7.15),and

by equation (7.16).Butthefunction p(z)isprescribed and may ormay not satisfyequations(7.17)and (7.18).Thenaturalboundary conditionsthustell usthattheproblem has a solution only ifp is such thatthetotalim pressed force and totalim pressed m om entiszero.A gain,the naturalboundary conditionslead usto physically sensible requirem ents. T he unsupported beam affords a glim pse ofa resultknow n asthe Fredholm alternative.The Fredholm alternative is usually encountered in the contextofintegralequations,butitisageneralresultapplicabletolinearoper-

ators.Briefly,themajormathematicaldifferencebetween CaseIV and theear-

liercasesisthatthehomogeneousequation: 4/(f' &)(z)= 0withthegivenboundary conditions has only the trivialsolution in the first three cases,whereas,

in thefourth case,therearenontrivialsolutionsoftheform ylz)= c1z+ ct l available.S In particular, w ehavethe nontrivialsolutions Fl= z and ' l' r tl= 1. TheFredholm alternativestatesthattheoriginalboundary-valueproblem w ill havesolutionsonly ifp isorthogonalto 130th ' l'r t land F1;i.e., 3 ln general, the Fredholm alternativeisconcernedwith solutionstothe adjointof

thehomogeneousequation.Here thelinearoperatorisself-adjoint.SeeHochstadt

(391orKreyszig(461formoredetailsontheFredholm alternative.

7 Problem swith Variable Endpoints '

(yt l,p)-jvp(z)clz-( ) and

Exercises 7.1:

1.Forthebrachystochroneproblem (Example2.3.4),letzt l= 0,zl = 1. Giventhecondition3/(0)= 1show thattheextremalsatisfiesthecondition 3//(1)= 0 and find an implicitequation for3/(1). 2.A sim plified version ofthe Ram sey grow th m odelin econom ics concerns a functionalofthe form

H ere,J corresponds to the tttotalproduct,''M is the capital,and the ck are positive constants.T he problem is to find the best use ofcapital

suchthatJ isminimizedinagivenplanningperiod (0,T1.Now,theinitial capital. JUf(0)= M ç jisknown,butthefinalcapital. JUf(T)isnotprescribed. U se the naturalboundary conditionsto find the extrem alforJ and the

finalcapital. JUf(T). 3.Derivethenaturalboundaryconditions(7.9)to(7.12)forf-unctionalsthat involve second-orderderivatives.

4.Letq = (t 71,...,qnland tl

J(q)-

te

Llt,q,ù)dt.

D erive the naturalboundary conditions that an extrem alm ust satisfjrif

neitherq(fo)norq(f1)areprescribed. 7.2 T he G eneral C ase ln thelastsection we considered problem swhereperhapsno boundary conditionsare prescribed.A lthough thevariationsneed notsatisfjrthesam econdi-

tionsattheendpoints,thez coördinatesoftheendpointsremained fixed (cf. figure 7.1).Even thisrestriction isnotsuitableforcertain variationalproblem s.ln thissection we considerthe generalcasew here130th theindependent and the dependentendpoint coördinatesm ay bevariable.

Lety :gzt),z11--+R bea s11100th function thatdescribesa curvey with endpoints Ptl= (zo,y(tl and P1 = (z1,3/1),and let : ' 2 :(:/0,J @?11 --+ R be

7.2 The GeneralCase

Fig. 7.6.

a s11100th function that describes a curve y m with endpoints Pn() =

(:0,: ' kl and Pn1= (:1,j1)(figure 7.6).Fortheensuing analysiswewish to compare curvesthatare ttclose''to each other' ,however,the functions y and :' èare not necessarily defined on the sam e interval,and the norm s discussed in Section

2.2 are not suitable.W e can nonetheless extend the definitions of y and : ' 2

so thatthey aredefined overa common interval.Let:tl= mintzt),J @?()jand : /1= maxtzl,:1/.Asweareinterestedinsmallvariationsony,wecanextend thefunctionsy and : ' 2to the common interval/0,J iillusing a Taylorseries approxim ation w here necessary.For exam ple,ifJ iit l= zt l and zl < :1,then we can extend the definition ofy asfollow s,

to geta function y% CECzg: @t ),:1j.Weassume thatallsuch extensions have been madeand retainthesymbolsyand (.Wedefinethedistancebetween y and : ' èas

whereIPk- iz'kI= lzk- J îlklz+ (yk- V)2.l-lere?11'1isthenorm defined by

7 Problem swith Variable Endpoints

sup I?/(z)I,

z(Eye,t ' é1j Or

11: v11- sup I: v(z)I+ sup I?/'(z)I,

zeEt éo,t ' ' é1q zelt ' éo,t ' é1q whicheverisappropriate totheproblem underconsideration.LetJ be afunctionalofthe form

:1

J(5)-

:o

/(z,b,7)dz.

b= !/+ 6r/,

wherepCE(72g tijl),J iilj.Noconditionsasidefrom thissmoothnesscondition are prescribedon p,butthecondition d(L,y)= 0/)requiresthatthequantities kk- zkand Lk- T/kbeoforder6.Let kk = zk + 6. Xk,

bk= VkV 6Vk, fork = 0,1.Then,

and since t îissm all

7.2 The GeneralCase

W e thereforehave

+ 0/2).

b- ylkù)- ylzù+ 67/0)+ 6ptzo+ 67/0) =: % -1-6V .

r/lzol= ' l'r o- Xèy'(tro)+ 0(6). Sim ilarly atthe otherendpoint

r/(z1)= 1'r l- .X1!/'(z1)+ 0(6). Substituting relations(7.20)and (7.21)intoequation (7.19)yields

ThefunctionalJ isstationary aty and thereforethe term sofordert îm ustbe zeroforallvariationsin theaboveexpression.W ecan alw ayschoosevariations

such thatXk= Fk= 0 (i.e.,lkxed endpointvariations).Arguingasin Section 7.1we therefore deducethaty m ustsatisfjrthe equation

7 Problem swith Variable Endpoints

dt 9/ t9/ 2V % ? tg?/ = 0. -

ln addition,y m ustsatisfy theendpointcondition 21

p6y - fftîz

= 0, 20

where

:/

p = ôy?,

H - y'p- /,

tîy(zk)= Fk, Jztzk)= Xk. Equation (7.23)isthestartingpointformorespecializedproblems.These problem sconcern variationsw heretheendpointssatisfy relationsofthe form

gklalo,#o,trl,#1)= 0. Evidently,no m orethanfoursuch relationscan beprescribed,sincefourequa-

tionswould determinetheendpoints(assuming therelationsarefunctionally independent).Thefixedendpointproblem thuscorrespondstothecasewhere foursuch relationsare given.T he naturalboundary problem softhe previous

section correspond tothree (ortwo)such relationsimposed on theproblem. Forexample,thecaseofonefixed endpoint,say (zo,y(j),ischaracterized by the three conditionszt l= ct azzsf.,yç j= ctm sf.,and zl = const.These equa-

tionsarethen supplementedby thenaturalboundarycondition at(z1,3/1)to providethefourth equation.ln thisproblem only thevariation ( qyat(z1,3/1) isarbitrary. Typically,variationalproblem s com ew ith relationsofthe form

gklrj,#. j)= 0, forj= 1,2sothattheendpointvariationsof(zo,y(tlarenotlinkedtothoseof (z1,3/1).ln thiscasewecanalwaysincludevariationsthatleaveoneendpoint fixed,and thisleadsto thetwo conditions

ln the next section we focus on variational problem s with endpoint re-

lations ofthe form (7.24).Geometrically such relations correspond to the

7.2 The GeneralCase

requirementthatan endpoint(zk,yk)1ieon thecurvedefinedbytheimplicit equation glzk,yk)= 0. lt is worth noting that,in general,som e relationship m ust be im posed am ong the endpoints to getcom patible boundary conditions.O therwise,the situation ism uch likethe unsupported beam in the previoussection.ln particular,suppose that no relations are im posed on the endpoints.C ertainly

equations (7.25)and (7.26)are satisfied,butsince éz and ( qy are independentand arbitrary ateach endpoint we have thatp = 0 and . bI = 0 at each endpoint.Hencewe have the boundary conditions

t' ?/ ç' ly?

=

0,

J = 0, thatm ustbe satisfied ateach endpoint.Since any extrem alm ust also satisfy

theEuler-lsagrangeequation (7.22),theboundary conditions(7.27)imply zto.

Jzo' gy ft f''-0* ln addition,we know that

dH

t 9/

dz

dz

(cf.Section 2.3)4hence,theboundary conditions(7.28)alsogive 21(' ?/dz = 0.

20 t' ?z

Equations(7.29)and (7.30)poseadditionalrestrictionsonythataregenerally notcom patiblew ith theEuler-lsagrange equations.Forinstance,supposethat

/ doesnotdepend on z explicitly.Thenweknow that. bI= const.along any

extremal(Section2.3).Since. bI= 0attheendpointswehavethat. bI= 0for allz and hence

Jly,y')- A(y)y'. Finally,we note thatthe aboveargum entscan beextended to copew ith functiona. lsthat depend on severaldependentvariables.Let

whereq = (t 71,q,,...,qnland L isas11100th f -unction.lfJ isstationary atq then itcan be shown that

7 Problem swith Variable Endpoints

@@ô4k *vk = O -

fork = 1,...,zz,and n

E rkt î' ?k-Hôt-0 k= 1 at the endpointstç jand f1.H ere,the quantitiespk and f' f are defined as DL

Pk= ô4k,

(7.33)

and 6qk,éfaredefined in a m anner analogous to ( qy and éz. Exercises 7.2:

r ransversality C onditions

and consider the problem offinding s11100th f-unctions y such that J is sta-

tionary,atoneend ylzçj)= yç j,and attheotherend yisrequired to lieon a curve F described param etrically by

r($)- tzr($),yr($)), forï CER.We know from Section 7.2 thatany candidate for a solution to

thisproblem mustbea solution totheEuler-lsagrangeequation (7.22)that passesthrough thepoint (zo,y(tland intersectsthecurveF (figure7.7).A solution to thisproblem,however,willalso have to satisfjrequation (7.26), and thismay (andgenerally does)limitthechoiceofextremals.lfwereturn to the analysisoftheprevioussection forthisproblem,we know that? )(z1) and ;(z1)arerelated through equation (7.34)4i.e.,allvariationsmusthave an endpoint on the curve F .This m eans that we can associate the ttvirtual

displacement''(qy atz = zlwith dyr dl and the tt virtualdisplacement''Jz

with dzr d1.Condition (7.26)thusbecomes

7.3 Transversality Conditions

dyr

dzr

dlp- dl H = 0,

wherep and . bI are evaluated atz = z1.ln thisfram ework we do notknow a

prioriwhatvaluetoassign to z1,butwedo know thatthepoint(z1,!/(z1)) lieson thecurveF.lfweknow eitherzlor!/(z1),then wewould alsoknow atwhich value ofï to evaluatethe derivativesin equation (7.35).We can thusregard equation (7.35)as eitheran equation for ï oran equation for z1.Geometrically,thevector (t/z. r d1,dyr d1)isa tangentvectoron F.lf v = (p,-f. f),we seethatequation (7.35)correspondsto thecondition that v beorthogonalto thetangentvector.Equation (7.35)issometimescalled a transversality condition. Evidently,the above analysis can be readily extended to cope w ith the problem offinding extrem a for J w hen one endpoint is required to be on a curve Ft land the otherendpointon a curve F1.lfthecurveFtlisdescribed by

(zro(c),t vro(c)),c e Eco,czlandthecurvez-lby (zzj($),yn ($)),. i'CE( $(),&j,

then

dyr, t/zz'o d nLI = 0 a ' da t/t vz' t

t/zz't

dl p- dl LI = 0.

Exam ple 7.. 3.1; Let

7 Problem swith Variable Endpoints

(x1,y1) r

y

A .

=

r

I,

r(o =(xst(y),yst(y)) > 1

(0,0) Fig. 7.8.

andconsidertheproblem offindingthefunctionts)yforwhich J isstationary subjectto thecondition that3/(0)= 0 and that(z1,!/(z1))lieson thecurve described by (7.34). G eom etrically,we are finding the distance ofa plane curve F from the origin.The extrem alsfor thisproblem willbe line segm entsthrough the origin,and we seek am ong the segm entsthat intersectF the one forw hich the

arclength isan extremum (figure7.8). Forthisproblem ,

t' ?/

#= t g#? =

W e thus have

dyr

dl

y'

y? 1+ #?2 '

dzr

1 + + y?2 dl

1

1+ y?2

= 04

7.3 Transversality Conditions

(dd zr yr)'(1,d dy)- 0. l , ddl z Geometrically,equation (7.39)impliesthatfora stationary value ofJ,the tangent to the extremal(i.e.,the line segment)must be orthogonalto the tangent to F .

A bitofreflection showsthatcondition (7.39)can yield any numberof

solutionsdepending on the curve F .lfforinstanceF isan arc ofacirclecentred atthe origin then any extrem alwillsatisfy the orthogonality condition. Forillustration,letussuppose thatF correspondsto the curve described by 1

a r($)- ($- 1,. i' + j.),

forï CE R.We know that the extremals forthisproblem are ofthe form y = zrzz.Now ,

(d( zr y y y ,d( yr)'(1,d (s )= (1,20(1,zrzl= 04 21m,+ 1= 04 ï=

-

1 2w

.

Theextremaland F havethepoint(z1,!/(z1))in common and therefore

The above relation provides two equationsforzl and zrz.A fter som ealgebra we see that?rzm ust satisfjrthe relation 4w 3+ 1 = 0.

and hencetheonlyextremalsatisfyingcondition (7.39)is

7 Problem swith Variable Endpoints

ro(c)- (-a2,c), r,($)- ($,($- 1)2), wherec,ïCER,respectively.W econsidertheproblem offindingan extremal

y forJ subjectto thecondition that(zo,y(tllieson Ft land (z1,3/1)lieson F1.W e know from Exam ple 7.3.1 that the extrem a. ls m ust be of the form y = zrzz + b forsom e constants?rzand 5.The f-unctionsp and . bI are given by

equations(7.37)and (7.38),respectively;hence, p=

and

S

=

1+ w 2 ,

1 . 1+ w 2

-

Thetransversality conditions(7.36)thusimply ?rz- 2c# = 0

2za(ï#- 1)H-1==0,

(7.40) (7.41)

and sim ilarly

and the relations

j'#= 1 - 1 . 2m Transversality conditionscan be derived forproblem sthatinvolve several dependent variables.Consider,for exam ple,the problem offinding s11100th

functionsq = (t 71,q,)suchthatthef-unctional tl

J(q)-

te

Llt,q,il)dt

7.3 Transversality Conditions

isstationary subjectto the condition thatq(fo)= qt l(flxed endpoint)and q(f1)isrequiredtolieonasurfaceE givenbyt= #(q).Evidently,anextremal to theproblem mustsatisfytheEuler-lsagrangeequations(7.31)and thetwo boundaryconditionsgivenbythefixedendpoint.W ecan glean theappropriate

boundaryconditionsattheotherendpointfrom equation (7.32). Equation(7.32)mustbesatisfied forallvariationsnearq withanendpoint on E .ln particular,w ecan considervariationsw ith endpointsq, = const.on

E.Forthisspecialclassofvariationsequation (7.32)gives 3?lét ?l- V éf= 0,

DL # :# = 0. t %2 ç' kz -

Exam ple 7.3.3:

q(0)- 0, and

W eseekanextremalforJsubjecttotheconditionthattheendpointq(f1)lies on thesurfacedefined by ' ?).Geometrically,theproblem amountstofinding

thecurvein1:. 3from theorigintothesurfacedefinedby' ?û(aconewithvertex at(1,1,0))such thatarclength isminimum. T he Euler-lsagrange equationsshow thatq isofthe form

(j= ctf+ p,

Now,

7 Problem swith Variable Endpoints

Pk =

1

kk + (1 2+ jz 2,

and

ff = r1I1+ /212- f 1

1+ 11 2+ (2

so thatthetransversality conditions(7.45)and (7.46)yield kk + t' ?' ? ) = 0, ç' kk fork = 1,2.G eom etrically,the above condition indicatesthatat the flend-

pointthetangentvectorto the extremal(11,t %)is parallelto V' ? );i.e.,the extrem alisnorm alto the surface.The transversality condition reducesto

t ?k(f1)- 1 (vz(fz)- 1)2:-(v2(f1)- 1)2

ctkfl- 1 fl

hence,

Equation (7.47)implies

2 = j. , ct2 1+ (22

The extrem alisthusgiven by the line t

qklt)- W ,

Exercises 7.3 :

1.The functionalfor the brachystochrone is zl j. y.yt2

J(t v)-jv y dz. Find an extremalfor J subject to the condition that 3/(0) (z1,!/(z1))lieson thecurvey= z - 1.

7.3 Transversality Conditions

2.Let

21

J(t v)-

o

(y'2+t v2)dz.

Find an extremalfor J subject to the condition that 3/(0) = 0 and (z1,!/(z1))lieson thecurvey = 1- z.Determine theappropriate con-

stantsin term sofim plicitrelations. 3.Lagrangem ultipliersprovidean alternativeapproach to derivingtransver-

salityconditions.Considertheproblem wherethe(z1,3/1)endpointisrequired tobeon thecurvedefined byglz,y)= 0,and let

where , L isa Lagrangemultiplierand :' è= y+ 6p.Derivethetransversality condition

t ' ?g t ' ? J.= 0 P +V t' ?z ç ' )y ...

at(z1,3/1).(1n thisproblem,theJz and ( qyvariationsareindependent.) 4.Letq = (t 71,q,)and consideraf-unctionaloftheform tl

J(q)-

zztf,q) 1+ 41+ t 11dt,

to

alongwith the boundary condition q(fo)= qtl(flxed endpoint)and the condition that the fl endpointlieon a surface E defined by t= #(q). Show thattheextrem als m ustbe orthogonalto E .

5.Letq = (t 71,q,)and tl

J(q)-

0

(4 +4 +2qïq,4dt.

Giventheconditionthatq(0)= 0andthatfl= t 71(f1)determinetheform

oftheextrem alforJ and derive the im plicitequationsforthe integration constantsand f1.

6.Letq = (t 71,...,qn).Derive the generaltransversality conditionsfora functionalJ to havean extremum subjectto one endpointfixed and the otherendpointon a hypersurfacedefined im plicitly by .t y(q,t)= 0.

Thispage intentionally Jc.# blank

T he H am iltonian Form ulation

G iven the existence ofa certain transform ation,the n,Euler-lsagrange equationsassociated with avariationalproblem can beconverted intoan equivalent system of2/,first-order ordinary differentialequations.These equations are called Ham ilton'sequations,and theyhavesom especialproperties.ln particular,thederivativesin thissystem areuncoupled,and thedifferentialequations

can bederived from asingle (scalar)f-unction called theHamiltonia. n.Given

a Ham iltonian system ,anotherH am iltonian system can be constructed by a specialtype oftransform ation called a sym plectic m ap.ltm ay be possibleto find a sym plecticm ap thatproducesa Ham iltonian system thatcan besolved and thereby used to solve the originalproblem .The search for such a m ap leadsto a partialdifferentialequation called the Ham ilton-lacobiequation. ln this chapter w e discuss the connexions between the Euler-lsagrange equations and Ham ilton's equations.W e first discuss a certain transform ation,theLegendretransform ation,and then useittoderiveH am ilton'sequations.Sym plectic m aps are discussed briefly in the third section,and the H am ilton-lacobiequation isthen derived.T he m otivation in this chapterfor the alternative form ulation issolving the Euler-lsagrange equations.W e thus focus on the use of the Ham ilton-lacobiequation as a toolfor solving certain variational problem s.ln particular,we discuss the m ethod ofadditive separation forsolving the H am ilton-lacobiequation.Thism ethod hasm any lim itations,butthere is a paucity ofanalytical techniques forsolving variationalproblem s,and the H am ilton-lacobiequation provides one additional

(albeitspecialized)toolthathasapplicationstoproblemsofinterest.Beyond

being sim ply a toolforsolving the Euler-lsagrange equations,the H am iltonJacobiequation is im portant in its own right.lt plays a centralrôle in the theory underlying the calculusofvariations.

8 The Ham iltonian Form ulation

8.1 T he Legendre r ransform ation The readerhasdoubtless encountered point transform ations and used them to solvedifferentialequationsorevaluateintegrals.A point transform ation

from onepair(z,!/(z))toanotherpair(. X,F(. X))consistsofrelationsofthe form

x = x tz,y) F = F(z,y). Anothertype oftransform ation that playsan im portant part in differential equationsand geom etry iscalled a contact transform ation.Contacttransform ations differ from point transform ations in that the functions defining the transform ation depend on the derivatives ofthe dependentvariable.O ne ofthe sim plest and m ostusefulcontacttransform ationsis called the Legendre transform ation. T histransform ation has som e rem arkableproperties and provides the link betw een the Euler-lsagrange equationsand Ham ilton's equations.W e considerfirst the sim plest Legendre transform ation involving one independentvariable. by

Lety :gzt),z11--+R bea s11100th function,and definethenew variablep

r --v'(z). (8.1) Equation (8.1)can be used to define thevariablez in termsofp provided !/??(z)# 0.Fordefiniteness,letussupposethat (//(z)> 0 : (8.2) forallz CE gzt ),zll.lnequality (8.2) implies that the curve y described by r(z)= (z,!/(z)),z CEgzt ),z11isstrictlyconvexupwardsinshape.Thenew variablepcorrespondstotheslopeofthetangentline(figure8.1).Geometrically, onecan seethatundertheseconditionsany pointon y isdeterm ined uniquely by theslope ofitstangentline.Suppose now thatwe introduce the f-unction

/ (r)- -t v(z)+ vz. (8.3) Here,we regard z as a function ofp.Equations (8.1) and (8.3)provide a transformationfrom thepair(z,!/(z))tothepair(p,# (p)).Thisisanexample ofa Legendre transform ation.A rem arkable property ofthis transform ation isthatitisan involution' ,i.e.,the transform ation isitsow n inverse.To see this,note that

8.1 The Legendre Transformation

161

wherewehaveused equation (8.1).Notealso that H(p)+ zp= - (-!/(z)+ pz)+ pz = !/(z). -

Thesecalculationsshow thatifwe apply theLegendretransform ation to the

pair(p,# (p))werecovertheoriginalpair(z,!/(z)). Example 8.1.1: Letylz)= z4 4.Then

s?y)-4 j.ty . s. s, z/s()-pk/z-z, and that

8 The Ham iltonian Form ulation

M any of the functiona. ls studied so far have integrands of the form

/(z,y,!/?).Supposethatweregard theargumentsof/ asthreeindependent variables,and define a new variablep as

:/

p = ç' ly?.

ln thistransformationweregard z and yaspassive variables(i.e.,notparticipating directly in the transformation) and y?asan active variable. ln otherwords,wearelooking foratransformation from thezl-tuple (z,y,!/?,/) tothez l-tuple(z,y,p,. bI4.Asinthepreviousexample,wecanregard equation (8.4)asa relation fory?intermsofp,provided t ' ?2/ / 0

.

(' )y?2

The function LI isdefined by

V lz,y,3?)= -/(z,y,!//)+ Vv. U sing the sam e argum ents asbefore,we see that this transform ation isalso an involution.

Example 8.1.2: Let/(z,y,!/?)= 1+ y?2.Then t' ?/

y?

# = ôy?= ? # =

1+ y?2 ; P

,

1- 2/ since y?and p m ustbe ofthe sam esign.Thefunction LI isthus

T he quantitiesp and . bI defined by the Legendretransform ation have already com e into prom inence in the theory.For exam ple,it is precisely the

quantity . bI thatisconstantalong extremalswhen / doesnotcontain z ex-

plicitly (Section2.3).Moreover,. bI appearsasaterm in thegeneralendpoint condition derived in Section 7.2.N ote that,for the passive variables in the transform ation,

8.1 The Legendre Transformation

DII

t9/

dz

dz

*#

*/

ôy

dp

Let us now consider a Legendre transform ation involving a f-unction

L(t,q,ù),whereL isas11100th functionand q = (t 71,q,,...,ty,zl.ln thistransform ation the variables tand q are regarded aspassive.Let

pk= A k,

fork = 1,2,...,zz.Equations (8.7)connectthe active variables t kk and pk. The implicitf-unction theorem can be invoked to show thatequations(8.7) can (in principle)besolved forthet kk provided then,x n,Hessian matrix 192L

()2L

192L

()2L

0%L

:2l

:2z,

î,t 91. î,t )1 ' î't jti î't ja ''' ' î't iti gla

.

X' lza=

:2z,

' gt iapt it ' gt japt ja ''' ' gt iapt ia p2o

:(a:(1 plapla ''' :(a:(a

isnonsingular,i.e.,satisfiesthe Jacobian condition

wherethevariables(kareregarded asfunctionsofp = (p1,pa,...,p,z),q and t.

TheLegendretransformation defined by equations(8.7)and (8.9)isalso

an involution.ln particular,

n -

ff(f,q,p)+

kkpk- L(t,q,ù). k=1

The function . bI in theabovetransform ation is called a H am iltonian function and the f -unction L is called a Lagrangian. T he ttnew'' coördinates

8 The Ham iltonian Form ulation

(f,q,p)are sometimescalled generalized coördinates.The setofpoints defined by thepairs(q,p)iscalled thephase space. ln m echanicsthevariablespk are called the generalized m om enta.The

namestemsfrom f-unctionalsmodellingthemotion ofparticles(Section 1.3). The integrand in thiscase isofthe form

Llt,q,ù)- T(f,q,ù)- t-tf,q), whereT isthe kinetic energy,U'isthe potentialenergy,and q representsthe positions ofthe particles attim e f.For a single particle ofm ass?rzin space,

ifq = (t 71,q,,t ya)aretheCartesian coördinatesoftheparticle,then T(f,q,ù)- 1 1 j.z'zltklz- . jzrzt/ + 4 + t 11); pk =

DL

=

:T

t' A k t' Xk

= Trzlk,

and pk isthusa componentofthem omentum vector.Forj particlesin space, we have n,= 3j and each pk isa component ofa mom entum vector. Exercises 8.1:

8.2 H am ilton's Equations Let J be a functionalofthe form

8.2 Ham ilton's Equations

@@ô4k *vk = O -

DL

Pk= A k, and

t kk = & - , /?k

where. bI isdefined by equation (8.9).Now tand q arepassivevariablesin thistransform ation so that

( ').FI ç ' )t

t' ?fv t ' ?f'

DII

t gfv

8qk

t' Xk

Sinceq isan extremalwehavefrom equations(8.13)that DL

d tgfv

t' kk = 70ç ''kk = * '

and thereforeequation (8.17)impliesthat >

DH

k= -gy . The solutionsq to the Euler-lsagrangeequations(8.13)are thusmapped to solutions(q,p)totheequations(8.15)and (8.18)undertheLegendretransformation.Conversely,supposethat(q,p)isa solutio:2 n to equations(8.15) and (8.18),andthatthen,xn,matrixwith elements(o ouq'k )isnonsingular;1 .

i.e.,

t ' ?(11,ti2,...,kn. l = detlklx ,# 0. t g(r1,r2,...,ra) Then equations (8.15)definea Legendretransformation from (f,q,p,.bI)to (f,q,(k,L)withL asdefined byequation (8.11).W ethushavethatequations (8.14)and (8.17)aresatisfied,andhenceequation (8.18)impliesthat (' ).FI t' M dpk (' ?. L d (' ?. L ( ')L 8 + =+ = - ws + = (),

qk t xk

dt

t xk

JfA k t xk

1Note that

0(t i1,t iz,...,(. al0(y'1,p2,...,y' . a) 1 0(y' 1,p,,...,pn)0(t i1,(2,...,t iw.)= sothatcondition(8.19)issatisfiedifand onlyifcondition (8.8)issatisfied.

8 The Ham iltonian Form ulation

so thatq isasolutiontotheEuler-lsagrangeequations(8.13).Theinvolutive characterofthe Legendre transform ation thus indicates that the problem of solving thesystem ofn,Euler-lsagrangeequationsisequivalentto theproblem

ofsolvingthesystem of2/,equations (8.15)and (8.18). Prim, aJacie,itseemsthatwehavegained little by exchangingn,secondorder differentialequations for 2/,first-order differential equations,but the new system of equations has som e attractive features.The Euler-lsagrange equationsare second order and generically nonlinearin the firstderivatives. M oreover,the firstderivatives in this system are generally coupled.The new equationsareoffirstorder.Thederivativesareuncoupled,and thesystem can be derived from a singlegenerating function,theH am iltonian.Thesystem of

of2/,equations(8.15)and (8.18)iscalled aHamiltonian system ,and the equationsarecalled H am iltonhs equations. ln the transition from the Euler-lsagrange equations to Ham ilton's equationsn,new variables,thegeneralized m om enta,areintroduced.ln Ham ilton's equations,theposition variablesq and the generalized m om enta variables p are on the sam e footing and regarded as independent.T hisapproach can be

somewhatconf-usingatfirstencountergiventhatq and(' l(henceq and p)are dependentin theoriginalproblem.zTheconcernhereisthatq and(karenot

independentand therefore cannotbevaried independently.ln contrast,q and p are independent in Ham ilton's equations and can thus be varied independently.ln fact,the Legendre transform ation thatdefinesthe new variables p also ensuresthatthesevariablescan be varied independentofq.To see this,

weintroducethef -unctionalj definedby tj

n

/(q,p)-

ptk - H lt,q,p) dt,

-

to

jzzz1

whereq and p areregarded as independentvariablesand t kk is the derivative

ofqk (i.e.,notregarded asafunctionofp).Evidently theintegrandsdefining

J and . 1areequivalentundertheLegendretransformation and hence J = J.

Suppose now thatwevary thepk butleave the qk fixed.Let 1 3 = p + 6/7,

tl

J-(q,1 3)- J-(q,p)-

n

H (t,q,p)- H (t,q,13)+ to

k (>f- pj) dt .

jzzz1

tl

=t ît

o

N.

L r).yg

pj - ç'èpj + t kj

dt+ (?(62).

j=ï

2 W e are not alone. lnhi sbookAppliedDIj-ferentialGeometryBurke(201addresses the dedication asfollows: t' i' o allthose who like me have wondered how in the

hellyou can change ( kwithoutchanging t ?.''

8.2 Ham ilton's Equations

167

J'(q,l3)- J'(q,p)- 0/2). Thiscalculation shows thatvariations on p do not affectvariations on j. Although q and p are independentvariablesin j,only thevariationsofq affectthevariation ofj.Thissituation isalso reflected in the derivation of

Hamilton'sequations.Specifically,equation (8.15)isvalid forany pair(ù,p) asitisapropertyoftheLegendretransformation;incontrast,equation(8.18) com es directly from the Euler-lsagrange equation.O nly functions q and p corresponding to an extrem alforJ w illsatisfy theseequations. Exam ple 8.2.1: Sim ple Pendulum Considerthe pendulum ofExam ple 1.3.1.The kinetic energy is 1

z

z

1

z

T=j.zrz(: il(f)+#(f))=j.zr? . zrp(j;, and the potentialenergy is

H am ilton's Principle im plies that the m otion ofthe pendulum is such that the f -unctional

J(/)isan extremum.Letq= 4 and

Then

p=

(' )L

( %

z

= w tyj,

so that

1= p

zrzéz.

The Ham iltonian . bI isgiven by

8 The Ham iltonian Form ulation

H am ilton'sequations arethus

For this exam ple .bI corresponds to the totalenergy ofthe pendulum .The Euler-lsagrangeequation is d tgfv

)@(%

-

DL

z

ôq = zz?f 4+ vrtgfsin q= 04

( = 0> 2 = vrtgf sin q = - g sin qt zrzé2 é in agreem entwith the Euler-lsagrange equation. Exam ple 8.2.2: G eom etrical O ptics

Let (z(z),ylz),z),z CE gzt),z11describe a space curve y.The opticalpath length ofy in amedium with refractiveindex zztz,y,z)isgiven by

Zlf,q,tk)= zzlf,q) 1+ 1ù12 iscalledtheopticalLagrangian.Hereq = (t 71,q,),zztf,q)= zztz,y,z),and 'denotesd dt.Thegeneralized m om enta are given by

8.2 Ham ilton's Equations

and hence

1't2 -

Pk 2 /)2 1 - pj j

and llam ilton'sequationsare

W ehavealreadyencounteredtheHamiltonian (inaslightlydifferentguise)

in Section 2.3.W eknow from C hapter2 that ifL does not contain thevariable texplicitly,then .bI is constantalong any extrem al.lt isclearfrom the Legendretransform ation thatL containstexplicitly ifand only if. bI contains texplicitly.Ham iltonian system sthatdo notdepend on texplicitly arecalled conservative.Thependulum in Exam ple8.2.1 isan exam ple ofa conservativesystem .ln thisexam ple. bI = const.correspondstothecondition thatthe totalenergy ofthe system is conserved.The H am iltonian system derived in Exam ple8.2.2isnotconservativeunlesstherefractive index isindependentof z.N otethata nonconservative system such asthisonem ay stillhaveconservation laws.N ote also thata nonconservativesystem can be converted into a conservativeoneby theintroduction ofa new ttposition''variablecorresponding to tin the originalform ulation and using a new variable for tttim e.''For

8 The Ham iltonian Form ulation

solving specificproblem s,however,thisobservation isoflim ited value because thefirstintegralafforded bya conservativesystem isoffsetby theintroduction ofa new dependentvariable.Nonetheless,it isa usefulobservation because, when convenient,w e can alw aysreform ulate a problem to get a conservative system and thususe generalresultsforconservative system s. Exercises 8.2:

1.The Lagrangian fora linearharm onic oscillatoris

where ?rzism assand k isa restoring force coefllcient.Show that

H lt, qtp)- 21 /2(ra+a?aqa), where = -

y

D erive and solvetheEuler-lsagrangeequation assum ing ?rzand kare constants.D eriveH am ilton'sequationsand verifjrthatthesolution obtained for the Euler-lsagrange equation is also a solution for Ham ilton's equations.

2.DeriveHamilton'sequationsforthecatenary (Exercises8.1-1).Verifjrthat thesolution found in Fuxam ple 2.3.3 isalso a solution ofH am ilton's equations.

3.Forany s11100th functions1(f,q,p)and 0 -(f,q,p),thePoisson bracket isdefined by

(*,

''

( )+ t '?t , ?l ( -)+ t '?t , ?l 81-/ ) ( ) ç ' k k t ' ? r k ô p k t ' k k. k=1

Let . bI be the Ham iltonian function associated w ith a functionalJ,and suppose thatalong the extrem alsfor J

*(q,P)= const. The function liisthen called a first integralofthe system .Show that

(*,V1= 0.

8.3 Sym plectic M aps

8.3 Sym plectic M aps

I

DH

k = 8pk,

fork = 1,2,...,zz.A sym plectic m apS is a transform ation ofthe form

Qk = Qklf,q,P), Pk = f? k(f,q,p), suchthattheHamiltoniansystem (8.21)transformsintoanotherHamiltonian system

Q .

k=

t :/z

,

( 3pk

where. H* isaf-unction off, Q ,and P. ln short,a symplecticm ap isa transform ation on the generalized coördinates thatpreservesthe Ham iltonian structure.Sym plectic m aps are also called canonical transform ations.T hese m apsloom largein theclassicalm echanicslore.Thereaderisdirected toAbra-

ham andMarsden (11,Arnold (61,Goldstein (351,Lanczos(481,andW hittaker (731among numerousotherworkson classicalmechanics.ln thissection we

briefly discusssym plectic m aps prim arily asa herald to the H am ilton-lacobi equation. W e know from the previous section that Ham iltonian system s such as

(8.21)and (8.23)can beassociatedwiththeextremalsto thef-unctionals

3 Theword sym plectic com esfrom the Greek word sumplektikos m eaning tt intertwined.' There isalso a bone in the skullofa fish by this name.

8 The Ham iltonian Form ulation

respectively,w here

lfw e regard q and p as independentvariablesand (' las the derivative ofq, then theEuler-lsagrangeequations forthef-unctional tl

J(q,p)-

to

n

pkkk- H (t,q,p) dt

kccc1

arepreciselyHamilton'sequations(8.21),andthesolutionstoequations(8.21) correspond to the extrem alsforJ.A sim ilarrem ark holdsforthe otherfunc-

tionalj W e say thattwo functiona. ls J and j are variationally equivalent if they produce the sam e set ofextrem als. A sym plectic m ap is essentially a

transformation from the (q,p)phase spaceto the (Q,P)phase space such thattheresultingf-unctionalsJ and j arevariationally equivalent. ln Section 2.5 w eshowed thatany nonsingular coördinate transform ation leadsto a variationally equivalentf-unctional.Thisresult can be extended to functiona. ls involving severaldependent variables.Transform ations that involve only position coördinates lead to variationally equivalent functionals and hence this class of transform ations is sym plectic.B ut transform ations of this type are too restrictive,and, in the spirit of the H am iltonian approach,we should let the m om enta variables participate in transform ations as independentvariables.The problem is,ifthe pk transform ,the resulting transform ation need notbe sym plectic. O ne m ethod for constructing sym plectic m aps involves the introduction of a generating f-unction.The m ethod is based on the observation that two functiona. lsare variationally equivalent iftheir integrandsdifferby a perfect

differential(cf.Exercises3.2-4).Suppose thatthereisa sm00th function li such that

Then thecorresponding functiona. lsJ and j arevariationally equivalentand thetransformation (8.22)issymplectic.Wecan usethetransformation (8.22) to convertlito a function off,q,and Q,and equation (8.24)can thusbe recastin the form

8.3 Sym plectic M aps

d+ - .x . ?om om - x. om 1--,ttx k'ik+ oQk

dt k=1

Qk)+ot'

-

vk = 8 , Pk = -

qk

,

t ' ?ok

and

W(f,Q,P)- Hlt,q,p)+ ot .

(8.26) Equations (8.25)providerelations forthe symplectic map.Equation (8.26) providesan expression forthe transform ed Ham iltonian f-unction. Exam ple 8.3.1: H arm onic O scillator The Ham iltonian for a linear harm onic oscillatorin one dim ension isofthe form 1 2 22

bI = 2w (p + a?q ),

.

whereq correspondsto theposition ofa particle ofm ass?rzat tim e f,p isthe

momentum,anda?isaconstantrelatingtotherestoringforce (seeFuxercises 8.2-1).TheHamiltonian system fortheequationofmotion is

Let

Xq2

*(t ?,Q)- 2 cotQ. Thispeculiar generating f-unction ischosen so thattheresulting H am iltonian system isparticularly sim ple.' lThem om enta coördinatesare

p=

= (vqcotQ,

and

p = -( ')Q = 2sin= aQ .

(y.2y)

Equations (8.27)and (8.28)can be used to determine the symplectic map (8.22).ltismoreconvenient,however,togivetheinversetransformationequations

q=

2P

sinQ,

p= 2a?# cosQ. 4Goldstein ( 351,p.389providesaderivationofthetransformation.

8 The Ham iltonian Form ulation

Thisisa particularly sim ple system to solve.5W e have

2ca

a? sin -

t+c1), p= 2=cacosYt+c1). q=

s,

w

Exercises 8.3 :

D erivesim ilarequationsforasym plecticm apifliisregarded asaf-unction

ofp and Q .

2.Letli= S k ''=1qkok.Show thatthisgenerating function leadsto asymplectic m ap that essentially interchanges the spatialvariables with the m om enta variables. This further shows that these variables are on the sam e footing in the Ham iltonian fram ew ork. 5 Ofcourse,itiseven easierto solvethe Euler-luagrangeequation directly,butthis example givesa sim ple illustration ofhow a symplectic m ap can beused to reduce Ham ilton's equationsto a sim ple form .

8.4 The Ham ilton-lacobiEquation

3.Letli= S k ''=1qkpk.Show thatthisfunctionmerelygeneratestheidentity transform ation.

4.Let li= S k ''=1gk(f,ql#k.Show that thisgenerating function leadsto pointtransformations;i.e.,theQkdepend onlyon q and f. 8.4 T he H am ilton-lacobi Equation A lthough sym plectic m aps are of intrinsic interest,they are also ofpractical interest because they m ay lead to sim pler Ham iltonian system s.ln this section we targeta particularly sim pleHam iltonian system thatcan bereadily solved.The problem is to derive a generating f -unction that produces a sym plectic m ap leading to thesim plersystem .ltturnsoutthatthe generat-

ingfunctionmustsatisfjrafirst-order(generally nonlinear)partialdifferential equation called the H am ilton-lacobi equation.Once a general solution is found to the Ham ilton-lacobiequation,the solution to the H am iltonian system can be derived by solving a set ofim plicit equations.The problem of solving a H am iltonian system can thus be exchanged for the problem of solving a single partialdifferentialequation.From a practicalstandpoint,a single partial differential equation is generally at least as difllcult to solve as a system ofordinary differentialequations,and in this sense the victory m ay seem Pyrrhic.Nonetheless,there are specialcasesofinterestw hen the H am ilton-lacobiequation can be solved.W e discuss som e of these cases in the next section.A lthough our m otivation here is to solve Ham ilton's equations,itturns out that the H am ilton-lacobiequation plays a pivotalrôle in the theory.Therealprofitfrom thisreform ulation isa deeperunderstanding ofvariationalprocesses. 8.4.1 T he G eneral Problem

Supposethata generatingfunction lican befound such thatthe transform ed

Hamiltonian isa constant,say X = 0.Thesymplecticmap produced by li then yieldsthe H am iltonian system

thatcan easily be solved to get

wherethectkand X areconstants.SinceX = 0,equation (8.26)impliesthat lim ustsatisfjr

8 The Ham iltonian Form ulation

I' Ilt,q,p)+

ç ' )t - 0.

ln the above equation,the function liis regarded as a f-unction ofq and Q .

To eliminatethepk variablesin thisexpression wecan use equation (8.25), and thus get

ff f,v1,...,va,ô ,..., + - 0. qk :va ôt

Equation (8.31)isafirst-orderpartialdifferentialequation forthegenerating function licalled the H am ilton-lacobi equation.H am ilton derived the equation in 18344in 1837 Jacobim ade a precise connexion between solutions

ofthedifferentialequation (8.31)and thecorrespondingHamiltonian system (Theorem 8.4.1). Exam ple 8.4.1: G eom etrical O ptics The Ham iltonian derived forthepath ofa lightray in Exam ple8.2.2 is

LI(f,q,p)= - n?- ,, 2 1- p2 :. The Ham ilton-lacobiequation forthisproblem is -

/,2

(' )112 -

%k

-

(' )1i2 (' )+ + 4 t' ?t ?z ç ' )t = ()

The H am ilton-lacobi equation has tw o notable features.Firstly,the function lidoesnotappearexplicitly in the differentialequation.O nly thepartial derivativesofliarepresentin theequation.Secondly,the differentialequation

doesnotdepend on any ofthe Qk variablesorpartialderivativesofliwith respecttotheQk.ln essence,thismeansthatifliisa solution to equation

(8.31)then so isany function oftheform li+ /(Q),where / isan arbitrary function.Thef-unction lidependsontheQk,and onemightrightfully query exactly how thesevariablesenterinto theproblem given no Qk dependence in thedifferentialequation.TheansweristhattheQkenterintotheproblem as initialdata forthe differentialequation.Typically partialdifferentialequa-

tionssuch as(8.31)aresolved subjectto acondition thatlitakeprescribed

8.4 The Ham ilton-lacobiEquation

valuesalong a curvein the q space.Ourproblem isto find a generating function that produces the sim plified H am iltonian system ,and this am ounts to

findinga generalsolutionto (8.31)containing n,arbitrary f-unctionsofQ.No uniqueness is expected for solutions to this problem ' ,we need the arbitrary functionsin orderto invertthe transform ation to solveforthe qk.W eexplain thism ore precisely afterwe introduce the concept ofa com plete solution.

Although we speak ofarbitrary f-unctionsofQ entering into solutions of theHamilton-lacobiequation,we know by construction thatthe Qk are in factconstantsand hence the arbitrary f-unctionsofQ are also constants.W e

can thusregardageneralsolutiontoequation (8.31)asafunctionoftheform 1(f,q,ct),where ct= (ctl,...,ctzzl,and the ctk are parametersthatcan be thoughtofas the Qk when convenient.A solution li= 1(f,q,ct)iscalled com plete6 iflihascontinuoussecond derivativesw ith respectto the qk, the ctk and tvariables,and the m atrix M defined by *2m *2( f) :21 5 t?1i5't5t1 i5't11i5'(22 ''' i5't11i5't5tzw i'

M

.

.

.

*2m *2( f) :21 'qvvi 5't2l 5 i'qvvi 5't22 ''' i5'qvvi 5'tzzw i5

isnonsingular;i.e.,

detl kl, # 0, (8.33) i ntherelevantq,ctdomain oftheproblem.Thecondition (8.33)isaJacobian condition forthesolvability ofthe qk given thef-unctions 0* qa k . The nextresult isfundam entalto the theory:itconnectsa com pletesolutlon to theH am iltonJacobiequation with the generalsolution to H am ilton'sequations.

Theorem 8.4.1 (Ham ilton-lacobi) Snpposetltat1(f,q,ct) is a complete solution tot/JcHamilton-lacobiequation (8.31).Tlten tltegeneralsolution to t/JcH amiltonian systevt t ' ?#

( ' ). FI

t ik= 8pk, f' k= - t'Ak

(8.34)

isgiven by t/Jc equations

ôak

=

-

pk,

:vk = pkt wltere tlte, dk arezzarbitrary constants.

Proof:Supposethat1(f,q,ct)isacompletesolutiontotheHamilton-lacobi equation (8.31).Then lisatisfiescondition (8.33),and theimplicitf-unction 6 Som e authorscallthese solutions com plete integrals.

8 The Ham iltonian Form ulation

theorem impliesthatequations(8.35)can besolved fortheqkin termsoff, thectk,and the, 3k.Oncethisisaccomplished equations(8.36)definethepk in termsofthesevariablesaswell.Hence,equations(8.35)and (8.36)define the f -unctions

qk- qklt,c.,p), pk- pk(f,c.,p), fork= 1,2,...,zz.Here,, d = (, d1,...,, 3n.).Toestablish theresultweneed to show that the qk and pk defined by equations (8.35)and (8.36)satisfjrthe Hamiltonian system (8.34). Substituting the solution liinto theH am ilton-lacobiequation and differentiating with respectto ct1yields

( ' ?( t' ?ct,

( -81At

( %+

x' X (% + ( ' ?. rz

qV + ot)-ocvïot+k-:J-,, ocvïoqkopk - 0. .

N ow theequation = - ,d1 t ' ?ctl m ust be satisfied identically,and therefore differentiating w ith respect to t J'ields; d( * ( *21 SL (921 dqk

2/t '?ctl = ç'ltôcvï+ k t'kkt:ctl dt = 0. = 1

(8.38)

By hypothesis,the solution iscom plete and thusallthesecond-order derivatives ofliarecontinuous;hence, ô2+ :21 = ç' ltôaï t' ?ctlt ' ?f,

Ya ôqokôaj ,+ X -to g =0 gpk

k= 1

V Strictly spealdng, we should use 0 otinstead ofd dtto denote partialdifferenti-

ationwith respectto fholdingtheakandpk constant.W enonethelessused dt or'to denote thisdifferentiation to avoid confusion with the operator0 otin the Ham ilton-lacobiequation w hich denotesdifferentiation with respecttof holding

theqk(aswellastheakandp dklconstant.

8.4 The Ham ilton-lacobiEquation

M = 0.

qk = 0

pk

forallk = 1,2,...,zz. To getthesecond setofHam ilton equationswe again substitute the solution liinto the H am ilton-lacobiequation butnow differentiate w ith respect

to qj.Forj = 1, ( $+

t :. r. f

n ( $+ ( :.r .r

ôqïôt+t Al+S ôqïôqkt gpk=0. k=1 The equation 11 = ç' èqk

m ustbe satisfied identically and therefore d om

.. ?az

o2+

-

-

o,+

>z- ù-ioqk- otoqï +11 2oqkoqïbk k=1

-

o2+ ou

oqïot+1 1E)oqïoqkopk' k= 1

(8.42)

wherewehaveused equations(8.40)and therelations ( $+

ç ' ltôqk

=

( :21

t ' Akt ' ?f,

($ +

=

( :21

ôqkôqk tk ltkk,

thatfollow from thecontinuity ofthe second derivatives.Subtracting equation

(8.41)from equation (8.42)gives DH

/1= - ç'zqk. Sim ilarargum entscan be used to show that

>

ôH

k = - -c- , Uqk

for k = 2,3,...,' n,.

8 The Ham iltonian Form ulation

From the standpointofsolving H am ilton's equations,the above theorem show s that we need not be concerned with the absence of initialdata for

equation (8.31)and the resulting nonuniqueness.Any complete solution to the Ham ilton-lacobiequation sufllces to enable us to construct a solution to Ham ilton's equations involving 2/,arbitrary constants and hence a generalsolution to the underlying Euler-lsagrangeequations.G iven a variational problem we can thus outline a procedure to get the solution based on the H am ilton-lacobiequation asfollow s:

(a) determinetheHamiltonian . bIforthegiven problem; (b) form theHamilton-lacobiequation; (c) finda completesolution litotheHamilton-lacobiequation; (d) form theequations X = - ç'lak, wherethepkareconstants;and

(e) solve the n,equations in part (d)forthe qk to get a generalsolution qlf,ct,/ $). Exam ple 8.4.2: G eom etrical O ptics Suppose that the optical m edium in Exam ple 8.4.1 has a refractive index

n,= ysfqï,where p.> 1 isa constant.The relevant dom ain for this problem

isqïk 1 (sothatn,k 1).8TheHamilton-lacobiequation forthisproblem is

(t o ' kmly2.y(( oq 9 mzy2.y(o t ' ? + f)2zycg. The readerm ay verifjrdirectly that

isa solution to equation (8.43).Thematrix M isgiven by

where-4 = j. tzqï- tctz y+ ct2 zl?and hence detl kl= G2.

W

8 Recallthat the refractive index isthe ratio ofthe speed of light in ' vacuo to the speed oflightin the m edium .

8.4 The Ham ilton-lacobiEquation

ôI

181

a1d

/71= - t'?ctl = p,z - qz

pz = - ( ')+ = ctazt- f, t' ?ctz v,2

,2 % + f)a+ c.l+zc.l t ?1(f,c.,p)- / c.1(/ p, G1

qzlt,c.,/ t 7)- G2 t+ /% - , t?1. Although part(d)in the above plan may in itselfbea formidable task to accomplish in practice,thecrux ispart (c).First-ordernonlinearpartial differentialequations are generally harder to solve than system sofordinary differentialequations.The only generalsolution technique for these partial differentialequations involvesthe useofcharacteristics,w hich aredefined by a system ofordinary differentialequations.lt turns out that the system of differentialequations defining the characteristics is equivalent to the origi-

nalHamiltonian system.ln general,to implementpart(c)oftheaboveplan

we first have to solve Ham ilton's equations in order to solve the H am iltonJacobiequation.Thisratherdefeatsthepurposeofusing theH am ilton-lacobi equation to solve the originalproblem .G enerically,the Ham ilton-lacobiform ulation does not actually help to find solutions.There are cases,how ever, when thesolution to the Ham ilton-lacobiequation can be found w ithout resorting to characteristics.Solution techniques such asseparation ofvariables do notrely on know ledge ofthe characteristics and therefore circum ventthe problem ofintegrating Ham ilton'sequationsfirst.The successofthe solution technique dependscrucially on thetypeofH am iltonian,butitturnsoutthat a num ber ofproblem s ofinterest have Ham iltonians thatallow a separation ofvariables.W e discussthistechnique in thenext section. 8.4.2 C onservative System s

A specialbutim portantcaseconcernsconservativeH am iltonian system s.The H am iltonian does not depend on explicitly tfor such system s,and we know from Section 3.2 that. bI isconstantalong any extrem al.W e can exploitthis situation because we know that the variable tcan be separated out in the com pletesolution to theH am ilton-lacobiequation.ln otherwords,w eknow thatthereis a com plete solution ofthe form

where . bI = J(a) = const.along the extremalq(f,ct,, d).W e can simplify mattersfurtherby identifying one ofthe coördinates,say Qn = ctzz,with

J(a).Thisapproach producesthepartialdifferentialequation

8 The Ham iltonian Form ulation

ff v1,...,va,ô ,..., = aa, qk :va which wecallthe reduced H am ilton-lacobiequation. T hefunction ! #'in thesolution to theHam ilton-lacobiequationisevidently a solution to thereduced H am ilton-lacobiequation.M oreover,itisclearthat liis a com plete solution if! #'is a com plete solution' ,i.e.,!#'has continuous derivativesofsecond orderand the m atrix

isnonsingular. T hefunction ! #'isofinterestasa generating function fora sym plecticm ap

in itsown right.The symplectic map S :(q,p) --+ (Q,P)produced by ! #' transformstheHamiltonian # tq,p)totheHamiltonian X(Q,P)= Qzz.The new position and m om enta coördinatesm ustsatisfy theequations a u' m

Q'k= -- = O, ôlok

b

t' ?H-

0, if1 s;k s;n 1

(' )ok

-1,ifk = zz,

-

and hencetheQkareconstantsfork = 1,2,...,zz,and thePkareconstants fork= 1,2,...,n,- 1.Theanomalouscoördinateisfk becausefk = pvv- f, wherepvvisaconstant. T hesym plecticm ap S hasan interesting geom etricalinterpretation.Fora

given constantE,thecondition # tq,p)= E producesahypersurfacein the zzz-dimensionalphasespaceand a (hyperlcylinderin the2/,+ l-dimensional (f,q,p)space.Theextremalscorrespondtoafamily ofcurvesthatlieonthe cylinder.The sym plectic m ap S transform s the picture dram atically.ln the

(f,q,p)space,the cylinderis transformed to a hyperplane and,even more rem arkable,the fam ily ofextrem a. ls in the originalspace istransform ed into a fam ily ofextrem alsin thenew space,whereeach extrem alisa straightline inclined at an angle r 4 to the f-axis.Roughly speaking,thesym plecticm ap S ttflattensout''the cylinders. bI = E and ttstraightensup''the extrem als. lf the Ham iltonian system is conservative,we generally start with the reduced H am ilton-lacobiequation.O ncea com plete solution ! #'isdeterm ined wecan return tothesolution li= W - ctzzfand proceed asbefore.T hisam ounts to solving the equations k = 1,2,...,n,- 1,

8.4 The Ham ilton-lacobiEquation

for the qk in term s of t and the constants ct,, 3. The absence of t in the H am iltonian sim plifiesthe problem slightly,but allthe com m ents aboutthe difllculty ofsolving the H am ilton-lacobiequation stillapply to the reduced equation. Exam ple 8.4.3: H arm onic O scillator The linear harm onicoscillatorofExam ple 8.3.1 hastheH am iltonian

H

1

a

aa

2. ,,z(p +a?q).

-

The reduced Ham ilton-lacobiequation isthus

21))o nv q)2 ( '?! P

zzrzct- (v2( )2

=

wherect> 0 isa constant.T he generating function istherefore ofthe form

W eneed only one arbitrary constantfora com plete solution so wecan ignore theintegration constant.A solution tothereduced H am ilton-lacobiequation isthus

! ff(q,ct)=S 2qt z2-q2+a2sin-1a V1)' where

q(L)= -

2m a sj

a? sin w

( , d+t)1,

which isequivalentto thesolution found in Exam ple 8.3.1.

8 The Ham iltonian Form ulation

Exercises 8.4 :

21

J(t v)-

t v/2dz. 20

D erive the H am ilton-lacobi equation corresponding to this f-unctional. Solve the Euler-lsagrange equation for this f -unctional and construct a solution to the Ham ilton-lacobiequation.

2.Letn,= vtly)denotetherefractiveindex in an opticalmedium.Fermat's Principleimpliesthatthepath ofa lightray from a point (zo,y(tlto a point(z1,3/1)isanextremalofthef-unctional 21

J'(: 4/)=

Tz(#) 1+ : v/2dz. 20

where ctand , d are constants.Use thissolution to find the corresponding extrem als im plicitly. 3.The Lagrangian for the m otion ofa particle ofunit m ass in the plane under the action ofa uniform field is

where q denotesthe Cartesian coördinatesofthe particle and g isa con-

stant (Example 3.2.2).Derive the Hamilton-lacobiequation and show thata solution to this equation is

8.5 Separation of V ariables The only chance we have ofsolving a problem using the H am ilton-lacobi equation withoutessentially solving theH am iltonian system firstisifa solution to the partialdifferentialequation can be obtained withoutresorting to characteristics.O nesolution techniquethatavoidscharacteristicsiscalled the

8.5 Separation ofVariables

m ethod ofadditive separation or sim ply separation of variables.g In thissection we presentthe m ethod and give som e exam ples.W e also discuss conditionsunderwhich weknow thata separablesolution exists.W e lim itour discussion to conservativesystem s. 8.5.1 T he M ethod of A dditive Separation

ff

:W :W v1,...,qn,ô ,..., - ;? - 0, qk :va

where E is a constant.Suppose that the term s qï and o' F appear in this V1 equation only through the com bination .t y1(t 71,oV F1 ),wheregïissomeknown '

function.Equation (8.46)could bethen recastin theform

W= ! l%(t ?1,Q)+ X1(t ?2,...,qn,Q). Substituting theaboveexpression for!P'intoequation (8.47)gives JP

t ' 9l p' l

(' IR ï

t ' 9f/1

p1(v1,ôqk),v2,...,va,:v2 ,...,ôqn - ;?- 0,

and thisequation m ustbesatisfied fora continuum ofqïvalues.Assum ing gï isa differentiablefunction,this m eansthat

DF = :F % ï= % k tgpl% k 04

t ?,(t ?,,! P' ,(t ?,))- (7,(Q), .

whereC1isan arbitraryfunction ofQ and /denotesd dqï.Equation (8.50) isa first-orderordinary differentialequation for!P' 1.

Thebestscenarioisifeachpairqk,(' ?!P 0qkenterintoequation (8.46)only through a combination gklqk,( ' ?! P oqk).ln thiscase,the partialdifferential equation can be written in the form 9 The latter term also includes the m ethod ofm ultiplicative separation, which we do notuse.

8 The Ham iltonian Form ulation

JP

:W

:W

p1(v1,ôqk),...,pa(va,:va) - ;?- 0,

! #'= !& (t ?1,Q)+ *2(t ?2,Q)+ '''+ Snlqn,Q).

gk(qk,IP' J(qk))- (7k(Q). Here,the Ck are functionsofQ satisfjring the equation

F(C1(Q),...,Czz(Q))- E - 0, but are otherw ise arbitrary.A reduced H am ilton-lacobi equation is called

separableifthereexistsacompletesolution ! #'oftheform (8.51).Thefunction ! #'iscalled a separable solution. T he m ethod ofadditive separation am ountsto thesteps:

(a) assumeasolutionoftheform (8.51)andsubstituteitintoequation (8.46). , (b) identify thegkandform thedifferentialequations(8.52). ,and (c) solvetheordinarydifferentialequationsforthe!&. O ncetheseparable solution !#'is determ ined,a com plete solution li= !#'- Et can be obtained forthe Ham ilton-lacobiequation andw ecan proceed tofind the position functions qk asdiscussed in theprevioussection. T he obviousweaknesswith the above m ethod isthata separable solution need not exist: there is no guarantee that the requisite gk can be found. The existence ofseparable solutionsdependson theH am iltonian and even a sim plecoördinate transform ation ofthe position variablescan affectwhether a separable solution isavailable.C onditionsunder which w ecan predictthe existence ofa separablesolution arediscussed in the nextsubsection. Exam ple 8.5.1: ln Cartesian coördinates,them otion ofa particlein space undertheaction ofgravityactingin theqsdirection producestheH am iltonian H

-

1

a

a

a

2. ,,z(r1+p,+ra)+nzgqz,

where ?rzis the m ass ofthe particle and g is a gravitationalconstant.The reduced H am ilton-lacobiequation is

(% ( 3$k)2.y(t ( ' :t ? l ? p 2 'yj2.y(ç ô ' èq b pzyj2.yggygogy.-rys, . s.( y whereE ,the totalenergy ofthe particle,isa constant.Supposethatequation

(8.54)hasasolution oftheform

8.5 Separation ofVariables

187

!#'- !&(t ?1,Q)+ !& (t ?2,Q)+ Watva,Q).

W e can thus take

ç' lbp' k.

t ' ?lp' k

gk(qk,oqk )= oqk , fork = 1,2,and t' ?lpa

t ' ?!p 2

z

)-joqs)+2,,zgqz.

gzlqz,oqs W e have that

ç 'èô' k

ôqk = ck(Q),

fork = 1,2,so that

Wk= Ck(Q)t 2k-1-Nk(Q), where the K k are arbitrary f-unctions.The differentialequation involving gs yields

t 'ka = =

(7a(Q)- zm,zgqs

zzrz. &- c?(Q)- c#(Q)- zvrtzgqs,

whereffaisanotherarbitraryf-unction.A separablesolutiontoequation(8.54) istherefore

i= zzrz. &- c?(Q)- c#(Q)-zvtzzgqs.

.

Fora com plete solution w eneed only three arbitrary constants presentin !P'.

LetC1(Q)= Q1= ct1andC2(Q)= Q2= ctz.TheconstantE isalsoarbitrary in theabove solution so wemay takeE = Qa= ctaand letff(Q)= 0.The solution !#'is then ofthe form

8 The Ham iltonian Form ulation

J1=

zzrzcta- a2 122- zvtzzgqs. 1- (

,2, .5 Thematrix M with entries(oq op .. sgiven by a k )i ,

so thatdetlkl, # 0,and thesolution isthuscomplete. Equipped with a com plete solution to the reduced H am ilton-lacobiequation,wecan proceed to determ inetheposition f-unctionsq from the equations

T he above exam ple is a som ew hat com plicated m ethod for obtaining a solution thatcan readily be obtained from theEuler-lsagrangeequations.The nextproblem isnotquite so easy to solve usingthe Euler-lsagrangeequations. Exam ple 8.5.2: T he m otion ofa particle in a plane under a centralforce field whose potentialperunitm assis U'leads to a H am iltonian ofthe form

H ere,qï= r and q, = 0 arepolarcoördinates.T he reduced H am ilton-lacobi equation leadsto thedifferentialequation

qî)(o .q eï)2+z( v(. ? ,)-s))+(( . ,q e,)2-( ,, and hencewe m ay take

8.5 Separation ofVariables

t 9!& ç ' èq,

= (22,

and

A solution to the reduced H am ilton-lacobiequation isthus

N ow ,

so that

8 The Ham iltonian Form ulation

and hencethe angularm om entum oftheparticleisconserved.

8.5.2 C onditions for Separable Solutions*

The m ethod ofadditive separation enables us to circum vent the problem of integrating H am ilton'sequations orthe equivalentsystem ofEuler-lsagrange equations.U nfortunately,them ethod isnotgenerally applicable toH am iltonJacobiequationsand itssuccessdepends largely on the form ofthe Ham iltonian.Asrem arked earlier,even coördinatetransform ationsaffecttheprocess. W e saw in Exam ple 8.5.2 that the reduced H am ilton-lacobiequation forthe centralforce problem is separable in polar coördinates.The sam e problem ,

however,isnotseparablewhen itisformulated in Cartesian coördinates(see Example 8.5.3).Some problems have more than one coördinate system in which the H am ilton-lacobiequation isseparable' ,others have no coördinate system sthatlead to separable solutions.lo T he im portance ofidentifying Ham ilton-lacobiequations thatare sepa-

rablewasrecognized soon aftertheequationwasfirstderived.Liouville (ca. 1846)studied the problem for thecasen,= 2.Fora specialbutimportant class ofH am iltonians he established necessary and sufllcient conditions for

separability.Later,Stàckel(ca.1890)generalized theresultsofLiouvillefor systemswhere n,k 3.80th Liouvilleand Stàckelwere concerned with Hamil-

tonianswheretheunderlyingcoördinatesystem isorthogonal.Levi-civita(ca. 1904)generalizedtheresultsfornonorthogonalcoördinatesystems.Thereare stillm any unanswered questionsconcerning the separability oftheH am ilton-

Jacobiequation.Themonograph by Kalnins(431detailssomeofthenewer, m orespecialized resultsin thisfield.Kalninsalso discussessom e ofthebasic

questions and provides a number ofkey references on the subject.Here,we lim itourdiscussion to a few elem entary resultsw ith exam ples. A significantclassofproblem sin m echanicshasa Ham iltonian oftheform

# tq,p)- T(q,p)+ t-(q), whereU'isapotentialenergy term ,and T isakineticenergy term oftheform 1 N'

T(q,p)-; -) éck(q)r2, k=1 10 The famous tthree body problem ''is am ong these.

8.5 Separation ofVariables

191

wherethe Ck are positive f-unctions.Thefeatureto note in theaboveform for

bI isthatthepk appearonly in the combination p2 k (thisindicatesthatthe underlyingcoördinate system isorthogonal).Hamiltoniansofthisform lead .

to a reduced Ham ilton-lacobiequation oftheform 1 Tz

*!P 2

Vk=1 S Ck ôqk +U=ctl, wherectlisa constant.TheresultsofLiouvilleand Stàckelconcern essentially

equationsoftheform (8.55). Theorem 8.5.1 (Liouville) .4 necessary and s' tz //cïczz/ condition Jor tlte H amilton-lacobiequation

2 ! .toLoq vkl.

C1 =

Q =

81 , cl+ c2

/22

,

cl+ c2

and

U

=

vk+ vz. cl+ c2

P roof: W e first show that the equations for C1, C2,and U'are necessary

conditionsforseparability.Supposethatequation (8.56)isseparable.Then there exists a com plete solution !#'ofthe form

! P'(q,c.)- ! & (t ?1,c.)+ !&(t ?2,c.),

' itC1(t ' ô ? Z qï12 Forsim plicity,let

yt-)(o ' 9q ! P k)2, r?-)(o ' 9q ! P 2)2.

.

8 The Ham iltonian Form ulation

Notethat-4dependsonlyonqïandB dependsonly onq,.Equation (8.57)is satisfied fora continuum ofct1and0,2values,and differentiating thisequation with respectto ct1yields

Q XI+ C2O1= 1,

(71-42+ (72. D2= 0.

(8.59)

t ' 9l p' lt 9! ' &' A 'ç 5 . b , ' pL . q 1,p a' zr q aj, z

4l. !?z- -4z. !?l= %

-

kt X 2 (9(ctl,ct2)

wherethe finalfactoron the right-hand side isa Jacobian term .The solution

!#'is complete and therefore the terms ç ' lô' j t' Ak cannotvanish identically. M oreover,detlkl, # 0,sothattheJacobian term cannotvanish identically.W e m ay thus choose a particular setofvaluesctl,(12 such thatthese term s are

nonzeroand solveequations(8.58)and (8.59)toget c1 =

+ A2 Aj

c =

->

/2

8.5 Separation ofVariables

where0,2is a constant.Hence, 1

2

1=

tcta+ ctlt zl- zzlldqï, p,1

2=

(-ct2+ ctltzl- v,)t/t o. /22

and 1

2

(Note thatp,l,/. :2 > 0 since C1,C2 > 0 by hypothesis.)ltremainsonly to show thatthe solution ! #'determ ined in thism anneriscom plete.Thesolution !#'hascontinuoussecond-order derivativesprovided

M oreover, -

crj

1

Mtv' r Mtv'r

M =

, Ga

1

Mav' f Mav'f

and thus

detlkl=

-

(( r1+ (rzl , MzMavoq # 0.

The solution is thus com pletein thedom ain defined by D > 0,E > 0. Exam ple 8.5.3: The reduced Ham ilton-lacobiequation ofFuxam ple 8.5.2

can bereadily putin theform (8.61).Thereduced Hamilton-lacobiequation from thisexam pleisequivalent to

1

2

q1 2

2 :g? 2

t ?1

( 41

+

:g? 2 (' lq,

+

qk 2Tz(v,) q1 2

= ct1,

and wem ay take p,1= v2 1? y,= 1,t zl= ql?(rz= 0,vïqz gU(t y1)?andv,= 0.We could have thusconcluded thata separablesolutionexistsbeforeweem barked on finding it. Suppose, how ever, that the problem was initially posed in Cartesian

coördinates(z,y).Thereduced Hamilton-lacobiequation in thiscoördinate system is

1

V

( ' ?!P 2 (' ?!P 2 t' ?z + ç')y

+ V( z2+ y2)= ctl.

Forthecentralforceproblem ,thepotentialfunction U'm ustdepend on z and y only through the com bination z2+ y2.Thism eansthatw ecannotgetU'

in theseparated form required by Liouville'stheorem (unlessU'isconstant) and henceno separable solution existsfor thisequation.

8 The Ham iltonian Form ulation

T hesufllciency partoftheabove theorem can beeasily extended to higher dim ensions.Specifically,itcan beshow n that a Ham ilton-lacobiequation of the form 1

( ' ?!P 2

(' ?!P 2

2(c1+ ...+ c,zl p,1 ç ' L kï + '''+ p,z t ' ?t yzz

vï+ . . . + vn

+ cl+ ...+ czz= ct1, (8.62) wherethe functionsvk,Jzk,and ck depend only on qktV k rL =1O' k > 0,and p. k > 0,adm itsa com plete solution ofthe form

W(q,ct)= !l%(t ?1,ct)+ '''+ Tvblqn.,ct). ln fact,!P'isgiven by

e,-j and W

k=

2

(ctlck- L/k- Ctkl(lqk,

p,k

fork = 2,...,zz.Reduced Hamilton-lacobiequationsofthetype (8.62)are said to be in Liouville form . lfn,= 2,then a Ham ilton-lacobiequation m ust be reducible to Liouville

form for a separable solution to exist.lfn,k 3,however,there are equations thatarenotreducibletoLiouvilleform thatarenonethelessseparable.Stàckel studied this problem and arrived atthe following characterization.

Theorem 8.5.2 (St:ckel) . 4 necessary ands' tz //cïczz/condition Jor tlte rcduced Hamilton-lacobiequation (8.55)to beseparable ïstltattltere ct c/s/s a nonsingnlarmt z/rïzU witltentriesukj,whereJorj= 1,...,n,,ukjisaX zzction o.fqk onl y, and a ct alqzmzzvtatriz w

= (t&1,...,tcalT',where wk is a

Junction o. fqk only,s' tzc/zthat

(8.63) j = 2 ... n,

P roof: T he proofofStàckel'stheorem issim ilar to thatgiven forLiouville's theorem .W e give only a sketch of the proof here. W e first establish that

8.5 Separation ofVariables

equations(8.63)to(8.65)arenecessaryforaseparablesolution.Supposethat theHamilton-lacobiequation(8.55)isseparable.Thenthereexistsacomplete solution ofthe form

!#'= !;%(t 71,ct)+ ...+ ô' vv(qn.,ct). Substituting theabovesolution into equation (8.55)and differentiatingwith respectto the aj givestheequations ''

ïl zckt ?lp' ' k?( %Wk -1 k ôqkt' ctltxk 1

=

and

X

/é)Ckt ' ?l ffk t' ?z! J% =1 k= 1

,

ç 'kkt' ?ctjt xk

j = 2,...,zz.

glpl ...t gl /k detlkl, zl = t 8

qk

t 'ka

where M isthe matrix with entries(t ' 92!P'( 'lcvjôqkj.Since W isa complete solution wehavethat. :1cannotvanish identically and thereforewem ay choose

a particularsetofctsuch that.:1, # 0.Wecanthustake ôbïrk *21 ' Ctkj = ôqkt M jtxk, '

andsubstitutingtheseexpressionsintoequations(8.67)and (8.68)yieldsequations(8.63)and (8.64).Notethatthematrix U thusdefined isnonsingular since. :1# 0forourchoiceofct.TheHamilton-lacobiequation (8.55)implies thatthe potentialterm U'can be written in the form 1 Tz

t' Vk 2

t-=ct1-j.k=1 S Ck 0qk , and usingequation (8.63)thisequationisequivalentto -

t'-k :2ckc.z' t zkz-. )(8 ç ' k *k)2, =

1

so thatequation (8.65)issatisfiedwith

8 The Ham iltonian Form ulation

A lthough the Ck m ay depend on t 71,...,qn,the coefllcientofeach Ck in the aboveequation involvesonly ( ' ?! P ôqkand qk.W ecan thusconstructa solution

oftheform (8.66)from theequations

(op qk()2

Theequations(8.63)to (8.65)aresometimescalled theSt:ckelconditions.Thematrix U isnonsingular and hence equations(8.63)and (8.64) can be solved for the Ck.The inverse m atrix S = U -1 is called a St:ckel m atrix. lfthe reduced Ham ilton-lacobiequation isseparable,then the underlying H am iltonian and Lagrangian system scan be solved by quadratures.ltis interesting to note that in this case the H am ilton-lacobiform ulation and the

Stàckelmatrix can also beused toderiven,conservationlaws(firstintegrals) forthesystem.lndetail,ifthereexistsacompletesolution to equation (8.55) oftheform (8.66)then theWksatisfjrequations(8.70).Now DH

Ik = : = Ckpk, pk

and therefore

k

:W

k- Ckôqk - Ck h, k(qk,c.).

(Note thatthe Ck depend on t 71,...,qn and so thet kkdepend on thesevariables.)Rearrangingequation (8.71)andusing thedefinition ofh,kgives

8.5 Separation ofVariables

197

and using the inverse m atrix S,thisexpression yieldstheconservation laws -

E1 zc 1(. i z( )2,-wj-ctk, skj -

j=

f

.

where,in familiarsphericalcoördinatenotation,qï= r,q,= 0,and qs= 4,

(z = vlcosq,sin qs,y = vlsin q,sin qs,z = vlcosqs).The corresponding

reduced H am ilton-lacobiequation isofthe form

è 'y'()ck(o ' î ' q ! l k -)2+v(. ?,)-ct,, k=1

wherect1is a constant,and C

1 1= - , C2= ?rz

1 z, Trtqk

Clzlll+ C2' tzal+ Cazlal= 1, C1' t z12+ (72u22+ Cazlaa= 0,

The first equation is satisfied if ztll = ?rzand ' tzzl = ztal = 0.The second equation is satisfied if' tzlz = 0,' t zzz = -1 sin2q,, and ' tzaz = 1.The third

equationissatisfiedifztla= -1 ql,ztza= 1,andttaa= 0.Hencewehavethe m atrix

()- W 1

and since

a

8 The Ham iltonian Form ulation

detu = -?rz, # 0,

U isnonsingularforqï > 0,0 < q, < r.The choices' tt;1= ?rzutt 7ll,w, = zt?a = 0 sufllce to meettheStàckelcondition (8.65).W ecan thus conclude thatthe reduced H am ilton-lacobiequation isseparable.Now,

and hence

The Stàckelm atrix S isgiven by

and it can be shown that the three conservation law s associated with the system correspond to the conservation ofenergy,angular m om entum about

thepolaraxis(thez-componentoftheangularmomentum),andtheangular m om entum .

Finally,itisinteresting to note that ifwe generalize the problem to allow

forageneralpotentialf-unctionU(q),theStàckelcondition (8.65)impliesthat U'm ustbe oftheform

t-(q)- t1( t ?1)+ q1lt2(t ?2)+ qlsi1 ?al nzq,Uatt in orderthatthe corresponding reduced Ham ilton-lacobiequation beseparable. T he results ofLiouville and Stàckelapply to H am iltoniansw herethe un-

derlyingt 71,...,qn coördinatesystem isorthogonal(thereareno crossterms t kjt kk,j: /:k,in thekineticenergyfunction).ltisnaturaltoenquirewhethera

8.5 Separation ofVariables

characterization ofseparable system sexistsfornonorthogonalsystem s.The follow ing result,w hich westatew ithoutproof,llappliestocoördinate system s notnecessarily orthogonal.

Theorem 8.5.3 (Levi-civita) .4 necessary and s' tz/lcïczz/conditionJortlte reducedHamilton-lacobiequation dejined by

-

*. I.J*. I.J (92/. f d.; .Jd.; .J (%/. f 8vkt grjnqkçgqj - 8vktgo nqkç gvj *. I. J*.I .J ( 92/. f d.; .Jd.;. J ( % /. f + :

ç 'kkt' ?rjôpkôqj t' kkç ''qjt '?rkt -?rj

T heorem 8.5.3 is a launching point for m uch ofthe w ork on separable system s,particularlyw ith the characterization ofcoördinate system sinw hich certain H am ilton-lacobiequations can be separated.The reader is directed

to Kalnins(431forfurtherresultsand references. ln closing this section (and chapter) we comment again that although the Ham iltonian form ulation and the Ham ilton-lacobiequation are often of lim ited practicalvalueforactually solvingtheEuler-lsagrangeequations,they are usefulin developing the underlying theory and m aking connexionsacross seem inglydisparatetheoriessuch aselectrom agnetism and geom etricaloptics. ln defenseofthe Ham ilton-lacobiequation as a toolforsolving a variational problem ,the sobering reality is that there is no generalm ethod for finding solutionsanalytically.Fora lim ited but im porta. ntclassofH am iltonians,the H am ilton-lacobiequation is separable and produces generalsolutions.ln its wake italso bringsa wealth ofbyproductssuch asconservation laws. Exercises 8.5:

zztz,y)- /(z)+ . q(y),

200

8 The Ham iltonian Form ulation

D eterm ine the associated Ham ilton-lacobiequation and use Liouville's theorem to show that it m ustbe separable.Reduce theproblem offinding the extrem als to quadratures.Note that this functionalalso m odels geodesicson a classofsurfaces called Liouville surfaces. 2.The m otion ofa particle under gravity on a s11100th sphericalsurface of radiusR givesa kinetic energy term 1

2 2

2

Ttp,4)- j.zrz. n (ti + sin p(/, 2), U(p)= vrtgltcos0. Here,0and 4 arepolarangles,with 0 being measured from theupward vertical.D erive the Ham ilton-lacobiequation and show thatitisseparable.Reduce theproblem offinding the extrem a. lsto quadratures.

3.Themotion ofa particle ofmass?rzin paraboliccoördinates(ï,p,4)= (t ?1,q2,t ?a)givesa Hamiltonian oftheform 2 v1r1 2+ gzp2 2 p2 a + qk+ q, 2:- 1:2 + v(q).

H -?rz Let

t-(q)- /(t ?1)+ glqz) .

qk + q2 U se Stàckel's theorem to show that the corresponding H am ilton-lacobi equation isseparable.Find a separablesolution.

N oether's T heorem

9.1 C onservation Law s

forallextrema. lsofJ then relation (9.2)iscalleda kth order conservation law forJ (and theassociated Euler-lsagrangeequation).Forexample, LI= y?% t 9/?- /

(Theorem 2.3.1).Thedefinitionofa conservation law can beadapted to cope

with f-unctionalsthat involveseveraldependentvariables.The definition can also be generalized forfunctiona. lsthatinvolveseveralindependentvariables.

ln thiscase4 isavectorfunction,andconservationlawsarecharacterized by the divergencecondition

202

9 Noether'sTheorem

C onservationlawsusually havean im portant interesting physicalinterpre-

tation (e.g.,conservationofenergy).lnaddition,theycan materiallysimplify theproblem offindingextrem a. lsw hen theorderoftheconservation law isless than thatforthecorrespondingEuler-lsagrangeequation.TheEuler-lsagrange

equation forthef-unctionaldefined by (9.1)isoforder2zz,andequation (9.2) im pliesthat

$(I #,#/,...,: 4/12))= covtst. lfk < 2/,then theaboverelation isa differentialequation oflow erorderthat each extrem alm ust satisfjr.Such relations are called a first integralto the Euler-lsagrangeequation.The right-hand sideofthe relation is a constantof integration that isdeterm ined by boundary conditions.

Givenafunctionaloftheform (9.1),itisnotobvioushow onemightderive

a conservation law ,or for that m atter, if it even has a conservation law . lfthe functional arises from som e m odel,then the application itselfm ight

suggest the existence ofa conservation law (e.g.,conservation ofenergy). Som e f-unctionals m ay have several conservation law s;others m ay have no conservation laws.The problem isthus to develop a system atic m ethod to identifjrfunctionalsthathave conservation law s and derive an algorithm for theirconstruction. A centralresult called N oether's theorem links conservation law s w ith certain invariance properties ofthe f -unctional,and it provides an algorithm forfinding the conservation law .ln this chapter,we presenta sim pleversion ofNoether'stheorem that ism otivated prim arily by thepragm atic desire to find firstintegrals.W elim it ourdiscussion m ostly to the sim plestcase w hen n,= 1.A m orecom plete study ofN oether'stheorem can be found in Blum an

and Kumei(111and Olver(571especially forthecaseofseveralindependent variables.

9.2 V ariational Sym m etries

W e considera one-param eter fam ily oftransform ationsofthe form

X = 0(z,y;6), F = #(z,y;6), where0 and ' ?ûares11100th f-unctionsofz,y,and theparam eterE.ln addition, we require

0(z,y;0)- z, ' ? /'(z,y;0)- y,

(9.8)

so thattheparam etervalue t î= 0 correspondstotheidentity transform ation. Exam ples ofsuch fam iliesaregiven by the translation transform ations

9.2 VariationalSym metries

X = z + 6, F = y, X = z, F = y+ t î, and a rotation transform ation X = zcos6+ ysin 6,

( r)(X,F)- ( pz 0y) ! t ' ?tz,y)

with determinant

h-

4z4,,)'

zl(z,y;6)- oz' l)y- oy'l)z.

zl(z,y;0)- 1, Z(z,y;6)/ 0 forI6Isufllciently small.Relation (9.12)impliesthatthetransformation (9.7) hasa unique inverse

z = &(. X' ,F;6), y = !P'(. X',F;6), provided I6Iissmall(Theorem A.2.2).Forexample,theinverse oftransformation (9.9)is

z = X cost î- F sin6, y = X sint î+ F cosE.

Foragiven function ylz)wecan use relations (9.13)to eliminatez and determ ine F asa f-unction ofX .ln the following discussion we have occasion to considerF asa function ofX and som e confusion m ightarise.W ethususe

thesymbol' 1z)(.X)todistinguish thiscasefrom F(z).Consider,forexample, thetransformation (9.9).Herez= X - t îand henceforany y : v(z)- F(z)- y(X - 6)- X (A'). 1f,forinstance,ylz)= costz),then' l1(. X)= cost. x -6).Foranotherexample, considertransformation (9.11)with ylz)= z.Then,

204

9 Noether'sTheorem

X (A-)= cos6- sin6X . sin 6+ cos6

W e also use the notation

t(A-)- dx Y)(A-). Notethat,fortransformation (9.7),(9.13), dz - (ex + /y.f)(A-))dx,

dy- (ex + !py.t(A-))dx,

(9.14) (9.15)

and hence

ex + lpvt (x). y'lz)- e

x + mwt t-v-l W e studied the effectof point transformations such as (9.7) on variationalproblemsin Section 2.5 (forflxed valuesof6).Theorem 2.5.1 shows thatthe transform ed problem isvariationally equivalentto the originalproblem .G enerically,however,the integrand defining the f-unctionalchanges undera transform ation.Ofspecialinterestherearetransform ationsthatdo not change theform ofthe integrand.

Theintegrand /(z,y,!/?)ofthefunctionalJ issaid to be variationally invariantovertheintervalgzt ),z11underthetransformation (9.7)if,forallt î sufllcientlysmall,in anysubintervalh,51f;lgzo,z11wehave b G

b-

/' -?(x,u (x),t(x))-/x

/(z,t vtzl,t v'(z))dz - .

foralls11100th functionsyon h,51.Here, aE- 0(a,: v(tz);6), bE- 0(b,: v(5);6). ln thiscasethetransformation (9.7)iscalled avariationalsym metry ofJ. Exam ple 9.2.1: Letzt l= 0,zl= 1,

/(z,y,!//)= !/Q(z)+ / (z),

v'(z)- t(. t X').

9.2 AfariationalSyrnrnetries

Exam ple 9.2.2: Letzt l= 0,zl= 1,

/(z,y,y')- y'2(z)+ z?/(z), y'2(z)+ zyzlz)- f)2(.v)+ (.v - 4y)2(. v) -

/ x,u(A' ),t(A' ))-. î' l' 7(A' ),

lnfact,itcanbeshownthattransformation (9.9)isavariationalsymmetry forany functionaloftheform (9.4).ltcanalsobeshownthattransformation (9.10)isavariationalsymmetry forany functionaloftheform

W e considertransform ationsofthe form

T = P(f,q;6), Qk= bbk(f,q;6),

206

9 Noether'sTheorem

0(t,q;0)- f, bbk(f,q;0)- qk. Sim ilar argum ents to those used for the previous case can be used to show

thatthetransformation (9.19)isinvertible.Theintegrand L(t,q,(k)isvariationally invariantover(f(),f11underthetransformation (9.19)if,forallI6I small,in any subintervalh,/71f;l(f(),f11,wehave '

/-z-(f,q,k)/f-/-'-o(z' ,q-(z' ),q-(z' ))t /z' , .

forallsm00thf-unctionsq ongct,, d1.Here,ct:= 0(a,qtctl;6),/ % = 0@ ,q(, d);6), and /denotesd dT . Exam ple 9.2.3:

Qk= qk

Q? E1= q'kcost î+ q'Ljsint î, Q? :2= -q'ksint î+ q'Ljcos6, so that

LIT,

1

,

,

K

QE,Q' E)= j.?rz(Q2+QZ)+ Qz za E1+ QE

9.3 Noether'sTheorem

Hence,forany gct,/71f;l(0,11wehave

andconsequentlyL isvariationallyinvariantundertransformation (9.22).W e thusseethatthisf-unctionalhasatleast two variationalsym m etries.

Exercises 9.2:

21

J(t v)-

zt v'zdz. 20

Show thatthetransform ation

X = z+ 62z ln z, F = (1+ E)y isa variationalsyrnrnetry forJ.

9.3 N oether's T heorem

W e know from Section 2.3 that the quantity . bI (the Hamiltonian)defined by equation (9.3)isconstantalong any extremalforf -unctionalsoftheform (9.4)andfrom Section9.2thatsuchf-unctionalshavethevariationalsymmetry (9.9).lnaddition,weknow (Section2.3)thatanyf-unctionaloftheform (9.18) hasa conservation law ,viz.,

and thatthetransformation (9.10)isavariationalsymmetry forsuchafunctional.Although this is a special selection,we m ay suspect that the existence ofa conservation law islinked w ith thatofa variationalsym m etry.ln thissection we present a result called Noether'stheorem ,w hich shows that each variationalsym m etry fora functionalcorrespondsto a conservation law . N oether'stheorem also provides theconservation law . Before we state N oether's theorem ,we need to introduce another term .

Taylor'stheorem showsthattransformation (9.7)can bewritten X

F

=

00

z

ptz,y;0)+ t î( % (z,?7;o)+ O(6) t' ?' ? )

2

=. ?)(z,y;0)+ t î( % (z,?7;o)+ O(6 ),

208

9 Noether'sTheorem

provided I6Iissmall.Let

1( z,; ( , ), plz,#)=( :9 4 65((z:.?;()). $(z,y)- oc

,

Then,relation (9.8)gives x - z+ ,$+ 0(,2), y-- y+ ,,y+ 0(,2), k-asz+ E1

.

Y'ssy + Erl.

Thef-unctionsïand parecalled theinfinitesim algeneratorsforthetrans-

formation (9.7).Similarly,theinfinitesimalgeneratorsforatransformation of theform (9.19)aregiven by

Theorem 9.3.1 (Noether) Supposetltat/(z,y,!/?)isvariationallyïzzwtzrïanton gzt),z11undertransjbnvtation (9.7)nitltïzz/zzï/csïmtzlgeneratorsï and p. Tlten

t ' ?/ t' ?/ ? r/t'93/?+ ï' /- ç'ly?!/ = covtst. along any eztrevtalo.f

r/r- SV = covtst. Theleft-hand side ofthisequation isprecisely thesam equantity encountered

in thegeneralvariation condition (7.23).Noether'stheorem can be proved using a calculation sim ilarto thatleading to thiscondition.Them ain differenceisthatwe areno longerdealing with arbitrary variations' ,instead,w eare restricted totheone-param eterfam ily off-unctionsdefined by transform ation

(9.7).

9.3 Noether'sTheorem

p roor: Let

1(y)-

/(z,y,: v')dz,

.

G

wherea and b are numberssuch thata < 5,gtz,51f;lgzt),z11,butotherwise

arbitrary.Here,weregard. 1asaf-unctionalwithvariablelimitsofintegration. By hypothesis/ isvariationallyinvaria. ntand henceforany X,' l1 defined by

transformation (9.7)wehave

Suppose thaty isan extremalfor j.For I6Ismallwecan regard ' l1 along with the lim itsaE,5,asa specialcaseofavariation with freeendpoints,since

condition (9.27)isstrongerthansimplyrequiringthatJ(X )- . 1(y)= 0(62). W emaythususethecalculationsleadingtoequation (7.23).Here, -k -- z+ .1+ (?(62)= z+ EXv, l1 - y+ 6p+ 0(62)= z+ 6' ' II. W ecannotargueasin Section 7.2thattheEuler-lsagrangeequation issatisfied because weare notfree to choosethe specialclassofendpointvariationsthat vanish.W ecan nonethelessassertthattheEuler-lsagrangeequation issatisfied

by' l1 becauseextrema. lsmaptoextremalsunderpointtransformations(Theorem 2.5.1),and theinvarianceof/impliesthattheEuler-lsagrangeequation isunchanged forthese transform ations.W eare thusled to therelation

zp - I. FI G = 0.

(9.28)

Exam ple 9.3.1: W e can rapidly recover T heorem 2.3.1 from Noether's

theorem.LetJ bea functionaloftheform (9.4).W eknow from Section 9.2 thatthe translationaltransformation (9.9)isa variationalsymmetry forJ. N ow ,

$(z,y)= ô0 :

6 E-o

:#

plz,y)= :6

E-o

= 1, = 0.

Equation (9.25)thusimplies '?/ - /= H = const. y?t t ' A/ along any extrem al.

9 Noether'sTheorem

$(z,y)- -2zlnz, ptz,y)- y. N oether'stheorem indicates that zyy?- !/?2z21n z = const.

(9.29)

along theextrem alsforJ.W e can verify thattheaboveexpression issatisfied

forallextremalsby differentiatingtheleft-hand side ofequation (9.29)and applying the Euler-lsagrangeequation,

(t r#/)/= 0

zy/= const.,

N oether's theorem can be generalized to accom m odate f-unctionals that involve severaldependentvariables.ln this case,it takesthe follow ing form .

Theorem 9.3.2 (Noether) Suppose tltatL(t,q,(k) isvariationally ïzzwtzrïanton (f(),f11undertlte transjbnvtation (9.19),wltereq = (t 71,...,qnl.Letï andpk betlteïzz/zzï/csïzrzt zlgeneratorsJort/zïstransjbrvtation, Pk = n4k, and

tlte fftzm ïl/tazzïtzzz .Tlten

(9.30)

9.3 Noether'sTheorem

along any eztrevtalo.f

tl

J(q)-

te

Llt,q,ù)dt.

Exam ple 9.3.3: Consider the Lagrangian L of Exam ple 9.2.3.W e know

thatthetranslationaltransformation (9.22) isa variationalsymmetry.For thistransformation ï= 1,pk= 0,and Noether'stheorem gives

bI = const. (9.31) along extremals.The rotationaltransformation (9.23)is also a variational .

symmetry.ln thiscaseï= 0,and

ra = ( 96 Ulcos6+ Asin6) E=o =

(-vlsint î+ qzcos6) 6=:0

and

Equation (9.30)thusgives kkqz - qzqk= const. along extrem als. T he Lagrangian in the above exam ple com es from the K epler problem

(Example1.3.2),whereq denotestheposition oftheplanet.Equation (9.31) indicatesthatenergyisconservedalongtheorbitoftheplanet(seeExample 3.2.2).Thesecondconservation law (9.32)islessobvious.Thisequation corresponds to K epler's second law of planetary m otion,viz.,the conservation of tt arealvelocity.''ln the nextexam plewe explore connexionsbetween som e well-known conservation lawsfrom classicalm echanicsand the corresponding variationalsym m etries.

T((k)- j 1. ?wz(ti/+t 11+t i1), where?rzdenotesthe massoftheparticle.Let U(f,q)denote the potential energy.H am ilton'sPrinciple im pliesthatq isan extrem alfor

9 Noether'sTheorem tl

J(q)-

te

Llt,q,ù)dt,

wheretheLagrangian is

L(t,q,ù)- T((k)- t-tf,q). The well-known conservation laws ofenergy,m om entum ,and angular m om entum correspond to translationalorrotationalvariationalsym m etriesfor J. A . C onservation of E nergy Suppose thatL isvariationally invariant under the transform ation

LI = T + U = covtst., so thatthe totalenergy isconserved along an extrem al. B . C onservation of M om entum Suppose thatL isvariationally invariant under the transform ation

Q1= qk+ 6,

rl= *11 = r2ll= c/asf., which indicatesthatthe qï com ponentofm om entum isconserved. C . C onservation of A ngular M om entum Suppose thatL isvariationally invariant under the transform ation

(9.36) (9.37) Then ï= 0,pl= q,,m = -t 71,and pa= 0.ForthiscaseNoether'stheorem yields Pkqz - Pzqk = covtst.

Now,themomentum vectorisp = (p1,yu,pa),and theangularmomentum abouttheorigin isp A q.Evidently,the term pïq,- pnqïisthe qs com ponent ofthe angularm om entum vector' ,hence,Noether'stheorem im plies thatthis com ponentofthe angularm om entum isconserved.

9.4 Finding VariationalSym metries

N oether's theorem thus providesa nice m echanism for interpreting wellknow n conservation law sin term sofvariationalsym m etries.ForLagrangians

oftheform (9.33)wecan readilydeducetheappropriatevariationalsymmetriesfrom thepotentialenergy f-unction U .1f,forinstance,U'doesnotcontain texplicitly then w ehavethe conservation ofenergy.lfU'doesnotdepend on one ofthe qk then we know that thecorresponding com ponentofm om entum

i sconserved.lfU'correspondsto acentralforce,i.e.,U'= U(f,r),wheresay r2 = q2 ï+ ( 7z 2? then we have thatthe qs com ponent oftheangularm om entum isconserved.

9.4 Finding V ariational Sym m etries Thereaderwillappreciateatthisstagethatthecrttx with Noether'sTheorem is finding thevariationalsym m etry.ln fact,itis clear from the statem entof N oether's theorem thatw e need find only the infinitesim algenerators ofa variationalsym m etry in order to construct the corresponding conservation law .ln thissection wegive a m ethod forfinding variationalsym m etries.The m ethod isbased on thefollow ing result,the proofofw hich we can be found

in W an (711,orin amoregeneralform,in Giaquintaand Hildebrandt(321. T heorem 9.4.1 Let

t' ?/

t '?/

? ?? t' ?/

?

i't' ?z + vloy + (p - y. i')t'??/,+ . i'/= 0,

.

t gp

. y ? T?= (:z + ..% ( :y ,

and

j'?= tgï + % yt ...

tgz X

.

Equation (9.38)can beused to find the infinitesimalgeneratorsp and ï. Prim, aJacie,itseemsthatwehaveonedifferentialequation fortwounknown functions,but the equation must hold for any y notjustextrem als,and it

isthis condition thatyieldsadditionalequations.The condition (9.38)isa relation oftheform

l/ lslz,#,!//)= 0,

9 Noether'sTheorem

thatmusthold forally.Theunknown functionsp and ï depend only on z and y,sothatwedoknow how W'dependson y?in termsofï and p.Now,

equation (9.39)isanidentitythatmustholdpointwiseon gzt lzllforanychoice ofy.Sincewemay alwayschoose!/(z#)and !/?(z#)independentlyatanypoint ztCEg ztlzll,wecanregardy?asanindependentvariableforthisidentity.This meansthatwecansupplementequation (9.39)withequationsoftheform (:kW ' ôy?k = 0,

(9.40)

l4S(z,y,y')- Ay'z+ By'+ C where-4,. B,andC aref -unctionsthatdependexplicitlyon z,y,1,palongwith

thepartialderivativesofthe generators.Then,equation (9.40)impliesthat thecoefllcientsof3//2 y?m ustvanish' .e.,. , 4= . B = 0,and hence C = 0.T hese ,i

threeequationscan then beused todetermineï and p.Notethatweexpect an overdeterm ined system ,since variationalsym m etriesare specialand not

every functionalhasthem.Moreover,ifthereexistïand pthatsatisfy these equations we expect these f-unctions to be determ ined to within a constant ofintegration because no initialdata are specified.Theabove com m entsare perhapsbestillustrated through specific exam ples. Exam ple 9.4.1: C onsiderthe f-unctionalofExercise 9.2-1.For this func-

tional,equation (9.38)is

$: 4//2+2zy'tpz+y'vly-y'L -: 4//2ï#)+z!//2(#z+yl1#) = -I1yy'3+ (ï+ 2zvly- zïz)3//2+ zzpzl// whereïz = 01 t ' ?z etc.Thecoefllcientsof3//3 y?2 and y?mustvanish' ,hence, z$y= 0, $+ 2z% - zïz= 0, zpz = 0, 1Of course, we could argue that sim ilar expressions can be ascertained by differentiating ' w-with respect to the other independentvariables z and p butwe do

notknow r/orïatthisstage(thisisthepurposeofstudyingtheequation)and hencewe do notknow how ' w-depends on z and p.

9.4 Finding VariationalSym metries

forallz CEgzt ),z11andy.Equation (9.41)impliesthatly= 04hence,ï= ï(z). Similarly,equation (9.43)impliesthatp = 1)y.Sinceïdependsonlyon z,and p dependsonlyon y,equation (9.42)issatisfied onlyifvly= ctmsf.;therefore, 07/)= c1#-1-C2, wherecland czareconstants.Equation (9.42)now givesthefirst-orderdifferentialequation

(9.45) ï= -2c1zlnz+ caz,

where ca is a constant.Equations (9.44) and (9.46) thus define a threeparam eter fam ily of infinitesim algenerators that correspond to variational sym m etries.T he infinitesim algenerators for the transform ation ofExercise 9.2-2 correspond to the choice c2= ca= 0,cl= 1. Exam ple 9.4.2: LetJ be thefunctionaldefined by

Forthisfunctional,equation (9.38)leadstotherelations z 2ï#= O,

2z$+ 2z207#- z2#2 = O, 2z2072+ y'sl#= 0,

4078/3+ lzy' i= 0,

(9.50)

which theinfinitesimalgeneratorsï and pmustsatisfjrforthecorresponding

transformation to be a variationalsymmetry.Equation (9.47) implies that ï = ï(z);hence,equation (9.49)impliesthatp = p(!/).Equation (9.48)thus shows thatvly = covtst.,and equation (9.50)shows thatïz = const.The functionsïand pmustthereforebeoftheform ï'= clz+ c2, p= czy+ c4.

2z(clz+ ca)+ 2z2ca- z2c1= 0,

9 Noether'sTheorem

ca = 0, 2ca + cl = 0.

Substituting expressions(9.51)intoequation (9.50)gives

4#3(csy+ c4)+c1/ = 0, which m ust be satisfied for a11y;hence, 4ca + cl = 0.

Theonlychoiceofconstantsthatsatisfies1 30th (9.52)and (9.53)iscl= ca= 0,sothatï= 0,p = 0istheonlysolution to (9.38)forallz CEgzt),z11and all y.ln thiscase there are no variationalsym m etriesforthe f-unctional.

whereq = (t 71,...,qnland n,> 1,issomewhatmorecomplicated than that given in Theorem 9.4.1.Letï and pl,...,pzzbe theinfinitesimalgenerators

forthetransformation (9.19),andlet

where

Theorem 9.4.2 Tltetransjbrvtation (9.51)isavariationalspmmc/r?/Jortlte Junctionaldehned by (9.54)z/and only zl /

pr(1)vtfvl+ Lj= 0 Jorallsmtata//zq on (f(),f11.

9.4 Finding VariationalSym metries

Exam ple 9.4.3: K epler Problem Let

(' )L = 0, ç ')t

t gfv

oqk

qkK

(. ??+ qk)z/,

Now,

and

l ($1 t ' ?$ t ' ?rp t' ?rp k= oqk, lt= t' j,k= oqk, r/ j,t= ot. ?f, r/

Equation (9.55)isthus

9 Noether'sTheorem

A -

K'I,

.

zrzt ia p2,t+

+K

ql+ qq

lt q2

plt ?l+ 072:2

-

2

2 3/2 ï+ q, (t ?21+ q, )

The sam e argum ents used in the n,= 1 case can be leveled at the above

equation,which is an identity for allq.The coefllcients oft J1 3 and t )z 3 must vanish,and this meansthatï dependsonly on f.Since ï1 = 1, = 0,the coefllcientsofthe othercubicterm sin thederivativesalso vanish.ln a sim ilar

way we argue that the coefllcients of (k 2 etc.vanish,and this leads to the followingsystem ofequationsfor1,pland w.

(9.57) (9.58) (9.59) (9.60) (9.61) Equations (9.59)and (9.60)show thatthe pk do notdepend on f.Sinceï dependsonlyonf,equations(9.56)and (9.57)indicatethatthereisaconstant cl such that

(9.62) r/11 = C1, 1)22 = C1;

'?l- clt ?l+ g(q,), '?2- clt o + h. lqk), whereg and /Jare f-unctionsto be determ ined.Substituting theaboveexpres-

sionsforthepkinto equation (9.58)gives (' zqz

ç 'kz

+ - = 0, &t yl

= -

t ' ?t ?l

= cz.

9.4 Finding VariationalSym metries

r/l= clt ?l+ c24 72+ ca, 072 = -c2t?1+ clA + c4,

which,usingequation (9.62),reducesto

cl(t ?2 1+(72)-caq1-c4q,=0.

(9.63)

Now,equation (9.63)isan identity thatmustbesatisfied forallq.We thus concludethatcl= ca= c4 = 0.

Equation (9.55)thusshowsthatthe infinitesimalgeneratorsofa varia-

tionalsym m etry ofJ m ustbe ofthe form

V1 ==C2q2t wx == --czgy?

$XXC5, where cs is another constant ofintegration.W e thus have a two-param eter

family ofgeneratorsthatleadtovariationalsymmetries.lfcz= 0andcs, # 0, then thetransformation isatimetranslation.lfcz, # 0 and cs= 0,then the

transformation isarotation (cf.caseC,Example9.3.4).Theorem 9.4.2shows thatthe only variationalsym m etries ofJ are com binations ofrotations and tim e translations.

N oether'stheorem asgiven in thischapter along with conditionsforvariationalsym m etries can be further generalized to accom m odate functionals involving higher-orderderivatives and or severalindependentvariables.The

quantityy, r(1)v(fv)iscalledthefirstprolongation ofthevectorfield v tdefined bytheinfinitesimalgenerators)actingon L.lfafunctionalhasan integrandthatinvolvesderivativesofordern,thenahigherprolongationpr('')v(fv) isneeded.Theexpression forpr('')v(fv)escalatesincomplexityasn,increases, butthecondition foravariationalsym m etry rem ainsdeceptively sim ple,viz.,

pr (,z)v(z,)+ L1'- 0. ThereaderisdirectedtoBlumanandKumei(111andOlver( 571forthegeneral expressionofpr('')v(fv)intermsofthegeneratorsand theirderivatives.Here, we have given only a basic ttno frills''version of Noether's theorem and the readeris encouraged to consultthe abovereferences fordeeperinsightsinto thisresult.

220

9 Noether'sTheorem

Exercises 9.3:

1.Considerthe f-unctionalofExam ple 9.4.1.Show that the transform ation

F = y,X = (1+ 6lzisavariationalsymmetryandfind thecorresponding conservation law . 2.The Em den-Fow ler equation ofastrophysicsis #??+.2 .y?+.y5 . (), I

which arisesasthe Euler-lsagrange equation forthe functional

J(y)=

z1z2

j

-2 y'2- 3 -y6 . dz.

20

Find theinfinitesim algeneratorsthatlead to a variationalsym m etry for this functionaland establish the conservation law z

2Ly'y+2: 71(: 4/2+ yö))= const.

3.The T hom as-Ferm iequation

correspondsto theEuler-lsagrangeequation forthefunctional 21 1 ?z

J(,)-,

23/5/2

(L. y+g.,s)dz.

Show thatthisf-unctionaldoesnothavea variationalsymmetry.(Exact solutionstotheThom as-Ferm iand Em den-Fow lerequationsarediscussed

in detailin (81.)

T he Second V ariation

The Euler-lsagrange equation form s the centrepiece ofthe necessary condition for a functionalto have an extremum .T he Euler-lsagrange equation is

analogousto thefirstderivative (orgradient)testforoptimization problems in finite dim ensions,and we know from elem entary calculusthata vanishing firstderivative isnotsufllcient fora localextrem um .Likew ise,satisfaction of theEuler-lsagrange equation isnotasufllcientcondition foralocalextrem um fora functional.ln essence,we need a result analogous to thesecond derivative test in order to assert that a solution to the Euler-lsagrange equation produces a localextrem um .ln this chapter we investigate the next term in

theexpansion ofJ(L)- J(y),thesecond variation,and develop morerefined necessary conditions for localextrem a.W e also develop sufllcient conditions fora function y to produce a localextrem um fora functionalJ.W e restrict our attention alm ost exclusively to the basic flxed endpoint problem in the plane.

10.1 T he Finite-llim ensional C ase The reader is doubtless aware of the second derivative test for determ ining whethera stationary pointisa localextrem um fora function ofonevariable. ln this section w e review a few concepts from the finite-dim ensional case, prim arily to m otivate ourstudy ofconditionsforfunctiona. lsto haveextrem a. W e begin w ith thefam iliarcaseoftwo independentvariables.

Let/ :J2 --+ R be a s11100th function on theregion J2 ( :z R2.Letx =

(z1,z2)CEJ2and letk = x + 6p,wheret î> 0 and p = (p1,w )CER2.Ift îis sm all,Taylor'stheorem im plies that

/(k)-/ (x)+6tnzt f:z t' v+' 1 ?20:2 : 7' z Y/(a) ) + ,2 :2/(x) x) :2/(x) jït TI:zt+2p1p2ozzoz,+pl:zj)

10 The Second Variation

+ 0/3). lf/ hasa stationary pointatx then V/(x) = 0 (cf.Section 2.1),and the Taylor expansion reducesto 62

/(2)- /(x)+ -j-Q('?)+ (?/2), where

Q('?)- '?, ' ?2/( %/(x)+ '?, t % /( x). ,t ,x)+ 2p1072t z t ' ?zl t ' ?zltr ?zz a t ' ?zz Thenatureofa stationary pointisdeterm ined by thelow est-orderderivatives that are nonzero atx.lfone ofthe second derivativesis nonzero,then the

sign ofJ(k)- /(x)iscontrolled bythesign ofQ.

Fornonzero p CER2, thequadratic term m ay be always positiveor always negative,but it m ay be that this term is positive for som e p and negative

forothers.Thecharacteroftheextremum revolvesaroundwhatsignsQ may haveforvariouschoicesofp,and wecan track thesign changesby exam ining

when Q(p) = 0.lfp , # 0,then eitherplorm is nonzero.W ithoutlossof generality suppose thatw , # 0.Now,Q isacontinuousf-unction ofp,and if

Q changessign,theremustbesomep, # 0 such thatQ(p)= 0.Hence,there

m ustbe a realsolution to thequadratic equation

p. l 2. % /(x)

p

(w) ozî+2p: ?-o . %zï ?o (x z) ,+. %t ' s ?( y x)-0. ..

Thenatureofthesolutionsto thisequation isdeterm ined by thediscrim inant,

Z = t%/d2/ t ' ?z2 1( ' ?zz 2

(92/

(t'?z1t'?zz12'

ofthe quadratic form Q atx.There can be at mosttwo solutions to this quadraticequation.lfrealsolutionsexist,then Q may changesign.lfthere are no realsolutionsto thequadratic equation then Q,being a continuous function,willneverchange sign.W hether Q = 0 has a nontrivialsolution

dependson thesign of. 4:if1(x)< 0and :2/ ( ' ?zz y: /:O (or(92/ ( ' )zz zL , ;lk0)at x then Q isindefinite and vanishesalong two distinctlines.Forthiscase,a

smallneighbourhood ofx,B(x;6),canbedividedintofoursets,twoinwhich Q > 0and twoinwhich Q < 0.lnthiscaseQ iscalled indefinite.Evidently,

x cannotproducealocalextremum becausethesign ofJ(k)- /(x)depends on thechoiceofp.Stationarypointsx forwhich 1(x)< 0arecalled saddle points.

ln contrast,if1(x)> 0,then therearenorealsolutionsto thequadratic equation;consequently,Q cannotchangesign.lnthiscaseQ iscalleddefinite, and x correspondstoa localextrem um .Thetypeofextrem um can bededuced from the exam ination ofany particular curve through x.T he sim plest such

curvescorrespond to ')4(p1) = (p1,0)and y,(w ) = (0,w).lfx isa local

m axim um m inim um ,then pl= 0 correspondstoa localm axim um m inim um

for&1(p1)(and w = 0correspondstoalocalmaximum minimum for' p (w )). Thus/(x)isalocalmaximum ift' ?2/(x) (' ?zz y< O (or( ' ?2/(x)(' ?zz z< 0)?and a localminimum if:2/(x) ( ' ?zz y> 0 (or(' ?2/(x) ( ' ?zz> 0). ltmay be that1(x)= 0 even though the second derivatives of/ atx are notallzero.ln thiscasethere isa line (z1(f),z2(f))in B(x;6)through x = (z1(0),z2(0))whereQ vanishes.Thenatureofthispointisdetermined by the third-order (orhigher) derivatives.lf1 (x) = 0,then x is called a degenerate stationary point (orparabolicpoint). Wemustexaminethe cubicterms(orhigher-orderterms)intheexpansion to discern thenatureof the stationary point.

Notethatifallthesecondderivativesof/vanish atx thenitisclearthat

thesign of/(x)- /(k)isdetermined by the third-order derivatives.These m ustalso vanish atx fora localextrem um ,and ifthisisthecase,thequartic term scontrolthe sign. T he above approach can be adapted to f -unctions ofthree or m ore independentvariables,although the increasein variablesescalatesthe numberof

possibilitiesand the complexity ofthecomputations.Let/ :J2 --+R be a

s11100th f-unction on theregion J2( :z R?zand supposex = (z1,z2,...,z,zlis a stationary point.Then,as in thetwo-variablecase,we have V/(x) = 0, andthesignof/(x)- Jlk)iscontrolled bythequadratictermsintheTaylor expansion.LetEk= x-h6p,wheret î> 0and p= (p1,pz,...,p,zl.Thequadratic term sin the Taylorexpansion m ay bew ritten in theform

( p(,y)- pwlltxlzy, *2.J(x) *2. J(x) é)2. j(x) é)zj igzligza ''' i gzligza

*2/(x) *2/(x)

l1(X)==

: :

:

*2*1 *2

*2/(x)

''' pz pz

2O

92.flx) *2. 1 flx) 1)2. j(x) ç'l2a i921 ' çl2a i 922 ''' ç'l2

The nature ofa stationary point depends on whether H is definite.lf H is

definite,then / hasa localextremum atx;ifH isindefinite,then x correspondsto som e type ofsaddlepoint.T heM orselem m a can beused to classify the typesofstationary pointsprovided the H essian m atrix at the stationary

pointhasthe samerank asthe numberofindependentvariables (i.e.,H is nondegenerate).Stationarypointssatisfyingthisconditionarecallednondegenerate.

Lemm a 10.1.1 (M orse Lemm a) Letxt lbeanondegeneratestationarypoint Jor tlte smtat a//zJunction J. Tlten tltere ezistsa smt at a//zinvertiblecoordinate

transjbrvtationzj--+zj(v),wlterev = (z?1,zu,...,r,zldehnedïzzaneigltbourItood.; V(xo)o. fxt lsuclttltattlteidentity

10 The Second Variation

/(X)= .1(V)= /(Xo)- ' &2 1- ' &z 2- ...- z? z 2 + tlà 2-hl+ '''+ vn 2

Itoldst/zrt atw/zt a' tz/Nlx(j).Tlteinteger, Liscalledtlteïzzde:ro. f/ atxo.

A proofofthislemmacanbefoundin (531.Thef-unction z?2 1+ zj+ ...+

z?z2 -

zu2+y- ...- zu2 is called a M orse à-saddle. The index is an invariantunder s1100th invertible coördinate transform ations' ,therefore,itcan be used to classifjrnondegenerate stationary points.A M orse zz-saddle is a local m axim um ;a M orse G saddle isa localm inim um .lf,L isnot0 orzz,then the

M orse à-saddleindicatesthatthe difference /(x)- /(k)can be positiveor negative depending on the choice ofx.The M orse lem m a has another consequence:the stationary points ofthe saddle are isolated,and since s11100th invertiblecoördinatetransform ationsleaveisolated stationary pointsisolated, allnondegenerate stationary points mustbe isolated. T here isa wealth ofresultsregarding conditionsunderwhich a quadratic form isdefinite.An exam pleisprovided by thefollow ing theorem .

Theorem 10.1.2 (Sylvester Criterion) LetX = (. X1,A3,...,Ak)andlet A denote an zzx n,spmmc/rïcvtatriz' ttlï//zentriestzu.. 4 necessary and suh jicientcondition Jor a quadratic/brm XT'AX to bepositive clc/zzï/c,istltat cwcr!/principalm ïzzt?r determinant o.fA ispositive.In y' tzràïctltzr,detA > 0 and every diagonalelevtentajj ïspositive.

Supposethatx isa stationary pointfor/and let /z11 /z12 '''/àl,z /z21 /z22 '''ltzn. H =

,

/à,zlltvbz '''ltvbn. denote the Hessian m atrix at x. The above theorem indicates that the quadratic form is positive definite if/z11 > 0 and the determ inants ofthe m atrices /z11/z12 /z21/à22 '

10.2 T he Second V ariation Letusreturn to the basic flxed endpointvariationalproblem .R ecallthatwe

seek as11100thf-unctiony :gzt),z11--+R such that!/(zo)= yçj,!/(z1)= 3/1,and the f -unctional

10.2 The Second Variation 21

/(z,ïl,t v')dr 20

is an extrem um .W e m ake the blanket assum ption throughout this chapter

that/ iss1100th in theindicated arguments.M orespecifically,weneed / to havederivativesofatleastthird orderforsom e ofourargum ents.W eassum e

thatforany given extremaly,/(z,!/(z),!/?(z))iss11100thin aneighbourhood ofzt land in a neighbourhood ofz1.

Supposethat J hasan extremum aty,and let:' èbe a tnearby''f -unction of

theform : ' è= yj-evl,wheret î> 0isasmallandpisas11100thfunctionon gzt),z11 such thatp(zo)= p(z1)= 0.Ourbriefforay intothefinite-dimensionalcase indicatesthatwe need to considerthe0(62)termsofJ(L)- J(y)in order to glean inform ation regarding the nature ofthe extrem al.Taylor's theorem im plies

h y= d2 / yy'= od2 / y/y'= od2//2, ô 2, l' (. y?, l' y y y y where,unlessotherwisenoted,thepartialderivativesareevaluatedat(z,y,!/?). Thus,

62

J(5)- J(: v)- 6JJ('?,y)+ -2J2J('?,y)+ O(6a), 21

J2J('?,y)-

20

(z?2&,+2p07/&,,+z?/2&,,,)dz.

Theterm J2J(p,y)iscalledthesecond variation ofJ.Thesecondvariation playsarôleanalogousto the Q(p)term in thefinite-dimensionalcase.Since y isan extremalforJ wehavethatJJ(p,y)= 0,and hence 62

J(5)- J(t v)- -j-J2J('?,y)+ 0/3). The sign ofJ(L)- J(y)thus dependson thesign ofJ2J(p,y),and consequently the nature ofthe extrem althusdependson whetherthesecond variation changes sign fordifferentchoicesofp.N ote that,atthisstage,w e have already solved theEuler-lsagrangeequationsso thaty isknow n and hencethe

functionshy,/:/://,and Jy/y'areknown intermsofz.Thissituation parallels

226

10 The Second Variation

that forthe finite-dim ensionalcase,w here we know the num ericalvalues of the entries in the H essian m atrix. U sing the notation of Section 2.2,let S denote the set of f-unctions y

s11100th on gzt),z11such that!/(zo)= yç jand !/(z1)= yï.Let. bI denotethe setoffunctionsp s11100th on gzt),z11such thatp(zo)= p(z1)= 0.Theabove argum entsyield the following necessary condition.

J2J(p,y)k 0

(10.3)

JorallpCEH ;zl /y ïsalocalmtzzïzzwm,tlten

J2J(p,y)< 0 JorallpCEH . T heaboveresultisoflim ited valueatpresent,becausewe haveno m ethod

totestthesecondvariationforsignchanges.lnequality (10.3)isanalogoustoa positive sem idefinitecondition on a Hessian m atrix,and wehaveseen how the

addition ofindependentvariablesescalatesthenumberofpossibilities(types ofsaddles)andthecomplexityofverifjringwhetherthematrix isdefinite.For theinfinite-dim ensionalcasewethusexpecttestsforestablishing sign changes in thesecond variation to be com plicated.ltisthus a pleasantsurprise that certain conditionscan be derived thataretractableand sim ple to im plem ent. Exercises 10.2 : 1.SupposethatJ isa f-unctionaloftheform

J(t v)-

'î3

+ y! ',53J(p,y)+ 0/4), where

é3J(p,y)= zt(p/3osj + 307/2,7 (. ?a/ 2 ç' ly'3 ( '?: 4//24 '7?y 0

?3/ ' 93/ V 307/072t'9t + 079t )dI. 2 //t ' 92 /2 (13

ThefunctionalJ3J(p,y)iscalled thethird variation ofJ.

10.3 The Legendre Condition

10.3 T he Legendre C ondition ln this section we develop a necessary condition for a f-unctionalto have a localextrem um .This result is called the Legendre condition.U nlike T heorem 10.2.1,it is straightforward to apply and hence useful for filtering out extrem als thatdo not produce a localm inim um orm axim um . T he second variation can be recastin a m ore convenientform thatseparatesthe p term sfrom the p?term sin the integrand.N otethat

2r/r///yy/= (r/2)//l/!//, and hence 21 zj 21 d /r///? 7?7/dr= Vhy'zo - zo 072-jyy/)dz 20 2r dzL z1 cl

=-

20

?72-dz Ljyy/)dz,

where we have used the conditions thatp(zo) = variation can thusbe written z1

é2J('?,t v)-

20

= 0.The second

d

'/ hy- -dz (S,') + p/2/,/,/ dz.

T he essence ofthe Legendre condition is that the second variation m ust

change sign for certain choices ofp CE . bI if/? y/://changes sign.Before we launch into a statem entand proofofthe Legendrecondition,a few com m ents to m otivatetheproofare in order.W ereiterate thatthecoefllcientsof072and p?2 areknow n f-unctionsofz. Let. , 4 and . B be the f-unctionsdefined by

B (z)- Jyylz,: , v(z),: v'(z))- (d s (hy'(z,ylz),y (z))), forz CEgzt),z11.Since / and y are sm00th,130th 4 and B are continuous functionsontheintervalgzt ),zll.Thereasonthatthesign of-4 playsapivotal rôlein thistheory isthattherearefunctionsp CE. bIforwhich Ip(z)Iissmall forallz CEgzt),z11,but Ip?(z)Iisnot.ln contrast,ifIp?(z)Iissmallfor all z CEgzt ),z11,then,sincep mustbes11100th andsatisfjrtheconditionsp(zo)= p(z1) = 0 for membership in .bI,we have that Ip(z)Iis also smallfor all z CEgzt ),z11.Thesimplemollifier

p(z)= exp (),

-. y-

()-c)),ifzCE( c-y,c+/1 ifz ( gc- y,c+ aj,

10 The Second Variation

wherecCE(zo,z1),and y > 0issomesuitably smallnumber,illustratesthis scenario.ltcan be show n that this function is s1100th 1 and vanishes at zt l and zl so thatitisin theset. bI.N ow them axim um value ofp is1 c,butthe

meanvaluetheorem showsthatthereisaleastonevalueofz CE(c- y,c)such thatp?(z)= 1 (&c),and thecontinuity ofthe derivative thusimpliesthat thereisasubintervalf ( :z(c- y,c)forwhichp?(z)> 1 (2&c)forallzCEf. Theissueatstake,ofcourse,isnotthepointwisebehaviourofIp?(z)Ibut rathertheinfluenceofthisfunction on theintegraldefining the second varia-

tion.Thesubintervalf,afterall,mightberathersmall,and although p?2(z) islarge forz CEf compared to 072(z)the overalleffecton the value ofthe integralm ightbesm allow ing tothesm alllength off.W e m ustkeep in m ind, how ever,thatw ecan constructafunction p CE. bI thatisessentially asuperpo-

sition ofany numberofmollifiers,each with adifferentvalueforcCE(zo,z1) with y > 0 chosen sufllciently smallsothattheirsupports (i.e.,intervalsin which they arenonzero)do notintersect.Theneteffectisthatf -unctionsin bI can alwaysbe found such thatthe derivative term s dom inate the second variation.A superposition ofm ollifiersm akesthe im portance ofthe sign of -4 t ransparent,at least conceptually.Rather than chase the above m ollifiers .

any f-urther,however,we opt (in the interests ofcomputationalsimplicity) fora sim pler function that capturesthesam e behaviourfor the proofofthe Legendrecondition.

Theorem 10.3.1 (Legendre Condition) LetJ beaJunctionalo. ftlteArm (10.2),wltere/ isa smtat a//zjhnction o. fz,y,and!/?,and supposetltatJ Itas a localm ïzzïm ' tzm in S aty.Tlten,

Jy/y'k 0

Jorallz CEgzo,z11. P roof: Using the notation introduced above,suppose that there is a c CE

gzt ),z11suchthatdtcl< 0.SinceJ hasalocalminimum aty,Theorem 10.2.1 impliesthatJ2J(p,y)k 0forallpCE. bI.Thetheorem isthusestablishedifit can beshown thatthereisan v CE. bIsuch thatJ2J(zz,y)< 0. Since4 iscontinuouson gzt ),z11thereisany > 0suchthatdtzl< dtcl 2 forallz CE(c- y,c+ &).Weconstructa f -unctionin . bI thateffectively filters outtheinfluenceof-4 and B forallz notin (c- y,c+ ')')and magnifiesthe contribution ofthe derivative.Let

ztzl= Sk yj 4 xtz n'c)( )'kfzsy-y,c+m j 0,

ifz ( gc- y,c+ &' 1.

10.3 The Legendre Condition

v'lz)= 0< ' è , . ' jsi yja' vtz ' è 'c)( jcos' r( z ' è 'c)( ),i fz(Eg c-y,c+' yj , ifz( gc- y,c+ &1. W ecan getarough upperbound on thev?contribution tothesecondvariation as follows.

SinceB iscontinuouson gc- y,c-h:1thereisanN > 0suchthat1. D(z)I< N forallz CEgc- y,c+ :1;hence,a rough upperbound forthev contribution to the second variation is given by

N ow ,

J2J(zz,y)< 16-4(c)r2+ 2Ny, so thatthe second variation isnegativeforv,ify is chosen arbitrarily sm all. W ehavethefreedom to choosey sm all,sothattherearefunctionsin . bI that m ake thesecond variation negative.T hiscontradicts Theorem 10.2.1and we

concludethatdtzlk 0forallzCEgzt ),z11.

En

Evidently,theaboveresultcan bereadily m odified forthecasewhereJ has

a localmaximum aty.lnequality (10.7)iscalled theLegendre condition. Aside from the theoreticalbenefits (which wereap later)the Legendre condition is a practical tool for deciding w hether a solution to the EulerLagrange equation is even in the running fora solution. Exam ple 10.3.1: Let

For a given set of boundary conditions at z = - 1 and z = 1 we can find

thecorrespondingextremalexplicitly forthisf -unctional(seeExercises2.3-1).

230

10 The Second Variation

H owever,w eneed noteven do thisto deducethattheextrem alscannotgivea localextrem um for J.W ecan use elem entary argum ents to show that J can be m ade arbitrarily sm all,butthe Legendre condition conveniently answers the question.ln particular,we have forany s11100th y,

Jy/y'(z,y,y')- (1 z -

F7/213/2,

Exam ple 10.3.2: C atenary Consider the catenary problem ofExam ple 4.2.1.W e consider the problem here as an unconstrained flxed endpoint problem w ith the appropriate Lagrange m ultiplier,so thatin the presentnotation

/(z,y,!//)= (!/- à) 1+ : v/2, where

Here,/Jistheheightofthepolessupporting thecableand ïisoneoftwo possible nonzero solutions to

r,j.= sinhty,

.

whereL is the arclength ofthe cable.Asdiscussed in Fuxam ple 4.2.1,we can distinguish which solution isrelevantfrom sim plephysicalconsiderations.The Legendrecondition also m akes a distinction.N ow ,forany y,

J

,L y/y'(z,y,y')- (1 yF7/213/2, -

so thatthe sign ofthisderivative isthe sam easthesign ofy- à.Thesolution

to theEuler-lsagrangeequationsisgivenbyequation (4.32)and therefore

Recallthatthereispreciselyonepositivesolutionandonenegativesolution1%

toequation (10.8).Onlythepositivesolution satisfiestheLegendrecondition fora localm inim um . T he Legendre condition cannot be converted into a sufllcient condition

even ifwereplaceinequality (10.7)by thestrengthened Legendre condition

Jy/y'> 0. The following well-worn butsim ple exam pleillustratesthis com m ent.

10.3 The Legendre Condition

Exam ple 10.3.3:

J(t v)and theboundary conditions3/(0)= 0,y(f)= 0.Wehaveseen in Chapter5 thatproblem softhissortlead toSturm -lsiouvilletypeproblem s,and thatthe

existenceofanontrivialsolutiondependson thechoiceoff.(ModifjrExample 5.1.1by fixing , L= 1and replacetheupperlimitr withf.)Supposethatwe choose f> r. Forthis sim ple problem the strengthened Legendre condition isevidently satisfied forany y,and the second variation is given by

Let

TI

p = sinttr). Clearly p CE.bI ,and

The trivial solution y = 0 is an extrem alfor the problem ,but the above calculation showsthatitcannot give a localm inim um for J. Legendre him selftried to fram e a sufllcientcondition fora localm inim um

around the inequality (10.9)but ran into various snags such as the above exam ple,and it becam e apparent that m ore inform ation was needed.The essenceoftheproblem isthatthestrengthened Legendrecondition and EulerLagrange equation place only pointwise restrictions on the functions.The

greatcircleson asphere (i.e.,thegeodesics)givean intuitiveexampleofwhy globalinformation isneeded.Considerthree pointson the earth (which we assumeisaperfectsphere)allon thesamegreatcircle,say theNorth Pole, London,and theSouth Pole.Theshortestdistancefrom London to theN orth Poleisalong the m eridian connecting thesepoints.Butthereare two choices: one can proceed directly north,or one can go initially south,through the South Pole,and then turn northwardsto eventually arriveatthe North Pole. Evidently,the latteroption producesa longerpath,butpointw isethe EulerLagrange equation is satisfied on the m eridian as is the Legendre condition. ltisonly when we look at ttthe big picture''thatwe realize the latteroption cannotbe even a localm inim um :there arepaths near the South Pole route

thatareshorter(they avoid theSouth Poleatthe ttlastminute''and jump ontoasuitably closelineoflongitudeand head north).

232

10 The Second Variation

Exercises 10.3:

0L

J(4)=

1+ sin2p4/2d0, Po

where4 isthepolarangle,0istheazimuth angle,and 4/denotesd4 d0.

Show thatJ satisfiestheLegendrecondition (10.7).

2.Let

10.4 T he Jacobi N ecessary C ondition

The major shortcoming ofthe Legendre condition is that it is a pointwise restriction.Exam ple10.3.3m akesitclearthatotherconsiderationsareneeded before a sufllcient condition can be form ulated.ln this section w e present a necessary condition that builds on the Legendre condition,but is distinctly

globalin character.The key conceptofa conjugate pointisintroduced,and itturnsoutthatthisnecessary condition pavesthew ay fortheform ulation of a sufllcient condition.W e focuson localm inim a trusting the readerto m ake

the necessary adjustm ents to getanalogousresultsforlocalmaxim a. 10.4.1 A R eform ulation of the Second V ariation

For a finite-dim ensionaloptim ization problem we can appealto the M orse Lem m a 10.1.1 to argue that the relevant quadratic form can bew ritten asa sum difference ofsquares.This specialtransform ation ofthe quadratic form allow susto classifjrcriticalpoints as described in Section 10.1.The infinitedim ensionalanalogue ofthis process for a M orse O-saddle is to convert the second variation into a functionaloftheform

whereT isafunction ofz,p,and (indirectly)theextremal.zIdeally,weseek a f-unction T such thatT isidentically zero on gzt),z11onlyifp isidentically zero on gzt ),z11.ln thiscase,the sign ofthesecond variation would depend on thatof/y/y/. 2Theanalogyismademoreprecisein Gelfand andFomin g 31j,pp.125-129.See also ( 221,pp.571-572.

Theideaoftransformingthesecond variation intotheform (10.10)dates back to Legendre,w ho tried to establish the existence ofT by com pleting the square of the quadratic form .Although he failed to achieve a sufllcient condition,his idea ofcom pleting the square proved fruitful. W e know thatthesecond variation can be written in the form

consequently,wecan alwaysadd aterm oftheform (zcp2)?totheintegra. nd of the second variation without changing the value of the functional.The strategy isto selecta function w such that

Jy/y'rl/2 +sr/2+ (1L)rl2)?=Jy/y',. 2, .

forsomeT.We know from theLegendrecondition thatS/://cannotchange sign in gzt),z11ify producesa localextremum forJ.We assume thatthe strengthened Legendrecondition (10.9)issatisfied.Thus, ?

w

B + w'

/y/y/r//2+. /,72+ Lwyzl=/y/y/ zy?2+2/,/,/))))?+ Jy'y' ,72 . Suppose thatw satisfies the differentialequation 2 /. . /J

=

/y/y/(. / + w?)

(10.11)

forallz CEgzo,z11.Then

so thatthesecondvariationcould berecastin theform (10.10).

Following the analogy with the finite-dim ensionalcase,the second varia-

tion iscalled positive definite ifJ(p,y)> 0 forallp CE. bI- (0/..Given a solution to (10.11),wehave 1-- p?+

p

/' Ll /' !// '

p?+ Jy/y'p = 0

.

(10.12)

234

10 The Second Variation

Since p CE. bI,however,the above differentialequation is accom panied by the

conditionsp(zo)= p(z1)= 0.Now,/ isassumed s11100th in z,y,and y?and y isa s1100th extremal;hence,Jy/y'isa s11100th f-unction.Picard'stheorem

showsthatthereexistsauniquesolution toequation (10.12)thatsatisfiesthe initialcondition p(zo)= 0,andaquickinspectionshowsthatthismustbethe trivialsolution p(z)= 0 fora11z CEgzt),z11.W ethusseethatT isidentically zero on gzt ),z11only ifp isidentically zeroon thisinterval.Otherwise,under thestrengthened Legendrecondition,theintegraldefiningthesecondvariation m ustbe positive.ln sum m ary,we have the following lem m a.

Lemm a 10.4.1 Let/ be a szrztat a//zJunction o.fz,y,and !/?,and lety be a

szrztat a//zeztrevtalJortlteJunctionalJ dehned by(10.2)suclttltat/?y/://> 0Jor allz CEgzt ),z11.f/ tltere ïsa solution w to tlte clz#crczz/ït zlequation (10.11) valid on tlteintervalgzt ),z11,tlten tltesecondvariation ïspositivedehnite. T he ttfly in the ointm ent''isthe existence ofthe solution ' t;.W e can appealto Picard'stheorem to assertthe existenceoflocalsolutionsto equation

(10.11),butthisisnotgood enough.W eneedsolutionsthataredefined over theentire intervalgzt ),z11ratherthan in somesmallsubinterval.The reformulation ofJ2J(p,y)thus hinges on whethera globalsolution w existsto equation (10.11). 10.4.2 T he Jacobi A ccessory E quation

Relation (10.11)isanexampleofawell-knownclassofequationscalled Riccatiequations.A standard solution techniqueforsuch equationsentailsconverting thenonlinearfirst-orderequation to asecond-orderlinearequation by use ofthe transform ation N/

w=/?y/:// (10.13) N (cf.( 411,(611).Underthistransformation,theRiccatiequation becomes dd (Jy'y'' u,)- B' u - 0; z

ddz (/p/p/' t zt)- Xy- cd u /pp/ ' tz= 0.

(10.14)

Equation (10.14)is called the Jacobiaccessory equation.lfthere is a solution zttothisequationthatisvalid on gzt ),z11andsuchthatutzl, # 0for allz CEgzt),z11,then transformation (10.13)impliesthattheRiccatiequation (10.11)hasasolutionvalid forz CEgzt ),zll. C ertainly one advantage ofworking with a second-order linear ordinary differentialequation asopposed to a first-ordernonlinearequation isthatthe theory underlying the linear equation is well developed and perhaps m ore tractable.lt is beyond the scope ofthis book to recount the theory in any

detail.Thereaderisdirected to textbookssuch asBirkhoffand Rota (91for a fulldiscussion.ltsufllceshere to m ention a few results concerning theexistenceofsolutionsto the Jacobiaccessory equation.Note thatthesm oothness

assumptionson / and y mean thatthecoefllcients Jy/y'and B ares11100th

functionsofz on gzt ),z11,and thestrengthened Legendre condition ensures thattheproblem isnotsingular.SW ecan thususePicard'stheorem todeduce

that,given initialvaluesutzol= ' t zt landzl/tzol= ' ? . t/ ( ;,thereexistsauniquesolution zt= zttz,' tzo,' tt/ ()ltoequation (10.14)thatsatisfiestheinitialconditions. Picard's theorem guaranteesonly a localsolution near zo,but we can now appealto standard resultsconcerning the extension ofsuch solutions to the

intervalgzt ),z11.lnfact,itcan beshown thatthereexistlinearlyindependent solutionsztlandu,toequation (10.14)such thatany solution to (10.14)can be represented in the form

z'(z)- ctzlztzl+ / t 7'. 'atzl,

(10.15)

wherectand , d are constants.Finally,anotherresultfrom the generaltheory show sthat the solution to the initial-valueproblem depends continuously on

theinitialdata;i.e.,zttz,' tzo,' ? . t/ ( ;liscontinuouswith respecttotheparameters zto and ' tz( ?). W e need m ore than a global existence result for solutions to equation

(10.14):in orderto asserttheexistenceofasolution totheRiccatiequation, we need to show thatthere are solutions ztthatdo notvanish on theinterval

gzt ),zll.Thisproblem leadsustotheimporta. ntconceptofconjugatepoints. Lets CER - (zo).lfthereexistsa nontrivialsolution ztto equation (10.14) thatsatisfiesutzol= utsl= 0,then s iscalled aconjugate pointtozo. Lemm a 10.4.2 Let/ satisj' y tlte conditionso.ffcmmtz10.4.1,and snppose

tltattlterearenoconjugatepointstoztlin (zo,z11.Tlten,tltere ezistsasolw tion ztto equation (10.14)suclttltatutzl, # 0Jor allz CEgzt ),z11. Proof:(Sketch)Giventhat,foranyinitialconditionsutz()l= zo,zl/tz()l= zt/ (), thereexistsa solution to equation (10.14)valid in gzt ),z11,weneed to show thattheabsenceofaconjugatepointtozt lin (zo,z11impliestheexistenceof a solutionztthatdoesnotvanish on g zt),z11. Considera family ofinitialconditions oftheform utzol = 6,zl/tzol = 1,where tîis a smallparameter.For each t îthere is a solution zttz,6) to equation (10.14)valid in gzt ),zll,and ztis a continuous f-unction oft înear t î= 0.Moreover,theinitialcondition zl/tzol= 1 precludesthepossibility of zttz,6)beingatrivialsolution.Now,' tztzt ),0)= 0,andtheabsenceofconjugate pointstoztlin (zo,z11impliesthatzttz,0), # 0forallz CE(zo,zll.Thuseither zttz,0)> 0forallz CE(zo,z11orzttz,0)< 0forallz CE(zo,z11,becauseztis a continuousfunction ofz.Supposethatzttz,0)> 0.We know thatzttz,6) is continuous with respect to t î and hence for t î sufllciently sm allwe have

zttz,6) > 0 exceptperhapsin a smallneighbourhood ofzo.To constructa 3 ln particular, the coefficient of' tz//is not zero.

236

10 The Second Variation

solution thatisnonzero nearztlwe choose t î> 0 so that' tztzt ),6)= t î> 0, and the condition zl/tzol = 1 ensuresthatztis nonzero near zo.A similar argumentcan be used forthe case zttz,0)< 0.Sometechnicaldetailsneed to betightened up,buttheessenceoftheargumentisthatthezeros (hence conjugatepoints)mustchangecontinuouslywith theparameterE.lfweare carefulw ith our choiceofsign for 6,theinitialconditionsim ply thatthezero

atztlwillshif' toutoftheintervalgzt),z11.

En

Theorem 10.4.3 Let/ be aszrzt at a//zJ' unction o. fz,y,and!//,andlety be a

szrztat a//zeztrevtalJortltejhnctionalJ dehned by(10.2)suclttltat/?y/? y/> 0Jor allz CE(z(),z11.f/tlterearenopointsïzztlteinterval(zo,z11conjngatetoztl, tlten tltesecondvariation ïspositivedehnite. Exam ple 10.4.1: Let

zt??(z)= 0' , hence the generalsolution isutzl= ct+ ? (7z,where ctand , d areconstants. Clearly,only the trivialsolution can satisfjrtheconditionsutzol = 0 and utsl= 0forany sCER - (zo).Hence,therearenopointsconjugatetozo.

u//tzl+ utzl= 0. Now,utzl = sintz)is a nontrivialsolution to this equation,and zt(0) = utrl= 0.Hence,r isapointconjugateto0.Wethusseethatthereisapoint conjugateto 0in theinterval(0,t1. Finally,w e note that ifwe consider the second variation as a functional

(ofp)in itsown right,the Jacobiaccessory equation istheEuler-lsagrange equation forthis f -unctional.There is,however,a distinction to bem ade concerning solutions.Specifically,the f-unctionsp thatsolve the Euler-lsagrange equation m ust vanish at the endpoints.ln contrast,we are actively seeking

solutionstotheJacobiaccessory equation thatdonotvanish in (zo,z11.

10.4.3 T he Jacobi N ecessary C ondition

Theorem 10.4.3givesa sufllcientcondition forthesecond variation to bepos-

itive definite.W e show thatthe absence ofconjugatepointsisalso necessary for positive definiteness.Before w e launch into a statem ent of this result, how ever,we establish tw o sm alllem m as.

Lemm a 10.4.4 Let' tzbesolution totlteJacobiaccessory equation (10.14)in gzt ),z11.f/tltereïsapointcCEgzt ),z11suclttltatutcl= 0 andu/tcl= 0,tlten ztvtustbe tlte trivialsolution.

Proof:SupposethereisapointcCEgzt),z11suchthatutcl= 0and u/tcl= 0. Consider the initial-value problem form ed by the Jacobiaccessory equation and these conditionsatz = c.Picard'stheorem im pliesthatthere isa unique solution to this problem in a neighbourhood ofc;hence,this solution m ust be the trivial solution.From the theory of linear differentialequations we

know thatthissolution hasauniqueextension intotheintervalgzt),zll,and consequentlyutzl= 0forallz CEgzo,z11. En Lemm a 10.4.5 Letztbea solution to tlte Jacobiaccessory equation (10.14) in gzt),z11suclttltatzltzol= utzll= 0.Tlten (10.16) P roof: lntegration by partsgives thisresultim m ediately.Specifically,since

ztisa solution to (10.14),

N ow ,

therefore,equation (10.17)impliesequation (10.16).

238

10 The Second Variation

P roof: W e beginw ith aproofofthefirststatem ent.Supposethatthesecond variation is positive definite.W e can quickly elim inate the possibility of a

conjugatepointatzlusing Lem ma 10.4.5.lfthereexistsa nontrivialsolution

totheJacobiaccessoryequation ztsuchthatutzol= utzll= 0,thenwemay take p = zt.Lem m a 10.4.5 shows that there is a nontrivialp CE. bI such that the second variation vanishescontradicting the assum ption that the second

variation is positive definite.W e thus conclude thatzl cannotbe conjugate to ztl.

To show thatthereisno pointconjugateto zt lin (zo,z1)wefollow the proofgiven byGelfand andFomin (4311,p.109).Thestrategy isto construct a familyofpositive-definitefunctionalsK (p. ),thatdependson theparameter p.CE(0,11,such thatff(1)isthesecond variation and ff(0)isfreefrom conjugate pointsto zo.Any solution to the Jacobiaccessory equation associated with K w illthusbe a continuousfunction ofp. .W e exploitthiscontinuity to

show thattheabsence ofa conjugatepointforff(0)impliesthatforK (p.), and inparticularff(1). LetK be thefunctionaldefined by

fftp, l- p, t P J('?,y)+ (1- p,)f'('?),

W e know from Example 10.4.1 that P has no pointsconjugate to zo,and it

isclearthatP ispositivedefinite.SinceJ2J(p,y)isalsopositivedefinitewe seethatK ispositivedefiniteforallp.CE(0,11.TheJacobiaccessory equation associated w ith K is

ddz Lllx%'y'+ (1- p,))z',).- lxB' u- 0. .

(10.18)

Now,any solution zttz,p,)to (10.18)iscontinuouswith respecttop.CE(0,11. lndeed,wecan assertthatzttz,p,)hasacontinuousderivativewith respectto p,forallp,in an open intervalcontaining (0,11,becausepth/y'+ (1- p. )> 0 forallp,CE(0,11,and thiscoefllcient (along with thatforzt)iss11100th with respect to the param eter p..Thus,zthascontinuous partialderivativesw ith respectto 130th z and p..Letztz and ug denotethese derivatives.

Let U denote the family of nontrivial solutions to (10.18) such that ' tztzt ),p,)= 0forallp,CE(0,11.LetX = gzt ),z11x(0,11andR = (zo,z1)x (0,1). Supposethatff(1)hasaconjugatepoints CE(zo,z1).Then thereisaztCEU such thatuts,1)= 0.Now,zthascontinuousderivativesinX,andbyLemma 10.4.4 (applied toK (p. ))wehavethatztz(s,1): # 0.Wecan thusinvokethe

implicitf-unction theorem in aneighbourhood of(s,1)to assertthatthereis a uniquefunction z = z(p,)such thatzt(z(p,),p. )= 0and z(1)= s.ln fact, since ug iscontinuous,w e have thatzg hasa continuousderivative,and ?

Mg

z(p,)- - Nz .

(10.19)

Thefunction (z(p,),p. )thusdescribesparametrically acurveyin someneighbourhood of (s,1) with a continuous tangent that is nowhere horizontal; hence,the intersection of y w ith the line p.= 1 m ust be transverse.Con-

sequently,yrnR , #( )4i.e.,y musthavepointsin theinteriorofX.Although

theimplicitfunction theorem givesonly theexistenceofacurvenear(s,1), itisstraightforw ard to see thaty cannotsim ply term inatein R .Specifically,

supposethaty did terminateatsomepoint@,5)CER.Theconditionsofthe im plicitf -unction theorem are stillsatisfied and w ethusconcludethatthere is

a uniquenontrivialcontinuation ofy inaneighbourhood of(c&,5)contradictingourassumption that(c&,5)isaterminusfory.Weconcludethaty must continueto the boundary ofR .

Relation (10.19)makesitclearthatany continuation ofy cannotinclude a point where the tangent is horizontal,since the conditions ofthe im plicit function theorem are satisfied at any pointon this curve and this theorem guarantees that z? is finite and continuous.W e thus see y cannot ttdouble

back''and intersectthelinep.= 1.Sinceff(0)doesnothaveany conjugate points itis also clear that y cannot intersectthe line p.= 0 except perhaps

atthepoint(zo,0).ln any event,the only possibleboundary curvesthaty mightintersectarethelinesz = zland z = zt l(figure10.1).Supposethat y intersects the line z = z1. Then,there is a p,1 and a zt CE U such that

' tztzt ),p,1)= utzl,p,1)= 0.Butthe argumentsused to proveLemma 10.4.5 can beappliedtofftp, lltoshow thatfftp, ll= 0forthechoicep= ztandthis contradictsthe factthatK (p. )ispositive definite.Hencey cannotintersect the linez = z1.

Considernow thelinez = zo.By construction wehave ' tztzt ),p. )= 0for allp.CE(0,11and hence the function z(p,)= ztlis a solution ofzttz,p.)= 0 forallp.CE(0,11.Theconditionsoftheimplicitf-unction theorem aresatisfied atany pointon thislineand thereforez(p, )= zt listheuniqn,esolution that intersects this line.Evidently,y is distinct from this line because s > zo. H ence,y cannot intersect the line z = zo.W e thus conclude that no such

curvey can existand hencethatff(1)hasno pointsconjugatetoztlin the interval(zo,z1). To prove the second statem ent,note that even ifJ2J k 0,thefunctional

K (p. )isstillpositivedefiniteforallp.CE(0,1).Theproofthatthereisnopoint conjugatetoztlintheinterval(zo,z1)givenaboveisthusvalid.Lemma10.4.5, however,doesnotprecludezlfrom being a conjugatepointsinceff(1)can be zero.

En

240

10 The Second Variation

R

Fig.10.1.

T heorem 10.2.1 and the second partofTheorem 10.4.6 com bine to form a m ore refined necessary condition for a localm inim um known asJacobihs necessary condition.

Theorem 10.4.7 (Jacobi) Lety beasmt ata//zeztrevtalJortlteJunctional

suclttltatJorallzCEgzt ),z11 Jy/y'> 0 alongy.IJyproducesalocalzrzïzzïzrz' tzzrzJorJ tltentlterearenopointsconjngate

tozt lin tlteïzz/crrtzl(zo,z1). N ote that Jacobi's necessary condition does notpreclude the possibility

thatzl is conjugate to zo. Exercises 10.4:

1.DerivetheRiccatiequation (10.11)associatedwiththef -unctionalofFuxam ple 10.3.3.Solve the Riccatiequation directly and show that there are

nosolutionsw defined forallz CE(0,tr 1iff> r.

10.5 A SufficientCondition

2.Let

/(z y y')- y'2- y'y+ y2. Show,using elementary arguments,thatJ2J(p,y) k 0 for a11p (E. bI. D erive the Jacobiaccessory equation and show by solving this equation thatany nontrivialsolution ztcan have atm ostone zero.

3.Supposethat/ doesnotdepend on y explicitly and that/ satisfiesthe

strengthenedLegendreconditionalonganextremal!/(z).Provethatthere are no points conjugate to zo. 4.Theproofofthefirststatem entofTheorem 10.4.6 isconsiderably sim pler

ifwettopen up''thespace. bItoamoregeneraloneX thatincludespiecewises11100th f -unctionsongzt ),z11thatvanishatztlandz1.lfJ2J(p,y)> 0 fora11p CEX,p , # 0,provethatthere are no pointsconjugate to ztlin (z(),z1)(cf.(711,p.91). 10.5 A Sullicient C ondition ln Section 10.2 w ederived the expression

(10.20) which an extremaly for J satisfiesfor any tneighbouring curve'':' è= y+ 6p.

ThenecessityoftheconditionJ2J(p,y)k 0foralocalminimum isclear,but thesufllciency ofthiscondition issuspect.lndeed,weknow from ourbrieftour of finite-dim ensionaloptim ization that sem idefinite quadratic form s do not necessarily lead to localextrem a.T heproblem isthatifthereisa nontrivialp

such thatJ2J(p,y)= 0,then the0(63)termsintheaboveexpansion control the sign ofJI()- J(y).Certainly,wecan avoid thisproblem by requiring thatJ2J(p,y)be positive definite,but even thisstrengthened requirement hassnagsbecausetheremay be nontrivialp such thatJ2J(p,y)> 0 butof order6,in which case the sign ofJ(L)- J(y)dependson thehigherorder term sfort îsm all.H arsherrestrictionsareneeded to controlthem agnitudeof

J2J(p,y)relativetotheremainderterms. Let11.111bethenorm on thespace(72(gzt),zlj)defined by 11:111- sup I: v(z)I+ sup I?/'(z)I. z(E( zo,z1j

z(Egze,z1j

W esaythatafunctionalJ :(72(gzt ),zlj)--+R hasaweak localminimum atyCES ifthereisa.:1> 0such thatJ(L)- J(y)k 0forall: ' èCES such that 115- : v111< .A.Similarly,J issaidtohaveaweak localm aximum atyCES if-J has a weak localminimum at y.The adjective ttweak''creeps into the definition to distinguish such extrem a from strong extrem a,the definition

ofwhich isidenticaltotheweak extremaexceptthatthenorm 11.10,defined by

10 The Second Variation

11: v110- sup I: v(z)I, zeg ze,z1j

isused.Clearly,115- : v111< .. : 1impliesthat11f/- : v10< .. A,buttheconverseis nottrue.Hence,ttweak''signifiesthatwe arerestricting the com petition to a subset ofthe setoffunctionsthatare candidates fora strong extremum . W eneed to getan expression forthe rem ainderterm in theTaylorexpan-

sion (10.20)in orderto develop a sufllcientcondition.Now,itcan beshown thatthereisafunction?rztplsuch that

l(b)-J' (: 4 / )=62(1( i2. J(r/?y)+j20 *3(zr/2+pr/2)dz(? (10. 21) and v,p--+0as IpI I1--+0. T he m ain resultofthis section isthe follow ing theorem ,which appliesto the basicfixed endpointproblem .

Theorem 10.5.1 Lety CES bean cz/rcmtzlJortlteJunctional

and supposet/lcg along t/zïs eztrevtalt/Jc strengtltened Legendre condition

Jy/y'(al,#lal),!//(al))> 0 issatishedJorallz CEgzt ),z11.Suppose/' urà/zcrtltattlterearenopointsconjw gatetoztlin (zo,z11.Tlten J Itasaweaklocalmïzzïm' tzm in S aty. P roof: Letp.CER beasm allparam eterand considerthe fam ily offunctionals K defined by

4Taylor'stheorenlhasan extension to operatorson generalBanach spaces(cf. E221). 5seeg31j,pp.101-102.

10.5 A SufficientCondition

The Jacobiaccessory equation forK is

ddz((%'y'-/z,)z',)-B' u-0. .

(10.22)

The smoothness conditions on / and the strengthened Legendre condition

implythatthereisapositivenumbera.such thatJy/y'k t zforallz CEgzt ),z11. Thus,forall/ . :2< a, we have

/://://- / .:2> 0,

(10.23)

fora11z CEgzt ),z11,and thisin turn meansthatthesolutionszttz,p. )to equation (10.22)arecontinuousf-unctionsof1 30thzandp,provided Ip,lissmall.By hypothesiswehavethattherearenopointsconjugatetozt lin (zo,z11forthe casep.= 0,and the continuity ofztw ith respectto p.im pliesthatthereare no

pointsconjugatetoztlin (zo,z11forallIp,lsufllcientlysmall.Therefore,there isa p,1> 0 such thatforallp. ,with Ip,l< p,l,thefunctionalK (p. )satisfies thestrengthened Legendrecondition (10.23)and hasnopointsconjugateto zt lin (zo,z11.Theorem 10.4.3thereforeimpliesthatK (p. )ispositivedefinite forallIp,l< p,l.Wethusconcludethatthereisanumberp > 0such that J

J2J('?,y)> p

21

n/2dz.

20

W e now considerthe rem ainderterm

N ow ,

(10.24)

10 The Second Variation

Thus,

ztr/2(jI !V (z1- z( ;)2 zlr//2(jI 2

20

,

20

and hence

I

(z,- zo)2

p

vI(1+ z j
consequently,

Thus,JIL)- J(y)> 0 forallnontrivialp,provided IIpII1sufllciently small, and thereforey isa weak localm inim um forJ.

En

Exercises 10.5:

1.Thesecond variation J2J(p,y)iscalled positive and nondegenerate (orstrongly positive)ifthereisaconstant. 4 > 0such that (1 2J(p,y)k ., 4IIpII 2 1.

10.6 M ore on Conjugate Points The Jacobinecessary condition and the sufllcient condition of Section 10.5

130th requireverification thattherearenopointsconjugatetozt lintheinterval

(zo,z1).lnthissectionwediscussasimplemethodforfindingconjugatepoints and a geom etricalinterpretation ofthesepoints.

10.6.1 Finding Conjugate Points Supposethatyisan extrem alforthefunctionalJ.Recallthatapoints CER -

(z())isconjugatetozt lifthereisanontrivialsolutionzttotheJacobiaccessory equation (10.14)such thatzltz()l= utsl = 0.ln orderto testwhether an extremalhasa conjugatepointin theinterval(zo,z11wearethusobliged to somehow procureageneralsolution zttoequation (10.14)andcheckwhether there isa zero ofztin the interval(zo,z11.Although the Jacobiaccessory equation is linear,finding a generalsolution to such equations can prove a

formidabletask.ltisthusarelieftodiscoverthatsolutionstoequation (10.14) can be derived from the generalsolution to the Euler-lsagrangeequation. Supposethaty is a generalsolution to the Euler-lsagrangeequation

t ' ?/ t' ?/ = Xd t ' A / ç ' )y 0, -

(10.25)

associated w ith the functional

Thegeneralsolution to a second-orderordinary differentialequation contains

two parametersc1,ca (constantsofintegration)and itcan be shown thaty dependssm oothly on theseparam eters.Sincey dependson cland c2,so does

/ in theEuler-lsagrangeequation,and the smoothnessof/ with respectto y and y?impliesthat/ alsodependssmoothly on cland c2.Differentiating

equation (10.25)withrespecttoc1,notingthatthesmoothnessassumptions on / allow theordersofdifferentiation tobechanged,gives

d t' / :t'?c,(l k' ) ' ;-ô?y ?: t ' ?: /. q ) 7-' t ) d ' 7L ( vôt ??l ' y ( kt ç ' ?y ) /: ç ' ) c y,-' , -ô t ' ? y /?: ç ' lc y, ?( j( j ( ' ? (().f( ??/ (' ?/ ôy? 3 ; l . o y : c , + o y ? . % , ) d // :2/ ç ' )y :2/ ôy? ( ' ?/ -' z;'b:2oy o y ? : c , + o v o y ? : c , + o v ) / ç ' )y :2/ ôy? ( ' ?/ (oyoy:c,+oyoy?:c,+oy) .

-

-

Letztl = ç ' )y t gcl.T hen,

246

10 The Second Variation

ln this m anner w e see that ztl m ust be a solution to the Jacobiaccessory

equation (10.14).A similarargumentshowsthatu,= ç' )y t ' ?czisalsoasolution to equation (10.14). Wrc can t/z' tzs obtain solutions to tlte Jacobi accessory equation by sïzrzy' l?/dtjferentiating tlte generalsolution to tlte Enler-Lagrange

equation ' tl ï//zrespecttocl and cz.ln fact,itcan be shown (4151,pp.68-72) thatztl and u, form a basis forthe solution space.

Letc = (c1,c2),k = (/c1,k,),and suppose thatylz,k)is a solution to equation (10.25)thatsatisfiestheboundary conditionsoftheflxed endpoint problem .Let

ôy

dp

' t zllt r,k)= t'?cl c-k, ? .12(al,k)= t'?cz c=k. Then,thegeneralsolution zttz,k)toequation (10.14)isgiven by zttz,k)= ctztltz,k)+ , 3u,(z,k), wherectand , d are constants.W e are interested in nontrivialsolutions zt,so

thatctand , d are not 1 30th zero.lfs isconjugate to zo,then therearevalues ofctand , d such that

z'tzo,k)= ctzlltzo,k)+ , dzlztzo,k)= 0, and hence,

z'2(s,klzlltzo,k)= uatzo,klzllts,k).

(10.26)

N otethatztland u,cannot130th vanish atthesam evalueofz,because this

would imply that the Wronskian W'(z) = ultzlu/ z(z)- u/ ltzluz(z) = 0 for allvaluesofz and hencethat ztland u,would be linearly dependent.6 Thus,

relation (10.26)isan equation fora conjugate point s.lf' t zltzt ),k), # 0 and ults,k), # 0,theaboverelation isusually written intheform uatzo,k) :.12(s,k) z'ltzo,k) zllts,k) 6Seeg9j,pp.42-45formoredetails.

7(tr,Cl,C2)= CICOSIV tosin1. The above argum entsshow that

aresolutionstotheJacobiaccessory equation.(Thiscan beverified directly, sincetheEuler-lsagrangeequation is equivalent to theJacobiaccessory equa-

tion forthisintegrand.)Hence,any conjugatepointsto0 mustsatisfy ua(s)uz(0)- u,(s)ua(0);

The points conjugate to 0 are therefore of the form s = ul uzzr,where n,=

1,2,....ln Example 10.3.3 we chose f > r and hence the interval (0,f) included the pointr,which isconjugate to 0. Exam ple 10.6.2: G eodesics in the P lane

Consider the arclength functionalof Example 2.2.1,where /(z,y,!/?) 1+ y?2.The generalsolution ofthe Euler-lsagrange equation is

#ltr,c1,c2)= clt r+ c2. The corresponding solutionsto the Jacobiaccessory equation are ztl= z and

' ? . tz = 1.Any points thatisconjugate to zt lmustsatisfy equation (10.26). The only solution to thisequation,how ever,is s = zo,and therefore there

arenopointsconjugateto zo.Since S/://> 0,Theorem 10.5.1impliesJ has a weak localminimum aty.ln fact,wecandobetterthan this(cf.Fuxample 10.7.1). Exam ple 10.6.3: C atenary

Considerthecatenary problem ofSection 1.2,where/(z,y,!/?)= y 1+ y?2 and z CE (0,11.We showed in Example2.3.3 thata generalsolution to the Euler-lsagrangeequation isofthe form

where cl and cz are constants.Solutions ztl and u, to the Jacobiaccessory equation forthisproblem are thus

248

10 The Second Variation

Considernow the extrem alsassociated w ith theboundary values

v(0)- 1, v(1)= 1. ln Exam ple2.6.1we studied thegeneralproblem ofdeterm iningthecland cz to satisfjrsuch boundary conditions.ln thatexam ple we argued thatfor any

value of3/(1)> 0.6thereareprecisely two solutionsfortheintegration constants.Forthisexam ple,we thusknow thatthere aretwo sets ofparam eters

thatsatisfytheboundary conditions.Equation (2.42)implies coshcz= coshtcoshcz+ c2); ulucz = cosh cz+ c2. Since coshcz > 0 forallczCER ,we m usthave

and cosh cz = - 2c2.

ztl= coshtcztl- 2z))- cztl- zzlsinhtcz(1- 2z)), ztz= - sinhtcztl- 2z)), whereczisasolutiontoequation (10.27).Letï= c2(1-2z).Equation (10.26) showsthat a conjugate pointto 0 m ustsatisfjrtherelation

(70thcz- cz= (70thï- ï. (10.28) Now,equation (10.27)hastwosolutionsrls: s-0.6and r2s: s-2.1,andfor anyfixed czequation (10.28)alsohasprecisely twosolutions,oneofwhich is simplyï= c2;i.e.,z = 0 (seefigure 10.2).Forthechoicec2= r1,thesecond solutionto (10.28)correspondstoz s: s2.4( (0,11.ThestrengthenedLegendre

1

1

z.

250

10 The Second Variation

II=cII- (.. : lc,)2+ (zlc,)2 is sm all.Since y is s1100th with respectto the ck,Taylor's theorem im plies that

and hence,foranyz CEgzt),z11,

Theaboverelation showsthatatanyz CEgzt),z11thedistancebetweenneighbouringextremalsisoforder I l=cllasIlzlc11--+0,and theleading orderterm is the m odulus ofa solution to the Jacobiaccessory equation.Suppose now

that there is a point s conjugate to zo.Then there exists a nontrivialsolu-

tion zt= ctztl+ , 3u, to equation (10.14)such thatzltz()l= utzll= 0,and hencetherearelcl,zlczsuch thatIl=cll# 0 and zlclztltzl,)+ zlcaztatzl,)t: îclzlltsl+ x t: îczzlztsl= 0.Relation (10.29)thusindicatesthatatzo,and any x

conjugatepointsto zo,thedistance between the neighbouring extrem alsisof

order II=cII2 asIl=cl l--+0,sincewecan alwaysscaleourchoiceofzlcla. nd zlcz.Roughlyspeaking,equation (10.29)indicatesthattheneighbouringextremals ttnearly intersect''atconjugate points.Atany rate,conjugatepoints bearthe distinctivehallm ark ofan envelope forthe fam ily ofextrem als.

Let/z(z,y,c)= 0 describea one-parameterfamily.F ofcurvesin thezy-

plane param etrized by c.A curve v iscalled an envelope ofthe fam ily . F if:

(a) ateach point(;,L)CEv thereisa y CE.F thatistangenttozz;and (b) thereareinfinitely manycurvesin . F tangentto each arcofv. Supposethat.F hasan envelopev.Then ateach point(:,L)CEv thereisa c= ctJ @,L)such thatthecurvedescribed by/z(z,y,ctJ @,())= 0 istangentto v at(:,L).Forsimplicity,weassumeherethat/Jisas11100thfunction ofz,y, and c,and that the arc under consideration issuch that c can be regarded

asas11100th f-unction oft@?.(Thelatterassumption istrue,forinstance,ifthe arc ofv can bedescribed parametrically in theform (;,?)(:))fork in some intervalf,where:' èisdifferentiable with respectto t @?,and L V(k), # 0 forall k CEf.)On theenvelopev,thef-unction/Jmustthereforesatisfy ltlk,: 9,c(:))- 0. (10.30)

ddkhlk,5,c(:))= :& :: + :& djy,(:)+ :& dc c,(:).

(10.31)

ç ' )k + ( %? )(:)= 0.

(10.32)

Equations(10.31)and (10.32)thusgivethecondition

tgc

=

0.

(10.33)

Equations(10.30)and (10.33)canberegarded asapairofimplicitequations involvingthethreevariablest @?,L,and c.Undertheassumption that ( -?5 / 0,

wecan invoketheimplicitfunction theorem tosolveequation (10.30)forL, and regard equation (10.33)asan implicitequation forc asa f-unction ofk (orviceversa).Once ciseliminated,equation (10.30)can then be used to determineL (tasafunction ofk and hencethecurvev. Exam ple 10.6.4:

/z(z,y,c)= y- (z+ c)3= 0.

(10.34)

Thefam ily ofcurvesdescribed by /Jconsistsofthecubiccurvey = z3shifted parallelto thez-axis by thevaluec.Now , t' ?/à=

t ' ?c

-

z

3(z+ c) ,

so thatequation (10.33)givesc = -z.Equation (10.34)thus implies that ylz)= 0,so thatifthereisan envelopev,itmustbethez-axis.ln thiscase the z-axisisan envelope forthe fam ily ofcurves. Exam ple 10.6.5:

/z(z,y,c)= y2+ (z+ c)2- 1= 0.

(10.35)

Thefamily. F correspondstocirclescentred at(-c,0)ofradius1.Here,

10 The Second Variation

dc

=

2(z+ c),

Notethatsatisfaction ofequations(10.30)and (10.33)isa necessary but

notsufllcientcondition foran envelope.A sim plecounterexam ple is given by

thefamily oflinesdefined by /z(z,y,c)= y- cz = 0.Equations (10.30)and (10.33)aresatisfied onlyifz = y = 0.Thefamily hasa ttfocus''at(0,0),but thissingularity isnotan envelope.

Returning to the calculus ofvariations,ifs is conjugate to zt l and zt= ctztl+ , 3u, is a nontrivialsolution to the Jacobiaccessory equation,then ct and , d cannot 1 30th be zero,and it is clear that we can choose zlcz such

thatzlcz= Acï, 3 ct(orzlclsuch thatzlcl= zlczct/7)and geta nontrivial solution flto equation (10.14)thatvanishesatztland s.ln otherwords,for the purposes of studying conjugate points we can let cz = cl/7 ctand thus

regard ylz,c1,c2)asaone-parameterfamily ylz,c).Now, ç' )y cl &= d = - .tt?

.

c

a

so thatatany conjugatepoint tçwe have --

X - 04

dc

consequently,the necessary condition foran envelope is satisfied atconjugate points. Envelopes abound in nature,and m any arise through variationalprinciples.For exam ple,causticsare form ed w hen lightraysform an envelope.The extrem als are the solutions to the Euler-lsagrange equations that arise from Ferm at'sPrinciple.A convenientexam pleisthebrightcurveform ed in a partially fulltea cup on a sunny day as a result ofthe sun's rays reflecting on the inside ofthecup.'/ Exam ple 10.6.6: Parabola of Safety A nother prom inent exam ple of an envelope is the so-called ttparabola of

safety'' familiar to artillery gunners (and combat pilots).The path ofthe projectile is governed by Ham ilton's Principle.Suppose that the cannon is fixedattheorigin,butthatitmay beelevated atany angle4,0< 4 < r 2. Theresulting trajectoriesforprojectilesleaving thecannon aretheparabolas given by % True, these extremalsare certainly notsr1100th,but ifneeded,we can restrictour attention to the family oflightraysafterthereflection.The curvethatform sthe caustic iscalled a nephroid.

(10.36)

so that

(10.37) Equations(10.36)and (10.37)implythat 2

2

ylz)= 2g - 2z?(a ;.

(10.38)

Equation (10.38)definestheparabolaofsafety.Eachextremalinthefamily definedbyrelation (10.36)liesbelow thisparabolaexceptatonepointwhere the extrem aland the parabola intersect and have a com m on tangent.The ttfiring zone''isthe space between the parabola ofsafety and the z-axis.The

projectiles neverexit thiszone,so a pilotcan fly safely above the parabola.

Anintroduction toenvelopesandapplicationsisgivenby Boltyanskii(131. A morerigorousandadvanced (butstillquiteaccessible)accountofenvelopes and othersingularitiesisgiven by Bruceand Giblin (181. Conjugate pointsneed notalwaysyield envelopesforafam ily ofextrem als. The fam ily ofgeodesics on a sphere through the N orth Pole,for instance,

definesa familyofextremals(linesoflongitude)thatintersectata common point (theSouth Pole),which is a conjugate point thatiscertainly notan envelope.Thereis,in fact,an opticaldevicethatm im icsgeodesicson asphere. The lensis called the M mxw ellfisheye.The refractiveindex forthis lensis

wherer denotesthedistancefrom a flxed point,and a and ro areconstants.

Born and W olf(141,pp.147-149and Luneberg (501pp.197-214discussthis rem arkable lens and certain generalizations.

10 The Second Variation

10.6.3 Saddle Points*

Theorem 10.4.7 show s that ifan extrem alproduces a localm inim um for a

functionalJ,then it cannot have conjugate points in the interval(zo,z1). Theorem 10.5.1 shows that extrema. ls that do not have conjugate points in

theinterval(zo,z11produceweaklocalminimaforJ.Geodesicson thesphere from the North Pole to the South Pole show that if zl is conjugate to zt l

then theextremal(albeitnotuniquely determined)maystillproducealocal m inim um .These results can be easily adapted to the case oflocalm axim a of a f -unctionalJ by sim ply applying the results to the functionalK = - J instead ofJ.W ecan thusconclude thatan extrem aly correspondsto neither

a localm inimum nor a localmaxim um if there is a point conjugate to zt l

in theinterval(zo,z1).Thequestion ariseswhetheritispossibleto classify

extrem alswith conjugatepointsin a manneranalogousto that used in finite dim ensionsforsaddle points.A lthough such aclassification m ay be oflim ited interestin m anyphysicalapplications,itturnsoutthatitiscertainlya fruitful line ofenquiry in topology and differentialgeom etry.

The classification ofextrem alswith conjugate pointsisthestarting point fora broad subjectcalled ttr l'heCalculusofVariationsin theLarge''pioneered by M .M orse.ltiswellbeyond thescope ofthisbook to give even a rudim en-

tary accountofthistopic,butwedogiveafew comments(noproofs),which we hope w illwhet the appetite ofthe reader to look at a serious study of

thissubject.A standardreferenceisthebook by Morse(551.Milnor(531and Spivak (651alsogiveaccountsofthetheory asitappliestogeodesics.Aswith thepreviousm ateria. l,wefocusexclusively on candidatesforlocalm inim aand

thestrengthened Legendrecondition (10.9)isassumed tobesatisfied. T he key to obtaining a classification of extrem a. ls lies in extending the M orse index to infinite-dim ensionalspaces.ln finite-dim ensional spaces,the M orseindex countsthenum berofm inussignsin thecanonicalrepresentation

of/nearxtl(Lemma10.1.1).Onaslightlydeeperlevel,theMorseindexcorrespondsto them axim um dim ension ofa subspaceofR?zw herein theHessian m atrix is negative definite. T his idea can be transferred to infinite dim ensionalspaces.lfthe function space is a H ilbertspace,then a decom position ofquadratic functionalssuch asJ2J analogousto thatgiven in Lem m a 10.1.1

ispossible (cf.(221,pp.571-572).Herewetakethedirectapproach.Lety be

an extrem alforthe f -unctionalJ and letJ2J :bIx .rf --+R bethecorresponding second variation.The M orse index , L ofy isdefined to bethe m axim al dim ension ofthe subspace of. bI on which J2J is negative definite. T heproblem w ith theabovedefinition isthatitisnotclearhow onem ight determ ine à,orforthatm atter,if, L iseven finite.ltturnsoutthatthereisa tractableway to calculate , L thanksto the M orse index theorem .Thegeneral statem entofthisresult concerns functiona. ls that involve severaldependent .

variables,and the notion ofmultiplicity for conjugate points isneeded.The m ultiplicity ofa point s conjugate to ztlis defined to be the number of linearly independentsolutionsu to the Jacobiaccessory equation thatsatisfy

utzt))= u(s)= 0.Forourcase,Sthegeneralsolution totheJacobiaccessory equation (10.14)isoftheform zt= ctztl+ , dzlz,whereztland u,arelinearly independentsolutionsto (10.14)and ct,, d areconstants.ltisthusclearthat them ultiplicity cannotexceed two.Theproblem offindingnontrivialsolutions to thisboundary-valueproblem isa thinly disguised Sturm -lsiouvilleproblem , and weknow thatallthe eigenvaluesassociatedw ith such problem saresim ple

(see Section 5.1).ln short,the multiplicity ofconjugate pointsis one,for functiona. lsofthetype considered here. A generalstatem ent of the M orse index theorem along w ith a proof is

given in Milnor(531.Here,wegivea simple tno frills''version forextremals to functionalsofthe form

Theorem 10.6.1 (M orse lndex Theorem ) Lety be an eztrevtalJor J. Tlte indez ,L ofJ2J is equalto tlte zz' tzmùcr ofpoints i n (zo,z1) conjugateto .

.

zt l.Thisindezïsalwaysjinite. T he above resultallow s usto classify extrem alsin a spiritsim ilarto that used to classify critical points in finite-dim ensional spaces.For instance,if

L = 0 for an extremal,and zl is not conjugate to zo,then Theorem 10.5.1

,

indicatesthatJ hasaweaklocalm inim um aty.ForthefunctionalofFuxam ple

10.3.3,theindex ,Lisatleast1sincer CE(0,f);if,say f= 7r 2,then , L= 3, since the conjugate pointsr,2r,and 3r are allin (0,f).Forthisexample the coefllcientsofthe Jacobiaccessory equation do notdepend on y,so that allextrem als with the sam e endpoints have the sam e index.G eodesics on the sphere can have an index of0 or 1 depending on whether they contain antipodalpoints.Sim ilarly,extrem alsforthecatenary can haveaM orse index of 0 or 1 depending on the choice ofsolutions for the integration constants

(Example10.6.3). Exercises 10.6:

1.D erive the Jacobiaccessory equation forthecatenary and verify directly thatthef-unctionsztland u,in Exam ple10.6.3are solutionsto thisequation.

2.ln Fuxample 10.6.3 suppose that the boundary valuesare 3/(0)= 1 and 3/(1)= coshtl).Find a solution (c1,c2)such thatthecorresponding extrem alproduces a weak localm inim um .

8 TheJacobiaccessoryequation isa vectordifferentialequation when thefunctional involves severaldependentvariables.Forour case the Jacobiaccessory equation isscalar.

10 The Second Variation

3.Let

x/4

J(y)=

(y2-y?2-2ycoshz)dz.

o

Find the extrem als for J and show that for the flxed endpoint problem these extrem alsproduce weak localm axim a. 4.Let 21

J(t v)-

20

y'(1+z2!//)dz,

where 0 < ztl< z1.Find the extrem als for J and the general solution

to the Jacobiaccessory equation.Find any conjugate points to ztland determ ine the nature ofthe extrem alsforthe flxed endpointproblem . 5.Let J be the functionalofExercises10.3-2.

(a) Derive a two-parameterfamily offunctionsylz,c1,c2) thatare extrem alsforJ.

(b) Find thegeneralsolution totheJacobiaccessoryequation and show thattherearenopointsconjugatetozt lforanychoiceofzt land!/(zo). D eterm ine the nature ofthe extrem alsfor thisproblem .

6.LetJ bethef-unctionalofFuxercises10.3-1(geodesicson thesphere).The extrem als for J satisfjrthe im plicitequation

tan0cost/+ c2)= tanc1, where cl and cz are constants.Find the generalsolution to the Jacobi

accessoryequation.lf-4isthepointwith sphericalcoordinates(R,)o,%) show thatthepointsconjugateto-4 havecoordinates(R,40,% ulur). 7.Let

h(1,#,c)= 3/5- (t r+ c)3. Find thecurvealongwith thepoints(z,y)thatsatisfy/z(z,y,c)= 0and equation (10.33).Doesthiscurveform an envelope?

8.Let c and é be constants such that c > f > 0,and consider the oneparam eterfam ily ofcircles given by

/z(z,y,ct)= a2 1-

:2

- 2ctz+ (z2+ y2+ é2)= ().

Solvetheequations/z(z,y,ct)= 0 and (10.33),and show thatthisfamily ofcurvesform s an envelope corresponding to thehyperbola

I2

F2

= 1. :2 :2 Thisfam ily ofcirclesarisesin thestudy ofthesound m ade bya supersonic aircraft.ln the m odel,f istheheightofthe aircraftand c= fv zt,w here

c2 -

-

- .

z?isthespeed ofthe aircraftand ztisthe speed ofsound in air (hence c> fforsupersonicaircraft).Therightbranch ofthehyperbola encloses a region known asthezone ofaudibility.See (131formoredetailsand otherapplications.

10.7 Convex lntegrands

10.7 C onvex lntegrands ln thissection we presenta sufllcientcondition fora m inimum thatdoes not

involve conjugatepoints.The condition exploitsthe casewherethe integra. nd isaconvex f-unction ofy andy?,andusesa basicresultaboutconvex functions to establish therequisiteinequalities.Therequirem entofconvexity,how ever, is harsh:m any functionals such as that for the catenary do not have convex integrands.N onetheless,the testforconvexity isstraightforward and the resultissim ple to use.

Recallthat a set J2 f;l1:. 2 is convex ifthe line segmentsconnecting any two pointsz1,z2 CEJ2lie in . Q.ln otherwords,ifz1,zzCEJ2 then

w(f)= (1- flzl+ tz,CEJ2 foralltCE(0,11.ThesetsR2,((: (/,. ? z;)(E1:.2:y2+ 07 2< ljand ttt/?. ? z;)(ER. 2: 1p1< 1 and Izt ?l< 1).areexamplesofconvex sets. LetJ2f;l1:. 2beaconvexset.A function / :J2--+R issaidtobeconvex if

(10.39) forallz1,zzCEJ2and alltCE(0,11.Geometrically,inequality (10.39)implies thatthe set M = ((z,z) CEJ2 x R :z k /(z)) is a convex set.Roughly speaking,M isthesetofpointsthat ttliesabove''thegraphof/. ln finite-dim ensionaloptim ization,convexity is a desirable property because one can proceed directly to the classification ofa criticalpointwithout resorting to the Hessian m atrix.M oreim portantly,the m inim um thus found

isglobalinthesensethatitisa minimum of/ fora11zCE. Q.Thecrucialinequality thatleadsto thisresultcom esdirectly from the m ean valuetheorem .

LetJ2f;l1:. 2bea convex setand let/ :J2--+R beaconvex function that hascontinuouspartialderivatives on . Q.Letz1,zzCE. Q.Then

/(w(f))- /(z,)+ ftza- z,).V/(w (r)). Equation (10.40)andinequality (10.39)imply

(10.40)

for allt CE (0,1).Now,zl and z2 are fixed points in . 62,but r depends on t and 0 < r < f.Since the partial derivatives of / are continuous,

limt--soV/(w (r))= V/(z1);hence,foranyz1,zzCE. 62,

10 The Second Variation

(za- z,).V/(z,)s;/(za)- /(z,). (10.41) Suppose now thatzl is a stationary point for/ so thatV/(z1) = 0.The above inequality show s that

/(z1):é /(z2) forallzz CE.Q.ln this m annerw e see thatstationary pointsforconvex func-

tionslead toaminimum for/in .Q. W e can exploit the above result to develop a sufllcient condition for a functionalto have a m inim um atan extrem al.Let 21

J(t v)-

/(z,y,t v')dz, 20

Jzz- ((: v,y')e 1:. 2:(z,y,? /?)CEDgj.

21(b' y'l. %'(z,y,y')dz - - 2021d(j (.%'(z,y,y'))(5- y)dz, z -

20

and hence --

J(5)-J(t v)kzo(, î-,)(?,(z,? ,,?, ?)-c dI(?? , , -(z,?,,? ,?)))dz. N ow ,y satisfies the Euler-lsagrange equation,so that the integrand in the above inequality iszero;therefore,

J(5)- J(: v)k 0, and consequently J hasa m inim um at y.ln sum m ary w e have the follow ing result.

10.7 Convex lntegrands

Theorem 10.7.1 Snppose tltatJor eacltz CEgzt),z11tlte setJzz is convez, andtltat/ isaconvezjhnction o. ftltevariables(y,!/?)CEJzz.f/y ïsasmt ata//z eztrevtalJorJ,tlten J Itasamïzzïm' tzm atyJortltehzed endpointproblem, . ln order to apply the above theorem we need a m ethod for discerning whether a given f -unction oftwo variables is convex.Fortunately,there is a

tractablecharacterizationwhen / isas1100th function.W eomittheproofof the nextresult.g

Theorem 10.7.2 LetJ2f;l1R. 2beaconvezsetandlet/:J2--+R beaJ' unction ttl ' ï//zcontinuon, shrst-and second-orderpartialderivatives.TlteJunction / is

convezz/andonlyzl //t?reaclt(y,' tt;)CEfî.

T he finalinequality in the above theorem is sim ply the requirem ent that

thequadraticform Q introducedinSection 10.1bepositivesemidefinite.Geom etrically,thisinequality ensuresthatthe Gaussian curvatureisnonnegative

andhenceeach pointon thesurfacedescribedby(y,' tt;,Jly,zc)),(y,' tt;)CEJ2is eitherellipticorparabolic.Elliptic and parabolicpointsare characterized by

theproperty thatthetangentplaneat(y,w,Jly,zc))doesnotintersectthe surface in a neighbourhood of(y,' t;,Jly,zc)).The otherinequalitiesensure thatthesurface always ttliesabove''the tangentplane.A paradigm fora con-

vex function istheparaboloiddescribedby Jly,' tt;)= y2+ w2for(y,. tt;)(ER2. A convexity condition for functions ofthree or m ore independent variables

can bederivedusing theHessian matrix ofSection 10.1.Forexample,if/ is a s1100th function ontheconvex setJ2f;lR?zand theHessian matrix for/ is

positivedefinite,then/isaconvex f -unction(cf.Theorem 10.1.2forconditions on theHessian matrix). Exam ple 10.7.1: G eodesics in the P lane Let J be the arclength functional

J(t v)-J 1+y F 20

hence,/ isconvex.Theorem 10.7.1thusimpliesthat(amongsm00th curves) line segm entsare the curvesofshortestarclength between tw o pointsin the plane.

9A proofcanbefound in g15j,pp.41-43.

260

10 The Second Variation

Exam ple 10.7.2: C atenary

The integrand ofthe catenary problem (Fuxample 2.3.3) is /(z,y,!/?) y 1+ y?.Now,hy= 0,but

so that.:1 < 0 forally?, # 0.The integrand isthus notconvex.We could have deduced thisfrom Example 10.6.3,because there are conjugate points forcertain solutions ofthe boundary value equations.

Exam ple 10.7.3: Consideran integrand oftheform

/(z,y,y')= (c1!/- y'- c2)2 wherecland czareconstants.(TheRamseygrowthmodelofExercises7.1-2 hasan integra. nd ofthisform.)HereJzz= R2,yy= 2c2 1> 0,/? y/? y/= 2 > 0, and /yy/= -2c1> 0.Hence,. :1 = 0,and theintegra. nd isconvex.Extremals to the flxed endpointproblem thuscorrespond to m inim a. Exercises 10.7:

1.LetJ2bea convex setand supposethat/andg areconvex functionson .Q.S how thatthefunction /+ g isalsoconvex.

2.Determinewhethertheintegrandforthebrachystochronef -unctional(Fuxample2.3.4)isconvex. 3.Show thatthefunctions y2+ y?2and c!/ 1+ y?2 are convex. 4.lstheintegrand convex forthe f-unctionalofExercises 10.3-2? 5.D evelop a resultanalogousto Theorem 10.7.1 for f -unctionalsofthe form

and apply itto theflxed endpointproblem forthebeam ofExam ple7.1.3.

A nalysis and D ifferential E quations

ln thisappendix w ereview som eelem entary analyticalconceptsthatare used frequently in the book.The review isintended to be sim ply a briefsum m ary ofa few key resultsfrom analysis and differentialequations thatare relevant to m aterialpresented in thetext.ltis notintended asa ttquick introduction'' to these topics:it ism erely a budgetofhandy results collected for the convenienceofthe reader.Thefirsttwo sectionsconcern Taylorpolynom ialsand theim plicitfunction theorem .A fullaccountofthese topicsresplendentw ith

proofscan befound in any book on realanalysisoradvanced calculus(e.g., (191,(561,(291).Thethird section dealswith thetheory ofordinary differentialequations.Here,onecanconsultBirkhoffand Rota (91,Coddington and Levinson (241,orPetrovski(601fordetailed presentations. A .1 Taylor's T heorem A good dealofthe m athem atics in thisbook relieson an exceedingly useful result known as Taylor's theorem .W e com m only encounter transcendental functionssuch as c2 oralgebraic functions such as 1+ z2,that need to be approxim ated neara given pointin term sofa polynom ia. l.Taylor's theorem provides the analytical fram ework to do such approxim ations.Let us first warm up with the m ean valuetheorem .

Theorem A.1.1 (M ean Value Theorem ) Letzt landzl berealzz' tzmôcrs suclttltatztl< z1.Let/ beaJunction continuon,sïzzgzt ),z11anddt ferentiable ïzz(zo,z1).Tlten tltereïsazz' tzmscrï suclttltatztl< ï< zl and T he m ean value theorem iseasy to explain geom etrically.Theslopeofthe

linesegmentthatconnectsthepoints(zo,/(zo))and (z1,/(z1))is /(z1)- /(zo). W = I 1 - I ()

262

A Analysis and DifferentialEquations

Themean valuetheorem assertsthatsomewherein the interval(zo,z1)the graph ofthefunction / hasatangentparalleltothelinesegment;i.e.,?rz=

//(ï)(figureA.1).ltisclearthattheremay beseveralvaluesofïCE(zo,z1)

Fig.Jt.1.

forwhichequation (A.1)isvalid.Thettcatch''isthatthemeanvaluetheorem doesnotgiveusanyvalueforï.Weknow only thatitisintheopeninterval

(zo,z1),sothatalltheuncertaintyoftherepresentation liesin thederivative term.Wecannonethelessusethisresulttoapproximate/ nearzt lwithsome controlovertheerrorthrough the derivative. N otethatthem ean valuetheorem can easily be interpreted to be arepre-

sentation of/ atzt lintermsof/(z1)and thederivativeterm.lnotherwords, therelation issym m etric and itdoesnotm atterwhetherzt l< zlorztl> z1.

Thepointisthatthereisanumberïbetweenthesenumberssuch that(A.1) issatisfied provided / satisfiesthecontinuity and differentiability conditions in therelevantinterval.

Themeanvaluetheorem can beextended toproviderepresentationsof/ in term sofanonlinearpolynom ial.Thisextension goesbyvariousnam essuch as the ttgeneralized m ean value theorem ,''the tthigherm ean value theorem ,'' ttrih aylor'stheorem with rem ainder,''orsim ply Taylor'stheorem .

Theorem A.1.2 (TaylorhsTheorem) Let/ beajhnctions' uclttltatitshrst zzderivativesarecontinuon, sïzztlteintervalgzt),zll,and /(r'-1 -1)(z)ezistsJor allzCE(zo,z1).Tlten,tltereïsazz' tzmscrïCE(zo,z1)suclttltat /(z1)- /(zo)+ (z1- zo)/'(zo)(z1-zzo)2/tt(zo)+ ...

A.1 Taylor'sTheorem

+ (z1- zol?z/(zz)(z())+ (z1- zo)''-F1/('z+1)($). ,z! (,z+ 1)! The polynom ial

iscalled thesàthdegreeTaylorpolynom ialof/ atzo.Theterm Xz z-hl= ('tr1(- tln Ul SrL'f'l/IXXZI($)

zz+ 1)! iscalled the remainder.lfthereisa numberM such thatM k I/''-F1(z)I forallzCE(zo,z1),then wecan approximate/ by theTaylorpolynomialfk with an errorbound ofthe form

I/(zz)- F,ztzzll< (t 7l1 'X1M . (- I0 1))K! zz+

Notethatif/''-F1(z)iscontinuousin theintervalgzt),z11,then thisf-unction m ustbebounded.Hence,forthiscasethere isalwaysa num ber M such that

M k I/''-F1(z)Iforallz CE(zt ),zll. Taylor's theorem can be generalized to f-unctions ofseveralindependent variables.To keep thingssim plew egiveaversion fortwo variablesand restrict the geom etry to discs in the plane.First,how ever,to avoid swim m ing in notation we introduce the operator

Forexam ple,

W e also use the notation

264

A Analysis and DifferentialEquations

to indicatethattheoperatoractson / and thederivativesareevaluated at

thepoint(z,y)= (c,d). Theorem A.1.3 LetDR = ((z,y)CE1R. 2 :(z - z())2+ Ly- yvlz < . J?2j ancl supposetltat/:Dp --+1:. 2Itascontinuonspartialderivativesnp tltrough order

zz+ 1 in DR.TltenJoranypoint(z1,3/1)CEDR)tltereïsapoint(c&,5)on tlte linesegrrtentconnecting (z1,3/1)to (zo,y(tlsuclttltat

Asintheone-variablecasewecan useTaylor'stheorem toapproximate/ and controltheerror by finding suitableboundsforthe n,+ lth-orderpartial derivatives.W euse thisversion ofTaylor'stheorem severaltim esin thebook. O ne m ightdraw com fort,however,from the factthat we seldom need term s beyond thesecond order. T he reader isdoubtlessfam iliarwith Taylor series or atleast the special case ofM aclaurin series.For exam ple,we are fam iliarw ith the seriesrepresentations z CER,

and itisnaturalto enquireifthe Taylorpolynomialtendsto / asn,--+0o, assuming / hasderivativesofallorders.ln otherwords,if/ hasderivatives

ofallordersin aneighbourhoodofztldoesF,ztzl--+/(z)asn,--+l xtforallz sufllciently closeto zo? .T he answerisno.TheCauchy f-unction

c 1/zaifz , # () -

/(z)= 0

ifz = 0,,

providesacounterexam ple.ltcan beshown using thedefinition ofa derivative

that / hasderivativesofallordersat z = 0 and that /('')(0) = 0 for all n,= 1,2,....The Taylorpolynom ialisthus

#,z(z)= 0, fora11zz.Hence limzz-sx F,ztzl= 0.ltisclearthat/(z)> 0 fora11z , # 04 consequently,limzz-sx F,ztzl# /(z)exceptatz = 0.Functionsthatcan be represented by a convergentpowerserieswith a nonzeroradiusofconvergence

arethusspecial.lfthereexistsarepresentationof/ oftheform X

/(z)-

avblz- zol'', n=0

validforallIz- ztlI< p,wherepissomepositivenumber,then/ issaidtobe (real)analyticatz().Suchf-unctionsalwayshaveTaylorseriesrepresentations at ztland Gn =

j(sz)(z( ;;. af

Sim ilarstatem entscan be m ade concerning functionsofseveralindependent variables. Finally,w e note that the C auchy function can also be used to construct a m ollifier.Roughly speaking,a m ollifier is a s11100th function that is zero outside a bounded intervalf and nonzero within f.Specifically,choose any

a CER,a , # 0,and considerthefunction

za-z2)k W Y)= 0 Y 1/(t if fj Izjsj(z, zI> a. -

lt can be show n that ?rzhas derivatives ofa11orders for allz CER ,and that

?rztzl> 0,ifz CE(-tz,tz)' ,otherwise,?rztzl= 0.Evidently,wecan modifjrthis function so thatgiven any two pointstz,bw ith a < bwegeta m ollifierthat is

zerooutside@,5)and positiveinside(c&,5).Such f-unctionsarealwaysuseful. ln this book, how ever,we tend to use sim pler functions that have sim ilar properties,butare notassm 00th.Such f-unctionshavethem eritofsim plicity forourcalculations.

A .2 T he lm plicit Function T heorem Frequently,we areconfronted w ith equations oftheform

q(I,!/)= 0,

.

which we need to eithersolve forz or y,or atleast discern w hether such an

equation definesy asa function ofz (orviceversa).Often,wecannotsolve im plicit equations,but it is im portant to know qualitative details such as whethera solution existsand isunique.W ealso usually need to know certain analyticalpropertiessuch ascontinuity ofsolutions.W hen wecannotfind an

explicit solution (orneed only qualitative properties),the implicitf-unction theorem com es to our rescue.

266

A Analysis and DifferentialEquations

t ?lalo,#0)= 0,

.

andsupposetltatg ïs dtferentiable' tlï//zrespecttoy andtltatt ' ?. t yç ' )y ïscontïzzut?t? , sin . 6 2.f/

glz,4(z))- 0' , 3.tlteJunction 4 ' ttl ï//ztlteabovepropertiesïsunique;and 4.4 ïscontinuon, sïzzfzo Moreover,zl /ç ' )g t ' ?z ezists and ïs continuon, s ïzz. 62,tlten tlteJunction 4 ïs dt jferentiableJor allz CEfzo,and

Looselyspeaking,givenapoint(zo,y(tlatwhichglzç j,: 4/0)= 0,theimplicit functiontheorem guaranteesthatimplicitrelationssuch as(A.2)aresolvable fory,provided g satisfiestherequisiteconditions.Thesolution 4 islocalin character:wedonotknow fzoexplicitly.Also,itisworth notingthattheabove

theorem doesnotprecludetheexistenceofanothersolution0(z)to (A.2),but itdoesprecludetwodistinctsolutions0,4 suchthat0(z(j)= 4(zo)= yçj.For example,letglz,y) = z - y2 and ztl= yçj= 1.Evidently,g satisfiesthe conditions ofthe implicit function theorem and 4(z) = xfz is the unique

solution w ith the propertieslisted in 1 and 2 oftheim plicitf-unction theorem .

Thefunction 0(z)= -' k/z isalso asolutiontoglz,y)= 0,butp(zo), # yç j.

T heim plicitfunction theorem can beextended to system sofim plicitequationssuch as

J(1,#,' tz,' &)= 0, q(1,#,' tz,' &)= 0.

.

Supposethatwewish to solvethe abovesystem for,say,ztand z?in term sofz

andyinaneighbourhoodofapoint(zo,yç j,' t zo,z?())thatsatisfiestheequations. ln thiscase,condition (A.4)generalizestotheJacobian condition

wherethepartialderivativesareevaluatedat(zo,yçj,' tzo,zo).Underconditions analogous to those given in Theorem A .2.1 it can be shown that there is a

uniquesolution 4tz,y),#tz,y)with propertiesanalogousto thoseof4(z)in Theorem A .2.1.An im portant specialcase ofthis result concernscoordinate transform ationsofthe form

z = ztzt,z?), y= yl' u,' &).

Theorem . A..2.2 (lnverse rrransform ations) Letztl= ztztt),zo)and3/0= yluçj,z?().SupposetltattlteJnnctionsztzt,z?) andylu,z?)Itavecontinuon,spartialderivatives of order 1 in a neigltbonrltood J2 ( :z 1:. 2 oftl tepointtzo,z?( ;). .

.

SupposeJnrtltertltattlteJacobian condition

J= t ' ?tz,y)# 0 t ' ?tu,r)

1.' tztztl,y(tl= ' tztlandz?(zo,y(tl= zo, '

2. tlte identities

ztutz,y),'Jtz,: v))- z, : v(z'(z,y),rtz,y))- y, arevalid througltoutN (zo,y(tl, ' 3.tlteJunctionszttz,y)andrtz,y)tltatsatisj' ytlteabovepropertiesare' tzzzït . wc, ' and

4.zttz,y)andrtz,y)Itave continuon,sy' tzràït zlderivativesïzzN lzç j,3/0),and 1 t:z t gu = ' 1' J @ ôu t gz or, V - -1 /t ?U' ôv 1 ny ç ')v 1 t '?z ..

dz

7:u %' L- J'ôu'

268

A Analysis and DifferentialEquations

I = tlCOS '&

y= ' tzsin v.

hence,the Jacobian is nonzero provided zt, # 0.Clearly ztzt,z?)and ylu,z?) satisfy therequisitesmoothnessconditionsforany (u,z?)CER2.W ethusconcludethatgiven anypointt' ? . to,zo),with ' t zt l, # 0,thereisa neighbourhood of t' ? . to,zo)wherein thetransformation isinvertible.W eseethat zt= (z2+ #2)1/2,

z?=arctan V). z A lthough theabove expressionsforztand z?involvem ultif-unctions,note that

the conditions' tztzt ),y(tl = ' tztland rtzt),y(tl = ' t?t ldetermine the branches ofthese f-unctions.The exceptionalpoint forthis transform ation isthe pole

t' ? . t(),zb)= (0,zb).Here,we know thatthe equationsztl= ztocos' t?tl= 0 and yçj= ztosinzo = 0 place no restrictionson r(),so thattherecannotbea unique inverse.

A .3 T heory of O rdinary D ifferential Equations M uch ofour study ofthe calculus ofvariations revolves around the EulerLagrange equation,which is a second-order nonlinear ordinary differential equation.W e also need to study ordinary differentialequationsarising from

constraintsand sufllcientconditions (theJacobiaccessory equation).Sufllce itto say thatordinary differentialequationsloom largein the subject.Som e ofthe theory underlyingtheseequationsisdeveloped asneeded in thecontext ofitsapplication.There aresom e results,however,thatwe use severaltim es and itisperhapsbest to collectthem in a single section forreference. G iven an equation ofthe form

t ?(al,!/,!//)= 0, .

wherey?denotesdy dz,w efacea m oreform idableproblem than thatposed by im plicitequations.A ssum ingg satisfiestheconditionsoftheim plicitf-unction

theorem with respecttoy?,andglzç j,: o ,y6)= 0,wecan (atleastin principle) solveequation (A.9)fory?and thusstudy an equationoftheform

A .3 Theory ofOrdinary DifferentialEquations

y'lz)- /(z,y)

(A.10)

along with the condition

#(alo)= #0. (A.11) Equation (A.10)alongwiththecondition (A.11)isan exampleofaninitialvalue problem .T here are no system atic solution techniques available for

solvingsuchproblemsexplicitlyinclosedform.lf/hassomespecialproperties

(e.g.,if/ isseparable)then thereare specialmethodsforsolution,butfor thegeneral/wemustconcededefeat.Aswith theimplicitfunction theorem, we often do not need to know the solution explicitly, but we do need to know w hethera solution existsand perhaps som equalitative propertiessuch as uniqueness and sm oothness.The follow ing result is basic to the theory of differentialequations and plays a rôle analogous to the im plicit f-unction theorem in thatitguaranteesthe existenceofa unique localsolution.

Theorem A.3.1 (PicardhsTheorem ) Snppose tltat/(z,y) ïs continuon, s ttl ' ï//zrespecttoz ïzzaneigltbonrltood . ; Vtzo,y(tl( : z1:.2 o. f(z( ;,yv))andtltereis a constantK > 0 suclttltatJor all(z,3/1),(z,n)CEN lzç j,: 4/0) I/(z,: va)- /(z,: vz)I< A' -lt va- t vzl.

(A.12)

: v'(z)- /(z,y) Jorallz CENlz(j),and v(zo)- yù. lnequality (A.12)is called the Lipschitz condition,and if/ satisfies thisinequalityforsomeK then / iscalled Lipschitz continuousin y.The requirementofLipschitzcontinuityisstrongerthan thatofcontinuity.lf/ is Lipschitz continuous in y then it is continuous in y,butthe converse is not

true.Wenotethatifweloosentherequirementon / tocontinuity iny,then

we stillhavetheexistenceofa solution (Peano'sexistence theorem,( 601,p. 29),butuniquenessisnotguaranteed.Forexample,thesimpleproblem #/(al)= #1/9, : v(0)- 0 3/2

y(z)-(( ?z) ,?,(z)-o. j Forourpurposeswe seldom need the generality afforded by the Lipschitz

condition.Usually,/ isdifferentiable in y,and this isa stronger condition

A Analysis and DifferentialEquations

than Lipschitzcontinuity.Suppose,forexample,thatt ' ?/ ç ' )y iscontinuousin

thediscXtzo,y(tl= ((z,y):(z- zo)2+ (y- yv)2< . J?2j?whereR > 0.For anychoiceof(z,3/1),(z,n )CEXtzo,y(tlwecan apply themeanvaluetheorem to assertthatthereisanumberï between yïand y,suchthat

/(z,y,)- /(z,: v1)+ (: v2- : v1):/ o (

y z, y).

Sincet ' ?/ ç ' )y iscontinuousin X(zo,y(tlitisbounded in thisdisc and hence there isa K > 0 such that

Thus,if/(z,y)hasacontinuouspartialderivativewithrespecttoyinaneighbourhood of(zo,y(tlthen wecan find a suitablediscX tzo,y(tland conclude thatin thisdisc/ isLipschitzcontinuousin y.

A generalsolution toequation (A.10)containsa parameter(theconstant ofintegration)thevalue ofwhich isdetermined by the condition (A.11).lt

isnaturalthusto enquirew hetherthe generalsolution depends continuously on this param eter.The nextresult gives conditions underw hich differential equationscontaining param eters havesolutionsthatare s11100th w ith respect to theparam eters.W eshow thattheinitialcondition param etersarea special Case.

Theorem . A..3.2 (Dependence ofSolutionson Param eters) Letct = (ct1,...,ctzzl and cîcyzzc tlteset. â?- (o,CER,z:1c.11< p1,...,IctzzI< pzz)p wltere tlte , dk arepositive zz' tzm ôcrs.LetJ2( J 1R. 2 be an open set' ttlï//zclosnreJ2

anddehnetltesetT = L x . B( : z R''-F2.Supposethat/ :T --+R hascontinn,ous derivatives ' ttlï//zrespectto y,ctl,...,ctzzoforderk k 0 on T' ,and tltat/ satishestlteLipscltitzcondition .

I/(z,y2,c.)- /(z,: v1,c.)I!;A' -lt va- t vzl

v'(z)- /(z,y,c.) :

(A.13)

4(zo,c.)- yù.

Basically,theabovetheorem show sthatthesolutionsto differentialequations

thatcontainparametersares11100thintheseparameters,provided /iss11100th

A .3 Theory ofOrdinary DifferentialEquations

in these parameters.Note that if k k 1 in the theorem then the Lipschitz condition willbe satisfied autom atically.

Returning toequation (A.10),considerthetransformation ' &J= # - yù

Z = I - 2l:.

Underthistransformation equation (A.10)is zc?(z)= /(z+ z(),w + y(j). A n im m ediateconsequence ofthe above theorem isthatthegeneralsolution

to (A.10)dependssmoothly on theinitialdatazo,yç j,provided / iss11100th in z andy near(zo,yv). T he results concerning the existence and uniqueness of solutions along with the continuous dependence on param eters can be extended to initialvalue problem s oftheform

y'(z)- f(z,y), ytzo)- yo,

whereI.Iisdefined by IyI= 1:112+ ...+ I : vzzI2 forally = (3/1,...,!/zz). A higher-orderdifferentialequation can bereadily converted into a system ofdifferentialequations.Forexam ple,given the differentialequation

!/?? = /(z,y,y?), let yï = y and n = y?.Theabove differentialequation can then be recastas the system 8/1 / = #2,

3/1= /(z,8/1,#2). ln this m annerwe can tackle questions concerning existenceand uniqueness forhigher-order equations.The results,however,are localin character and concern the initial-value problem ,w here y and y?are specified ata pointzo. Thecalculusofvariationsisim pregnated w ith second-orderdifferentialequations,butm ostoftheproblem sareboundary-valuenotinitial-valueproblem s. Boundary-value problem s consist of determ ining solutions to a differential

thatsatisfjrconditionsoftheform !/(zo)= yçjand !/(z1)= 3/1,whereztl< z1. These problem s are globalin character becausew e require a solution to be

validthroughouttheintervalgzt ),z11andsatisfytheboundaryconditions.The theory behind such problem sis m ore com plicated than thatfor initial-value problem s.A sam ple ofoneexistence result is given in Section 2.6.

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Function Spaces

W e give here a briefsynopsis ofsom e concepts from functionalanalysis.Although w edo notrely heavily on thism aterial,itisincluded because a deeper understanding ofthe calculusofvariations requiresat least a nodding fam iliarity w ith functionalanalysis.Ata m inim um w eneed a sensibledefinition of ttneighbouring functions,''and certain conceptsfrom Hilbertspaceare helpful fortopics such aseigenvalue problem s.T hissaid,the book has been w ritten so thatit is not essentialthatthereaderknow f-unctionalanalysis.A person

ignorantofthe subject can nonetheless progress through the book and read virtually every section with profit.Com plete accountsofthism aterialcan be

found in anybookon f-unctionalanalysissuch asKreyszig (461orHutsonand Pym (401.A concentratedaccountfrom aphysicist'sstandpointcanbefound in Choquet-Bruhatetal.(221. B .1 N orm ed Spaces The calculus ofvariations isessentially optim ization in spaces offunctions. lt is thus usefulto introduce som e concepts from functionalanalysis,and basicam ong these conceptsisthatofa norm ed vectorspace.The reader has probably encountered the concept ofa finite-dim ensionalvector space in a course on linearalgebra.These spacesare m odelled afterthe setofvectorsin R''.Vector spaces,how ever,can be defined m ore generally and need not be finite dim ensional.ln fact,m ost the vectorspacesofinterestin the calculus ofvariations are not finite dim ensional.A vector space is a nonem pty set X equipped with the operations ofaddition (t+ ''and scalar m ultiplication.

Forany elements/,g,/Jin X and any scalarsct,, d theseoperationshavethe properties:

(i) /+ ge . X'; (ii) /+ g - g+ /; (iii) /+ (. q+ /z)- (/+ . q)+ h.;

B Function Spaces

Example B.1.1: The set ofvectors ((z1,z2,...,z,zl : zk CE R,k = 1,2,...,zzjisdenotedbyRrL.Letx = (z1,z2,...,z,zlandy = (3/1,n ,...,ynl be vectorsin R ''.lfaddition isdefined by

x+ y = (zl+ #1,z2+ :2,...,Ivb+ ynl, and scalarm ultiplication by

(AX = lt AJ r1,t Al rz?...?Ctzszl?

Example B.1.2: LetCgzt ),z11denotethesetofallf-unctions/ :gzt),z11--+ R thatare continuouson the interval ( z(),z11.1f,for any /,g CE Cgzt ),z11, addition is defined by

Example B.1.3: Letfkdenote the setofsequences(tzszjin R such that theseriesS n C'=1 D Itzzzlisconvergent,and defineaddition so thatfora. ny two elements-4 = ttzzzl,B = (5zz), ,4+ B = t tzzz+ s,zl, .

and scalarm ultiplication so that

ct. zt= tcttzzz).. , Then é1is also a vectorspace.

Normed Spaces

T he above exam plesshow thattheelem entsin differentvectorspacescan be quite different in nature.M ore im portant,however,there is a significant

differencebetween avectorspacesuch asR?zand onesuch asCgzt l,z11having to do with ttdim ension. ''T he space R?z has a basis:any setofn,linearly

independentvectorsin R?zsuchasel= (1,0,...,0),e2= (0,1,...,0)...ezz= (0,0,...,1)formsa basis.The conceptofdimension istied to the number ofelem entsin a basisfor spaces such as R'',but it isnotclearw hat a basis

would befora space like Cgzt ),z11.ln ordertomakesomeprogresson generalizing the concept ofdim ension we need first to define w hat is m eant by a linearly independentsetwhen the setitselfm ight contain an infinite num-

berofelements.Wesay thata setislinearly independentifeveryhnite subsetislinearly independent' ,otherwise it is called linearly dependent.lf there exists a positive integer n,such thata vector space X has n,linearly independentvectorsbutany set ofn,+ 1vectors islinearly dependent,then X is called finite dim ensional.lfno such integer exists,then X is called infinite dim ensional. A subspace ofa vectorspace X isa subset ofX w hich is itselfa vector space under the sam e operations ofaddition and scalar m ultiplication.For

example,the set offunctions/ :gzt ),z11--+R such that / isdifferentiable on gzt ),z11isa subspace ofCgzt ),zll.Given any vectorsz1,z2,...,zn in a vectorspace X ,a subspacecan alwaysbe form ed by generating allthe linear com binationsinvolving the zk,i.e.,allthevectorsoftheform ctlzl+ ctzzz+ . ..+ c tzzzzz,w herethectksarescalars,isasubspaceofX .G iven any finiteset S( :zX the subspace ofX form ed in thism anneriscalled thespan ofS and

denoted by (SI.lfS ( : z X hasan infinitenumberofelementsthen thespan of S isdefined tobethesetofallhnitelinearcombinationsofelementsofS.

Vectorspacesoff-unctionssuch asCgzt ),z11arecalled function spaces. W e areconcerned prim arily w ith f-unction spaces,and to avoid repetition we agree here that for any function space the operations ofaddition and scalar

multiplication are defined pointwise aswasdone forthe space Cgzt),z11in Exam ple B.1.2.

Vector spaces are purely algebraic objects.ln order to do any analysis m ore structure is needed.ln particular,basic concepts such as convergence

requiresom emeansofmeasuring the tdistance''between objectsin thevector space.This leads us to the conceptofa norm .A norm on a vectorspace X

isareal-valued f-unction on X whosevalueat/ CEX isdenoted by 11/1a. nd which hasthe properties:

(i) 11/11k 0; (ii) 11/1 1- 0ifand onlyif/ = 04 (iii) 1ct/11- Ic.l11/11; (iv) II/+ . v11< 1/11+ 11 . v11(thetriangleinequality).

B Function Spaces

Example B.1.4: Foranyx CER?zlet11.IIcbedefined by I lxllc- ((z2 ,+ (za)2+ ...+ (zzz)2).'/2. Then 1.IIcisa norm on R,z.Thisfunction iscalled theEuclidean norm on R''.A nothernorm on R?zisgiven by

Exam ple B .1.5:

11/11x - sup I/(z)I, zeye,z1j

iswelldefined forany / CECgzt ),zll,and itcan be shown that 11.Ix isa norm forCgzt),zll.Alternatively,sinceany f-unction / in thisvectorspaceis continuous,thefunction I/1isintegrablea. ndthusthefunction 11.11 J?given by 21

II/IIz-

I/(z)Idz 20

Example B.1.6: Letn,beapositiveintegerand letC'zgzt ),z11denotethe setoffunctionsthat have atleastan sàth ordercontinuousderivative on the

intervalgzt ),z11.Sinceanyf-unctionthatisdifferentiableon theintervalgzt),z11 mustalsobecontinuousonthisintervalwehavethatCzbgzo,z11( :zCgzt l,z11for n,= 1,2,....ln fact,wehavethehierarchy Czbgzt ),z11( :zC''-1gzt ;,z1j( : z ...( : z Clg zt),z1j( : z Cgzt ),zlj.W eleaveittothereadertoshow thatforn,= 1,2,... C'zgzt),z11isavectorspaceand thatthenormsdefinedin ExampleB.1.5are alsonormsforC'zgzt),zll.Othernorms,however,canbedefinedforthespace C'zgzt),z11which takeadvantageoftheextrapropertyofdifferentiability.For example,supposen,= 1.Then thef-unction 11.IIx,1givenby II/IIx,z- sup I/(z)I+ sup I/'(z)I zeE zo,z1q

ze(zo,zlj

forthespaceC'zgzt ),z11.Here,Jkdenotesthekthderivativeof/and /(0)= /.

Normed Spaces

Theaboveexam plesindicatethatagiven vectorspacem ayhaveseveralnorm s

leadingtodifferentnormedvectorspaces.Forthisreason,thenotation(. X,1.II) isoften usedtodenotethevectorspaceX equipped with thenorm 11.I. Onceavectorspaceisequipped with anorm 11.1,ageneralized distance function (calledthemetricinduced bythenorm 11.II)can bereadilydefined. The distance dlJ,g)ofan element/ CEX from anotherelementg CEX is defined to be

dlJ,. q)- II/- . VI. The distance function forthe normed vectorspace (R'',1.IIc)corresponds to the ordinary notion ofEuclidean distance.The distance f -unction for the

normedvectorspace (C(zo,zll,11.IIx)measuresthemaximum verticalseparation ofthegraph of/ from thegraph ofg.

Neighbourhoodsofan elementin anormed vectorspace (. X,11.II)can be defined as in the fam iliar finite-dim ensionalcase.Specifically,for t î > 0 we

definean 6-neighbourhoodofan element/ CEX as

B(J,',11'II)- L. qe X :II/- . vII< t î4.. W esuppressthe 11.11in theabovenotation.

C onvergence can be defined for sequences in a norm ed vector space in a

mannerwhich mimicsthefamiliardefinition in realanalysis.Let(. X,11.II)be a normed vectorspace and let (/,zj denote an infinitesequencein X.The sequence (/,zj issaid to converge in the norm ifthere existsan / CEX such thatfor every t î> 0 an integer N can be found w ith the property that

/zz CE B (/,6)whenever n,> N .The element / iscalled the lim it ofthe sequence(/zz),and therelationship isdenoted by limzz-sx /zz= / orsimply /zz--+/.Notethatconvergencedependson thechoiceofnorm:a sequence may convergein onenorm and divergeinanother.Notealso thatthelimit/ m ustalso be an elem entin X . ln a sim ilar spirit,w e can define C auchy sequences for a norm ed vector

space.A sequence(/,zjin X isa Cauchy sequence (in thenorm 11.II)iffor any t î> 0 thereisan integerN such that

IIJm - X II< 6, whenever?rz> N and n,> N .Cauchy sequencesplay avitalrôle in thetheory

ofnormedvectorspaces.Aswithconvergence,asequence(/,zjin X may be a Cauchy sequence for one choice of norm but not a Cauchy sequence for anotherchoice. ltm ay be possibleto define any num ber ofnorm son a given vectorspace X .Two differentnorm s,however,m ay yield exactly thesam eresultsconcern-

ingconvergenceandCauchysequences.Twonorms11.Ilrzand 1.II:onavector space X are said to be equivalent ifthere exist positive num bers ctand , d

such thatforall/ CEX,

c.11/1a< 1/11b< /t ?11/11a.

B Function Spaces

lfthenorms11.Ilrzand 11.Ibareequivalent,thenitisstraightforward toshow thatconvergence in one norm im pliesconvergence in theother,and thatthe

setofCauchysequencesin (. X,11.IIrz)isthesameasthesetofCauchysequences in (. X,11.IIb).Equivalentnormsleadto thesameanalyticalresults. ldentifjringnorm sas equivalent can bedifllcult.ln finite-dim ensionalvector spaces,how ever,the situation is sim ple:all norm s defined on a finitedim ensionalvector space are equivalent.Thus thetwo norm sdefined in Fuxam ple B.1.4 are equivalent.Thesituation isdifferent forinfinite-dim ensional

spaces.Forexample,itcan beshown thatthenorms11.11J?and 11.Ix defined on thespaceCgzt),z11in ExampleB.1.5arenotequivalent. B .2 B anach and H ilbert Spaces The definitions for convergence and C auchy sequences for norm ed vector spacesareform ally analogousto thosegiven in elem entary realanalysis.Various results such as the uniqueness of the lim it can be proved for general norm ed vector spacesby essentially the sam e techniquesused to proveanal-

ogousresultsin realanalysis.The space (R'',11.IIc),however,hasa special property not inherent in the definition of a norm ed vector space.lt is well

known thatasequencein (R,11.IIc)convergesifand onlyifitisaCauchy sequence.Thisresultdoesnotextend to thegeneralnorm ed vectorspace.Every convergent sequence in a norm ed vector space m ust be a Cauchy sequence, but the converse is nottrue. A norm ed vector space is called com plete ifevery Cauchy sequence in the vector space converges.Com plete norm ed vector spaces are called B anach spaces.ln finite-dim ensionalvector spaces,com pletenessin one norm im pliescom pletenessin any norm sinceallnorm sare equivalent.T hus,spaces

such as(R'',11.IIc)and (R'z,1.II z')areBanach spaces.Forfinite-dimensional vectorspaces,com pletenessdependsentirely on thevectorspace' ,forinfinitedim ensionalvector spaces com pleteness depends also on the choice ofnorm .

Thespace(C(-1,11,11.II x ),forinstance,isaBanach space,whereasthespace (C(-1,11,11.11)isnot.lfthenorms1.I lr zand 11.IIbareequivalent,thenthecorresponding norm ed vectorspacesare either130th Banach or130th incom plete sincethesetofCauchy sequencesisthesam e foreach space,and convergence

in onenorm impliesconvergencein theother.Thetwonorms1.111and 11.IIx on C(-1,11areevidently notequivalent.

ln passing w e note that if a norm ed vector space is not com plete,it is possible to ttenlarge'' the vector space and redefine the norm so that the resulting space is com plete,and thevalue ofthe norm in theoriginalspace is preserved.ln finite dim ensions,the paradigm is the com pletion ofthe setof rationalnum bers to form the set of realnum bers.An exam ple involving an

infinite-dimensionalspaceisgivenbythespace(C( z(),z11,1.11).Thisnormed spaceisnotcomplete.lfthevectorspaceCgzt ),z11isexpandedtoincludeall functionsthatareLebesgueintegrableovertheintervalgzt ),z11,andthenorm

B.2 Banach and HilbertSpaces

isreplaced by

where the Lebesgue integralisnow used,then it can be shown that the resulting space iscom plete.l A specialtype ofBanach spacethatplaysa largerôle in analysisiscalled a H ilbert space.Hilbertspacesare sim plerthan the generalBanach space

owing to an additionalstructure called an innerproduct.Briefly,a (real) inner producton avectorspaceX isa function (.,.)on X x X such that forany /,g,/JCEX and anyctCEC thefollowingconditionshold.

(i) (/,/)k 0; (ii) (/,/)- 0 ifand only if/= 04 (iii) (/+ g,/z)- (/,/z)+ (. q,/z); (iv) (/,gs- (. q,/); (v) (c./,. v)- c.4/,. v). A vectorspace X equipped with an innerproduct (.,.)iscalled an inner product space and denoted by (. X,(.,.)).Notethatcondition (i)indicates that(/,/)isalwaysarealnonnegativenumber.Notealsothatconditions(iii) and (iv)imply that (/,. v+ /z)- (/,gs+ (/,/z).

Then (.,.)definesan innerproducton R''.ln fact,thedefinition oftheinner productismodelledafterthefamiliarinnerproduct(dotproduct)definedfor R,z

Example 8.2.2: Letf2 denote the setofsequences (tzszj such thatthe seriesS n C'D =1a2 nisconvergent.lfaddition and scalarmultiplicationaredefined the sam eway asforthe spacefkin Exam ple B.1.3,then f2 isa vectorspace.

Supposethata = ttzzz).,b = (5zz) CEf2,and letcsz= maxttzsz,5sz).Then the series E n C'=1 D (cn 2 isconvergentand hence the series E n C'D =1tzszszzisabsolutely convergent.An innerproducton thisvectorspace isdefined by 1Strictly speaking the function replacing the norm is not even a norm because

11/IIol= 0doesnotimply that/ = 0.Thisproblem iseasilyremedied using equivalence classes;i.e. two functions / and g are equivalentif/ = g almost everm here.

280

B Function Spaces

11/11- (/,/), is a norm on the vector space X .Thus,any inner productspace leads to a norm ed vectorspace.The specialnorm defined above is called the norm induced by the innerproduct.Thenorm ed vectorspaceform ed by theinduced

norm mayormaynotbecomplete.lf(. X,N)isaBanachspacethen theinner productspace(. X,(.,.))iscalledaHilbertspace.A Hilbertspaceisthusan innerproductspace that iscom pletein the norm induced by theinner prod-

uct.Theinnerproductspace (R'',(.,.))isan exampleofafinite-dimensional Hilbertspace.ltcan beshown thattheinnerproductspace (f2,(.,.))ofFuxam ple8.2.2 isalsoa Hilbertspace.A notherinfinite-dim ensionalH ilbertspace

ofimportancein analysisisthespacefvzgzt l,zlj. Example B.2.3: Letfvzgzt l,z1jdenotethesetoffunctions2/ :gzt ),z1j--+R

suchthattheLebesgueintegraljE zo,ztq/2(z)dzexists(i.e.,thesetofttsquare integrable''functions),and let(.,.)bedefined by

ltcanbeshownthatforany/,g CEL2gzt),z1jtheabovef-unctioniswelldefined and satisfiestheaxiom sofan innerproduct.Theresulting innerproductspace isa Hilbertspace. H ilbert spaces have found w idespread applications in pure and applied m athem atics.T he extra structureafforded by an innerproductgivesriseto a generalization ofgeom etricalconceptsin R''.ln particular,thereisa straightfolavard extensionoftheorthogonality based on theinnerproduct.Recallthat

in R?ztwononzerovectorsu,v areorthogonalifand only if(u,v)= 0.For thegeneralHilbertspace,wesay thattwo elements /,g are orthogonalif

(/,gj= 0.Thus,forexample,in thespaceL2g zt),z1jtwo f-unctions/,g are orthogonalif

/(z). v(z)dz - 0.

E zo,zt1 A sin the finite-dim ensionalcase,given a setofelem entsin a generalH ilbert spaceitispossibletoform an orthogonalsetby an algorithm analogoustothe 2 Strictly speaking, the elem ents of this set are equivalence classes of functions m odulo equality alm osteverywhere.

B.2 Banach and HilbertSpaces

281

G ram -schm idt process.lt is also possible to construct orthogonalbases for H ilbertspaces.Although basesforgeneralinfinite-dim ensionalBanach spaces

play a somewhatnominalrôle (in contrastwith finite-dimensionalspaces), bases play a significantrôle in the theory ofHilbertspaces.

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R eferences

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Index

action integral 11 Aeneid,15 allowable variation 120 arealvelocity,211

Carthage,15 catenary,4,39,51,93,130, 138, 164, 170,230,247,260 as an isoperim etric problem 87 on a cylinder 125 catenoid 5 Cauchy function 264 Cauchy sequence,277 Cauchy-Euler equation 109 caustics 252

conjugatepoint,235,244,252 multiplicity,255 conservation law 41 definition,201 constraint finite,119 holonom ic 119 integrable,119,126, 131 isoperim etric 6 83 multiple isoperim etric 96 nonholonom ic 119 rheonom ic 124 scleronom ic 124

convergence

in the norna 277 COnVeX

function,257 sets 257 cycloid 9 41 d'Alem bert's Principle,132 Dido's problem ,14,89 in polar coördinates 94 Zenodorus'solution 15 Dirichlet problem ,69 eigenfunction 104 expansion, 105 eigenvalue,104 higher,115 simple, 105 eikonalequation 176 Em den-Fowler equation 220 envelope,250 Euler-luagrange equation 33 degenerate case,43 hrstintegralof 57 for functionals w ith higher-order derivatives 59 for functionals with second derivam tives 56 severaldependentvariables 62 two independent variables 67 extrernal 30 abnormal 128 normal 128 rigid,84,86,127

288

Ferm at's Principle,168,184,252 firing zone,253 firstfundam entalform 17 firstintegral,170,202 firstprolongation 219 firstvariation 30 35 for functionals with second derivatives 56 severaldependentvariables 61 fixed endpoint problem 28 severaldependentvariables 60 Fourier series 107 generalized 105 Fredholm alternative 143 function spaces,275 functional 1 variationally equivalent 172 Gaussian curvature 259 generalized coördinates 11 164 generalized m om enta 164 geodesics 16 124 in the plane,33,247,259 on a cylinder 125 on a sphere,18,37,231,232,253,256 geom etricaloptics,168,176,180 globalextrema ofa function,24,26 Halm 's equation,114,118 Ham ilton-lacobiequation 176 conlplete integral 177 com plete solution 177 Liouville form ,194 reduced 182 separable 186 separation ofvariables 184 Ham ilton-lacobitheorem ,177 Ham iltonian,163 Ham iltonian system 166 conservative 14 169 181 Ham ilton'sequations 166 Ham ilton's Principle,11,63,132,252 harm onic oscillator,170,173,183 Helm holtz conditions 72 Hengist and Horsa 15 Hessian m atrix 223 254 Hilbert 231.d problem 1

space,104 280

implicit function theorem 266 inverse transform ations 267 infinitesim algenerators 208 initial-value problem 269 inner productspace 279 intrinsic geornetry, 18 inverse problern 70 isochrone,10 isoperim etric problem 83 abnormal 86 as a Sturm-luiouville problem 106 duality of,93 for functionals w ith higher-order derivatives 95 normal,86 severaldependentvariables 99 Jacobi accessory equation 234 246 necessary condition 240 Kepler problem ,13,206,211,217 Kepler's second law 211 kinetic energy,10 Lagrange equationsofm otion 64 Lagrange m ultiplier 75 abnormal 79 as an eigenvalue,106 as rate ofchange,93 extended rule 82 normalproblem 79 rule 77 rule for abnormalisoperim etric problem s 86 rule for normal isoperim etric problem s 86 rule for severalconstraints 78 Lagrange problena 125 abnormalcase 128 Lagrangian,11,163 optical 168 Lebesgue integral 104 Legendre condition 227-229 strengthened 230 Legendre transform ation 160 Levi-civita conditionsforseparability 199

Liouville condition forseparable system s 191 form for Hamilton-lacobiequation 194 norm alform 116 surface,200 Lipschitz condition,269,271 continuity,269,270 localextrem a ofa function,23,26 ofa functional 28 localm aximum weak 241 localm inimum weak 241 M athieu functions 113 M athieu'sequation,113, 114,118 eigenvalues,118 M aupertuis'Principle 14 M axwellfisheye,253 m ean curvature,68 m ean value theorem 261 m etric tensor 17 m inim alsurface 20 m ollifier,227,265 m onkey saddle 27 M orse lem m a,223,232 saddle,224 M orse index,224,254 for an extremal 254 theorem ,254,255 naturalboundary condition 137 nephroid 252 Newton's equation 64 Noether's theorem ,41,208,210 n orm

definition,275 equivalent 104 277 Euclidean,276 norm ed vectorspace com plete 278 definition,275 opticalcosines 168 parabola ofsafety,252

parabolic point 223 Peano's existence theorem 269 pendulum ,12,123,167 phase space,164 Picard's theorem 269 Poisson bracket 170 potentialenergy,11 Pygm alion 14 Pyrrhic victory,175 quadratic form definite,222 discrim inantof 222 indefinite,222 Sylvester criterion 224 Ramsey growth m odel 144 260 Rayleigh quotient,110,115 refractive index 168 180 Riccatiequation,234,240 Riem annian manifold 19 rigid extrernal 127 saddle,222 m onkey,27 M orse,224 second variation 225 positive and nondegenerate 244 positive definite,233,236 strongly positive 244 separable solution conditionsfor 190 to the Ham ilton-lacobiequation 186 span,275 spectrum ,104 Stiickel conditions 196 m atrix, 196 theorem 194 stationary point degenerate,223 for a function 25 26 for a functional 30 nondegenerate 223 strengthened Legendre condition 230 strong extrema 241 Sturm -luiouville problem ,104,231,255 as an isoperimetric problem 106 eigenvalues,104

290

first eigenvalue 105 109 Sylvester criterion 224 symplectic m ap,171 Taylorpolynom ial 263 Taylor's theorem 262 rem ainder 263 third variation,226 Thomas-Fermiequation 220 three body problem 190 Titchm arsh equation 114 transform ation active variables 162 canonical 171 contact,160 involution 160 Legendre,160 nonsingular,44 passive variables 162

point,160 rotation 203 sym plectic 171 translation 202 transversality condition 151

variational equivalence 172 invariance,204 206 sym metry,204 vector space

definition,273 finite dim ensional 275 infinite dim ensional 275 nornaed 275 subspace,275 Voltaire,14 zone ofaudibility,256