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Mathematics for Physics - Goldbart: Home Page

MathematicsforPhysicsA guidedtourforgraduatestudentsMichaelSton eandPaulGoldbartPIMANDER-CASAUBONA lexandria Florence LondoniiCopyrightc ,P. thismaterialcanbe reproduced,storedortransmittedwithoutthe writtenpermissionof informationcontact:MichaelStoneor PaulGoldbart,Department of Physics ,Universityof Illinoisat Urbana-Champaign,1110 WestGreenStreet,Urbana,Illinois61801-308 0, thememoryof Mike'smother,AileenStone:9 9 = Paul'smotherandfather, greatmany peoplehave encouragedus alongtheway:Ourteachersat theUniversity of Cambridge,theUniversity of California-LosAngeles, { yourquestionsandenthusiasmhave helped shape |faculty andsta |attheUniversity of Illinoisat Urbana-Champaign{ how fortunatewe areto have a community sorich in bothaccomplishment :KyreandSteve andGinna.}}

Mathematics for Physics A guided tour for graduate students Michael Stone and Paul Goldbart PIMANDER-CASAUBON Alexandria Florence London

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1 MathematicsforPhysicsA guidedtourforgraduatestudentsMichaelSton eandPaulGoldbartPIMANDER-CASAUBONA lexandria Florence LondoniiCopyrightc ,P. thismaterialcanbe reproduced,storedortransmittedwithoutthe writtenpermissionof informationcontact:MichaelStoneor PaulGoldbart,Department of Physics ,Universityof Illinoisat Urbana-Champaign,1110 WestGreenStreet,Urbana,Illinois61801-308 0, thememoryof Mike'smother,AileenStone:9 9 = Paul'smotherandfather, greatmany peoplehave encouragedus alongtheway:Ourteachersat theUniversity of Cambridge,theUniversity of California-LosAngeles, { yourquestionsandenthusiasmhave helped shape |faculty andsta |attheUniversity of Illinoisat Urbana-Champaign{ how fortunatewe areto have a community sorich in bothaccomplishment :KyreandSteve andGinna.}}

2 AndJenny, OllieandGreta{ we hope to be moreattentive now thatthisbookis CambridgeUniversity Press{ of Energy, whohave supportedourresearch over is basedona two-semestersequenceof coursestaught to incominggraduatestudents at theUniversity of Illinoisat Urbana-Champaign,pri-marilyphysicsstuden ts butalsosomefromotherbranchesof introducestudents to someof themathematicalmethods andconceptsthattheywill ndusefulin havesought to enliven thematerialby thereforeprovideillustrative theseillustrationsareclassicalbutmany aresmallpartsofcontemporaryresearch thetextandat theendof each chapterweprovidea collectionof materialpresentedin thetext;thelatterareintendedto be interesting,andtake rathermorethought devotethe rst,andlongest,part(Chapters1 to9, andthe rstsemesterin theclassroom)to exploretheanalogybetweenlinearoperatorsa ctingonfunctionspacesandmatricesactingon nitedimensionalspaces,andusetheoperatorl anguageto pro-videa uni edframeworkforworkingwithordinarydi erentialequations,partialdi erentialequations, soundgraspof undergraduatecalculus(includingthevector calculusneededforelectricity andmagnetismcourses),elementarylinearal- gebra,andcompetenceat ,aswell as basicordinarydi erentialequationtheory, receive a quick review,butit wouldhelpif thereaderhadsomepriorexperienceto notrequiredforthispartof (Chapters10to 14)}}

3 Focusesonmoderndi erentialge-ometryandtopology, withaneye to itsapplicationto ,especiallytheexteriorcalculus,areintrod uced,andviiviiiPREFACE usedto investigateclassicalmechanics,electromag netism,andnon-abeliangauge homologyandcohomologyis introducedandis usedto investigatethein uenceof theglobaltopologyof a manifoldonthe eldsthatlive in it andonthesolutionsof di erentialequationsthatconstrainthese grouprepresentationsandtheirapplications to (Chapters17 to 19)exploresthetheoryof complexvariablesandits of thematerialis standard,we make useof theexteriorcalculus,anddiscussrathermore of thetopologicalaspectsofanalyticfunctions thanis cursoryreadingof theContents of thebookwillshow thatthereismorematerialherethancanbe comfortablycoveredin two thebasisforlecturesin theclassroom,we have founditusefulto tailorthepresentedmaterialto theinterestsof Calculusof it good for?

4 Minimum?.. 422 .. 853 LinearOrdinaryDi solutions.. 108ixxCONTENTS4 LinearDi .. operator.. eigenfunctions.. 1455 .. Lagrange'sidentity.. Greenfunctions.. andtheGelfand-Dikiiequation.. 1856 PartialDi cationof PDE's.. data.. equation.. 2497 TheMathematicsof waves.. 2898 .. 340 CONTENTSxi9 .. cationof integralequations.. 38210 andcontravariant vectors.. 41511 Di elds.. erentiatingtensors.. derivatives.. 45612 .. 'theorem.. 49113 AnIntroductionto Di eomorphism.. 'stheorem.. e duality .. 55614 .. 58815 .. SU(2).. 64216 TheGeometryof .. thetotalspace.. 66417 .. :Cauchy andStokes.. Cauchy'stheorem.. 74318 Applicationsof .. ectionprinciple.. 788 CONTENTS xiii19 .. erentialequations.

5 'sviacontourintegrals.. 831A .. erencesof vectorspaces.. 859B .. 888C Bibliography893xivCONTENTSC hapter1 Calculusof VariationsWe beginourtourof usefulmathematicswithwhatis calledthecalculusofvariations. Many physicsproblemscanbe formulatedin thelanguageof thiscalculus,andoncetheyarethereareusefu ltoolsto thetextandassociatedexerciseswe willmeetsomeof theequationswhosesolutionwilloccupy us formuch of it good for?Theclassicalproblemsthatmotivatedthe creatorsof thecalculusof varia-tionsinclude:i)Dido'sproblem: In Virgil'sAeneidwe readhow QueenDidoof Carthagemust ndlargestareathatcanbe enclosedby a curve (a stripof bull'shide)of )Plateau'sproblem: Findthesurfaceof minimumareafora given soap lmona wireframewilladoptthisminimal-areacon )JohannBernoulli'sBrachistochrone: A beadslidesdowna curve with +V(x) is constant, ndthecurve thatgives )Catenary: Findtheformof a hangingheavychainof ndingmaximaor minima,andhenceequatingsomesortof derivative to thenextsectionwe de nethisderivative,andshow how to variationalproblemswe areprovidedwithanexpressionJ[y] that\eats"wholefunctionsy(x) andreturnsa objectsarecalledfunctionalsto a mapf:R!

6 R. A functionalJis a mapJ:C1(R)!RwhereC1(R) is thespaceof smooth(havingderivatives of allorders) ndthefunctiony(x) thatmaximizesor minimizesa givenfunctionalJ[y] we needto de ne,andevaluate, restrictourselves to expressionsof theformJ[y] =Zx2x1f(x;y;y0;y00; y(n))dx;( )wherefdependson thevalueofy(x) andonly nitelymany of its functionalsaresaidto rsta functionalJ=Rfdxin whichfdependsonlyx,yandy0. Make a changey(x)!y(x) +" (x), where"is a (small)x-independentconstant. Theresultant changeinJisJ[y+" ] J[y] =Zx2x1ff(x;y+" ;y0+" 0) f(x;y;y0)gdx=Zx2x1 " @f@y+"d dx@f@y0+O("2) dx= " @f@y0 x2x1+Zx2x1(" (x)) @f@y ddx @f@y0 dx+O("2):If (x1) = (x2) = 0, thevariation y(x) " (x) iny(x) is saidto have\ xedendpoints."For such variationstheintegrated-outpart[:::] ning Jto be theO(") partofJ[y+" ] J[y], we have J=Zx2x1(" (x)) @f@y ddx @f@y0 dx=Zx2x1 y(x) J y(x) dx:( ) J y(x) @f@y ddx @f@y0 ( )is calledthefunctional(orFr echet) derivative ofJwithrespecttoy(x).

7 Wecanthinkof it as a generalizationof thepartialderivative@J=@yi, wherethediscretesubscript\i" onyis replacedby a continuouslabel \x," andsumsoveriarereplacedby integralsoverx: J=Xi@J@yi yi!Zx2x1dx J y(x) y(x):( ) have a di erentiablefunctionJ(y1;y2;:::;yn) ofnvariablesandseekitsstationarypoints| thesebeingthelocationsat whichJhasitsmaxima, a stationarypoint (y1;y2;:::;yn) thevariation J=nXi=1@J@yi yi( )mustbe zeroforallpossible yi. Thenecessaryandsu cient conditionforthisis thatallpartialderivatives@J=@yi,i= 1;:::;nbe ,we expectthata functionalJ[y] willbe stationaryunder xed-endpoint vari-ationsy(x)!y(x)+ y(x), whenthefunctionalderivative J= y(x) vanishesforallx. In otherwords,when@f@y(x) ddx @f@y0(x) = 0; x1< x < x2:( )Thecondition( )fory(x) to be a stationarypoint is J= y(x) 0 is asu cientconditionfor Jto be zerois clearfromitsde nitionin ( ).

8 To seethatit is anecessaryconditionwe mustappealto theassumedsmoothnessofy(x). Considera functiony(x) at whichJ[y] is stationarybutwhere J= y(x) is non-zeroatsomex02[x1;x2].Becausef(y;y0;x ) is smooth,thefunctionalderivative J= y(x) is alsoasmoothfunctionofx. Therefore,by continuity, it willhave thesamesignthroughoutsomeopen interval containingx0. Bytaking y(x) =" (x) to (x) :Soap lmbetweentwo ,andof onesignwithinit, we obtaina non-zero J|in contradictionto stationarity. In makingthisargument, we seewhy itwas essentialto integrateby partsso as to take thederivative o y: whenyis xedat theendpoints,we haveR y0dx= 0, andso we cannot nda y0thatis zeroeverywhereoutsideaninterval andof , thenstation-arity underallpossiblevariationsrequiresoneequ ation J yi(x)=@f@yi ddx @f@y0i = 0( )foreach functionyi(x).

9 If thefunctionfdependsonhigherderivatives,y 00,y(3),etc., thenwehave to integrateby partsmoretimes,andwe endupwith0 = J y(x)=@f@y ddx @f@y0 +d2dx2 @f@y00 d3dx3 @f@y(3) + :( ) we useournewfunctionalderivative toaddresssomeof theclassicproblemsmentionedin :Soap lmsupportedby a pairof coaxialrings( )Thisa simplecaseof Plateau' thesoap lmisequalto twice(onceforeach liquid-airinterface)thesurfacetension of thesoapsolutiontimestheareaof the lmcanthereforeminimizeitsfreeenergyby minimizingitsarea, a surfaceof thereforeseekthepro ley(x) thatmakes theareaJ[y] = 2 Zx2x1yq1 +y02dx( )of thesurfaceof revolutiontheleastamongallsuch surfacesboundedbythecirclesof radiiy(x1) =y1andy(x2) =y2. Becausea minimumis astationarypoint, we seekcandidatesfortheminimizingpro ley(x) by settingthefunctionalderivative J= y(x) to beginby formingthepartialderivatives@f@y= 4 q1 +y02;@f@y0=4 yy0p1 +y02( )andusethemto writedowntheEuler-Lagrangeequationq1 +y02 ddx yy0p1 +y02!

10 = 0:( )Performingtheindicatedderivative withrespecttoxgivesq1 +y02 (y0)2p1 +y02 yy00p1 +y02+y(y0)2y00(1 +y02)3=2= 0:( )Aftercollectingterms,thissimpli esto1p1 +y02 yy00(1 +y02)3=2= 0:( )Thedi erentialequation( )stilllooksa tri simplifyfurther,we multiplybyy0to get0=y0p1 +y02 yy0y00(1 +y02)3=2=ddx yp1 +y02!:( )Thesolutionto theminimizationproblemthereforereducesto solvingyp1 +y02= ;( ) is anas yet , rstorder,di erentialequationis elementary. We recastit asdydx=ry2 2 1( )andseparatevariablesZdx=Zdyqy2 2 1:( )We now make thenaturalsubstitutiony= cosht, whenceZdx= Zdt:( )Thus we ndthatx+a= t, leadingtoy= coshx+a :( )We selecttheconstants andato t theendpointsy(x1) =y1andy(x2) = + :HangingchainExample:HeavyChainover Pulleys. We cannotyet considertheformofthecatenary, a hangingchainof xedlength,butwe cansolve a simplerproblemof a heavy exiblecabledrapedover a pairof pulleyslocatedatx= L,y=h, andwiththeexcesscablerestingona horizontalsurfaceasillustratedin ht/Ly=coshtt= :Intersectionofy=ht=Lwithy= thesystemisP:E:=Xmgy= gZL Lyp1 + (y0)2dx+ const.


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