Lecture Notes in Classical Mechanics (80751)

51936_7LectureNotesMechanics.pdf

LectureNotesin ClassicalMechanics(80751)

RazK upferman

InstituteofMathematics

TheHebrew University

July14,2008

2

Contents

1Pr eliminaries1

1.1Vector calculus..... ... ... .. ... ... ... ... ..1

1.2Dimensionalanalysis .. ... ... ... ... ... ... ... .5

2NewtonianMechanics 15

2.1Kinematics. ... ... ... ... .. ... ... ... ... ..15

2.2Dynamics:Ne wton'sla ws........ ..... ... ... ..18

2.3Work andmechanicalenergy.. ... ... ..... ... ... .23

2.4Angularmomentum andtorque. ... ... ... .. ... ... .26

2.5Systemsof pointparticles ... ... ... ... ... .. ... ..27

2.6What ismissing?. ... ... ... ... ... .. ... ... ..33

2.7d'Alembert's principleofvirtualwork. ... ... .. ...... 37

2.8Examples. .. ... ... ... ... ... ... ... ... .. .41

3Variational Calculus45

3.1Thebrachistochrone ... .. ... ... ... ... ... ... .. 45

3.2Functionalso ver normedspaces....... ... ... .. ... 47

3.3Functionalderi vati ves............. ... ... ... .. 48

3.4Necessaryconditions foralocal extremum. ... ... .. ....50

3.5TheEuler -Lagrangeequations. ..... ... ... ... ... .52

iiCONTENTS

4Lagrangianmechanics 55

4.1Hamilton's principle..... ... ... .. ... ... ... ..55

4.2Conservation theorems..... ... ... .. ... ... ... .56

5Thetw o-bodycentralf orceproblem59

5.1Reductionto aone-particlesystem ... ... .. ... ... ... 59

5.2Analysisof thereduced one-particleproblem. ... ... ... .61

5.3Classification oforbits. ... ... ... ... ... .. ... ..64

5.4Orbit equations.. ... ... ... ... ... .. ... ... ..66

5.5Gravitational systems..... ... ... .. ... ... ... ..67

6Smalloscillations 71

6.1Equilibriaand deviations. ... ..... ... ... ... ... .71

6.2Normalmode equations.. ... ... ... .. ... ... ... .73

6.3Oscillationsof atri-atomicmolecule ... ... ... .. ... ..74

7Hamiltonianmechanics 77

7.1TheLe gendretransformation. ...... .. ... ... ... ..77

7.2Hamilton's equations..... ... ... ... .. ... ... ..79

7.3Symmetriesand conservationla ws.. ...... ..... ... .86

7.4Deriv ationthroughavariationalprinciple.. ... ... .....87

7.5Finalcomments ... ... ... ... ... .. ... ... ... .88

8Canonicaltransf ormations89

8.1Motiv ation....... ... ... ... ... ... ... .. ... 89

8.2Canonicaltransformations ... ... ... ... .. ... ... .90

8.3Choicesof generatingfunctions ... ... ... ... ... .. .91

8.4Thesymplectic approach.. ... ... .. ... ... ... ... 94

8.5Poissonbrack ets.. ...... ... ... .. ... ... ... ..96

8.6Poissonbrack etformulationof mechanics..... .. ... ... 99

8.7Differentiabletransformations. ... ... ... ... .. ... .99

CONTENTSiii

8.8Liouville's theorem..... ... ... ... .. ... ... ... 101

9TheHamilton-J acobitheory103

9.1TheHamilton-Jacobi equation. ... ... ... ... ... .. .103

9.2Ane xample.. ...... ... ... .. ... ... ... ... .105

10Quantummechanics 107

10.1Matterand wav es.. ........... ... ... ... ... .107

10.2Thepostulates ofquantum mechanics.. ... ... ... ... .110

10.3Aparticle inapotential well.. ... ... ... .. ... ... .112

10.4Theharmonic oscillator.. ... ... ... .. ... ... ... .115

ivCONTENTS

Chapter1

Preliminaries

1.1Vector calculus

Accordingtoclassical physics,"reality" takesplace inaproductspaceR 3

×R,

whereR 3 representsspaceandRrepresentstime.Thenotions ofspace andtime areaxiomaticin classicalph ysics,meaningthat theydo notdeserveadefinition. (Inrelativisticphysics,thenotions ofspaceand timeareintermingled, andone ratherspeaksabout afour dimensionalspace-time.) Becausetheph ysicalspaceis athree-dimensionalvectorspace,we willhav eto dealextensivelywithvectorsinR 3 .We willdenotevectorsbyboldfacecharacters, e.g.,a,b,c.Afterchoosing anan orthonormalbasis,the entries,orcomponents ofav ectorinR 3 arecommonlydenoted by a=(a 1 ,a 2 ,a 3 ). Attimeswe willusedi fferentnotations,such asr=(x,y,z).

Letaandbbetwo vectorsinR

3 .Theirscalarproduct ,ordotproduct isareal numberdefinedby a·b=a 1 b 1 +a 2 b 2 +a 3 b 3 = 3 ? i=1 a i b i . Physicistsoftenadoptanotational convention, theEinsteinsummationcon ven- tion,wherebyinde xesthat appeartwiceinanexpressionaresummed over ,with- outneedof thesummationsign. Thatis, a·b=a i b i .

2Chapter1

Whileconfusingat first,thisnotation endsupbeing veryuseful. Ofcourse,what physicistscallascalarproduct isnothingb utthestandard innerproduct inR 3 . Thelength,ormagnitudeofav ectora,isits Euclideannorm , |a|=(a·a) 1/2 = ⎷ a i a i . Thescalarproduct oftwo vectors,can beattributed ageometricmeaningthat involvestheangle,θ,betweenthe twov ectors, a·b=|a||b|cosθ. Two(non-zero)vectorsaresaid tobeorthogonal(denoteda?b)iftheir dot productvanishes, i.e.,ifθ=±π/2.

Wewilloftendenotethe standardbasis inR

3 by e 1 =(1,0,0),e 2 =(0,1,0),ande 3 =(0,0,1).

Thus,av ectorcanbe alsowrittenas

a=a 1 e 1 +a 2 e 2 +a 3 e 3 =a i e i .

Othertimes, wewillrather denotee

1 =ˆx,e 2 =ˆy,ande 3 =ˆz. Proposition1.1Thedotpr oductsatisfiesthe followingproperties: ?Itiscommutative ,a·b=b·a. ?Itisbilinear ,(αa)·b=α(a·b). ?Itisdistrib utive,a·(b+c)=a·b+a·c. ?TheCauc hy-Schwarzinequality,|a·b|≤|a||b|. Wedefineanotherproductbetween pairsof vectors,the crossproduct,orthe vectorproduct.Unlike thedotproduct,thecrossproduct resultsina vector. It canbedefined insev eralways, forexample, a×b= ? ? ? ? ? ? ? ? e 1 e 2 e 3 a 1 a 2 a 3 b 1 b 2 b 3 ? ? ? ? ? ? ? ? =(a 2 b 3 -a 3 b 2 )e 1 +(a 3 b 1 -a 1 b 3 )e 2 +(a 1 b 2 -a 2 b 1 )e 3 . Theotherw ayisto definethecrossproductsofall pairsofbasis vectors,e.g., e 1 ×e 1 =0,e 1 ×e 2 =e 3 ,etc.,

Preliminaries3

andimposebilinearity anddistributi vity.A thirddefinitionintroduces theLevi-

Civitatensor,

? ijk = ? ? ? ? ? ? ? ? ? ? ? ? ? +1if( i,j,k)is(1 ,2,3),(2,3,1),or(3 ,1,2) -1if( i,j,k)is(1 ,3,2),(2,1,3),or(3 ,2,1)

0otherwise.

.

Then,thecross producttakes thesimpleform,

a×b=? ijk e i a j b k . Avery usefulpropertyoftheLe vi-Civitatensor (easilychecked byexplicit sub- stitution)is ? ijk ? imn =δ jm δ kn -δ jn δ km .(1.1) Thecross productalsohas ageometricinterpretation. First,weclaim thatfor everya,b, a·(a×b)=b·(a×b)=0, thatis,the crossproduct oftwo vectorsis perpendiculartoboth. Whilethiscan be verifiedbya(relativ ely)tedioussubstitution, thecleanw ayofderivingthisresult isusingour index notation,e.g., a·(a×b)=(a i )(? ikl a k b l )=? ikl a i a k b l . Now,? ikl changessignwhen apairof indexes isswitched,hence, a·(a×b)=-? kil a i a k b l =-? ikl a i a k b l , wherewe have justrenamedthesummationindexesi→kandk→i.Thus, this tripleproductequals minusitself,hence itiszero.

Whataboutits magnitude?Notethat

|a×b| 2 =(a×b)·(a×b)=? ijk a j b k ? imn a m b n . Usingtheproduct formula(1.1)for theLevi-Ci vitatensor, |a×b| 2 =(δ jm δ kn -δ jn δ km )a j b k a m b n =a j b k a j b k -a j b k a k b j =|a| 2 |b| 2 -(a·b) 2 =|a| 2 |b| 2 (1-cos 2

θ),

4Chapter1

Figure1.1:V isualizationofthe crossproduct.

i.e., |a×b|=|a||b|sinθ. Thus,thecross productoftw ovectors isav ectorperpendicularto both,whose magnitudeequalsto theareaof theparallelepipedformed bythetw ov ectors.Its directionisdetermined bytherighthandrule (seeFigure1.1). Proposition1.2Thecross productsatisfiesthefollowing algebraicproperties: ?Anti-symmetry:a×b=-b×a. ?Distributivity:a×(b+c)=a×b+a×c. ?Bilinearity,(αa)×b=α(a×b). ?Itisnot associative. ?Jacobi'sidentity:a×(b×c)+c×(a×b)+b×(c×a)=0. ?Thevectortriple product, or"BA CminusCAB"formula:a×(b×c)= b(a·c)-c(a·b). ?Thescalartriple product formula,a·(b×c)=b·(c×a)=c·(a×b). ?Exercise1.1Proveit.

Preliminaries5

1.2Dimensionalanalysis

Unitsofmeasur ementAllphysical quantitiesareexpressedinterms ofnum- bers,obtained throughmeasurements,duringwhich comparisonismade witha standard,or, aunitofmeasur ement.For example,thelengthofa rulerismea- suredbycomparison toastandard, saythe meter;themass ofarock ismeasured bycomparisonwith aunitmass, say,the gram;theduration oftheday ismea- suredbycomparison withaunit time,saythe timeittak esastandard hour-glass toempty;the (mean)velocity ofacar ismeasuredby measuringthedistanceit went,sayin inches,thetime ittook,say inmilliseconds,and dividingthose two numbers. Fundamentalandderi vedunits Theunitsof measurementsare dividedinto twocategories:fundamentalunitsandderivedunits.Supposewe wantto study aclass ofphenomena,for example,the motionofbodies. Wemay listallthe quantitieswhichwill ever bemeasured,and forcertainofthemchooseunitsof measurement,whichwe willcallfundamental units;thechoice isarbitrary. For example,wemaychoosefundamental unitsofmass, length,andtime (butalso force,lengthand time).Deriv edunitsof measurementsarebased uponthefun- damentalunitsvia somemethodof measurement(perhapsonly conceptual).For example,themeasurementofv elocity,whose (derived) unitsusesthe (fundamen- tal)unitsof lengthandtime. Systemsof unitsAsetof fundamentalunitsthat issufficientformeasuringthe propertiesof aclassof phenomenaiscalled asystemofunits ,fore xample, cgs=centimeter,gram,second inmechanics. Classesofsystems ofunitsTwosystemsofunitswhichdi fferonlyin themag- nitudeoftheir standards,but notin theirphysical natures,aresaidtobelongto the sameclass.F orexample, allsystemsofunitswhichare oftheform unitoflength =cm/L unitofmass =gram/M unitoftime =second/T,

6Chapter1

withL,M,T>0,belongto thesameclass (theLMTclass). Anotherclass,the

LFTclass consistsofsystems ofunitsof theform

unitoflength =cm/L unitofforce =kg-f/F unitoftime =second/T. (Forthetimebeing,ignore thefact wehav enoclue what"force"means. Allyou needtounderstand, isthatit isa physicalquantity thatcanbe measuredwiththe appropriateapparatus.) Note,howe ver,thatwhilebothclasses,theLMTandLFTclasses,formasystem ofunitsfor aclassof physicalphenomena whichwecall mechanics,they arenot asufficientsetof unitsifwe want,in addition,tomeasure, say,temperature, oran electriccharge. DimensionsSupposewechoose aclassof systemofunits, forexample, the LMTclass inmechanics,and supposewechange oursystemof unitswithinthe sameclass, bydecreasingthe lengthunitby afactor L,themass unitbya factor M,andthe timeunitby af actorT(e.g.,weuse centimetersratherthan meters, ouncesratherthan grams,andweeks ratherthanseconds). Howwill thisa ffect themagnitude oftheoutcome ofmeasurements? Lengthmeasurementswill bemagnified byaf actorofL;massmeasurements will bemagnifiedby afactor ofM;timemeasurements willbe magnifiedbya factorof T;velocity measurementswillbemagnifiedbya factorof LT -1 ,andso on.Ev ery physicalquantitywillbemagnified byaf actorwhichdepends onL,M,T.That is,ev eryphysicalquantityhasanassociatedfunction ofL,M,T,whichwe callits dimension.Thedimension ofmassdensity (massper unitvolume), forexample, isML -3 .Aquantity whichremainsin variantunder achangeof unitsiscalled dimensionless,andits dimensionisby definitionone.Note thatthedimensionof aphysicalquantity dependsonthe classofthe systemofunits! ?Exercise1.2Considerthe MVTclasswhere thefundamentalunits arethatof mass,velocity andtime.Whatkindofe xperimentmeasureslength? Whatisthe dimensionoflength? Whatisthe dimensionofmass density? Physicsisusuallynotthought interms ofanaxiomatic theory.One ratherspeaks aboutfundamentalprinciples .Themost fundamentalprinciplein allbranches ofphysics isprobablythefollowing:

Preliminaries7

Thelawsof physicsare invariantunder thechoice ofsystems ofunits. Whatarelawsofph ysics?They arerelationsbetweenmeasurablequantities.F or example,aphysicalla wstatesthat theforceofattractionbetweentw ophysical bodies(whichcan bemeasured)is proportionaltothe massesofeach ofthem (whichcanbe measured),andin verselyproportional tothesquare oftheir separa- tion(whichcan bemeasuredindependently). Thefirstobserv ationtobe madeisthatifaph ysicallaw isofthe form expression1=expression2, thenbothe xpressionmustha vethesamedimension,otherwise theidentitycould notholdindependently onthesystem ofunits. Thedimensionis alwaysapo wer-lawmonomial Sofar ,alldimensionfunc- tionswere always power-lawmonomials.Isitacoincidence? Supposewework withintheLMT class,andwe areinterestedin somequantitya.Byassumption (thattheLMT classiscomplete), thedimensionof aonlydependson L,M,T: [a]=f(L,M,T). Whatdoesit mean?That ifwechange unitsbydi vidingtheunits oflength,mass andtimeby L 1 ,M 1 ,T 1 ,themeasured valuea 1 willincreaseby f(L 1 ,M 1 ,T 1 ). Similarly,ifwechange unitsbydi vidingtheunits oflength,mass andtimeby L 2 , M 2 ,T 2 ,themeasured valuea 2 willincreaseby f(L 2 ,M 2 ,T 2 ).Thatis, a 2 a 1 = f(L 2 ,M 2 ,T 2 ) f(L 1 ,M 1 ,T 1 ) . Theunderlyingassumption isthatall systemsofunits withinagi ven classare equivalent.Thus,wemaythinkofthe system1as theoriginalsystem, andsystem

2asobtained bydecreasingthe fundamentalunitsby L

2 /L 1 ,M 2 /M 1 ,andT 2 /T 1 . Then, a 2 =a 1 f(L 2 /L 1 ,M 2 /M 1 ,T 2 /T 1 ), fromwhichimmediately followsthe functionalequation f(L 2 ,M 2 ,T 2 ) f(L 1 ,M 1 ,T 1 ) =f ? L 2 L 1 , M 2 M 1 , T 2 T 1 ? .

8Chapter1

Whatcanbe learnedfromsuch afunctionalrelation?

Assumingthatthe dimensionfunctionis smooth,wedi fferentiatebothsides with respecttoL 2 andsetL 2 =L 1 =L,M 2 =M 1 =M,andT 2 =T 1 =T, 1 f(L,M,T) ∂f ∂L (L,M,T)= 1 L ∂f ∂L (1,1,1)≡ α L , fromwhichwe concludethat f(L,M,T)=L α g(M,T).

Substitutingintothe functionalequationwe get

g(M 2 ,T 2 ) g(M 1 ,T 1 ) =g ? M 2 M 1 , T 2 T 1 ? , andbythe sameprocedureget that g(M,T)=M β h(T).

Repeatingthisfor athird timeweget h(T)=cT

γ ,i.e., f(L,M,T)=cL α M β T γ .

Sincef(1,1,1)=1,weconclude thatc=1.

Quantitieswith independentdimensionsThephysical quantitiesa 1 ,...,a k aresaid tohav eindependentdimensionsifnoneof thesequantitiesha vea di- mensionwhichcan bepresentedas apower monomialofthe dimensionsof the remainingquantities.It isperhapseasier togiv ealog arithmicformulation. Inthe LMTsystem,for example,e veryset ofphysicalquantities(a i ),i=1,...,k,hasa dimensionfunctionsof theform, log[a 1 ]=α 1 logL+β 1 logM+γ 1 logT . . .= . . . log[a k ]=α k logL+β k logM+γ k logT Thatis,the logarithms ofdimensionsform avectorspacewithbasis vectorslog L, logM,logT.Independence ofdimensionis linearindependenceo verthis space.

Preliminaries9

Obviously,inthiscasetherecanbe atmostthree physicalquantities ofindepen- dentdimension.

Supposenow physicalquantitiesa

1 ,...,a k andb 1 ,...,b m ,suchthat thefirst k haveindependentdimensions,andallthemadditionalquantities have dimensions thatdependon thedimensionsof thefirstkquantities.Thef actthatthe a i have independentdimensionsimplies thatifwe work,say ,withan L 1 L 2 ...L r system, thennecessarily, r≥kandif ( ( ( ( ( ( ( ( ( ( ( log[a 1 ] . . . log[a k ] ) ) ) ) ) ) ) ) ) ) ) = ( ( ( ( ( ( ( ( ( ( ( γ 11

···γ

1r . . . . . . . . . γ k1

···γ

kr ) ) ) ) ) ) ) ) ) ) ) ( ( ( ( ( ( ( ( ( ( ( logL 1 . . . logL r ) ) ) ) ) ) ) ) ) ) ) , thenthematrix ofcoefficientshasfull rank(i.e.,k).Thishas animportantimpli- cation:itis possibletochange unitsofmeasurement suchthat,as aresult,a single a j changesits value,while alltheotherremainfixed.

Nowwecometothe b

j .Sincetheir dimensionsdependon thedimensionsof the a j ,then thereexist constantsα ij ,suchthat log[b 1 ]=α 11 log[a 1 ]+···+α 1k log[a k ] . . .= . . . log[b m ]=α m1 log[a 1 ]+···+α mk log[a k ].

Considernow thefollowingnewquantities,

Π 1 = b 1 a α 11 1 ...a α 1k k . . .= . . . Π m = b m a α m1 1 ...a α mk k . Itiseasy toseethat thesequantities aredimensionless;their valuedoes not change,no matterhow wechangeour unitsofmeasurement. PhysicallawsAphysical lawalwaysconsists ofarelationshipbetweenphysical quantities, c=f(a 1 ,...,a k ,b 1 ,...,b m ).(1.2)

10Chapter1

Herecisthequantity beingdetermined anda

1 ,...,a k ,b 1 ,...,b m areallthe quan- titiesitdepends upon(asabo ve,the a j haveindependentdimensions).Itisim- portantto stressthata physicalla wisan equationthatrelates betweenmeasurable quantities.Theree xistsafunction f,suchthat ifone measuresthequantities a j , b j ,andc,thenumbersthusobtainedsatisfy theprescribedequation. Wefirstclaimthatthe dimensionofcmustdependon thedimension ofthea j . Whyisitso? Otherwise,wecould performachange ofunitsof measurement, suchthatthe valueof cchanges,whilethe valuesof allthea j (andconsequently alltheb j )remainunchanged. Thisw ouldcontradictthe assumptionthatfisonly afunctionof thosephysical quantities.Thus,there existnumbers β 1 ,..., β k ,such that [c]=[a 1 ] β 1 ...[a k ] β k , andtheph ysicalquantity Π = c a β 1 1 ...a β k k isdimensionless. TheΠtheoremUsingourdefinitionsofdimensionlessquantities,wecanrewrite (1.2)asfollo ws: Π = 1 a β 1 1 ...a β k k f ? a 1 ,...,a k ,a α 11 1 ...a α 1k k Π 1 ,...,a α m1 1 ...a α mk k Π m ? , andthiscan bebroughtinto analternativ eform, Π =F(a 1 ,...,a k ,Π 1 ,...,Π m ). Thatis,we rewrote(1.2) asarelation betweenkphysicalquantitiesofindepen- dentdimensionand mdimensionlessquantities, ononeside, andadimensionless quantity,ontheotherside. Nowrememberwecan performachange ofunitssuch thatonlya 1 changesits value.Thus,Fcannotdependon a 1 .Byrepeating thisar gumentktimesweend upwiththe conclusionthat Π =F(Π 1 ,...,Π m ). Thisiskno wnasthe "Π-theorem".Aphysicalr elationshipbetween somedimen- sionalparameter andseveraldimensional governingpar ameterscanberewritten

Preliminaries11

asar elationbetweensome dimensionlessparameterandsever aldimensionless productsofthegoverning parameters. Whatisthe immediategain? Alaw thatseemedto dependonk+mparameters reducesintoa lawthat onlydependson mparameters.Thismight beahuge reduction. Example:InEuclidean geometryitis knownthat theareaSofaright triangle dependsonlyon thelengthof itshypotenuse, c,andthe magnitudeφof,say, the smallerofits acuteangles.Thus,

S=f(c,φ).

Forthisproblemwe candowell withforonly fundamentalunitsthose oflength (anLsystem).In thiscase,chasdimensionL,φisdimensionlessand Shas dimensionL 2 .Bythe Π-theorem,thedimensionless quantityΠ=S/c 2 canonly dependonthe dimensionlessparameterΠ 1 =φ, Π =F(Π 1 )?S=c 2

F(φ).

Withoutcaringwhatthefunction Factuallyis,we hav eatonce, S 1 =a 2

F(φ),S

2 =b 2

F(φ),S

1 +S 2 =c 2

F(φ),

fromwhichfollo wsPythagoras' theorem. c b a ! ! S 1 S 2 ?

12Chapter1

Example:Amass misthrown verticallywithvelocityv.Whatis themaximal heighthitwill attain?We startwith h=f(m,v,g).

Since[ h]=L,[m]=M,[v]=L/Tand[g]=L/T

2 ,weconclude fromthe

Π-theoremthat

Π = h ? v 2 /g =const. ? ?Exercise1.3 Apendulum isamassi veparticle suspendedona string(which weassumeto have negligible mass).Ifoneperturbsthependulum(forexample byplacingit attheequilibrium point,but assigningitan initialvelocity), thenthe pendulumperformsa periodicmotion.Suppose wewant tofinda law topredict theperiod,τ,ofthe motion.We startbymaking alistof allquantitiesthatτmay dependupon:(1) themass mofofthe particle,(2)the initialvelocity v,(3)earth' s accelerationg,and(4) thelengthof thestring,?.Inthe LMTsystem wehav e [τ]=T[m]=M[v]= L T ? g ? = L T 2 [?]=L. ?Whatcanyou sayaboutthe function

τ=f(m,v,g,?)

basedondimensional analysis? ?Supposenow anexperimentalisttellsyouthat theperioddoes notdepend ontheinitial velocity. Howwill youranswerchange? Example:In anatomice xplosion,alar geamountof energyEisreleasedwithin asmallre gion,anda strongsphericalshockwav edev elopsat thepointof detona- tion.Inthe earlystages, thepressurebehind theshockw ave ishugecompared to theatmosphericpressure, whichiscompletely negligible.If rdenotestheradius ofthew ave andtistime,the radiusisonly expectedto dependon r=f(t,E,ρ), whereρisthedensity oftheair atequilibrium(no shockwa vewithout airdensity , andothercharacteristics ofair, suchassound wav espeedareirrelev antforshock waves).

Preliminaries13

Thedimensionsof thegov erningparametersin theLMTclass are [t]=T[E]=ML 2 T -2 ? ρ ? =ML -3 . Itiseasy toseethat thoseareall independent,i.e.,k=3andm=0.The dimensionofrisL,sothat Π = ρ 1/5 r E 1/5 t 2/5 isdimensionless,and dependson nothing,i.e.,must beaconstant. Thismeans that r=Cρ -1/5 E 1/5 t 2/5 , or logr=logC- 1 5 logρ+ 1 5 logE+ 2 5 logt. Thus,ifwe plotlog rversuslogt,theslope shouldbe2 /5,andthe graphshould intersectzeroat thev alue logC- 1 5 logρ+ 1 5 logE. Sinceρiskno wn,knowingtheconstantCwouldrevealthe energyoftheblast. Inthe1940' stheAmericans performednucleartests,whichwerephotographed byJ.Mack. Thosephotographswere notclassified.G.I. Tayloranalyzed those pictures.He knewthat theconstantCwasaboutO(1)(onecan finditfrom asmall explosion),andbasedonthe above analysispublishedthe valueof theenergy (about10 21
ergs)whichcausedahuge embarrassment.?

14Chapter1

Chapter2

NewtonianMechanics

Inthischapter wewillre viewthe basicsofNe wton's"old" classicalmechanics - oldinthe sensethatit islessgeneral andformalthan themorerecent formulations ofLagrangeand Hamilton - classical,todistinguish fromquantummechanics.

2.1Kinematics

Mechanicsprimarilydeals withthemotion ofso-calledpointparticles.Apoint particleisa modelofaph ysicalobjectwhose stateisfullyspecifiedbyits loca- tion,thelatter beingapoint inR 3 .Itis importanttodistinguish betweenmodels andreality. Thepointparticleisanidealization, whichcanonly bejustifiedempir - ically.Attimes,thesolar systemcanbe describedascomprised ofpointparticles, eventhoughthesunandtheplanets arecomplex objects.Inother instances,ev en aroundtennis ballcannotbe modeledas apointparticle (e.g.,ifrotation isim- portant). Physicsmakepredictionsabout howphysicalentitiese volve intime.Gi vena pointparticle,its ev olutioniscompletely determinedbyitspositionr=(x,y,z), asfunctionof timet,orits trajectory.Mathematically, thetrajectoryr(t)isa pathinR 3 .For thetimebeing,wewillassume thatthetrajectory isatleast twice differentiablewith respecttotime; thejustificationfor thisassumptionwill be giveninthenextsection. Notethatthe definitionofrassumesthechoice ofanorigin, withaspecific ori- entationofthe principalaxes. Inorderto beableto measurerwealso needaunit

16Chapter2

oflength.Centimeters andmetersare themostcommon choices(forthose who haveneverheardofinches andfeet...).Timealsorequiresanoriginand aunitof measurement,seconds beingthemost standardchoice. Therate-of-changeof theposition r(t),isthe vector v(t)= dr dt (t), whichiscalled thevelocityvectorofthepoint particle.Notethat thisisa vector identity,namely,v=(v 1 ,v 2 ,v 3 ),with v 1 = dx dt ,v 2 = dy dt ,andv 3 = dz dt . Themeasurementof velocityis somewhat"conceptual". Weimaginetwomea- surementsofposition attimetandtimet+Δt,andan approximationtov elocity intermsof thechange inrdividedbyΔt.Theactual velocity isthelimit whereΔt isinfinitelysmall. Despitethe factthat theexact velocitycan neverbemeasured, thisconceptualmeasurement sufficestoassign velocitydimensions ofL/T,where Listhe dimensionoflength andTthedimensionof time.Ifwe measurelength inunitsof metersandtime inunitsof seconds,thenthe ratioofΔr/Δtisassigned aunitof meters/seconds.Thisunit hastobe understoodinthe followingw ay: thevelocity wasdeterminedbya ratioofdisplacementandtimeinterv al,inan experimentsinwhichlengthw asmeasuredin metersandtime inseconds. Notethatif weknow thevelocity asfunctionof time,v(t),thenthe positioncan beretriev edbyintegration,butonlyupto aconstanttranslation, r(t)=r(t 0 )+ ? t t 0 v(s)ds. Ifthev elocityis constant,v(t)=v,thenthe displacementis alinearfunction of time, r(t)=r(t 0 )+v(t-t 0 ). Thetimederi vati veofthevelocity,whichisthesecondderivativ eofthe trajectory, iscalled theaccelerationofthepoint particle,andis denotedby a(t)= dv dt (t)= d 2 r dt 2 (t).

NewtonianMechanics17

Accelerationhasdimensions ofL/T

2 ,andis usuallymeasuredin unitsof m/sec 2 (again,wedon'treally meanthata meterisdividedbythe squareofa second).

Giventheacceleration,a(t),the velocityis givenby

v(t)=v(t 0 )+ ? t t 0 a(s)ds, andthe positionisgi venby r(t)=r(t 0 )+v(t 0 )(t-t 0 )+ ? t t 0 ? s t 0 a(τ)dτds.

Whenthe accelerationisconstant a(t)=a,then

v(t)=v(t 0 )+a(t-t 0 ), fromwhichwe obtainthetrajectory , r(t)=r(t 0 )+v(t 0 )(t-t 0 )+ 1 2 a(t-t 0 ) 2 , whichrequiresthe knowledgeof theinitialposition andvelocityofthepoint par- ticle. Example:Itis anempiricalfactthatbodiesexperiencingonlytheforcesofgravity haveaconstantacceleration, a=(0,0,-9.81) m sec 2 regardlessoftheirmass.Hereitis assumedthatthe z-axispointsupw ard,whereas thex,yplaneistangent tothesurf aceofearth. Supposeweare inaflying balloon, andwethro wamassi vebodyfromthepoint (0,0,100)mwithinitialv elocity (3,0,10)m/sec.Whenand wherewillthe bodyhitthe ground?

Bytheabo veformula, settingt

0 =0, r(t)=(0,0,100)+(3,0,10)t- 1 2 (0,0,9.81)t 2 .

Componentwise,we have

x(t)=3t,y(t)=0,andz(t)=100+10t- 9.81 2 t 2 .

18Chapter2

"When"isanswered bysolvingthe quadraticequationz(t)=0(theresult is measuredinseconds). "Where"isanswered byr(t)atthe timeof impact(the resultismeasured inmeters).Note alsothatthe curvethat thetrajectory makesis aparabola, z=100+10 x 3 - 9.81 2 ? x 3 ? 2 . ? CircularmotionConsideracircular trajectoryatconstant speed, r(t)=(Rcosωt,Rsinωt,0),R,ω>0.

Thev elocityisgivenby

v(t)=(-Rωsinωt,Rωcosωt,0), i.e.,|v(t)|=Rω,whereas theaccelerationis a(t)=(-Rω 2 cosωt,-Rω 2 sinωt,0)=-ω 2 r(t), i.e.,|a(t)|=R 2 ω.We tookaveryparticularcircular motion,inwhich thespeed|v| isconstant.The accelerationvector isdirectedto wardthe centerofthecircle,and inparticularis perpendiculartothe velocityv ector. Thisismore general:foran ymotion where|v(t)|isconstant, 0= d|v| 2 dt =2v· dv dt , i.e.,v?a.

2.2Dynamics: Newton'slaws

Sofar ,wehaveonlyseendefinitions,not physicalla ws.Physicallawsarenot inferredfrom asetof axioms,but areratherbased onempiricalobserv ations. Theirvalue isintheabilitytocapture awholeset ofphysical phenomenaina relativelysmallsetoflaws. Recallthata physicalla wisan equationthatrelatesbetweenphysical quantities, eachofwhich canbemeasured independently.A physicalbody ,fore xample,has

NewtonianMechanics19

Figure2.1: SirIsaacNe wton(1643-1727).

aphysical quantityassociatedwith - amass - orthe"amount ofmatter".Mass iscommonlymeasured inunits ofgramsor kilograms(unlessyou buyit bythe ounce),andis denotedbym.Thedimension ofmassis independentofthat of lengthandtime, andwedenote itbyM. Theproductof aparticle' smassand velocity, whichwasoncecalled"the amount ofmotion"is calledthemomentumoftheparticle, andisdenoted by p=mv.

Momentumisa vectorthat hasdimensionsof ML/T.

Newton,inhiswork PhilosophiaeNaturalis PrincipiaMathematica(1687),for- mulatedwhatis nowkno wnasNe wton'sfirstandsecondla wsofmotion:there existframesofr eferences inwhichparticles have fixedmomenta, unlessthere isforceexertedonthem.Whenobservedfrom suchareference frame,therate ofchangeof aparticle's momentumisproportional tothe(v ector)sumofallthe forcesactingon it.Ifthe totalforceis denotedbyf,thenthe firstandsecond laws statethat f? dp dt . Choosingthe constantofproportionality amountstochoosing unitsofmeasure- ments.Theunit offorceNe wtonis definedasthe amountofforce neededto changethe momentumofa particleata rateofone kgm/secpersecond, inwhich caseweha vean equality.Clearly,forcehasdimensionsofML/T 2 .Ifthe massof

20Chapter2

theparticledoes notchangein time,thenthis lawreduces tothecelebrated f=ma. Notethatforce, asthederi vativ eofa vectorquantity isitselfavector. Butwhatisforce?Even Newtonrealizedthathecould onlygiveacircular defini- tion.Force isthethingthatcausesmomentum tochange,and momentumchanges asaresult ofactingforces. (Ontheother hand,wecertainly have fromdailylife anintuitiv enotionofwhatforceis.)EquippedwithNe wton'ssecond law ,the modellingofa mechanicalsystem consistsofspecifying whattheforces are. Example:F orapointparticleofmass mkgunderonly theinfluence ofearth gravity,theforce(inNewtons)is f=-9.81me 3 ≡-mge 3 .

Inthiscase, Newton's secondlaw reducesto

ma=-mge 3 , i.e.,a=-ge 3 ,i.e.,a constantaccelerationindependent ofthe mass.? Example:Considera massmmotionlessona table.Sincethe bodydoesnot accelerate,wemust concludethatthe totalforceacting onitzero. Sinceearth' s gravityexertsa constantforce-mge 3 onit,there hastobe anotherforcecoun- teringit.It isthecontact withthetable thatex ertsthiscountering force,+mge 3 . ? Inman ycases,theforceona pointparticleis afunctionof itsposition,f=f(r), wheref:R 3 →R 3 isavectorfield.Inthis case,Newton' ssecondla wisan equationofthe form, m d 2 r dt 2 =f(r), i.e.,asecond-orderordinarydi fferentialsystem.Provided thatfissufficiently regular,thisequationhasaunique twice-differentiablesolution given theinitial datar(t 0 )andv(t 0 ).

Example:

NewtonianMechanics21

Consideran harmonicoscillator :amass msuspendedbya spring.Thespring hasarest length?;wefix theoriginat avertical distance?fromthetop ofthe spring.Assumingthat weonlyconsider verticalmotion, thetotalforce actingon themassis f=(-mg-kz)e 3 , wherezisthev erticaldistance fromtheorigin,andkisaconstant associatedwith thespring.Thus, thetrajectoryof themasssatisfies thesecondorder differential equation, m d 2 z dt 2 =-mg-kz, withinitialdata z(0)=z 0 anddz/dt(0)=v 0 .Thisis aninhomogeneouslinear equationwith constantcoefficients.To simplifythesolutionitisw orthtochange variables.Wefirstre writetheequationas d 2 z dt 2 =- k m ? z+ mg k ? . Wethensetw=z+mg/k,whichsatisfies thehomogeneousdi fferentialequation, d 2 w dt 2 =- k m w, withinitial dataw(0)=z 0 +mg/k,anddw/dt(0)=v 0 .Thesolution is w(t)=w(0)cos ωt+ v 0 ω sinωt, whereω 2 =k/m.Thatis, amasssuspended byaspring willperforma periodic sinusoidalmotion aboutthepoint z=-mg/k.?

22Chapter2

Example:Finally, asanexamplewhichwe arenotgoing tosolve now,wecon- siderthemotion ofaplanet arounda massive star.The massofthe planetism andthemass ofthestar isM.For thetimebeing,weassumethat thestar does notmov e(whichisapproximatelytrueifM?m).Newton' slawofgra vitation statesthatthe forceactingon theplanetis directedtow ardthe centerofthe star, andisgi ven bythewell-knowninverse-squarelaw, f(r)=-GMm r |r| 3 , wheretheorigin isatthe centerofthe star.Gistheuni versalconstant ofgrav- itation.Ifmasses aremeasuredin kilograms,lengthsin metersandforces in

Newtons,then

G=6.67.×10

-11 Nm 2 kg 2 .

Theresultingdi fferentialequation

m d 2 r dt 2 =-GMm r |r| 3 issolvable byanalyticalmeans,butwe have goodreasonsto deferits solutionto alaterstage. ? Comment:Howdoestheinv erse-squarelaw reconcilewiththe constantaccelera- tionon earth'ssurf ace?IfRisearth's radius(6370kilometers),and|r|-R?R, thenwecan expandthe forceinT aylorseries, f(r)=-GMm r-Rr/|r|+Rr/|r| |r| 3 =- GMm R 2 ?? 1- R |r| ? r/R |r/R| 3 + r/|r| |r/R| 3 ? . Byassumption,|r|/R-1?1,sothat uptocorrections ofthatorder ,theforce is wellapproximatedby f(r)≈- GMm R 2 r |r| .

Substituting

G=6.67.×10

-11 Nm 2 kg 2

M=5.97×10

24
kgR=6.37×10 6 m, weget f(r)≈-

6.67×5.97

6.37 2

×10

1 m r |r| =-9.81m r |r| .

NewtonianMechanics23

2.3Work andmechanicalenergy

Consideraph ysicalbodymo vingundertheinfluenceofa forcef(t)(which may beduea forcefieldf(r).Suppose thatthebody performsbetweentime t 1 andt 2 a trajectoryr(t).W edefinetheworkexertedonthebodyas 1 W= ? r f·dr= ? t 2 t 1 f(s)·dr(s)= ? t 2 t 1 f(s)·v(s)ds. Thatis,w orkis thepathintegraloftheforce alongthetrajectory .Work maybe eitherpositiv eornegative,wherein thelattercase wewillsaythatitisthebody thathasperformed work. Inmany cases,theforcefieldf(r)actingon aph ysicalbodyis conservative.We remindourselves thatavectorfieldis calledconservati veif integralsalongpaths onlydependon theend pointsandnot onthetrajectory .Moreov er,a vectorfield isconservati veifandonlyifitisthegradientofa scalarfield, f=-?Φ, whereΦ(r)is calledthepotentialfield,orthe potentialenergy ;thene gativ e signisa conv entionwherebythe forceisdirectedinthedirectionofdecreasing potential.For atrajectoryrconnectingthepoints r 1 andr 2 , ? r f·dr=-Φ(r 2 )+Φ(r 1 ). Comment:Recallthatthe potentialisdetermined onlyupto anadditiv econstant.

Workisindependentofthis constant.

Example:For amassundertheinfluenceof earthgravity

Φ(r)=mgz.

Foramasssuspended onaspring,

Φ(r)=

1 2 kz 2 . 1 Notethegeneralization comparedwiththe pathintegra lstudiedin advancedcalculus, the latteralw aysbeinganintegralofavector field.

24Chapter2

Foraplanetunderthe influenceofa star'sgra vity,

Φ(r)=-

GMm |r| . ? Considerno waparticleofmassmundertheinfluence ofaconserv ative force field.Ifthe particleperformsa trajectoryr(t)connectingthe pointsr(t 1 )andr(t 2 ) then -Φ(r(t 2 ))+Φ(r(t 1 ))= ? r f·dr=m ? r a·dr=m ? t 2 t 1 a(s)·v(s)ds, wherewe have usedthedefinitionofthepathintegral.Noting that a(t)·v(t)= 1 2 d dt |v(t)| 2 , weobtain -Φ(r(t 2 ))+Φ(r(t 1 ))=T(v(t 2 ))-T(v(t 1 )), where T(v)= m|v| 2 2 iscalledthe kineticenergy oftheparticle. Definingthetotalmechanicalener gy, orsimplythe energy,

E(r,v)=Φ(r)+T(v),

weconclude thatitassumes thesamev alueattime t 1 andt 2 ,i.e.,it isconserved (itisa functionofthe trajectorywhosev alueremainsconstant intime).Another waytoseeitis viathechain rule, d dt

E(r(t),v(t))=

d dt ?

Φ(r(t))+

m 2 |v(t)| 2 ? =v(t)·[?Φ(r(t))+ma(t)]=0. Theconservation oftotalmechanicalenergywhen forcesareconserv ative isuse- fulassho wsthefollo wingexample: Example:Abody isdropped (atrest)from aheightof hmeters.Atwhat speed willithit theground?

NewtonianMechanics25

Oneway tosolveitisvia theequationsof motion:since

d 2 z dt 2 =-g, withinitial dataz(0)=handdz/dt(0)=0,thesolution is z(t)=h- 1 2 gt 2 .

Thusithits thegroundafter timet

1 = ?

2h/g,i.e.,z(t

1 )=0,andits (vertical) velocitythenis v(t 1 )=-gt 1 =- ? 2hg. Theother wayof solvingthisexerciseis withenergies. TakingΦ(z)=mgz,the conservationofenergyimplies that

Φ(z(0))+

mv 2 (0) 2 =

Φ(z(t

1 ))+ mv 2 (t 1 ) 2 , i.e., mgh+0=0+ mv 2 (t 1 ) 2 , whichgi vestheexactsameanswer.? Example:Earthdoes notperforman ywork onthemoon becausethetrajectory of themoonis perpendiculartothe vectorthat connectsthemoon tothe earth,i.e.,it isperpendicularto theforce ofgravity (well,atleast approximately).? ?Exercise2.1 Showthatfora singleparticlewith constantmassm,Newton' s equationsimplythe followingequation forthekinetic energy, dT dt =f·v, andmoregenerally ,ifthe masscanvary,then d(mT) dt =f·p.

26Chapter2

?Exercise2.2 Theescape velocityof aparticleonearthisthe minimumveloc- ityrequiredin orderfor aparticleto escapefromearth' sgravitational field.Use theconservation ofenergytocalculatethe escapevelocity .Couldyouobtainthis resultbydirectly solvingNewton' sequations,which formotionalong theradial directiontake theform, m d 2 r dt 2 =- GMm r 2 , withinitialdata r(0)=Randv(0)=v 0 ,whereRistheradius ofearth?

2.4Angularmomentum andtorque

Foraparticleat positionrhavingmomentump,wedefine itsangularmomen- tumas

L=r×p.

Theimportanceof thisquantity willbeseen lateron.Note thatunlike themo- mentum,theangular momentumdependson theabsoluteposition oftheparticle. Therateof changeofthe angularmomentumcan beinferredfrom Newton's sec- ondlaw ,as dL dt =v×p+r× dp dt =r×f, whereweused thefact thatv×p=m(v×v)=0andNe wton'ssecond law.The quantityonthe righthand sideiscalled thetorque(ormomentof force),andis denotedby

τ=r×f.

Thus,the"angular" versionof Newton's secondlawis

dL dt =τ, i.e.,thetorque equalstothe rateofchange oftheangular momentum 2 . 2 Uptothe dependenceofmass, theangularmomentum isrelatedto thearealvelocity,which istherate atwhicharea issweptout bytheparticle.

NewtonianMechanics27

2.5Systemsof pointparticles

Untilnow,westudiedaworldcomprisingof asinglepointmass.Wenowconsider asystemcomprising ofnparticlesofmass m 1 ,...,m n .We denotethetrajectory ofthei-thparticleby r i (t),itsv elocitybyv i (t),andso on 3 .Also, weuserandv asshort-handnotation forthe collections{r 1 ,...,r n }and{v 1 ,...,v n }. Newton'ssecondlawstatesnow thatev eryparticlesatisfiestheequationof mo- tion 4 , m i a i = dp i dt =f i , wheref i isthetotal forceex ertedonthe i-thparticle. Onceagain, thisdoesnot saymuch unlesstheforces f i arespecified.In many cases,theforce actingonthe i-thparticlecan bewrittenas asum, f i =f e i + ? j?i f ji , wheref e i standsforan externalforceandf ji isthe"internal" forcethat thej-th particlee xertsonthei-thparticle.The implicitassumptionis thatthee xternal forcef e i onlydependson theposition(and perhapsvelocity) ofthei-thparticle, whereasthe internalforcef ji onlydepends onthepositions (andperhapsthe ve- locities)ofthe i-thandj-thparticles.Ne wton'sthird law,alsoknownasthela w ofactionandr eactionstatesthatthe forcethatthe i-thparticlee xerts onthej-th particleis equalinmagnitude butopposite insignto theforcethat thej-thparticle exertsonthei-thparticle f ji =-f ij . Thedistinctionbetween internalande xternalforcesis inasense arbitrary.It is one'schoicewhichpartsof theuniv erseareincluded inthemodel, andwhich are not.Externalforces representthe interactionsbetweenthe systemandparticles thathav ebeenleftoutofourmodel.Ane xternalforcecould becomeaninternal forcebye xtendingthemodel (e.g.,gravitycanbevie wedasan external force,but alsoasan internalforce ifweadd earthtoour system). 3

Sinceweuse thenotation r=(x,y,z),wealso user

i =(x i ,y i ,z i ).Onthe otherhand,we will usev i =(v 1i ,v 2i ,v 3i ). 4 Herethereis nosummation over i.Throughoutthis section,summationswill bedenoted explicitly.

28Chapter2

Figure2.2:Pierre SimonLaplace (1749-1827).

Notethatonce thefunctionsf

e i (r i ,v i )andf ij (r i ,r j ,v i ,v j )arespecified, weare leftwitha systemof3 nsecond-order???s,or, equivalently,asystem of6nfirst- order???s.Theirsolution isuniquelydetermined byinitial dataforthe positions andvelocities ofthenparticles.Itw astheFrench mathematicianPierre-Simon Laplacewhopointed outthataccording toNewton' slaws thepresent uniquely determinedboththe pastandthe future:"Wemayregar dthepr esentstateofthe universeastheeffectofits pastandthe causeofits future. Anintellectwhic h atacertain momentwouldknow allforces thatsetnatur einmotion, andall positionsofall itemsofwhic hnature iscomposed,if thisintellectwer ealsovast enoughtosubmit thesedata toanalysis,it wouldembrace inasingle formulathe movementsofthegreatest bodiesofthe universeand thoseofthetiniestatom; forsuch anintellectnothingwouldbeuncertain andthefutur ejustlik ethe past wouldbepr esentbefore itseyes". ConservationofmomentumWeturnnowto definitionsthatare pertinentto systemsofparticles. Thecenterofmass ofa systemofparticles isamean posi- tionweightedby themass, R= ? n i=1 m i r i ? n i=1 m i . Thecorrespondingv elocityv ector(center-of-massvelocity)is V= dR dt = ? n i=1 m i v i ? n i=1 m i .

NewtonianMechanics29

IfM= ? n i=1 m i denotesthe totalmass,then thecorrespondingmomentum, P=MV= n ? i=1 m i v i = n ? i=1 p i , issimplythe (vector) sumofmomenta. Therateofchangeofthe totalmomentum is dP dt = n ? i=1 dp i dt = n ? i=1 ( ( ( ( ( ( ( ( f e i + ? j?i f ji ) ) ) ) ) ) ) ) = n ? i=1 f e i , wherethe binaryinteractionscancel bythela wofaction andreaction.This amountstothe lawof conservationof momentum: Intheabsence ofexternal forcethe totalmomentumof the systemis conserved. ConservationofangularmomentumAnanalogousconserv ationla wcanbe derivedfortheangularmomentum.W edefinethe totalangularmomentum ofthe systemby n ? i=1 r i ×p i . Now, d dt n ? i=1 r i ×p i = n ? i=1 r i × ( ( ( ( ( ( ( ( f e i + ? j?i f ji ) ) ) ) ) ) ) ) .

Considertheterms

? i,j?i r i ×f ji . Sincewemay interchangethesymbols iandjandusethe lawof actionand reaction,thisequals 1 2 ? i,j?i (r i -r j )×f ji . Thestronglawofaction andreaction statesthatthe forcethat aparticlee xerts onanotheris directedalongthe linejoiningthe twoparticles, i.e.,r i -r j ?f ji .In suchcase wehav e d dt n ? i=1 r i ×p i = n ? i=1 r i ×f e i .

30Chapter2

Theright-handside isthetotal externaltorque. Thisleadsto thefollowing con- servationlaw, Intheabsence ofexternal toquethetotal angularmomentum ofthesystem isconserved. ConservationofmechanicalenergyThetotalw orkdoneby allforcesis W= n ? i=1 ? r i f i

·dr

i = n ? i=1 ? r i f e i

·dr

i + n ? i=1 ? j?i ? r i f ji

·dr

i .

Ontheother hand,

n ? i=1 ? r i f i

·dr

i = n ? i=1 ? t 2 t 1 m i a i ·v i dt=T(v(t 2 ))-T(v(t 1 )), where T(v)= 1 2 n ? i=1 m i |v i | 2 isthetotalkineticener gy.Thatis, thetotalw orkdoneby alltheforces equalsto theg ainintotalkineticenergy ,

W=T(v(t

2 ))-T(v(t 1 )). ConservativeexternalforcesConsidernow thecasewheretheexternal forces areconservati ve,i.e., f e i (r i )=-?Φ e i (r i ), whereΦ e i :R 3 →R.Then, n ? i=1 ? r i f e i

·dr

i =- n ? i=1 Φ e i (r i (t 2 ))+ n ? i=1 Φ e i (r i (t 1 )). ConservativeinternalforcesNext,considerthecase wheretheinternal (bi- nary)forcesare conservativ e.Morespecifically ,weconsiderthecasewherethe interactionbetweenthe i-thandj-thparticle isderiv edfroma potentialΦ ij = Φ ij (r i ,r j ),and f ji =-? 1 Φ ij (r i ,r j )f ij =-? 2 Φ ij (r i ,r j ),

NewtonianMechanics31

where? 1 standsforthe gradientwith respecttothe firstargument ofΦ.Bythe lawofactionandreaction f ij =-f ji ,thatis, ? 1 Φ ij (r i ,r j )=-? 2 Φ ij (r i ,r j ). Whatarethe implications?Whatcan wesayabout afunctionφ(x,y),satisfying ∂φ / ∂x=-∂φ/∂y?For everyx,ywehav ebythechainrule d ds

φ(x+s,y+s)=

∂φ ∂x (x+s,y+s)+ ∂φ ∂y (x+s,y+s)=0, fromwhichwe concludethatφisafunction ofx-y 5 .Similarly, inthevector case, Φ ij = Φ ij (r i -r j ), thatis, thepotentialonly dependsonthe separationbetweenthe twoparticles. ?Exercise2.3 Verifythatthela wofaction andreaction(Ne wton'sthirdlaw) impliesthatinternal potentialsareof theform Φ ij (r i ,r j )=g(r i -r j ). Wealsohav ethestronglawofactionandreaction, whichstatesthat ?Φ ij (r)=g(r)r, whereg:R 3 →Risascalar function.Ifthe gradientof afunctionis alongr, thenthe functionisnecessarily constantonspheres centeredatthe origin,i.e. 6 , Φ ij = Φ ij (|r i -r j |). Example:Neutralatoms andmoleculese xerton eachothertw omain forces:an attractiveforceatlargeseparations,due toso-calledv anderW aalsforces,anda repulsiveforceatshortseparations,due toquantume ffects.Asimple mathemati- calmodelthat representsthisbeha vioristhe Lennard-Jonespotential,

Φ(r

1 ,r 2 )=C ? ? ? ? ? ? ? ? σ |r 2 -r 1 | ? 12 - ? σ |r 2 -r 1 | ? 6 ? ? ? ? ? ? ? ,

32Chapter2

Figure2.3:The Lennard-Jonespotential.

whereC,σareconstantspertinent tothespecific molecules. ? Backtoener gyconservation. Thecontributionoftheinternal forcestothetotal workis n ? i=1 ? j?i ? r i f ji

·dr

i =- n ? i=1 ? j?i ? t 2 t 1 ? i Φ ij (|r i (t)-r j (t)|)·v i (t)dt =- ? pairsi,j ? t 2 t 1 ? i Φ ij (|r i (t)-r j (t)|)·(v i (t)-v j (t))dt.

Notingthat

? i Φ ij (|r i (t)-r j (t)|)·(v i (t)-v j (t))= d dt Φ ij (|r i (t)-r j (t)|), weconcludethat thattotal workis given by

W=-Φ(r(t

2 ))+Φ(r(t 1 )), 5 Sets=0ands=-ytogetthat φ(x,y)=φ(x-y,0)≡ψ(x-y). 6

Whatweha ve shownisthatthefunctionΦ

ij (r i ,r j ),which isafunction R 3 ×R 3 →R,can be representedas Φ ij (r i ,r j )=g ij (|r i -r j |), whereg ij :R→R.Byan abuse ofnotationwe denotethisg ij byΦ ij aswell.

NewtonianMechanics33

whereΦ(r)isthe totalpotentialener gy,giv enby

Φ(r)=

n ? i=1 Φ e i (r i )+ ? pairsi,j Φ ij (|r i -r j |). Sincethe totalwork equalstothe gainintotalkineticener gy,we obtainoncemore theconservation oftotalmechanicalenergy, E(r,v)=Φ(r)+T(v).

2.6Whatis missing?

Atthispoint,onemayfeelthatweha veacompletetheoryofmechanics.Asystem isacollection ofpointmasses; ifwe knowall theinternaland externalforces as functionofthe stateofthe system,thenall weneedis initialdata,and solvea set of???s, m i d 2 r i dt 2 =f e i (r i )+ ? j?i f ji (r i ,r j ). Thisisindeed thecasefor ,say, stellarmechanics,where f e i =0andthe interaction forcesarederi vablefrom apotential, Φ ij (r i ,r j )=- Gm i m j |r i -r j | , namely, f ji (r i ,r j )=- Gm i m j |r i -r j | 3 (r i -r j ). ?Exercise2.4Readaboutthe 3-bodyproblem (searchtheweb; muchofthat isassociated withPoincar´e). ConstraintsOnethingmissing inthe currentpictureare constraints.To clarify theissuewe considersev eralexamples: Example:Thefirst exampleis abodysliding downaslope,asdepicted inFig- ure2.4.By anappropriatechoice ofaxes, themotioncan berestrictedto the xz-plane,sothat r=(x,z)will beviewed astwo dimensional.Ifwetrytowrite theequationsof motion,weimmediately facea problem:earth's gravitation ex- ertsonthe bodyaforce -mge 2 ,but whataboutthecontactwith theslope?The

34Chapter2

Figure2.4:A bodysliding downa slope.

forceofcontact isnot a-priorigiv en;itacts suchtoimpose theconstraintthat the bodyremainsabo vethe plane. Welearnhowto solvesuch problemsalreadyinhigh-school.We assumethatthe contactforcesare always normaltothe inclinedplane.Wethenrotatetheax es byanangle θ,anddeclare thattheforces normaltothe planecanceleach other, whereasaforce componentofmgsinθtangenttothe planecauses anacceleration ofgsinθinthisdirection. Thedisturbingpoints arethead-hoc treatmentandthe extra-assumptionsthat one hastoincorporate. ? Example:Thesecond exampleis thatofa pendulum,showninFigure2.5. Abead issuspendedon arigid(massless) rodwhoseupper endisfix ed;thebead isfree to performcircularmotion. Herethe roleofthe rodisto imposetheconstraint that thetrajectoryremainsonacircle.RatherthanworkingwithCartesiancoordinates, itmakes sensetoparametrizethepositionof thebeadby asingleangle, say,θ. Thequestion iswhatequation θ(t)satisfies.Once again, theforcethat therod exertsonthebeadisnota-priori known.This isanothere xampleofa problem thathigh-schoolstudents cansolve, butne vertheless,lea vesuswitha feelingthat mechanicsis anincompletetheory . ? Wenowdiscussconstraints inageneralsetting.Constraintscan beofv arious type.Oneof themiswhere thepositionsr={r 1 ,...,r n }ofthev ariousparticles satisfyequationsof theform h(r,t)=0.(2.1) whereh:R 3n+1 →R k .Ane xampleof suchconstraintsarerigidbodies,where

NewtonianMechanics35

Figure2.5: Apendulum.

thedistancesbetween particlesarefix ed,thatis, foralli,j, |r i -r j | -d ij =0. Constraintsofthe type(2.1)are calledholonomic.Constraintsthat arenotof thatformare callednon-holonomic.Asimple exampleof anon-holonomiccon- straint,isa particleconstrainedto remaininsidethe unitsphere,i.e., |r|≤1. Thereisno systematicway todeal withnon-holonomicconstraints (whichisto saythatthe systemis notuniquelydefined bythekno wnforcesand thenon- holonomicconstraints).In thiscourse,we willonlydeal withholonomicones. Asdiscussed above, constraintsinducetwosortsofdifficulties:(i)the coordinates arenolonger independent,hencethe equationsofmotion have tobe compatible withtheconstraints; (ii)theforces thatmaintainthe constraintsarea-priori un- known.Theyareonly characterizedthroughtheireffect,whichis toimposethe constraints. "Generalizedcoordinates" Asystemof nparticleswithoutconstraints lives in a3n-dimensionalEuclidean space.Thepresence of,say, kholonomicconstraints oftheform h i (r,t)=0,i=1,...,k,

36Chapter2

wheretheh i aresufficiently"nice"functions, restrictsthe motionintoa (3n-k)- dimensionalmanifold,M,which mayitselfe volve intime(a manifoldisaset thatlocallylooks like aEuclideanspace). Thus,itispossible,atleastlocally ,to reparametrizetheconstrained spaceM,ate verygi ventimet,bya setof3 n-k coordinates,which wedenoteby q=(q 1 ,...,q 3n-k )?U?R 3n-k , suchthatthere exists aone-to-onemapping ?:U→M?R 3n .Mathematicians wouldcallthepair (U,?)localcoordinatesforthemanifold M.Physicist callthe coordinatesqgeneralizedcoordinates(generalized,todistinguish fromthecoor - dinatesinthe Euclideanspace).Thus, thefirstdi fficultyofdependent coordinates canbeo vercomeby theuseofgeneralizedcoordinates. Comment:Fromaph ysicist'spoint ofviewtheq'saregeneralizedcoordinatesin thesensethat they neednotha vedimensionsoflength. Example:Considera systemoftw opointparticles atbothends ofarigid rodof length?.The constrainedmanifoldis M= ? (r 1 ,r 2 )?R 6 :|r 1 -r 2 |=? ? .

Thismanifoldis homeomorphictoR

3 ×S 2 ,andwe maychoose asgeneralized coordinatesthepositionofthecenterofmassandthesphericalanglesofthevector r 2 -r 1 .? Theseconddi fficultyhowe verremains:theforcesthatrestrictthemotiontoMare unknown.Thegoalisto obtainane wformulationof mechanicsforconstained systemsthatwill notrequirethe knowledgeof thoseforces.I.e., thatthe sys- temwillbe completelydeterminedby thespecificationof the"applied"forces, alongwitha listofconstraints. Thiscannotbe done,obviously ,withoutassuming somethingaboutthe forcesofconstraints. FrictionFinally,thereisanotherf actormissingin ourcurrentformulation of mechanics - friction.Friction istreatedin high-schoolphysics inanad-hoc man- neraswell. Frictionisa forcethatcon vertsmechanical energyinto thermalen- ergy,whosenatureisoutsidethescope ofmechanics.Thermal energyis aterm describingthemicroscopic motionofthe moleculesthatform aphysical body. Conceptually,ifweincludedin ourmodele verysingle molecule(ignoring the lawsofquantummechanics),there wouldbe nosuchthing asthermal energy. We willdeliberatelylea ve frictionalforcesoutsidethescopeofthepresentcourse.

NewtonianMechanics37

2.7d'Alembert's principleofvirtualwork

First,letus rewriteNe wton'ssecond lawinamorecompact form.Ifwedefinea diagonal3n-by-3nmassmatrix,M,whosediagonal elementsare(m 1 ,m 1 ,m 1 ,m 2 ,...,m n ), thenNewton' slawcanbewritteninthe compactform, M d 2 r(t) dt 2 =f (a) (r(t),t)+f (c) (r(t),t), wherethe3 n-vectorsf (a) (r(t),t)andf (c) (r(t),t)arethe forces,whichwe have split into"applied"forces, andforcesof constraints.Theapplied forcesareassumed tobeprescribed, whereastheforces ofconstraintsare a-prioriunknown. Supposethenthat weha vekholonomicconstraints,in generaltimedependent, h i (r,t)=0,i=1,...,k, andthattheconstrainedmanifold M(t)isparametrizedwithd=3n-kgeneralized coordinates,q=(q 1 ,...,q d ),withthe (generallytime-dependent) mappings, r=?(q,t), where?:R d

×R→R

3n .Theassumption thatthethe constraintsdefinea d- dimensionalmanifold impliesthatthis relationcanbe inverted - that (r,t)defines uniquely( q,t). Ourgoal istoreformulate Newton's equationssystemas ?Asystemof equationsforthe generalizedcoordinatesq(t). ?Inaw aythatdoes notrequireanexplicitkno wledgeofthe forcesofcon- straint. Were-emphasizethatthiscannot followfrom anypre viouslaw .One needsto makenewassumptionsthat wouldtellushow tocopewith forcesof constraints. Example:As asimplee xample,considera masssuspendedon arodoflength ?,onlyallo wedto moveinthexz-plane,whilethe upperendof therodmo ves upwardwithvelocityu.Thus,r=(x,y,z)satisfiesthe constraints, h 1 (x,y,z,t)=y=0 h 2 (x,y,z,t)=x 2 +(z-ut) 2 -? 2 =0.

38Chapter2

Figure2.6:Jean leRond d'Alembert(1717-1783).

Hereweha ve n=1,k=2andd=3n-k=1,Atan ytimet,theparticle isconstrainedto move onacircle (butthecirclechangeswithtime),which isa one-dimensionalmanifold parametrizedbyan angleθ.Therelation betweenthe generalizedcoordinate θandtheCartesian coordinateris r=?(θ,t)=(?sinθ,0,ut-?cosθ). ? Thisadditional principlewas proposedbyJean leRondd'Alembert,andiskno wn astheprinciple ofvirtualwork .Hisprinciple statesthefollo wing.Atan ytimet, thev elocityvectorisconstrainedto beinad-dimensionalhyper -plane.Theforces ofconstrains,according tod'Alembert' sprinciple,are normaltothis hyper-plane. Notethatthere isonly oneactualvelocityvector;thed-dimensionalhyper -plane containsallthose directionsofmotion thatdonot violatetheconstrains. Such motionsarecalled virtualmotions.d'Alembert's principlestatesthateven ifthe systemadoptedan yofthose virtualmotions,ratherthantheactual motion,the forcesofconstraints wouldnot performany work.Forthatreason, d'Alembert's principleiscalled theprincipleof virtualwork. D'alembert'sprincipleisaph ysicist'sinterpretation ofageometric statement.

Givenapointqattimet,an yvector

∂? ∂q j (q,t),j=1,...,d, istangentto theconstrained manifold.D'alembert's principlestatesthat forev ery

NewtonianMechanics39

j=1,...,d, f (c) (r(t),t)? ∂? ∂q j (q(t),t). Thus,takingthe scalarproduct ofNewton' sequationwith eachofthose dtangent vectors,weobtain M d 2 r(t) dt 2 · ∂? ∂q j (q(t),t)=f (a) (r(t),t)· ∂? ∂q j (q(t),t), wherer(t)andq(t)arelink edbythe relationr(t)=?(q(t),t). Theseare dequations,which weview asequationsfor thedvariablesq(t).We havesuccessfullyusedd'Alembert'sprincipleto eliminatetheforces ofcon- straints.Whatremains tobedone ittoe xpresstheresulting equationssuchthat onlyq(t)appears.In principle,we candoit "brutally"byapplying thechainrule, butwithsomesubtlety ,wecan obtainanice-looking setofequations. First,letus rewrite theequationsin componentform.Also,letusplug inthe assumptionthatall appliedforcesare derived froma(possibly time-dependent) potential,Φ(r,t).Then, v k (t)= dr k (t) dt m ik dv k (t) dt ∂? i ∂q j (q(t),t)=- ∂Φ ∂r i (r(t),t) ∂? i ∂q j (q(t),t). Since r k (t)=? k (q(t),t), itfollows that v k (t)= ∂? k ∂t (q(t),t)+ ∂? k ∂q ? (q(t),t) dq ? (t) dt . Inthesame wayas wecanwrite the3nsecond-orderequationsfor r(t)as6 nfirst orderequationsfor (r(t),v(t)),wewill writethene wequationsas 2dfirst-order equationsfor( q(t),˙q(t)),wherewe viewthe variables˙qasentitiesindependent of q.3dofthose equationsarethen dq(t) dt =˙q(t).(2.2) Then,in thesamew ayasrandqareconnected by?,thepair (r,v)and( q,˙q)are connectedby r k =? k (q,t)v k = ∂? k ∂t (q,t)+ ∂? k ∂q ? (q,t)˙q ? .

40Chapter2

Itwillbe convenient toredefineno wvasafunction of(q,˙q,t),as above, inwhich casewecan writethe3 dequationsofmotion as m ij ? d dt v k (q(t),˙q(t),t) ? ∂? i ∂q j (q(t),t)=- ∂Φ ∂r i (?(q(t),t),t) ∂? i ∂q j (q(t),t),(2.3) whichtogetherwith (2.2)formsa closedsystemof 2dequationsfor( q(t),˙q(t)). Let'sstartwiththe right-handside,which canbebrought forthemore compact form - ∂Φ ∂q j (?(q(t),t),t).

Next,considertheleft-handside, andnotefirst that

∂v i ∂˙q j (q,˙q,t)= ∂? i ∂q j (q,t).

Theleft-handside takesthe form

m ik d dt ? v k (q(t),˙q(t),t) ∂v i ∂˙q j (q(t),˙q(t),t) ? -m ik v k (q(t),˙q(t),t) d dt ∂? i ∂q j (q(t),t).

Byintroducingthe function

T(q,˙q,t)=

1 2 m ik v i (q,˙q,t)v k (q,˙q,t), wecanre writethefirst termas d dt ? ∂T ∂˙q j (q(t),˙q(t),t) ? .

Asforthe secondtermwe notethat

d dt ∂? i ∂q j (q(t),t)= ∂ 2 ? i ∂q j ∂q ? (q(t),t)˙q ? (t)+ ∂ 2 ? i ∂q j ∂t (q(t),t) = ∂v i ∂q j (q(t),˙q(t),t), sothatthe secondtermis - ∂T ∂q j (q(t),˙q(t),t).

NewtonianMechanics41

Atthispoint, werewrote theequationsof motionas

d dt ? ∂T ∂˙q j (q(t),˙q(t),t) ? - ∂T ∂q j (q(t),˙q(t),t)=- ∂Φ ∂q j (?(q(t),t),t). Defininga functionof( q,˙q,t),which wecallthe Lagrangian,

L(q,˙q,t)=T(q,˙q,t)-Φ(?(q,t),t),

Thegeneralizedcoordinates satisfythedcoupledequations, d dt ? ∂L ∂˙q k ? - ∂L ∂q k =0.(2.4) Bythe endoftedious calculusmanipulations,we have obtainedaclosed setof equationsforonly thegeneralizedcoordinates; theyare knownas theEuler- Lagrangeequations.Writing theseequationse xplicitlyonlyrequires theknowl- edgeofa scalarvalued function - theLagrangian - interms ofthegeneralized coordinatesandthe generalizedvelocities.

2.8Examples

Example:Ourfirst exampleis apendulum:a massmhungona rigidrodof length ?,whose(upper) endisfix ed.Themass isfreeto move onacircleofradius ?on thexz-plane.We onlyneedonegeneralizedcoordinate:the anglethatthe rod makeswiththev erticalaxis.The relationbetweenrandθis r=?(θ)=(?sinθ,0,-?cosθ), hence v=(?cosθ,0,?sinθ) ˙ θ , andthe kineticenergy isgiv enby T= m? 2 2 ˙ θ 2 .

Thepotentialener gyisgi venby

Φ =mgz=-mg?cosθ,

42Chapter2

sothatthe Lagrangianis

L(θ,

˙ θ ,t)= m? 2 2 ˙ θ 2 +mg?cosθ.

Theev olutionofθ(t)isdetermined by(2.4),

d dt (m? 2 ˙

θ)+mg?sinθ=0,

or, ¨

θ=-

g ? sinθ. (Forsmallanglesitis standardtoapproximate sinθ≈θ.)? Example:Supposeno wthatthis pendulumisfreetomov eonthe spheredefined bythedistance ?fromtheanchor point(i.e.,a two-dimensionalmanifold). We parametrizethelocation ofthe massbythe twospherical angles(θ,φ)with r=?(θ,φ)=?(sinθcosφ,sinθsinφ,-cosθ), hence

˙r=?(cosθcosφ,cosθsinφ,sinθ)

˙

θ+?(-sinθsinφ,sinθcosφ,0)

˙ φ , andthekinetic energyis given by T= m? 2 2 ? ˙ θ 2 +sin 2 θ ˙ φ 2 ? . Thepotentialener gy,on theotherhand,isstillgivenby Φ =mgz=-mg?cosθ, sothatthe Lagrangianis

L(θ,

˙ θ , φ , ˙ φ ,t)= m? 2 2 ? ˙ θ 2 +sin 2 θ ˙ φ 2 ? +mg?cosθ.

Theequations ofmotionare therefore,

d dt (m? 2 ˙

θ)-

m? 2 2 sin2θ ˙ φ 2 +mg?sinθ=0 d dt (m? 2 sin 2 θ ˙

φ)=0.

?

NewtonianMechanics43

Figure2.7: AnAtwood machine

Example:Consider nowthe pendulumwheretheanchorpointis pulledupward withv elocitya.No wwehave ?(q,t)=(?sinθ,0,at-?cosθ).

Notethatwe endupwith theexact samedynamics.?

?Exercise2.5 Considera massmsuspectedbya rodoflength ?andfreeto moveinthexzplane.Theupper endofthe rodundergoes "forced"oscillationsin thev erticalaxis;itpositionisgi venby z(t)=cosω 0 t, whereω 0 isaconstant. Thegoalis todescribethe motionofthe suspendedmass. (i)Choosegeneralized coordinates.(ii)Write theLagrangian.(iii) Writethe equationsof motion.(iv) Tryto solvethem. ?Exercise2.6UsetheLagrangian formalismtowrite theequationsof motion oftwo pointparticlesofmassmandMconnectedbya rigidrod oflength?,falling undertheinfluence ofgravity . Example:TheAtwoodmachinewasinventedin 1784byGeorgeAtwoodas alaboratorye xperimenttov erifythemechanicallawsof uniformlyaccelerated motion.

44Chapter2

Considerasimple Atwoodmachine withamass m

1 atoneend ofthe stringand amassm 2 attheother end.Thestate ofthesystem maybeparametrized witha singlecoordinate,say ,z,whichis thevertical coordinateofthe massm 1 relative tothepoint wherebothare atthesame height.Herewe have r 1 =? 1 (z)=(-a,0,z) r 2 =? 2 (z)=(a,0,-z), where,withoutloss ofgenerality, wehav eassumedthat thehorizontalseparation is2aalongthex-axis.TheLagrangian is

L(z,˙z,t)=

m 1 2 ˙z 2 + m 2 2 ˙z 2 +m 1 gz-m 2 gz, andthecorresponding equationsofmotion are (m 1 +m 2 )¨z-(m 1 -m 2 )g=0, namely,

¨z=

m 1 -m 2 m 1 +m 2 g. Notethatthe accelerationisrelati veto thenormalizedmass difference.? ?Exercise2.7Considerabead sliding(withoutfriction) onarotating wirein aforce-freespace (i.e.,no gravity).Specifically ,considera straightwirerotated atconstantangular speedabout afixed axisperpendicularto thewire.(1) Choose generalizedcoordinates. (2)Establishthe relationbetweenthe generalizedcoor- dinatesandthe Cartesiancoordinates. (3)Writethe Lagrangian.(4)Write the correspondingequationsof motion.(5) Solvethe equations.(6)Pro videan"intu- itive"interpretationoftheresult.

Chapter3

VariationalCalculus

3.1Thebrachistochr one

Thenameof thisclassicalproblem comesfrom Greek(βραχιστoςmeansshortest andχρoνoςmeanstime).It isthefirst variationalproblem thathase venbeen formulated.Itw asfirstsolv edbyJohannBernoulli,whoposed ittoreaders of ActaEruditorumin June1696.F ourmathematiciansresponded withsolutions: IsaacNewton, JakobBernoulli,GottfriedLeibnizand Guillaumedel'Hˆopital.

Theproblemis thefollowing: LetP

1 =(0,0)andP 2 =(1,-1)betw opointsin R 2 .We connectthesetwopointsby acontinuouslydi fferentiablecurve y=f(x). Supposethat abeadw asallowed toslidfrictionless alongthiscurve,whatis the curveforwhichthetime thatittak estoreach thepointP 2 isminimal?W eassume thattheonly appliedforceis gravity. SeeFigure3.1foravisualization. Thebeadperforms amotion( x(t),y(t)),where y(t)=f(x(t)).Thev elocityis therefore v 1 (t)=˙x(t)v 2 (t)=˙y(t)=f ? (x(t))˙x(t), i.e., |v(t)|= ? 1+[f ? (x(t))] 2

˙x(t).

Conservationofmechanicalenergy

1 impliesthat 1 2 m|v(t)| 2 =-mgy(t), 1 Atthispoint, wearenot concernedaboutthe lawsof physics.The variationalproblem canbe posedirrespectiv elyofitsmechanicalorigin.

46Chapter3

Figure3.1:An experimentaldemonstration ofthebrachistochrone anditsinven- tor,JohannBernoulli(1667-1748). sothat ? 1+[f ? (x(t))] 2

˙x(t)=

? -2gf(x(t)), or, ? 1+[f ? (x(t))] 2 ? -2gf(x(t))

˙x(t)=1.

Integratingover timefromt=0tot=T,thetime ittakes toreachthe pointP 2 , weget T= ? 1 0 ? 1+[f ? (x)] 2 ? -2gf(x) dx. Thisis thequantityto beminimized.W earelooking forafunction f(x)satisfying f(0)=0,f(1)=-1,whichminimizes thisintegral.Forthosewhograduatedfrom ourcalculus course,thisis atotallyne wproblem.W eareask edtominimize a functionofa function,i.e.,the spaceov erwhichwe seekthe minimumisinfinite- dimensional.We willsoonhavethe toolstoanswer thisquestion. ?Exercise3.1 Formulatemathematicallytheproblem offindingthe shortest curvethatconnectstwo pointsinthe plane.

VariationalCalculus47

3.2Functionalso vernormed spaces

Inthepreceding sectionwewere requiredtominimize areal-valued functionofa function.Suchobjects arecalledfunctionals. Definition3.1LetXbeanormedlinearspace.AfunctionalonXisamapping

X→R,i.e., arealvaluedfunction.

Inthecourse onadvanced calculuswedeal withfunctionsbetween arbitrarymet- ricspaces.Recall thatany normedspacehas aninducedmetric. Inparticular,we haveawell-definednotionofcontinuityof suchmappings. Definition3.2LetXbeanormed linearspace .Theset ofallcontinuous linear functionalsX→Riscalledthe spacedualtoXandisdenoted byX ? .

Example:LetX=C

1 [0,1]be thespaceof continuously-differentiablefunctions on[0,1]withthe norm ?f?=sup

0≤x≤1

|f(x)|+sup

0≤x≤1

|f ? (x)|.

Thefollowing isafunctionalonX,

I[f]= ? 1 0 ? f 2 (x)+f(x)f ? (x) ? dx.

Wecanshow thatitis infactcontinuous.Let( f

n )bea sequenceof functionsthat convergestofinC 1 [0,1].Then, I[f n ]-I[f]= ? 1 0 (f n -f)(f n +f)dx+ ? 1 0 f n (f ? n -f ? )dx+ ? 1 0 (f n -f)f ? dx.

Becausethesequence f

n isconv ergent,itisbounded(callthecommonboundof f n andf,C),and wecanbound thisdifferenceby |I[f n ]-I[f]|≤4C?f n -f?→0. ? ?Exercise3.2Showthat ?f?=sup

0≤x≤1

|f(x)|+sup

0≤x≤1

|f ? (x)| isa normonC 1 [0,1].

48Chapter3

Example:Letag ainX=C[0,1].Thefollo wingfunctionalis calledtheδ- measureconcentratedat x=1/2,

I[f]=f(

1 2 ). Itisclearly alinearfunctional, andit iscontinuouson X.Physicists callitthe

Diracdelta-functionanddenoteit by

I[f]= ? 1 0

δ(x-

1 2 )f(x)dx. Itshould beemphasizedthat thereisno suchfunctionδ,andthat thisisa mere notation.?

3.3Functionalderi vativ es

Intheadv ancedcalculus courseweonlydeveloped differentialcalculusfor func- tionsbetweenfinite-dimensional normedspaces.W ewillno wgeneralize thecon- ceptofdi fferentiabilityforfunctionals over infinite-dimensionalnormedspaces. Definition3.3LetXbeanormed space.A functionalI:X→Rissaid tobeFr´echetdifferentiableatfif theree xistsa continuouslinearfunctional

T:X→R,such that

lim g→0

I[f+g]-I[f]-T[g]

?g? =0.

Naturally,wecandenoteT=D

I (f).

Example:LetX=C[0,1]andconsider thefunctional

I[f]= ? 1 0 f 2 (x)dx.

WeclaimthatIisFr´echetdi fferentiablewith

D I (f)[g]= ? 1 0 f(x)g(x)dx.

VariationalCalculus49

Toshowthatwe firstneedtoshowthat D

I (f)isa continuousfunctional(it is clearlylinear),that it,thatif g n →ginC[0,1],then D I (f)[g n ]→D I (f)[g]inR.

Indeed,

|D I (f)[g n ]-D I (f)[g]|= ? ? ? ? ? ? ? 1 0 f(x)[g n (x)-g(x)]dx ? ? ? ? ? ? ≤?f??g n -g?→0.

Finally,weneedto showthat

lim g→0

I[f+g]-I[f]-D

I (f)[g] ?g? =0, andindeed,

I[f+g]-I[f]-D

I (f)[g] ?g? = ? 1 0 g 2 (x)dx ?g? ≤?g?→0. ? ?Exercise3.3 Provethat I[f]= ? 1 0 f n (x)dx, isFr ´echetdifferentiableonC[0,1].Whatis thederiv ative? Recallthatfor functionsbetweenfinite-dimensional normedspacesdi fferentia- bilityimpliesthe existenceof directionalderivati ves,but theconverseisnot necessarilytrue.Thus, onecandefine aweaker notionofdi fferentiability,known asGˆateauxdifferentiability. Specifically,ifIisdifferentiableatf?X,thenfor every g?X, lim t→0

I[f+tg]-I[f]-tD

I (f)[g] t =0, i.e., D I (f)[g]=lim t→0

I[f+tg]-I[f]

t . Definition3.4LetXbeanormed space. AfunctionalI :X→Rissaid tobeGˆateauxdifferentiableatfif theree xistsacontinuous linearfunctional

T:X→R,such that