chapter 1 dynamics of particles and fieldsbt.pa.msu.edu/pub/papers/aiep108book/chap1.pdf · chapter...

ADVANCESIN IMAGING AND ELECTRON PHYSICS,VOL. 108

Chapter 1

Dynamicsof Particles and Fields

1.1 BEAMS AND BEAM PHYSICS

In a very generalsense,thefield of beamphysicsis concernedwith theanalysisof the dynamicsof certainstatevectors

�� Thesecomprisethe coordinatesofinterest,andtheirmotionis describedby adifferentialequation

�� Usually it is necessaryto analyzethe manifold of solutionsof the differentialequationnot only for onestatevector

�� but alsofor an entireensembleof statevectors.Differentfrom otherdisciplines,in thefield of beamphysicstheensembleof statevectorsis usuallysomewhatclosetogetherandalsostaysthatway overthe rangeof the dynamics.Suchan ensembleof nearbystatevectorsis calleda beam.

Thestudyof thedynamicsof beamshastwo importantsubcategories:thede-scriptionof themotionto veryhighprecisionovershorttimesandtheanalysisofthetopologyof theevolution of statespaceover long times.Thesesubcategoriescorrespondto the historical foundationsof beamphysics,which lie in the twoseeminglydisconnectedfields of opticsandcelestialmechanics.In the caseofoptics,in orderto makeanimageof anobjector to spectroscopicallyanalyzethedistributionsof colors,it is importantto guidetheindividual raysin a light beamto very clearly specifiedfinal conditions.In the caseof celestialmechanics,theprecisecalculationof orbitsover shorttermsis alsoof importance,for example,for thepredictionof eclipsesor theassessmentof orbitsof newly discoveredas-teroidsunderconsiderationof theuncertaintiesof initial conditions.Furthermore,celestialmechanicsis also concernedwith the questionof stability of a set ofpossibleinitial statesover long periodsof time, oneof thefirst andfundamentalconcernsof the field of nonlinear dynamics; it may not be relevant whereex-actly theearthis locatedin a billion yearsaslong asit is still approximatelythesamedistanceaway from thesun.Currently, werecognizethesefieldsasjust twospecificcasesof the studyof weakly nonlinearmotion which are linked by thecommonconceptof theso-calledtransfermapof thedynamics.

On thepracticalside,currentlythestudyof beamsis importantbecausebeamscanprovide andcarry two fundamentallyimportantscientificconcepts,namely,energy andinformation. Energyhasto beprovidedthroughacceleration,andthesignificanceof thisaspectis reflectedin thenameacceleratorphysics,whichis of-tenusedalmostsynonymouslywith beamphysics.Informationis eithergenerated

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1999by Martin BerzAll rightsof reproductionin any form reserved.

2 DYNAMICS OF PARTICLES AND FIELDS

by utilizing thebeam’senergy, in whichcaseit is analyzedthroughspectroscopy,or it is transportedat high ratesandis thusrelevant for the practicalaspectsofinformationscience.

Applicationsof thesetwo conceptsarewide.Thefield of high-energy physicsor particlephysicsutilizes both aspectsof beams,and theseaspectsareso im-portantthatthey aredirectly reflectedin their names.First,commonparticlesarebroughtto energiesfarhigherthanthosefoundanywhereelseon earth.Thenthisenergy is utilized in collisionsto produceparticlesthatdonot exist in ourcurrentnaturalenvironment,andinformationaboutsuchnew particlesis extracted.

In asimilarway, nuclear physicsusestheenergy of beamsto produceisotopesthatdo not exist in our naturalenvironmentandextractsinformationabouttheirproperties.It alsousesbeamsto studythedynamicsof the interactionof nuclei.Both particlephysicsandnuclearphysicsalsore-createthestateof our universewhenit wasmuchhotter, andbeamsareusedto artificially generatetheambienttemperatureat theseearliertimes.Currently, two importantquestionsrelateto theunderstandingof thetimeperiodscloseto thebig bangaswell astheunderstand-ing of nucleosynthesis,thegenerationof thecurrentlyexistingdifferentchemicalelements.

In chemistry andmaterial science, beamsprovide tools to studythe detailsof thedynamicsof chemicalreactionsanda varietyof otherquestions.In manycases,thesestudiesareperformedusingintenselight beams,which areproducedin conventionallasers,free-electronlasers,or synchrotronlight sources.

Also, greatprogresscontinuesto bemadein thetraditionalrootsof thefield ofbeamphysics,optics, andcelestialmechanics. Modernglasslensesfor camerasarebetter, moreversatile,andusednow morethaneverbefore,andmodernelec-tron microscopesnow achieveunprecedentedresolutionsin theAngstromrange.Celestialmechanicshasmadeconsiderableprogressin the understandingof thenonlineardynamicsof planetsandtheprospectsfor thelong-termstabilityof oursolarsystem;for example,while the orbit of earthdoesnot seemto be in jeop-ardyat leastfor themediumterm,thereareotherdynamicalquantitiesof thesolarsystemthatbehavechaotically.

Currently, the ability to transportinformation is beingappliedin the caseoffiber optics, in which short light pulsesprovide very high transferrates.Also,electronbeamstransportthe informationin thetelevision tube,belongingto oneof themostwidespreadconsumerproductsthat,for betteror worse,hasasignifi-cantimpacton thevaluesof our modernsociety.

1.2 DIFFERENTIAL EQUATIONS, DETERMINISM , AND MAPS

As discussedin theprevioussection,thestudyof beamsis connectedto theun-derstandingof solutionsof differentialequationsfor anensembleof relatedini-tial conditions.In thefollowing sections,we provide thenecessaryframework tostudythis question.

DIFFERENTIAL EQUATIONS,DETERMINISM, AND MAPS 3

1.2.1 ExistenceandUniquenessof Solutions

Let usconsiderthediffer ential equation�� (1.1)

where�� is astatevectorthatsatisfiesthe initial condition

�� In practicethevector

�� cancomprisepositions,momentaor velocities,or any otherquantityinfluencingthemotion,suchasaparticle’sspin,mass,or charge.Wenotethattheordinarydifferentialequation(ODE) beingof first order is immaterialbecauseany higher order differ ential equation�� (1.2)

canberewritten asa first-orderdifferentialequation;by introducingthe "!$#&% �

new variables�� '�� )(* � ( �'�� ( ��+�� '��,�� , we have

theequivalentfirst-ordersystem��.-///0�� ...��

1�2223 -///0�� (...��4 ��+� ��

1�2223 � (1.3)

Furthermore,it is possibleto rewrite the differentialequationin termsof anyother independentvariable 5 thatsatisfies� 5��687 (1.4)

andhencedependsstrictly monotonicallyon� � Indeed,in this casewe have�� 5 �� :9 �� 5 �� 5 �� 9 �� 5 � (1.5)

Similarly, thequestionof dependenceontheindependentvariableis immaterialbecauseanexplicitly ; -dependentdiffer ential equation canberewritten asa

�-

independentdifferentialequationby the introductionof an additionalvariable 5with

�� 5 % andsubstitutionof�

by 5 in theequation:�� 5 � ��< �� 5 �% � (1.6)

Frequently, onestudiesthe evolution of a particularquantitythat dependsonthephasespacevariablesalonga solutionof theODE.For a givendifferentiable


function = �� from phasespaceinto the realnumbers(sucha function is alsooftencalledanobservable), it is possibleto determineits derivativeandhenceanassociatedODEdescribingits motionvia�� = �� 9 �> =:?A@@ � = 8B$CD = �FE (1.7)

thedifferentialoperatorB CD is usuallycalledthevector field of theODE. Othercommonnamesaredir ectional derivative or Lie derivative. It playsan impor-tant role in the theoryof dynamicalsystemsandwill be usedrepeatedlyin theremainderof the text. Apparently, the differentialoperatorcan alsobe usedtodeterminehigherderivativesof = by repeatedapplicationof B CD , andwe have� �� = 8B � CD = � (1.8)

An importantquestionin the study of differential equationsis whetheranysolutionsexist and,if so,whetherthey areunique.Thefirst partof thequestionisansweredby thetheoremof Peano; aslong asthefunction

��is continuous in a

neighborhoodof ��HG� � G � , theexistenceof a solution is guaranteedat leastin a

neighborhood.The secondquestionis moresubtlethanonemight think at first glance.For

example,it is ingrainedin our thinkingthatclassicalmechanicsis deterministic,whichmeansthatthevalueof astateatagiventime

� G uniquelyspecifiesthevalueof thestateat any later time

� � However, this is not alwaysthecase;considertheHamiltonianandits equationsof motion:I KJ (L ?NM PO � � where M PO � #�Q O�Q RTS ( (1.9)UO J and

UJ sign VO ��WL Q O�Q � S ( � (1.10)

ThepotentialM VO � is everywherecontinuouslydifferentiableandhasanunstableequilibriumattheorigin; it is schematicallyshown in Fig.1.1.However, aswill beshown,thestationarypointattheorigin givesriseto nondeterministicdynamics.

To thisend,considerthesolutionsthroughtheinitial condition PO G� J G � 7+��7 �

at� &7+� We readilyconvinceourselvesthatthetrajectory PO � J � 7��T7 � for all

�(1.11)

is indeeda solutionof thepreviousHamiltonianequations.However, we observethatfor any

��X, thetrajectory PO � J � ZY 7��7 � for

�\[N��X]_^ �`�a � # ��Xb� a � �� ` � # ��Xb� RHc for�\de��X (1.12)

DIFFERENTIAL EQUATIONS,DETERMINISM, AND MAPS 5

FIGURE 1.1. A nondeterministicpotential.

is asolutionof thedifferentialequations!Thismeansthattheparticlehasachoiceof stayingon top of theequilibriumpoint or, at any time

��Xit pleases,it canstart

to leavetheequilibriumpoint,eithertowardtheleft or towardtheright. Certainly,thereis no determinismanymore.

In adeepersense,theproblemisactuallymorefar-reachingthanonemaythink.While the previous examplemay appearsomewhat pathological,onecanshowthat any potential M PO � canbe modifiedby addinga secondterm f�g PO � withg PO � continuousand

Q g PO � Q [ % suchthatfor any f , thepotentialM PO � ?�f�g PO �is nondeterministic.Choosingf smallerthantheaccuracy to whichpotentialscanbemeasured,we concludethatdeterminismin classicalmechanicsis principallyundecidable.

In a mathematicalsense,the secondquestionis answeredby the Picard-Lindel of Theorem, which assertsthatif theright-handside

��of thedifferential

equation(1.1) satisfiesa Lipschitz condition with respectto�� , i.e., thereis a h

suchthat Q ��< �� # ��4 ��)(� �� Q [ h 9 Q �� # ��)( Q (1.13)

for all�� , �� ( andfor all

�, thenthereis a unique solution in a neighborhoodof

any �� G � � G � �

Therefore,determinismof classicalmechanicscanbeassuredif oneaxiomat-ically assumesthe Lipschitznessof the forcesthat canappearin classicalme-chanics.Thisaxiomof determinismof mechanics,however, is moredelicatethanotheraxiomsin physicsbecauseit cannotbecheckedexperimentallyin a directway, asthepreviousexampleof thepotential g shows.


1.2.2 Mapsof DeterministicDifferentialEquations

Let usassumewe aregivenadeterministicsystem�� (1.14)

which meansthat for every initial condition�� at

� �Fithereis a uniquesolution�� with

�� For a given time� �

, this allows to introducea func-tion jlkVm i k"n that for any initial condition

�� determinesthevalueof thesolutionthrough

�� at thelatertime� ( E thereforewe have�� ( � jlkVm i k"n �� (1.15)

Thefunction j describeshow individual statespacepoints“flow” astime pro-gresses,andin thetheoryof dynamicalsystemsit is usuallyreferredto astheflowof thedifferentialequation.Wealsoreferto it asthepropagator (describinghowstatespacepointsarepropagated),the transfer map (describinghow statespacepointsare transferred),or simply the map. It is obvious that the transfermapssatisfytheproperty j k m i k"o j k n i k"oqp j k m i k n (1.16)

where“ p ” standsfor thecompositionof functions.In particular, we haver jlk"n i kVm p jlkVm i k"n � (1.17)

andsothetransfermapsareinvertible.An important caseis the situation in which the ODE is autonomous,i.e.,�independent. In this case,the transfermap dependsonly on the differences � � ( # �� , andwehavejut k mwvxt k n jut k n p jut k m � (1.18)

An importantcaseof dynamicalsystemsis the situationin which the motion isrepetitive,whichmeansthatthereis a

s �suchthatj k i k vyt k j k vyt k i k v ( t k � (1.19)

In this case,the long-termbehavior of the motion can be analyzedby merelystudyingthe repeatedactionof jlk i k vyt k . If the motion suitsour needsfor thediscretetime steps

s �andoccupiesa suitableset z of statespacesthere,thenit

is merelynecessaryto studyoncehow this set z is transformedfor intermediatetimesbetween

�and

� ? s � andwhetherthissatisfiesourneeds.Thisstroboscopicstudy of repetitive motion is an importantexampleof the methodof Poincare

LAGRANGIAN SYSTEMS 7

sections, of which thereare several varietiesincluding suitablechangesof theindependentvariablesbut thesealwaysamountto thestudyof repetitive motionby discretization.

In the following sections,we embarkon the studyof oneof the most impor-tantclassesof dynamicalsystems,namely, thosethatcanbeexpressedwithin theframeworksof thetheoriesof Lagrange,Hamilton,andJacobi.

1.3 LAGRANGIAN SYSTEMS

Let usassumewe aregivena dynamicalsystemof!

quantitiesO � �� O � whose

dynamicsis describedby a second-orderdifferentialequationof theform{|{�O �} � � �O � {�O � � (1.20)

If thereexistsa function B dependingon�O

and{�O, i.e.,B~�B �O � {�O � �� (1.21)

suchthattheequations�� @ B@ UO�� # @ B@ O�� 87+�y�� % �� ! (1.22)

areequivalentto the equationsof motion (1.20)of the system,then B is calleda Lagrangian for this system.In this case,equations(1.22) areknown asLa-grange’sequationsof motion.Thereareusuallymany advantagesif equationsofmotion canbe rewritten in termsof a Lagrangian;the onethat is perhapsmoststraightforwardly appreciatedis that the motion is now describedby a singlescalarfunction B insteadof the

!componentsof

�} � However, thereareotheradvantagesbecauseLagrangiansystemshave specialproperties,many of whicharediscussedin thefollowing sections.

1.3.1 ExistenceandUniquenessof Lagrangians

Beforeweproceed,wemayaskourselveswhetherafunction B �O � {�O � �� alwaysex-istsand,if so,whetherit is unique.Thelatterpartof thequestionapparentlyhas

to be negatedbecauseLagrangians are not unique. Apparentlywith B �O � {�O � ��also B �O � {�O � �� ?�� , where� is aconstant,is alsoaLagrangianfor

�} � Furthermore,a constant� canbe addednot only to B , but also moregenerallyto any func-

tion � �O � {�O � �� thatsatisfies��'��

( @ � � @ {�O � # @ � � @ �O �7 for all choicesof�O

and{�O � Suchfunctionsdo indeedexist; for example,the function � �O � {�O � �� O 9 {�O


apparentlysatisfiestherequirement.Trying to generalizethis example,onereal-

izesthat while the linearity with respectto{�O

is important,the dependenceon�O

canbe moregeneral;also,a dependenceon time is possible,if it is connectedin the properway to the positiondependence.Altogether, if

}is a threetimes

differentiablefunctiondependingon position�O

and�, then� �O � {�O � �� } �O � �� @ }@ O � 9 UO � ? @ }@ � (1.23)

satisfies��'�� @ � � @ {�O � # @ � � @ �O �7 ; indeed,studyingthe � th componentof the

conditionyields�� @@ UO � -0 �� @ }@ O � 9 UO � ? @ }@ � 13 # @@ O � -0 �� @ }@ O � 9 UO � ? @ }@ � 13 �� @ }@ O � # �� @ ( }@ O � @ O � 9 UO � # @ ( }@ O � @ � �� @ ( }@ O � @ O�� 9 UO � ? @ ( }@ � @ O�� # �� @ ( }@ O�� @ O � 9 UO � # @ ( }@ O�� @ � &7+� (1.24)

The questionof existencecannot be answered in a generalway, but fortu-natelyfor many systemsof practicalrelevanceto us,Lagrangianscanbe found.This usually requiressome“guessing”for every specialcaseof interest;manyexamplesfor this canbefoundin Section1.3.4.

One of the importantadvantagesof the Lagrangianformulation is that it isparticularlyeasyto performtransformationsto differentsetsof variables,asillus-tratedin thenext section.

1.3.2 CanonicalTransformationof Lagrangians

Givena systemdescribedby coordinates PO � �� O � � , its analysiscanoftenbe

simplifiedby studyingit in a setof differ ent coordinates. Oneimportantcaseismotion that is subjectto certainconstraints; in this case,onecanoftenperforma changeof coordinatesin sucha way that a given constraintconditionmerelymeansthat oneof the new coordinatesis to be kept constant.For example,if aparticleis constrainedto movealongacircularloop,thenexpressingits motioninpolarcoordinates

"� �b� � concentricwith theloop insteadof in Cartesianvariablesallows thesimplificationof theconstraintconditionto the simplestatementthat�

is kept constant.This reducesthedimensionality of the problemandleadstoa new systemfree of constraints,and thus this approachis frequentlyappliedfor the studyof constrainedsystems.Currently, thereis alsoan importantfield


thatstudiessuchconstrainedsystemsordiffer ential algebraicequationsdirectlywithout removal of the constraints(AscherandPetzold1998;GriepentrogandMarz1986;Brenan,Campbell,andPetzold1989;Matsuda1980).

Anotherimportantcaseis thesituationin which certaininvariants of themo-tion areknown to exist; if theseinvariantscanbe introducedasnew variables,it againfollows that thesevariablesareconstant,andthe dimensionalityof thesystemcanbereduced.Moresophisticatedchangesof coordinatesrelevantto ourgoalswill bediscussedlater.

It is worth noting that the choiceof new variablesdoesnot have to be thesameeverywherein variablespacebut may changefrom oneregion to another;this situationcanbedealtwith in a naturalandorganizedway with thetheoryofmanifolds in which the entireregion of interestis coveredby an atlas,which isa collectionof local coordinatesystemscalledchartsthatarepatchedtogethertocover the entireregion of interestandto overlapsmoothly. However, we do notneedto discusstheseissuesin detail.

Let a Lagrangian system with coordinates PO � �� O � � be given. Let V� � �� be anotherset of coordinates,and let � be the transformation

from theold to thenew coordinates,which is allowedto betimedependent,suchthat �� O � �� (1.25)

Thetransformation� is demandedto beinvertible,andhencewehave�O � � � �� FE (1.26)

furthermore,let bothof thetransformationsbecontinuouslydifferentiable.Alto-gether, the changeof variablesconstitutesan isomorphismthat is continuouslydifferentiablein bothdirections,aso-calleddiffeomorphism.

In thefollowing, we will oftennot distinguishbetweenthesymboldescribingthetransformationfunction � andthesymboldescribingthenew coordinates

��if it is clear from the context which oneis meant.The previous transformationequationsthenappearas �� O � �� and

�O �O� �� (1.27)

where��

and�O

on theleft sideof theequationsandinsidetheparenthesesdenote!-tuples,while in front of theparenthesesthey denotefunctions.While this latter

conventionadmittedlyhasthe dangerof appearinglogically paradoxicalif notunderstoodproperly, it oftensimplifiesthenomenclatureto bediscussedlater.

If the coordinatestransformaccordingto Eq. (1.25), then the corresponding

timederivatives{��

transformaccordingto{�� Jac(� � �O � �� 9 {�O ?K@ �@ � � (1.28)


andwe obtaintheentiretransformationj from (�O � {�O � to

�� {�� as� �� {��A� � � �O � ��Jac(� � �O � �� 9 {�O ? @ � � @ � j �O � {�O � �� (1.29)

It is a remarkableand most useful fact that in order to find a Lagrangianthat

describesthemotionin thenew variables��

and{��

from theLagrangiandescribing

themotionin thevariables�Oand

{�O, it is necessaryonly to insertthetransformation

rule (Eq. 1.26) aswell as the resultingtime derivative{�O {�O� �� {�� into the

Lagrangian,and thus it dependson��

and{�� We say that the Lagrangian is

invariant under a coordinate transformation. Therefore,the new Lagrangianwill begivenby � �� {�� B ^ �O� �� {�O� �� {�� c � (1.30)

or in termsof thetransformationmaps,� &B j � � �and B� � j � � (1.31)

In orderto provethisproperty, let usassumethat B~&B �O � {�O � �� is aLagrangiandescribingthemotionin thecoordinates

�O, i.e.,�� @ B@ UO�� # @ B@ O�� 87+�y�� % �� ! (1.32)

yield theproperequationsof motionof thesystem.DifferentiatingEqs.(1.25)and(1.26)with respectto time, asis necessaryfor

thesubstitutioninto theLagrangian,we obtain{�� {�� O � {�O � ��{�O {�O� �� {�� (1.33)

We now perform the substitutionof Eqs. (1.26) and (1.33) into (1.30) and let� �� {�� betheresultingfunction;thatis,� �� {�� B ^ �O� �� {�O� �� {�� c � (1.34)

Thegoalis now to show thatindeed�� @ �@ U� � # @�@ � � �7��4�� % �� ! � (1.35)


We fix a �q� %:� � �N! . Then @ �@ U� � �� @ B@ UO � @ UO �@ U� � (1.36)

and @ �@ � � �� @ B@ O � @ O��@ � � ? �� @ B@ UO � @ UO��@ � � � (1.37)

Thus, �� @ �@ U� � �� @ B@ UO�� @ UO �@ U� � � �� @ B@ UO�� @ UO �@ U� � ? �� @ B@ UO�� @ UO �@ U� � � �Notethatfrom UO�� @ O��@ � � U� � ? @ O��@ � � (1.38)

we obtain @ UO�� @ U� � @ O�� @ � � . Hence,�� @ �@ U� � �� @ B@ UO�� @ O �@ � � ? �� @ B@ UO�� @ O �@ � � � (1.39)

However, usingEq.(1.32),wehave��'�� @ B � @ UO � � @ B � @ O � . Moreover,�� @ O �@ � � �� @ ( O �@ � � @ � � U� � ? @ ( O �@ � � @ � @@ � � � �� @ O �@ � � U� � ? @ O �@ � � @@ � � � �� O � � @ UO��@ � � �


Therefore, �� @ �@ U� � �� @ B@ O�� @ O �@ � � ? �� @ B@ UO�� @ UO �@ � � @ �@ � � �andthus �� @ �@ U� � � # @

�@ � � �7 for � % �� ! � (1.40)

which we setout to prove. Therefore,in Lagrangiandynamics,the transforma-tion from onesetof coordinatesto anotheragainleadsto Lagrangiandynamics,andthe transformationof the Lagrangiancanbe accomplishedin a particularlystraightforwardway by mereinsertionof thetransformationequationsof theco-ordinates.

1.3.3 Connectionto a Variational Principle

Theparticularform of theLagrangeequations(1.22)is thereasonfor aninterest-ing cross-connectionfrom thefield of dynamicalsystemsto thefield of extremumproblemstreatedby the so-calledcalculusof variations.As will be shown, ifthe problemis to find the path

�O� ��with fixed beginning

�O� �� O �and end�O� � ( � �O ( suchthattheintegral� �� k"nk m B ^ �O� �� {�O� �� c �� (1.41)

assumesan extremum,then as shown by Euler, a necessarycondition is thatindeedthepath

�O� ��satisfies�� @ B@ UO�� # @ B@ O�� 7��4�< % �� ! � (1.42)

which arejust Lagrange’s equations.Note that theconverseis not true:Thefactthat a path

�O� ��satisfiesEq. (1.42) doesnot necessarilymeanthat the integral

(1.41)assumesanextremumat�O� ��

To show the necessityof Eq. (1.42) for the assumptionof an extremum,let�O� ��be a path for which the action � k nkVm B �O � �O � ��:�� is minimal. In particular,

this implies thatmodifying�O

slightly in any directionincreasesthe valueof theintegral.We now select

!differentiablefunctions� � �� , %�� e! , thatsatisfy� � ��Fi ( � 87 for

%�� e! � (1.43)


Then,we definea family of variedcurvesvia�O � � �� O� �� ? � � � �� 9 �� (1.44)

Thus,in�O � , a variationonly occursin the directionof coordinate� E the amount

of variationis controlledby the parameter� , andthe value ��Z7 correspondsto the original minimizing path.We note that usingEqs.(1.43) and(1.44),wehave

�O � ��bi (�� O �Fi ( for%~� � ��!

, and thus all the curves in the familygo throughthesamestartingandendingpoints,asdemandedby therequirementof the minimization problem.Because

�O � minimizes the action integral, when�O ? � � � 9 �� is substitutedinto theintegral,a minimumis assumedat � �7 , andthus 7� �� k nkVm B �O � � �T� � � {�O � � �T� � � ��4��'¡¡¡¡|¢ � G for all

%:� � �e! (1.45)

Differentiatingthe integral with respectto � , we obtainthe following accordingto thechainrule:�� k"nk m B �O � � �T� � � {�O � � �� 4�� _� k"nk m � @ B@ O � � � ?l@ B@ UO � U� � �� (1.46)

However, via integrationby parts,thesecondtermcanberewrittenas� k nkVm @ B@ UO � U� � �� @ B@ UO � � � ¡¡¡¡ k"nkVm # � k nkVm �� @ B@ UO � � � �� (1.47)

Furthermore,sinceall thecurvesgo throughthesamestartingandendingpoints,we have @ B@ UO � � � ¡¡¡¡ k"nkVm �7�� (1.48)

Thus,weobtain�� k nkVm B ^ �O � � �T� � � {�O � � �� c �� k nkVm � @ B@ O � # �� @ B@ UO � � � �� (1.49)

This conditionmustbetruefor everychoiceof � � satisfying� � � �Fi ( � _7 , whichcanonly bethecaseif indeed @ B@ O � # �� @ B@ UO � 87+� (1.50)


1.3.4 Lagrangiansfor Particular Systems

In thissection,wediscussthemotionof someimportantsystemsandtry to assesstheexistenceof a Lagrangianfor eachof them.In all cases,we studythemotionin Cartesianvariables,bearingin mind that the choiceof differentvariablesisstraightforwardaccordingto thetransformationpropertiesof Lagrangians.

NewtonianInteractingParticlesandthePrinciple of LeastAction

We begin our studywith the nonrelativistic motion of a single particle underNewtonianforcesderivablefrom time-independentpotentials,i.e.,thereis a M �£ �with

�} # �> M . TheNewtonianequationof motionof theparticleis�} 8¤ {|{�£$� (1.51)

We now try to “guess” aLagrangianB �£y� {�£4� �� It seemsnaturalto attemptto ob-tain ¤¦¥£ � from

�� @ B � @ U£ � � and} � from @ B � @ £ � for h % � L � W , andindeed

thechoice B �£y� {�£4� �� %L ¤ {�£ ( # M �£ � (1.52)

yieldsfor h % � L � W that�� @ B@ U£ � ¦¤¦¥£ � and @ B@ £ � # @ M@ £ � } � � (1.53)

Thus,Lagrange’sequations�� @ B@ U£ � # @ B@ £ � �7�� h % � L � W (1.54)

yield theproperequationsof motion.Next, we studythenonrelativistic motionof asystemof § interacting parti-

clesin which theexternalforce�} �

on the � th particleis derivablefrom apotentialM � �£ � � andtheforceonthe � th particledueto the � th particle,�} � � , is alsoderivable

from a potential M � � M � � �Q �£ � # �£ � Q � . Thatis,�} � # �> M � (1.55)�} � � # �> � M � � �> � M � � �> � M � � # �} � � � (1.56)

where�> � actson the variable

�£ � � Guidedby the caseof thesingleparticle,weattempt B�©¨� � � � %L ¤ � {�£ (� # ¨� � � � M �b �£ � � # � �Pª � M � �b �Q �£ �y# �£ � Q � � (1.57)


Denotingthefirst term,oftencalledthetotal kinetic energy, as « � andthesecondterm,oftencalledthetotal potential,as M , we haveB¬ « # M � (1.58)

Let £ � i � andU£ � i � denotethe h th componentsof the � th particle’s position and

velocityvectors,respectively. Then,@ B@ £ � i � # @ M �@ £ � i � # �� i �)� � @ M � �@ £ � i � } � i � ? � �)� � } � � i � �where

} � i � and} � � i � denotethe h th componentsof

�} �and

�} � � , respectively. Ontheotherhand, @ B@ U£ � i � 8¤ � U£ � i � � (1.59)

Thus, �� @ B@ U£ � i � # @ B@ £ � i � 87 (1.60)

is equivalentto ¤ � ¥£ � i � } � i � ? � �)� � } � � i � � (1.61)

Therefore, �� @ B@ U£ � i � # @ B@ £ � i � &7 for h % � L � W (1.62)

is equivalentto ¤ {|{�£ � �} � ? � �)� � �} � � � (1.63)

which is theproperequationof motionfor the � th particle.Consideringthediscussionin theprevioussectionandthebeautyandcompact-

nessof the integral relation(Eq. 1.41),onemaybe led to attemptto summarizeNewtonianmotionin theso-calledprinciple of leastaction by Hamilton:


All mechanicalmotion initiating at ®°¯�±b²³´ ±�µ andterminatingat ®°¯,¶�²³´ ¶Fµ occurssuchthatit minimizestheso-calledaction integral·¹¸~º.» n» m~¼¾½ ³´ ®°¯wµ�²�¿³´ ®°¯,µ�²À¯ÀÁ4Â)¯�Ã (1.64)

In this case,theaxiomaticfoundationof mechanicsis movedfrom the some-whattechnicalNewtonianview to Hamilton’s leastactionprinciplewith its mys-teriousbeauty. However, many caveatsarein orderhere.First,not all Newtoniansystemscannecessarilybe written in Lagrangianform; asdemonstratedabove,obtainingaLagrangianfor aparticulargivensystemmayor maynotbepossible.Second,asmentionedpreviously, Lagrange’s equationsalonedo not guaranteeanextremum,let aloneaminimum,of theactionintegralasstatedby Hamilton’sprinciple. Finally, it generallyappearsadvisableto selectthosestatementsthatareto haveaxiomaticcharacterin aphysicstheoryin suchaway thatthey canbecheckedasdirectly andascomprehensively aspossiblein the laboratory. In thisrespect,theNewtonapproachhasanadvantagebecauseit requiresonly acheckofforcesandaccelerationslocally for any desiredpoint in space.Ontheotherhand,the testof the principle of leastactionrequiresthe studyof entireorbits duringanextendedtime interval. Then,of all orbitsthatsatisfy

�O� �� O �, a practically

cumbersomepreselectionof only thoseorbitssatisfying�O� � ( � �O ( is necessary.

Finally, theverificationthat theexperimentallyappearingorbits really constitutethosewith theminimumactionrequiresanoftensubstantialandrathernontrivialcomputationaleffort.

NonrelativisticInteractingParticlesin General Variables

Now thatthemotionof nonrelativistic interactingparticlesin Cartesianvariablescanbe expressedin termsof a Lagrangian,we cantry to expressit in termsofany coordinates

��for which

��Ä �£ � is a diffeomorphism,which is useful for avery wide varietyof typical mechanicsproblems.All we needto do is to expressB� « # M in termsof thenew variables,andthuswewill haveB �� {�� « �� {�� # M �� (1.65)

Without discussingspecificexamples,we makesomegeneralobservationsabout

the structureof the kinetic energy term « �� {�� which will be very useful forlaterdiscussion.In Cartesianvariables,

« {�£ � R ¨� � � � ¤ �L U£ (� %L 9 {�£ k 9 -/0 ¤ � 7. . .7 ¤ R ¨

1�23 9 {�£�� (1.66)


wherethe ¤ � ’sarenow countedcoordinatewisesuchthat ¤ � ¦¤Å(Æ&¤ R areallequalto themassof thefirst particle,etc.Let thetransformationbetweenold andnew variablesbedescribedby themap j , i.e.,

��Ç �£ � j �£ � � Thenin anothersetof variables,we have {�� Jac

j � 9 {�£$� (1.67)

Since by requirementj is invertible, so is its Jacobian,and we obtain{�£È

Jac j � � � 9 {�� , andso « {�� %L 9 {�� k 9ÅÉ�¬Ê 9 {�� (1.68)

wherethenew “massmatrix” É� Ê hastheformÉ� Ê ^ Jac j � � � c k 9 -/0 ¤ � 7

. . .7 ¤ R ¨1�23 9 Jac

j � � � � (1.69)

We observe that the matrix É� Ê is againsymmetric,andif noneof the originalmassesvanish,it is also nonsingular. Furthermore,accordingto Eq. (1.68), inthe new generalizedcoordinatesthe kinetic energy is a quadratic form in the

generalizedvelocities{��.

Particlesin ElectromagneticFields

We studythe nonrelativistic motion of a particlein an electromagneticfield. Adiscussionof thedetailsof a backboneof electromagnetictheoryasit is relevantfor ourpurposesis presentedbelow. Thefieldsof mechanicsandelectrodynamicsarecoupledthroughtheLor entz force,�} 8� �Ë ? {�£�Ì �Í � � (1.70)

whichexpresseshow electromagneticfieldsaffect themotion.In thissense,it hasaxiomaticcharacterandhasbeenverifiedextensivelyandto greatprecision.Fromthetheoryof electromagneticfieldsweneedtwo additionalpiecesof knowledge,namely, thattheelectricandmagneticfields

�Ëand

�Ícanbederivedfrom poten-

tials � �£�� and�Î �£4� �� as �Ë # �> � # @ �Î@ �


and �Í �> Ì �Î �Thedetailsof why this is indeedthecasecanbefoundbelow. Thus,in termsofthepotentials� and

�Î, theLorentzforce(Eq.1.70)becomes�} &� � # �> � # @ �Î@ � ? {�£¬Ì �> Ì �Î � � � (1.71)

Thus,the h th componentof theforceis} � 8� � # @ �@ £ � # @ Î �@ � ?ÐÏ {�£�Ì Ï �> Ì �Î¹ÑÈÑ � � (1.72)

However, usingthecommonantisymmetrictensorÒ � � � andtheKronecker Ó � � , weseethatÏ {�£¬Ì Ï �> Ì �ÎÆÑÔÑ � � � i � Ò � � � U£ � Ï �> Ì �Î¹Ñ � � � i � Ò � � � U£ � � Õ i Ö Ò Õ Ö � @ Î Ö@ £ Õ �� i � i Õ i Ö Ò � � � Ò Õ Ö � U£ � @ Î Ö@ £ Õ �� i � i Õ i Ö Ò � � � � ( Ó � Õ Ó � Ö # Ó � Ö Ó � Õ � U£ � @ Î Ö@ £ Õ � � i � Ò � � � � ( � U£ � @ ÎÆ�@ £ � # U£ � @ Î �@ £ � � � � U£ � @ Î¹�@ £ � # U£ � @ Î �@ £ � @@ £ � {�£.9 �Î � #¦ {�£.9 �> � Î � � (1.73)

On theotherhand,thetotal timederivativeofÎ � is� Î �� @ Î �@ � ? {�£.9 �> � Î � � (1.74)

Thus,Eq. (1.73)canberewrittenasÏ {�£¬Ì Ï �> Ì �Î ÑÔÑ � @@ £ � {�£.9 �Î � # � Î �� ? @ Î �@ � � (1.75)

SubstitutingEq.(1.75)in Eq.(1.72),weobtainfor the h th componentof theforce} � &� � # @@ £ � � # {�£.9 �Î � # � Î �� (1.76)


While this initally appearsto be morecomplicated,it is now actuallyeasierto

guessa Lagrangian;in fact,thepartial @ � @ £ � suggeststhat � # {�£¬9 �Î is thetermresponsiblefor the force.However, becauseof the velocity dependence,thereisalsoacontributionfrom

��'�� @ � @ U£ � ��E thisfortunatelyproducesonly therequired

term� Î � �'�� in Eq. (1.76).Theterms ¤ {|{�£ canbeproducedasbefore,andalto-

getherwe arriveatB �£4� {�£�� %L ¤ {�£ ( # �� £4� �� ? � �Î �£�� 9 {�£.� (1.77)

Indeed,��'�� @ B � @ U£ � � # @ B � @ £ � ×7 for all h % � L � W is equivalentto

} � ¤¦¥£ � for all h % � L � W , andhenceLagrange’s equationsyield the right Lorentzforcelaw.

It is worthwhile to seewhat happensif we considerthe motion of an ensem-ble of nonrelativistic interactingparticlesin an electromagneticfield, wheretheinteractionforces

�} � � , � 6 � , arederivablefrom potentialsM � � M � � �Q �£ � # �£ � Q � .Fromthepreviousexamples,we areled to tryB~ ¨� � � � %L ¤ � {�£ (� # ¨� � � � � � � � �£ � � �� ? ¨� � � � � � �Î¹� �£ � � �� 9 {�£ � # %L ¨�� Ø�� M � � � (1.78)

Indeed,in this case��'�� @ B � @ U£ � i � � # @ B � @ £ � i � K7 is equivalentto ¤ � ¥£ � i � } � i � ?eÙ �)� � } � � i � , andtherefore�� @ B@ U£ � i � # @ B@ £ � i � �7 for all h % � L � W (1.79)

is equivalentto ¤ � {|{�£ � �} � ? � �)� � �} � � � (1.80)

which againyield theright equationsof motionfor the � th particle.Now weproceedto relativistic motion. In this case,werestrictourdiscussion

to themotionof asingleparticle.Thesituationfor anensembleis muchmoresub-tle for a varietyof reasons.First, theinteractionpotentialswould have to includeretardationeffects.Second,particlesmoving relativistically alsoproducecompli-catedmagneticfields;thereforetheinteractionis notmerelyderivablefrom scalarpotentials.In fact,thematteris socomplicatedthatit is not fully understood,andthereareeven seeminglyparadoxical situations in which particlesinteractingrelativistically keepaccelerating,gainingenergy beyond bounds(Parrott 1987;Rohrlich1990).


As a first step,let usconsidertherelativistic motionof a particleunderforcesderivablefrom potentialsonly. Theequationof motionis givenby�} �� -0 ¤ {�£Ú %# {�£ ( � � ( 13 � (1.81)

We try to find aLagrangianB �£y� {�£ � suchthat @ B � @ U£ � yields ¤ U£ � � Ú %# {�£ ( � � (and @ B � @ £ � givesthe h th componentof theforce,

} � , for h % � L � W . LetB �£4� {�£y� �� # ¤ � ( Ú %�# {�£ ( � � ( # M �£4� �� (1.82)

DifferentiatingB with respectto £ � � h % � L � W , we obtain@ B@ £ � # @ M@ £ � } � � (1.83)

DifferentiatingB with respecttoU£ � � h % � L � W , we obtain@ B@ U£ � # ¤ � ( � %L -0 %¾# {�£ (� ( 13 � � S ( � # LÛU£ �� ( ¤ U£ �Ú %# {�£ ( � � ( (1.84)

Thus,Lagrange’sequationsyield theproperequationof motion.Next we study the relativistic motion of a particle in a full electromagnetic

field. Basedon previousexperience,we expectthatwe merelyhave to combinethe termsthat lead to the Lorentz force with thosethat lead to the relativisticversionof theNewtonaccelerationterm,andhenceweareled to tryB¬ # ¤ � ( Ú %# {�£ ( � � ( ? � {�£.9 �Î # �'�4� (1.85)

where� is ascalarpotentialfor theelectricfield and�Î

is avectorpotentialfor themagneticfield. Sincethelasttermdoesnot contributeto

�� @ B � @ U£ � � , thever-ification thattheLagrangianis indeedtheproperonefollows like in thepreviousexamples.

1.4 HAMILTONIAN SYSTEMS

Besidesthecaseof differentialequationsthatcanbeexpressedin termsof a La-grangianfunction that wasdiscussedin the previous section,anotherimportant

HAMILTONIAN SYSTEMS 21

classof differentialequationsincludesthosethatcanbeexpressedin termsof aso-calledHamiltonian function.We begin our studywith theintroductionof theL ! Ì L ! matrix ÉÜ thathastheformÉÜ � É7 É�# É� É7 � (1.86)

where É� is the appropriateunity matrix. Beforeproceeding,we establishsomeelementarypropertiesof thematrix ÉÜ � We apparentlyhaveÉÜ k # ÉÜ �lÉÜ 9¹ÉÜ k �É� � and ÉÜ � � ÝÉÜ k # ÉÜ . (1.87)

From the last condition one immediatelyseesthat det ÉÜ � ] %

, but we alsohave det

ÉÜ � ? % , which follows from simplerulesfor determinants.We firstinterchangerow

%with row

"! ? % � , rowL

with row P! ? L � , etc.Eachof these

yieldsa factorof#�%

, for a total of �#�% �� We thenmultiply thefirst

!rows by ,#�% �

, which producesanotherfactorof ,#�% �,�

, andwe have now arrivedat theunity matrix; therefore,

det� É7 É�# É� É7 �#�% � � 9 det

� # É� É7É7 É� �#�% � � 9 �#�% � � 9 det� É� É7É7 É� % (1.88)

Wearenow readyto definewhenwecall asystemHamiltonian , or canonical.Let usconsidera dynamicalsystemdescribedby

L !first-orderdifferentialequa-

tions.For the sake of notation,let us denotethe first!

variablesby the columnvector

�Oandthesecond

!variablesby

�J , sothedynamicalsystemhastheform�� O �J �} � �O �J � (1.89)

The fact that we are apparentlyrestricting ourselves to first-order differentialequationsis no reasonfor concern:As seenin Eq. (1.3),any higherordersystemcanbe transformedinto a first-ordersystem.For example,any setof

!second-

orderdifferentialequationsin�O

canbe transformedinto a setofL !

first-order

differentialequationsby themerechoiceof�J {�O � Similarly, in principleanodd-

dimensionalsystemcanbemadeevendimensionalby the introductionof a newvariablethatdoesnot affect theequationsof motionandthatitself cansatisfyanarbitrarydifferentialequation.

We now call the system(Eq. 1.89) ofL !

first-order differential equationsHamiltonian, or canonical if thereexistsa function

Iof�O

and�J suchthat the


function�}

is connectedto thegradient @ I � @ �O � @ I � @ �J � of thefunction

Ivia�}Þ� �O �J ßÉÜ 9 @ I � @ �O � @ I � @ �J � k � (1.90)

In thiscase,wecall thefunctionI

aHamiltonian function or aHamiltonianof themotion.Theequationsof motionthenof courseassumetheform�� O �J ÝÉÜ 9 � @ I � @ �O@ I � @ �J � (1.91)

Writtenout in components,this reads�� O @ I@ �J (1.92)

and �� J # @ I@ �O � (1.93)

In the casethat the motion is Hamiltonian,the variables �O � �J � arereferredto

ascanonical variables, andtheL !

dimensionalspaceconsistingof theorderedpairs

��N �O � �J � is calledphasespace. Frequently, we alsosay that the�J are

the conjugate momenta belongingto�O, andthe

�Oarethe canonical positions

belongingto�J �

Note that in the caseof Hamiltonianmotion, different from the Lagrangiancase,the secondvariable,

�J , is not directly connectedto�O

by merelybeingthetime derivative of

�O, but ratherplaysa quite independentrole. Therefore,there

is oftenno needto distinguishthe two setsof variables,andfrequentlythey aresummarizedin thesinglephasespacevector

��à �O � �J � . While this makessomeof thenotationmorecompact,we oftenmaintaintheconventionof

�O � �J � for thepracticalreasonthat in this caseit is clearwhethervectorsare row or columnvectors,thedistinctionof which is importantin many of thelaterderivations.

We mayalsoaskourselveswhetherthefunctionI �O � �J � �� hasany specialsig-

nificance;while similar to B andmostly a tool for computation,it hasan inter-estingproperty. Let (

�O� �� J �� be a solutioncurve of the Hamilton differentialequations;thenthefunction

I �O� �� J �� evolving with thesolutioncurvesat-isfies � I�� @ I@ �O 9 � �O�� ?K@ I@ �J 9 � �J�� ?á@ I@ � @ I@ �O 9 @ I@ �J # @ I@ �J 9 @ I@ �O ?á@ I@ � @ I@ � � (1.94)


Therefore,in thespecialcasethatthesystemis autonomous, theHamiltonianisa preservedquantity.

1.4.1 Manipulationsof theIndependentVariable

As will beshown in this section,Hamiltoniansystemsallow variousconvenientmanipulationsconnectedto theindependentvariable.First, we discussa methodthatallows the independentvariable to bechanged. As shown in Eq. (1.5), inprinciple this is possiblewith any ODE; remarkably, however, if we exchange

�with a suitablepositionvariable,thentheHamiltonian structur e of themotioncanbe preserved. To seethis, let us assumewe aregiven a Hamiltonian

I I �O � �J � �� andsupposethat in the region of phasespaceof interestone of thecoordinates,e.g.,

O �, is strictly monotonicin time andsatisfies� O �� 687 (1.95)

everywhere.Theneachvalueof time correspondsto a uniquevalueofO �

, andthusit is possibleto use

O �insteadof

�to parametrizethemotion.

We begin by introducinga new quantitycalledJ k , which is thenegativeof theHamiltonian,i.e., J k # I � (1.96)

We thenobservethat @ J k@ J � # @ I@ J � # UO � 687�� (1.97)

which accordingto the implicit function theorem,entailsthat for every valueofthecoordinates

O � � O ( �� O � � J ( �� J � � � , it is possibleto invertthedependenceof J k on J � and to obtain J � asa function of J k � In this process,J k becomesavariable;in fact,thecanonicalmomentumJ k belongsto thenew positionvariable�. Also, themotionwith respectto thenew independentvariable

O �will beshown

to bedescribedby theHamiltonian� � � O ( �� O � � J k � J ( �� J � � O � � # J � � � O ( �� O � � J k � J ( �� J � � O � � �(1.98)

We observethatby virtue of Eq.(1.97),we have@ �@ J k # @ J �@ J k %UO � �� O � � (1.99)


Because� is obtainedby inverting the relationshipbetween J � and J k ,it follows that if we reinsert J k VO � �� O � � J � �� J � � �� # I PO � �� O � �J � �� J � � �� into � , all the dependencieson

� � O ( �� O � � J k � J ( �� J � � O �disappear;soif wedefine� � � � O ( �� O � � J k PO � �� O � � J k �� J � � �� J ( �� J � � O � � � (1.100)

thenin fact � # J � � (1.101)

In particular, this entailsthat7� @ �@ O � @ �@ O � ? @ �@ J k 9 @ J k@ O � for h L �� ! (1.102)7� @ �@ J � @ �@ J � ? @ �@ J k 9 @ J k@ J � for h L �� ! (1.103)

FromEq.(1.102),it followsunderutilization of Eq.(1.99)that@ �@ O � # @ �@ J k 9 @ J k@ O � %UO � 9 @ I@ O � # �� O � 9 � J �� # � J �� O � for h L �� ! � (1.104)

andsimilarly @ �@ J � # @ �@ J k 9 @ J k@ J � %UO � 9 @ I@ J � �� O � 9 � O �� O �� O � for h L �� ! � (1.105)

Sincewe alsohave 7$ @ � � @ � @ � � @ J k 9 @ J k � @ � ? @ � � @ � , it follows usingEq.(1.94)thatwe have@ �@ � # @ �@ J k 9 @ J k@ � �� O � 9 � I�� I� O � � (1.106)

Togetherwith Eq. (1.99), Eqs. (1.104), (1.105),and (1.106) representa set ofHamiltonianequationsfor theHamiltonian� � � O ( �� O � � J k � J ( �� J � � O � � , inwhichnow

O �playstheroleof theindependentvariable.Thus,theinterchangeof�

andO �

is complete.


Next, we observe that it is possibleto transformany nonautonomousHamil-toniansystemto anautonomoussystemby introducingnew variables,similar tothe caseof ordinarydifferentialequationsin Eq. (1.6). In fact, let

I �O � �J � �� betheL !

-dimensionaltime-dependentor nonautonomousHamiltonianof a dynam-ical system.For ordinarydifferentialequationsof motion,we remindourselvesthatall thatwasneededin Eq.(1.6)to makethesystemautonomouswasto intro-ducea new variable 5 thatsatisfiesthenew differentialequation

�� 5 % andthe initial condition 5 � G � � G , which hasthe solution 5 � andhenceallowsreplacementof all occurrencesof

�in theoriginaldifferentialequationsby 5 �

In the Hamiltonianpicture it is not immediatelyclear that somethingsimilaris possiblebecauseonehasto assertthat the resultingdifferentialequationscansomehow berepresentedby a new Hamiltonian.It is alsonecessaryto introducea matchingcanonicalmomentumJØâ to thenew variable 5 , andhencethedimen-sionality increasesby

L. However, now considertheautonomousHamiltonianofL P! ? % � variables ÉI �O � 5 � �J � J â � I �O � �J � 5 � ? J â � (1.107)

TheresultingHamiltonianequationsare�� O � @@ J � I @@ J � ÉI for �� % �� ! (1.108)�� J � # @@ O � I # @@ O � ÉI for �� % �� ! (1.109)�� 5 % @@ JØâ ÉI (1.110)�� JÈâ # �� I # @@ � I # @@ 5 ÉI (1.111)

Thethird equationensuresthat indeedthesolutionfor thevariable 5 is 5 � , asneeded,andhencethereplacementof

�by 5 in thefirst two equationsleadsto the

sameequationsaspreviously described.Therefore,theautonomousHamiltonian(Eq. 1.107)of

L "! ? % � variablesindeeddescribesthesamemotionastheorig-inal nonautonomousHamiltonianof

L !variables

I � The fourth equationis notprimarily relevantsincethe dynamicsof J â do not affect the otherequations.Itis somewhatilluminating,however, thatin thenew Hamiltoniantheconjugateofthe time canbechosenastheold Hamiltonian.

The transformationto autonomousHamiltonian systemsis performedfre-quentlyin practice,andthe

L "! ? % � dimensionalspace �O � 5 � �J � JÈâ � is referredto

astheextendedphasespace.In conclusion,weshow thatdespitetheunusualintermixedform characteristic

of Hamiltoniandynamics,thereis aratherstraightforwardandintuitivegeometricinterpretationof Hamiltonianmotion.To thisend,let

IbetheHamiltonianof an


FIGURE 1.2. Contoursurfacesof theHamiltonian.

autonomoussystem,eitheroriginally or after transitionto extendedphasespace,with theequationsof motion�� O �J ÝÉÜ 9 � @ I � @ �O@ I � @ �J � (1.112)

Then,we apparentlyhavethat @ I � @ �O � @ I � @ �J � 9 �� O �J (1.113) @ I � @ �O � @ I � @ �J � 9ãÉÜ 9 � @ I � @ �O@ I � @ �J �7�� (1.114)

Thus, the direction of the motion is perpendicular to the gradient ofI

andis constrainedto thecontoursurfacesof

I. Furthermore,thevelocity of motion

is proportionalto the magnitudeQä @ I � @ �O � @ I � @ �J � Q of the gradient.Sincethe

distancebetweencontoursurfacesis antiproportionalto thegradient,this is rem-iniscentto themotionof an incompressibleliquid within a structureof guidancesurfaces:Thesurfacescannever becrossed,andthecloserthey lie together, thefasterthe liquid flows. Figure1.2 illustratesthis behavior, which is oftenusefulfor anintuitiveunderstandingof thedynamicsof a givensystem.

1.4.2 ExistenceandUniquenessof Hamiltonians

Beforeweproceed,wefirst try to answerthequestionof existenceanduniquenessof a Hamiltonianfor a givenmotion.Apparently, theHamiltonianis not uniquebecausewith

I, also

I ?å� , where� is any constant,it yieldsthesameequationsof motion; asidefrom this, any othercandidateto be a Hamiltonianmusthavethesamegradientas

I, andthis entailsthat it differs from

Iby not morethan


a constant.Therefore,the questionof uniquenesscanbe answeredmuchmoreclearlythanit couldin theLagrangiancase.

Thesameis truefor thequestionof existence, at leastin astrictsense:In orderfor Eq.(1.90)to besatisfied,since ÉÜ � � # ÉÜ , it musthold that� @ I � @ �O@ I � @ �J �æ � �O �J # ÉÜ 9 �}Þ� �O �J � (1.115)

This, however, is a problemfrom potential theory, which is addressedin detaillater. In Eq. (1.300)it is shown that the condition @ æ�� @ £ � @ æ � � @ £ � for all�b� � is necessaryandsufficient for thispotentialto exist. Oncethese

! 9 "!ç#~% �� Lconditionsaresatisfied,theexistenceof aHamiltonianis assured,andit canevenbeconstructedexplicitly by integration,muchin thesameway asthepotentialisobtainedfrom theelectricfield.

In a broadersense,one may wonder if a given systemis not Hamiltonianwhetherit is perhapspossibleto make a diffeomorphictransformationof vari-ablessuchthat the motion in the new variablesis Hamiltonian.This is indeedfrequentlypossible,but it is difficult to classifythosesystemsthatareHamilto-nianup to coordinatetransformationsor to establisha generalmechanismto finda suitabletransformation.However, a very importantspecialcaseof suchtrans-formationswill bediscussedin thenext section.

1.4.3 TheDuality of HamiltoniansandLagrangians

As wementionedin Section1.4.2,thequestionof theexistenceof a Hamiltonianto a givenmotion is somewhat restrictive, andin generalit is not clearwhethertheremaybecertainchangesof variablesthatwill leadto a Hamiltoniansystem.However, for the very importantcaseof Lagrangiansystems,thereis a trans-formationthatusuallyallows theconstructionof a Hamiltonian.Conversely, thetransformationcanalsobeusedto constructaLagrangianto agivenHamiltonian.This transformationis namedafter Legendre,andwe will develop someof itselementarytheoryin thefollowing subsection.

LegendreTransformations

Let�

bea realfunctionof!

variables�£ thatis twice differentiable.Let

�æ �> �be thegradientof the function

�, which is a function from è � to è � � Let

�æbe

a diffeomorphism,i.e., invertibleanddifferentiableeverywhere.Then,we definetheLegendre transformation é V� � of thefunction

�asé V� � �æ � � 9 �� #¬� ^ �æ � � c � (1.116)

Here,��

is the identity function, and the dot “ 9 ” denotesthe scalarproductofvectors.


Legendretransformationsareausefultool for many applicationsin mathemat-icsnotdirectlyconnectedto ourneeds,for example,for thesolutionof differentialequations.Of all theimportantpropertiesof Legendretransformations,werestrictourselvesto theobservationthat if theLegendretransformationof

�exists,then

theLegendretransformationof é ê� � existsandsatisfiesé é V� �� (1.117)

Therefore,theLegendretransformation is idempotentor self-inverse.To showthis, we first observe thatall partialsof é V� � mustexist since

�æwasrequiredto

be a diffeomorphismand�

is differentiable.Using the productandchainrules,we thenobtainfrom Eq.(1.116)that�> é ê� � Jac

�æ � � � 9 �� ? �æ � � # �æ ^ �æ � � c 9 Jac �æ � � � �æ � �

(1.118)

andso é é V� �� ^ �æ � � c � � 9 �� # é ê� � ^)^ �æ � � c � � c �æ 9 �� # é V� � ^ �æ c �æ 9 �� # �� 9 �æ ? � � � (1.119)

asrequired.

LegendreTransformationof theLagrangian

We now apply the Legendretransformationto the Lagrangianof the motion in

a particularway. We will transformall{�O

variablesbut leave the original�O

vari-ablesunchanged.Let usassumethatthesystemcanbedescribedby aLagrangianB �O � {�O � �� , and the

!second-orderdifferential equationsare obtainedfrom La-

grange’sequationsas �� @ B@ UO � # @ B@ O � �7��4�< % �� ! � (1.120)

In orderto performthe Legendretransformationwith respectto the variables�O,

wefirst performthegradientof B with respectto{�O E

wecall theresultingfunction�J J � � J (�� J � � andthushave

J � @ B �O � {�O � ��@ UO � � (1.121)


The J � will play theroleof thecanonicalmomentabelongingtoO �

. TheLegendre

transformationexistsif thefunction�J {�O � is adiffeomorphism,andin particularif

it is invertible.In fact, it will bepossibleto obtaina Hamiltonianfor thesystemonly if this is the case.Thereare importantsituationswherethis is not so, forexample,for thecaseof therelativistically covariantLagrangian.

If�J {�O � is a diffeomorphism,then the Legendretransformationcan be con-

structed,andwe definethe functionI �O � �J � to be the Legendretransformation

of Bë� Therefore,we haveI �O � �J � é B � �O � �J � {�O� �J � 9 �J # B �O � {�O� �J � � �� (1.122)

As we shall now demonstrate,the newly definedfunctionI

actuallydoesplaythe role of a Hamiltonianof themotion.Differentiating

Iwith respectto anas

yetunspecifiedarbitraryvariablef , we obtainby thechainrule� I� f �� @ I@ O � � O �� f ? �� @ I@ J � � J �� f ?K@ I@ � �� f � (1.123)

On the otherhand,from thedefinitionofI

(Eq. 1.122)via the Legendretrans-formation,we obtain� I� f �� UO � � J �� f ? �� J � � UO �� f # �� @ B@ O�� O �� f # �� @ B@ UO�� UO �� f # @ B@ � �� f �� UO�� J �� f # �� @ B@ O � � O �� f # @ B@ � �� f �� UO � � J �� f # �� UJ � � O�� f # @ B@ � �� f � (1.124)

wherethedefinitionof theconjugatemomenta(Eq.1.121)aswell astheLagrangeequations(1.120)wereused.Choosingf O�� , J � , � , �ã % �� ! successivelyandcomparingEqs.(1.123)and(1.124),weobtain:UO � @ I@ J � (1.125)# UJ � @ I@ O�� (1.126)# @ B@ � @ I@ � � (1.127)

Thefirst two of theseequationsareindeedHamilton’sequations,showing thatthefunction

Idefinedin Eq.(1.122)is aHamiltonianfor themotionin thevariables

30 DYNAMICS OF PARTICLES AND FIELDS �O � �J � . The third equationrepresentsa noteworthy connectionbetweenthe timedependencesof HamiltoniansandLagrangians.

In somecasesof thetransformationsfrom Lagrangiansto Hamiltonians,it maybeof interestto performa Legendretransformationof only a selectgroupof ve-locities and to leave the other velocitiesin their original form. For the sake ofnotationalconvenience,let us assumethat the coordinatesthat are to be trans-formedarethefirst h velocities.Thetransformationis possibleif thefunctioninh variablesthatmaps

UO � �� UO � � intoJ � @ B@ UO�� for �< % �� h (1.128)

is a diffeomorphism.If this is thecase,theLegendretransformationwould leadto a functionè PO � �� O � � J � �� J � � UO � v � �� UO � � �� UO � �J � 9 J � # B PO � �� O � � UO � �J � �� UO � �J � � UO � v � �� UO � �

(1.129)

thatdependson thepositionsaswell asthefirst h momentaandthelast "!$# h �

velocities.Thisfunctionis oftencalledtheRouthian of themotion,andit satisfiesHamiltonianequationsfor thefirst h coordinatesandLagrangeequationsfor thesecond

P!à# h � coordinatesUO�� @ è@ J � � UJ � # @ è@ O � for �< % �� h7� �� @ è@ UO � # @ è@ O � for �� h�? % �� ! � (1.130)

LegendreTransformationof theHamiltonian

Sincethe Legendretransformationis self-inverse,it is interestingto studywhatwill happenif it is appliedto theHamiltonian.As is to beexpected,if theHamil-tonianwasgeneratedby a Legendretransformationfrom a Lagrangian,thenthisLagrangiancanberecovered.Moreover, evenfor Hamiltoniansfor which no La-grangianwasknownperse,aLagrangian canbegeneratedif thetransformationcanbeexecuted,which wewill demonstrate.

As thefirst step,wecalculatethederivativesof theHamiltonianwith respecttothemomentaandcall them

UO � � We getUO�� @ I@ J � (1.131)


(which,in thelight of Hamilton’sequations,seemssomewhattautological).If thetransformationbetweenthe

UO��andJ � is adiffeomorphism,theLegendretransfor-

mationcanbeexecuted;we call theresultingfunction B �O � {�O � �� andhaveB �O � {�O � �� {�O 9 �J �O � {�O � # I �O � �J �O � {�O � � �� (1.132)

In casethe Hamiltonianwasoriginally obtainedfrom a Lagrangian,clearly thetransformationbetweenthe

UO �andJ � is adiffeomorphism,andEq.(1.132)leadsto

theold Lagrangian,sinceit is thesameasEq.(1.122)solvedfor B �O � {�O � �� Ontheotherhand,if

Iwasnotoriginally obtainedfrom aLagrangian,wewantto show

that indeedthe newly createdfunction B �O � {�O � �� satisfiesLagrange’s equations.To this end,wecompute@ B@ O�� UO � @ J �@ O�� # @ I@ O�� # �� @ I@ J � @ J �@ O�� # @ I@ O�� (1.133)�� @ B@ UO�� J � � (1.134)

andhenceLagrange’sequationsareindeedsatisfied.Thisobservationis oftenimportantin practicesinceit offersonemechanismto

changevariables in a Hamiltonianwhile preservingthe Hamiltonianstructure.To this end,onefirst performsa transformationto a Lagrangian,thenthedesiredandstraightforwardcanonicalLagrangiantransformation,andfinally thetransfor-mationbackto thenew Hamiltonian.An additionalinterestingaspectis that theLagrangianB of asystemis notunique,but for any function

} �O � �� , accordingtoEq. (1.23),thefunction ÉB withÉB¬�B ? �� } �O � �� B ? �� @ }@ O � 9 UO � ? @ }@ � (1.135)

alsoyields the sameequationsof motion. Sucha changeof Lagrangiancanbeusedto influencethenew variablesbecauseit affectsthemomentaviaÉJ � @ ÉB@ UO � @ B@ UO � ? @ }@ O � J � ? @ }@ O � � (1.136)

Weconcludeby notingthatinsteadof performingaLegendretransformationof

themomentumvariable�J to avelocityvariable

{�O, becauseof thesymmetryof the

Hamiltonianequationsit is alsopossibleto computea Legendretransformation

of thepositionvariable�O

to a variable{�J � In this case,oneobtainsa systemwith

LagrangianB �J � {�J � ��


1.4.4 Hamiltoniansfor Particular Systems

In thissection,themethodof Legendretransformsis appliedto find Hamiltoniansfor severalLagrangiansystemsdiscussedin Section1.3.4.

NonrelativisticInteractingParticlesin General Variables

Weattemptto determineaHamiltonianfor thenonrelativistic motionof asystemof ì interacting particles. We begin by investigatingonly Cartesiancoordi-nates.Accordingto Eq. (1.57),theLagrangianfor themotionhastheformB~ ¨� � � � %L ¤ � {�£ (� # ¨� � � � M � �£ � � # � �Pª � M � � �Q �£ � # �£ � Q � � (1.137)

Accordingto Eq.(1.121),thecanonicalmomentaareobtainedvia�J � @ B � @ {�£ � ;here,we merelyhave �J � 8¤ � {�£ � � (1.138)

which is the sameas the conventionalmomentum.The relationsbetweenthecanonicalmomentaand the velocitiescan obviously be invertedand they rep-

resenta diffeomorphism.For the inverse,we have{�£ � �J � � ¤ � E accordingto Eq.

(1.122),theHamiltoniancanbeobtainedviaI �£ � � �J � � ¨� � � � �J � 9 {�£ � �£ � � �J � � # B �£ � � {�£ � �£ � � �J � �� ¨� � � � �J (�¤ � # ¨� � � � %L �J (�¤ � ? ¨� � � � M � �£ � � ? � �Pª � M � � �Q �£ � # �£ � Q �©¨� � � � �J (�L ¤ � ? ¨� � � � M �� £ � � ? � �"ª � M � �b �Q �£ �x# �£ � Q � � (1.139)

In this case,the entireprocessof obtainingthe Hamiltonianwasnext to trivial,mostlybecausetherelationshipbetweenvelocitiesandcanonicalmomentais ex-ceedinglysimpleandcaneasilybeinverted.Weseethatthefirst termis thekineticenergy « , andthesecondtermis thepotentialenergy M , sothatwehaveI «~?~M � (1.140)

Let usnow studythecaseof W ì generalizedvariables��; theLagrangianstill has

theform B~ « {�� # M �� (1.141)


where« {�� now hastheform« {�� %L 9 {�� k 9ÅÉ� Ê 9 {�� (1.142)

whereaccordingto Eq. (1.69), É� Ê is a nonsingularsymmetricmatrix. We thenhave �í @ B@ {�� @ «@ {�� É� Ê 9 {�� (1.143)

andsince É� Ê is nonsingular, thisrelationshipcanbeinverted;therefore,theLeg-endretransformationcanbeperformed,andweobtainI R ¨� � � � í � 9 U� � # B �í k 9ÅÉ� � �Ê 9 �í # %L 9 �í k 9 Ï É� � �Ê Ñ k 9ÅÉ� Ê 9$É� � �Ê 9 �í ?~M %L 9 �í k 9$É� � �Ê 9 �í ?NM «�?NM E (1.144)

Again, theHamiltonianis givenby «¬?eM for thisspecialcase.

Particlesin ElectromagneticFields

We now considerthe relativistic motion of a single particle in an electromag-netic field. Recall that the following suitableLagrangianfor this problemwasobtainedin Eq. (1.85)andfoundto beB �£�� {�£y� �� # ¤ � ( Ú %�# {�£ ( � � ( ? � {�£.9 �Î �£4� �� # �'� �£4� �� (1.145)

Now we proceedto find the Hamiltonianfor themotion of the particle.Let�J

denotethe canonicalmomentumof the particle; then,for h % � L � W , we haveJ � @ B � @ U£ � &îÆ¤ U£ � ? � Î � with îà % � Ú %�# {�£ ( � � ( , or in vectorform,�J &îØ¤ {�£ ? � �Î � (1.146)

In this case,thecanonicalmomentumis differentfrom theconventionalmomen-tum. However, utilizing theLegendretransformation,it is possibleto proceedina ratherautomatedway. We first studywhetherit is possibleto solve Eq. (1.146)


for{�£ . Apparently,

�J # � �Î � eîÈ¤ {�£ , andthus �J # � �Î � ( 8¤ ( {�£ ( � �%# {�£ ( � � ( � �

After somesimplearithmetic,we obtain{�£ ( �J # � �Î � ( � ( �J # � �Î � ( ? ¤ ( � ( � (1.147)

andnotingthataccordingto Eq. (1.146),{�£ is alwaysparallelto

�J # � �Î � , wecanevenconcludethat {�£È � �J # � �Î �Ú �J # � �Î � ( ? ¤ ( � ( � (1.148)

Indeed,the transformation(Eq. 1.146)could be solved for{�£ and it represents

a diffeomorphism,which ensuresthe existenceof the Hamiltonian.Proceedingfurther, weseethat Ú %�# {�£ ( � � ( ¤ �Ú �J # � �Î � ( ? ¤ ( � ( �Substitutingtheselast two resultsinto theLagrangian(Eq.1.145),we obtaintheHamiltonian:I �J 9 {�£ # B �£4� {�£ �£y� �J �� J 9 � �J # � �Î �Ú �J # � �Î � ( ? ¤ ( � ( ? ¤ ( � RÚ �J # � �Î � ( ? ¤ ( � ( # � � �J # � ( �Î � 9 �ÎÚ �J # � �Î � ( ? ¤ ( � ( ? �'� � 9 �J ( # L � �J �Î ? ¤ ( � ( ? � ( �Î (Ú �J # � �Î � ( ? ¤ ( � ( ? �'� � 9 Ú �J # � �Î � ( ? ¤ ( � ( ? �'� (1.149)

1.4.5 CanonicalTransformationof Hamiltonians

Forsimilarreasonsasthosediscussedin Section1.3.2for Lagrangians,for Hamil-toniansit is importantto studyhow theHamiltonianstructurecanbemaintainedunderachangeof coordinates. For example,if it canbedeterminedby achangeof variablethat the Hamiltoniandoesnot explicitly dependon

O �, then

UJ � ï7 ,andtheconjugatemomentumcorrespondingto

O �is a constant.

In thecaseof theLagrangian,it turnedout thatregardlessof whatnew coordi-nates

��areusedto describethesystem,in orderto obtaintheLagrangianfor the


new variables��

and{��, it is necessaryonly to insertthe functionaldependence

of�O

on��

aswell asthe functionaldependenceof{�O

on��

and{��

resultingfromdifferentiationinto theold Lagrangian.

Thequestionnow ariseswhetherthereis asimilarly simplemethodfor thecaseof Hamiltonians.In thecaseof theLagrangian,whenchangingfrom

�Oto��, the

velocity-like coordinates{�O

and{��

are directly linked to the respective positioncoordinates

�Oand

��by virtue of thefact that they arethetime derivativesof the

positions.In theHamiltoniancase,thesituationis morecomplicated:If we wantto transformfrom

�Oto��, it is not automaticallydeterminedhow the canonical

momentum�í

belongingto the new coordinate��

is to be chosen,but it seemsclearthatonly very specificchoiceswould preserve theHamiltonianstructureofthe motion. Furthermore,the fact that

�J is not directly tied to�O

and,asshownin the last section,the rolesof

�Oand

�J areratherinterchangeable,suggeststhatit may be possibleto changethe momentum

�J to our liking to a moresuitablemomentum

�íand,if wedo so,to choosea“matching”

��.

In orderto establishthetransformationpropertiesof Hamiltonians,it is in gen-eral necessaryto transform positions and momenta simultaneously, i.e., tostudy

L !dimensionaldiffeomorphisms �� í � j �O � �J � � (1.150)

We now studywhat happensif it is our desireto changefrom the old variables �O � �J � to the new variables �� í � � It is our goal to obtain the Hamiltonianfor

thevariables �� í � in thesamewayasoccurredautomaticallyin thecaseof the

Lagrangianin Eq. (1.34), i.e., by mereinsertionof the transformationrules,sothat � �� í � I �O� �� í � � �J �� í �� (1.151)

In termsof themap j , we want� I j � � �and

I � j � � (1.152)

If a transformationj satisfiesthis conditionfor every HamiltonianI

, we callthe transformationcanonical, indicatingits preservationof the canonicalstruc-ture.

Thesetof canonicaltransformationsof phasespaceform a group undercom-position.Obviously, the identity transformationis canonical.Furthermore,sinceeachtransformationis a diffeomorphism,it is invertible, and the inverseis acanonicaltransformationsinceit merelyreturnsto theoriginal Hamiltoniansys-tem.Finally, associativity is satisfiedbecausecompositionis associative.


ElementaryCanonicalTransformations

Thereare a few straightforward examplesfor canonicaltransformationscalledelementarycanonicaltransformations.First, it is apparentlypossibleto changethelabelingof thevariables;hence,for agiven h and ð , thetransformation� � �í � � O ÕJ Õ � � � Õí Õ � O �J � and � � �í � � O �J � for � 6 h � ð (1.153)

is canonical.Thereareapparently!ñ "!Å#N% �� L

suchtransformations.A moreinterestingtransformationis thefollowing one:for a given h , set� � �í � � J �#ëO � � and

� �ò�í � � O��J � for � 6 h � (1.154)

To verify thatthis is in factcanonical,consideranarbitraryHamiltonianI �O � �J �

with Hamiltonianequations� UO��UJ � � � @ I � @ J �# @ I � @ O�� ¬� (1.155)

In light of (1.153),we mayassumethat h % � Thenew Hamiltonian ÉI thenisgivenby ÉI ê� � � � (�� í � � í (�� í � � I J � � O ( �� O � � #ëO � � J ( �� J � � � (1.156)

andwe have @ ÉI@ í � # @ I@ O � UJ � U� �@ ÉI@ � � @ I@ J � UO � # Uí<� � (1.157)

which are the proper Hamilton equations.For the other coordinatesnothingchanges,andaltogetherthetransformationis canonical.While essentiallytrivial,thetransformationis of interestin thatit stressesthatpositionsandmomentaplayessentiallyidentical roles in the Hamiltonian;this is different from the caseof

theLagrangian,in which the{�O

variableis alwaysobtainedasa quantityderivedfrom

�O �


CanonicalTransformationandSymplecticity

We now derive a fundamentalcondition that canonicaltransformationsmustsatisfy—theso-calledsymplecticconditionor conditionof symplecticity. Let usassumewe aregivenadiffeomorphismof phasespaceof theform �� í � j �O � �J � � (1.158)

Let theJacobianof j begivenby

Jac j � � @ �� @ �O @ �� @ �J@ �í � @ �O @ �í � @ �J (1.159)

We want to study the motion in the new variables.Apparently, we haveU�� Ù � � � � @ �� @ O � 9 UO � ? @ �� @ J � 9 UJ � � and

Uí � Ù � � � � @ í � � @ O � 9 UO � ? @ í � � @ J � 9 UJ � �FEusingtheJacobianmatrix of j , wehave�� í Jac(j � 9 �� O �J Jac(j � 9:ÉÜ 9 � @ I � @ �O@ I � @ �J � (1.160)

Furthermore,becauseI � j �

, we have @ I � @ O�� Ù � � � � @ � � @ � � 9@ � � � @ O�� ? @ � � @ í � 9 @ í � � @ O�� and @ I � @ J � Ù � � � � @ � � @ � � 9 @ � � � @ J � ?@ � � @ í � 9 @ í � � @ J � . In matrix form this canbewrittenas� @ I � @ �O@ I � @ �J Jac j � k 9 � @ � � @ ��@ � � @ �í � (1.161)

andcombinedwith thepreviousequationwehave�� í Ï Jac(j � 9�ÉÜ 9 Jac(j � k Ñ 9 � @ � � @ ��@ � � @ �í � (1.162)

This equationdescribeshow themotionin thenew variablescanbeexpressedintermsof the gradientof the Hamiltonianexpressedin the new variables.If wewantto maintaintheHamiltonianstructure,it is necessaryto have�� í óÉÜ 9 � @ � � @ ��@ � � @ �í � (1.163)

Apparently, this requirementcanbemetif thetransformationmap j satisfiesÏ Jac(j � 9�ÉÜ 9 Jac(j � k Ñ ßÉÜ � (1.164)

which is commonlyknown as the symplectic condition or condition of sym-plecticity. A map j is calledsymplecticif it satisfiesthesymplecticcondition.


Thus,any symplecticmapproducesa transformationthatyieldsnew coordinatesin which themotionis againHamiltonian.

We have seenthat for thetransformationfrom theold to thenew coordinates,for a given Hamiltonian,symplecticityis sufficient to preserve the Hamiltonianstructureof the motion. Therefore,symplectic maps are canonical. However,therearesituationsin which symplecticityis not necessaryto providea transfor-mationresultingagainin a Hamiltonianform; if we considera two-dimensionalHamiltonian

I PO � � O (�� J � � J ( � that in fact doesdependon onlyO �

and J � , thenwhile it is importantto transform

O �andJ � properly, thetransformationof

O ( andJ ( is insignificantandhencetherearemany nonsymplecticchoicespreservingtheHamiltonianform.

On the other hand,if the demandis that the transformationj � �O � �J ��ô �� í � transformevery Hamiltoniansystemwith a givenHamiltonianI

into anew Hamiltoniansystemwith Hamiltonian � I j � � �

, thenit is alsonec-essarythat j be symplectic.Therefore,canonical transformations are sym-plectic. Indeed,consideringthe

L !Hamiltonians� � �� for �¾ % �� ! and� � v � í � for �� % �� ! , showsthatfor eachof the

L !columnsof therespec-

tivematricesin Eq. (1.162),wemusthave^Jac(j � 9 �Ü 9 Jac(j � k c ßÉÜ �

The symplecticconditioncanbe castin a variety of forms. Denoting É� Jac j �

, wearrivedat theform É� 9 ÉÜ 9 É� k ÉÜ � (1.165)

whichis alsoprobablythemostcommonform.Multiplying with ÉÜ � � # ÉÜ fromtheright, we seethat É� � � # ÉÜ 9ÅÉ� k 9ãÉÜ � (1.166)

which is aconvenientformulafor theinverseof theJacobian.Multiplying with ÉÜfrom theright or left, respectively, weobtainÉÜ 9ÅÉ� k õÉ� � � 9:ÉÜ and É� k 9ãÉÜ óÉÜ 9$É� � � . (1.167)

TransposingtheseequationsyieldsÉ� 9ãÉÜ ÝÉÜ 9 É� � � � k and ÉÜ 9ÅÉ� É� � � � k 9¹ÉÜ , (1.168)

while multiplying thesecondequationin Eq. (1.167)with É� from theright, wehave É� k 9:ÉÜ 9ÅÉ� óÉÜ � (1.169)

Thereis actuallyahighdegreeof symmetryin theseequations,whichalsogreatlysimplifiesmemorization:All equationsstayvalid if ÉÜ or É� arereplacedby theirtranspose.


Propertiesof SymplecticMaps

It is worthwhile to study someof the propertiesof symplecticmapsin detail.First,weobservethatsymplecticdiffeomorphismsform a group undercomposi-tion. Theidentity mapis clearlysymplectic.Furthermore,for a givensymplecticdiffeomorphismj , we haveby virtueof Eq. (1.166)that

Jac j � � � Jac

j � � � # ÉÜ 9 Jac j k � 9ãÉÜ � (1.170)

andweobtain

Jac j � � � 9¹ÉÜ 9 Jac

j � � � k �# ÉÜ 9 Jac j � k 9 ÉÜ � 9 ÉÜ 9 �# ÉÜ 9 Jac

j � k 9 ÉÜ � k # ÉÜ 9 Jac j � k 9 ÉÜ 9 Jac

j � 9 ÉÜ # ÉÜ 9¹ÉÜ 9¹ÉÜ óÉÜ � (1.171)

which shows that the inverseof j is symplectic.Finally, let j �and j ( be

symplecticmaps;thenwe haveJac(j � p j ( � Jac j � � 9 Jac

j ( � , andso

Jac j � p j ( � 9ãÉÜ 9 Jac

j � p j ( � k Jac j �� 9 Jac

j ( � 9:ÉÜ 9 Jac j ( � k 9 Jac

j �� k Jac j �� 9¹ÉÜ 9 Jac

j �� k óÉÜ � (1.172)

wherethesymplecticityof j �and j ( hasbeenused.

A particularly importantclassof symplecticmapsare the linear symplectictransformations,which form a subgroupof the symplecticmaps.In this case,theJacobianof themapis constanteverywhereandjust equalsthematrix of thelineartransformation,andthecompositionof mapscorrespondsto multiplicationof thesematrices.Thisgroup of

L ! Ì L ! symplecticmatrices is usuallydenotedby z J L ! � �

In a similar way astheorthogonaltransformationspreserve thescalarproductof vectors,thesymplecticmatricesin z J L ! � preservetheantisymmetric scalarproduct of thecolumnvectors

�}and

�æ, definedbyö �} � �æò÷ �} k 9¹ÉÜ 9 �æ � (1.173)

Indeed,for any matrix ÉÎùø z J L ! � , we haveö ÉÎ 9 �} � ÉÎ 9 �æò÷ Ï ÉÎ 9 �} Ñ k 9 ÉÜ 9 Ï ÉÎ 9 �æ Ñ (1.174) �} k 9 Ï ÉÎ k 9¹ÉÜ 9:ÉÎÆÑ 9 �æ �} k 9¹ÉÜ 9 �æ ö �} � �æò÷ � (1.175)


Below, we will recognizethat the antisymmetricscalarproductis just a specialcaseof thePoissonbracket for thecaseof linearfunctions.

It is also interestingto study the determinant of symplecticmatrices.Be-causedet

ÉÜ � %, we have from the symplecticcondition that

% �ú�û�ü ÉÜ � ú�û�ü Jac j � 9:ÉÜ 9 Jac

j � k � 8ú�û�ü Jac j �� (

, andsoú�û�ü Jac j �� ] % � (1.176)

However, we canshow that thedeterminantis always ? % � This is connectedto aquiteremarkableandusefulpropertyof theantisymmetricscalarproduct.

Let usconsiderL !

arbitraryvectorsin phasespace��+ý ��þ , � % �� L ! � Let us

arrangetheseL !

vectorsinto thecolumnsof a matrix Éÿ , sothatÉÿ -/0 � ý � þ� � ý ( � þ�. . .� ý � þ( � � ý ( � þ( �

1�23 � (1.177)

Let � denotea permutationof� % �� L !�� and � � � its signature.We now study

theexpression� � � � � � 9 ö �� ý � þ � �� ý ( þ ÷ 9�9�9�99 ö �� ý ( �� þ � �� ý ( � þ ÷ � (1.178)

Performingall productsexplicitly usingthematrix elements ÉÜ � i � of thematrix ÉÜ ,weobtain� �� m i {|{|{ i � n�� ÉÜ � m i � n 9ÈÉÜ � o i � 9V9�9�9V9ÈÉÜ � n�� m i � n � 9 � � � � � 9ä� � ý � þ� m � � ý ( þ� n 9�9�9T� � ý ( � þ� n�� (1.179)

Observethatthelasttermis thedeterminantof thematrixobtainedfrom Éÿ by re-arrangingtherows in theorder � � ��w('��w( � � Becausethis determinantvanishesif any two rowsareequal,only thosechoices� � ��w(��À( � contributewherethenumbersarepairwisedistinct,andhencea permutation�� of

� % ��F� L !�� In thiscase,thevalueof thelastproductequals� �� 9�ú�û�ü Éÿ � � andwehave� Ï � �� ÉÜ �� ý � þ i �� ý ( þ 9ãÉÜ �� ý R þ i �� ý a þ 99�9�9�9ãÉÜ �� ý ( �� þ i �� ý ( � þ 9 � �� Ñ 9�ú�û�ü Éÿ � � � 9Hú�û�ü Éÿ � (1.180)

Let usnow studythesum � � , which is independentof thevectors��+ý ��þ � To obtain

a contribution to the sum,all factors ÉÜ��i � mustbe nonzero,which requires� � ] ! � Onesuchcontribution is � � � ��w( � � � R �� a � �� w( �� Fi ( � �� ,% � ! ? % � � L � ! ? L � �� P! � L ! �� (1.181)


which hasa signatureof ,#�% �,�� ý �� þ S ( � It correspondsto the casein which for

all!

factors � �� ? ! andhencethe productof theÜ

matrix elementsis% �

Theothercontributionsareobtainedin the following way. First, it is possibletoexchangeentiresubpairs,which doesnot affect the signatureandstill yields aproductof ÉÜ matrix elementsof

%; thereapparentlyare

!��waysfor suchanex-

change.Secondly, it is possibleto flip two entrieswithin asubpair � � �<? ! � , each

of which changesboth the signatureandthe signof theproductof the ÉÜ matrixelements,thusstill yielding the contribution

%; thereareapparently

L �waysfor

sucha flip. Altogether, we haveL � 9 !�� permutationsthatyield contributions,all

of whichhavethesamemagnitude ,#�% �� ý �� þ S ( astheoriginalone,andwethus

obtain � � ,#�% � �� ý �� þ S ( 9 L � 9 !�� (1.182)

Therefore,the antisymmetricscalarproductallows for the determinationof thevalueof adeterminantmerelyby subsequentscalarmultiplicationsof thecolumnsof thematrix.

While interestingin its own right, this factcanbeusedasakey to ourargumentsincetheantisymmetricscalarproductis invariantunderasymplectictransforma-tion.Let É� besuchatransformation.For Éÿ wechoosethespecialcaseof theunitmatrix,andargueasfollows:ú�û�ü É� � �ú�û�ü É� 9ëÉÿ � &ú�û�ü Ï É� 9 �� É� 9 �� ( � Ñ � � � � �� 9 ö É� 9 �� ý � þ �ÇÉ� 9 �� ý ( þ ÷ 9�9�9�99 ö É� 9 �� ý ( �� þ �ÇÉ� 9 �� ý ( � þ ÷ � � � � �� 9 ö �� ý � þ � �� ý ( þ ÷ 9�9�9�9�9 ö �� ý ( �� þ � �� ý ( � þ ÷�ú�û�ü Éÿ � % � (1.183)

A very importantconsequenceof thefact that thedeterminantof theJacobianof a symplecticmapis unity is thatthis meansthatthemappreservesvolume inphasespace.We studyameasurableset z � of phasespace,andlet z ( j z � �be the setthat is obtainedby sendingz � throughthe symplecticmap j � Thenthevolumesof z � and z ( , denotedby M � and M ( , areequal.In fact,accordingtothesubstitutionruleof multidimensionalintegration,we haveM (¹_�� n � � �� í_�� m ú�û�ü Jac

j ��¾� � �O � � �J_�� m � � �O � � �J M � � (1.184)


PoissonBracketsandSymplecticity

Let�

and = beobservables,i.e.,differentiablefunctionsfrom phasespaceinto è .We definethePoissonbracket of

�and = as� � � =�� @ � � @ �O � @ � � @ �J � 9ãÉÜ 9 @ = � @ �O � @ = � @ �J � k �� @ �@ O�� @ =@ J � # @ �@ J � @ =@ O�� (1.185)

For example,considerthespecialchoicesfor thefunctions�

and = of theformsO �andJ � � We obtainthatfor all �b� � % �� ! ,� O�� O � � 87�� O�� J � � Ó � � � � J � � J � � 87+� (1.186)

Amongotheruses,thePoissonbracketsallow a convenientrepresentationof thevector field (Eq.1.7)of aHamiltoniansystem.Let = �� bea functionof phasespace;accordingto Eqs.(1.91)and(1.185)weobtainfor thevectorfield B CD thatB$CD = U= �� @ =@ O�� UO � ? @ =@ J � UJ � ? @ =@ � �� @ =@ O � @ I@ J � # @ =@ J � @ I@ O � ? @ =@ � � = � I ��? @ =@ � � (1.187)

We observe in passingthat replacing= byO �

and J � �Û�� % �� ! , we recoverHamilton’sequationsof motion,namely

UO � @ I � @ J � andUJ � # @ I � @ O � .Introducing �! å� asa “Poissonbracket waiting to happen,” i.e., asanoperator

on thespaceof functions = �O � �J � �� , thatactsas �" ~�Ø= � � =�� , the vectorfieldcanalsobewrittenas B$CD # � I ��? @ � @ � � (1.188)

Above, we recognizedthat the antisymmetricscalarproductis a specialcaseof thePoissonbracketwhenthefunctions

�and = arelinear. In this case,we can

write�< �� } k 9 �� and = �� æ k 9 �� , and� � � =�� } k 9¹ÉÜ 9 �æ � (1.189)

Similar to thewayin whichthesymplecticmatricespreservetheantisymmetricproduct,symplecticmapspreserve Poissonbrackets. By this we meanthat if

HAMILTONIAN SYSTEMS 43j is a symplecticmap,thenfor any observables,i.e.,differentiablefunctionsofphasespace

�and = , we have� � p j � = p j#� � � � =�� p j � (1.190)

This factfollowsquitereadilyfrom thechainruleandthesymplecticityof j :� � p j � = p j#� �> ê� p j � 9 ÉÜ 9 �> = p j �� k � �> � � p j � 9 Jac j � 9:ÉÜ 9 Jac

j � k 9 � �> = � p j � k � �> � � p j � 9:ÉÜ 9 � �> = � p j � k � � � =�� p j � (1.191)

In particular, this entailsthatif we take for�

and = thecomponentsO �

andJ � andwrite thesymplecticmapas j � �%$ � , thenwe seefrom Eq.(1.186)that� � � � � � � �7�� %$ � � Ó � � � � $ � �&$ � � 87 (1.192)

for all �b� � % �� ! � Thisis thesamestructureasEq.(1.186),andthuswespeakof thepreservationof theso-calledelementaryPoissonbrackets.Conversely, weshow now that preservation of all elementaryPoissonbrackets implies sym-plecticity.

Let j � �&$ � be an arbitrary diffeomorphismon phasespace;then weapparentlyhave� � p j � = p j#� �> V� p j � 9:ÉÜ 9 �> = p j �� k � �> � � p j � 9 Jac

j � 9:ÉÜ 9 Jac j � k9 � �> = � p j � k (1.193)

Let j preservetheelementaryPoissonbrackets.Now weconsidertheL ! 9 L !

casesof choosingO �

and J � for both�

and = . We observe thatO � p j � � andJ � p j '$ � � Furthermore,

�> O��and

�> J � areunit vectorswith a%

in coordinate� and � ? ! , respectively, and 7 everywhereelse.Eachof theL ! 9 L ! choicesfor�

and = , hence,projectout a differentmatrix elementof Jac j � 9�ÉÜ 9 Jac

j � kon theright of theequation,while on the left we have therespective elementaryPoissonbrackets.Sincethesearepreservedby assumption,we have

Jac j � 9¹ÉÜ 9 Jac

j � k ßÉÜ (1.194)

andthemap j is symplectic.To finalizeourdiscussionof thePoissonbracket,weeasilyseefromEq.(1.185)

thatfor all observables�

and = , we have� � � =�� # � = � � � (1.195)


and � � ?¬= � (� � � � ��? � = � �� and� � 9 � � =�� 9 � � � =�� (1.196)

Thus� � � is an antisymmetricbilinear form. Moreover,

� � � satisfiestheJacobiidentity � � � � = � (�)��? � = � � � � ��? � � � � � =��)� 87 for all

� � = � � (1.197)

which follows from a straightforwardbut somewhatinvolvedcomputation.Alto-gether, thespaceof functionson phasespacewith multiplication

�+* = � � � =�� ,formsa Lie Algebra.

1.4.6 UniversalExistenceof GeneratingFunctions

After having studiedsomeof thepropertiesof canonicaltransformations,in par-ticular their symplecticityand the preservation of the Poissonbracket, we nowaddressa representationof the canonicaltransformationvia a so-calledgener-ating function. Later, we will also show that any generatingfunction yields acanonicaltransformationwithout having to imposefurtherconditions;thus,gen-eratingfunctionsprovide a convenientandexhaustive representationof canoni-cal transformations.We begin by restatingthesymplecticconditionin oneof itsequivalentforms(Eq.1.167): ÉÜ 9 É� k É� � � 9 ÉÜ (1.198)

Writing out theequationsreads� 7 É�# É� 7 9 � @ �� @ �O � k @ �í � @ �O � k @ �� @ �J � k @ �í � @ �J � k � @ �O � @ �� @ �O � @ �í � @ �J � @ �� @ �J � @ �í � 9 � 7 É�# É� 7 (1.199)

andso� @ �� @ �J � k @ �í � @ �J � k#ò @ �� @ �O � k #ò @ �í � @ �O � k � #ò @ �O � @ �í � @ �O � @ �� #ò @ �J � @ �í � @ �J � @ �� (1.200)

Writing theconditionsexpressedby thesubmatricesexplicitly yields@ � �@ J � # @ O �@ í � � @ í �@ J � @ O �@ � �@ � �@ O�� @ J �@ í � � @ í �@ O�� # @ J �@ � � � (1.201)


At first glance,theseequationsmayappearto bea strangemixtureof partialsof the coordinatetransformationdiffeomorphismand its inverse.First, a closerinspectionshows that they areall related;theupperright conditionfollows fromtheupperleft if wesubjectthemapunderconsiderationto theelementarycanoni-cal transformation

�� í �ñô �í � # �� , andthelowerrow followsfrom theupperoneby thetransformationof theinitial conditionsvia

�O � �J �ñô �J � # �O � �Becauseof this fact we may restrict our attentionto only one of them, the

lower right condition.This relationshipcanbeviewedasa conditionon a map ,in which thefinal andinitial momenta

�íand

�J areexpressedin termsof thefinalandinitial positions

��and

�Ovia� �í �J � , C- �� O �, C. �� O � � , C-, C. � �� O � (1.202)

if suchamap , exists.In thiscase,theconditionsarereminiscentof theconditionfor theexistenceof a potential(Eq.1.300).Indeed,let usassumethat@ í �@ O�� # @ J �@ � � , (1.203)

and @ í �@ � � @ í �@ � � (1.204)

and @ J �@ O � @ J �@ O � � (1.205)

SettingO0/� #ëO � , Eqs.(1.203),(1.204),and(1.205)apparentlyrepresentintegra-

bility conditions(1.300)assertingtheexistenceof a function} / �� O / � suchthat, - i � @ } / � @ � � and , . i � @ } / � @ O0/� � Introducingthefunctionof theoriginal

variables} �� O � } / �� # �O / � , weobtain, - i � @ }@ � � and , . i � # @ }@ O � � (1.206)

In passingwe notethattheminussignappearingin theright equationis immate-rial, asthechoiceof

} �� O � # } /� �� # �O0/ � would leadto a minussignin theleft equation;clearly, however, oneof theequationshasto havea minussign.

Thisrepresentationof thecanonicaltransformationby asinglefunctionis rem-iniscentof thefact thatevery canonicaldynamicalsystemcanberepresentedbya singleHamiltonianfunction

I. However, herethe representationis somewhat

indirectin thesensethatold andnew coordinatesaremixed.


Of coursethe map , mentionedpreviously neednot exist a priori ; for ex-ample,if thecanonicaltransformationunderconsiderationis simply the identitymap,it is not possibleto find sucha , � However, in the following we establishthat , anda generatingfunction alwaysexists, aslongastheunderlyingmapisfirst subjectedto asuitablecombinationof elementarycanonicaltransformations.

Let j be a canonicaltransformation,let � and $ denotethe position andmomentumpartsof j , and let

r"1and

r . denotethe positionandmomentumpartsof theidentity map.Then,we have� �� í � � �O � �J �$ �O � �J � � �$ � �O �J (1.207)

andhence � �� O � �r"1 � �O �J � (1.208)

If themap � � r"1 � k canbeinverted,we have� �O �J � �r"1 � � � �� O � (1.209)

from which we thenobtainthedesiredrelationship� �í �J � $r . p � �r 1 � � � �� O (1.210)

whichexpressestheold andnew momentain termsof theold andnew positions.First, we have to establishthe invertibility condition for the map

� � r"1 � k �Accordingto the implicit function theorem,this is possiblein even a neighbor-hoodof a point as long as the Jacobiandoesnot vanishat the point. SincetheJacobianof

r"1containsthe identity matrix in the first

!columnsand the zero

matrix in thesecond!

columns,weconcludethatú�û�ü32 Jac� �r"1 54 ,#�% � � 9 ¡¡¡¡¡¡¡ @ �

�� @ J � �� @ � �� @ J �......@ � � � @ J � �� @ � � � @ J �

¡¡¡¡¡¡¡ � (1.211)

The determinanton the right is denotedby 6 � In general,it cannotbe ensuredthat 6 687 for agivencanonicaltransformation.In fact,asalreadymentioned,inthecaseof the identity transformationthefinal

��doesnot dependon the initial�J , andthedeterminantwill certainlybezero.However, aswe now show, for any

givenpoint�� in phasespaceit is possibleto subjectthecoordinatesto asequence

of elementarycanonicaltransformationsdiscussedpreviously, consistingonly of


rearrangementsof thepairs VO � � J � � of thecoordinatesandof exchangesof

PO � � J � �with

J � i #åO�� , suchthatthedeterminant is nonzero at��+�

We prove this statementby induction. As thefirst step,we observe thatsince� � �Fi $ � � % becauseof symplecticityof j , notall partialderivativesof � � withrespectto

O�� J � canvanish.We thenrearrange thevariables �O � �J � andpossibly

performanexchangeofO �

andJ � , suchthat @ � � � @ J � 6�7��We now assumethestatementis truefor ¤ [ !; i.e., thesubdeterminant6 Ö

definedby

6 Ö ¡¡¡¡¡¡¡ @ �� @ J � �� @ � �� @ J Ö...

...@ � Ö:� @ J � �� @ � Öã� @ J Ö¡¡¡¡¡¡¡ (1.212)

satisfies6 Ö 6 7 for all ¤ [ ! � We thenshow that, after suitableelementarycanonicaltransformations,wewill alsohave 6 Ö v � 6�7�� To thisend,weconsiderallL !à# ¤ determinants

6 ý . þÖ v � ¡¡¡¡¡¡¡¡¡@ � �� @ J � �� @ � �� @ J Ö @ � �� @ J �...

......@ � Ö�� @ J � �� @ � Ö:� @ J Ö @ � Ö�� @ J �@ � Ö v �� @ J � �� @ � Ö v �� @ J Ö @ � Ö v �� @ J �

¡¡¡¡¡¡¡¡¡ � (1.213)

6 ý 1 þÖ v � ¡¡¡¡¡¡¡¡¡@ � � � @ J � �� @ � � � @ J Ö @ � � � @ O �...

......@ � Ö�� @ J � �� @ � Ö:� @ J Ö @ � Ö�� @ O �@ � Ö v �� @ J � �� @ � Ö v �� @ J Ö @ � Ö v �� @ O �

¡¡¡¡¡¡¡¡¡ � (1.214)

wheretheindex h runsfrom ¤ ? % to! � Now thegoalis to show thatat leastone

of theseis nonzero.If this is thecase,thenby performingareorderingamongtheO � and J � andpossiblyan exchangeofO � and J � , indeed 6 ý . þÖ v � is nonzeroforh &¤ ? % .

To find thismatrixwith nonvanishingdeterminant,weconsiderthe ¤ ? % � Ì L !

matrix

ÉÎ -/////07�8 m7 . m �� 7�8 m7 .:9 7�8 m7 .;9=< m �� 7�8 m7 . � 7�8 m7 1 m �� 7�8 m7 1 �

......

......

......7�8 97 . m �� 7�8 97 . 9 7�8 97 . 9=< m �� 7�8 97 . � 7�8 97 1 m 7�8 97 1 �7�8 9>< m7 . m �� 7�8 9>< m7 . 9 7�8 9=< m7 . 9=< m �� 7�8 9>< m7 . � 7�8 9=< m7 1 m �� 7�8 9>< m7 1 �

1 222223 �(1.215)

Therankof thematrix ÉÎ mustthenbe ¤ ? % sinceit consistsof theupper¤ ? %rows of the Jacobianof the canonicaltransformation,which is known to have


determinantone.Now considerthe ¤ ? % � Ì L !¬# ¤ � matrix ÉÎ / obtained

by striking from ÉÎ the ¤ columnscontainingderivativeswith respecttoO � forh % ��¤å� So ÉÎ / hastheform

ÉÎ / -/////07�8 m7 . m �� 7�8 m7 . 9 7�8 m7 . 9>< m �� 7�8 m7 . � 7�8 m7 1 9>< m �� 7�8 m7 1 �

......

......

......7�8 97 . m �� 7�8 97 .:9 7�8 97 .:9>< m �� 7�8 97 . � 7�8 97 1 9>< mêm 7�8 97 1 �7�8 9>< m7 . m �� 7�8 9>< m7 .:9 7�8 9>< m7 .:9>< m �� 7�8 9=< m7 . � 7�8 9=< m7 1 9>< mêm �� 7�8 9>< m7 1 �

1 222223 �(1.216)

We now show that even the matrix ÉÎ / hasrank ¤ ? % . We proceedindirectly:Assumethattherankof ÉÎ?/ is not ¤ ? % . Becauseof 6 Ö 6�7 , it musthave rank¤ , andfurthermoreits last row canbe expressedasa linear combinationof thefirst ¤ rows.This meansthattherearecoefficients � � suchthat@ � Ö v �@ J � Ö� � � � � � 9 @ � �@ J � for � % �� !@ � Ö v �@ O � Ö� � � � � � 9 @ � �@ O � for � &¤ ? % �� ! (1.217)

Becauseof thesymplecticityof j � �%$ � , Poissonbracketsbetweenany twopartsof � vanish.Thisentails7� � � Ö v � � � � � �� @ � Ö v �@ O � @ � �@ J � # �� @ � Ö v �@ J � @ � �@ O � Ö�� @ � Ö v �@ O � @ � �@ J � ? �� Ö v � Ö� � � � � � 9 @ � �@ O � @ � �@ J � # �� Ö� � � � � � 9 @ � �@ J � @ � �@ O � Ö�� @ � Ö v �@ O � @ � �@ J � ? Ö� � � � � � 9 � � � � � � � # Ö�� Ö� � � � � � 9 @ � �@ O � @ � �@ J � Ö�� @ � Ö v �@ O � # Ö� � � � � � 9 @ � �@ O � � @ � �@ J � �Because6 Ö 6�7 , thisevenimpliesthat@ � Ö v �@ O � Ö� � � � � � 9 @ � �@ O � for � % ��¤�� (1.218)


Togetherwith Eq. (1.217),this entailsthatevenin thematrix ÉÎ , thelastrow canbeexpressedin termsof a linearcombinationof theupper¤ rows,which in turnmeansthat ÉÎ hasrank ¤å� But this is acontradiction,showing thattheassumptionof ÉÎ@/ not having rank ¤ ? % is false.Thus, ÉÎ@/ hasrank ¤ ? % �

Sincethefirst ¤ columnsof ÉÎ / arelinearly independentbecause6 Ö 6�7 , it ispossibleto selectoneof the

L !à# ¤ rightmostcolumnsthatcannotbeexpressedasa linearcombinationof thefirst ¤ columns.We now performareorderingandpossiblya

O � J exchangesuchthat this columnactuallyappearsasthe ¤ ? % � st

column.Thenindeed,thedeterminant

6 Ö v � ¡¡¡¡¡¡¡¡¡¡¡7�8 m7 . m �� 7�8 m7 . 9 7�8 m7 . 9>< m

......7�8 97 . m �� 7�8 97 .;9 7�8 97 .:9>< m7�8 9>< m7 . m �� 7�8 9=< m7 .;9 7�8 9>< m7 .:9>< m

¡¡¡¡¡¡¡¡¡¡¡ (1.219)

is nonzero.This completestheinduction,andwe concludethat

6 ¡¡¡¡¡¡¡7�8 m7 . m �� 7�8 m7 . �

......7�8 �7 . m �� 7�8 �7 . � ¡¡¡¡¡¡¡ 6�7�� (1.220)

assertingthatit is possibleto determine, by inversion.To show that it is possibleto representthe , - i � andthe , . i � via thegenerating

function}

as in Eq. (1.206),it is sufficient to show that the , - i � and the , . i �satisfy the integrability conditions(1.203),(1.204)and (1.205).To show theseconditions,weproceedasfollows.Fromthedefinitionof j � �%$ � and , , wehave � � � � �O � , C. �� O �� (1.221)

Recognizingthatboth sidesarenow expressionsin the variables �� O � andthe

left sideis independentofO � , we obtainby partialdifferentiationwith respecttoO � therelation 7ò @ � �@ O � ? �� @ � �@ J � 9 @ , .;A@ O � (1.222)

for all �T� h % �� ! � Multiplying this relationwith @ � Ö � @ J � andsummingover h , we have # �� @ � Ö@ J � 9 @ � �@ O � �� i � � � @ � Ö@ J � 9 @ � �@ J � 9 @ , .;A@ O � (1.223)


This relationis modifiedby interchanging¤ and � on bothsides,and h and � ontheright-handside.We obtain# �� @ � Ö@ O � 9 @ � �@ J � �� i � � � @ � Ö@ J � 9 @ � �@ J � 9 @ , .:B@ O � (1.224)

for all �b��¤ % �� ! � We now subtractthe two previous equationsfrom eachotherwhile observingthatbecauseof thesymplecticityof j ,7� � � � � � Ö � �� @ � �@ O � @ � Ö@ J � # @ � �@ J � @ � Ö@ O �andobtain 7� �� i � � � @ � Ö@ J � @ � �@ J � � @ , .CA@ O � # @ , .DB@ O � for all �b��¤l % �� ! � We now abbreviateE � � �� @ � �@ J � � @ , .;A@ O � # @ , .:B@ O � (1.225)

andwrite theconditionas 7ò �� @ � Ö@ J � 9 E � � � (1.226)

whichagainholdsfor all �b��¤ % �� ! �Now considerthis relationshipfor any fixed �b� Sincethefunctionaldeterminant6 in Eq. (1.220)is nonzero,this actuallyimpliesthatwe musthave

E � � �7 forall h % �� ! , andhencewe haveE � � &7 for all �b� h % �� ! � (1.227)

Now we considerthedefinitionofE � i � in Eq. (1.225)andagainusethefact that6 687 , whichshows thatwemusthave@ , . A@ O � @ , .DB@ O � for all � � h % �� ! � (1.228)

andhencewehaveshown Eq.(1.205).


Now considerthe relationship� � � � �O � �J � , which by using

�J , C. �� O �canbewrittenas

�� T �O � , C. �� O �� Partialdifferentiationwith respectto� �

shows thatfor all �T� h % �� ! , we haveÓ � � �� @ � �@ J � @ , .CA@ � � � (1.229)

We alsohaveí � , - B �� O � , which by using

�� O � �J � canbe written así � , - B � �O � �J � � �O � � We usethis factto obtain# Ó � � � $ � � � � � � , - B � �O � �J � � �O � � � � �O � �J � � �� Y � @ , - B@ O � ? �� Õ � � @ , - B@ � Õ @ �Õ

@ O � � @ � �@ J � # �� Õ � � @ , - B@ � Õ @ �Õ

@ J � @ � �@ O �GF �� @ , - B@ O � @ � �@ J � ? �� Õ � � @ , - B@ � Õ �� @ � Õ@ O � @ � �@ J � # @ �Õ

@ J � @ � �@ O � �� @ , - B@ O � @ � �@ J � (1.230)

for all �b� h % �� ! , wherein thelaststep,usehasbeenmadeof� � Õ � � � � �7��

Combiningthis resultandEq.(1.229)yields7� �� @ � �@ J � � @ , .;A@ � � ? @ , - B@ O � (1.231)

for all �b� h % �� ! . Again, becausethe functional determinant6 doesnotvanish,this canonly betrueif@ , . A@ � � # @ , - B@ O � for all � � h % �� ! � (1.232)

which is Eq. (1.203).In a similarmanner, we now write $ � , - A � �O � �J � � �O � andobtain7� � $ � �%$ � � � $ � � , - A � �O � �J � � �O � �� Y @ $ �@ O � � � Õ @ , - A@ � Õ @ �

Õ@ J � � # @ $ �@ J � � @ , - A@ O � ? �� Õ � � @ , - A@ � Õ @ �

Õ@ O � � F

52 DYNAMICS OF PARTICLES AND FIELDS # �� @ $ �@ J � @ , - A@ O � ? �� Õ � � @ , - A@ � Õ �� @ $ �@ O � @ �Õ

@ J � # @ $ �@ J � @ �Õ

@ O � # �� @ $ �@ J � @ , - A@ O � # �� Õ � � @ , - A@ � Õ 9 Ó � Õfor all �b� � % �� ! , whereweuse

� $ � i � � � # Ó � � � UsingEq.(1.232),thiscanbewrittenas @ , - A@ � � �� @ $ �@ J � @ , . B@ � � � (1.233)

On the other hand,from , -�H �� O � I$ � , C. �� O � � �O � we obtain after partialdifferentiationwith respectto

� � that@ , -�H@ � � �� @ $ �@ J � @ , .DB@ � � (1.234)

Comparingthesetwo equationsnow showsthat@ , -0H@ � � @ , - A@ �ò� for all �b� � % �� ! (1.235)

Theequations(1.228),(1.232),and(1.235)now asserttheexistenceof a function} �O � �� thatsatisfiesthecondition, - i � @ }@ � � and , . i � # @ }@ O � � (1.236)

Giventhefunctions , - i � and , . i � , thefunction}

canbecalculatedby mereinte-grationby virtue of Eq. (1.304).Furthermore,thenonvanishingof thefunctionaldeterminant6 describingthedependenceof

��on�J alsoimpliesthatthedetermi-

nantof thedependenceof�J on

��is nonzero.Therefore,accordingto theimplicit

functiontheorem,theconditionsJ � # @ }@ O � � í � @ }@ � � (1.237)

canbe solved for the coordinate��

in a neighborhood,andhencethe canonicaltransformation

�� í � canbedetermined.


1.4.7 Flowsof HamiltonianSystems

In the previous sections,we establishedvariousnecessaryandsufficient condi-tions for a mapto becanonical,andwe establisheda standardrepresentationforgivencanonicaltransformations.In thisandthefollowing section,weaddressthequestionof how to obtaincanonicaltransformations.

An important way of generatingcanonicaltransformationsis via the flowj �� G � �� of a Hamiltoniansystem,which describeshow initial coordinates�� G

and� G aretransformedto latertimes

�via�� j �� G � �� (1.238)

As wenow show, ifI

is threetimescontinuouslydifferentiable,theflow is sym-plectic for any choiceof

� G and�, andhencesatisfies

Jac j � 9¹ÉÜ 9 Jac

j � k ßÉÜ (1.239)

To this end,let É� Jac j � � SetÉí �� G � �� É� �� G � �� 9 ÉÜ 9 É� k �� G � �� (1.240)

Then, Éí �� G � � G � ßÉÜ . Considertheequationsof motion�� < �� (1.241)

which imply �� j �� G � �� < j �� G�� (1.242)

Now we have�� É� �� G � �� Jac j �� G � �� Jac

� �� j �� G�� Jac ��< j �� G � �� Jac

�� j �� G � �� 9 Jac j � �� G � �� Jac

�� j �� G � �� 9$É� �� G � �� (1.243)

wherein thesecondtransformationwe haveused�� @ � �@ � G � �� @ ( � �@ � G � @ � G � U� G � ? @ ( � �@ � G � @ � @@ � G � � �� @ � �@ � G � U� G � ?K@ � �@ � � @@ � G � � �� @ U� �@ � G � �


Becauseof this, it followsthat�� Éí �� G � �� É� �� G � �� 9ãÉÜ 9ÅÉ� k �� G � �� Jac �� j �� G � �� 9ÅÉ� �� G � �� 9 Ü 9$É� k �� G � ��? É� �� G � �� 9:ÉÜ 9$É� k �� G � �� 9 Jac

�� k j �� G � �� Jac �� j �� G � �� 9ÆÉí �� G � ��? Éí �� G � �� 9 Jac

�� k j �� G�� (1.244)

Now we utilize that the underlyingdifferentialequationis Hamiltonian.Wethenshow thatJac

�� 9 ÉÜ is symmetric.In fact,�� @ I � @ �J# @ I � @ �O (1.245)

andso

Jac �� 9¹ÉÜ � @ ( I � @ �O @ �J @ ( I � @ �J @ �J# @ ( I � @ �O @ �O # @ ( I � @ �J @ �O 9ãÉÜ � # @ ( I � @ �J @ �J @ ( I � @ �O @ �J@ ( I � @ �J @ �O # @ ( I � @ �O @ �O (1.246)

showing symmetryasclaimed.Because ÉÜ k # ÉÜ , wealsohavethatJac �� 9 ÉÜ ?ÉÜ 9 Jac

�� k u7+� This entailsthat onesolutionof the differentialequationforÉí �� G � �� thatsatisfiesthenecessaryinitial condition Éí �� G � � G � ÉÜ isÉí �� G � �� ßÉÜ for all�� G � � � (1.247)

However, becauseof theuniquenesstheoremfor ordinarydifferentialequations,hereappliedto thesystemconsistingof the J ! ( matrixelementsof Éí , this is alsotheonly solution,andwe haveprovedwhatwe wantedto show.

Thus,theflowsof Hamiltoniansystemsaresymplectic.Sincesymplectictrans-formationspreserve thevolumeof phasespaceaccordingto Eq. (1.184),sodoestheflow of a Hamiltoniansystem,which is known asLiouville’ s theorem.

1.4.8 GeneratingFunctions

In Section1.4.6,it wasshown thateverycanonicaltransformationandeverysym-plecticmapcanberepresentedby ageneratingfunction.Wenow addressthecon-versequestion,namely, whetherany mixed-variable function thatcanbesolvedfor the final variablesand that is twice continuouslydifferentiableprovides acanonicaltransformation.


Without lossof generalitywe assumethe functionhasthe form} �� O �FE if it

is representedin termsof othermixedvariables,it canbebroughtinto this formvia elementarycanonicaltransformations.We now assumethatJ � # @ }@ O � � í � @ }@ � � (1.248)

canbesolvedfor��

in termsof �O � �J � to providea diffeomorphism� �� í � � �O � �J �$ �O � �J � j � �O �J (1.249)

thatgeneratesa coordinatetransformation.Combiningthis with Eqs.(1.248),weobtaintherelationships J � ? } 1 H � �O � �J � � �O � �7 (1.250)$ �� O � �J � # } Ê H � �O � �J � � �O � �7�� (1.251)

We notethat accordingto the implicit function theorem,invertibility of the de-pendenceof

�J on��

in�J # @ } � @ �Oq �O � �� entailsthat for any valueof

�O, the! Ì ! Jacobianof thedependenceof

�J on��

is regular, i.e.,¡¡¡¡¡¡¡ @( } � @ O � @ � � �� @ ( } � @ O � @ � �...

...@ ( } � @ O � @ � � �� @ ( } � @ O � @ � �¡¡¡¡¡¡¡ 6�7�� (1.252)

Moreover, thenonvanishingof thedeterminantandthe fact that thedeterminantis continuousassertstheexistenceof aninverseevenin aneighborhood.

To show that thetransformationis canonical,we derive relationshipsbetweenvariousderivativesof

}and

� �&$ � , andusetheseto show that the elementaryPoissonbracket relations� � � �&$ � � Ó � � � � $ � �%$ � � 87à� � � � � � � � 87 (1.253)

aresatisfied,which accordingto Eq. (1.192)assertsthat j is indeedcanonical.We begin by partially differentiatingEq. (1.250)with respectto

O � and J Õ , re-spectively, andobtain }K1 H 1 A ? �� }�1 H Ê B @ � �@ O � 87 (1.254)

56 DYNAMICS OF PARTICLES AND FIELDSÓ � Õ ? �� } 1 H Ê B @ � �@ J Õ &7�� (1.255)

Writing} 1 H 1 A Ù �Õ � � } 1MLN1 A 9 Ó � Õ in the first equationand insertingfor Ó � Õ the

valuefrom thesecondequation,we obtain�� }�1 H Ê B 9 � @ � �@ O � # �� Õ � � }�1MLN1 A @ � �@ J Õ � �7��Becausethis equationis satisfiedfor all �< % �� ! andbecauseof Eq. (1.252),it followsthat @ � �@ O � # �� Õ � � }�1MLN1 A @ � �@ J Õ �7 for all � � h % �� ! � (1.256)

Next, wemultiply Eq. (1.255)by}�1ML Ê A andsumover ð to obtain7� �� Õ � � } 1 L Ê A Ó � Õ ? �� i Õ � � } 1 L Ê A } 1 H Ê B @ � �@ J Õ �� } 1 H Ê B Ó � � ? �� i Õ � � } 1 L Ê A } 1 H Ê B @ � �@ J Õ �� }�1 H Ê B � Ó � � ? �� Õ � � }�1ML Ê A @ � �@ J Õ � � (1.257)

Observingthattherelationis satisfiedfor all �< % �� ! , becauseof (1.252)wehave Ó � � ? �� Õ � � } 1ML Ê A @ � �@ J Õ 87 for all � � h % �� ! � (1.258)

To concludeour derivation of conditionsbetweenthe derivatives of}

and � �%$ � , we utilize Eq. (1.251)anddifferentiatewith respectto J Õ to obtain

@ $ �@ J Õ # �� } Ê H Ê B @ � �@ J Õ �7��


Multiplying with}K1ML Ê A , summingover ð , andconsideringEq. (1.258),we ob-

tain 7� �� Õ � � }K1ML Ê A Y @ $ �@ J Õ # �� } Ê H Ê B @ � �@ J Õ F �� Õ � � }K1ML Ê A @ $ �@ J Õ # �� } Ê H Ê B Y �� Õ � � }K1ML Ê A @ � �@ J Õ F �� Õ � � }K1ML Ê A @ $ �@ J Õ ? }Ê H Ê A for all �b� � % �� ! � (1.259)

Now we begin the computationof the elementaryPoissonbrackets.We takethe Poissonbracket between� � andbothsidesof Eq. (1.250),andutilizing Eq.(1.256),weobtain7� � � � � J � ? } 1 B � �O � �J � � �O � � @ � �@ O � ? �� Õ � �POQ R @ � �@ O Õ 9 �� }K1 B

Ê A 9 @ � �@ J Õ # @ � �@ J Õ -0 �� }K1 BÊ A @ � �@ O Õ ? }K1 B 1ML 13TS UV @ � �@ O � # �� Õ � � } 1 B 1 L @ � �@ J Õ ? �� } 1 B Ê A 9�Y �� Õ � � @ � �@ O Õ @ � �@ J Õ # @ � �@ J Õ @ � �@ O Õ F&7 ? �� }K1 B Ê A 9 � � � � � � � for all �b� h % �� ! � (1.260)

Becauseof Eq. (1.252),wehave� � � � � � � 87 for �b� � % �� ! � (1.261)

Further, takingthePoissonbracketbetween� � and $ � asgivenby Eq.(1.251),andconsideringEqs.(1.261)and(1.258),we have� � � �%$ � � � � � � } Ê A � �O � �J � � �O � �� Y @ � �@ O � � �� Õ � � }

Ê A Ê L @ � Õ@ J � � # @ � �@ J � � }Ê A 1 B ? �� Õ � � } Ê A Ê L @ � Õ@ O � � F # �� } Ê A 1 B @ � �@ J � ? �� Õ � � } Ê A Ê L � �� @ � �@ O � @ �Õ

@ J � # @ � �@ J � @ �Õ

@ O � � Ó � � ? �� Õ � � } Ê A Ê L 9 � � � � � Õ � Ó � � for all �T� � % �� ! (1.262)


As thelaststep,we infer from Eq.(1.251)while utilizing Eq.(1.262)and(1.259)that� $ � �%$ � � � $ � � } Ê A � �O � �J � � �O � � �� Y @ $ �@ O � � �� Õ � � }

Ê A Ê L @ � Õ@ J � � # @ $ �@ J � � }Ê A 1 B ? �� Õ � � } Ê A Ê L @ � Õ@ O � � F # �� } Ê A 1 B @ $ �@ J � ? �� Õ � � } Ê A Ê L 9 � �� @ $ �@ O � @ �

Õ@ J � # @ $ �@ J � @ �

Õ@ O � � # �� } Ê A 1 B @ $ �@ J � ? �� Õ � � } Ê A Ê L 9 � $ � � � Õ � # �� } Ê A 1 B @ $ �@ J � # }

Ê A Ê H 87 for all �b� � % �� ! � (1.263)

ThethreePoissonbracketrelations(Eqs.(1.261),(1.262),and(1.263))togetherassertthatthetransformationis indeedcanonical.

Of course,theentireargumentis notspecificto theparticularchoiceof thegen-eratingfunction

} �� O � thatwe havemadebecausethroughelementarycanoni-cal transformations(Eq.1.154),it is possibleto interchangebothinitial andfinalpositionandmomentumvariables.If onedoesthis for all positionsandmomentasimultaneously, oneobtainsa total of four differentgeneratingfunctionsthat tra-ditionally havebeenreferredto as

} �–} a andhavetheform} � �O � �� } ( �O � �í � �} R �J � �� } a �J � �í � � (1.264)

Folding in therespectiveelementarycanonicaltransformationsexchangingposi-tionsandmomenta,we obtainthefollowing relationships:�J @ }<�@ �O ,

�í # @ }<�@ �� (1.265)�J @ } (@ �O ,�� @ } (@ �í (1.266)�O # @ } R@ �J ,�í # @ } R@ �� (1.267)�O # @ } a@ �J ,�� @ } a@ �í � (1.268)

For eachof theabove generatingfunctions} �

, its negative is alsoa valid gener-atingfunction.Thisalsoentailsthatin eachof thefour equations(1.265)through


(1.268),it is possibleto exchangebothsignsin front of thepartialsof}

; andinfact, thegeneratingfunction (1.248)is thenegative of

}��. This facthelpsmem-

orizationof thefour equations;onenow mustrememberonly thatthegeneratingfunctionsdependingon positionandmomentahave equalsigns,and thosede-pendingonly on momentaor only onpositionshaveunequalsigns.

However, in light of our discussionit is clearthat this selectionof four typesis indeedsomewhat arbitrary, as it is not necessaryto perform the elementarycanonicaltransformationssimultaneouslyfor all componentsof thecoordinates.So insteadof only four casesshown here,therearefour choicesfor eachcom-ponent. Moreover, onecansubjecttheoriginal functionto otherlinearcanonicaltransformationsthatarenot merelycombinationsof elementarycanonicaltrans-formations,increasingthemultitudeof possiblegeneratingfunctionsevenmore.

Also, it is importantto pointout thatnot every canonicaltransformation canbe representedthroughoneof the four generators

} �–} a , but rather, asseenin

Section1.4.6,at leasta setof J � generatorsis needed.

1.4.9 Time-DependentCanonicalTransformations

The framework of canonicaltransformationsis intrinsically time independentinthe sensethat the new coordinatesandmomentadependon the old coordinatesandmomentabut noton time.In this respect,thereis anapparentdifferencefromthe Lagrangianframework of transformations,which were allowed to be timedependentfrom the outset.However, this is not of fundamentalconcernsinceaccordingto Eq. (1.107), in the

L "! ? % � # dimensionalextendedphasespace �O � 5 � �J � J â � with theextendedHamiltonianÉI JØâ ? I �O � �J � 5 � � (1.269)

time appearsin theform of thedynamicalvariable 5 , andhencecanonicaltrans-formationin extendedphasenaturallyincludespossibletimedependence.

While fundamentallythe issueis settledwith this observation,practicallythequestionoften ariseswhetherthe transformedHamiltoniansystemin extendedphasespacecanagainbeexpressedin termsof a nonautonomoussystemin con-ventionalphasespace.According to Eq. (1.107),this is the caseif in the newvariables,theHamiltonianagainhastheformÉI í � ? æÇ �� í � z � � (1.270)

This requirementis easilymet in the representationof canonicaltransforma-tions throughgeneratingfunctions,andthe interpretationof the resultingtrans-formationsin conventionalphasespaceis straightforward. Let us considerthegeneratingfunctionin extendedphasespaceÉ} �O � �í � 5 � ?N5 9 í � � (1.271)


whichproducesa transformationsatisfyingJ � @ É}@ O � , JØâ @ É}@ 5 ? í � (1.272)� � @ É}@ í � � z 5 �We observe that in the first equation,the J � do not dependon

í �since É} is

independentofí � E

likewise,aftersolvingthesecondequationforO �

, thesealsodonot dependon

í �. Therefore,insertingthis transformationinto theHamiltonian

(Eq.1.269)yields ÉI new í � ? @ É}@ 5 ? I new �� í � z � � (1.273)

which is indeedof the form of Eq. (1.270).Viewed entirely within the nonex-tendedphasespacevariables,the transformationsare describedby the time-dependentgeneratingfunction } �O � �í � ��FE (1.274)

theresultingtransformationis notcanonicalin theconventionalsensethatthenewHamiltoniancanbeobtainedby expressingold variablesby new ones.Rather, weobtaina Hamiltoniansystemwith thenew HamiltonianI �� í � �� @ }@ � ? I new �� í � �� (1.275)

1.4.10 TheHamilton–JacobiEquation

The solutionof a Hamiltoniansystemis simplified significantly if it is possibleto find a canonicaltransformationsuchthat the new Hamiltoniandoesnot de-pendon someof the positionsor momentasinceby Hamilton’s equations,thecorrespondingcanonicallyconjugatequantitiesareconstant.Certainly, themostextremesuchcasewouldbeif theHamiltonianin thenew variablesactuallyvan-ishescompletely;thenall new phasespacevariablesareconstant.

Apparently, this cannotbe achieved with a conventionalcanonicaltransfor-mation becauseif a Hamiltonianvarieswith the valuesof the old variables,itwill also vary with the valuesof the new variables.However, for the caseoftime-dependentcanonicaltransformations,it maybe possibleto achieve a com-pletelyvanishingHamiltonian.For thispurpose,we try a time-dependentgenera-tor} ( �O � �í � �� , which,whenviewedin conventionalphasespaceaccordingto the


previoussection,satisfies�J @ } ( �O � �í � ��@ �O � �� @ } ( �O � �í � ��@ �í � (1.276)�I �� í � I �O� �� í � � �J �� í �� ?á@ } ( �O � �í � ��@ � � (1.277)

Insertingthetransformationequationfor�J shows theexplicit form of theHamil-

tonianin thenew variablesto be�I �� í � I �O� �� í � � @ } ( �O � �í � ��@ �O � ? @ } ( �O � �í � ��@ � � (1.278)

But now we utilize that the new Hamiltonian �I is supposedto be zero,whichentailsthat the new momentaareall constants.Thereforethe function

} ( is in-deedonly a functionof thevariables

�O, andthemomentaappearastheconstants�� í � Thepreviousequationthenreducesto a conditionon

} ( , theHamilton–Jacobiequation, 7� I � �O � @ } ( �O � �� @ �O ?á@ } ( �O � �� @ � � (1.279)

This is an implicit partialdifferentialequationfor} ( �O � �� asa functionof the

!variables

�Oonly. Therefore,thenumberof unknownshasbeenreducedby half,

at the expenseof making the systemimplicit and turning it from an ODE to apartialdifferentialequation(PDE).

If theHamilton–JacobiPDEcanbesolvedfor} ( for eachchoiceof theparam-

eters�� , thenit oftenallowsfor thecompletesolutionof theHamiltonianequations

of motion.Indeed,if �I 87 , thenapparentlythefinal positionsareconstant,andwe have

�� WÅ const.Thentheequationsfor thegenerator} ( in Eq. (1.276)

read �J @ } ( �O � �� @ �O � �W @ } ( �O � �� @ �� (1.280)

Thesecondvectorequationis merelyasetof!

implicit algebraicequationsin�O �

If it canbesolvedfor�O, we obtain

�Oasa functionof

�W and�� . Thefirst equation

thenyields�J directly by insertionof

�O� �WØ� �� . Altogether, we obtain�O �O� �WØ� �� J �J �WØ� �� (1.281)

Theentiresolutionof theHamiltonianequationcanthenbeobtainedby algebraicinversionof therelationship,if suchinversionis possible.


Whetheror not theHamilton–Jacobiapproachactuallysimplifies thesolutionof theHamiltonianproblemgreatlydependsonthecircumstances,becausein gen-eralthesolutionof theHamilton–JacobiPDEis by nomeanssimpleandstraight-forward.However, perhapsmoreimportant,it providesaconnectionbetweenthetheoryof PDEsand ODEs, andsometimesalsoallowsfor theexpressionof PDEproblemsin termsof Hamiltoniandynamics.

An importantapproachthat often allows oneto solve or simplify PDEsis toassesswhethertheequationis separable,i.e.,whetherit canbebrokendown intoseparatepartsof lowerdimension.In ourcase,onemakesthespecialAnsatz} (Æ �� } ( i � PO � � �� (1.282)

andinsertsinto theHamilton–JacobiPDE.If it is thenpossibleto split thePDEinto

!separatepiecesof theformI � � O�� @ } �@ O � � � � �� (1.283)

wherethe � � areseparationconstants.Eachof theseequationsmerelyrequiresthesolutionfor @ } � � @ O�� andsubsequentintegration.

1.5 FIELDS AND POTENTIALS

In thissection,wepresentanoverview of conceptsof electrodynamics,with apar-ticular emphasison deriving all thetoolsneededin thefurtherdevelopmentfromthefirst principles.As in the remainderof thebook,we will utilize theSystemeInternationald’unites(SI system),thewell-known extensionof the MKSA sys-temwith its unitsof meter, kilogram,second,andampere.Thesymmetrybetweenelectricandmagneticphenomenamanifestsitself moredramaticallyin theGaus-sian system,which becauseof this fact is often the preferredsystemfor purelyelectrodynamicpurposes.Wewill notdwell ontheconversionrulesthatallow thedescriptionof onesetof units in termsof another, but ratherwe refer thereaderto therespectiveappendixin Jackson’sbook(Jackson1975)for details.

1.5.1 Maxwell’sEquations

It is oneof thebeautifulaspectsof electrodynamicsthatall thecommonlytreatedphenomenacan be derived from one set of four basic laws, the equationsofMaxwell. In our theory, they have axiomatic character in the sensethat theycannotbe derived from any other laws, andthey areassumedto be universallyvalid in all furtherdevelopments.Of course,likeother“good” axiomsof physics,

FIELDS AND POTENTIALS 63

they canbecheckeddirectly experimentally, andall suchchecksso far have notshown any deviation from them.

Thefirst of Maxwell’sequationsis Coulomb’s law, whichstatesthatthequan-tity W , anelementarypropertyof mattercalledthechargedensity, is thesourceof afield calledtheelectric flux density(sometimescalledelectricdisplacement)denotedby

�6 : �> 9 �6 XW�� (1.284)

There is anotherfield, the magnetic flux density (sometimescalled magneticinduction)called

�Í, andthis field doesnot haveany sources:�> 9 �Í &7+� (1.285)

In otherwords,thereareno magneticmonopoles.Faraday’s law of inductionstatesthat thecurl of theelectricfield

�Ëis connectedto changein themagnetic

flux density�Í: �> Ì �Ë ?á@ �Í@ � �7�� (1.286)

Finally, thereis theAmpere–Maxwell law, whichstatesthatthecurl of themag-netic field

�Iis connectedto changein the electric flux density

�6 as well asanotherelementarypropertyof mattercalledthecurrentdensity

�Ü:�> Ì �I �Ü ? @ �6@ � � (1.287)

Theelectricandmagneticflux densitiesarerelatedto theelectricandmagneticfields, respectively, throughtwo additionalelementarypropertiesof matter, theelectric and magneticpolarizations

�íand

��:�6 Ò G �Ë ? �í � (1.288)�Í � G �I ? �� (1.289)

Themagneticpolarizationis oftenreferredto asmagnetizationor magneticmo-mentdensity. ThenaturalconstantsÒ G and � G aretheelectricpermittivity constantor dielectricconstantof vacuumandthemagneticpermeabilityconstantof vac-uum, respectively. The relationswhich connect

�Ë,�6 ,�Í, and

�Iareknown as

constitutive relations. In thecaseof many materialsthatarehomogeneousandisotropic,the polarizationsareproportionalto the fields thatareapplied;in thiscase,theconstitutiverelationshave thesimplifiedforms�6 Ò �Ë (1.290)�Í � �I � (1.291)


wherethe quantitiesÒ and � dependon the materialat handandarereferredtoastherelativeelectricpermittivity or dielectricconstantandtherelativemagneticpermeabilityof thematerial,respectively.

Similar to theconstitutive relations,for many homogeneousandisotropicma-terials,thereis a relationbetweencurrentandelectricfield known asOhm’s law:�Ü � �Ë � (1.292)

wherethequantity � , theelectricconductivity, is anelementarypropertyof mat-ter.

In many cases,it is of interestto studyMaxwell’s equationsin theabsenceofmatterat thepointof interest.In this case,wehaveW �7�� Ü �7�� (1.293)

Thus,Maxwell’sequationstake thesimplifiedandverysymmetricforms�> 9 �6 87+� �> Ì �Ë # @ �Í@ � (1.294)�> 9 �Í 87�� > Ì �I @ �6@ � (1.295)

In many situations,it is alsoimportantto studythecaseof time independence,inwhichcaseall right-handsidesvanish:@ �6@ � �7�� @ �Í@ � �7�� (1.296)

An importantconsequenceof Maxwell’s equationsis that the charge densityandthecurrentdensityareconnectedto eachotherthrougha relationshipknownasthecontinuity equation: �> 9 �Ü ? @ W@ � 87+� (1.297)

This equationis derivedby applyingthedivergenceoperatoron Ampere’s law(Eq.1.287)andcombiningit with Coulomb’s law (Eq.1.284);wehave�> 9 �> Ì �I � �> 9 �Ü ? �> 9 @ �6@ �7� �> 9 �Ü ? @ W@ � �wherein the last stepusehasbeenmadeof the easilyderivablevectorformula�> 9 �> Ì �Y � 87 , whichis alsodiscussedlater(Eq.1.307).Similarly, theoperation


of thedivergenceon Eq.(1.286)showsagreementwith Eq. (1.285):�> 9 �> Ì �Ë � ? �> 9 @ �Í@ � 87@ �> 9 �Í@ � 87+�Thereare two importanttheorems,the integral relationsof StokesandGauss,thatareof importancefor many of thederivationsthatfollow:Z�[ �} 9 � �5 ��\ �> Ì �} 9 �! � Î (1.298)��\ �} 9 �! � Î ��] �> 9 �} � M � (1.299)

1.5.2 ScalarandVectorPotentials

In thissection,we introduceanew setof scalarandvectorfieldscalledpotentialsthatallow thedeterminationof theelectricandmagneticfieldsby differentiationandthatallow asimplificationof many electrodynamicsproblems.Webegin withsomedefinitions.

We call ^ a scalarpotential for the!

-dimensionalvectorfield�}

if�} �> ^ .

We call�Î

a vector potential for the three-dimensionalvectorfield�}

if�} �> Ì �Î

.

The questionnow arisesunderwhat conditiona given vectorfield�}

on è �hasa scalarpotentialor a vectorpotential.We first addressthecaseof thescalarpotentialand answerthis questionfor the general

!-dimensionalcase.For the

purposesof electrodynamics,the case! W is usuallysufficient, but for many

questionsconnectedto LagrangianandHamiltoniandynamics,the generalcaseis needed.

In thefollowing, westudytheexistenceand uniquenessof scalarpotentials.Let

�}bea continuouslydifferentiablevectorfield on è � �

1. If�} �> ^ , then@ } � � @ £ � @ } � � @ £ � for all �b� � % �� ! � (1.300)

2. If @ } � � @ £ � @ } � � @ £ � for all �b� � % �� ! , thenthereexists a scalarpotential suchthat �} �> ^ � (1.301)

3. A scalarpotential for thevector�}

is uniquelyspecifiedup to a constant.


Note the differenceof (1) and (2), which areactually the conversesof eachother. We alsonotethat in thethree-dimensionalcase,thecondition @ } � � @ £ � @ } � � @ £ � is equivalentto themorereadilyrecognizedcondition

�> Ì �} �7+�For theproof,we proceedasfollows.

1. If thereexists a twice continuouslydifferentiable suchthat�} �> ^ ,

then} � @ ^ � @ £ � and

} � @ ^ � @ £ � , andthus@ } �@ £ � @ ( ^@ £ � @ £ � @ ( ^@ £ � @ £ � @ } �@ £ � � (1.302)

2. Now assumethat�}

is continuouslydifferentiableand@ } �@ £ � @ } �@ £ � for all �b� � % �� ! � (1.303)

Define ^ asapathintegralof�}

from 7��7 � to

£ � ��£�(��£ � � alongapaththat first is parallelto the £ � axis from

7+��7��7 � to £ � �T7��7 � ,

then parallel to the £Ø( axis from £ � ��7��7 � to

£ � ��£�(��7+��7 � , andso on, and finally parallel to the £ � axis from

£ � ��£�(��£ �� T7 � to £ � ��£ ( ��£ �� £ � � � Thenwe have^ � ý)_ m i {|{|{ i _ � þý G i {|{|{ i G þ �} 9 � �� _ mG } � £ � �T7��T7 �� £ � ? � _ nG } ( £ � ��£Ø(��T7��T7 �� £�( ? 9�9�9? � _ �G } � £ � ��£Ø(��£ � �� £ � � (1.304)

In thefollowing, we show that ^ indeedsatisfiestherequirement�} �> ^ .

We first differentiatewith respectto £ � andobtain@ ^@ £ � }<� £ � �T7��7+��F��7 � ? � _ nG @ } ( £ � ��£Ø(��T7��T7 �@ £ � � £ ( ? 9�9�9? � _ �G @ } � £ � ��£ ( ��£ � �@ £ � � £ � }<� £ � �T7��7+��F��7 � ? � _ nG @ } � £ � ��£Ø(��T7��T7 �@ £�( � £ ( ? 9�9�9? � _ �G @ }<� £ � ��£ ( ��£ � �@ £ � � £ �

FIELDS AND POTENTIALS 67 } � £ � �T7��7+��T7 �? � }<� £ �bi £ ( ��7+��T7 � # }<� £ � �T7��7+��F��7 � ��? 9�9�9? � }<� £ �bi £ ( ��£ �� Fi £ � � # }<� £ � ��£ ( ��£ �� 7 � � }<� £ � ��£ ( ��£ � � �whereEq. (1.303)wasusedfor moving from the first to the secondline.Similarly, we have@ ^@ £Ø( } ( £ � ��£Ø(��7+��T7 � ? � _ oG @ } R £ � ��£ ( ��£ R ��7+��7 �@ £�( � £ R ? 9�9�9? � _ �G @ } � £ � ��£�(��£ � �@ £ ( � £ � } ( £ � ��£ ( ��7+��T7 � ? � _ oG @ } ( £ � ��£ ( ��£ R ��7+��7 �@ £ R � £ R ? 9�9�9? � _ �G @ } ( £ � ��£�(��£ � �@ £ � � £ � } ( £ � ��£Ø(��£ � � �We continuein thesameway for @ ^ � @ £ R and @ ^ � @ £ a , andultimatelywederive thefollowing for @ ^ � @ £ � :@ ^@ £ � } � £ � ��£Ø(��£ � � � (1.305)

which shows that indeedthe scalarfield ^ definedby Eq. (1.304)satisfies�} �> ^ .3. Assumethatthereexist two scalars

�and ^ ( which satisfy�} �> ^ ��} �> ^ (��

Then�> ^ � # ^ ( � �7 , which canonly bethecaseif^ � # ^ (Æ constant�

Therefore,ascalarpotential is specifiedupto aconstant,whichconcludestheproof.

We note that in the three-dimensionalcase,the peculiar-looking integrationpath in the definition of the potential ^ in Eq. (1.304)canbe replacedby anyotherpathconnectingtheorigin and

£y�a`Ø�T� � , because�> Ì �} �7 ensurespath


independenceof the integral accordingto the Stokestheorem(Eq. 1.298).Fur-thermore,thechoiceof theorigin asthestartingpoint is by nomeansmandatory;indeed,any otherstartingpoint

�£�G leadsto theadditionof aconstantto thepoten-tial, becausedueto pathindependencethe integral from

�£+G to�£ canbereplaced

by one from�£�G to

�7 andone from�7 to

�£ , and the first onealwaysproducesaconstantcontribution.

Next we studytheexistenceand uniquenessof vector potentials.Let�}

beacontinuouslydifferentiablevectorfield.

1. If�} �> Ì �Î

, then �> 9 �} 87�� (1.306)

2. If�> 9 �} &7 , thereexistsavectorpotential

�Îsuchthat�} �> Ì �Î � (1.307)

3. A vectorpotential�Î

for thevector�}

is uniquelydeterminedupto agradientof ascalar.

For theproof,we proceedasfollows.

1. If thereexists�Î

suchthat�} �> Ì �Î

, then�> 9 �} �> 9 �> Ì �Î �&7+�asdirectevaluationof thecomponentsreveals.

2. Since�> 9 �} 87 , @ } _@ £ ? @ }Kb@ ` ? @ }=c@ � 87+� (1.308)

Considering �> Ì �Î � _ @ Î c@ ` # @ Î b@ � �> Ì �Î � b @ Î _@ � # @ Î c@ £ �> Ì �Î �%c @ Î b@ £ # @ Î _@ ` �define

�Îas Î _ _� cG } b £4�&`È��

FIELDS AND POTENTIALS 69Î b # � cG } _ £y�&`È�T� �� ? � _G }Kc £4�&`È��7 �� £Î c &7+� (1.309)

Then�Î

indeedsatisfiestherequirement�} �> Ì �Î , asshownby computing

thecurl: �> Ì �Î � _ } _ £y�a`È�� > Ì �Î � b }Kb £y�&`È�T� � �> Ì �Î � c # � cG @ } _ £4�&`È�� @ £ � � ? }Kc £4�&`È��7 � # � cG @ } b £y�a`È�� @ ` � � # � cG 2 @ } _ £4�&`È�� @ £ ? @ } b £4�&`È�� @ ` 4 � � ? } c £y�a`È��7 �� cG @ } c £y�a`È�� @ � � � ? } c £y�a`Ø�T7 � }Kc £4�&`È�� # }Kc £4�&`È��7 � ? }Kc £y�a`È��7 � }Kc £4�&`È�� whereEq. (1.308)wasusedin moving from thesecondto thethird line ofthe � component.Indeed,thereexistsavector

�Îwhichsatisfies

�} �> Ì �Î .3. Assumethatthereexist two vectors

�Î �and

�Î ( which satisfy�} �> Ì �Î ��} �> Ì �Î (��then �> Ì �Î � # �Î ( � �7��Thus,accordingto theprevious theoremsabouttheexistenceof scalarpo-tentials,thereexists d which is uniqueup to a constantsuchthat�Î � # �Î ( �> d �So,thescalarpotential

�Îis uniqueupto theadditional

�> d , whichconcludestheproof.

A closerstudyof theproof revealsthatinsteadof havingÎ c �7 , by symmetry

it is possibleto constructvectorpotentialsthatsatisfyeitherÎ _ 87 or

Î b �7��We now addressthequestionof nonuniquenessof scalarandvectorpotentials

in moredetail.As wasshown, differentchoicesof theintegrationconstantin the


caseof the scalarpotentialand specificchoicesof d in the caseof the vectorpotentialall lead to valid potentials.Thesetransformationsbetweenpotentialsaresimpleexamplesof so-calledgaugetransformations which leadto thesamephysicalsystemvia differentpotentials.The variouspossiblechoicesthat yieldthesamefield arecalleddifferentgauges.Wewill discussthematterin detailafterdevelopinga morecompleteunderstandingof the electrodynamicsexpressedintermsof potentials.

It is illuminating to notethatthedifferentgaugesbasedon thenon-uniquenessof thepotentials,whichatfirst mayappearsomewhatdisturbing,canbeaccountedfor very naturallywith the conceptof an equivalencerelation. An equivalencerelation is a relationship“ e ” betweentwo elementsof a given set that for allelementsof thesetsatisfies fGe8f �fGegfihjfie8f � andfGegf � fke&�5h f�e8� � (1.310)

For thecaseof scalarpotentials, therelationshipe â onthesetof scalarfunctionsis that M � e â M ( if M � # M ( is a constant.For thecaseof vector potentials, therelationship e@l on the setof vector functionsis that

�Î � e@l �Î ( if andonly if�Î � # �Î ( canbe written asthe gradientof a scalar. In termsof gauges,in bothcasespotentialsare relatedif one can be obtainedfrom the other via a gaugetransformation.Onecanquite readily verify thatboth theserelationssatisfytheconditionin Eq.(1.310).

For afixedelementf , all elementsrelatedto it constitutetheequivalenceclassof f , denotedby

� f�� E therefore, � f0� � £ Q £ e8f � � (1.311)

Apparently, the equivalenceclasses� � â of scalarpotentialsand

� �ml of vectorpotentialsdescribethe collectionof all functions ^ ,

�Îsatisfying

�} �> ^ and�} �> Ì �Î , respectively, whicharethosethatcanbeobtainedthroughvalidgaugetransformationsfrom oneanother. Theconceptof classesallows the descriptionof scalarand vector potentialsin a very naturalway; indeed,we can say thatthescalarpotential ^ to a vectorfunction

�}is really anequivalenceclassunder

the relationshipe â of all thosefunctionswhosedivergenceis�}, andthe vector

potential�Î

to�}

is theequivalenceclassunder e l of all thosewhosecurl is�} �

Within this framework, both scalarand vectorpotentialsare unique,while theunderlyingfreedomof gaugeis maintainedandaccountedfor.

Thetheoremsontheexistenceof scalarand vector potentialsallow usto sim-plify the equationsof electrodynamicsandmake themmorecompact.Supposewearegiven

�ÜandW , andthetaskis to describethesystemin its entirety. Utilizing

Maxwell’sequations,we canfind�Ë

and�Í

of thesystem,andfor thedescriptionof theelectrodynamicphenomenawe needa totalof six components.


By usingscalarandvectorpotentials,wecandecreasethenumberof variablesin theelectromagneticsystem.TheMaxwell equation

�> 9 �Í &7 impliesthatthereexists

�Îsuchthat

�Í �> Ì �Î � Ontheotherhand,Faraday’slaw�> Ì �Ë ? @ �Í � @ � �7 canthenbewrittenas �> Ì � # �Ë # @ �Î@ � � �7�� (1.312)

Thisensuresthatthereis a scalarpotential n for# �Ë # @ �Î � @ � thatsatisfies# �Ë # @ �Î@ � �> n � (1.313)

Thus,the two homogeneousMaxwell’s equationsaresatisfiedautomaticallybysetting �Í �> Ì �Î

and�Ë # �> n # @ �Î@ � � (1.314)

The othertwo inhomogeneousMaxwell equationsdeterminethe space-timebe-havior of

�Îand n . Usingtheconstitutive relations,

�6 Ò �Ë and�Í � �I , we

expressthesetwo Maxwell equationsin termsof�Î

and n . TheAmpere–Maxwelllaw takestheform�> Ì � %� �> Ì �Î �Ü ? @@ �po Ò � # �> n # @ �Î@ � �@q�> ( �Î # Ò�� @ ( �Î@ � ( �>ï� �> 9 �Î ? Ò�� @ n@ � # � �Ü � (1.315)

On theotherhand,Coulomb’s law appearsas�> 9 o Ò � # �> n # @ �Î@ � �@q rW�> ( n�? @@ � �> 9 �Î � # W Ò � (1.316)

Altogether, we have obtaineda coupledsetof two equations.To summarize,thefollowing setof equationsareequivalentto thesetof Maxwell’sequations:�Í �> Ì �Î

(1.317)�Ë # �> n # @ �Î@ � (1.318)

72 DYNAMICS OF PARTICLES AND FIELDS�> ( �Î # Òs� @ ( �Î@ � ( �> � �> 9 �Î ?¬Òs� @ n@ � # � �Ü (1.319)�> ( n ? @@ � �> 9 �Î � # W Ò � (1.320)

Thefields�Í

and�Ë

canbedeterminedby thesolutions�Î

and n of Eqs.(1.319)and(1.320).However, asseenin thetheoremsin thebeginningof thissection,thepotentialshave thefreedomof gaugein that

�Îis uniqueup to thegradient

�>Gtof

a scalarfieldt

, and n is uniqueup to a constant.For thecoupledsituation,it ispossibleto formulatea generalgaugetransformationthatsimultaneouslyaffects�Î

and n without influenceon thefields.Indeed,ift

is anarbitrarysmoothscalarfield dependingon spaceandtime, thenthetransformation�Î ô �Îvu �Î ? �>wt

(1.321)n ô n u n # @ t@ � (1.322)

doesnot affect thefields�Ë

and�Í

andthetwo inhomogeneousequations.Equa-tion (1.317)for themagneticfield hastheform�Í u �> Ì �Î u �> Ì �Î ? �>xt � �> Ì �Î ? �> Ì �>wt �Í �whereasthecorrespondingone(Eq.1.318)for theelectricfield hastheform�Ë u # �> n u # @ �Î u@ � # �>ï� n # @ t@ � # @@ � �Î ? �>wt � # �> n # @ �Î@ � ? �> @ t@ � # @ �>xt@ � �Ë �Furthermore,Eq. (1.319)transformsas�> ( �Î u�# Òs� @ ( �Î u@ � ( # �>ï� �> 9 �Î u ?¬Ò�� @ n u@ � ?y� �Ü �> ( �Î ? �> ( �>xt � # Òs� @ ( �Î@ � ( # Ò�� @ ( �>Gt@ � (# �> � �> 9 �Î ? �> ( t ?¬Òs� @ n@ � # Ò�� @ ( t@ � ( ?z� �Ü �> ( �Î # Ò�� @ ( �Î@ � ( # �>ï� �> 9 �Î ?¬Ò�� @ n@ � ?y� �Ü �


andEq.(1.320)reads�> ( n u ? @@ � �> 9 �Î u°� ? W Ò �> ( n # �> ( @ t@ � ? @@ � �> 9 �Î � ? @@ � �> ( t ? W Ò �> ( n�? @@ � �> 9 �Î � ? W Ò �andaltogetherthesituationis thesameasbefore.

In thegaugetransformation(Eqs.1.321and1.322),thefreedomof choosingt

is left for theconveniencedependingon theproblem.When�Î

and n satisfytheso-calledLorentzcondition, �> 9 �Î ?¬Ò�� @ n@ � 87�� (1.323)

thenthegaugeis calledtheLor entz gauge. Supposetherearesolutions�Î G andn G to Eqs.(1.319)and(1.320),then�Î|{ �Î G ? �>wt {

(1.324)n { n G # @ t {@ � (1.325)

arealsosolutionsasshown previously. Now,�> 9 �Îv{ ?¬Ò�� @ n {@ � �> 9 �Î G ? �>Gt { � ? Ò�� @@ � � n G # @ t {@ � �> 9 �Î G ?¬Òs� @ n G@ � ? �> ( t { # Ò�� @ ( t {@ � ( �By choosing

t {asa solutionof�> ( t { # Ò�� @ ( t {@ � ( # � �> 9 �Î G ? Ò�� @ n G@ � � (1.326)

obviously theLorentzgaugecondition�> 9 �Î { ? Ò�� @ n {@ � 87 (1.327)


is satisfied.Thenthetwo inhomogeneousequationsaredecoupled,andweob-tain two symmetricinhomogeneouswave equations�> ( �Î {à# Ò�� @ ( �Î {@ � ( # � �Ü (1.328)�> ( n {ç# Ò�� @ ( n {@ � ( # W Ò � (1.329)

Anotherusefulgaugeis theCoulomb gauge, whichsatisfiestheCoulombcon-dition �> 9 �Î �7�� (1.330)

Supposing�Î G and n G aresolutionsto Eqs.(1.319)and(1.320),then�Î|} �Î G ? �>Gt }

(1.331)n } n G # @ t }@ � (1.332)

arealsosolutions.Now observe�> 9 �Î } �> 9 �Î G ? �>xt } � �> 9 �Î G ? �> ( t } �By choosing

t }asa solutionof�> ( t } # �> 9 �Î G� (1.333)

weobtainthat �> 9 �Î } �7 (1.334)

holds.Thenthetwo inhomogeneousequationsread�> ( �Î|} # Ò�� @ ( �Îv}@ � ( Ò�� @ �> n }@ � # � �Ü (1.335)�> ( n } # W Ò � (1.336)

andwhile thereis no symmetry, it is convenientthat the scalarpotential n } isthe“instantaneous”Coulombpotentialdueto thetime-dependentchargedensityW �£y� �� .


Sometimesa gaugewhereÎ c

, the � componentof�Î, is setto 0, is useful.In

theproofof theexistenceof vectorpotentialswe saw thatit is possibleto choose�Îin sucha way; we now assume

�Îdoesnot satisfythis conditionoutright,and

we attemptto bring it to this form. Thegaugeconditionin this caseisÎ c 87�� (1.337)

Supposing�Î G and n G aresolutionsof Eqs.(1.319)and(1.320),then�Î � �Î G ? �>wt �

(1.338)n � n G # @ t �@ � (1.339)

arealsosolutions.In particular, we haveÎ � c Î G c ? @ t �@ � �andby choosing

t �asa solutionof@ t �@ � # Î G c � (1.340)

we obtainthat,asneeded, Î � c &7�� (1.341)

In a similar way we canof coursealsoconstructvectorpotentialsin which the £or ` componentvanishes.

We conclude the discussionof potentials with the special case of time-independentfree space,in which the whole argumentbecomesvery simple.With theconstitutiverelations

�6 Ò G �Ë and�Í � G �I , Maxwell’sequationsare�> 9 �Ë �7�� > Ì �Ë �7 (1.342)�> 9 �Í &7+� �> Ì �Í �7+� (1.343)

Fromthecurl equationswe infer thattherearescalarpotentialsn and ni~ for�Ë

and�Í, respectively, suchthatwehave�Ë # �> n (1.344)�Í # �> ni~ � (1.345)


Applying thedivergenceequationsleadsto Laplaceequationsfor theelectricandmagneticscalarpotential: �> ( n �7 (1.346)�> ( ni~ �7�� (1.347)

1.5.3 BoundaryValueProblems

TheMaxwell theoryyieldsavarietyof partialdifferentialequationsspecifyingthefieldsandpotentials,andthepracticaldeterminationof theserequiresmethodstosolvePDEs. Oneof theimportantcasesis thesolutionof thePoissonequation�> ( n # W Ò � (1.348)

which in theregionwhereno chargeexistsreducesto theLaplaceequation�> ( n 87+� (1.349)

It is importantto studyunderwhat conditionsthesePDEs have solutions andin what situationsthesesolutionsareunique. This questionis similar in natureto the questionof existenceanduniquenessof solutionsof ordinarydifferentialequations,in which caseeven if existenceis ensured,the specificationof initialconditionsis usuallyneededto assertuniqueness.However, in thecaseof PDEs,the situationis morecomplicatedsinceusually conditionshave to be specifiednot only at a point but alsooverextendedregions.For example,a solutionof thePoissonequationfor W O const.is givenbyn �f 9 £y�&`È�T� � ? �f 9 £ ( �a` ( �� ( � (1.350)

which solvesthe equationas long as f � ?�f ( ?�f R #ëO � L Ealso, it is easyto

constructmoresolutions.However, apparentlyit is not sufficient to specify thevalueof n at just onepoint to determinethe exact valuesfor

�f and�f E indeed,

sincetherearefive freeparameters,at leastfivepointswouldbeneeded.Therefore,it is necessaryto study in detail underwhat conditionssolutions

exist andin what situationsthey areunique.Of particularinterestis the caseofthe boundaryvalue problem,in which conditionsare formulatedalong certain“boundaries,” by which we usuallymeansimply connectedsetsthatenclosetheregionof interest.

As we now show, if the valuesof the potentialare specifiedover an entireboundary, thenany solutionsof the Poissonequationareunique.Let usassumewe have given a region M which is enclosedby a simply connectedboundarysurfacez � We considerGreen’sfirst identity ,��] � �> ( ^�? �> �ç9 �> ^ �,� R £à�� @ ^@ ! � f � (1.351)


where � and ^ arescalarfunctions.Green’sfirst identity is a directconsequenceof theGausstheorem, � ] �> 9 �Î � R £*��(� �Î 9 �! � f � (1.352)

where�Î

is avectorfunctionand�!

is theunit outwardnormalvectoratthesurfacez . Choosingthespecialcaseof�Î 8� �> ^ , andobserving�> 9 � �> ^ � 8� �> ( ^å? �> �Ä9 �> ^� �> ^ 9 �! 8� @ ^@ ! �

yieldsGreen’sfirst identity.Now let us assumethereexist two solutions n � and n ( for the samebound-

ary valueproblemof the Poissonequation.Definethe scalarfunction � as thedifferenceof n ( and n � , � n � # n (� (1.353)

Since n � and n ( arebothsolutions,�> ( n � # W Ò and�> ( n (Æ # W Ò

aresatisfiedinsidethevolume M , andwe thushave�> ( � 87 inside M � (1.354)

Now applyGreen’sfirst identity (Eq.1.351)to �à ^ � ; we have��] � �> ( �N? �> � 9 �> � �� R £*�� à@ �@ ! � f (1.355)� ] Q �> � Q ( � R £* �(� � @ �@ ! � f � (1.356)

where(Eq. 1.354)is usedin moving from the left-handsideof thefirst equationto theleft-handsideof thesecondequation.

Now assumethatateverypointonthesurfacez , either n � n ( or @ n �� @ ! @ n ( � @ ! holds.This entailsthat the right-handsidevanishes,andhence�> �

const.,eveneverywhereinside M , andn � n ( ? const.inside M � (1.357)


FIGURE 1.3. Methodof images:A point chargeput closeto aconductorplane.

which meansthatthesolutionof thePoissonequationis uniqueup to a constant.Furthermore,if at leastatonepointontheboundaryeven n � n ( , then n � n (in all of M andhencethesolutionis completelyunique.

The two mostimportantspecialcasesfor the specificationsof boundarycon-ditions arethosein which n is specifiedover the whole surfaceand @ n �� @ ! isspecifiedoverthewholesurface.Thefirst caseis usuallyreferredto astheDirich-let boundary condition, whereasthe latter is usuallycalledthe von Neumannboundary condition. As discussedpreviously, however, a mixtureof specifica-tion of valueandderivativeis sufficient for uniqueness,aslongasateachpointatleastoneof themis given.

In somesituations,it is possibleto obtainsolutionsto boundaryvalueproblemsthroughsimplesymmetryargumentsby suitablyplacingso-calledimagechargesto obtaintheproperpotentialon theboundaryof theproblem.A simpleexampleis a point charge

Oput in thefront of an infinite planeconductorwith a distance�

, asshown in Fig. 1.3.Theimagecharge

#ëO, anequalandoppositecharge,putata distance

�behind

theplaneensuresthat thepotentialon thesurfaceassumesthepropervalue,andthusthat theuniquepotentialhasbeenfound.Anotherexampleis a point chargeO

put in the region betweentwo infinite conductorplaneswhich form the angle� � h , asshown in Fig. 1.4.

FIGURE 1.4. Methodof images:apoint chargeput in a region betweentwo conductorplanesofangle�(�� .


The problemis simulatedbyL h #ù% imagechargesput in the region out of

interestto form mirror images.AllL h chargesform potential 7 at eachedgeline

in thepicture.As discussedin subsequentchapters,themethodof imagesis oftenpractically

usefulin orderto numericallysolvepotentialproblems.

chapter 1 dynamics of particles and fieldsbt.pa.msu.edu/pub/papers/aiep108book/chap1.pdf · chapter...

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