Two-degree-of-freedom Hamiltonian for the time-sym

arXiv:math-ph/0303044v118Mar2003Two-degree-of-freedomHamiltonianforthetime-symmetrictwo-bodyproblemoftherelativisticaction-at-a-distanceelectrodynamicsEfrainBuksmanHollanderandJaymeDeLuca∗UniversidadeFederaldeS˜aoCarlos,DepartamentodeF´ısicaRodoviaWashingtonLuis,km235CaixaPostal676,S˜aoCarlos,S˜aoPaulo13565-905(Dated:February5,2008)AbstractWefindatwo-degree-of-freedomHamiltonianforthetime-symmetricproblemofstraightlinemotionoftwoelectronsindirectrelativisticinteraction.Thistime-symmetricdynamicalsystemappeared100yearsagoanditwaspopularizedinthe1940’sbytheworkofWheelerandFeynmaninelectrodynamics,whichwasleftincompleteduetothelackofaHamiltoniandescription.TheformofourHamiltonianissuchthattheactionofaLorentztransformationisexplicitlydescribedbyacanonicaltransformation(withrescalingoftheevolutionparameter).ThemethodisclosedanddefinestheHamitonianinimplicitformwithoutpowerexpansions.Weoutlinethemethodwithanemphasisonthephysicsofthiscomplexconservativedynamicalsystem.TheHamiltonianorbitsarecalculatednumericallyatlowenergiesusingaself-consistentsteepest-descentmethod(astablenumericalmethodthatchoosesonlythenonrunawaysolution).Thetwo-degree-of-freedomHamiltoniansuggestsasimpleprescriptionforthecanonicalquantizationoftherelativistictwo-bodyproblem.∗correspondingauthor;emailaddress:[e-mail]uivariantdynamicalsystemsunderthePoincar´egrouphasenormousrele-vancetophysicsandyet,todate,onlytheone-bodyrelativisticmotionisfullyunderstood.Alreadywithtwobodiesinrelativisticmotion,oneencounterstheno-interactiontheorem:agrouptheoreticalobstacletotheHamiltoniandescriptionofrelativistictwo-particlemotion[1].Theno-interactiontheoremcanbeovercomebycovariantconstraintdynamics[2],butoneisleftwiththefewcaseswheretheconstraintschemecloses.Forexample,foranequiv-ariantphysicaltheorylikethetime-symmetricelectrodynamics[3],aconstraintdescriptionisunknown.Inthispaperwepresentareductionofthetime-symmetrictwo-bodyproblemoftherelativisticaction-at-a-distanceelectrodynamicstoatwo-degree-of-freedomHamilto-niansystemalongthenonrunawaysolutions.TheformoftheHamiltonianissuchthataLorentztransformationisexplicitlydescribedbyacanonicaltransformationwithrescalingoftheevolutionparameter.TheHamiltonianorbitsarecalculatednumericallybyanumer-icallystableself-consistentmethodthatusessteepest-descentquenchingandchoosesonlythenonrunawayorbits.In1903,Schwarzchildproposedarelativistictypeofinteractionbetweenchargesthatwastimereversiblepreciselybecauseitinvolvedretardedandadvancedinteractionssym-metrically[4].Thesamemodelreappearedinthe1920sintheworkofTetrodeandFokker[5]anditfinallybecameaninterestingphysicaltheoryafterWheelerandFeynmanshowedthatthisdirect-interactiontheorycandescribealltheclassicalelectromagneticphenomena(i.e.theclassicallawsofCoulomb,Faraday,Amp`ere,andBiot-Savart)[3,6].Anotherac-complishmentwasthatWheelerandFeynmanshowedin1945thatinacertainlimitwheretheelectronpracticallyinteractswithacompletelyabsorbinguniverse,theresponseofthisuniversetotheelectron’sfieldisequivalenttothelocalLorentz-Diracself-interactiontheory[7]withouttheneedofmassrenormalization[3].Itisamusingtounderstandthattheclas-sicalradiativephenomenaofMaxwell’selectrodynamicscanbedescribedasalimitingcaseofthisdirect-interactiontheory(completeabsorptionisaddedtothetheoryasasimplemodeltouncoupleitfromthedetailedneutral-delaydynamicsoftheotherchargesoftheuniverse;forotherlimitsseeRef.[8]).Fortherelativistictwo-bodysystemoftheaction-at-a-distanceelectrodynamics,generalsolutionsarenotknownandtheonlyknownspecialsolutionisthecircularorbitforthe2attractivetwo-bodyproblem,firstfoundinRef.[9]andlaterrediscoveredinRef.[10](seealso[11]and[12]).Ourproblemhasalreadybeenstudied:thesymmetricmotionoftwoelectronsalongastraightline[−x2(t)=x1(t)≡x(t)],whichhasthefollowingequationintheaction-at-a-distanceelectrodynamicsmddt(vq1−(v/c)2)=e22r21−v(t−r)/c1+v(t−r)/c!+e22q21+v(t+q)/c1−v(t+q)/c!,(1)wherev(t)≡dx/dtisthevelocityofthefirstelectron,ofmassmandchargee,andrandqarethetime-dependentdelayandadvance,respectively,whichareimplicitlydefinedbythelight-coneconditionscr(t)=x(t)+x(t−r),(2)cq(t)=x(t)+x(t+q),wherecisthespeedoflight.Ingeneral,aneutral-delayequationsuchasEqs.(1)and(2)requiresaninitialfunctionastheinitialcondition,butforthespecialcaseofequations(1)and(2)itwasprovedin1979thatforsufficientlylowenergiestheNewtonianinitialcondition[x(0)=xoandv(0)=vo]determinesauniquesymmetricsolutionthatisgloballydefined(i.e.,thatdoesnotrunawayatsomepoint)[13,14].Thissurprisinguniquenesstheoremreducingtheinitialconditionfromanarbitraryfunctiontotwosimplerealnumbers(initialpositionandvelocity)alreadysuggeststhatthephysicalphasespacecouldbeisomorphictoatwo-degree-of-freedomHamiltonianvectorfield,atleastforlowvelocities(whichiswhatwefindhere).ThefirstnumericalmethodtosolveEqs.(1)and(2)wasgivenin[15]andconvergedtosolutionsuptov/c=0.94.Lateranothermethod[16]convergeduptov/c=0.99.InthefollowingwepresentamethodtofindthenonrunawaysolutionofEqs.(1)and(2)withatwo-degree-of-freedomHamiltoniansystem.Ourmethod