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1、GENERALIZEDHAMILTONIANDYNAMICSP.A.M.DIRAC1.Introduction.TheequationsofdynamicswereputintoageneralformbyLagrange,whoexpressedthemintermsofasetofgeneralizedcoordinatesandvelocities.AnalternativegeneralformwaslatergivenbyHamilton,intermsofcoordinatesandmomenta.Letusc
2、onsidertherelativemeritsofthetwoforms.WiththeLagrangianformtherequirementsofspecialrelativitycanveryeasilybesatisfied,simplybytakingtheaction,i.e.thetimeintegraloftheLagrangian,tobeLorentzinvariant.ThereisnosuchsimplewayofmakingtheHamiltonianformrelativistic.Forth
3、epurposeofsettingupaquantumtheoryonemustworkfromtheHamiltonianform.Therearewell-establishedrulesforpassingfromHamilton'sdynamicstoquantumdynamics,bymakingthecoordinatesandmomentaintolinearoperators.Therulesleadtodefiniteresultsinsimplecasesand,althoughtheycannotbe
4、appliedtocomplicatedexampleswithoutambiguity,theyhaveprovedtobeadequateforpracticalpurposes.Thusbothformshavetheirspecialvaluesatthepresenttimeandonemustworkwithboth.Thetwoformsarecloselyconnected.StartingwithanyLagrangianonecanintroducethemomentaand,inthecasewhen
5、themomentaareindependentfunctionsofthevelocities,onecanobtaintheHamiltonian.Thepresentpaperisconcernedwithsettingupamoregeneraltheorywhichcanbeappliedalsowhenthemomentaarenotindependentfunctionsofthevelocities.AmoregeneralformofHamiltoniandynamicsisobtained,whichc
6、anstillbeusedforthepurposeofquantization,andwhichturnsouttobespeciallywellsuitedforarelativisticdescriptionofdynamicalprocesses.2.Strongandweakequations.WeconsideradynamicalsystemofNdegreesoffreedom,describedintermsofgeneralizedcoordinatesqn(n=1,2,...,N)andvelocit
7、iesdqn/dtorqn.WeassumeaLagrangianL,whichforthepresentcanbeanyfunctionofthecoordinatesandvelocities(1)L-L(q,q).Wedefinethemomentaby(2)pn=dL/dqn.Forthedevelopmentofthetheoryweintroduceavariationprocedure,varyingeachofthequantitiesqn,qnypnindependentlybyasmallquantit
8、yèqn,dqn,opnofordereandworkingtotheaccuracyofe.AsaresultofthisThispaperisbasedonthefirsthalfofacourseoflecturesgivenattheCanadianMath•ematicalSeminarinV
