资源描述：
《Hamilton dynamics.pdf》由会员上传分享，免费在线阅读，更多相关内容在学术论文-天天文库。
1、Chapter7HamiltoniandynamicsConservativemechanicalsystemshaveequationsofmo-tionthataresymplecticandcanbeexpressedinHamilto-nianform.Thegenericpropertieswithintheclassofsym-plecticvectorfieldsarequitedifferentfromthosewithintheclassofallsmoothvectorfields:thesystemalwayshasafirstintegral
2、(“energy”)andapreservedvolume,andequilibriumpointscanneverbeasymptoticallystableintheirenergylevel.—JohnGuckenheimeroumightthinkthatthestrangenessofcontractingflows,flowssuchastheYR¨osslerflowoffigure2.6isofconcernonlytochemistsorbiomedicalengineersortheweathermen;physicistsdoHamiltoni
3、andynamics,right?Now,that'sfullofchaos,too!Whileitiseasiertovisualizeaperiodicdynam-icswhenaflowiscontractingontoalower-dimensionalattractingset,thereareplentyexamplesofchaoticflowsthatdopreservethefullsymplecticinvarianceofHamiltoniandynamics.ThewholestorystartedwithPoincar´e'srestr
4、icted3-bodyproblem,arealizationthatchaosrulesalsoingeneral(non-Hamiltonian)flowscamemuchlater.Herewebrieflyreviewpartsofclassicaldynamicsthatwewillneedlateron;symplecticinvariance,canonicaltransformations,andstabilityofHamiltonianflows.Ifyoureventualdestinationareapplicationssuchascha
5、osinquantumand/orsemiconductorsystems,readthischapter.Ifyouworkinneuroscienceorfluiddynamics,skipthischapter,continuereadingwiththebilliarddynamicsofchapter8whichrequiresnoincantationsofsymplecticpairsorloxodromicquartets.fasttrack:chapter8,p.143125CHAPTER7.HAMILTONIANDYNAMICS1267.1
6、Hamiltonianflows(P.Cvitanovi´candL.V.Vela-Arevalo)AnimportantclassofflowsareHamiltonianflows,givenbyaHamiltonianappendixCH(q,p)togetherwiththeHamilton'sequationsofmotionremark2.1∂H∂Hq˙i=,p˙i=−,(7.1)∂pi∂qiwiththed=2Dphase-spacecoordinatesxsplitintotheconfigurationspacecoordinatesandthec
7、onjugatemomentaofaHamiltoniansystemwithDdegreesoffreedom(dof):x=(q,p),q=(q1,q2,...,qD),p=(p1,p2,...,pD).(7.2)Theequationsofmotion(7.1)foratime-independent,D-dofHamiltoniancanbewrittencompactlyas∂x˙i=ωijH,j(x),H,j(x)=H(x),(7.3)∂xjwherex=(q,p)∈Misaphase-spacepoint,andtheaderivativeof
8、(·)withrespecttoxjisdenote
