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1、IntroductiontoDeterministicChaosandNonlinearDynamicsPhysicistsmakemanyapproximations.Wehavemadeanumberthisterms,e.g.,frictionlessplanes,masslessandfrictionlesspulleys,masslessandnon-extendiblestrings.Mostoftheseapproximationscanbehandled,atleastapproximately,withoutunduedifficulty.Forexam
2、ple,weknowhowinprinciplehowtoincludefrictionmacroscopicallyevenifwedon'tknowmuchaboutitonamicroscopiclevel.Nexttermwewilllearnhowtogiveswheelsandpulleyssomemass.Theapproximationsaboutstringsareconvenientbutnotnecessary.Thereisoneassumptionthatwehavemadefromtheoutsetthatissosubtlewehavenot
3、reallydiscusseditmuch.Thatis,wehaveassumedthatifweknowalltheforcesactingonasystemofbodiesandwealsoknowtheinitialpositionandvelocityofeachbody,wecouldinprincipledeterminethepositionsandvelocitiesatallothertimes.ThiswouldrequireapplyingNewton'slawstoeachbodyindependentlyandslowlyintegrating
4、theequationsofmotion.Thisassumptionof'deterministic'behaviorseemsquitereasonable.Ontheotherhand,therecertainlyarephysicalphenomenathatseem'chaotic':•theatmosphere;•thestockmarket;•theorbitofoneofSaturn'smoons.Youmightshrugyourshouldersatthefirsttwoofthese.Theseareincrediblycomplexsystemsw
5、hichareimpactedbya)divineinterventionandb)humanintervention.Ontheotherhand,youmightgetmoreconcernedaboutthethird.Theorbitofamooninvolvesafairlysmallnumberofparameters--itseemsasthoughthisshouldsimple,butapparentlyitisnot.5.1ThethreebodyproblemTheso-calledthreebodyproblemhasalonghistoryinp
6、hysics,bothclassicallyandquantum-mechanically.Itisusuallyeasytosolveatwo-bodyproblem.Youknowsomeofthesolutions.Themoonorbitsaroundtheearthwithanapproximatelycircularorbit.Theelectroninthehydrogenatomorbitstheprotonwithwell-definedandstationaryprobability.Nosimple,generalsolutionsexistfort
7、hethree-bodyproblem,eveninclassicalmechanics.Twomoonsmightorbitaplanetforalongtime,andthenan'anomaly'mightoccurwhereintheycometooclosetooneanotherandoneofthetwoeitherleavestheplanetcompletely,orhasitsorbitstronglyperturbed.Itseemsweird,butit'snottoosurprisingwhenyou
