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1、AnIntroductiontoKAMTheoryC.EugeneWayneJanuary22,20081IntroductionOverthepastthirtyyears,theKolmogorov-Arnold-Moser(KAM)theoryhasplayedanimportantroleinincreasingourunderstandingofthebehaviorofnon-integrableHamiltoniansystems.Ihopetoillustrateintheselecturesthatthecentralideasofthetheoryare,infa
2、ct,quitesimple.Withthisinmind,Iwillconcentrateontwoexamplesandwillforegogeneralityforconcretenessand(Ihope)clarity.TheresultsandmethodswhichIwillpresentarewell-knowntoexpertsinthefieldbutIhopethatbycollectingandpresentingtheminassimpleacontextaspossibleIcanmakethemsomewhatmoreapproachabletonewco
3、mersthantheyareoftenconsideredtobe.Theoutlineofthelecturesisasfollows.Afterashorthistoricalintroduction,IwillexplainindetailoneofthesimplestsituationswheretheKAMtechniquesareused–thecaseofdiffeomorphismsofacircle.Iwillthengoontodiscussthetheoryinitsoriginalcontext,thatofnearly-integrableHamilton
4、iansystems.TheproblemwhichtheKAMtheorywasdevelopedtosolvefirstaroseincelestialmechanics.Morethan300yearsago,Newtonwrotedownthedif-ferentialequationssatisfiedbyasystemofmassivebodiesinteractingthroughgravitationalforces.Ifthereareonlytwobodies,theseequationscanbeex-plicitlysolvedandonefindsthattheb
5、odiesrevolveonKeplerianellipsesabouttheircenterofmass.Ifoneconsidersathirdbody(the“three-body-problem”),noexactsolutionexists–evenif,asinthesolarsystem,twoofthebodiesaremuchlighterthenthethird.Inthiscase,however,oneobservesthatthemutualgravitationalforcebetweenthesetwo“planets”ismuchweakerthant
6、hatbe-tweeneitherplanetandthesun.Underthesecircumstancesonecantrytosolvetheproblemperturbatively,firstignoringtheinteractionsbetweentheplanets.Thisgivesanintegrablesystem,oronewhichcanbesolvedexplicitly,witheachplanetrevolvingaroundthesunobliviousoftheother’sexistence.Onecanthentrytosystematical
7、lyincludetheinteractionbetweentheplanetsinaperturbativefashion.Physicistsandastronomersusedthismethodexten-sivelythroughoutthenineteenthcentury,developingseriesexpansionsforthesolutionsoftheseequationsinthesmallparameterrepresente
