a lecture on the classical kam theory,

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1、Version1.3–2009SmallcorrectionsandupdatedreferencesALectureontheClassicalKAMTheoremJürgenPöschel1TheClassicalKAM-Theorema.ThepurposeofthislectureistodescribetheKamtheoreminitsmostbasicformandtogiveacompleteanddetailedproof.Thisproofessentiallyfollowsthe

2、traditionallineslaidoutbytheinventorsofthistheory,Kolmogorov,ArnoldandMarXiv:0908.2234v1[math.DS]16Aug2009oser(whencetheacronym‘Kam’),andtheemphasisismoreontheunderlyingideasthanonthesharpnessofthearguments.Afterall,Kamtheoryisnotonlyacollectionofspecifi

3、ctheorems,butratheramethodology,acollectionofideasofhowtoapproachcertainproblemsinperturbationtheoryconnectedwith‘smalldivisors’.b.TheclassicalKamtheoremisconcernedwiththestabilityofmotionsinhamiltoniansystems,thataresmallperturbationsofintegrablehamilt

4、oniansystems.Theseintegrablesystemsarecharacterizedbytheexistenceofactionanglecoordinatessuchthatthehamiltoniandependsontheactionvariablealone–see[2,14]fordetails.ThuswearegoingtoconsiderhamiltoniansoftheformH(p;q)=h(p)+f"(p;q);f"(p;q)=f(p;q;)forsmal

5、l,wherep=(p1;:::;pn)aretheactionvariablesvaryingoversomedomainDRn,whileq=(q;:::;q)aretheconjugateangularvariables,whosedomain1nistheusualn-torusTnobtainedfromRnbyidentifyingpointswhosecomponents2Section1:TheClassicalKAM-Theoremdifferbyintegermultipleso

6、f2.Thus,f"hasperiod2ineachcomponentofq.Moreover,allourhamiltoniansareassumedtoberealanalyticinallarguments.Theequationsofmotionare,asusual,p_=Hq(p;q);q_=Hp(p;q)instandardvectornotation,wherethedotindicatesdifferentiationwithrespecttothetimet,andthesub

7、scriptsindicatepartialderivatives.TheunderlyingphasespaceisDTnRnTnwiththestandardsymplecticstructureX=dpj^dqj:16j6nThehamiltonianvectorfieldXHassociatedwiththeequationsofmotionsthensatisfies(XH;
)=dH.Weassumethatthenumbernofdegreesoffreedomisatleast

8、2,sinceonedegreeoffreedomsystemsarealwaysintegrable.c.For=0thesystemisgovernedbytheunperturbed,integrablehamilton-ianh,andtheequationsofmotionreducetop_=0;q_=!with!=hp(p):Theyareeasilyintegrated–hencethenameintegrablesystem–andt