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1、Portugal.Math.(N.S.)PortugaliaeMathematicaVol.xx,Fasc.,200x,xxx–xxxcEuropeanMathematicalSociety(Quasi)periodicsolutionsin(in)finitedimensionalhamiltoniansystemswithapplicationstoCelestialMechanicsandwaveequationLucaBiasco,EnricoValdinoci∗Abstract.Wedescribeagen
2、eralmethod,basedonaLyapunov–Schmidtreductionandperturbativetechniques,recentlyusedbytheauthorstofindperiodicandquasi–periocidsolutionsbothinfiniteandininfinitedimensionalhamiltoniansystems.WealsoillustratesomeconcreteapplicationstoCelestialMechanicsandtononlinear
3、waveequation.MathematicsSubjectClassification(2000).Primary34C25,35L05,70F10,34C27;Secondary37K50,37J40,70K43.Keywords.Nearly–integrableHamiltoniansystems,periodicsolutionslowerdimen-sionalelliptictori.N–bodyproblem,waveequation.IntroductionInthisnote,wedealwit
4、hfourtopics:Spatialplanetarythree-bodyproblem.Weconsiderone“star”andtwo“planets”,modelledbythreemassivepoints,interactingthroughgravityinathree-dimensionalspace.Nearthelimitingsolutionsgivenbythetwoplanetsrevolv-ingaroundthestaronKeplerianellipseswithsmallecce
5、ntricityandsmallnon-zeromutualinclination,thesystemisprovedtohavetwo-dimensional,elliptic,quasiperiodicsolutions,providedthemassesoftheplanetsaresmallenoughcom-paredtothemassofthestarandprovidedtheosculatingKeplerianmajorsemiaxesbelongtoatwo-dimensionalsetofde
6、nsityclosetoone.Planarplanetarymany-bodyproblem.Asabove,butone“star”andN“planets”,theinteriortwoonesbiggerthantheothers(asintheexteriorsolarsystem).NearthelimitingsolutionsgivenbytheNplanetsrevolvingaroundthe∗SupportedbyMIURVariationalMethodsandNonlinearDiffere
7、ntialEquations.2BiascoValdinocistaronKeplerianellipseswithsmalleccentricityandzeromutualinclination,thesystemisprovedtohaveN-dimensional,elliptic,quasiperiodicsolutions.PeriodicorbitsapproachinglowerdimensionalellipticKAMtori.ByageneralBirkhoff-Lewis-Conley-Zeh
8、nder-typeresult,weprovetheexistenceofinfinitelymanyperiodicsolutions,withlargerandlargerminimalperiod,accumu-latingontoellipticinvarianttoriofHamiltoniansystems.Asanapplication,peri
