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1、Portugal.Math.(N.S.)PortugaliaeMathematicaVol.xx,Fasc.,200x,xxx–xxxcEuropeanMathematicalSociety(Quasi)periodicsolutionsin(in)?nitedimensionalhamiltoniansystemswithapplicationstoCelestialMechanicsandwaveequationLucaBiasco,EnricoValdinoci?Abstract.Wedescribeagen
2、eralmethod,basedonaLyapunov–Schmidtreductionandperturbativetechniques,recentlyusedbytheauthorsto?ndperiodicandquasi–periocidsolutionsbothin?niteandinin?nitedimensionalhamiltoniansystems.WealsoillustratesomeconcreteapplicationstoCelestialMechanicsandtononlinear
3、waveequation.MathematicsSubjectClassi?cation(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?SupportedbyMIURVariationalMethodsandNonlinearDi?ere
7、ntialEquations.2BiascoValdinocistaronKeplerianellipseswithsmalleccentricityandzeromutualinclination,thesystemisprovedtohaveN-dimensional,elliptic,quasiperiodicsolutions.PeriodicorbitsapproachinglowerdimensionalellipticKAMtori.ByageneralBirkho?-Lewis-Conley-Zeh
8、nder-typeresult,weprovetheexistenceofin?nitelymanyperiodicsolutions,withlargerandlargerminimalperiod,accumu-latingontoellipticinvarianttoriofHamiltoniansystems.Asanapplication,peri
