Mould Calculus for Hamiltonian Vector Fields

arXiv:0801.2953v1[math.DS]18Jan2008MOULDCALCULUSFORHAMILTONIANVECTORFIELDSbyJakyCresson&GuillaumeMorinAbstrat.Wepresentthegeneralframeworkofalle’smouldsintheaseoflineariza-tionofaformalvetoreldwithoutandwithinresonanes.Weenlightenthepowerofmouldsbytheiruniversality,andalulability.Wemodifythenalle’stehniquetotintheseekofaformalnormalformofaHamiltonianvetoreldinartesianoordinates.WeprovethatmouldalulusanalsoproduesuessiveanonialtransformationstobringaHamiltonianvetoreldintoanormalform.WethenproveaKolmogorovtheoremonHamiltonianvetoreldsnearadiophantinetorusination-angleoordinatesusingmouldstehniques.Contents1.Introdution...........................................................12.Reminderaboutmoulds...............................................23.Continuousprenormalformsofavetoreld..........................74.Eetiveaspetsofontinuousprenormalforms.......................125.ArstapproahtothePoinarØ-Dulanormalform...................146.Thetrimmedform.....................................................207.TheHamiltonianase.................................................238.KolmogorovTheorem..................................................279.Conlusion.............................................................29Referenes................................................................291.IntrodutionWedealinthistextwithformalnormalformsforformalvetoreldsonCν.WeusethemouldformalismbyJeanalletoobtainthose.TheideainthisformalismistoonsidervetoreldsasderivationsonthealgebraofformalpowerseriesC[[x]]andworkinthegeneralfreeLiealgebrasframeworkassoiatedtothealgebrabuildonthesederivations.Itwasdevelopedbyalle(see[5,6,7℄)butdidn’tgetthesuessitdeservedyet.Thistextomesbakonalle’sideawithsomepreisealuluswedidn’tndinhisworks,althoughitwassaidtoberight.TheHamiltonianparts(setions7and8)werealsoevoked2000MathematisSubjetClassiation.37G05,37J40,17B40,17A50,17B66,17B70.Keywordsandphrases.normalform,ontinuousprenormalform,mould,mouldalulus,Hamil-toniansystems,Kolmogorovtheorem.2JACKYCRESSON&GUILLAUMEMORINbyallein[9℄butstillnotwritten:wehopetogiveherealittleontributiontohisworkandaneduationalaspet.Inordertomakethereaderfamiliarwithmouldsandmouldalulusinthesearhforformalnormalforms,wereallinsetions2to6someofalle’swork,andsetaglobalframeworkformoulds,whihisthegeneralfreeLiealgebrasframework.Then,insetions7and8wepresentalreadyknownresults,withthenewtehniquesofmoulds.Thesearhforformalnormalformsofvetoreldshasagreatrsttheoremfromagreatmathematiian:thelinearizationtheorembyPoinarØ.Wegivehereamouldsproofofthistheorem,whihobviouslymakethesmalldivisorsappear,andmoreover,arousesauniversalharaterofmoulds:thelinearizationmouldonlydependonthegraduation(i.e.thedeomposition)ofthevetoreldX.Thisisofgreatinterest,beausewhenthevetoreldismodied,thelinearizationmouldisstilthesame,aslongasthegraduationofthevetoreldisthesame.Theplanofthistextisthefollowing:setions2to6areofpedagogialinterest,andsummarizethemaindenitions,resultsandtehniquesofalle’smouldsweneed.Mostofitanbefoundin[3,4,5,6,7,8℄.Theoriginalworkwedidanbefoundinthelasttwosetions.Morepreisely:Setion2reallssomebasidenitionsandresultsaboutmouldformalism.Insetion3wedenethemainobjetofouronern:aprenormalform.Thatis,avetoreldXbeinggivenwithaxeddiagonallinearpartXlin,welookforahangeofvariableswhihbringsXintoXnor,suhthat[Xlin,Xnor]=0.Setion3dealswithontinuousprenormalforms,followingalle’sterminology;thatis,howdoesaprenormalformXnorbehavewhenthevetoreldXismodied,itslinearpartbeinguntouhed?Wegiveherearstappliationofthepowerofthemouldformalism,alulatingadirettransformoflinearizationofX,aordingtoPoinarØ’slinearizationtheorem.Theaseofresonantvetoreldsrisesinthenextsetion4:weobtainananalogousresultofthelassialPoinarØ-Dulatheorem;neverthelesstheprenormalformalulatedhereisnotthePoinarØ-Dulanormalform;alleallsitthetrimmedform.ThelasttwosetionsfousonHamiltonianvetorelds,whihwastheoriginalgoalofthistext:wemakehereaslightlymodiationinalle’sformalism:wherehomogeneousdierentialoperatorswereused,weneedanothergraduation(i.e.deomposition)ofthevetoreldXtoprovethatitispossibletomakesuessiveanonialtransformationstobringaformalHamiltonianwitharesonantlinearpartinartesianoordinatesintoatrimmedform,preservingtheHamiltonianharaterateahstep.Then,insetion7,following[11℄weproveaformalKolmogorovtheoremonaformalHamiltoniannearadiophantinetorususingtehniquesshowninsetion6.Westudyhereperturbationsination-angleoordinatesastrigonometripolynomialsfortehnialreasons.2.ReminderaboutmouldsAllproofsanddetailsaboutthissetionanbefoundin[4℄.WedenotebyAanalphabet,niteornot,whihisasemigroupforalaw⋆.Inthissetion,aletterofAisdenotedbya.A∗denotesthesetofallwordsabuildonAi.e.thetotallyorderedsequenesa1···ar,r0,withaiinAandr=ℓ(a)thelengthoftheworda.WesettheMOULDCALCULUSFORHAMILTONIANVECTORFIELDS3onventionthatawordoflength0istheemptyword∅.Moreover,A∗rdenotesthesetofwordsofexatlengthr.ThenaturaloperationonA∗istheusualonatenationoftwowordsaandbofA∗,whihgluesthewordatothewordb,i.e.a•b,oroftensimplyabwhenthere