"Solution périodique." . . "Kritischer Punkt." . . "Kritischer Punkt" . "Problème n-corps." . . "Sistemi dinamici differenziabili." . . "Point critique, théorie du (Analyse mathématique)" . . "Oscillations non linéaires." . . "Periodic solutions of singular Lagrangian systems"@en . "Periodic solutions of singular Lagrangian systems" . . . "Periodic Solutions of Singular Lagrangian Systems" . "Periodic solutions of singular lagrangian systems" . . . . . . . . . . "Thismonographdealswiththeexistenceofperiodicmotionsof Lagrangiansystemswith ndegreesoffreedom ij + V'(q) =0, where Visasingularpotential.Aprototypeofsuchaproblem, evenifitisnottheonlyphysicallyinterestingone,istheKepler problem .. q 0 q+yqr= . This,jointlywiththemoregeneralN-bodyproblem,hasalways beentheobjectofagreatdealofresearch.Mostofthoseresults arebasedonperturbationmethods,andmakeuseofthespecific featuresoftheKeplerpotential. OurapproachismoreonthelinesofNonlinearFunctional Analysis:ourmainpurposeistogiveafunctionalframefor systemswithsingularpotentials,includingtheKeplerandthe N-bodyproblemasparticularcases.PreciselyweuseCritical PointTheorytoobtainexistenceresults,qualitativeinnature, whichholdtrueforbroadclassesofpotentials.Thishighlights thatthevariationalmethods,whichhavebeenemployedtoob tainimportantadvancesinthestudyofregularHamiltonian systems,canbesuccessfallyusedtohandlesingularpotentials aswell. Theresearchonthistopicisstillinevolution,andtherefore theresultswewillpresentarenottobeintendedasthefinal ones. Indeedamajorpurposeofourdiscussionistopresent methodsandtoolswhichhavebeenusedinstudyingsuchprob lems. Vlll PREFACE Partofthematerialofthisvolumehasbeenpresentedina seriesoflecturesgivenbytheauthorsatSISSA,Trieste,whom wewouldliketothankfortheirhospitalityandsupport. We wishalsotothankUgoBessi,PaoloCaldiroli,FabioGiannoni, LouisJeanjean,LorenzoPisani,EnricoSerra,KazunakaTanaka, EnzoVitillaroforhelpfulsuggestions. May26,1993 Notation n 1.For x, yE IR , x. ydenotestheEuclideanScalarproduct, and IxltheEuclideannorm. 2. meas(A)denotestheLebesguemeasureofthesubset Aof n IR - 3.Wedenoteby ST =[0,T]/{a,T}theunitarycirclepara metrizedby t E[0,T].Wewillalsowrite SI= ST=I. n 1 n 4.Wewillwrite sn = {xE IR + : Ixl =I}andn = IR \\{O}. n 5.Wedenoteby LP([O,T], IR ),1~ p~+00,theLebesgue spaces,equippedwiththestandardnorm lIulip. l n l n 6. H (ST, IR )denotestheSobolevspaceof u E H ,2(0, T; IR ) suchthat u(O) = u(T).Thenormin HIwillbedenoted by lIull2 = lIull~ + lIull~· 7.Wedenoteby(·1·)and11·11respectivelythescalarproduct andthenormoftheHilbertspace E. 8.For uE E, EHilbertorBanachspace,wedenotetheball ofcenter uandradiusrby B(u,r) = {vE E: lIu- vii~ r}.Wewillalsowrite B = B(O, r). r 1 1 9.WesetA (n) = {uE H (St,n)}. k 10.For VE C (1Rxil,IR)wedenoteby V'(t, x)thegradient of Vwithrespectto x. l 11.Given f E C (M,IR), MHilbertmanifold,welet r = {uEM: f(u) ~ a}, f-l(a,b) = {uE E : a~ f(u) ~ b}. x NOTATION 12.Given f E C1(M,JR), MHilbertmanifold,wewilldenote by Zthesetofcriticalpointsof fon Mandby Zctheset Z U f-l(c, c). 13.Givenasequence UnE E, EHilbertspace,by Un --\"\" Uwe willmeanthatthesequence Unconvergesweaklyto u. 14.With £(E)wewilldenotethesetoflinearandcontinuous operatorson E. 15.With Ck''''(A,JR)wewilldenotethesetoffunctions ffrom AtoJR, ktimesdifferentiablewhosek-derivativeisHolder continuousofexponent0:. Main Assumptions Wecollecthere,forthereader'sconvenience,themainassump tionsonthepotential Vusedthroughoutthebook. (VO) VEC1(lRXO,lR),V(t+T,x)=V(t,X) V(t,x)ElRXO, (VI) V(t,x)"@en . . . . . . . . . . . . . . . . . . . . . . . . . . . . "Electronic books"@en . . . . . . . . . . . . . . "Sistemes dinàmics diferenciables." . . "Système Lagrange." . . . . "Differenzierbares dynamisches System." . . "Differenzierbares dynamisches System" . "Oscillazioni non lineari." . . "Oscillations nonlinéaires." . . "Points critiques, Théorie des (analyse mathématique)" . . "Système dynamique différentiable." . . "Periodische Lösung." . . "Periodische Lösung" . "Systèmes dynamiques différentiables." . . "Mehrkörperproblem." . . "Punto critico (Analisi matematica)" . . "Kritischer Punkt (Math.)" . . "Oscil·lacions." . . "Dynamique différentiable." . .