Periodic Solutions of Singular Lagrangian Systems: 10 (Progress in Nonlinear Differential Equations and Their Applications) - Tapa dura

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)

Nonlinear functional analysis has proven to be a powerful alternative to classical perturbation methods in the study of periodic motions of regular Hamiltonian systems. The authors of this monograph present a summary and synthesis of recent research demonstrating that variational methods can be used to successfully handle systems with singular potential, the Lagrangian systems. The classical cases of the Kepler problem and the N-body problem are used as specific examples. Critical point theory is used to obtain existence results, qualitative in nature, which hold true for broad classes of potentials. These results give a functional frame for systems with singular potential. The authors have provided some valuable methods and tools to researchers working on this constantly evolving topic. At the same time, they present the new approach and results that they have shared over recent years with their colleagues and graduate students.