Periodic Solutions of Singular Lagrangian Systems |
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Author:
| Ambrosetti, A. Coti-Zelati, V. |
Series title: | Progress in Nonlinear Differential Equations and Their Applications Ser. |
ISBN: | 978-1-4612-6705-8 |
Publication Date: | Mar 2013 |
Publisher: | Birkhäuser Boston
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Imprint: | Birkhäuser |
Book Format: | Paperback |
List Price: | USD $109.99 |
Book Description:
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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...
More DescriptionThismonographdealswiththeexistenceofperiodicmotionsof 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)