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Periodic Solutions of Singular Lagrangian Systems

A. Ambrosetti

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ISBN 10: 0817636552 / ISBN 13: 9780817636555
Published by Birkhäuser
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158 pages. Dimensions: 9.3in. x 6.2in. x 0.7in.Thismonographdealswiththeexistenceofperiodicmotionsof Lagrangiansystemswith ndegreesoffreedom ij V(q) 0, where Visasingularpotential. Aprototypeofsuchaproblem, evenifitisnottheonlyphysicallyinterestingone, istheKepler problem . . q 0 qyqr . 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, Ta, Ttheunitarycirclepara- metrizedby t E0, T. Wewillalsowrite SI STI. n 1 n 4. Wewillwrite sn xE IR : Ixl Iandn IR O. n 5. Wedenoteby LP(O, T, IR ), 1 p00, 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)and1111respectivelythescalarproduct 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 GBP(E)wewilldenotethesetoflinearandcontinuous operatorson E. 15. With Ck(A, JR)wewilldenotethesetoffunctions ffrom AtoJR, ktimesdifferentiablewhosek-derivativeisHolder continuousofexponent0: . Main Assumptions Wecollecthere, forthereadersconvenience, themainassump- tionsonthepotential Vusedthroughoutthebook. (VO) VEC1(lRXO, lR), V(tT, x)V(t, X) V(t, x)ElRXO, (VI) V(t, x) This item ships from multiple locations. Your book may arrive from Roseburg,OR, La Vergne,TN. Bookseller Inventory # 9780817636555

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Title: Periodic Solutions of Singular Lagrangian ...

Publisher: Birkhäuser

Binding: Hardcover

Book Condition:New

Book Type: Hardcover

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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)

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