A. Ambrosetti; V. Coti-Zelati Periodic Solutions of Singular Lagrangian Systems 9781461267058

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Автор: A. Ambrosetti; V. Coti-Zelati
Название: Periodic Solutions of Singular Lagrangian Systems
ISBN: 9781461267058
Издательство: Springer
Классификация:

Основы высшей математики и математический анализ

Дифференциальное исчисление и дифференциальные уравнения

ISBN-10: 1461267056
Обложка/Формат: Paperback
Страницы: 160
Вес: 0.25 кг.
Дата издания: 01.03.2013
Серия: Progress in Nonlinear Differential Equations and Their Applications
Язык: English
Размер: 234 x 156 x 10
Основная тема: Mathematics
Ссылка на Издательство: Link
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Поставляется из: Германии
Описание: 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, forthereadersconvenience, themainassump- tionsonthepotential Vusedthroughoutthebook. (VO) VEC1(lRXO, lR), V(t+T, x)=V(t, X) V(t, x)ElRXO, (VI) V(t, x)

Periodic Solutions of Singular Lagrangian Systems, A. Ambrosetti; V. Coti-Zelati