Periodic solutions of singular Lagrangian systems
Periodic solutions of singular Lagrangian systems
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DOI：
10.1007/978-1-4612-0319-3
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发表时间：
1993
期刊：
影响因子：
--
通讯作者：
A. Ambrosetti;V. C. Zelati
中科院分区：
文献类型：
--
作者：
A. Ambrosetti;V. C. Zelati
Thismonographdealswiththeexistenceofperiodicmotionsof拉格朗日系统，n阶离散度为ij+V‘(Q)=0，其中粘性单势。开普勒问题是这样一个问题的原型，即使它不仅仅是一个物理上令人感兴趣的问题。Q 0 q+yqr=。这一点，连同其他再生的N-Body问题，一直是许多研究的目标。这些结果大多基于微扰法，并利用了开普勒势的特殊性质。OurapproachismoreonthelinesofNonlinearFunctional分析：我们的主要目的是给出单势系统的函数框架，包括特例的Kepler和N体问题。我们用临界点理论来获得存在的结果，本质上是定性的，这对于更广泛的潜能类别来说是正确的。这表明，tainimportantadvancesinthestudyofregularHamiltonian系统和canbesuccessfallyusedtohandlesingularpotentials系统都采用了变分方法。这种研究仍然是进化的，因此也是theresultswewillpresentarenottobeintendedasthefinal进化的。实际上，我们讨论的一个主要目的就是讨论当前的methodsandtoolswhichhavebeenusedinstudyingsuchprob问题。Vll对作者在的里雅斯特SISSA给出的Partofthematerialofthisvolumehasbeenpresentedina系列文章作了序，wewouldliketothankfortheirhospitalityandsupport.我们也要感谢乌戈贝西、保罗·卡尔迪罗里、法比奥·詹诺尼、路易斯·让让、洛伦佐·皮萨尼、恩里科·塞拉、田中和中、伊索·维拉罗的帮助建议。1993年5月26日记号n 1.对于x，Ye IR，x表示欧几里得和标度积，以及Ixl欧几里得范数。2.meas(A)不是指IR-3的子集的勒贝格度量。T=[0，T]/{a，T}由t E[0，T]度量的么正圆参数。Wewillalsowite SI=ST=I.n 1 n 4.Wewillwrite sn={Xe IR+：Ixl=I}和n=IR\{O}。N5.由Lp([O，T]，IR)，1~p~+00所定义的Lebesgu型空间，具有标准的赋范郁金香。H(ST，IR)表示u EH，2(0，T；IR)的体波列夫空间，使得u(O)=u(T)。7.我们分别用(·1·)和(11·11)表示阶乘和移动希尔伯特空间E。8.对于UE，我们用B(u，r)={ve E：Liu-vii~r}表示中心球和半径球。Wewillalsowite B=B(O，r)。R1 1 9.WesetA(N)={UE H(ST，n)}.K10.对于V‘(t，x)记为V’(t，x)的VEC(1Rxil，IR)关于X的梯度。11.给定f EC(M，IR)，M Hilbert流形，使得r={uem：F(U)~a}，fl(a，b)={ue：A~f(U)~b}。X记号12.给定f E c1(M，Jr)，M Hilbert流形，我们用Z表示Fon Mandby Zctheet zu fl(c，c)的临界点的集合。13.给出一个序列unE，EHilbert空间，通过Un--“我们将表示该序列弱收敛于u.14.对于GB(E)，我们将定义E.15上的直线和连续算子.对于k‘(A，Jr)，我们将定义来自Ato Jr的k次可微的函数集，k次可微的k次导数持有指数为0的连续.为方便读者，我们在这里收集了贯穿全书的电势的主要假设：(Vo)VEC1(lRx0，Lr)，V(t+T，x)=V(t，X)V(t，x)ElRx0，(Vi)V(t，x)
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. lnln 6. H (ST, IR) denotestheSobolevspaceof u EH, 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 EC (M, IR), MHilbertmanifold, welet r={uEM: f (u)~ a}, fl (a, b)={uE E: a~ f (u)~ b}. x NOTATION 12. Given f E C1 (M, JR), MHilbertmanifold, wewilldenote by Zthesetofcriticalpointsof fon Mandby Zctheset ZU fl (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)