Periodic Solutions of Singular Lagrangian Systems PDF
Preview Periodic Solutions of Singular Lagrangian Systems
Progress in NonlinearDifferential Equations and TheirApplications Volume 10 Editor HaimBrezis UniversitePierreetMarieCurie Paris and RutgersUniversity NewBrunswick,N.J. EditorialBoard A.Bahri,RutgersUniversity,NewBrunswick JohnBall,Heriot-WattUniversity, Edinburgh LuisCafarelli,InstituteforAdvancedStudy,Princeton MichaelCrandall,UniversityofCalifornia,SantaBarbara MarianoGiaquinta,UniversityofFlorence DavidKinderlehrer,Carnegie-MellonUniversity,Pittsburgh RobertKohn,NewYorkUniversity P. L.Lions,UniversityofParisIX LouisNirenberg,NewYorkUniversity LambertusPeletier,UniversityofLeiden PaulRabinowitz,UniversityofWisconsin,Madison Antonio Ambrosetti Vittorio Coti Zelati Periodic Solutions of Singular Lagrangian Systems Springer Science+Business Media, LLC Antonio Ambrosetti Vittorio Coti Zelati Scuola Nonnale Superiore Facoltli di Architettura 1-56100 Pisa 80134 Naples Italy Italy Library of Congress Cataloging In-Publication Data Ambrosetti, A. (Antonio) Periodic solutions of singular Lagrangian systems 1 Antonio Ambrosetti, Vittorio Coti Zelati. p. cm. -- (Progress in nonlinear differential equations and their applications ; v. 10) Includes bibliographical references. ISBN 978-1-4612-6705-8 ISBN 978-1-4612-0319-3 (eBook) DOI 10.1007/978-1-4612-0319-3 1. Differentiable dynamical systems. 2. Nonlinear oscillations. 3. Critical point theory. I. Coti Zelati, Vittorio, 1956- QA614.8.A45 1993 93-24376 515'.355--dc20 CIP Printed on acid-free paper © Springer Science+Business Media New York 1993 Originally published by Birkhliuser Boston in 1993 Softcover reprint oft he hardcover 1st edition 1993 Copyright is not claimed for works of U.S. Government employees. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopy ing, recording, or otherwise, without prior permission of the copyright owner. Permission to photocopy for internal or personal use of specific clients is granted by Birkhauser Boston for libraries and other users registered with the Copyright Clearance Center (CCC), provided that the base fee of $6.00 per copy, plus $0.20 per page is paid directly to CCC, 21 Congress Street, Salem, MA 01970, U.S.A. Special requests should be addressed directly to Springer Science+Business Media, LLC. ISBN 978-1-4612-6705-8 Typeset by the Authors in LATEX. 9 8 7 6 5 432 1 Contents I Preliminaries 1 1 Lagrangian systems with smooth potentials. 1 2 Models involving singular Lagrangians 3 2.a Kepler's problem . . . . . . 3 2.b A class of model potentials. . . 6 2.c The N-body problem . . . 8 2.d Other problems arising in Celestial Me- chanics . . . . . 9 2.e Electrical forces 10 3 Critical point theory 10 II Singular Potentials 19 4 The functional setting 19 4.a Prescribed period 20 4.b Fixed energy .. 21 5 The Strong Force assumption 25 6 Collision solutions. . . . . . . 27 III The Strongly Attractive Case 31 7 The abstract setting ... 31 7.a The (PS) condition .. 35 r. . . 7.b The topology of 38 7.c A critical point theorem 40 7.d Another critical point theorem. 41 8 Existence of periodic solutions . . . 43 8.a Even and planar potentials . 45 8.b The general case . 47 VI CONTENTS 9 Repulsive potentials. . . . . . . . . . . . . . . .. 51 IV The Weakly Attractive Case 57 10 Weak solutions . 57 11 Existence of weak solutions 64 12 Regularity of weak solutions 68 13 Local assumptions 73 14 Global assumptions ..... 79 V Orbits with Prescribed Energy 87 15 Strongly attractive potentials . 88 16 Weakly attractive potentials .. 92 16.a A modified variational principle 92 16.b Existence of closed orbits. 95 17 Symmetric potentials . . . . . . . 104 VI The N-Body Problem 113 18 The N-body equation. 113 19 Even potentials . 115 20 The general case 119 21 Fixed energy. . . 122 VII Perturbation Results 129 22 A perturbation result in critical point theory 130 23 T-periodic solutions. . . . . . 131 24 First order systems . . . . . . 137 25 Solutions of prescribed energy 142 26 Restricted N-Body problems. 144 Preface This monograph deals with the existence of periodic motions of Lagrangian systems with n degrees of freedom ij +V'(q) = 0, where V is a singular potential. A prototype ofsuch a problem, even ifit is not the only physically interesting one, is the Kepler problem q..+yqqr= 0. This, jointly with the moregeneral N-body problem, has always been the object of a great deal of research. Most ofthose results are based on perturbation methods, and makeuseofthe specific features of the Kepler potential. Our approach is more on the lines of Nonlinear Functional Analysis: our main purpose is to give a functional frame for systems with singular potentials, including the Kepler and the N-body problem as particular cases. Precisely we use Critical Point Theory to obtain existence results, qualitative in nature, which hold true for broad classes of potentials. This highlights that the variational methods, which have been employed to ob tain important advances in the study of regular Hamiltonian systems, can be successfally used to handle singular potentials as well. The research on this topic is still in evolution, and therefore the results we will present are not to be intended as the final ones. Indeed a major purpose of our discussion is to present methods and tools which have been used in studying such prob lems. Vlll PREFACE Part of the material of this volume has been presented in a series of lectures given by the authors at SISSA, Trieste, whom we would like to thank for their hospitality and support. We wish also to thank Ugo Bessi, Paolo Caldiroli, Fabio Giannoni, Louis Jeanjean, Lorenzo Pisani, Enrico Serra, Kazunaka Tanaka, Enzo Vitillaro for helpful suggestions. May 26, 1993 Notation 1. For x,y E IRn, x .y denotes the Euclidean Scalar product, Ixl and the Euclidean norm. 2. meas(A) denotes the Lebesgue measure of the subset A of IRn• 3. We denote by ST = [0,T]/{a,T} the unitary circle para metrized by t E [0,T]. We will also write SI =ST=I. 4. We will write sn = {x E IRn+1 : Ixl = I} and n= IRn\{O}. 5. We denote by LP([O,T],IRn), 1 ~ p ~ +00, the Lebesgue lIulip. spaces, equipped with the standard norm 6. Hl (ST,IRn) denotes theSobolev spaceofu E Hl,2(0,T;IRn) such that u(O) = u(T). The norm in HI will be denoted by lIull2 = lIull~ + lIull~· (·1·) 11·11 7. We denote by and respectively the scalar product and the norm of the Hilbert space E. 8. For u E E, E Hilbert or Banach space, we denote the ball of center u and radius r by B(u,r) = {v E E : lIu- vii ~ r}. We will also write B = B(O,r). r 9. We set A1(n) = {u E H1(St, n)}. 10. For V E Ck(1R x il,IR) we denote by V'(t, x) the gradient of V with respect to x. r 11. Given f E Cl(M,IR), M Hilbert manifold, we let = {u EM: f(u) ~ a}, f-l(a,b) = {u E E : a ~ f(u) ~ b}. x NOTATION 12. Given f E C1(M,JR), M Hilbert manifold, we will denote by Z the set of critical points of f on M and by Zc the set Z U f-l(c, c). 13. Given a sequence E E, E Hilbert space, by we Un Un ---"" U will mean that the sequence converges weakly to u. Un 14. With £(E) we will denote the set of linear and continuous operators on E. 15. With Ck''''(A,JR) we will denote the set offunctions f from A to JR, k times differentiable whose k-derivative is Holder continuous of exponent 0:.
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