London Mathematical Society Lecture Note Series ManagingEditor: ProfessorN.J.Hitchin, Mathematical Institute, 24-29 St. Giles, Oxford OX1 3DP, UK AllthetitleslistedbelowcanbeobtainedfromgoodbooksellersorfromCambridge UniversityPress. Foracompleteserieslistingvisit http://publishing.cambridge.org/stm/mathematics/lmsn/ 285. Rationalpointsoncurvesoverfinitefields,H. Niederreiter & C. Xing 286. Cliffordalgebrasandspinors,2ndedn,P. Lounesto 287. TopicsonRiemansurfacesandFuchsiangroups, E. Bujalance, A.F. Costa & E. Martinez(eds) 288. Surveysincombinatorics,2001,J.W.P. Hirschfeld(ed) 289. AspectsofSobolev-typeinequalities,L. Saloffe-Coste 290. QuantumgroupsandLietheory,A. Pressley 291. Titsbuildingsandthemodeltheoryofgroups,K. Tent 292. Aquantumgroupsprimer,S. Majid 293. SecondorderpartialdifferentialequationsinHilbertspaces,G. da Prato & J. Zabczyk 294. Introductiontooperatorspacetheory,G. Pisier 295. Geometryandintegrability,L. Mason & Y. Nutku(eds) 296. Lecturesoninvarianttheory,I. Dolgachev 297. Thehomotopytheoryofsimply-conected4-manifolds, H.J. Baues 298. Higheroperads,highercategories,T. Leinster 299. Kleiniangroupsandhyperbolic3-manifolds,Y. Komori, V. Markovic & C. Series(eds) 300. IntroductiontoM¨obiusdifferentialgeometry,U. Hertrich-Jeromin 301. StablemodulesandtheD(2)-problem,F.A.E. Johnson 302. DiscreteandcontinuousnonlinearSchr¨odingersystems,M. Ablowitz, B. Prinari & D. Trubatch 303. NumbertheoryandalgebraicgeometryM. Reid & A. Skorobogatov 304. GroupsStAndrews2001inOxfordvol. 1,C.M. Campbell, E.F. Robertson & G.C. Smith(eds) 305. GroupsStAndrews2001inOxfordvol. 2,C.M. Campbell, E.F. Robertson & G.C. Smith(eds) 306. Geometricmechanicsandsymmetry: thePeyresqlectures,J. Montaldi & T. Ratiu(eds) 307. Surveysincombinatorics,2003,C.D. Wensley(ed) 308. Topology,geometryandquantumfieldtheory,U.L. Tillmann(ed) 309. Coringsandcomodules,T. Brzezinski & R. Wisbauer 310. Topicsindynamicsandergodictheory,S. Bezuglyi & S. Kolyada(eds) 311. Groups: topological,combinatorialandarithmeticaspects,T.W.Mu¨ller(ed) 312. Foundations of computational mathematics: Mineapolis, 2002, F. Cucker et al. (eds) 313. Transcendental aspects of algebraic cycles, S. Mu¨ller-Stach & C. Peters (eds) 314. Linearlogicincomputerscience,T. Ehrhardet al. (eds) London Mathematical Society Lecture Note Series: 306 Geometric Mechanics and Symmetry The Peyresq Lectures Edited by James Montaldi University of Manchester Tudor Ratiu Ecole Polytechnique F´ed´erale de Lausanne cambridge university press Cambridge,NewYork,Melbourne,Madrid,CapeTown,Singapore,S˜aoPaulo CambridgeUniversityPress TheEdinburghBuilding,CambridgeCB22RU,UK www.cambridge.org Informationonthistitle: www.cambridge.org/9780521539579 (cid:1)C CambridgeUniversityPress2005 Thisbookisincopyright. Subjecttostatutoryexception andtotheprovisionsofrelevantcollectivelicensingagreements, noreproductionofanypartmaytakeplacewithout thewrittenpermissionofCambridgeUniversityPress. Firstpublished2005 PrintedintheUnitedKingdomattheUniversityPress,Cambridge A catalogue record for this book is available from the British Library ISBN-13 978-0-521-53957-9paperback ISBN-10 0-521-53957-9paperback CambridgeUniversityPresshasnoresponsibilityforthepersistenceoraccuracyof URLsforexternalorthird-partyinternetwebsitesreferredtointhisbook,and doesnotguaranteethatanycontentonsuchwebsitesis,orwill remain,accurateorappropriate. Contents List of contributors page vii Preface ix I Stability in Hamiltonian Systems: Applications to the restricted three-body problem (K.R. Meyer) B. Rink, T. Tuwankotta 1 II A Crash Course in Geometric Mechanics (T.S. Ratiu) T.S. Ratiu, E. Sousa Dias, L. Sbano, G. Terra, R. Tudoran 23 III The Euler-Poincar´e variational framework for modeling fluid dynamics (D.D. Holm) D.D. Holm 157 IV No polar coordinates (R.H. Cushman) D. Sadovskif, K. Efstathiou 211 V SurveyondissipativeKAMtheoryincludingquasi-periodic bifurcation theory (H. Broer) H. Broer, M.-C. Ciocci, A. Litvak-Hinenzon 303 VI Symmetric Hamiltonian Bifurcations (J.A. Montaldi) P.-L. Buono, F. Laurent-Polz, J. Montaldi 357 Each set of lectures begins with its own table of contents. III II VI I IV V Interrelationships between the courses. v Contributors Henk Broer, Department of Mathematics, University of Groningen, P.O. Box 800, 9700 AV Groningen, The Netherlands. Pietro-Luciano Buono, Faculty of Science, University of Ontario Institute of Technology, 2000 Simcoe St. North, Oshawa, ON L1H 7K4, Canada. Richard Cushman, Mathematics Institute, University of Utrecht, Budapestlaan 6, 3508TA Utrecht, The Netherlands. Maria-Cristina Ciocci, Department of Pure Mathematics and Com- puter Algebra, Ghent University, Krijgslaan 281, 9000 Gent, The Netherlands. Konstantinos Efstathiou, Department of Physics, University of Athens, Panepistimiopolis Zografos, Athens, Grece. Fr´ed´eric Laurent-Polz, Institut Nonlin´eaire de Nice, 1361 route des Lucioles, 06560 Valbonne, France. Ana Litvak-Hinenzon , Mathematics Institute, University of Warwick, Coventry CV4 7AL, UK. Ken Meyer, Department of Mathematical Sciences, University of Cincinnati, PO Box 210025, Cincinnati, OH 45221-0025, USA. James Montaldi, School of Mathematics, University of Manchester, Manchester, UK. Tudor Ratiu, D´epartment de Math´ematiques, Ecole Polytechnique F´ed´erale de Lausanne, 1015 Lausanne, Switzerland. vii viii Contributors Esmeralda Sousa Dias,DepartmentodeMatema´tica,InstitutoSupe- rior T´ecnico, Av. Rovisco Pais 1049-001, Lisbon, Portugal. Bob Rink, Department of Mathematics, Imperial College of Science Technology and Medicine, London SW7 2AZ, UK. Dmitri´ı Sadovski´ı, Universit´e du Littoral, MREID, 145 av. Maurice Schumann, 59140 Dunkerque, France. Luca Sbano, Mathematics Institute, University of Warwick, Coventry CV4 7AL, UK. Glaucio Terra, Instituto de Matematica e Estatistica da Universidade de Sao Paulo, Rua do Mata˜o 1010, Cidade Universita´ria, CEP 05508- 090, Sa˜o Paulo, SP, Brazil. Rˇazvan Tudoran, Departamentul de Matematicaˇ, Universitatea de Vest, RO1900 Timi¸soara, Romania. Johan Matheus Tuwankotta,DepartmentofMathematics,Bandung InstituteofTechnologyJl. Ganesano. 10,Bandung,40132JawaBarat, Indonesia. Preface In the summers of 2000 and 2001, we organized two European Summer Schools in Geometric Mechanics. They were both held in the wonderful environmentprovidedbythevillage-cum-internationalconferencecentre at Peyresq in the Alpes de Haute Provence in France, about 100km North of Nice. Each school consisted of 6 short lecture courses, as well asnumerousshorttalksgivenbyparticipants,ofwhomtherewereabout 40ateachschool. ThemajorityofparticipantswerefromEuropewitha few coming from West of the Atlantic or East of the Urals, and we were pleased to se a number of participants from the first year returning in the second. Several of the courses and short talks led to collaborations betwen participants and/or lecturers. The summer schools were funded principally by the European Com- mission under the High-Level Scientific Conferences section of the Fifth Framework Programme. Additional funding was very kindly provided by the Fondation Peiresc. The principal aim of the two schools was to provide young scientists with a quick introduction to the geometry and dynamics involved in geometric mechanics and to bring them to a level of understanding where they could begin work on research problems. The schools were also closely linked to the Mechanics and Symmetry in Europe(MASIE)researchtrainingnetwork,organizedbyMarkRoberts, and several of the participants went on to become successful PhD stu- dents or postdocs in MASIE. Of the lecture courses, seven have ben written up for this book— mostly by the participants themselves with varying degres of collab- oration from the lecturers. The book is divided into 6 chapters, each representing a course of 5 or 6 lectures, with the exception of Ratiu’s whicharetakenfromtwocourses. ThenotesonStabilityinHamiltonian systems by Rink and Tuwankotta based on Meyer’s lectures on N-body ix x Preface problems have ben placed first as they require the least background knowledge. They cover not only Lyapounov’s and Dirichlet’s stability theorems but also the instability theorem of Chetaev, with applications to the restricted 3-body problem. Second are the notes from Ratiu’s courses which give an introduction to the mathematical formalism of geometric mechanics, beginning with the Hamiltonian, Lagrangian and Poissonformalisms,andcontinuingwithaspectsofreductionandrecon- struction,thewholebeinglacedwithnumerousexamples,andincluding some meterial on Euler-Poincar´e equations. This last topic is the basis of the third set of lecture notes: Holm’s course on the Euler-Poincar´e approach to fluid dynamics, showing especially how this approach helps to model the multiscale physics involved. ThefourthchaptercontainsCushman’slecturesontheglobalgeome- tryofintegrablesystems,describingparticularlythemonodromyinsuch systems,whichhasrecentlyprovedtobesoimportantinexplainingsome features of molecular spectra. When integrability breaks down, one re- quireskamtheorywhichisdescribedinBroer’slecturespresentedinthe followingchapter. Thetheoryisdescribedtherefordissipativesystems, showinghowquasiperiodicattractorspersistandbifurcateinfamiliesof systems, but applies also to conservative systems as is described in the appendix to that chapter. The final chapter consists of (a slightly expanded version of) Mon- taldi’s lecture course on Hamiltonian bifurcations in symmetric sys- tems. These deal firstly with bifurcations near equilibria including Hamiltonian-Hopf bifurcation, and then with bifurcations of relative equilibria. We believe all the participants and lecturers would like to join us in thanking Mme. Mady Smets and the staff of the Peyresq Foyer d’Humanisme for their warmth, generosity and hospitality, and for the smoothrunningofthecentrewithoutwhichtheSchoolswouldnothave had the academic success they did. I Stability in Hamiltonian Systems: Applicationstotherestrictedthree-bodyproblem BobRink&TheoTuwankotta BasedonlecturesbyKenMeyer 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . 1 2. Restrictedthree-bodyproblem . . . . . . . . . . . . . . . . 2 3. Relativeequilibria . . . . . . . . . . . . . . . . . . . . . 5 4. LinearHamiltonianSystems . . . . . . . . . . . . . . . . . 7 5. Liapunov’sandChetaev’stheorems . . . . . . . . . . . . . . 9 6. Applicationstotherestrictedproblem . . . . . . . . . . . . . 10 7. Normalforms . . . . . . . . . . . . . . . . . . . . . . . 13 8. Poincare´ sections . . . . . . . . . . . . . . . . . . . . . 16 9. ThetwistmapandArnold’sstabilitytheorem . . . . . . . . . . 17 References . . . . . . . . . . . . . . . . . . . . . . . . . 22 1 Introduction As participants in the MASIE-project, we attended the summer school Me- chanicsandSymmetryinPeyresq,France,duringthefirsttwoweeksofSeptem- ber2000. TheselecturenotesarebasedonthenoteswetooktherefromPro- fessorMeyer’slectureseries“N-BodyProblems”. TheN-bodyproblemisafamousclassicalproblem.Itconsistsindescribing the motion of N planets that interact with a gravitational force. Already in 1772,Eulerdescribedthethree-bodyprobleminhisefforttostudythemotion of the moon. In 1836 Jacobi brought forward an even more specific part of the three body problem, namely that in which one of the planets has a very smallmass. Thissystemisthetopicofthispaperandisnowadayscalledthe restrictedthree-bodyproblem. Itisaconservativesystemwithtwodegreesof freedom,whichgainedextensivestudyinmechanics. 1 2 I StabilityinHamiltonianSystems The N-body problem has always been a major topic in mathematics and physics. In 1858, Dirichlet claimed to have found a general method to treat any problem in mechanics. In particular, he said to have proven the stability oftheplanetarysystem. Thisstatementisstillquestionablebecausehepassed awaywithoutleavinganyproof. Nevertheless,itinitiatedWeierstrassandhis studentsKovalevskiandMittag-Lefflertotryandrediscoverthemethodmen- tioned by Dirichlet. Mittag-Leffler even managed to convince the King of SwedenandNorwaytoestablishaprizeforfindingaseriesexpansionforco- ordinatesoftheN-bodyproblemvalidforalltime,asindicatedbyDirichlet’s statement. In 1889, this prize was awarded to Poincare´, although he did not solvetheproblem. Hisessay,however,producedalotoforiginalideaswhich laterturnedouttobeveryimportantformechanics. Moreover,someofthem even stimulated other branches of mathematics, for instance topology, to be born and later on gain extensive study. Despite of all this effort, the N-body problemisstillunsolvedforN>21. This paper focuses on the relatively simple restricted three-body problem. Thisdescribesthemotionofatestparticleinthecombinedgravitationalfield of two planets and it could serve for instance as a model for the motion of a satelliteintheEarth-MoonsystemoracometintheSun-Jupitersystem. The restricted three-body problem has a number of relative equilibria, which we compute. TheremainingtextwillmainlybeconcernedwithgeneralHamilto- nianequilibria. Stabilitycriteriafortheseequilibriawillbederived,aswellas detection methods for bifurcations of periodic solutions. Classical and more advanced mathematical techniques are used, such as spectral analysis, Lia- punov functions, Birkhoff-Gustavson normal forms, Poincare´ sections, and Kolmogorov twist stability. All help to study the motion of the test particle neartherelativeequilibriaoftherestrictedproblem. 2 Therestrictedthree-bodyproblem Before introducing the restricted three-body problem, let us study the two- body problem, the motion of two planets interacting via gravitation. Denote by X1,X2 ∈ R3 the positions of the planets 1 and 2 respectively. Let us assume that planet 1 has mass 0 < µ < 1, planet 2 has mass 1−µ and the gravitationalconstantisequalto1. Theseassumptionsarenotveryrestrictive, becausetheycanalwaysbearrangedbyarescalingoftime. Theequationsof 1 Summarizedfrom[10],[11]and[8]