Geometric Mechanics and Symmetry: From Finite to Infinite Dimensions PDF
Preview Geometric Mechanics and Symmetry: From Finite to Infinite Dimensions
Geometric Mechanics and Symmetry OXFORDTEXTSINAPPLIEDANDENGINEERINGMATHEMATICS Booksintheseries ∗ G.D.Smith:NumericalSolutionofPartialDifferentialEquations,thirdedition ∗ R.Hill:AFirstCourseinCodingTheory ∗ I.Anderson:AFirstCourseinCombinatorialMathematics,secondedition ∗ D.J.Ancheson:ElementaryFluidDynamics ∗ S.Barnett:Matrices:Methodsandapplications ∗ L.M.Hocking:OptimalControl:Anintroductiontothetheorywithapplications ∗ D.E.Ince:AnIntroductiontoDiscreteMathematics,FormalSystemSpecification, andZ,secondedition ∗ O.Pretzel:Error-CorrectingCodesandFiniteFields ∗ P.Grindrod:TheTheoryandApplicationsofReaction-DiffusionEquations: Patternsandwaves,secondedition 1. AlwynScott:NonlinearScience:Emergenceanddynamicsofcoherentstructures 2. D.W. Jordan and P. Smith: Nonlinear Ordinary Differential Equations: An introductiontodynamicalsystems,thirdedition 3. I.J.Sobey:IntroductiontoInteractiveBoundaryLayerTheory 4. A.B.Tayler:MathematicalModelsinAppliedMechanics,reissue 5. L.RamdasRam-Mohan:FiniteelementandBoundaryElementApplicationsin QuantumMechanics 6. Bernard Lapeyre, Étienne Pardoux, and Rémi Sentis: Introduction to Monte- CarloMethodsforTransportandDiffusionEquations 7. IsaacElishakoffandYongjianRen:FiniteElementMethodsforStructureswith LargeStochasticVariations 8. AlwynScott:NonlinearScience:Emergenceanddynamicsofcoherentstructures, secondedition 9. W.P. Petersen and P. Arbenz: Introduction to Parallel Computing: A practical guidewithexamplesinC 10. D.W.JordanandP.Smith:NonlinearOrdinaryDifferentialEquations, fourthedition 11. D.W.JordanandP.Smith:NonlinearOrdinaryDifferentialEquations: ProblemsandSolutions 12. DarrylD.Holm,TanyaSchmah,andCristinaStoica:GeometricMechanicsand Symmetry:FromFinitetoInfiniteDimensions Titlesmarkedwith(*)appearedinthe‘OxfordAppliedMathematicsandComputing ScienceSeries’,whichhasbeenfoldedinto,andiscontinuedby,thecurrentseries. Geometric Mechanics and Symmetry From Finite to Infinite Dimensions Darryl D. Holm Imperial College London Tanya Schmah Macquarie University and The University of Toronto Cristina Stoica Wilfrid Laurier University With solutions to selected exercises by David C. P. Ellis Imperial College London 1 3 GreatClarendonStreet,OxfordOX26DP OxfordUniversityPressisadepartmentoftheUniversityofOxford. ItfurtherstheUniversity’sobjectiveofexcellenceinresearch,scholarship, andeducationbypublishingworldwidein Oxford NewYork Auckland CapeTown DaresSalaam HongKong Karachi KualaLumpur Madrid Melbourne MexicoCity Nairobi NewDelhi Shanghai Taipei Toronto Withofficesin Argentina Austria Brazil Chile CzechRepublic France Greece Guatemala Hungary Italy Japan Poland Portugal Singapore SouthKorea Switzerland Thailand Turkey Ukraine Vietnam OxfordisaregisteredtrademarkofOxfordUniversityPress intheUKandincertainothercountries PublishedintheUnitedStates byOxfordUniversityPressInc.,NewYork ©DarrylDHolm,TanyaSchmah,andCristinaStoica2009 Themoralrightsoftheauthorshavebeenasserted DatabaserightOxfordUniversityPress(maker) Firstpublished2009 Allrightsreserved.Nopartofthispublicationmaybereproduced, storedinaretrievalsystem,ortransmitted,inanyformorbyanymeans, withoutthepriorpermissioninwritingofOxfordUniversityPress, orasexpresslypermittedbylaw,orundertermsagreedwiththeappropriate reprographicsrightsorganization.Enquiriesconcerningreproduction outsidethescopeoftheaboveshouldbesenttotheRightsDepartment, OxfordUniversityPress,attheaddressabove Youmustnotcirculatethisbookinanyotherbindingorcover andyoumustimposethesameconditiononanyacquirer BritishLibraryCataloguinginPublicationData Dataavailable LibraryofCongressCataloginginPublicationData Dataavailable TypesetbyNewgenImagingSystems(P)Ltd.,Chennai,India PrintedinGreatBritain onacid-freepaperby CPIAntonyRowe,Chippenham,Wiltshire ISBN 978–0–19–921290–3 (Hbk) ISBN 978–0–19–921291–0 (Pbk) 10 9 8 7 6 5 4 3 2 1 Preface Thisisatextbookforgraduatestudentsthatintroducesgeometricmechan- icsinfiniteandinfinitedimensions,usingaseriesofarchetypalexamples. Classicalmechanics,oneoftheoldestbranchesofscience,hasundergone alongevolution,developinghandinhandwithmanyareasofmathematics, including calculus, differential geometry and the theory of Lie groups and Lie algebras. The modern formulations of Lagrangian and Hamiltonian mechanics, in the coordinate-free language of differential geometry, are elegantandgeneral.Theyprovideaunifyingframeworkformanyseemingly disparate physical systems, such as N-particle systems, rigid bodies, fluids andothercontinua,andelectromagneticandquantumsystems. The first part of this book concerns finite-dimensional conservative mechanicalsystems.ThemodernformulationsofLagrangianandHamilto- nianmechanicsusethelanguageofdifferentialgeometry.Someadvantages ofthisapproachare:(i)itappliestosystemsongeneralmanifolds,includ- ing configuration spaces defined by constraints; (ii) it is coordinate-free, or at least independent of a particular choice of coordinates; (iii) the geo- metrical structures have analogues in infinite-dimensional systems. Just as importantly, the geometric approach provides an elegant and suggestive viewpoint.Forexample,rigidbodymotioncanbeseenasgeodesicmotion on the rotation group. Symmetries of mechanical systems are represented mathematically by Lie group actions. The presence of symmetry allows a reductioninthenumberofdimensionsofamechanicalsystem,intwobasic ways: by grouping together equivalent states; and by exploiting conserved quantities(momentummaps)associatedwiththesymmetry.Thebookdis- cusses Lie group symmetries, Poisson reduction and momentum maps in a general context before specializing to systems where the configuration space is itself a Lie group, or possibly the product of a Lie group and a vector space. For systems, such as the free rigid body, whose symmetry groupisalsoitsconfigurationspace,anespeciallypowerfulreductionthe- oremexists,calledEuler–Poincaréreduction.Anextensionofthistheorem covers systems where the Lie group configuration space is augmented by a vector space describing certain ‘advected quantities’, such as the gravity covectorintheheavytopexample. vi Preface Thesecondpartofthebooktreatswhatmightbeconsideredtheinfinite- dimensional versions of the rigid body and the heavy top by replacing the action of the rotation group by the action of a group of diffeomorphisms. Roughly speaking, passing from finite to infinite dimensions in geometric mechanicsmeansreplacingmatrixmultiplicationbycompositionofsmooth invertiblefunctions.ThebookdevelopstheseideasinthesettingofEuler– Poincarétheory,basedonreductionbysymmetryofHamilton’svariational principle.Theinfinite-dimensionalresultscorrespondingtorigid-bodyand heavy-top dynamics are exemplified, respectively, in geodesic motion on thediffeomorphismsgovernedbytheEPDiffequationandintheactionof thediffeomorphismsonvectorspacesof‘advectedquantities’governedby theequationsofcontinuumdynamics. EPDiffarisesinonespatialdimensionasthezero-dispersionlimitofthe Camassa–Holm(CH)equationforshallowwaterwaves.TheCHequation is an approximate model of shallow water waves obtained at one order intheasymptoticexpansionbeyondthefamousKorteweg–deVries(KdV) equation. KdV and CH are both nonlinear partial differential equations. They each support remarkable solutions called ‘solitons’ that interact in fullynonlinearwavecollisionsandwhoseexactsolutionmaybeobtainedby thelinearmethodoftheinversescatteringtransform.Inthezero-dispersion limit of shallow water wave theory in which the EPDiff equation arises, these solitons become singular particle-like solutions carrying momentum supported on Dirac delta measures. The EPDiff equation for the Euler– Poincarédynamicsofgeodesicmotiononthediffeomorphismsalsoapplies inimageanalysis.Inparticular,EPDiffappliesinthecomparisonofshapes inmorphologyandcomputationalanatomy. The Euler–Poincaré approach that generalizes the heavy-top problem fromtheactionoftherotationsonvectorsinR3 totheactionofthediffeo- morphisms on vector spaces produces yet another rich array of results. In particular,itproducestheextensivefamilyofequationsforidealcontinuum dynamics, whose applications range from nanofluids to galaxy dynamics. Amongthemanyavailablevariantsofidealcontinuumdynamics,weselect a single class for a unified treament. Namely, we treat a class of approxi- matemodelsofglobaloceancirculationthatareusedinclimateprediction. Thus, the theoretical development of these parallel ideas in finite and infi- nite dimensions is capped by the explicit application in the last chapter to derive a unified formulation of the family of approximate equations for oceandynamicsandclimatemodellingfamiliartomoderngeoscientists. One may think of moving from the first part of the book to the second part as moving from finite-dimensional to infinite-dimensional geometric mechanics. The analogies between the two types of problems are very close. The first part of the book deals with systems of nonlinear ordinary Preface vii differentialequations(ODE),whosequestionsofexistence,uniquenessand regularityofsolutionsmaygenerallybeansweredbyusingstandardmeth- ods. The second part of the book deals with nonlinear partial differential equations(PDE)wheretheanswerstosuchquestionsareoftenquitechal- lenging and even surprising. For example, these particular PDE possess coherent excitations and even singular solutions that emerge from smooth initialdataandwhosenonlinearinteractionsexhibitparticle-likescattering behaviour reminiscent of solitons. Unlike many other PDE investigations, geometricmechanicstreatstheemergenceofthesemeasure-valued,particle- like solutions in the initial-value problem for some of the models as a challengetobecelebrated,ratherthanacauseforregret. Prerequisitesandintendedaudience The reader should be familiar with linear algebra, multivariable calculus, andthestandardmethodsforsolvingordinaryandpartialdifferentialequa- tions. Some familiarity with variational principles and canonical Poisson bracketsinclassicalmechanicsisdesirablebutnotnecessary.Readerswith anundergraduatebackgroundinphysicsorengineeringwillhavetheadvan- tage that many of the examples treated here, such as the motion of rigid bodies and the dynamics of fluids, will be familiar. In summary, the pre- requisitesarestandardforanadvancedundergraduatestudentorfirst-year postgraduatestudentinmathematicsorphysics. Howtoreadthisbookandwhatisnotinit Part I is meant to be used as a textbook in an upper-level course on geo- metric mechanics. It contains many detailed explanations and exercises. Although a wide range of topics is treated, the introduction to each of them is meant to be gentle. Part II addresses a more advanced reader and focuses on recent applications of geometric mechanics in soliton theory, image analysis and fluid mechanics. However, the mathematical prerequi- sites for rigorous treatments of these applications are not provided here. Readers interested in a more technical mathematical approach are invited to consult some of the many citations in the bibliography that treat the subjectinthatstyle. The book focuses on Euler–Poincaré reduction by symmetry, which is a broadly applicable theory, but it excludes many other important topics, such as Lagrangian reduction of general tangent bundles and symplec- tic reduction. Likewise, it omits many other standard mechanics topics, including integrability, Hamilton–Jacobi theory, and more generally, any questionsinthemoderntheoryofdynamics,bifurcationandcontrol. viii Preface MostofthenotationisconsistentwiththatoftheMarsden–Ratiuschool. This was a deliberate choice, since one of the intentions in writing the first part was to bridge the gap between the standard classical mechanics bookssuchasClassicalMechanicsbyGoldstein[Gol59]andMechanicsby Landau and Lifshitz [LL76] and the more advanced books such as Foun- dationsofMechanicsbyAbrahamandMarsden[AM78]andIntroduction toMechanicsandSymmetrybyMarsdenandRatiu[MR02]. Description of contents PartI The opening chapter briefly presents Newtonian, Lagrangian and Hamil- tonianmechanicsinthefamiliarsettingofN-particlesystemsinEuclidean space.Ashortclassicaldescriptionofrigidbodymotionisalsogiven.Chap- ters 2 and 3 build up the prerequisites for the extension of Lagrangian and Hamiltonian mechanics to systems on manifolds. Chapter 2 intro- ducesmanifolds,withemphasisonsubmanifoldsofEuclideanspace,which appear in mechanics as configuration spaces defined by constraints. This chapteralsointroducesmatrixLiegroups(coveredinmoredetailinChap- ter 5). Chapter 3 gives a minimal introduction to differential geometry, including a taste of Riemannian and symplectic geometry. Chapter 4 presents Lagrangian and Hamiltonian mechanics on manifolds, and ends withabrieflookatsymmetry,reductionandconservedquantities. Inordertostudysymmetryinmoredepth,oneneedsLiegroupsandtheir actions,whicharethesubjectofChapters5and6.Liegroupsandalgebras areintroducedinbothmatrixandabstractframeworks.Liegroupactions on manifolds, and the resulting quotient spaces, provide sufficient tools to introducePoissonreduction. TheremainingchaptersfocusonmechanicalsystemsonLiegroups,that is,mechanicalsystemswheretheconfigurationspaceisaLiegroup.Chap- ter 7 covers Euler–Poincaré reduction, emphasizing the examples of the freerigidbodyandtheheavytop.Chapter8introducesmomentummaps. Chapter 9 covers Lie–Poisson reduction, which is the Hamiltonian coun- terpart of Euler–Poincaré reduction. Chapter 10 applies the results of the preceding three chapters to the example of a pseudo-rigid body. Pseudo- rigidmotionsprovidealinkbetweentherigidmotionsstudiedinPartIand thefluidmotionsthatarethesubjectofPartII. PartII In a famous paper, Arnold [Arn66] observed that Euler ideal fluid motion may be identified with geodesic flow on the volume-preserving Preface ix diffeomorphisms, with a metric determined by the fluid’s kinetic energy. This observation was further developed in a rigorous analytical setting by Ebin and Marsden [EM70]. The methods of geometric mechanics sys- tematically develop this result from the Euler–Poincaré (EP) variational principle for the Euler fluid equations. These methods also generate their Lie–Poisson Hamiltonian structure, Noether theorem, momentum maps, etc. As Arnold observed, the configuration space for the incompressible motionofanidealfluidisthegroupG=Diff (D)ofvolume-preserving Vol diffeomorphisms (smooth invertible maps with smooth inverses) of the region D occupied by the fluid. The tangent vectors in TG for the maps inG=DiffVol(D)representthespaceoffluidvelocities,whichmustsatisfy appropriate physical conditions at the boundary of the region D. Group multiplication in G = DiffVol(D) is composition of the smooth invertible volume-preservingmaps.Oneofthepurposesofthistextistoexplainhow theEulerequationsoffluidmotionmayberecognizedastheEuler–Poincaré equationsEPDiffVol definedonthedualofthetangentspaceattheidentity TeG=TeDiffVol(D)oftheright-invariantvectorfieldsoverthedomainD. Applications of EPDiff in Part II • In the motivating example, Euler’s fluid equations emerge as EPDiffVol when the diffeomorphisms are constrained to preserve volume so that G=DiffVol(D),andthekineticenergynormistakentobetheL2 norm (cid:2)u(cid:2)2 of the spatial fluid velocity. From the viewpoint of constrained L2 dynamics,thefluidpressurepmayberegardedastheLagrangemultiplier thatimposespreservationofvolume. • Other choices of the kinetic energy norm besides the L2 norm of veloc- ity also produce interesting continuum equations as geodesic flows on DiffVol(D). For example, the choice of the H1 norm (L2 norm of the gradient) of the spatial fluid velocity yields the Lagrangian- averaged Euler-alpha (LAE-alpha) equations when incompressibility is imposed. For more discussion of the LAE-alpha equations, see, e.g., [HMR98a,Shk00]. • TheH1 normalsoyieldsoneofafamilyofinterestingEPDiffequations when incompressibility is not imposed, so that the motion takes place on the full diffeomorphism group. In one spatial dimension on the real line (and also in a periodic domain), the EPDiff equation for the H1 norm of the spatial fluid velocity is completely integrable in the Hamil- tonian sense and possesses soliton solutions. This equation is the limit of the Camassa–Holm (CH) equation for shallow-water wave motion when its linear dispersion coefficients tend to zero [CH93]. The CH equation and its peaked-soliton solutions – called peakons – that exist
The list of books you might like
