Table Of Content

CAMBRIDGEMONOGRAPHSON APPLIEDANDCOMPUTATIONAL MATHEMATICS SeriesEditors P.G.CIARLET,A.ISERLES,R.V.KOHN,M.H.WRIGHT 14 Simulating Hamiltonian Dynamics The Cambridge Monographs on Applied and Computational Mathematics reflects the crucial role of mathematicalandcomputationaltechniquesincontemporaryscience.Theseriespublishesexpositions onallaspectsofapplicableandnumericalmathematics,withanemphasisonnewdevelopmentsinthis fast-movingareaofresearch. State-of-the-artmethodsandalgorithmsaswellasmodernmathematicaldescriptionsofphysical andmechanicalideasarepresentedinamannersuitedtograduateresearchstudentsandprofessionals alike.Soundpedagogicalpresentationisaprerequisite.Itisintendedthatbooksintheserieswillserve toinformanewgenerationofresearchers. Alsointhisseries: 1. APracticalGuidetoPseudospectralMethods,BengtFornberg 2. DynamicalSystemsandNumericalAnalysis,A.M.StuartandA.R.Humphries 3. LevelSetMethodsandFastMarchingMethods,J.A.Sethian 4. TheNumericalSolutionofIntegralEquationsoftheSecondKind,KendallE.Atkinson 5. OrthogonalRationalFunctions,AdhemarBultheel,PabloGonza´lez-Vera,ErikHendiksen,and OlavNja˚stad 6. TheTheoryofComposites,GraemeW.Milton 7. GeometryandTopologyforMeshGenerationHerbertEdelsfrunner 8. Schwarz–ChristoffelMappingTofinA.DriscollandLloydN.Trefethen 9. High-OrderMethodsforIncompressibleFluidFlow,M.O.Deville,P.F.Fischerand E.H.Mund 10. PracticalExtrapolationMethods,AvramSidi 11. GeneralizedRiemannProblemsinComputationalFluidDynamics,MataniaBen-Artziand JosephFalcovitz 12. RadialBasisFunctions,MartinD.Buhmann 13. IterativeKrylovMethodsforLargeLinearSystems,HenkA.vanderVorst. 15. CollocationMethodsforVolterraIntegralandRelatedFunctionalEquations,HermannBrunner. Simulating Hamiltonian Dynamics BENEDICT LEIMKUHLER UniversityofLeicester SEBASTIAN REICH ImperialCollege,London    Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo Cambridge University Press   The Edinburgh Building, Cambridge , UK Published in the United States of America by Cambridge University Press, New York www.cambridge.org Information on this title: www.cambridge.org/9780521772907 © Cambridge University Press 2004 This book is in copyright. Subject to statutory exception and to the provision of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published in print format 2005 - ---- eBook (NetLibrary) - --- eBook (NetLibrary) - ---- hardback - --- hardback Cambridge University Press has no responsibility for the persistence or accuracy of  s for external or third-party internet websites referred to in this book, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate. Contents Preface ix Acknowledgements xvi 1 Introduction 1 1.1 N-body problems . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2 Problems and applications . . . . . . . . . . . . . . . . . . . . 3 1.3 Constrained dynamics . . . . . . . . . . . . . . . . . . . . . . 6 1.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2 Numerical methods 11 2.1 One-step methods . . . . . . . . . . . . . . . . . . . . . . . . 13 2.2 Numerical example: the Lennard–Jones oscillator . . . . . . . . 18 2.3 Higher-order methods . . . . . . . . . . . . . . . . . . . . . . 20 2.4 Runge–Kutta methods . . . . . . . . . . . . . . . . . . . . . . 22 2.5 Partitioned Runge–Kutta methods . . . . . . . . . . . . . . . . 25 2.6 Stability and eigenvalues . . . . . . . . . . . . . . . . . . . . . 27 2.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 3 Hamiltonian mechanics 36 3.1 Canonical and noncanonical Hamiltonian systems . . . . . . . . 36 3.2 Examples of Hamiltonian systems . . . . . . . . . . . . . . . . 39 3.3 First integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 3.4 The flow map and variational equations . . . . . . . . . . . . . 48 3.5 Symplectic maps and Hamiltonian flow maps . . . . . . . . . . 52 3.6 Differential forms and the wedge product . . . . . . . . . . . . 61 3.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 4 Geometric integrators 70 4.1 Symplectic maps and methods . . . . . . . . . . . . . . . . . . 74 v vi Contents 4.2 Construction of symplectic methods by Hamiltonian splitting . . 76 4.3 Time-reversal symmetry and reversible discretizations . . . . . . 81 4.4 First integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 4.5 Case studies . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 4.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 5 The modified equations 105 5.1 Forward v. backward error analysis . . . . . . . . . . . . . . . . 107 5.2 The modified equations . . . . . . . . . . . . . . . . . . . . . 117 5.3 Geometric integration and modified equations . . . . . . . . . 129 5.4 Modified equations for composition methods . . . . . . . . . . 133 5.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 6 Higher-order methods 142 6.1 Construction of higher-order methods . . . . . . . . . . . . . . 143 6.2 Composition methods . . . . . . . . . . . . . . . . . . . . . . 144 6.3 Runge–Kutta methods . . . . . . . . . . . . . . . . . . . . . . 149 6.4 Generating functions . . . . . . . . . . . . . . . . . . . . . . . 159 6.5 Numerical experiments . . . . . . . . . . . . . . . . . . . . . . 161 6.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 7 Constrained mechanical systems 169 7.1 N-body systems with holonomic constraints . . . . . . . . . . . 170 7.2 Numerical methods for constraints . . . . . . . . . . . . . . . 173 7.3 Transition to Hamiltonian mechanics . . . . . . . . . . . . . . 184 7.4 The symplectic structure with constraints . . . . . . . . . . . . 186 7.5 Direct symplectic discretization . . . . . . . . . . . . . . . . . 188 7.6 Alternative approaches to constrained integration . . . . . . . 191 7.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195 8 Rigid body dynamics 199 8.1 Rigid bodies as constrained systems . . . . . . . . . . . . . . . 201 8.2 Angular momentum and the inertia tensor . . . . . . . . . . . 210 8.3 The Euler equations of rigid body motion . . . . . . . . . . . . 212 8.4 Order 4 and order 6 variants of RATTLE for the free rigid body 223 8.5 Freely moving rigid bodies . . . . . . . . . . . . . . . . . . . . 224 8.6 Other formulations for rigid body motion . . . . . . . . . . . . 228 8.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230 9 Adaptive geometric integrators 234 9.1 Sundman and Poincaré transformations . . . . . . . . . . . . . 235 Contents vii 9.2 Reversible variable stepsize integration . . . . . . . . . . . . . . 238 9.3 Sundman transformations . . . . . . . . . . . . . . . . . . . . . 246 9.4 Backward error analysis . . . . . . . . . . . . . . . . . . . . . . 249 9.5 Generalized reversible adaptive methods . . . . . . . . . . . . . 251 9.6 Poincaré transformations . . . . . . . . . . . . . . . . . . . . . 253 9.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255 10 Highly oscillatory problems 257 10.1 Large timestep methods . . . . . . . . . . . . . . . . . . . . . 258 10.2 Averaging and reduced equations . . . . . . . . . . . . . . . . 269 10.3 The reversible averaging (RA) method . . . . . . . . . . . . . 271 10.4 The mollified impulse (MOLLY) method . . . . . . . . . . . . 276 10.5 Multiple frequency systems . . . . . . . . . . . . . . . . . . . . 279 10.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 280 11 Molecular dynamics 287 11.1 From liquids to biopolymers . . . . . . . . . . . . . . . . . . . . 290 11.2 Statistical mechanics from MD trajectories . . . . . . . . . . . 296 11.3 Dynamical formulations for the NVT, NPT and other ensembles . . . . . . . . . . . . . . . . . . . . . . . . . . 299 11.4 A symplectic approach: the Nosé–Poincaré method . . . . . . . 305 11.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313 12 Hamiltonian PDEs 316 12.1 Examples of Hamiltonian PDEs . . . . . . . . . . . . . . . . . . 316 12.2 Symplectic discretizations . . . . . . . . . . . . . . . . . . . . 325 12.3 Multi-symplectic PDEs . . . . . . . . . . . . . . . . . . . . . . 335 12.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 357 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 374 Preface About geometric integration This book is about simulating dynamical systems, especially conservative systems such as arise in celestial mechanics and molecular models. We think of the integrator as the beating heart of any dynamical simulation, the scheme which replaces a differential equation in continuous time by a difference equation defin- ing approximate snapshots of the solution at discrete timesteps. As computers grow in power, approximate solutions are computed over ever-longer time inter- vals, and the integrator may be iterated many millions or even billions of times; in suchcases,thequalitativepropertiesoftheintegratoritselfcanbecomecriticalto the success of a simulation. Geometric integrators are methods that exactly (i.e. up to rounding errors) conserve qualitative properties associated to the solutions of the dynamical system under study. The increase in the use of simulation in applications has mirrored rising interest in the theory of dynamical systems. Many of the recent developments in mathematics have followed from the appreciation of the fundamentally chaotic nature of physical systems, a consequence of nonlinearities present in even the simplest useful models. In a chaotic system the individual trajectories are by def- inition inherently unpredictable in the exact sense: solutions depend sensitively on the initial data. In some ways, this observation has limited the scope and usefulness of results obtainable from mathematical theory. Most of the common techniquesrelyonlocalapproximationandperturbationexpansions,methodsbest suitedforunderstandingproblemswhichare“almostlinear,”whilethenewmath- ematics that would be needed to answer even the most basic questions regarding chaotic systems is still in its infancy. In the absence of a useful general theoreti- calmethodforanalyzingcomplexnonlinearphenomena,simulationisincreasingly pushed to the fore. It provides one of the few broadly applicable and practical means of shedding light on the behavior of complex nonlinear systems, and is now a standard tool in everything from materials modeling to bioengineering, from atomic theory to cosmology. ix