LAGRANGIAN & HAMILTONIAN DYNAMICS Lagrangian & Hamiltonian Dynamics Peter Mann UniversityofStAndrews 1 3 GreatClarendonStreet,Oxford,OX26DP, UnitedKingdom OxfordUniversityPressisadepartmentoftheUniversityofOxford. ItfurtherstheUniversity’sobjectiveofexcellenceinresearch,scholarship, andeducationbypublishingworldwide.Oxfordisaregisteredtrademarkof OxfordUniversityPressintheUKandincertainothercountries ©PeterMann2018 Themoralrightsoftheauthorhavebeenasserted FirstEditionpublishedin2018 Impression:1 Allrightsreserved.Nopartofthispublicationmaybereproduced,storedin aretrievalsystem,ortransmitted,inanyformorbyanymeans,withoutthe priorpermissioninwritingofOxfordUniversityPress,orasexpresslypermitted bylaw,bylicenceorundertermsagreedwiththeappropriatereprographics rightsorganization.Enquiriesconcerningreproductionoutsidethescopeofthe aboveshouldbesenttotheRightsDepartment,OxfordUniversityPress,atthe addressabove Youmustnotcirculatethisworkinanyotherform andyoumustimposethissameconditiononanyacquirer PublishedintheUnitedStatesofAmericabyOxfordUniversityPress 198MadisonAvenue,NewYork,NY10016,UnitedStatesofAmerica BritishLibraryCataloguinginPublicationData Dataavailable LibraryofCongressControlNumber:2017960667 ISBN 978–0–19–882237–0(hbk.) ISBN 978–0–19–882238–7(pbk.) Printedandboundby CPIGroup(UK)Ltd,Croydon,CR04YY DOI10.1093/oso/9780198822370.001.0001 LinkstothirdpartywebsitesareprovidedbyOxfordingoodfaithand forinformationonly.Oxforddisclaimsanyresponsibilityforthematerials containedinanythirdpartywebsitereferencedinthiswork. For my beautiful daughter Hallie. Contents Preface xiii PART I NEWTONIAN MECHANICS 1 Newton’s Three Laws 3 1.1 Phase Space 7 1.2 Systems of Particles 8 1.3 The N-body Problem 11 Chapter summary 13 2 Energy and Work 15 Chapter summary 22 3 Introductory Rotational Dynamics 24 Chapter summary 33 4 The Harmonic Oscillator 34 Chapter summary 38 5 Wave Mechanics & Elements of Mathematical Physics 44 PART II LAGRANGIAN MECHANICS 6 Coordinates & Constraints 55 Chapter summary 59 7 The Stationary Action Principle 61 7.1 The Inverse Problem 70 7.2 Higher-Order Theories & the Ostrogradsky Equation 72 7.3 The Second Variation 73 7.4 Functions & Functionals 74 7.5 Boundary Conditions 76 7.6 Variations 78 7.7 Weierstrass-Erdmann Conditions for Broken Extremals 79 7.8 Hamilton-Suslov Principle 79 Chapter summary 80 8 Constrained Lagrangian Mechanics 89 8.1 Holonomic Constraints 89 viii Contents 8.2 Non-Holonomic Constraints 93 Chapter summary 96 9 Point Transformations in Lagrangian Mechanics 100 Chapter summary 103 10 The Jacobi Energy Function 107 Chapter summary 112 11 Symmetries & Lagrangian-Hamilton-Jacobi Theory 115 11.1 Noether’s Theorem 115 11.2 Gauge Theory 120 11.3 Isotropic Symmetries 122 11.4 Caratheodory-Hamilton-Jacobi theory 123 Chapter summary 124 12 Near-Equilibrium Oscillations 130 12.1 Normal Modes 137 Chapter summary 140 13 Virtual Work & d’Alembert’s Principle 147 13.1 Gauss’s Least Constraint & Jourdain’s Principle 153 13.2 The Gibbs-Appell Equations 156 Chapter summary 158 PART III CANONICAL MECHANICS 14 The Hamiltonian & Phase Space 167 Chapter summary 172 15 Hamilton’s Principle in Phase Space 174 Chapter summary 178 16 Hamilton’s Equations & Routhian Reduction 179 16.1 Phase Space Conservation Laws 181 16.2 Routhian Mechanics 183 17 Poisson Brackets & Angular Momentum 190 17.1 Poisson Brackets & Angular Momenta 195 17.2 Poisson Brackets & Symmetries 197 Chapter summary 200 18 Canonical & Gauge Transformations 202 18.1 Canonical Transformations I 202 18.2 Canonical Transformations II 206 Contents ix 18.3 Infinitesimal Canonical Transformations 211 Chapter summary 214 19 Hamilton-Jacobi Theory 217 19.1 Hamilton-Jacobi Theory I 217 19.2 Hamilton-Jacobi Theory II 224 Chapter summary 229 20 Liouville’s Theorem & Classical Statistical Mechanics 237 20.1 Liouville’s Theorem & the Classical Propagator 237 20.2 Koopman-von Neumann Theory 244 20.3 Classical Statistical Mechanics 246 20.4 Symplectic Integrators 255 Chapter summary 259 21 Constrained Hamiltonian Dynamics 267 Chapter summary 274 22 Autonomous Geometrical Mechanics 277 22.1 A Coordinate-Free Picture 284 22.2 Poisson Manifolds & Symplectic Reduction 291 22.3 Geometrical Lagrangian Mechanics 296 22.4 Elements of Constrained Geometry 300 Chapter summary 303 23 The Structure of Phase Space 309 23.1 Time-Dependent Geometrical Mechanics 313 23.2 Picturing Phase Space 319 Chapter summary 322 24 Near-Integrable Systems 325 24.1 Canonical Perturbation Theory 325 24.2 KAM Theory & Elements of Chaos 333 PART IV CLASSICAL FIELD THEORY 25 Lagrangian Field Theory 345 Chapter summary 350 26 Hamiltonian Field Theory 353 27 Classical Electromagnetism 357 Chapter summary 365