UNITEXT 109 Andreas Knauf Mathematical Physics: Classical Mechanics UNITEXT - La Matematica per il 3+2 Volume 109 Editor-in-chief Alfio Quarteroni Series editors L. Ambrosio P. Biscari C. Ciliberto C. De Lellis M. Ledoux V. Panaretos W.J. Runggaldier More information about this series at http://www.springer.com/series/5418 Andreas Knauf Mathematical Physics: Classical Mechanics 123 Andreas Knauf Department ofMathematics Friedrich-Alexander University Erlangen-Nürnberg Erlangen Germany Translated byJochen Denzler,Department ofMathematics, University of Tennessee, Knoxville, Tennessee,USA ISSN 2038-5714 ISSN 2532-3318 (electronic) UNITEXT- La Matematica peril3+2 ISSN 2038-5722 ISSN 2038-5757 (electronic) ISBN978-3-662-55772-3 ISBN978-3-662-55774-7 (eBook) https://doi.org/10.1007/978-3-662-55774-7 LibraryofCongressControlNumber:2017952919 MathematicsSubjectClassification(2010): 70-01,37-01,37J05,37J40 TranslationfromtheGermanlanguageedition:MathematischePhysik:KlassischeMechanikbyAndreas Knauf,Springer-LehrbuchMasterclass,©Springer,2ndedition2017.AllrightsReserved. ©Springer-VerlagGmbHGermany2018 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpart of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission orinformationstorageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilar methodologynowknownorhereafterdeveloped. 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Printedonacid-freepaper ThisSpringerimprintispublishedbySpringerNature TheregisteredcompanyisSpringer-VerlagGmbH,DE Theregisteredcompanyaddressis:HeidelbergerPlatz3,14197Berlin,Germany Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2 Dynamical Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.1 Iterated Mappings, Dynamical Systems . . . . . . . . . . . . . . . . . . 12 2.2 Continuous Dynamical Systems. . . . . . . . . . . . . . . . . . . . . . . . 16 2.3 Differentiable Dynamical Systems . . . . . . . . . . . . . . . . . . . . . . 25 3 Ordinary Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 3.1 Definitions and Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 3.2 Local Existence and Uniqueness of the Solution. . . . . . . . . . . . 37 3.3 Global Existence and Uniqueness of the Solution . . . . . . . . . . . 44 3.4 Transformation into a Dynamical System. . . . . . . . . . . . . . . . . 47 3.5 The Maximal Interval of Existence . . . . . . . . . . . . . . . . . . . . . 50 3.6 Principal Theorem of the Theory of Differential Equations . . . . 53 3.6.1 Linearization of the ODE Along a Trajectory. . . . . . . . 54 3.6.2 Statement and Proof of the Principal Theorem . . . . . . . 56 3.6.3 Consequences of the Principal Theorem. . . . . . . . . . . . 59 4 Linear Dynamics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 4.1 Homogeneous Linear Autonomous ODEs. . . . . . . . . . . . . . . . . 62 4.2 Explicitly Time Dependent Linear ODEs . . . . . . . . . . . . . . . . . 70 4.3 Quasipolynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 5 Classification of Linear Flows. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 5.1 Conjugacies of Linear Flows. . . . . . . . . . . . . . . . . . . . . . . . . . 80 5.2 Hyperbolic Linear Vector Fields . . . . . . . . . . . . . . . . . . . . . . . 82 5.3 Linear Flows in the Plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 5.4 Example: Spring with Friction. . . . . . . . . . . . . . . . . . . . . . . . . 91 v vi Contents 6 Hamiltonian Equations and Symplectic Group . . . . . . . . . . . . . . . . 97 6.1 Gradient Flows and Hamiltonian Systems . . . . . . . . . . . . . . . . 97 6.1.1 Gradient Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 6.1.2 Hamiltonian Systems . . . . . . . . . . . . . . . . . . . . . . . . . 101 6.2 The Symplectic Group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 6.2.1 Linear Hamiltonian Systems . . . . . . . . . . . . . . . . . . . . 103 6.2.2 Symplectic Geometry . . . . . . . . . . . . . . . . . . . . . . . . . 104 6.2.3 The Symplectic Algebra . . . . . . . . . . . . . . . . . . . . . . . 110 6.3 Linear Hamiltonian Systems . . . . . . . . . . . . . . . . . . . . . . . . . . 112 6.3.1 Harmonic Oscillators . . . . . . . . . . . . . . . . . . . . . . . . . 112 6.3.2 Harmonic Oscillations in a Lattice. . . . . . . . . . . . . . . . 119 6.3.3 Particles in a Constant Electromagnetic Field. . . . . . . . 122 6.4 Subspaces of Symplectic Vector Spaces. . . . . . . . . . . . . . . . . . 125 6.5 *The Maslov Index. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 7 Stability Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 7.1 Stability of Linear Differential Equations . . . . . . . . . . . . . . . . . 138 7.2 Lyapunov Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 7.3 Bifurcations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144 7.3.1 Bifurcations from Equilibria . . . . . . . . . . . . . . . . . . . . 145 7.3.2 Bifurcations from Periodic Orbits . . . . . . . . . . . . . . . . 148 7.3.3 Bifurcations of the Phase Space . . . . . . . . . . . . . . . . . 152 8 Variational Principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155 8.1 Lagrange and Hamilton Equations . . . . . . . . . . . . . . . . . . . . . . 156 8.2 Holonomic Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161 8.3 The Hamiltonian Variational Principle . . . . . . . . . . . . . . . . . . . 164 8.4 Geodesic Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 8.5 The Jacobi Metric. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177 8.6 Fermat’s Principle. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181 8.7 Geometrical Optics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184 9 Ergodic Theory. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191 9.1 Measure Preserving Dynamical Systems. . . . . . . . . . . . . . . . . . 191 9.2 Ergodic Dynamical Systems . . . . . . . . . . . . . . . . . . . . . . . . . . 195 9.3 Mixing Dynamical Systems. . . . . . . . . . . . . . . . . . . . . . . . . . . 198 9.4 Birkhoff’s Ergodic Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 205 9.5 Poincaré’s Recurrence Theorem. . . . . . . . . . . . . . . . . . . . . . . . 212 10 Symplectic Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215 10.1 Symplectic Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216 10.2 Lie Derivative and Poisson Bracket . . . . . . . . . . . . . . . . . . . . . 222 10.3 Canonical Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . 227 10.4 Lagrangian Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235 10.5 Generating Functions of Canonical Transformations . . . . . . . . . 237 Contents vii 11 Motion in a Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241 11.1 Properties of General Validity . . . . . . . . . . . . . . . . . . . . . . . . . 242 11.1.1 Existence of the Flow. . . . . . . . . . . . . . . . . . . . . . . . . 242 11.1.2 Reversibility of the Flow . . . . . . . . . . . . . . . . . . . . . . 243 11.1.3 Reachability. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245 11.2 Motion in a Periodic Potential . . . . . . . . . . . . . . . . . . . . . . . . . 245 11.2.1 Existence of Asymptotic Velocities . . . . . . . . . . . . . . . 246 11.2.2 Distribution of Asymptotic Velocities . . . . . . . . . . . . . 249 11.2.3 Ballistic and Diffusive Motion. . . . . . . . . . . . . . . . . . . 252 11.3 Celestial Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256 11.3.1 Geometry of the Kepler Problem. . . . . . . . . . . . . . . . . 256 11.3.2 Two Centers of Gravitation. . . . . . . . . . . . . . . . . . . . . 265 11.3.3 The n-Body Problem . . . . . . . . . . . . . . . . . . . . . . . . . 270 12 Scattering Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277 12.1 Scattering in a Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 278 12.2 The Møller Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . 287 12.3 The Differential Cross Section. . . . . . . . . . . . . . . . . . . . . . . . . 294 12.4 Time Delay, Radon Transform, and Inverse Scattering Theory. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 298 12.5 Kinematics of the Scattering of n Particles . . . . . . . . . . . . . . . . 306 12.6 * Asymptotic Completeness. . . . . . . . . . . . . . . . . . . . . . . . . . . 311 13 Integrable Systems and Symmetries . . . . . . . . . . . . . . . . . . . . . . . . 325 13.1 What is Integrability? An Example . . . . . . . . . . . . . . . . . . . . . 326 13.2 The Liouville-Arnol’d Theorem. . . . . . . . . . . . . . . . . . . . . . . . 330 13.3 Action-Angle Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . 336 13.4 The Momentum Mapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343 13.5 * Reduction of the Phase Space. . . . . . . . . . . . . . . . . . . . . . . . 351 14 Rigid and Non-Rigid Bodies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 365 14.1 Motions of Euclidean Space . . . . . . . . . . . . . . . . . . . . . . . . . . 366 14.2 Kinematics of Rigid Bodies. . . . . . . . . . . . . . . . . . . . . . . . . . . 367 14.3 Solution of the Equations of Motion . . . . . . . . . . . . . . . . . . . . 373 14.3.1 Force Free Top . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 375 14.3.2 Heavy (Symmetric) Tops . . . . . . . . . . . . . . . . . . . . . . 381 14.4 Nonrigid Bodies, Nonholonomic Systems. . . . . . . . . . . . . . . . . 384 14.4.1 Geometry of Flexible Bodies . . . . . . . . . . . . . . . . . . . 384 14.4.2 Nonholonomic Constraints . . . . . . . . . . . . . . . . . . . . . 387 15 Perturbation Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 391 15.1 Conditionally Periodic Motion of a Torus . . . . . . . . . . . . . . . . 392 15.2 Perturbation Theory for One Angle Variable . . . . . . . . . . . . . . 401 15.3 Hamiltonian Perturbation Theory of First Order . . . . . . . . . . . . 403 15.4 KAM Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 412 viii Contents 15.4.1 * A Proof of the KAM Theorem. . . . . . . . . . . . . . . . . 413 15.4.2 Measure of the KAM Tori . . . . . . . . . . . . . . . . . . . . . 426 15.5 Diophantine Condition and Continued Fractions. . . . . . . . . . . . 430 15.6 Cantori: In the Example of the Standard Map. . . . . . . . . . . . . . 436 16 Relativistic Mechanics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 441 16.1 The Speed of Light. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 442 16.2 The Lorentz- and Poincaré Groups. . . . . . . . . . . . . . . . . . . . . . 444 16.3 Geometry of Minkowski Space . . . . . . . . . . . . . . . . . . . . . . . . 449 16.4 The World from a Relativistic Point of View . . . . . . . . . . . . . . 455 16.5 From Einstein to Galilei—and Back. . . . . . . . . . . . . . . . . . . . . 460 16.6 Relativistic Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 466 17 Symplectic Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 469 17.1 The Symplectic Camel and the Eye of a Needle. . . . . . . . . . . . 470 17.2 The Theorem by Poincaré-Birkhoff . . . . . . . . . . . . . . . . . . . . . 474 17.3 The Arnol’d Conjecture. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 478 Appendix A: Topological Spaces and Manifolds .. .... .... ..... .... 483 Appendix B: Differential Forms ... .... .... .... .... .... ..... .... 505 Appendix C: Convexity and Legendre Transform. .... .... ..... .... 537 Appendix D: Fixed Point Theorems, and Results About Inverse Images .... .... .... .... .... ..... .... 543 Appendix E: Group Theory.. ..... .... .... .... .... .... ..... .... 547 Appendix F: Bundles, Connection, Curvature .... .... .... ..... .... 563 Appendix G: Morse Theory.. ..... .... .... .... .... .... ..... .... 579 Appendix H: Solutions of the Exercises . .... .... .... .... ..... .... 595 Bibliography .. .... .... .... ..... .... .... .... .... .... ..... .... 657 Table of Symbols... .... .... ..... .... .... .... .... .... ..... .... 667 Image Credits . .... .... .... ..... .... .... .... .... .... ..... .... 669 Index of Proper Names.. .... ..... .... .... .... .... .... ..... .... 671 Index .... .... .... .... .... ..... .... .... .... .... .... ..... .... 675 Remarks on Mathematical Physics Themes and Goals “Thelawsofnatureareconstructedinsuchawayastomake theuniverseasinterestingaspossible.”FREEMANDYSON,in ImaginedWorlds(1997) In mathematical physics, one attempts, beginning with the fundamental equations andassumptionsofphysics(likeNewton’sequation,theBoltzmanndistribution,or the Schrödinger equation), to derive facts of physics by means of mathematics. So it is the problem of physics that is center stage (for instance, the question aboutthestabilityofthesolarsystem,thereasonfortheexistenceofcrystals,orthe localization of electrons in an amorphous solid). The methods needed to solve the respective problems can, in their majority, be classified as analysis or geometry, but algebraic techniques also play a role. Roughly, the mathematical correspondence (cid:129) to classical mechanics is the theory of ordinary differential equations, (cid:129) to quantum mechanics is functional analysis, and (cid:129) to (classical) statistical mechanics is probability theory. However, a series on theoretical physics should also include electrodynamics and thus, mathematically speaking, the theory of Maxwell’s equations, which are linear partial differential equations. The general theory of relativity, one of the foundations of modern physics, leads, like many other questions, to a nonlinear partial differential equation. Quantum field theory relies on a variety of analytic, geometric, as well as algebraic methods. Giventhevastextentofthearea,thequestionariseshowonecouldpossiblygain some ground within a reasonable time and whether the study of mathematical physics is worthwhile. The present first volume of a projected three-volume course on mathematical physics offers an answer to the first question. ix