Mathematical Physics: Classical Mechanics

683 Pages·2018·18.14 MB·English

by  Andreas Knauf (auth.)

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
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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