Mechanics Florian Scheck Mechanics From Newton's Laws to Deterministic Chaos With 144 Figures Springer-Verlag Berlin Heidelberg GmbH Florian A. Scheck Professor of Theoretical Physics Institut fur Physik, Fachbereich Physik, Johannes Gutenberg-Universitiit Postfach 39 80, D-6500 Mainz 1, Fed. Rep. of Germany Title ofthe original German edition: Mechanik, 2.,erweiterte Auflage ISBN 978-3-540-52715-2 ISBN 978-3-540-52715-2 ISBN 978-3-662-02630-4 (eBook) DOI 10.1007/978-3-662-02630-4 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. Duplication of this publication or parts thereof is only permitted under the pro visions of the German Copyright Law of September 9, 1965, in its current version, and a copyright fee must always be paid. Violations fall under the prosecution act of the German Copyright Law. ©Springer-Verlag Berlin Heidelberg 1990 Originally published by Springer-Verlag Berlin Heidelberg New York in 1990 The use ofr egistered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. 2156/3140-543210-Printed on acid-free paper Preface Purpose and Emphasis. Mechanics not only is the oldest branch of physics but was and still is the basis for all of theoretical physics. Quantum mechanics can hardly be understood, perhaps cannot even be formulated, without a good knowledge of general mechanics. Field theories such as electrodynamics borrow their formal framework and many of their building principles from mechanics. In short, throughout the many modem developments of physics where one fre quently turns back to the principles of classical mechanics its model character is felt. For this reason it is not surprising that the presentation of mechanics reflects to some extent the development of modem physics and that today this classical branch of theoretical physics is taught rather differently than at the time of Arnold Sommerfeld, in the 1920s, or even in the 1950s, when more emphasis was put on the theory and the applications of partial-differential equations. Today, symme tries and invariance principles, the structure of the space-time continuum, and the geometrical structure of mechanics play an important role. The beginner should realize that mechanics is not primarily the art of describing block-and-tackles, collisions of billiard balls, constrained motions of the cylinder in a washing ma chine, or bicycle riding. However fascinating such systems may be, mechanics is primarily the field where one learns to develop general principles from which equations of motion may be derived, to understand the importance of symmetries for the dynamics, and, last but not least, to get some practice in using theoretical tools and concepts that are essential for all branches of physics. Besides its role as a basis for much of theoretical physics and as a training ground for physical concepts, mechanics is a fascinating field in itself. It is not easy to master, for the beginner, because it has many different facets and its structure is less homogeneous than, say, that of electrodynamics. On a first assault one usually does not fully realize both its charm and its difficulty. Indeed, on returning to various aspects of mechanics, in the course of one's studies, one will be surprised to discover again and again that it has new facets and new secrets. And finally, one should be aware of the fact that mechanics is not a closed subject, lost forever in the archives of the nineteenth century. As the reader will realize in Chap. 6, if he or she has not realized it already, mechanics is an exciting field of research with many important questions of qualitative dynamics remaining unanswered. Structure of the Book and a Reading Guide. Although many people pre fer to skip prefaces, I suggest that the reader, if he or she is one of them, make VI Preface an exception for once and read at least this section and the next. The short introductions at the beginning of each chapter are also recommended because they give a summary of the chapter's content. Chapter 1 starts from Newton's equations and develops the elementary dy namics of one-, two-, and many-body systems for unconstrained systems. This is the basic material that could be the subject of an introductory course on the oretical physics or could serve as a text for an integrated (experimental and theoretical) course. Chapter 2 is the "classical" part of general mechanics describing the princi ples of canonical mechanics following Euler, Lagrange, Hamilton, and Jacobi. Most of the material is a MUST. Nevertheless, the sections on the symplectic structure of mechanics (Sect. 2.28) and on perturbation theory (Sects. 2.38-40) may be skipped on a first reading. Chapter 3 describes a particularly beautiful application of classical mechan ics: the theory of spinning tops. The rigid body provides an important and highly nontrivial example of a motion manifold that is not a simple Euclidean space R2f, where f is the number of degrees of freedom. Its rotational part is the manifold of S0(3), the rotation group in three real dimensions. Thus, the rigid body illustrates a Lie group of great importance in physics within a framework that is simple and transparent. Chapter 4 deals with relativistic kinematics and dynamics of pointlike objects and develops the elements of special relativity. This may be the most difficult part of the book, as far as the physics is concerned, and one may wish to return to it when studying electrodynamics. Chapter 5 is the most challenging in terms of the mathematics. It develops the basic tools of differential geometry that are needed to formulate mechanics in this setting. Mechanics is then described in geometrical terms and its underlying structure is worked out. This chapter is conceived such that it may help to bridge the gap between the more "physical" texts on mechanics and the modern mathematical literature on this subject. Although it may be skipped on a first reading, the tools and the language developed here are essential if one wishes to follow the modern literature on qualitative dynamics. Chapter 6 provides an introduction to one of the most fascinating recent developments of classical dynamics: stability and deterministic chaos. It defines and illustrates all important concepts that are needed to understand the onset of chaotic motion and the quantitative analysis of unordered motions. It culminates in a few examples of chaotic motion in celestial mechanics. Chapter 7, finally, gives a short introduction to continuous systems, i.e. sys tems with an infinite number of degrees of freedom. Exercises and Practical Examples. In addition to the exercises that follow Chaps.l-6, the book contains a number of practical examples in the form of exercises followed by complete solutions. Most of these are meant to be worked out on a personal computer, thereby widening the range of problems that can be solved with elementary means, beyond the analytically integrable ones. I have Preface VII tried to choose examples simple enough that they can be made to work even on a programmable pocket computer and in a spirit, I hope, that will keep the reader from getting lost in the labyrinth of computional games. Length of this Book. Clearly there is much more material here than can be covered in one semester. The book is designed for a two-semester course (i.e., typically, an introductory course followed by a course on general mechanics). Even then, a certain choice of topics will have to be made. However, the text is sufficiently self-contained that it may be useful for complementary reading and individual study. Mathematical Prerequisites. A physicist must acquire a certain flexibility in the use of mathematics. On the one hand, it is impossible to carry out all steps in a deduction or a proof, since otherwise one will not get very far with the physics one wishes to study. On the other hand, it is indispensable to know analysis and linear algebra in some depth, so as to be able to fill in the missing links in a logical deduction. Like many other branches of physics, mechanics makes use of many and various disciplines of mathematics, and one cannot expect to have all the tools ready before beginning its study. In this book I adopt the following, somewhat generous attitude towards mathematics. In many places, the details are worked out to a large extent; in others I refer to well-known material of linear algebra and analysis. In some cases the reader might have to return to a good text in mathematics or else, ideally, derive certain results for him- or herself. In this connection it might also be helpful to consult the appendix at the end of the book. General Comments and Acknowledgements. In writing this English version I have closely followed the second, enlarged edition of the German original. Mark Seymour from Springer-Verlag kindly took care of the linguistic fine tuning. There exists a booklet by R. Schoepf and myself containing the solutions to all exercises (and a few more), written in German. I am hoping this will be translated into English soon. This book was inspired by a two-semester course on classical physics that I have taught on and off over the last fifteen years at the Johannes Gutenberg University at Mainz and by a seminar on geometrical aspects of mechanics. I thank my collaborators, colleagues, and students for many helpful remarks, stimulating questions, and profitable discussions. I am grateful for the many lively and encouraging reactions I have received since the first appearance of the first German edition in 1988. Among those to whom I owe special gratitude are P. Hagedorn, K. Hepp, D. Kastler, H. Leutwyler, N. Papadopoulos, J.M. Richard, G. Schuster, J. Smith, M. Stingl, N. Straumann, W. Thirring, and V. Vento. I thank J. Wisdom for his kind permission to use four of his figures illustrating chaotic motions in the solar system. Finally, I owe special thanks to Dorte, whose encouragement, unlimited tolerance, and practical help were essential in the task of writing this English version. Mainz, August 1990 Florian Scheck Contents 1. Elementary Newtonian Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 Newton's Laws (1687) and Their Interpretation . . . . . . . . . . . 1 1.2 Uniform Rectilinear Motion and Inertial Systems . . . . . . . . . . 4 1.3 Inertial Frames in Relative Motion . . . . . . . . . . . . . . . . . . . . . . 6 1.4 Momentum and Force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.5 Typical Forces. A Remark about Units . . . . . . . . . . . . . . . . . . 8 1.6 Space, Time, and Forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.7 The Two-Body System with Internal Forces . . . . . . . . . . . . . . 11 1.7.1 Center-of-Mass and Relative Motion . . . . . . . . . . . . . . 11 1.7.2 Example: The Gravitational Force between Two Celestial Bodies (Kepler's Problem) . . . . . . . . . . 12 1.7.3 Center-of-Mass and Relative Momentum in the Two-Body System . . . . . . . . . . . . . . . . . . . . . . . 18 1.8 Systems of Finitely Many Particles . . . . . . . . . . . . . . . . . . . . . 18 1.9 The Principle of Center-of-Mass Motion . . . . . . . . . . . . . . . . . 19 1.10 The Principle of Angular-Momentum Conservation . . . . . . . . 20 1.11 The Principle of Energy Conservation . . . . . . . . . . . . . . . . . . . 20 1.12 The Closed n-Particle System . . . . . . . . . . . . . . . . . . . . . . . . . . 21 1.13 Galilei Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 1.14 Space and Time with Galilei Invariance . . . . . . . . . . . . . . . . . . 25 1.15 Conservative Force Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 1.16 One-Dimensional Motion of a Point Particle . . . . . . . . . . . . . . 30 1.17 Examples of Motion in One Dimension . . . . . . . . . . . . . . . . . . 30 1.17 .1 The Harmonic Oscillator . . . . . . . . . . . . . . . . . . . . . . . 30 1.17 .2 The Planar Mathematical Pendulum . . . . . . . . . . . . . . 33 1.18 Phase Space for the n-Particle System (in R3) • • • • • • • • • • • • 34 1.19 Existence and Uniqueness of the Solutions of~=~(~, t) . . . 35 1.20 Physical Consequences of the Existence and Uniqueness Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 1.21 Linear Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 1.21.1 Linear, Homogeneous Systems . . . . . . . . . . . . . . . . . . 38 1.21.2 Linear, Inhomogeneous Systems . . . . . . . . . . . . . . . . . 39 1.22 Integrating One-Dimensional Equations of Motion . . . . . . . . . 40 1.23 Example: The Planar Pendulum for Arbitrary Deviations from the Vertical . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 1.24 Example: The Two-Body System with a Central Force . . . . . 44 X Contents 1.25 Rotating Reference Systems: Coriolis and Centrifugal Forces 50 1.26 Examples of Rotating Reference Systems . . . . . . . . . . . . . . . . 51 1.27 Scattering of Two Particles that Interact via a Central Force: Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 1.28 Two-Particle Scattering with a Central Force: Dynamics . . . . 57 1.29 Example: Coulomb Scattering of Two Particles with Equal Mass and Charge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 1.30 Mechanical Bodies of Finite Extension . . . . . . . . . . . . . . . . . . 65 1.31 Time Averages and the Vtrial Theorem . . . . . . . . . . . . . . . . . . 69 Appendix: Practical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 2. The Principles of Canonical Mechanics . . . . . . . . . . . . . . . . . . . . . 77 2.1 Constraints and Generalized Coordinates . . . . . . . . . . . . . . . . . 77 2.1.1 Definition of Constraints . . . . . . . . . . . . . . . . . . . . . . . 77 2.1.2 Generalized Coordinates . . . . . . . . . . . . . . . . . . . . . . . . 79 2.2 D' Alembert's Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 2.2.1 Definition of Virtual Displacements . . . . . . . . . . . . . . 79 2.2.2 The Static Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 2.2.3 The Dynamical Case . . . . . . . . . . . . . . . . . . . . . . . . . . 80 2.3 Lagrange's Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 2.4 Examples of the Use of Lagrange's Equations . . . . . . . . . . . . 83 2.5 A Digression on Variational Principles . . . . . . . . . . . . . . . . . . . 85 2.6 Hamilton's Variational Principle (1834) . . . . . . . . . . . . . . . . . . 88 2.7 The Euler-Lagrange Equations . . . . . . . . . . . . . . . . . . . . . . . . . 88 2.8 Further Examples of the Use of Lagrange's Equations . . . . . . 89 2.9 A Remark about Nonuniqueness of the Lagrangian Function 91 2.10 Gauge Transformations of the Lagrangian Function . . . . . . . . 92 2.11 Admissible Transformations of the Generalized Coordinates 93 2.12 The Hamiltonian Function and Its Relation to the Lagrangian Function L . . . . . . . . . . . . . . . . . . . . . . . . . . 94 2.13 The Legendre Transformation for the Case of One Variable 95 2.14 The Legendre Transformation for the Case of Several Variables 97 2.15 Canonical Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 2.16 Examples of Canonical Systems . . . . . . . . . . . . . . . . . . . . . . . . 99 2.17 The Variational Principle Applied to the Hamiltonian Function 100 2.18 Symmetries and Conservation Laws . . . . . . . . . . . . . . . . . . . . . 101 2.19 Noether's Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 2.20 The Generator for Infinitesimal Rotations about an Axis . . . . 104 2.21 More about the Rotation Group . . . . . . . . . . . . . . . . . . . . . . . . 105 2.22 Infinitesimal Rotations and Their Generators . . . . . . . . . . . . . . 108 2.23 Canonical Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 2.24 Examples of Canonical Transformations . . . . . . . . . . . . . . . . . 113 2.25 The Structure of the Canonical Equations . . . . . . . . . . . . . . . . 114 2.26 Example: Linear Autonomous Systems in One Dimension . . . 116 Contents XI 2.27 Canonical Transformations in Compact Notation ......... . 117 2.28 On the Symplectic Structure of Phase Space ............. . 119 2.29 Liouville's Theorem ................................. . 122 2.29.1 The Local Form ............................. . 123 2.29.2 The Global Form ............................ . 124 2.30 Examples of Liouville's Theorem ...................... . 125 2.31 Poisson Brackets ................................... . 128 2.32 Properties of Poisson Brackets ........................ . 131 2.33 Infinitesimal Canonical Transformations ................ . 133 2.34 Integrals of the Motion .............................. . 134 2.35 The Hamilton-Jacobi Differential Equation .............. . 137 2.36 Examples of the Hamilton-Jacobi Equation .............. . 138 2.37 The Hamilton-Jacobi Equation and Integrable Systems .... . 140 2.37.1 Local Rectification of Hamiltonian' Systems ....... . 140 2.37.2 Integrable Systems ........................... . 144 2.37.3 Angle and Action Variables ................... .. 149 2.38 Perturbing Quasiperiodic Hamiltonian Systems ........... . 150 2.39 Autonomous, Nondegenerate Hamiltonian Systems in the Neighborhood of Integrable Systems .............. . 153 2.40 Examples. The Averaging Principle .................... . 154 2.40.1 The Anharmonic Oscillator .................... . 154 2.40.2 Averaging of Perturbations ..................... . 156 Appendix: Practical Examples ............................. . 159 3. The Mechanics of Rigid Bodies ........................... . 165 3.1 Definition of Rigid Body ............................. . 165 3.2 Infinitesimal Displacement of a Rigid Body ............. . 167 3.3 Kinetic Energy and the Inertia Tensor .................. . 169 3.4 Properties of the Inertia Tensor ........................ . 171 3.5 Steiner's Theorem .................................. . 174 3.6 Examples of the Use of Steiner's Theorem .............. . 175 3.7 Angular Momentum of a Rigid Body ................... . 178 3.8 Force-Free Motion of Rigid Bodies .................... . 180 3.9 Another Parametrization of Rotations: The Euler Angles ... . 182 3.10 Definition of Eulerian Angles ......................... . 184 3.11 Equations of Motion of Rigid Bodies ................... . 185 3.12 Euler's Equations of Motion .......................... . 188 3.13 Euler's Equations Applied to a Force-Free Top ........... . 190 3.14 The Motion of a Free Top and Geometric Constructions ... . 195 3.15 The Rigid Body in the Framework of Canonical Mechanics . 197 3.16 Example: The Symmetric Children's Top in a Gravitational Field .............................. . 201 Appendix: Practical Examples 204 XII Contents 4. Relativistic Mechanics 207 4.1 Failures of Nonrelativistic Mechanics ................... . 208 4.2 Constancy of the Speed of Light ...................... . 211 4.3 The Lorentz Transformations ......................... . 212 4.4 Analysis of Lorentz and Poincare Transfonnations ........ . 218 4.4.1 Rotations and Special Lorentz Transfonnations ("Boosts") .................................. . 220 4.4.2 Interpretation of Special Lorentz Transfonnations .. . 223 4.5 Decomposition of Lorentz Transformations into Their Components .............................. . 224 4.5.1 Proposition on Orthochronous, Proper Lorentz Transfonnations ................ . 224 4.5.2 Corollary of the Decomposition Theorem and Some Consequences ...................... . 226 4.6 Addition of Relativistic Velocities ..................... . 229 4.7 Galilean and Lorentzian Space-Time Manifolds .......... . 232 4.8 Orbital Curves and Proper Time ....................... . 236 4.9 Relativistic Dynamics ............................... . 237 4.9.1 Newton's Equation ........................... . 237 4.9.2 The Energy-Momentum Vector ................. . 239 4.9.3 The Lorentz Force ........................... . 242 4.10 Time Dilatation and Scale Contraction .................. . 244 4.11 More About the Motion of Free Particles ............... . 246 4.12 The Confonnal Group ............................... . 249 5. Geometric Aspects of Mechanics ......................... . 251 5.1 Manifolds of Generalized Coordinates .................. . 252 5.2 Differentiable Manifolds ............................. . 255 5.2.1 The Euclidean Space Rn ...................... . 255 5.2.2 Smooth or Differentiable Manifolds ............. . 257 5.2.3 Examples of Smooth Manifolds ................. . 259 5.3 Geometrical Objects on Manifolds ..................... . 263 5.3.1 Functions and Curves on Manifolds ............. . 264 5.3.2 Tangent Vectors on a Smooth Manifold .......... . 266 5.3.3 The Tangent Bundle of a Manifold .............. . 268 5.3.4 Vector Fields on Smooth Manifolds ............. . 269 5.3.5 Exterior Fonns .............................. . 273 5.4 Calculus on Manifolds ............................... . 275 5.4.1 Differentiable Mappings of Manifolds ........... . 276 5.4.2 Integral Curves of Vector Fields ................ . 278 5.4.3 Exterior Product of One-Forms ................. . 279 5.4.4 The Exterior Derivative ....................... . 281 5.4.5 Exterior Derivative and Vectors in R3 •••••••..•••• 283 5.5 Hamilton-Jacobi and Lagrangian Mechanics ............. . 285