Lecture Notes on Classical Mechanics for Physics 106ab Sunil Golwala Revision Date: January 15, 2007 Introduction These notes were written during the Fall, 2004, and Winter, 2005, terms. They are indeed lecture notes – I literally lecture from these notes. They combine material from Hand and Finch (mostly), Thornton, and Goldstein, but cover the material in a different order than any one of these texts and deviate from them widely in some places and less so in others. The reader will no doubt ask the question I asked myself many times while writing these notes: why bother? There are a large number of mechanics textbooks available all covering this very standard material, complete with worked examples and end-of-chapter problems. I can only defend myself by saying that all teachers understand their material in a slightly different way and it is very difficult to teach from someone else’s point of view – it’s like walking in shoes that are two sizes wrong. It is inevitable that every teacher will want to present some of the material in a way that differs from the available texts. These notes simply put my particular presentation down on the page for your reference. These notes are not a substitute for a proper textbook; I have not provided nearly as many examples or illustrations, and have provided no exercises. They are a supplement. I suggest you skim them in parallel while reading one of the recommended texts for the course, focusing your attention on places where these notes deviate from the texts. ii Contents 1 Elementary Mechanics 1 1.1 Newtonian Mechanics ................................... 2 1.1.1 The equation of motion for a single particle ................... 2 1.1.2 Angular Motion .................................. 13 1.1.3 Energy and Work .................................. 16 1.2 Gravitation ......................................... 24 1.2.1 Gravitational Force ................................. 24 1.2.2 Gravitational Potential .............................. 26 1.3 Dynamics of Systems of Particles ............................. 32 1.3.1 Newtonian Mechanical Concepts for Systems of Particles ........... 32 1.3.2 The Virial Theorem ................................ 47 1.3.3 Collisions of Particles ............................... 51 2 Lagrangian and Hamiltonian Dynamics 63 2.1 The Lagrangian Approach to Mechanics ......................... 64 2.1.1 Degrees of Freedom, Constraints, and Generalized Coordinates ........ 65 2.1.2 Virtual Displacement, Virtual Work, and Generalized Forces ......... 71 2.1.3 d’Alembert’s Principle and the Generalized Equation of Motion ........ 76 2.1.4 The Lagrangian and the Euler-Lagrange Equations ............... 81 2.1.5 The Hamiltonian .................................. 84 2.1.6 Cyclic Coordinates and Canonical Momenta ................... 85 2.1.7 Summary ...................................... 86 2.1.8 More examples ................................... 86 2.1.9 Special Nonconservative Cases .......................... 92 2.1.10 Symmetry Transformations, Conserved Quantities, Cyclic Coordinates and Noether’s Theorem ................................. 96 2.2 Variational Calculus and Dynamics ............................ 103 2.2.1 The Variational Calculus and the Euler Equation ................ 103 2.2.2 The Principle of Least Action and the Euler-Lagrange Equation ....... 108 2.2.3 Imposing Constraints in Variational Dynamics ................. 109 2.2.4 Incorporating Nonholonomic Constraints in Variational Dynamics ...... 119 2.3 Hamiltonian Dynamics ................................... 123 2.3.1 Legendre Transformations and Hamilton’s Equations of Motion ........ 123 2.3.2 Phase Space and Liouville’s Theorem ...................... 130 2.4 Topics in Theoretical Mechanics ............................. 138 2.4.1 Canonical Transformations and Generating Functions ............. 138 2.4.2 Symplectic Notation ................................ 147 iii CONTENTS 2.4.3 Poisson Brackets .................................. 149 2.4.4 Action-Angle Variables and Adiabatic Invariance ................ 152 2.4.5 The Hamilton-Jacobi Equation .......................... 160 3 Oscillations 173 3.1 The Simple Harmonic Oscillator ............................. 174 3.1.1 Equilibria and Oscillations ............................ 174 3.1.2 Solving the Simple Harmonic Oscillator ..................... 176 3.1.3 The Damped Simple Harmonic Oscillator .................... 177 3.1.4 The Driven Simple and Damped Harmonic Oscillator ............. 181 3.1.5 Behavior when Driven Near Resonance ...................... 186 3.2 Coupled Simple Harmonic Oscillators .......................... 191 3.2.1 The Coupled Pendulum Example ......................... 191 3.2.2 General Method of Solution ............................ 195 3.2.3 Examples and Applications ............................ 204 3.2.4 Degeneracy ..................................... 211 3.3 Waves ............................................ 215 3.3.1 The Loaded String ................................. 215 3.3.2 The Continuous String ............................... 219 3.3.3 The Wave Equation ................................ 222 3.3.4 Phase Velocity, Group Velocity, and Wave Packets ............... 229 4 Central Force Motion and Scattering 233 4.1 The Generic Central Force Problem ........................... 234 4.1.1 The Equation of Motion .............................. 234 4.1.2 Formal Implications of the Equations of Motion ................. 240 4.2 The Special Case of Gravity – The Kepler Problem ................... 243 4.2.1 The Shape of Solutions of the Kepler Problem ................. 243 4.2.2 Time Dependence of the Kepler Problem Solutions ............... 248 4.3 Scattering Cross Sections ................................. 252 4.3.1 Setting up the Problem .............................. 252 4.3.2 The Generic Cross Section ............................. 253 1 4.3.3 r Potentials ..................................... 255 5 Rotating Systems 257 5.1 The Mathematical Description of Rotations ....................... 258 5.1.1 Infinitesimal Rotations ............................... 258 5.1.2 Finite Rotations .................................. 260 5.1.3 Interpretation of Rotations ............................ 262 5.1.4 Scalars, Vectors, and Tensors ........................... 263 5.1.5 Comments on Lie Algebras and Lie Groups ................... 267 5.2 Dynamics in Rotating Coordinate Systems ........................ 269 5.2.1 Newton’s Second Law in Rotating Coordinate Systems ............. 269 5.2.2 Applications .................................... 278 5.2.3 Lagrangian and Hamiltonian Dynamics in Rotating Coordinate Systems ... 280 5.3 Rotational Dynamics of Rigid Bodies ........................... 282 5.3.1 Basic Formalism .................................. 282 5.3.2 Torque-Free Motion ................................ 296 iv CONTENTS 5.3.3 Motion under the Influence of External Torques ................. 313 6 Special Relativity 323 6.1 Special Relativity ...................................... 324 6.1.1 The Postulates ................................... 324 6.1.2 Transformation Laws ................................ 324 6.1.3 Mathematical Description of Lorentz Transformations ............. 333 6.1.4 Physical Implications ................................ 339 6.1.5 Lagrangian and Hamiltonian Dynamics in Relativity .............. 346 A Mathematical Appendix 347 A.1 Notational Conventions for Mathematical Symbols ................... 347 A.2 Coordinate Systems .................................... 348 A.3 Vector and Tensor Definitions and Algebraic Identities ................. 349 A.4 Vector Calculus ....................................... 354 A.5 Taylor Expansion ...................................... 356 A.6 Calculus of Variations ................................... 357 A.7 Legendre Transformations ................................. 357 B Summary of Physical Results 359 B.1 Elementary Mechanics ................................... 359 B.2 Lagrangian and Hamiltonian Dynamics ......................... 363 B.3 Oscillations ......................................... 371 B.4 Central Forces and Dynamics of Scattering ....................... 379 B.5 Rotating Systems ...................................... 384 B.6 Special Relativity ...................................... 389 v Chapter 1 Elementary Mechanics This chapter reviews material that was covered in your first-year mechanics course – Newtonian mechanics, elementary gravitation, and dynamics of systems of particles. None of this material should be surprising or new. Special emphasis is placed on those aspects that we will return to later in the course. If you feel less than fully comfortable with this material, please take the time to review it now, before we hit the interesting new stuff! The material in this section is largely from Thornton Chapters 2, 5, and 9. Small parts of it are covered in Hand and Finch Chapter 4, but they use the language of Lagrangian mechanics that you have not yet learned. Other references are provided in the notes. 1 CHAPTER 1. ELEMENTARY MECHANICS 1.1 Newtonian Mechanics References: • Thornton and Marion, Classical Dynamics of Particles and Systems, Sections 2.4, 2.5, and 2.6 • Goldstein, Classical Mechanics, Sections 1.1 and 1.2 • Symon, Mechanics, Sections 1.7, 2.1-2.6, 3.1-3.9, and 3.11-3.12 • any first-year physics text Unlike some texts, we’re going to be very pragmatic and ignore niceties regarding the equivalence principle, the logical structure of Newton’s laws, etc. I will take it as given that we all have an intuitive