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Analytical Mechanics

Book Description

Analytical Mechanics, first published in 1999, provides a detailed introduction to the key analytical techniques of classical mechanics, one of the cornerstones of physics. It deals with all the important subjects encountered in an undergraduate course and prepares the reader thoroughly for further study at graduate level. The authors set out the fundamentals of Lagrangian and Hamiltonian mechanics early on in the book and go on to cover such topics as linear oscillators, planetary orbits, rigid-body motion, small vibrations, nonlinear dynamics, chaos, and special relativity. A special feature is the inclusion of many 'e-mail questions', which are intended to facilitate dialogue between the student and instructor. Many worked examples are given, and there are 250 homework exercises to help students gain confidence and proficiency in problem-solving. It is an ideal textbook for undergraduate courses in classical mechanics, and provides a sound foundation for graduate study.

Table of Contents

  1. Cover
  2. Title
  3. Copyright
  4. Contents
  5. Preface
  6. 1 Lagrangian Mechanics
    1. 1.1 Example and Review of Newton’s Mechanics: A Block Sliding on an Inclined Plane
    2. 1.2 Using Virtual Work to Solve the Same Problem
    3. 1.3 Solving for the Motion of a Heavy Bead Sliding on a Rotating Wire
    4. 1.4 Toward a General Formula: Degrees of Freedom and Types of Constraints
    5. 1.5 Generalized Velocities: How to “Cancel the Dots”
    6. 1.6 Virtual Displacements and Virtual Work – Generalized Forces
    7. 1.7 Kinetic Energy as a Function of the Generalized Coordinates and Velocities
    8. 1.8 Conservative Forces: Definition of the Lagrangian L
    9. 1.9 Reference Frames
    10. 1.10 Definition of the Hamiltonian
    11. 1.11 How to Get Rid of Ignorable Coordinates
    12. 1.12 Discussion and Conclusions – What’s Next after You Get the Eom?
    13. 1.13 An Example of a Solved Problem
    14. Summary of Chapter 1
    15. Problems
    16. Appendix A. About Nonholonomic Constraints
    17. Appendix B. More about Conservative Forces
  7. 2 Variational Calculus And Its Application To Mechanics
    1. 2.1 History
    2. 2.2 The Euler Equation
    3. 2.3 Relevance to Mechanics
    4. 2.4 Systems with Several Degrees of Freedom
    5. 2.5 Why Use the Variational Approach in Mechanics?
    6. 2.6 Lagrange Multipliers
    7. 2.7 Solving Problems with Explicit Holonomic Constraints
    8. 2.8 Nonintegrable Nonholonomic Constraints – A Method that Works
    9. 2.9 Postscript on the Euler Equation with More Than One Independent Variable
    10. Summary of Chapter 2
    11. Problems
    12. Appendix. About Maupertuis and What Came to Be Called “Maupertuis’ Principle”
  8. 3 Linear Oscillators
    1. 3.1 Stable or Unstable Equilibrium?
    2. 3.2 Simple Harmonic Oscillator
    3. 3.3 Damped Simple Harmonic Oscillator (Dsho)
    4. 3.4 An Oscillator Driven by an External Force
    5. 3.5 Driving Force Is a Step Function
    6. 3.6 Finding the Green’s Function for the Sho
    7. 3.7 Adding up the Delta Functions – Solving the Arbitary Force
    8. 3.8 Driving an Oscillator in Resonance
    9. 3.9 Relative Phase of the Dsho Oscillator with Sinusoidal Drive
    10. Summary of Chapter 3
    11. Problems
  9. 4 One-Dimensional Systems: Central Forces and the Kepler Problem
    1. 4.1 The Motion of a “Generic” One-Dimensional System
    2. 4.2 The Grandfather’s Clock
    3. 4.3 The History of the Kepler Problem
    4. 4.4 Solving the Central Force Problem
    5. 4.5 The Special Case of Gravitational Attraction
    6. 4.6 Interpretation of Orbits
    7. 4.7 Repulsive Forces
    8. Summary of Chapter 4
    9. Problems
    10. Appendix. Tables of Astrophysical Data
  10. 5 Noether’s Theorem and Hamiltonian Dynamics
    1. 5.1 Discovering Angular Momentum Conservation from Rotational Invariance
    2. 5.2 Noether’s Theorem
    3. 5.3 Hamiltonian Dynamics
    4. 5.4 The Legendre Transformation
    5. 5.5 Hamilton’s Equations of Motion
    6. 5.6 Liouville’s Theorem
    7. 5.7 Momentum Space
    8. 5.8 Hamiltonian Dynamics in Accelerated Systems
    9. Summary of Chapter 5
    10. Problems
    11. Appendix A. A General Proof of Liouville’s Theorem Using the Jacobian
    12. Appendix B. Poincaré Recurrence Theorem
  11. 6 Theoretical Mechanics: From Canonical Transformations to Action-Angle Variables
    1. 6.1 Canonical Transformations
    2. 6.2 Discovering Three New Forms of the Generating Function
    3. 6.3 Poisson Brackets
    4. 6.4 Hamilton–Jacobi Equation
    5. 6.5 Action–Angle Variables for 1-D Systems
    6. 6.6 Integrable Systems
    7. 6.7 Invariant Tori and Winding Numbers
    8. Summary of Chapter 6
    9. Problems
    10. Appendix. What Does “Symplectic” Mean?
  12. 7 Rotating Coordinate Systems
    1. 7.1 What Is a Vector?
    2. 7.2 Review: Infinitesimal Rotations and Angular Velocity
    3. 7.3 Finite Three-Dimensional Rotations
    4. 7.4 Rotated Reference Frames
    5. 7.5 Rotating Reference Frames
    6. 7.6 The Instantaneous Angular Velocity ω
    7. 7.7 Fictitious Forces
    8. 7.8 The Tower of Pisa Problem
    9. 7.9 Why Do Hurricane Winds Rotate?
    10. 7.10 Foucault Pendulum
    11. Summary of Chapter 7
    12. Problems
  13. 8 The Dynamics Of Rigid Bodies
    1. 8.1 Kinetic Energy of a Rigid Body
    2. 8.2 The Moment of Inertia Tensor
    3. 8.3 Angular Momentum of a Rigid Body
    4. 8.4 The Euler Equations for Force-Free Rigid Body Motion
    5. 8.5 Motion of a Torque-Free Symmetric Top
    6. 8.6 Force-Free Precession of the Earth: The “Chandler Wobble”
    7. 8.7 Definition of Euler Angles
    8. 8.8 Finding the Angular Velocity
    9. 8.9 Motion of Torque-Free Asymmetric Tops: Poinsot Construction
    10. 8.10 The Heavy Symmetric Top
    11. 8.11 Precession of the Equinoxes
    12. 8.12 Mach’s Principle
    13. Summary of Chapter 8
    14. Problems
    15. Appendix A. What Is a Tensor?
    16. Appendix B. Symmetric Matrices Can Always Be Diagonalized by “Rotating the Coordinates”
    17. Appendix C. Understanding the Earth’s Equatorial Bulge
  14. 9 The Theory Of Small Vibrations
    1. 9.1 Two Coupled Pendulums
    2. 9.2 Exact Lagrangian for the Double Pendulum
    3. 9.3 Single Frequency Solutions to Equations of Motion
    4. 9.4 Superimposing Different Modes; Complex Mode Amplitudes
    5. 9.5 Linear Triatomic Molecule
    6. 9.6 Why the Method Always Works
    7. 9.7 Tv Point Masses Connected by a String
    8. Summary of Chapter 9
    9. Problems
    10. Appendix. What Is a Cofactor?
  15. 10 Approximate Solutions To Nonanalytic Problems
    1. 10.1 Stability of Mechanical Systems
    2. 10.2 Parametric Resonance
    3. 10.3 Lindstedt–Poincaré Perturbation Theory
    4. 10.4 Driven Anharmonic Oscillator
    5. Summary of Chapter 10
    6. Problems
  16. 11 Chaotic Dynamics
    1. 11.1 Conservative Chaos – The Double Pendulum: A Hamiltonian System with Two Degrees of Freedom
    2. 11.2 The Poincaré Section
    3. 11.3 Kam Tori: The Importance of Winding Number
    4. 11.4 Irrational Winding Numbers
    5. 11.5 Poincaré–Birkhoff Theorem
    6. 11.6 Linearizing Near a Fixed Point: The Tangent Map and the Stability Matrix
    7. 11.7 Following Unstable Manifolds: Homoclinic Tangles
    8. 11.8 Lyapunov Exponents
    9. 11.9 Global Chaos for the Double Pendulum
    10. 11.10 Effect of Dissipation
    11. 11.11 Damped Driven Pendulum
    12. 11.12 Fractals
    13. 11.13 Chaos in the Solar System
    14. Student Projects
    15. Appendix. The Logistic Map: Period-Doubling Route to Chaos; Renormalization
  17. 12 Special Relativity
    1. 12.1 Space–Time Diagrams
    2. 12.2 The Lorentz Transformation
    3. 12.3 Simultaneity Is Relative
    4. 12.4 What Happens to y and z if We Move Parallel to the X Axis?
    5. 12.5 Velocity Transformation Rules
    6. 12.6 Observing Light Waves
    7. 12.7 What Is Mass?
    8. 12.8 Rest Mass Is a Form of Energy
    9. 12.9 How Does Momentum Transform?
    10. 12.10 More Theoretical “Evidence” for the Equivalence of Mass and Energy
    11. 12.11 Mathematics of Relativity: Invariants and Four-Vectors
    12. 12.12 A Second Look at the Energy–Momentum Four-Vector
    13. 12.13 Why Are There Both Upper and Lower Greek Indices?
    14. 12.14 Relativistic Lagrangian Mechanics
    15. 12.15 What Is the Lagrangian in an Electromagnetic Field?
    16. 12.16 Does a Constant Force Cause Constant Acceleration?
    17. 12.17 Derivation of the Lorentz Force from the Lagrangian
    18. 12.18 Relativistic Circular Motion
    19. Summary of Chapter 12
    20. Problems
    21. Appendix. The Twin Paradox
  18. Bibliography
  19. References
  20. Index