Structure and Interpretation
of Classical Mechanics

Short Table of Contents

Table of Contents

• Dedication
• Preface
• Acknowledgments
• 1 Lagrangian Mechanics
  • 1.1 Configuration Spaces
  • 1.2 Generalized Coordinates
  • 1.3 The Principle of Stationary Action
  • 1.4 Computing Actions
  • 1.5 The Euler–Lagrange Equations
    • 1.5.1 Derivation of the Lagrange Equations
    • 1.5.2 Computing Lagrange's Equations
  • 1.6 How to Find Lagrangians
    • 1.6.1 Coordinate Transformations
    • 1.6.2 Systems with Rigid Constraints
    • 1.6.3 Constraints as Coordinate Transformations
    • 1.6.4 The Lagrangian Is Not Unique
  • 1.7 Evolution of Dynamical State
  • 1.8 Conserved Quantities
    • 1.8.1 Conserved Momenta
    • 1.8.2 Energy Conservation
    • 1.8.3 Central Forces in Three Dimensions
    • 1.8.4 The Restricted Three-Body Problem
    • 1.8.5 Noether's Theorem
  • 1.9 Abstraction of Path Functions
  • 1.10 Constrained Motion
    • 1.10.1 Coordinate Constraints
    • 1.10.2 Derivative Constraints
    • 1.10.3 Nonholonomic Systems
  • 1.11 Summary
  • 1.12 Projects
• 2 Rigid Bodies
  • 2.1 Rotational Kinetic Energy
  • 2.2 Kinematics of Rotation
  • 2.3 Moments of Inertia
  • 2.4 Inertia Tensor
  • 2.5 Principal Moments of Inertia
  • 2.6 Vector Angular Momentum
  • 2.7 Euler Angles
  • 2.8 Motion of a Free Rigid Body
    • 2.8.1 Computing the Motion of Free Rigid Bodies
    • 2.8.2 Qualitative Features
  • 2.9 Euler's Equations
  • 2.10 Axisymmetric Tops
  • 2.11 Spin-Orbit Coupling
    • 2.11.1 Development of the Potential Energy
    • 2.11.2 Rotation of the Moon and Hyperion
    • 2.11.3 Spin-Orbit Resonances
  • 2.12 Nonsingular Coordinates and Quaternions
    • 2.12.1 Motion in Terms of Quaternions
  • 2.13 Summary
  • 2.14 Projects
• 3 Hamiltonian Mechanics
  • 3.1 Hamilton's Equations
    • 3.1.1 The Legendre Transformation
    • 3.1.2 Hamilton's Equations from the Action Principle
    • 3.1.3 A Wiring Diagram
  • 3.2 Poisson Brackets
  • 3.3 One Degree of Freedom
  • 3.4 Phase Space Reduction
    • 3.4.1 Lagrangian Reduction
  • 3.5 Phase Space Evolution
    • 3.5.1 Phase-Space Description Is Not Unique
  • 3.6 Surfaces of Section
    • 3.6.1 Periodically Driven Systems
    • 3.6.2 Computing Stroboscopic Surfaces of Section
    • 3.6.3 Autonomous Systems
    • 3.6.4 Computing Hénon–Heiles Surfaces of Section
    • 3.6.5 Non-Axisymmetric Top
  • 3.7 Exponential Divergence
  • 3.8 Liouville's Theorem
  • 3.9 Standard Map
  • 3.10 Summary
  • 3.11 Projects
• 4 Phase Space Structure
  • 4.1 Emergence of the Divided Phase Space
  • 4.2 Linear Stability
    • 4.2.1 Equilibria of Differential Equations
    • 4.2.2 Fixed Points of Maps
    • 4.2.3 Relations Among Exponents
  • 4.3 Homoclinic Tangle
    • 4.3.1 Computation of Stable and Unstable Manifolds
  • 4.4 Integrable Systems
  • 4.5 Poincaré–Birkhoff Theorem
    • 4.5.1 Computing the Poincaré–Birkhoff Construction
  • 4.6 Invariant Curves
    • 4.6.1 Finding Invariant Curves
    • 4.6.2 Dissolution of Invariant Curves
  • 4.7 Summary
  • 4.8 Projects
• 5 Canonical Transformations
  • 5.1 Point Transformations
  • 5.2 General Canonical Transformations
    • 5.2.1 Time-Dependent Transformations
    • 5.2.2 Abstracting the Canonical Condition
  • 5.3 Invariants of Canonical Transformations
  • 5.4 Generating Functions
    • 5.4.1 F₁ Generates Canonical Transformations
    • 5.4.2 Generating Functions and Integral Invariants
    • 5.4.3 Types of Generating Functions
    • 5.4.4 Point Transformations
    • 5.4.5 Total Time Derivatives
  • 5.5 Extended Phase Space
    • 5.5.1 Poincaré–Cartan Integral Invariant
  • 5.6 Reduced Phase Space
  • 5.7 Summary
  • 5.8 Projects
• 6 Canonical Evolution
  • 6.1 Hamilton–Jacobi Equation
    • 6.1.1 Harmonic Oscillator
    • 6.1.2 Hamilton–Jacobi Solution of the Kepler Problem
    • 6.1.3 F₂ and the Lagrangian
    • 6.1.4 The Action Generates Time Evolution
  • 6.2 Time Evolution is Canonical
    • 6.2.1 Another View of Time Evolution
    • 6.2.2 Yet Another View of Time Evolution
  • 6.3 Lie Transforms
  • 6.4 Lie Series
  • 6.5 Exponential Identities
  • 6.6 Summary
  • 6.7 Projects
• 7 Canonical Perturbation Theory
  • 7.1 Perturbation Theory with Lie Series
  • 7.2 Pendulum as a Perturbed Rotor
    • 7.2.1 Higher Order
    • 7.2.2 Eliminating Secular Terms
  • 7.3 Many Degrees of Freedom
    • 7.3.1 Driven Pendulum as a Perturbed Rotor
  • 7.4 Nonlinear Resonance
    • 7.4.1 Pendulum Approximation
    • 7.4.2 Reading the Hamiltonian
    • 7.4.3 Resonance-Overlap Criterion
    • 7.4.4 Higher-Order Perturbation Theory
    • 7.4.5 Stability of the Inverted Vertical Equilibrium
  • 7.5 Summary
  • 7.6 Projects
• 8 Appendix: Scheme
• 9 Appendix: Our Notation
• References
• List of Exercises
• Index