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Structure and Interpretation
of Classical Mechanics

Second Edition
Unofficial HTML Version
Gerald Jay Sussman and Jack Wisdom

(c)2014 by The Massachusetts Institute of Technology

SVG

This work is licensed under a Creative Commons
Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-SA
3.0). Based on a work at mitpress.mit.edu.

The MIT Press
Cambridge, Massachusetts
London, England

Title page image credit: Wellcome Library, London. Licensed under a
Creative Commons Attribution only license (CC BY 4.0).

Short Table of Contents

  * Dedication
  * Preface
  * Acknowledgments
  * Lagrangian Mechanics
  * Rigid Bodies
  * Hamiltonian Mechanics
  * Phase Space Structure
  * Canonical Transformations
  * Canonical Evolution
  * Canonical Perturbation Theory
  * Appendix: Scheme
  * Appendix: Our Notation

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
          o 1.5.1 Derivation of the Lagrange Equations
          o 1.5.2 Computing Lagrange's Equations
      + 1.6 How to Find Lagrangians
          o 1.6.1 Coordinate Transformations
          o 1.6.2 Systems with Rigid Constraints
          o 1.6.3 Constraints as Coordinate Transformations
          o 1.6.4 The Lagrangian Is Not Unique
      + 1.7 Evolution of Dynamical State
      + 1.8 Conserved Quantities
          o 1.8.1 Conserved Momenta
          o 1.8.2 Energy Conservation
          o 1.8.3 Central Forces in Three Dimensions
          o 1.8.4 The Restricted Three-Body Problem
          o 1.8.5 Noether's Theorem
      + 1.9 Abstraction of Path Functions
      + 1.10 Constrained Motion
          o 1.10.1 Coordinate Constraints
          o 1.10.2 Derivative Constraints
          o 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
          o 2.8.1 Computing the Motion of Free Rigid Bodies
          o 2.8.2 Qualitative Features
      + 2.9 Euler's Equations
      + 2.10 Axisymmetric Tops
      + 2.11 Spin-Orbit Coupling
          o 2.11.1 Development of the Potential Energy
          o 2.11.2 Rotation of the Moon and Hyperion
          o 2.11.3 Spin-Orbit Resonances
      + 2.12 Nonsingular Coordinates and Quaternions
          o 2.12.1 Motion in Terms of Quaternions
      + 2.13 Summary
      + 2.14 Projects
  * 3 Hamiltonian Mechanics
      + 3.1 Hamilton's Equations
          o 3.1.1 The Legendre Transformation
          o 3.1.2 Hamilton's Equations from the Action Principle
          o 3.1.3 A Wiring Diagram
      + 3.2 Poisson Brackets
      + 3.3 One Degree of Freedom
      + 3.4 Phase Space Reduction
          o 3.4.1 Lagrangian Reduction
      + 3.5 Phase Space Evolution
          o 3.5.1 Phase-Space Description Is Not Unique
      + 3.6 Surfaces of Section
          o 3.6.1 Periodically Driven Systems
          o 3.6.2 Computing Stroboscopic Surfaces of Section
          o 3.6.3 Autonomous Systems
          o 3.6.4 Computing Henon-Heiles Surfaces of Section
          o 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
          o 4.2.1 Equilibria of Differential Equations
          o 4.2.2 Fixed Points of Maps
          o 4.2.3 Relations Among Exponents
      + 4.3 Homoclinic Tangle
          o 4.3.1 Computation of Stable and Unstable Manifolds
      + 4.4 Integrable Systems
      + 4.5 Poincare-Birkhoff Theorem
          o 4.5.1 Computing the Poincare-Birkhoff Construction
      + 4.6 Invariant Curves
          o 4.6.1 Finding Invariant Curves
          o 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
          o 5.2.1 Time-Dependent Transformations
          o 5.2.2 Abstracting the Canonical Condition
      + 5.3 Invariants of Canonical Transformations
      + 5.4 Generating Functions
          o 5.4.1 F[1] Generates Canonical Transformations
          o 5.4.2 Generating Functions and Integral Invariants
          o 5.4.3 Types of Generating Functions
          o 5.4.4 Point Transformations
          o 5.4.5 Total Time Derivatives
      + 5.5 Extended Phase Space
          o 5.5.1 Poincare-Cartan Integral Invariant
      + 5.6 Reduced Phase Space
      + 5.7 Summary
      + 5.8 Projects
  * 6 Canonical Evolution
      + 6.1 Hamilton-Jacobi Equation
          o 6.1.1 Harmonic Oscillator
          o 6.1.2 Hamilton-Jacobi Solution of the Kepler Problem
          o 6.1.3 F[2] and the Lagrangian
          o 6.1.4 The Action Generates Time Evolution
      + 6.2 Time Evolution is Canonical
          o 6.2.1 Another View of Time Evolution
          o 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
          o 7.2.1 Higher Order
          o 7.2.2 Eliminating Secular Terms
      + 7.3 Many Degrees of Freedom
          o 7.3.1 Driven Pendulum as a Perturbed Rotor
      + 7.4 Nonlinear Resonance
          o 7.4.1 Pendulum Approximation
          o 7.4.2 Reading the Hamiltonian
          o 7.4.3 Resonance-Overlap Criterion
          o 7.4.4 Higher-Order Perturbation Theory
          o 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

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