Contents Preface Acknowledgments 1 Lagrangian Mechanics 1.1 The Principle of Stationary Action Experience of motion Realizable paths 1.2 Configuration Spaces 1.3 Generalized Coordinates Lagrangians in generalized coordinates 1.4 Computing Actions Paths of minimum action Finding trajectories that minimize the action 1.5 The Euler-Lagrange Equations Lagrange equations 1.5.1 Derivation of the Lagrange Equations Varying a path Varying the action Harmonic oscillator Orbital motion 1.5.2 Computing Lagrange's Equations The free particle The harmonic oscillator 1.6 How to Find Lagrangians Hamilton's principle Constant acceleration Central force field 1.6.1 Coordinate Transformations 1.6.2 Systems with Rigid Constraints Lagrangians for rigidly constrained systems A pendulum driven at the pivot Why it works More generally 1.6.3 Constraints as Coordinate Transformations 1.6.4 The Lagrangian Is Not Unique Total time derivatives Adding total time derivatives to Lagrangians Identification of total time derivatives 1.7 Evolution of Dynamical State Numerical integration 1.8 Conserved Quantities 1.8.1 Conserved Momenta Examples of conserved momenta 1.8.2 Energy Conservation Energy in terms of kinetic and potential energies 1.8.3 Central Forces in Three Dimensions 1.8.4 Noether's Theorem Illustration: motion in a central potential 1.9 Abstraction of Path Functions Lagrange equations at a moment 1.10 Constrained Motion 1.10.1 Coordinate Constraints Now watch this Alternatively The pendulum using constraints Building systems from parts 1.10.2 Derivative Constraints Goldstein's hoop 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 Representation of the Angular Velocity Vector Implementation of angular velocity functions 2.7 Euler Angles 2.8 Vector Angular Momentum 2.9 Motion of a Free Rigid Body Conserved quantities 2.9.1 Computing the Motion of Free Rigid Bodies 2.9.2 Qualitative Features of Free Rigid Body Motion 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 2.12 Euler's Equations Euler's equations for forced rigid bodies 2.13 Nonsingular Generalized Coordinates A practical matter Composition of rotations 2.14 Summary 2.15 Projects 3 Hamiltonian Mechanics 3.1 Hamilton's Equations Illustration Hamiltonian state Computing Hamilton's equations 3.1.1 The Legendre Transformation Legendre transformations with passive arguments Hamilton's equations from the Legendre transformation Legendre transforms of quadratic functions Computing Hamiltonians 3.1.2 Hamilton's Equations from the Action Principle 3.1.3 A Wiring Diagram 3.2 Poisson Brackets Properties of the Poisson bracket Poisson brackets of conserved quantities 3.3 One Degree of Freedom 3.4 Phase Space Reduction Motion in a central potential Axisymmetric top 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 Hénon-Heiles background The system of Hénon and Heiles Interpretation 3.6.4 Computing Hénon-Heiles Surfaces of Section 3.6.5 Non-Axisymmetric Top 3.7 Exponential Divergence 3.8 Liouville's Theorem The phase flow for the pendulum Proof of Liouville's theorem Area preservation of stroboscopic surfaces of section Poincaré recurrence The gas in the corner of the room Nonexistence of attractors in Hamiltonian systems Conservation of phase volume in a dissipative system Distribution functions 3.9 Standard Map 3.10 Summary 3.11 Projects 4 Phase Space Structure 4.1 Emergence of the Divided Phase Space Driven pendulum sections with zero drive Driven pendulum sections for small drive 4.2 Linear Stability 4.2.1 Equilibria of Differential Equations 4.2.2 Fixed Points of Maps 4.2.3 Relations Among Exponents Hamiltonian specialization Linear and nonlinear stability 4.3 Homoclinic Tangle 4.3.1 Computation of Stable and Unstable Manifolds 4.4 Integrable Systems Orbit types in integrable systems Surfaces of section for 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 Implementing point transformations 5.2 General Canonical Transformations 5.2.1 Time-Independent Canonical Transformations Harmonic oscillator 5.2.2 Symplectic Transformations 5.2.3 Time-Dependent Transformations Rotating coordinates 5.2.4 The Symplectic Condition 5.3 Invariants of Canonical Transformations Noninvariance of p v Invariance of Poisson brackets Volume preservation A bilinear form preserved by symplectic transformations Poincaré integral invariants 5.4 Extended Phase Space Restricted three-body problem 5.4.1 Poincaré-Cartan Integral Invariant 5.5 Reduced Phase Space Orbits in a central field 5.6 Generating Functions The polar-canonical transformation 5.6.1 F1 Generates Canonical Transformations 5.6.2 Generating Functions and Integral Invariants Generating functions of type F1 Generating functions of type F2 Relationship between F1 and F2 5.6.3 Types of Generating Functions Generating functions in extended phase space 5.6.4 Point Transformations Polar and rectangular coordinates Rotating coordinates Two-body problem Epicyclic motion 5.6.5 Classical ``Gauge'' Transformations 5.7 Time Evolution Is Canonical Liouville's theorem, again Another time-evolution transformation 5.7.1 Another View of Time Evolution Area preservation of surfaces of section 5.7.2 Yet Another View of Time Evolution 5.8 Hamilton-Jacobi Equation 5.8.1 Harmonic Oscillator 5.8.2 Kepler Problem 5.8.3 F2 and the Lagrangian 5.8.4 The Action Generates Time Evolution 5.9 Lie Transforms Lie transforms of functions Simple Lie transforms Example 5.10 Lie Series Dynamics Computing Lie series 5.11 Exponential Identities 5.12 Summary 5.13 Projects 6 Canonical Perturbation Theory 6.1 Perturbation Theory with Lie Series 6.2 Pendulum as a Perturbed Rotor 6.2.1 Higher Order 6.2.2 Eliminating Secular Terms 6.3 Many Degrees of Freedom 6.3.1 Driven Pendulum as a Perturbed Rotor 6.4 Nonlinear Resonance 6.4.1 Pendulum Approximation Driven pendulum resonances 6.4.2 Reading the Hamiltonian 6.4.3 Resonance-Overlap Criterion 6.4.4 Higher-Order Perturbation Theory 6.4.5 Stability of the Inverted Vertical Equilibrium 6.5 Summary 6.6 Projects 7 Appendix: Scheme Procedure calls Lambda expressions Definitions Conditionals Recursive procedures Local names Compound data -- lists and vectors Symbols 8 Appendix: Our Notation Functions Symbolic values Tuples Derivatives Derivatives of functions of multiple arguments Structured results Bibliography List of Exercises Index