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
                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
                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
            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
        5.10  Lie Series
                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
                Recursive procedures
                Local names
                Compound data -- lists and vectors

    8  Appendix: Our Notation
                Symbolic values
                Derivatives of functions of multiple arguments
                Structured results


    List of Exercises