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2001-01-01

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 はじめに (1/01) /Preface

 謝辞 /Acknowledgments

 1. ラグランジュ力学 (1/03) /Lagrangian Mechanics

  1.1 停留作用の原理 (1/03) /The Principle of Stationary Action

        運動に関する見識 /Experience of motion

        実現可能な経路 /Realizable paths

  1.2 配置空間 (1/08) /Configuration Spaces

  1.3 一般化座標 (1/08) /Generalized Coordinates

        一般化座標におけるラグランジアン /Lagrangians in generalized coordinates

  1.4 作用を計算する (1/09) /Computing Actions

        最小作用経路 /Paths of minimum action

        作用を最小化する軌道を探す /Finding trajectories that minimize the action

  1.5 オイラー・ラグランジュ方程式 (1/15) /The Euler-Lagrange Equations

        ラグランジュ方程式 /Lagrange equations

   1.5.1 ラグランジュ方程式の微分 /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:記法についての解説 (1/02) /Our Notation

        関数 /Functions

        Symbolic values

        タプル /Tuples

        微分 /Derivatives

        多変数関数微分 /Derivatives of functions of multiple arguments

        Structured results

 Bibliography

 List of Exercises

 Index