内容简介

This textbook takes an innovative approach to the teaching of classical mechanics, emphasizing the development of general but practical intellectual tools to support the analysis of nonlinear Hamiltonian systems. The development is organized around a progressively more sophisticated analysis of particular natural systems and weaves examples throughout the presentation. Explorations of phenomena such as transitions to chaos, nonlinear resonances, and resonance overlap to help the student to develop appropriate analytic tools for understanding. Computational algorithms communicate methods used in the analysis of dynamical phenomena. Expressing the methods of mechanics in a computer language forces them to be unambiguous and computationally effective. Once formalized as a procedure, a mathematical idea also becomes a tool that can be used directly to compute results.The student actively explores the motion of systems through computer simulation and experiment. This active exploration is extended to the mathematics. The requirement that the computer be able to interpret any expression provides strict and immediate feedback as to whether an expression is correctly formulated. The interaction with the computer uncovers and corrects many deficiencies in understanding.

作品目录

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