1. Introduction
1.1 The beginning of dynamics

1.2 From calculus to chaos

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2. A brief review of calculus

2.1 Introduction

2.2 Some elementary results

2.3 Taylor series

2.4 The functions exp x and log x

Exercises

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3. Ordinary differential equations

3.1 Introduction

3.2 First-order linear equations

3.3 First-order nonlinear equations

3.4 Second-order linear equations

3.5 Second-order nonlinear equations

3.6 Phase space

Exercises

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4. Computer solution methods

4.1 Introduction

4.2 Euler's method

4.3 Computer implementation of Euler's method

4.4 Systems of differential equations

4.5 More accurate step-by-step methods

Exercises

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5. Elementary oscillations

5.1 Introduction

5.2 The linear oscillator

5.3 Multiple modes of oscillation

5.4 Coupled oscillators

5.5 Nonlinear oscillations

Exercises

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6. Planetary motion

6.1 Introduction

6.2 Equations of motion under a central force

6.3 The equal-area rule

6.4 Differential equation for the orbit

6.5 Orbits under an inverse-square law

6.6 A numerical approach

6.7 The two-body problem

6.8 The three-body problem

Exercises

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7. Waves and diffusion

7.1 Introduction

7.2 Wave motion

7.3 The diffusion equation

Exercises

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8. The best of all possible worlds?

8.1 Introduction

8.2 The concept of 'action'

8.3 The calculus of variations

8.4 Lagrange's equations of motion

Exercises

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9. Fluid flow

9.1 Introduction

9.2 The geometry of fluid motion

9.3 The equations of viscous flow

9.4 Very viscous flow

9.5 The case of small viscosity

Exercises

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10. Instability and catastrophe

10.1 Introduction

10.2 Linear stability theory

10.3 Multiple solutions; bifurcation

10.4 Sudden changes of state

10.5 Imperfection and catastrophe

10.6 Instability of motion

Exercises

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11. Nonlinear oscillations and chaos

11.1 Introduction

11.2 Limit cycles; the van der Pol equation

11.3 Conditions for chaos

11.4 The Lorenz equations

11.5 Chaotic mixing: stretch and fold

11.6 One route to chaos: period-doubling

11.7 Multiple solutions and 'jumps'

Exercises

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12. The not-so-simple pendulum

12.1 Introduction

12.2 Pendulums from the past

12.3 A vibrated pendulum

12.4 Chaotic pendulums

12.5 Not quite the Indian Rope Trick

Exercises

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Further reading
Appendix A: Elementary programming in QBasic

A.1 Introduction

A.2 Getting started

A.3 Mathematical variables, operations and functions

A.4 Program loops

A.5 Graphics

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Appendix B: Ten programs for exploring dynamics
Solutions to the exercises

Index

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"From Calculus to Chaos" page