# Python for Classical Mechanics

**Newly Released**

Designed to accompany John Taylor’s internationally best-selling Classical Mechanics, this text provides a series of interactive computational exercises in Python that analyze classical mechanical systems from both analytical and numerical perspectives. No pre-existing experience with Python is required, as this book integrates scientific programming instruction directly into the standard undergraduate classical mechanics physics course.

### Summary

Contemporary physics relies heavily on computer programming for analyzing data and modeling systems, yet time constraints often prevent undergraduate physics students from taking the computer science courses needed to develop these skills. This textbook integrates scientific programming instruction directly into a standard undergraduate classical mechanics physics course. Built to accompany John Taylor’s popular Classical Mechanics, but compatible with other books, Python for Classical Mechanics provides a series of interactive Python computational exercises that analyze classical mechanical systems from both analytical and numerical perspectives. The exercises guide students chapter-by-chapter through modeling classical physics systems such as the simple pendulum at high angle, two or more gravitational bodies in orbit, and damped, driven oscillators leading to period-doubling and chaos.

Using guided instruction in critical programming techniques such as loops, logic, array manipulation, numerical integration, and data analysis and plotting to help intermediate physics, students gain proficiency in both analytical and computational methods. This book assumes no prior knowledge of programming on the part of the student and includes step-by-step instructions for starting the student programming in Python with the interactive Jupyter Notebook interface.

### Table of Contents

Preface

Acknowledgements

1 Introduction

1.2 Who Is This Book For?

1.3 A Word of Encouragement to the Student

2 How To Use This Book

2.2 Scheduling

2.3 Working Through the Units

2.4 Relationship between Units

3 Description of Units

Introduction to Python

Taylor Series with Loops and Functions

Numerical Integration Applied to Projectile Motion

Projectile Motion with Drag

Launching a Rocket

Simple Pendulum with Large Angle Release

Comparing Data and Theory for Simple Pendulum

Oscillations in a Potential Well

Damped and Undamped Harmonic Oscillators

Driven Damped Harmonic Oscillator and Resonance

Brachistochrone Problem

Spherical Pendulum

Three-Body Problem

Orbits, Keplerian and Not

Motion on a Turntable

Coriolis Force on Earth

Principal Axes of a Cuboid

Precession of a Cuboid

Masses Connected with Springs

Damped Driven Pendulum

Bifurcation Diagram

State-Space Orbits and Poincaré Sections

Appendix 1: Using solve_ivp for Numerical Integration

Appendix 2: Basic Input/Output of Files in Python

Appendix 3: Creating Animations with FuncAnimation

4 Preparing to Use Jupyter Notebooks with Python

4.1 Installation on a Personal Computer/p>

4.1.2 Running Jupyter Notebook

4.1.3 Closing Jupyter Notebook

4.2 Using a Web Server to Run Jupyter Notebooks

4.2.2 Google Colab

5 Introduction to Python

5.1 Objectives

5.2 Working with Jupyter Notebook

5.2.2 The kernel

5.3 Introduction to Python Programming

5.3.2 Variables

5.3.3 Using Python for calculations

5.3.4 Order of operations, and left-right associativity

5.3.5 Data types

5.3.6 Formatting output

5.4 Lists and Arrays

5.4.2 Indexing and slicing

5.4.3 An aside on additional data types

5.5 Plotting

5.6 Check-out

6 Taylor Series with Loops and Functions

6.1 Objectives

6.2 Taylor Series

6.2.2 Taylor series for sin(x)

6.2.3 Taylor series about an arbitrary x value

6.3 New Programming Tools

6.3.2 Conditional statements

6.3.3 Loops

6.3.4 An aside on speed

6.3.5 Functions and other subroutines

6.4 Taylor Series using Loops, Functions, and Conditional Statements

6.5 Check-out

7 Numerical Integration Applied to Projectile Motion

7.1 Objectives

7.2 Simple Euler Integration of 2-D Projectile Motion

7.2.2 Accuracy of simple Euler integration

7.3 Improving the Simple Euler Integration Method: Euler Half-Step Integration

7.3.2 Applying improved Euler to projectile motion

7.4 Check-out

7.5 Challenge Problem

8 Projectile Motion with Drag

8.1 Objectives

8.2 Improved Euler Numerical Integration

8.3 Projectile Motion with Linear Drag

8.4 Quadratic Drag

8.4.2 Energy loss in one-dimensional motion

8.5 Linear and Quadratic Drag

8.6 Check-out

8.7 Challenge Problem

9 Launching a Rocket

9.1 Objectives

9.2 Motion of Rocket without Gravity

9.3 Introducing solve_ivp for a Rocket

9.3.2 Finding the solution with solve_ivp()

9.3.3 Accessing the solution from solve_ivp

9.3.4 Comparison between solve_ivp() and Improved Euler

9.4 Motion of a Rocket in a Constant Gravitational Field

9.4.2 Computational solution

9.5 Launching to the International Space Station

9.6 Check-out

9.7 Challenge Problem

10 Simple Pendulum with Large-Angle Release

10.1 Objectives

10.2 Simple Pendulum, Theory

10.2.2 Simple pendulum equation of motion

10.3 Numerical Solution for θ(t)

10.3.2 Finding the solution with solve_ivp()

10.4 Comparing Solutions for Simple Pendulum at Different Initial Angles

10.5 Period of Simple Pendulum as a Function of Release Angle

10.6 Check-out

10.7 Challenge Problems

11 Comparing Data and Theory for Simple Pendulum

11.1 Objectives

11.2 Computational Solution

11.3 Analytical Solution

11.4 Experimental Solution

11.4.2 Experimental data

11.5 Check-out

12 Oscillations in a Potential Well

12.1 Objectives

12.2 Potential Energy Function for an Oscillating Wheel

12.3 Analytical Approximation Using Taylor Series

12.3.2 Theoretical analysis of oscillations

12.4 Computational Solution for φ(t)

12.4.2 Computational solution for different mass ratios

12.4.3 Computational solution for different release angles

12.5 Check-out

12.6 Challenge Problems

13 Damped and Undamped Harmonic Oscillators

13.1 Objectives

13.2 Harmonic Motion without Damping

13.2.2 Numerical solution

13.3 Damped Harmonic Oscillator

13.3.2 Case 2: Large damping, β2 > ω20

13.3.3 Case 3: Critical damping, β2 = ω20

13.3.4 Case 4: Underdamping, β2 < ω20

13.4 Check-out

13.5 Challenge Problems

14 Driven Damped Harmonic Oscillator and Resonance

14.1 Objectives

14.2 Motion of a Driven Damped Harmonic Oscillator

14.3 Amplitude of Driven Damped Oscillations at Resonance

14.4 Phase of Driven Damped Oscillations

14.5 The Phase Dependence

14.6 Amplitude as a Function of Driver Frequency

14.7 Creating a Numerical Resonance Curve by Plotting the Maximum Amplitude vs. Frequency

14.7.2 Now for the geometric interpretation

14.8 Check-out

14.9 Challenge Problems

14.9.2 Challenge Problem 2, Effect of β on the resonance curve

14.9.3 Challenge Problem 3, Multiprocessing for multiple resonance curves

14.10 Contents of resonance_for_multi_processing_solvers.py

15 Brachistrochrone Problem

15.1 Objectives

15.2 Introducing the Cycloid

15.3 Cycloidal Path Length

15.4 Travel Time

15.4.2 Time taken for an arbitrary starting angle

15.4.3 Numerical solution

15.5 An Animation

15.6 Check-out

15.7 Challenge Problem

16 Spherical Pendulum

16.2 Equations of Motion for a Spherical Pendulum

16.3 Validating Your Code with Special Cases

16.4 Varying Initial Velocity

16.5 Conservation of Generalized Momentum

16.6 Near-Conical Motion

16.7 Check-out

16.8 Challenge Problem

17 The Three-Body Problem

17.1 Objectives

17.2 Solving the Two-Body Problem Numerically for a Circular Orbit

17.3 Numerical Integration of the Three-Body Problem

17.3.2 Mystery system

17.3.3 Plotting in the center-of-mass frame of reference

17.4 Making a Movie

17.5 Three Dimensions

17.6 Check-out

17.7 Challenge Problem

18 Orbits, Keplerian and Not

18.1 Objectives

18.2 Circular Keplerian Orbit (ϵ = 0)

18.3 Numerical Solution

18.4 An Elliptical Orbit

18.5 Hyperbolic Orbits

18.6 Comparing with Pluto Ephemeris

18.7 Non-Keplerian Orbits

18.8 Check-out

18.9 Challenge Problems

19 Motion on a Turntable

19.1 Objectives

19.2 Preliminary Questions

19.3 Framing the Problem

19.3.2 Setting the initial conditions and other physical parameters

19.3.3 Numerically solving the equations of motion

19.3.4 Comparing the trajectory in the rotating reference frame to that in a fixed inertial reference frame

19.3.5 Explore different initial conditions

19.4 Animating the Trajectory

19.5 Check-out

19.6 Challenge Problems

20 Coriolis Force on Earth

20.1 Objectives

20.2 Equations of Motion

20.3 Solving the Equations of Motion

20.3.2 Solving the equations of motion for a set of initial conditions

20.3.3 Plotting on the surface of a sphere

20.4 Verifying Your Code

20.5 Explore Different Initial Conditions

20.6 Integrate with Realistic Parameters

20.7 Check-out

20.8 Challenge Problems

21 Principal Axes of a Cuboid

21.1 Objectives

21.2 Coding Techniques Preamble

21.2.2 Classes

21.3 Moment of Inertia Tensor and Angular Momentum

21.3.2 Creating a class

21.4 Finding the Principal Axes

21.5 Check-out

21.6 Challenge Problem

22 Precession of a Cuboid

22.1 Objectives

22.2 Creating and Importing Modules

22.3 Euler’s Equations

22.4 Check-out

22.5 Template Code for principal_axes.py

23 Masses Connected with Springs

23.1 Objectives

23.2 Two Carts Connected and Attached between Two Walls by Three Springs

23.3 Identical Masses and Springs

23.3.2 Numerical solution

23.3.3 Solving the eigenvalue problem numerically

23.4 Weak Coupling, an Illustration of Beats

23.5 Fourier Analysis

23.6 Check-out

23.7 Challenge Problems

24 Damped Driven Pendulum

24.1 Objectives

24.2 Equation of Motion

24.3 Numerical Solution

24.3.2 Numerically solving the equation of motion

24.3.3 Coding style

24.4 Behavior with Increasing Values of Driving

24.4.2 Intermediate and strong driving

24.5 Sensitivity to Initial Conditions

24.6 Check-out

24.7 Challenge Problem

25 Bifurcation Diagram

25.2 Setup

25.3 One-Point Bifurcation Diagram

25.4 Creating the Full Bifurcation Diagram

25.5 Features of the Bifurcation Diagram

25.6 Class Parallel Programming Project

25.7 Check-out

25.8 Challenge Problem

25.9 Template Code for damped_driven_pendulum.py

26 State-Space Orbits and Poincaré Sections

26.2 Setup

26.3 State-Space

26.4 Poincaré Sections

26.5 Strange Attractors

26.6 Check-out

27 Appendix 1: Using solve_ivp for Numerical Integration

27.1 Using solve_ivp()

27.1.2 Accuracy

27.1.3 Checking for events

28 Appendix 2: Basic Input/Output of Files in Python

28.1 Read/Write to a File Handle

28.1.2 Read

28.2 Using NumPy for Reading and Writing Data Arrays

28.3 Pandas

28.4 Binary Data

28.4.2 Reading binary data

29 Appendix 3: Creating Animations with FuncAnimation

29.1 Overview

29.2 Walk-through Example

29.2.2 Frame functions

29.3 Call the Animator

Index