PTCT

Physics Through Computational Thinking

Chapter 1: Visual Thinking

1.1 Introduction to Visual Thinking 1.2 Visually Understanding Functions 1.3 Vector Fields 1.4 Flow Visualization 1.5 Equipotentials and Field Lines 1.6 Non-Dimensionalization

Chapter 2: Data Analysis

2.1 Hello Data: Reading, Plotting, Saving 2.2 Random Variables & Combinations 2.3 From Samples to Populations 2.4 Galileo's Experiment 2.5 The χ² Criterion 2.6 Manual Parameter Exploration 2.7 Automated Fitting 2.8 Model Selection 2.9 Error Bars in Fitting (Optional) 2.10 The Superposition Principle 2.11 Same Frequency Interference 2.12 Beats: Near-Resonance Behavior 2.13 Lissajous Figures

Chapter 3: Oscillations

3.1 What Makes Things Oscillate? 3.2 The Math Behind the Motion 3.3 Energy and Phase Space 3.4 One Equation, Many Systems 3.5 Introducing Damping 3.6 The Three Regimes of Damping 3.7 Critical Damping in Engineering 3.8 Adding External Forces 3.9 Resonance: When Small Forces Win 3.10 Beats and Near-Resonance 3.11 Resonance Disasters and Designs 3.12 Numerical Methods: Euler's First Steps 3.13 Better Algorithms: Runge-Kutta

Chapter 4: Numerical Methods

4.1 Why Can't We Solve All Equations? 4.2 The Euler Method Formula 4.3 Programming Loops for Iteration 4.4 Coding Euler's Method 4.5 Testing and Validation 4.6 From Second-Order to First-Order 4.7 Solving Coupled Systems 4.8 LCR Circuits: Electrical Oscillations 4.9 Mechanical Oscillators with Damping 4.10 Error Analysis in Practice (Optional) 4.11 Improved Euler Method (RK2) 4.12 Runge-Kutta 4th Order (RK4) 4.13 Choosing Your Method 4.14 Escape Velocity: The Physics 4.15 Non-dimensionalization 4.16 RK4 for Escape Velocity 4.17 Adding Air Resistance (Optional)

Chapter 5: Dynamical Systems

5.1 What is Phase Space? 5.2 Vector Fields & Streamlines 5.3 Finding Fixed Points 5.4 The Big Three: Centers, Sources, Saddles 5.5 Trajectory Exploration 5.6 Romeo & Juliet: Love in Phase Space 5.7 Linear Systems & Eigenvalues 5.8 The τ-Δ Classification Framework 5.9 Damped Harmonic Oscillator Revisited 5.10 Cautious Lovers Model (Optional) 5.11 1D Linearization: The Basics 5.12 Jacobian Matrix for 2D Systems 5.13 Stability from Linearization 5.14 When Linearization Fails 5.15 Polar Coordinates (Optional)

Chapter 6: Monte Carlo

6.1 Why Use Randomness? 6.2 Monte Carlo Integration Basics 6.3 Error Analysis: The 1/√n Rule 6.4 Importance Sampling (Optional) 6.5 Estimating π: Throw Darts at a Circle 6.6 Probability via Sampling 6.7 Ising Model Introduction (Optional) 6.8 The Metropolis Algorithm 6.9 Ising Model Simulation & Results 6.10 Monte Carlo in Science (Optional)

Chapter 7: Random Walks

7.1 The Drunkard's Walk 7.2 Statistics of Random Walks 7.3 The Binomial Distribution 7.4 Biased Walks & Drift (Optional) 7.5 From Discrete to Continuous 7.6 The Diffusion Equation & Gaussian Solutions 7.7 Higher Dimensions: 2D & 3D Diffusion 7.8 Numerical Random Walk Simulations 7.9 Stirling's Approximation 7.10 Applications: Diffusion in the Real World

Chapter 8: Central Force

8.1 The Two-Body Problem Reduction 8.2 Conservation Laws: Energy & Angular Momentum 8.3 Polar Coordinates & The Radial Equation (Optional) 8.4 The Effective Potential 8.5 The Kepler Problem: Planetary Orbits 8.6 Numerical Integration of Orbits 8.7 Other Central Potentials (Optional)

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