The Computational Pendulum: From Classical Mechanics to Neural Networks

The simple pendulum is usually relegated to the first few weeks of an introductory physics course, quickly simplified with the small-angle approximation and left behind. However, when you strip away that approximation, the pendulum reveals itself as a rich theoretical and computational laboratory.

The Computational Pendulum takes you on a progressive journey from AP-level mechanics to modern machine learning. By keeping the physical system constant, you can master advanced mathematical, numerical, and AI techniques without the cognitive overload of constantly changing theoretical contexts.

• Classical Mechanics & Exact Mathematics: Start with kinematics, Newton's laws, and rigid-body dynamics, then push beyond standard textbooks into exact large-angle solutions using Jacobi elliptic functions, phase portraits, and exact Fourier power spectra.
• Computational Physics in Python: Continuous periodic systems accumulate numerical errors rapidly. Learn how to simulate dynamical systems accurately using 4th-order Runge-Kutta (RK4) integration, event-driven solvers, and Fast Fourier Transforms (FFT).
• Machine Learning for Physics: Use the pendulum as a transparent baseline for modern AI. Train Physics-Informed Neural Networks (PINNs) in JAX, explore Koopman operator theory, and build Physical Reservoir Computing models.

Each theory chapter is tightly integrated with companion Python scripts and is immediately followed by a dedicated chapter of 100% fully solved exercises that act as an open-ended laboratory.

• Part I: Foundations and Simulations (Available Now) — Bridges introductory mechanics, rigid-body dynamics, and energy conservation into exact analytical solutions, Fourier series, and foundational Python visualizations.
• Part II: Exact Solutions and Machine Learning (In Progress) — Introduces heavy numerical integration (RK4) for non-idealized systems and takes the first leap into machine learning by training neural networks to discover physical laws.
• Part III: Explorations in Modern Dynamics (In Progress) — Investigates non-inertial reference frames, parametric pumping, Koopman theory, and Reservoir Computing in an open-ended research sandbox.

Leanpub Early-Access Pricing Advantage: This textbook is being published iteratively and will reach approximately 600 pages ($69.99 final retail price) upon completion. Purchasing the early-access digital edition today locks in a lower introductory price and gives you immediate access to Part I, along with 100% free lifetime updates as Parts II and III are released and the price increases.

Every script referenced in the text is open-source and available on the Companion Website & Code Directory. You can view the raw Python files on GitHub or launch them with a single click in Google Colab—no local installation or setup required.

• Physics Educators: Teachers of AP Physics or introductory university mechanics looking for rigorous, ready-to-use honors extensions, computational labs, or student capstone projects anchored to a familiar curriculum topic.
• Students & Self-Learners: Readers who want to see every step of a derivation and learn how textbook equations translate into working Python simulations.
• Developers & Data Scientists: Programmers and ML practitioners seeking a focused, math-first bridge into computational physics, differential equations, and scientific machine learning (SciML).