gni-refactor
📐 Geometric Numerical Integration
A Rigorous Extended Introduction
Based on Blanes & Casas (2026), with Expanded Exposition
🌟 Overview
This repository contains a rigorous, self-contained introduction to Geometric Numerical Integration (GNI). It is designed to bridge the gap between elementary principles and graduate-level understanding, unifying geometric, algebraic, and analytic perspectives.
The content has been refactored from a single monolithic document into a modular, multi-chapter structure for better readability and maintainability.
📚 Table of Contents
The material is organized into the following modules. You can start reading from the Main Table of Contents.
| Chapter | Title | Status |
|---|---|---|
| 01 | Foundations of GNI | ✅ Complete |
| 02 | Symplectic Geometry & Hamiltonian Dynamics | ✅ Complete |
| 03 | Symplectic Integrators | ✅ Complete |
| 04 | Splitting & Composition Methods | ✅ Complete |
| 05 | Exponential Integrators & Magnus Expansions | ✅ Complete |
| 06 | Backward Error Analysis (BEA) | ✅ Complete |
| 07 | Non-Autonomous Systems | 🚧 Placeholder |
| 08 | Lie-Group Methods | 🚧 Placeholder |
| 09 | Geometric Methods for PDEs | 🚧 Placeholder |
| 10 | Applications (HMC, Quantum, N-Body) | 🚧 Placeholder |
| 11 | Advanced Topics | 🚧 Placeholder |
| 12 | Appendix A: Mathematical Tools | 🚧 Placeholder |
| App | Appendix B: Brachistodynamic Holonomy | ✅ Complete |
🚀 Key Features
- Structure-Preserving: Focuses on methods that preserve symplectic forms, invariants, and symmetries.
- Interactive: Includes JavaScript-based visualizations for key concepts (e.g., phase portraits, energy drift).
- Modern: Covers recent developments in non-autonomous systems and applications to quantum computing.
🛠️ Usage
Option 1: View Online (Recommended)
Once GitHub Pages is enabled and DNS propagates, you can view the rendered book here: https://gni.walkerjian.com/toc.html
Option 2: Run Locally
# Clone the repository
git clone https://github.com/walkerjian/gni-refactor.git
# Open the Table of Contents
open toc.html
⚙️ Setup GitHub Pages
To view the rendered HTML on GitHub:
- Go to Settings > Pages in your repository.
- Under Source, select
Deploy from a branch. - Under Branch, select
mainand/ (root). - Click Save.
Maintained by walkerjian