Geometric Numerical Integration — Rigorous Extended Introduction

Based on Blanes & Casas (2026), with Expanded Exposition

Table of Contents

• 1. Foundations of Geometric Numerical Integration
  • 1.1 What is Geometric Numerical Integration?
  • 1.2 Fundamental Objects in Differential Equations
  • 1.3 The Geometry of Flows
• 2. Symplectic Geometry and Hamiltonian Dynamics
  • 2.1 Symplectic Manifolds
  • 2.2 Hamiltonian Vector Fields and Poisson Brackets
  • 2.3 Symmetries, Momentum Maps, and Noether’s Theorem
  • 2.4 Poisson Manifolds and Poisson Brackets
  • 2.5 Darboux’s Theorem and Canonical Coordinates
  • 2.6 Lie–Poisson Reduction and Coadjoint Orbits
• 3. Symplectic Integrators for Hamiltonian Systems
  • 3.1 Why Symplectic Integrators?
  • 3.2 Symplectic Euler, Störmer–Verlet, and Basic Symplectic Schemes
  • 3.3 Variational Integrators and Discrete Lagrangian Mechanics
  • 3.4 Splitting Methods and Composition Techniques
  • 3.5 Backward Error Analysis and Modified Equations
  • 3.6 Symmetric Methods, Processing, and Modified Flow Techniques
  • 3.7 Symplectic Runge–Kutta and Partitioned Runge–Kutta Methods
  • 3.8 Energy-Preserving and Volume-Preserving Integrators
• 4. Splitting and Composition Methods
  • 4.1 Lie–Trotter and Strang Splitting
  • 4.2 Higher-Order Splitting via Composition
  • 4.3 Optimised Splitting Methods
  • 4.4 Splitting Methods for Time-Dependent and Non-Autonomous Systems
  • 4.5 High-Order Composition & Optimised Non-Autonomous Schemes
• 5. Exponential Integrators and Magnus Expansions
  • 5.1 Symplectic Runge–Kutta Methods
  • 5.2 Algebraic Order Conditions and B-Series for Symplectic RK Methods
  • 5.3 Symmetric, Symplectic, and Energy-Preserving RK Methods
  • 5.4 Partitioned and Symplectic Partitioned Runge–Kutta Methods
  • 5.5 Exponential Runge–Kutta (ERK), Lawson, and Integrating Factor Methods
  • 5.6 Magnus–Runge–Kutta, Commutator-Free Magnus Methods, and Lie–Algebraic Exponential Integrators
• 6. Backward Error Analysis (BEA)
  • 6.0 Overview: Why Splitting & Composition Methods?
  • 6.1 Lie and Strang Splitting
  • 6.2 Symmetric Composition Methods: From Strang to High Order
  • 6.3 Optimised High-Order Splitting Methods: Real & Complex Coefficients
  • 6.4 Backward Error Analysis for Splitting Methods
  • 6.5 Splitting Methods for PDEs
  • 6.6 Relation of Splitting to Magnus Methods and Exponential Integrators
• 7. Geometric Integrators for Non-Autonomous Systems (NEW) [Placeholder]
• 8. Lie-Group Methods and Manifold Integrators [Placeholder]
• 9. Geometric Methods for PDEs (EXPANDED in 2nd Edition) [Placeholder]
• 10. Applications Highlighted in 2nd Edition [Placeholder]
• 11. Advanced Topics and Research Directions [Placeholder]
• 12. Appendix A: Mathematical Tools [Placeholder]
• Appendix B: Brachistodynamic Holonomy