Introduction
Symplectic geometry is a branch of differential geometry that studies smooth manifolds equipped with a symplectic form—a closed, nondegenerate 2-form. This structure naturally arises in classical mechanics, where it serves as the mathematical foundation for Hamiltonian dynamics. In this setting, the symplectic form encodes information about conserved quantities and the evolution of systems over time. Unlike Riemannian geometry, which focuses on distances and angles, symplectic geometry is concerned with area-like properties and invariants under canonical transformations, leading to deep connections with dynamical systems, algebraic geometry, and even quantum mechanics.
Contents
Symplectic Structures
Poisson Bracket
Integrable Systems
Linear Symplectic Geometry
Local Forms
Symplectic Reduction
Toric Symplectic Manifold
Duistermaat-Heckman Theorems
Almost Complex Structures
Acknowledgements
This part is mainly based on a special topic course at ANU in 2025, convened by Dr. Brett Parker. We refer to the textbook Mathematical Methods of Classical Mechanics by V.I. Arnold.