Symplectic manifold

A symplectic manifold is a mathematical structure used to describe the geometry of systems in classical mechanics, especially phase space where positions and momenta are tracked. It is a smooth space equipped with a special, non-degenerate, closed 2-form called the symplectic form, which encodes information about how these physical quantities relate and evolve over time. Think of it as a geometric framework that preserves the structure of physical laws during transformations, allowing precise modeling of motion and dynamics in complex systems. It provides the foundation for understanding how systems change and conserve properties like energy.