Symplectic Manifolds

A symplectic manifold is a mathematical space used to describe systems in classical physics, especially those involving motion and energy, like planets or particles. It’s a smooth, even-dimensional space equipped with a special structure called a symplectic form—a non-degenerate, closed 2-form—that allows the measurement of areas in a way that respects the system’s geometric properties. This structure provides a framework for understanding how systems evolve over time, emphasizing the geometric relationships between position and momentum, and is fundamental in the study of Hamiltonian mechanics.