A symplectic manifold is a special type of geometric space that arises in the field of mathematics, particularly in the study of Hamiltonian mechanics. It is defined by a symplectic form, which is a non-degenerate, closed differential 2-form. This structure allows for the definition of concepts like area and volume in a way that is compatible with the dynamics of physical systems. Symplectic manifolds are essential in understanding the behavior of systems in classical mechanics and are closely related to phase space, where the positions and momenta of particles are represented. They provide a framework for analyzing how systems evolve over time, making them a fundamental concept in both mathematics and physics.