A symplectic manifold is a special type of geometric space used in mathematics and physics, particularly in the study of classical mechanics. It is defined as a smooth manifold equipped with a closed, non-degenerate 2-form, known as the symplectic form. This structure allows for the formulation of Hamiltonian mechanics, where the symplectic form encodes information about the system's phase space. In a symplectic manifold, the symplectic form provides a way to measure areas and volumes, which is crucial for understanding the dynamics of systems. Key concepts related to symplectic manifolds include Hamiltonian systems, Lagrangian submanifolds, and Poisson brackets, all of which play significant roles in the mathematical framework of mechanics and dynamical systems.