Symplectic geometry is a branch of mathematics that studies geometric structures on smooth manifolds, focusing on symplectic manifolds. A symplectic manifold is a smooth, even-dimensional space equipped with a closed, non-degenerate 2-form called a symplectic form. This structure allows for the analysis of systems in classical mechanics, where it provides a natural framework for understanding the phase space of dynamical systems. In symplectic geometry, the concepts of Hamiltonian mechanics and Lagrangian mechanics are often explored. The symplectic form encodes information about the conservation laws and dynamics of physical systems, making it essential for understanding the behavior of physical systems in both classical and modern physics.