Symplectic geometry is a branch of mathematics that studies geometric structures called symplectic manifolds. These manifolds are equipped with a closed, non-degenerate 2-form, which allows for the definition of concepts like area and volume in a way that is compatible with the underlying geometry. This field is particularly important in understanding the mathematical foundations of classical mechanics. In symplectic geometry, the focus is often on the relationships between geometry and physics, especially in areas like Hamiltonian mechanics. The symplectic structure provides a framework for analyzing the motion of systems and their conserved quantities, making it a vital tool in both mathematics and theoretical physics.