Symplectic manifold

Summary

A symplectic manifold is a smooth manifold with a closed nondegenerate $2$-form, its symplectic form. It is the geometric setting of Hamiltonian mechanics: phase spaces of mechanical systems are symplectic manifolds. By Darboux's theorem all symplectic manifolds of the same dimension look alike near a point, so the subject is global and close to topology. Each tangent space is a symplectic vector space, so the dimension is even.