 Cotangent bundle

The cotangent bundle has the canonical one-form ${\lambda }_{(q,p)}(v)=p(d\pi (v))$ and the canonical symplectic form $d\lambda$ up to a conventional sign. Its zero section is a Lagrangian submanifold.

The cotangent bundle is a fundamental construction in differential geometry and symplectic geometry. It is particularly important in the study of manifolds and classical mechanics. Given a smooth manifold \( M \), the cotangent bundle, denoted \( T^*M \), is the vector bundle whose fibers at each point consist of the cotangent vectors (or covectors) at that point.