Any cotangent bundle is endowed with a canonical structure of exact symplectic manifold. Then it becomes “natural” to apply what we have designed for the symplectic manifolds in general in the previous chapter to the special case of cotangent bundles. The material associated with this reduction becomes very rich. Among the various terms that arise spontaneously, the main one is the “generating function” of a Lagrangian submanifold, which is extended to the more general notion of “generating family”. This notion is in fact the
around which the entire analysis of the following chapters is built up.