Generating function of a canonical transformation

Summary

A generating function of a canonical transformation is a single function of half of the old and half of the new variables. The whole transformation is obtained from it by partial derivatives. It gives a convenient way to write down canonical transformations, since any nondegenerate function produces one. There are four standard types, according to which variables it depends on. The solution of the Hamilton–Jacobi equation is a generating function that makes the new Hamiltonian zero. See the definition.