Generation of solutions of the Hamilton-Jacobi equation

Abstract: It is shown that any function $G(q_{i}, p_{i}, t)$, defined on the extended phase space, defines a one-parameter group of canonical transformations which act on any function $f(q_{i}, t)$, in such a way that if $G$ is a constant of motion then from a solution of the Hamilton-Jacobi (HJ) equation one obtains a one-parameter family of solutions of the same HJ equation. It is also shown that any complete solution of the HJ equation can be obtained in this manner by means of the transformations generated by $n$ constants of motion in involution.