ABSTRACT
Canonical transformations form a group. A given canonical transformation can be derived by differentiation with respect to either set of complementary variables. Transformations in the infinitesimal neighborhood of the identity - infinitesimal canonical transformations - must be described by an action operator that differs infinitesimally from the one producing the identity transformation. Differently oriented coordinate systems are intrinsically equivalent and should expect that the kinematical term in the Lagrangian operator presents the same appearance in terms of the variables appropriate to any coordinate system. With a single parameter t and generator -H, the auuthor regain the original action principle, appearing as the characterization of a canonical transformation - the Hamiltonian - Jacobi transformation - from a description at time t to the analogous one referring to another time t0. For a canonical transformation involving several parameters, the last term of displays the effect of altering the path along which the parameters evolve.
