  The Hamilton-Jacobi equation in the sense of Poincar\'e, i.e. formulated in
the extended phase space and including regularization, is revisited building
canonical transformations with the purpose of Hamiltonian reduction. We
illustrate our approach dealing with orbital and attitude dynamics. Based on
the use of Whittaker and Andoyer symplectic charts, for which all but one
coordinates are cyclic in the Hamilton-Jacobi equation, we provide whole
families of canonical transformations, among which one recognizes the familiar
ones used in orbital and attitude dynamics. In addition, new canonical
transformations are demonstrated.
