Exact oscillation-centre transformations

The topological constraints imposed on a canonical transformation by the requirement of continuous connection to the identity are considered. In the case of a particle interacting with a periodic potential, it is shown that a transformation to normal form (new Hamiltonian independent of position) is in general impossible using a transformation satisfying this requirement; the best that can be done being a transformation to a Hamiltonian which is almost everywhere in normal form, but which retains potential barriers of infinitesimal width to reflect trapped particles. A new Hamiltonian-Jacobi theory based on Lie operator techniques is presented and its relation to the usual theory established. It is suggested that the new method enables the motion of a particle in a random potential to be transformed into an almost Markovian random walk.