Dual transformations in one-dimensional classical and quantum mechanics

Dual transformations in two-dimensional classical and quantum mechanical systems have been widely studied using conformal mapping techniques but one-dimensional systems have been largely ignored. In this paper we study dual transformations in one-dimensional mechanical systems, both classical and quantum mechanical, using some previously developed methods. A number of examples, mostly involving periodic motion or bound states, are presented. Dual transformations provide interesting connections between hitherto unconnected problems.