  This paper contains results on geometric Routh reduction and it is a
continuation of a previous paper where a new class of transformations is
introduced between Lagrangian systems obtained after Routh reduction. In
general, these reduced Lagrangian systems have magnetic force terms and are
singular in the sense that the Lagrangian does not depend on some velocity
components. The main purpose of this paper is to show that the Routh reduction
process itself is entirely captured by the application of such a new
transformation on the initial Lagrangian system with symmetry.
