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With a view to apply it to the secular evolution of rotating stars, we base it on the Lagrangian description, i.e. we solve the force-balance equations\u00a0to find the spatial positions of fluid elements endowed individually with a mass, specific entropy and angular momentum. The system of non-linear equations\u00a0obtained by finite-differencing the basic equations\u00a0is solved with the W4 method, which is a new multidimensional root-finding scheme of our own devising. We augment it with a remapping scheme to avoid distortions of the Lagrangian coordinates. In this first one of a series of papers, we will give a detailed description of these methods initially. We then present the results of some test calculations, which include the construction of both rapidly rotating barotropic and baroclinic equilibrium states. We gauge their accuracies quantitatively with some diagnostic quantities as well as via comparisons with the counterparts obtained with an Eulerian code. 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