Poincar\'e-like approach to Landau Theory. I. General theory

We discuss a procedure to simplify the Landau potential, based on Michel’s reduction to orbit space and Poincar'e normalization procedure; and illustrate it by concrete examples. The method makes use, as in Poincar'e theory, of a chain of near-identity coordinate transformations with homogeneous generating functions; using Michel’s insight, one can work in orbit space. It is shown that it is possible to control the choice of generating functions so to obtain a (in many cases, substantial) simplification of the Landau polynomial, including a reduction of the parameters it depends on. Several examples are considered in detail.