Lie-Poincare' transformations and a reduction criterion in Landau theory

In the Landau theory of phase transitions one considers an effective potential U whose symmetry group G and degree d depend on the system under consideration; generally speaking, U is the most general G-invariant polynomial of degree d. When such a U turns out to be too complicate for a direct analysis, it is essential to be able to drop unessential terms, i.e., to apply a simplifying criterion. Criteria based on singularity theory exist and have a rigorous foundation, but are often very difficult to apply in practice. Here we consider a simplifying criterion (as stated by Gufan) and rigorously justify it on the basis of classical Lie-Poincaré theory as far as one deals with fixed values of the control parameter(s) in the Landau potential; when one considers a range of values, in particular near a phase transition, the criterion has to be accordingly partially modified, as we discuss. We consider some specific cases of group G as examples, and study in detail the application to the Sergienko–Gufan–Urazhdin model for highly piezoelectric perovskites.