We discuss a procedure to simplify the Landau potential, based on Michel’s reduction to orbit space and Poincaré normalization procedure, and illustrate it by concrete examples. The method makes use, as in Poincaré 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.

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31.

In that paper, they will however be considered via a “brute force” approach; this will avoid to enter into the mathematical details needed for further normalization, which are therefore not discussed here.

32.

Note that if G is a continuous group, some of the x variables will actually be inessential physically, as can be quotiented out; in the present note, we only consider discrete groups, while in the companion paper8 we will meet this situation.

33.

Actually, their transformation can be described by means of the Baker-Campbell-Haussdorff formula;5 but this is inessential here.

34.

One could actually consider a “higher order normalization,” e.g., following the steps of Ref. 6, but we prefer not to enter into such details.

35.

With z = x + iy, these correspond to J1=z2, J2 = Re[z3], J3 = − Im[z3].

36.

Here and in the other examples, the generating function has a minus sign for convenience in writing the homological equation and the final results.

37.

If some term in ΦN is resonant and cannot be eliminated, then we should enter into details of the term, and see how we can guarantee convexity for large x.

38.

The nilpotent part could (and will most often) vanish, but we are not guaranteed this will be the case in general.

39.

Note again here we should not attempt to eliminate the maximal order terms as these are needed to guarantee the thermodynamic stability.

40.

It is necessary to consider this as being second order in ε for the method to be viable.21,22

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