Kolmogorov variation: KAM with knobs (`a la Kolmogorov)

In this paper we reconsider the original Kolmogorov normal form algorithm with a variation on the handling of the frequencies. At difference with respect to the Kolmogorov approach, we do not keep the frequencies fixed along the normalization procedure. Besides we select the frequencies of the final invariant torus and determine a posteriori the corresponding starting one. In particular, we replace the classical translation step with a change of the frequencies. The algorithm is based on classical expansion in a small parameter and particular attention is paid to the constructive aspect: we produce an explicit algorithm that can be effectively applied, e.g., with the aid of an algebraic manipulator, and that we prove to be absolutely convergent.