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In this paper we reconsider the original Kolmogorov normal form algorithmwith a variation on the handling of the frequencies. At difference with respectto the Kolmogorov approach, we do not keep the frequencies fixed along thenormalization procedure. Besides, we select the frequencies of the finalinvariant torus and determine a posteriori the corresponding starting ones. Inparticular, we replace the classical translation step with a change of thefrequencies. The algorithm is based on the original scheme of Kolmogorov, thusexploiting the fast convergence of the Newton-Kantorovich method.

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