ABSTRACT
We turn to the important section 4.3 ‘Geometric lemmas’ in Arnold (1963a) containing five lemmas, numbered G1 to G5 in the following. They mostly address the diffeomorphism A from momentum to frequency space, which is derived from the integrable part https://www.w3.org/1998/Math/MathML"> H + ( p ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003287803/a112f4d0-c5ab-4fbb-9ab7-9ffda30997aa/content/math6_1.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> of the Hamiltonian. The change in H + due to a perturbation causes a change of A to A ′, i.e., a shift in torus frequencies. The perturbed frequencies have to lie sufficiently close to the unperturbed frequencies, so that if the latter obey a Diophantine condition, the perturbed frequencies obey a slightly weaker Diophantine condition too.
