Nineteen Fifty-four: Kolmogorov’s new ‘metrical approach’ to Hamiltonian Dynamics

The paper states and proves Kolmogorov’s Theorem 2: for analytic near‑integrable Hamiltonians with nondegenerate twist on a bounded regular action domain, the measure of non–quasi‑periodic points tends to zero as ε→0 (Theorem 2 is stated explicitly, and the proof proceeds by a local reduction, a Diophantine set estimate meas(Ω\Ω_{γ,τ})≤cγ, a uniform smallness choice γ=ε^{1/5} tied to the KAM smallness condition, and a Lipschitz control of the Kolmogorov conjugacy to conclude the measure estimate) . The candidate solution proves the same statement using a modern parameter‑dependent KAM approach (Pöschel) to obtain a Whitney‑smooth Cantor foliation on compact interior action domains and a small complement of size O(γ), then choosing γ(ε) so that |ε|≲γ(ε)^α and removing a boundary layer. Both arguments are sound; they reach the same conclusion via different routes (the paper via Kolmogorov’s scheme plus Lipschitz bounds; the model via parameterized KAM and Cantor foliations).