Luigi Chierchia

This paper continues the discussion started in [10] concerning Arnold's legacy on classical KAM theory and (some of) its modern developments. We prove a detailed and explicit ''global'' Arnold's KAM theorem, which yields, in particular, the Whitney conjugacy of a non-degenerate, real-analytic, nearly-integrable Hamiltonian system to an integrable system on a closed, nowhere dense, positive measure subset of the phase space. Detailed measure estimates on the Kolmogorov set are provided in case the phase space is: (A) a uniform neighbourhood of an arbitrary (bounded) set times the $d$-torus and (B) a domain with $C^2$ boundary times the $d$-torus. All constants are explicitly given.

We review V.I. Arnold's 1963 celebrated paper [1] Proof of A.N. Kolmogorov's Theorem on the Conservation of Conditionally Periodic Motions with a Small Variation in the Hamiltonian, and prove that, optimising Arnold's scheme, one can get ''sharp'' asymptotic quantitative conditions (as $\varepsilon \to 0$, $\varepsilon$ being the strength of the perturbation). All constants involved are explicitly computed.

Following closely Kolmogorov’s original paper [1], we give a complete proof of his celebrated Theorem on perturbations of integrable Hamiltonian systems by including few "straightforward" estimates.

Birkhoff periodic orbits associated to spin-orbit resonances in Celestial Mechanics and in particular to the Moon–Earth and Mercury–Sun systems are considered. A general method (based on a quantitative version of the Implicit Function Theorem) for the construction of such orbits with particular attention to "effective estimates" on the size of the perturbative parameters is presented and tested on the above mentioned systems. Lyapunov stability of the periodic orbits (for small values of the perturbative parameters) is proved by constructing KAM librational invariant surfaces trapping the periodic orbits.