Luigi Chierchia

We review Kolmogorov's 1954 fundamental paper On the persistence of conditionally periodic motions under a small change in the Hamilton function (Dokl. akad. nauk SSSR, 1954, vol. 98, pp. 527–530), both from the historical and the mathematical point of view. In particular, we discuss Theorem 2 (which deals with the measure in phase space of persistent tori), the proof of which is not discussed at all by Kolmogorov, notwithstanding its centrality in his program in classical mechanics.
In Appendix, an interview (May 28, 2021) to Ya. Sinai on Kolmogorov’s legacy in classical mechanics is reported.

In this note, we discuss the topology of Diophantine numbers, giving simple explicit examples of Diophantine isolated numbers (among those with the same Diophantine constants), showing that Diophantine sets are not always Cantor sets.
General properties of isolated Diophantine numbers are also briefly discussed.

We discuss the holomorphic properties of the complex continuation of the classical Arnol’d – Liouville action-angle variables for real analytic 1 degree-of-freedom Hamiltonian systems depending on external parameters in suitable Generic Standard Form, with particular regard to the behaviour near separatrices. In particular, we show that near separatrices the actions, regarded as functions of the energy, have a special universal representation in terms of affine functions of the logarithm with coefficients analytic functions. Then, we study the analyticity radii of the action-angle variables in arbitrary neighborhoods of separatrices and describe their behaviour in terms of a (suitably rescaled) distance from separatrices. Finally, we investigate the convexity of the energy functions (defined as the inverse of the action functions) near separatrices, and prove that, in particular cases (in the outer regions outside the main separatrix, and in the case the potential is close to a cosine), the convexity is strictly defined, while in general it can be shown that inside separatrices there are inflection points.

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.