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 consider the infinite-dimensional vector of frequencies $\omega(\mathtt{m})=( \sqrt{j^2+\mathtt{m}})_{j\in \mathbb{Z}}$, $\mathtt{m}\in [1,2]$ arising from a linear Klein – Gordon equation on the one-dimensional torus and prove that there exists a positive measure set of masses $\mathtt{m}'$s for which $\omega(\mathtt{m})$ satisfies a Diophantine condition similar to the one introduced by Bourgain in [14], in the context of the Schrödinger equation with convolution potential. The main difficulties we have to deal with are the asymptotically linear nature of the (infinitely many) $\omega_{j}'$s and the degeneracy coming from having only one parameter at disposal for their modulation. As an application we provide estimates on the inverse of the adjoint action of the associated quadratic Hamiltonian on homogenenous polynomials of any degree in Gevrey category.

We study the KAM-stability of several single star two-planet nonresonant extrasolar systems. It is likely that the observed exoplanets are the most massive of the system considered. Therefore, their robust stability is a crucial and necessary condition for the longterm survival of the system when considering potential additional exoplanets yet to be seen. Our study is based on the construction of a combination of lower-dimensional elliptic and KAM tori, so as to better approximate the dynamics in the framework of accurate secular models. For each extrasolar system, we explore the parameter space of both inclinations: the one with respect to the line of sight and the mutual inclination between the planets. Our approach shows that remarkable inclinations, resulting in three-dimensional architectures that are far from being coplanar, can be compatible with the KAM stability of the system. We find that the highest values of the mutual inclinations are comparable to those of the few systems for which the said inclinations are determined by the observations.

In this paper we consider the persistence of elliptic lower-dimensional invariant tori with prescribed frequencies in Hamiltonian systems with small parameters. Under the Brjuno nondegeneracy condition, if the prescribed frequencies satisfy a Diophantine condition, by the KAM technique we prove that for most of small parameters in the sense of Lebesgue measure, the Hamiltonian systems admit a lower-dimensional invariant torus whose frequency vector is a dilation of the prescribed frequencies.

This paper considers a class of nearly integrable reversible systems whose unperturbed part has a degenerate frequency mapping and a degenerate equilibrium point. Based on some KAM techniques and the topological degree theory, we prove the persistence of multiscale degenerate hyperbolic lower-dimensional invariant tori with prescribed frequencies.

We consider time-dependent perturbations of integrable and near-integrable Hamiltonians. Assuming the perturbation decays polynomially fast as time tends to infinity, we prove the existence of biasymptotically quasi-periodic solutions. That is, orbits converging to some quasi-periodic solutions in the future (as $t \to +\infty$) and the past (as $t \to -\infty$).
Concerning the proof, thanks to the implicit function theorem, we prove the existence of a family of orbits converging to some quasi-periodic solutions in the future and another family of motions converging to some quasi-periodic solutions in the past. Then, we look at the intersection between these two families when $t=0$. Under suitable hypotheses on the Hamiltonian's regularity and the perturbation's smallness, it is a large set, and each point gives rise to biasymptotically quasi-periodic solutions.

In the last years substantial mathematical progress has been made in KAM theory for quasi-linear/fully nonlinear Hamiltonian partial differential equations, notably for water waves and Euler equations. In this survey we focus on recent advances in quasi-periodic vortex patch solutions of the $2d$-Euler equation in $\mathbb R^2 $ close to uniformly rotating Kirchhoff elliptical vortices, with aspect ratios belonging to a set of asymptotically full Lebesgue measure. The problem is reformulated into a quasi-linear Hamiltonian equation for a radial displacement from the ellipse. A major difficulty of the KAM proof is the presence of a zero normal mode frequency, which is due to the conservation of the angular momentum. The key novelty to overcome this degeneracy is to perform a perturbative symplectic reduction of the angular momentum, introducing it as a symplectic variable in the spirit of the Darboux – Carathéodory theorem of symplectic rectification, valid in finite dimension. This approach is particularly delicate in an infinite-dimensional phase space: our symplectic change of variables is a nonlinear modification of the transport flow generated by the angular momentum itself.

We study the existence of infinite-dimensional invariant tori in a mechanical system of infinitely many rotators weakly interacting with each other. We consider explicitly interactions depending only on the angles, with the aim of discussing in a simple case the analyticity properties to be required on the perturbation of the integrable system in order to ensure the persistence of a large measure set of invariant tori with finite energy. The proof we provide of the persistence of the invariant tori implements the renormalisation group scheme based on the tree formalism, i.e., the graphical representation of the solutions of the equations of motion in terms of trees, which has been widely used in finite-dimensional problems. The method is very effectual and flexible: it naturally extends, once the functional setting has been fixed, to the infinite-dimensional case with only minor technical-natured adaptations.