Ugo Locatelli

Kolmogorov theorem and classical perturbation theory

Abstract. We reconsider the original proof of Kolmogorovs theorem in the light of classical perturbation methods based on expansions in some parameter. With a careful analysis of the accumulation of small divisors we prove that their effect is bounded by a geometrically increasing numerical sequence. This allows us to achieve the proof without using the so called quadratic method.

Invariant Tori in the Secular Motions of the Three-body Planetary Systems

We consider the problem of the applicability of KAM theorem to a realistic problem of three bodies. In the framework of the averaged dynamics over the fast angles for the Sun–Jupiter–Saturn system we can prove the perpetual stability of the orbit. The proof is based on semi-numerical algorithms requiring both explicit algebraic manipulations of series and analytical estimates. The proof is made rigorous by using interval arithmetics in order to control the numerical errors.

On the construction of the Kolmogorov normal form for the Trojan asteroids

In this paper, we focus on the stability of the Trojan asteroids for the planar restricted three-body problem, by extending the usual techniques for the neighbourhood of an elliptic point to derive results in a larger vicinity. Our approach is based on numerical determination of the frequencies of the asteroid and effective computation of the Kolmogorov normal form for the corresponding torus. This procedure has been applied to the first 34 Trojan asteroids of the IAU Asteroid Catalogue, and it has worked successfully for 23 of them. The construction of this normal form allows computer-assisted proofs of stability. To show this, we have implemented a proof of existence of families of invariant tori close to a given asteroid, for a high order expansion of the Hamiltonian. This proof has been successfully applied to three Trojan asteroids.

Kolmogorov and Nekhoroshev theory for the problem of three bodies

We investigate the long time stability in Nekhoroshev’s sense for the Sun– Jupiter–Saturn problem in the framework of the problem of three bodies. Using computer algebra in order to perform huge perturbation expansions we show that the stability for a time comparable with the age of the universe is actually reached, but with some strong truncations on the perturbation expansion of the Hamiltonian at some stage. An improvement of such results is currently under investigation.

Improved estimates on the existence of invariant tori for Hamiltonian systems

The existence of invariant tori in nearly integrable Hamiltonian systems is investigated. We focus our attention on a particular one-dimensional, time-dependent model, known as the forced pendulum . We present a KAM algorithm which allows us to derive explicit estimates on the perturbing parameter ensuring the existence of invariant tori. Moreover, we introduce some technical novelties in the proof of the KAM theorem which allow us to provide results in good agreement with the experimental breakdown threshold. In particular, we have been able to prove the existence of the golden torus with frequency ½((5)1/2 -1) for values of the perturbing parameter equal to 92% of the numerical threshold, thus significantly improving the previous calculations.

On the stability of the secular evolution of the planar Sun–Jupiter–Saturn–Uranus system

We investigate the long time stability of the Sun–Jupiter–Saturn–Uranus system by considering the planar, secular model. Our method may be considered as an extension of Lagrange’s theory for the secular motions. Indeed, concerning the planetary orbital revolutions, we improve the classical circular approximation by replacing it with a torus which is invariant up to order two in the masses; therefore, we investigate the stability of the elliptic equilibrium point of the secular system for small values of the eccentricities. For the initial data corresponding to a real set of astronomical observations, we find an estimated stability time of 107 years, which is not extremely smaller than the lifetime of the Solar System (∼5Gyr).

A CLASSICAL SELF-CONTAINED PROOF OF KOLMOGOROV'S THEOREM ON INVARIANT TORI

The celebrated theorem of Kolmogorov on persistence of invariant tori of a nearly integrable Hamiltonian system is revisited in the light of classical perturbation algorithm. It is shown that the original Kolmogorov’s algorithm can be given the form of a constructive scheme based on expansion in a parameter. A careful analysis of the accumulation of the small divisors shows that it can be controlled geometrically. As a consequence, the proof of convergence is based essentially on Cauchy’s majorants method, with no use of the so called quadratic method. A short comparison with Lindstedt’s series is included.

On the relationship between the Bruno function and the breakdown invariant tori

We study the ratio

Secular dynamics of a planar model of the Sun-Jupiter-Saturn-Uranus system; effective stability in the light of Kolmogorov and Nekhoroshev theories

\epsilon_c(\omega)/\exp(-\eta B(\omega))\,

Canonical perturbation theory for nearly integrable systems

, where