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

Abstract

We investigate the long-time stability of the Sun-Jupiter-Saturn-Uranus system by considering a planar secular model, that can be regarded as a major refinement of the approach first introduced by Lagrange. Indeed, concerning the planetary orbital revolutions, we improve the classical circular approximation by replacing it with a solution that is invariant up to order two in the masses; therefore, we investigate the stability of the secular system for rather small values of the eccentricities. First, we explicitly construct a Kolmogorov normal form, so as to find an invariant KAM torus which approximates very well the secular orbits. Finally, we adapt the approach that is at basis of the analytic part of the Nekhoroshev's theorem, so as to show that there is a neighborhood of that torus for which the estimated stability time is larger than the lifetime of the Solar System. The size of such a neighborhood, compared with the uncertainties of the astronomical observations, is about ten times smaller. *

Hamiltonian control of magnetic field lines: Computer assisted results proving the existence of KAM barriers

“…Applications of approaches based on other CAP techniques to the Hamiltonian model considered here could be interesting, also to compare the performances of different methods. In this context, let us emphasize that our approach is in a good position to obtain rigorous results about the stability in the vicinity of an invariant "chief torus", because our algorithm is designed to construct Kolmogorov normal forms (see [16] and [17]).…”

Sharp Nekhoroshev estimates for the three-body problem around periodic orbits

“…In the sequel, we will see that this phenomenon arises dramatically when considering the same computations for the restricted, circular, planar problem. Lastly, as we have already stressed, this study relies on rigorous estimates on the domain of analyticity for hamiltonian (18) which are contained in [6].…”

“…In the case of the 5 : 2 resonance, stability holds for a time comparable with the age of the Solar System if the ratio for the mass of the greater planet on the Sun mass does not exceed 10 −13 (the real value is actually 10 −3 in the Solar System). On the other hand, numerical-assisted studies on Nekhoroshev stability, with realistic magnitudes for the perturbation, have been achieved by Giorgilli, Locatelli and Sansottera in [17] and [18] for a suitably truncated three or even four body hamiltonian in the neighborhood of an invariant torus. An application leading to a remarkably good upper bound on the perturbative parameter (ε < 10 −6 ) in the non-resonant restricted, circular, planar case has also been considered by Celletti and Ferrara in [8].…”

Exponential Stability in the Perturbed Central Force Problem

“…Here we will proceed by an explicit construction using a symbolic manipulator. We recall that explicit computations of Birkhoff normal form have already been implemented numerically in many situations (see, e.g., [SLG13], [GLS14], [SLL14], [SLL15] and [GLS17]).…”