KAM stability and celestial mechanics

Celletti, Alessandra; Chierchia, Luigi

doi:10.1090/memo/0878

Cited by 69 publications

(98 citation statements)

References 0 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…For this particular case, we produce a strong evidence of the existence of KAM tori, which are close to the actual orbits of the two biggest planets of our solar system. Our work is inspired to the method of proof known as "computer assisted" applied to nearly integrable Hamiltonian systems, as suggested by Gallavotti and 378 U. LOCATELLI, A. GIORGILLI firstly applied in [6] (see [4] for a recent application of this kind of approach to a problem in Celestial Mechanics). The goal is typically to improve the threshold of the applicability of some rigorous results on perturbation theory, in particular Kolmogorov's theorem, so that the statement applies to physically interesting systems.…”

mentioning

confidence: 99%

Invariant tori in the Sun--Jupiter--Saturn system

Locatelli¹,

Giorgilli²

2007

Discrete &Amp; Continuous Dynamical Systems - B

View full text Add to dashboard Cite

Abstract. We discuss the applicability of Kolmogorov's theorem on existence of invariant tori to the real Sun-Jupiter-Saturn system. Using computer algebra, we construct a Kolmogorov's normal form defined in a neighborhood of the actual orbit in the phase space, giving a sharp evidence of the convergence of the algorithm. If not a rigorous proof, we consider our calculation as a strong indication that Kolmogorov's theorem applies to the motion of the two biggest planets of our solar system. Introduction.We reconsider the problem of the applicability of KAM theory (see [13], [22] and [1]) to physical systems and, in particular, to planetary systems. As it is well known, the announcement of Kolmogorov's theorem has been immediately considered by the scientific community as a major step towards the solution of the classical problem of the stability of planetary motions. On the other hand, the results of several numerical calculations of the orbits seem to indicate that the applicability of the theorem to the whole solar system is an hopeless task. Indeed, Kolmogorov's theorem states that for most initial conditions the orbits of the system should be quasi-periodic with fixed frequencies, while the numerical calculations strongly indicate that the actual orbits exhibit a limited chaotic behavior which is superimposed to the quasi-periodic motion (see, e.g., [28], [15] and [23]). As a matter of fact, a rigorous check of the applicability of Kolmogorov's theorem at least to the main part of our solar system is still lacking.In this paper, we make a further step in this research field by considering the SunJupiter-Saturn (hereafter SJS) system in the framework of the general problem of three bodies. For this particular case, we produce a strong evidence of the existence of KAM tori, which are close to the actual orbits of the two biggest planets of our solar system.

show abstract

mentioning

confidence: 99%

Invariant tori in the Sun--Jupiter--Saturn system

Locatelli¹,

Giorgilli²

2007

Discrete &Amp; Continuous Dynamical Systems - B

View full text Add to dashboard Cite

show abstract

“…For circular motions of Jupiter the problem reduces to two degrees of freedom (see Section 1.8) and plausibly there are invariant 2-dimensional tori separating the three dimensional energy surfaces; see for example [GDF + 89, Féj02b,CC07]. If the eccentricity of Jupiter is not zero, the system has two and a half degrees of freedom and then KAM tori do not prevent drastic changes in the eccentricity.…”

Section: Kirkwood Gaps and Wisdom's Ejection Mechanismmentioning

confidence: 99%

Kirkwood gaps and diffusion along mean motion resonances in the restricted planar three-body problem

Fejoz¹,

Guàrdia²,

Калошин³

et al. 2016

J. Eur. Math. Soc.

View full text Add to dashboard Cite

We study the dynamics of the restricted planar three-body problem near mean motion resonances, i.e. a resonance involving the Keplerian periods of the two lighter bodies revolving around the most massive one. This problem is often used to model Sun-Jupiter-asteroid systems. For the primaries (Sun and Jupiter), we pick a realistic mass ratio µ = 10 −3 and a small eccentricity e0 > 0. The main result is a construction of a variety of non local diffusing orbits which show a drastic change of the osculating (instant) eccentricity of the asteroid, while the osculating semi major axis is kept almost constant. The proof relies on the careful analysis of the circular problem, which has a hyperbolic structure, but for which diffusion is prevented by KAM tori. In the proof we verify certain nondegeneracy conditions numerically.Based on the work of Treschev, it is natural to conjecture that the time of diffusion for this problem is ∼ − ln(µe 0 ) µ 3/2 e 0 . We expect our instability mechanism to apply to realistic values of e0 and we give heuristic arguments in its favor. If so, the applicability of Nekhoroshev theory to the three-body problem as well as the long time stability become questionable.It is well known that, in the Asteroid Belt, located between the orbits of Mars and Jupiter, the distribution of asteroids has the so-called Kirkwood gaps exactly at mean motion resonances of low order. Our mechanism gives a possible explanation of their existence. To relate the existence of Kirkwood gaps with Arnol'd diffusion, we also state a conjecture on its existence for a typical ε-perturbation of the product of the pendulum and the rotator. Namely, we predict that a positive conditional measure of initial conditions concentrated in the main resonance exhibits Arnol'd diffusion on time scales − ln ε ε 2 .

show abstract

“…For example, an extensive study of the phase space of the related Hill's problem was completed by Simó and Stuchi [ST]; computer assisted proofs of the existence of KAM tori (and hence of stability) for realistic (Sun-Jupiter-Ceres or Sun-Jupiter-Victoria), but truncated to a trigonometric polynomial, Restricted Three-Body Problems, were given by Alessandra Celletti and Luigi Chierchia in [CC1,CC2].…”

Section: Computer Experiments and The Geometrization Of Phase Spacementioning

confidence: 99%

Poincaré and the Three-Body Problem

Chenciner

2014

Henri Poincaré, 1912–2012

View full text Add to dashboard Cite

Abstract. The Three-Body Problem has been a recurrent theme of Poincaré's thought. Having understood very early the need for a qualitative study of "nonintegrable" differential equations, he developed the necessary fundamental tools: analysis, of course, but also topology, geometry, probability. One century later, mathematicians working on the Three-Body Problem still draw inspiration from his works, in particular in the three volumes of Les méthodes nouvelles de la mécanique céleste published respectively in 1892, 1893, 1899.

show abstract

KAM stability and celestial mechanics

Cited by 69 publications

References 0 publications

Invariant tori in the Sun--Jupiter--Saturn system

Invariant tori in the Sun--Jupiter--Saturn system

Kirkwood gaps and diffusion along mean motion resonances in the restricted planar three-body problem

Poincaré and the Three-Body Problem

Contact Info

Product

Resources

About