Papers by Luigi Chierchia
Regular & chaotic dynamics/Regular and chaotic dynamics, Jul 5, 2024
Annali Della Scuola Normale Superiore Di Pisa-classe Di Scienze, 1989
Annali di Matematica Pura ed Applicata, Apr 29, 2019
We consider geometric properties of 3-jet non-degenerate functions in connection with Nekhoroshev... more We consider geometric properties of 3-jet non-degenerate functions in connection with Nekhoroshev theory. In particular, after showing that 3-jet non-degenerate functions are "almost quasi-convex", we prove that they are steep and compute explicitly the steepness indices (which do not exceed 2) and the steepness coefficients.
Planetary and Space Science, Nov 1, 1998
A review of KAM stability estimates in Celestial Mechanics is presented. Rotational and libration... more A review of KAM stability estimates in Celestial Mechanics is presented. Rotational and librational invariant surfaces are constructed to ensure confinement in the phase space of a model obtained in the spin-orbit coupling between the revolutional and rotational motions of a satellite around a primary body. Stability of invariant tori for the restricted, circular, planar three body problem is also presented. Finally, an application of Arnold's theorem to the inclined planetary problem is briefly discussed.
Journal of Mathematical Physics, Sep 1, 1987
Nonautonomous Hamiltonian systems of one degree of freedom close to integrable ones are considere... more Nonautonomous Hamiltonian systems of one degree of freedom close to integrable ones are considered. Let ε be a positive parameter measuring the strength of the perturbation and denote by εc the critical value at which a given KAM (Kolmogorov–Arnold–Moser) torus breaks down. A computer-assisted method that allows one to give rigorous lower bounds for εc is presented. This method has been applied in Celletti–Falcolini–Porzio (to be published in Ann. Inst. H. Poincaré) to the Escande and Doveil pendulum yielding a bound which is within a factor 40.2 of the value indicated by numerical experiments.
Archive for Rational Mechanics and Analysis, Jul 4, 2008
We prove that KAM tori smoothly bifurcate into quasi-periodic attractors in dissipative mechanica... more We prove that KAM tori smoothly bifurcate into quasi-periodic attractors in dissipative mechanical models, provided external parameters are tuned with the frequency of the motion. An application to the dissipative "spin-orbit model" of celestial mechanics (which actually motivated the analysis in this paper) is presented.
Springer eBooks, 2018
the behavior of "most" initial data. In general, this is not the case in infinite dimensional sys... more the behavior of "most" initial data. In general, this is not the case in infinite dimensional systems and PDEs. Still, the search for periodic and quasiperiodic solutions is obviously an interesting and challenging task.
Duke Mathematical Journal, Dec 1, 1987
Ergodic Theory and Dynamical Systems, Jun 1, 2009
In 2004 J. Féjoz [7], completing investigations of M. Herman's [9], gave a complete proof of "Arn... more In 2004 J. Féjoz [7], completing investigations of M. Herman's [9], gave a complete proof of "Arnold's Theorem" [1] on the planetary many-body problem, establishing, in particular, the existence of a positive measure set of smooth (C ∞) Lagrangian invariant tori for the planetary many-body problem. Here, using Rüßmann's 2001 KAM theory [16], we prove the above result in the real-analytic class. * Acknowledgments. We are indebted with Jacques Féjoz for many enlightening discussions. This work was partially supported by the Italian MIUR project "Metodi variazionali e equazioni differenziali nonlineari". where q j = q j (t) ∈ R 3 denotes the position at time t of the j th body, '| • |' denotes the euclidean norm and '˙' denotes time derivative. In [1, Chapter III, p. 125], V.I. Arnold made the following 1 : Arnold's statement: "In the n-body problem there exists a set of initial conditions having a positive Lebesgue measure and such that, if the initial positions and velocities belong to this set, the distances of the bodies from each other will remain perpetually bounded." As well known, such statement solves a fundamental problem considered, for several centuries, by astronomers and mathematicians. However, Arnold considered in details only the planar three-body case 2 and it appears that his indication for extending the result to the general case contains a flaw; compare end of §1.2, p. 1524 in [7]. A complete general proof of Arnold's statement was given only in 2004, when J. Féjoz, completing the work of M. Herman, proved the following 3 Theorem 1 (Arnold, Herman, Féjoz [7, §1.2, p. 1523, THÉORÈME 1]). Si le maximum ǫ = max{m j /m 0 } j=1,...,n des masses des planètes rapportées à la masse du soleil est suffisamment petit, les équations (1) admettent, dans l'espace des phases au voisinage des mouvements képlériens circulaires et coplanaires, un ensemble de mesure de Lebesgue strictement positive de conditions initiales conduisant à des mouvements quasipériodiques. The beautiful proof of this result given in [7] (see also [8]) relies, on one side, on the elegant C ∞ KAM theory worked out by Herman (§ 2÷5 in [7]), and, on the other side, on the analytical celestial mechanics worked out, especially, by Poincaré and clarified and further investigated in Paris in the late 1980's by A. Chenciner and J. Laskar in the Bureau des Longitudes 4 and later by Herman himself.
Celestial Mechanics and Dynamical Astronomy, Aug 17, 2006
Regular & Chaotic Dynamics, Feb 1, 2009
Regular & chaotic dynamics/Regular and chaotic dynamics, Jul 5, 2024
Springer eBooks, 1988
ABSTRACT
Journal of Differential Equations, Feb 1, 1995
Discrete and Continuous Dynamical Systems - Series S, 2010
Celestial Mechanics and Dynamical Astronomy, May 1, 2008
Communications in Mathematical Physics, Mar 1, 1988
A class of analytic (possibly) time-dependent Hamiltonian systems with d degrees of freedom and t... more A class of analytic (possibly) time-dependent Hamiltonian systems with d degrees of freedom and the "corresponding" class of area-preserving, twist diffeomorphisms of the plane are considered. Implementing a recent scheme due to Moser, Salamon and Zehnder, we provide a method that allows us to construct ""explicitly" KAM surfaces and, hence, to give lower bounds on their breakdown thresholds. We, then, apply this method to the Hamiltonian H = y 2 /2 + ε(cosx J rcos(x-t)) and to the map (y, x)->(y + ε sinx, x + y + εsinx) obtaining, with the aid of computer-assisted estimations, explicit approximations (within an error of ~10~5) of the golden-mean KAM surfaces for complex values of ε with |ε| less or equal than, respectively, 0.015 and 0.65. (The experimental numerical values at which such surfaces are expected to disappear are about, respectively, 0.027 and 0.97.) A possible connection between breakdown thresholds and singularities in the complex ε-plane is pointed out.
Inventiones Mathematicae, Feb 23, 2011
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Papers by Luigi Chierchia