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Antonio Giorgilli

Publications and source records attributed to Antonio Giorgilli.

14 recordsLinked to original sources
arXiv2017

Improved convergence estimates for the Schröder-Siegel problem

We reconsider the Schröder-Siegel problem of conjugating an analytic map in $\mathbb{C}$ in the neighborhood of a fixed point to its linear part, extending it to the case of dimension $n>1$. Assuming a condition which is equivalent to Bruno's one on the eigenvalues $λ_1,\ldots,λ_n$ of the linear part we show that the convergence radius $ρ$ of the conjugating transformation satisfies $\ln ρ(λ)\geq -CΓ(λ)+C'$ with $Γ(λ)$ characterizing the eigenvalues $λ$, a constant $C'$ not depending on $λ$ and $C=1$. This improves the previous results for $n>1$, where the known proofs give $C=2$. We also recall that $C=1$ is known to be the optimal value for $n=1$.

math.DS↗
arXiv2017

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

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.

math-ph↗
arXiv2014

An extensive adiabatic invariant for the Klein-Gordon model in the thermodynamic limit

We construct an extensive adiabatic invariant for a Klein-Gordon chain in the thermodynamic limit. In particular, given a fixed and sufficiently small value of the coupling constant $a$, the evolution of the adiabatic invariant is controlled up to times scaling as $β^{1/\sqrt{a}}$ for any large enough value of the inverse temperature $β$. The time scale becomes a stretched exponential if the coupling constant is allowed to vanish jointly with the specific energy. The adiabatic invariance is exhibited by showing that the variance along the dynamics, i.e. calculated with respect to time averages, is much smaller than the corresponding variance over the whole phase space, i.e. calculated with the Gibbs measure, for a set of initial data of large measure. All the perturbative constructions and the subsequent estimates are consistent with the extensive nature of the system.

math.DS↗
arXiv2014

A Kepler's note on secular inequalities

I discuss the problem of secular inequalities in Kepler by giving account of a manuscript note that has not been published until 1860. In his note Kepler points out the need for a model, clearly inspired by the method of epicycles, that describes the secular inequalities as periodic ones. I bring attention to this point, that seems to have been underestimated, since the references to Kepler's work usually report only that he observed a decreasing mean motion for Saturn and an increasing one for Jupiter.

physics.hist-ph↗
arXiv2014

Normal Form and Energy Conservation of High Frequency Subsystems Without Nonresonance Conditions

We consider a system in which some high frequency harmonic oscillators are coupled with a slow system. We prove that up to very long times the energy of the high frequency system changes only by a small amount. The result we obtain is completely independent of the resonance relations among the frequencies of the fast system. More in detail, denote by $ε^{-1}$ the smallest high frequency. In the first part of the paper we apply the main result of [BG93] to prove almost conservation of the energy of the high frequency system over times exponentially long with $ {ε^{-1/n}} $ ($n$ being the number of fast oscillators). In the second part of the paper we give e new self-contained proof of a similar result which however is valid only over times of order $ε^{-N}$ with an arbitrary $N$. Such a second result is very similar to the main result of the paper [GHL13], which actually was the paper which stimulated our work.

math.DS↗
arXiv2014

On the convergence of an algorithm constructing the normal form for lower dimensional elliptic tori in planetary systems

We give a constructive proof of the existence of lower dimensional elliptic tori in nearly integrable Hamiltonian systems. In particular we adapt the classical Kolmogorov's normalization algorithm to the case of planetary systems, for which elliptic tori may be used as replacements of elliptic keplerian orbits in Lagrange-Laplace theory. With this paper we support with rigorous convergence estimates the semi-analytical work in our previous article (2011), where an explicit calculation of an invariant torus for a planar model of the Sun-Jupiter-Saturn-Uranus system has been made. With respect to previous works on the same subject we exploit the characteristic of Lie series giving a precise control of all terms generated by our algorithm. This allows us to slightly relax the non-resonance conditions on the frequencies.

math-ph↗
arXiv2013

On the representation of maps by Lie transforms

The problem of representing a class of maps in a form suited for application of normal form methods is revisited. It is shown that using the methods of Lie series and of Lie transform a normal form algorithm is constructed in a straightforward manner. The examples of the Scrhöder--Siegel map and of the Chirikov standard map are included, with extension to arbitrary dimension.

math.DS↗
arXiv2013

On a theorem of Lyapounov

It is shown that a Hamiltonian system in the neighbourhood of an equilibrium may be given a special normal form in case the eigenvalues of the linearized system satisfy non--resonance conditions of Melnikov's type. The normal form possesses a two dimensional (local) invariant manifold on which the solutions are known. If the eigenvalue is pure imaginary then these solutions are the natural continuation of a normal mode of the linear system. The latter result was first proved by Lyapounov. The present paper completes Lyapounov's result in that the convergence of the transformation of the Hamiltonian to a normal form is proven and the condition that the eigenvalues be pure imaginary is removed.

math.DS↗
arXiv2013

Su un'estensione della teoria di Lagrange per i moti secolari

La teoria di Lagrange per i moti secolari delle eccentricità ed inclinazioni delle orbite planetarie si fondava su un'approssimazione, dettata in larga misura dalla complessità dei calcoli necessari, che consisteva nel considerare solo equazioni lineari. In questa memoria riprendiamo in considerazione i metodi di Lagrange alla luce della teoria della stabilità esponenziale di Nekhoroshev. Grazie agli algoritmi sviluppati negli ultimi anni e alle tecniche di manipolazione algebrica possiamo tener conto anche dei contributi non lineari alle equazioni. Come applicazione cerchiamo di determinare i tempi di stabilità per il problema dei tre corpi nel caso del Sole e dei due pianeti maggiori, Giove e Saturno, mostrando che si possono ottenere risultati realistici, ancorché non ottimali. Lagrange's theory for the secular motion of perihelia and nodes of the planetary orbits was based on consideration of a linear approssimation of the dynamical equations, compatible with the complexity of the calculations. We extend Lagrange's investigations in the light of Nekhoroshev's theory of exponential stability. Using effective algorithms recently developed and computer algebra we investigate the non linear problem. We apply our methods to the problem of three bodies in the Sun-Jupiter-Saturn case, thus showing that realistic results, although not optimal, can be obtained.

math-ph↗
arXiv2013

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.

math-ph↗
arXiv2013

A Semi-Analytic Algorithm for Constructing Lower Dimensional Elliptic Tori in Planetary Systems

We adapt the Kolmogorov's normalization algorithm (which is the key element of the original proof scheme of the KAM theorem) to the construction of a suitable normal form related to an invariant elliptic torus. As a byproduct, our procedure can also provide some analytic expansions of the motions on elliptic tori. By extensively using algebraic manipulations on a computer, we explicitly apply our method to a planar four-body model not too different with respect to the real Sun--Jupiter--Saturn--Uranus system. The frequency analysis method allows us to check that our location of the initial conditions on an invariant elliptic torus is really accurate.

math.DS↗
arXiv2013

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 $10^7$ years, which is not extremely smaller than the lifetime of the Solar System ($\sim 5$ Gyr).

math-ph↗
arXiv2012

A series expansion for the time autocorrelation of dynamical variables

We present here a general iterative formula which gives a (formal) series expansion for the time autocorrelation of smooth dynamical variables, for all Hamiltonian systems endowed with an invariant measure. We add some criteria, theoretical in nature, which enable one to decide whether the decay of the correlations is exponentially fast or not. One of these criteria is implemented numerically for the case of the Fermi-Pasta-Ulam system, and we find indications which might suggest a sub-exponential decay of the time autocorrelation of a relevant dynamical variable.

math-ph↗