Papers by Antonio Giorgilli
Nonlinearity, 2000
The existence of invariant tori in nearly{integrable Hamiltonian systems is investigated. We focu... more 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 KAM theorem which allow us to provide results in good agreement with the experimental break{down threshold. In particular, we have been able to prove the existence of the golden torus with frequency p 5?1 2 for values of the perturbing parameter equal to 92 % of the numerical threshold, thus signi cantly improving the previous calculations. (?) Hereafter, all informations about the CPU{time are referred to an AlphaServer 8200/440 EV5 with 4 Gb of RAM.
Celestial Mechanics and Dynamical Astronomy, 2010
Modern Celestial Mechanics gains new momentum as its methods and results enter more and more into... more Modern Celestial Mechanics gains new momentum as its methods and results enter more and more into different academic fields and are, in turn, influenced by them. This is exactly what happens now: perturbation theories are a continuous source of inspiration for novel applications in spaceflight dynamics, while planetary dynamics must keep pace with the rapidly expanding field of exoplanets as well as with more traditional yet unresolved problems concerning the Solar System. In the short time span of a few years, significant results have been obtained on the subject of the long-standing questions on the stability of the inner planets. New and classical problems related to periodic orbits and chaotic diffusion have been investigated under a new light, as mission design is increasingly entering the realm of the three-body problem for computing station keeping strategies and transfer trajectories.
ZAMP Zeitschrift f�r angewandte Mathematik und Physik, 1988
A recently proposed algorithm for the estimate of the threshold above which a certain torus disap... more A recently proposed algorithm for the estimate of the threshold above which a certain torus disappears, which combines classical Birkhoff normalization procedure with KAM theory, is reconsidered and improved. This is done by studying the particular case of the forced pendulum Hamiltonian <img src="/fulltext-image.asp?format=htmlnonpaginated&src=W76066RT17N30756_html\33_2004_Article_BF00948734_TeX2GIFIE1.gif" border="0" alt=" $$H(y,x,t) = \frac{{y^2 }}{2} + \varepsilon [\cos x + \cos (x - t)]$$ "
Planetary and Space Science, 1998
Two methods for constructing quasiperiodic solutions as expansion in a small parameter are discus... more Two methods for constructing quasiperiodic solutions as expansion in a small parameter are discussed. The first one is the classical Lindstedt's method; the second one an algorithm based on Kolmogorov's paper . Besides a complete formulation of the algorithms, an overview of the main ideas leading to the proof of convergence of the expansions is given. Some comparison is also made, including in particular the analysis of the effectiveness of the algorithms.
Centennial of Georges David Birkhoff, 1986
ABSTRACT We report on preliminary investigations, of numerical and analytical character, on the d... more ABSTRACT We report on preliminary investigations, of numerical and analytical character, on the dynamical properties of the classical Hamiltonian model for the interaction of electromagnetic radiation with a nonrelativistic charged point particle. We investigate the distribution of energy among the field normal modes when the energy is initially given to the particle, and we find that the high frequency modes have a tendency to be frozen, a fact that is in agreement with the qualitative trend expected from Nekhoroshev's theorem for systems of weakly coupled harmonic oscillators. Moreover we point out that, as a consequence of the dynamical interaction with the radiation field, the charged particle appears to have a highly fluctuating motion.
Seminar on Dynamical Systems, 1994
Physics Letters A, 2000
We consider the problem of removing the islands of stability in the phase space of the standard m... more We consider the problem of removing the islands of stability in the phase space of the standard map by means of tuning the parameter. A possible construction which gives a classi cation of periodic chains of islands in terms of a symbolic dynamics and predicts the values of the parameters for which the island with a given symbolic code exists is suggested and discussed.
Il Nuovo Cimento B, 1980
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Bollettino dell Unione Matematica Italiana
We consider a Hamiltonian system in a neighbourhood of an elliptic equilibrium which is a minimum... more We consider a Hamiltonian system in a neighbourhood of an elliptic equilibrium which is a minimum for the Hamiltonian. With appropriate non-resonance conditions we prove that in the neighbourhood of the equilibrium there exist low dimensional manifolds that are exponentially stable in Nekhoroshev’s sense. This generalizes the theorem of Lyapounov on the existence of periodic orbits. The result may be meaningful for, e.g., the dynamics of non-linear chains of FPU type.
Discrete and Continuous Dynamical Systems - Series B, 2005
We discuss the use of the maximal Lyapunov Characteristic Number as a stochasticity indicator in ... more We discuss the use of the maximal Lyapunov Characteristic Number as a stochasticity indicator in connection with the persistence of the FPU paradox in the thermodynamic limit. We show that the positiveness of the LCN does not imply that the dynamic is ergodic in statistical sense. On the other hand, our numerical exploration suggests that the energy surface may be separated into different chaotic regions that may trap the orbit for a long time. This is compatible with the existence of exponentially long times of relaxation to statistical equilibrium in the sense of Nekhoroshev's theory. Thus, the relevance of the FPU phenomenon for large systems remains a still open problem.
We consider a system in which some high frequency harmonic oscillators are coupled with a slow sy... more 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 ǫ −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 ǫ −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 ǫ −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.
Some recent applications and extensions of Nekhoroshev's theory on exponential stability are pres... more Some recent applications and extensions of Nekhoroshev's theory on exponential stability are presented. Applications to physical system concern on the one hand realistic evaluations of the regions where exponential stability is e ective, and, on the other hand, the relaxation time for resonant states in large, possibly in nite systems. Extension of the theory concern the phenomenon of superexponential stability of orbits in the neigbourhood of invariant KAM tori.
It is shown that a Hamiltonian system in the neighbourhood of an equilibrium may be given a speci... more 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.
We investigate the long time stability in Nekhoroshev's sense for the Sun-Jupiter-Saturn problem ... more 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.
We give a constructive proof of the existence of lower dimensional elliptic tori in nearly integr... more 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.
Lecture Notes in Physics, 2000
The stability problem in Hamiltonian dynamics is discussed in the light of Nekhoroshev's the... more The stability problem in Hamiltonian dynamics is discussed in the light of Nekhoroshev's theorem. This guarantees a form of weak stability, namely referred to finite (rather than infinite) times. Applications are discussed for the restricted problem of three bodies and for the problem of energy equipartition in statistical mechanics.
We give a short introduction to the methods of representing polynomial and trigonometric series t... more We give a short introduction to the methods of representing polynomial and trigonometric series that are often used in Celestial Mechanics. A few applications are also illustrated.
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Papers by Antonio Giorgilli