1,354,618 research outputs found
Stability and Chaos in Celestial Mechanics
No full textThis book presents classical celestial mechanics and its interplay with dynamical systems in a way suitable for advance level undergraduate students as well as postgraduate students and researchers. First paradigmatic models are used to introduce the reader to the concepts of order, chaos, invariant curves, cantori. Next the main numerical methods to investigate a dynamical system are presented. Then the author reviews the classical two-body problem and proceeds to explore the three-body model in order to investigate orbital resonances and Lagrange solutions. In rotational dynamics the author details the derivation of the rigid body motion, and continues by discussing related topics, from spin-orbit resonances to dumbbell satellite dynamics.
Perturbation theory is then explored in full detail including practical examples of its application to finding periodic orbits, computation of the libration in longitude of the Moon. The main ideas of KAM theory are provided including a presentation of long-term stability and converse KAM results. Celletti then explains the implementation of computer-assisted techniques, which allow the user to obtain rigorous results in good agreement with the astronomical expectations. Finally the study of collisions in the solar system is approached
Rigorous estimates for a Computer-assisted KAM theory
No full textNonautonomous 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 ε c 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 ε c 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
Celletti: Un’eredità controversa
No full textA cento anni dalla nascita, viene fatto il punto sulla eredità intellettuale di Rodolfo Celletti (1917-2004) come critico musicale, con particolare riferimento al suo stile linguistico e retorico nel formulare recensioni discografiche di registrazioni operistiche a tutt'oggi insuperate. Contemporaneamente viene esaminata la sua statura come storico della vocalità
Periodic and quasi-periodic attractors of weakly-dissipative nearly-integrable systems
No full textWe consider nearly-integrable systems under a relatively small dissipation. In particular we investigate two specific models: the discrete dissipative standard map and the continuous dissipative spin-orbit model. With reference to such samples, we review some analytical and numerical results about the persistence of invariant attractors and of periodic attractors
On rigorous stability results for low-dimensional KAM surfaces
No full textThe stability of invariant (KAM) surfaces for nonintegrable dynamical systems with few degrees of freedom, as a nonlinearity parameter is increased, is considered. A rigorous method, which allows one to construct explicitly such surfaces, is discussed. A byproduct of this method allows one to give lower bounds on breakdown thresholds and applications to the standard map and to a two wave hamiltonian system yield results that agree within 60% with the numerical expectations
Perturbation expansions around elliptic fixed points in the spin-orbit problem
No full textThe reader will find in this volume the Proceedings of the NATO Advanced Study Institute held in Cortina d' Ampezzo, Italy, between July 25 and August 6, 1993, under the title From Newton to Chaos: Modem Techniques for Understanding and Coping With Chaos inN-Body Dynamical Systems. This institute was the latest in a series of meetings held every three years from 1972 to 1990 in dynamical astronomy, theoretical mechanics and celestial mechanics. The proceedings from these institutes have been well-received in the international community of research workers in these disciplines. The present institute was well attended with 15 series of lectures being given by invited speakers: in addition some 40 presentations were made by the other participants. The majority of these contributions are included in these proceedings. The all-pervading influence of chaos in dynamical systems (of even a few variables) has now been universally recognised by researchers, a recognition forced on us by our ability, using powerful computer hardware and software, to tackle dynamical problems that until twenty-five years ago were intractable. Doubtless it was felt by many that these new techniques provided a break-through in celestial mechanics and its related disciplines. And so they were
Some aspects of conservative and dissipative KAM theorems
No full textWe discuss some aspects of conservative and dissipative KAM theorems, with particular reference to a comparison between the main assumptions needed to develop KAM theory in the two settings. After analyzing the qualitative behavior of a paradigmatic model (the standard mapping), we study the existence of quasi–periodic tori in the two frameworks, paying special attention to the occurrence of small divisors and to the non–degeneracy (twist) condition in the conservative and in the dissipative case. These conditions are the main requirements for the applicability of KAM theorem, which is then stated for invariant tori as well as for invariant attractors. We proceed to discuss a criterion for the determination of the breakdown threshold of invariant tori and invariant attractors through approximating periodic orbits. These results can be applied to a wide set of physical problems; concrete applications to Celestial Mechanics are discussed with particular reference to the rotational and orbital motion of celestial bodies
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