56,807 research outputs found
Areali visioni
No full textLa realtà percettivo-visiva viene considerata sulla base della chiave di lettura "Areale", secondo i principi teorici del ricercatore visuale Ugo Locatelli
CAP4KAM_nDOF: Computer-Assisted Proofs of existence of KAM tori in Hamiltonian systems with n (>=2) Degrees Of Freedom
No full textIn the folder you can produce by uncompressing the attached zippedfile (namely, that folder is called "CAP4KAM_nDOF"), you should findeverything you need in order to perform a complete computer-assistedproof of existence of invariant tori for a Hamiltonian that satifiesthree assumptions: (i) it describes a (Hamiltonian) system with n>=2 degrees of freedom and its canonical coordinates are n pairs of action-angle variables; (ii) it is close enough to a Kolmogorov normal form (so fulfilling also both the non-resonance and the non-degeneracy conditions usually adopted in the framework of KAM theory); (iii) its expansion in Taylor series (with respect to the actions) is finite, while its Fourier expansions (in the angles) can be infinite.The software included in the present folder "CAP4KAM_nDOF" is anextension of a first public release that was a sort of supplementarymaterial of the paper [VL], i.e.,Locatelli, Ugo (2021), “CAP4KAM2D: Computer-Assisted Proofs Fordemonstrating the existence of 2-Dimensional KAM tori”, MendeleyData, V1, doi: 10.17632/jdx22ysh2s.1The software included in the folder "CAP4KAM_nDOF" is designed tobe in a "easy-to-use" layout. Moreover, it is probably not too difficultto be modified for people expert in programming (in C).Everything about the files included in the folder "CAP4KAM_nDOF" iswidely described in the README.txt, that contains also carefulexplanations that should be useful for running the codes, monitoringthe results, modifying the input files, etc.Eventual corrections or remarks about the software package includedin the folder "CAP4KAM_nDOF" are more than welcome and can besent to the author (Ugo Locatelli) at the following e-mail address: [email protected]
New Hamiltonian expansions adapted to the Trojan problem
No full textA number of studies, referring to the observed Trojan asteroids of various planets in our Solar System, or to hypothetical Trojan bodies in extrasolar planetary systems, have emphasized the importance of so-called secondary resonances in the problem of the long term stability of Trojan motions. Such resonances describe commensurabilities between the fast, synodic, and secular frequency of the Trojan body, and, possibly, additional slow frequencies produced by more than one perturbing bodies. The presence of secondary resonances sculpts the dynamical structure of the phase space. Hence, identifying their location is a relevant task for theoretical studies. In the present paper we combine the methods introduced in two recent papers (Páez and Efthymiopoulos in Celest Mech Dyn Astron 121(2):139, 2015; Páez and Locatelli in MNRAS 453(2):2177, 2015) in order to analytically predict the location of secondary resonances in the Trojan problem. In Páez and Efthymiopoulos (2015), the motion of a Trojan body was studied in the context of the planar Elliptic Restricted Three Body or the planar Restricted Multi-Planet Problem. It was shown that the Hamiltonian admits a generic decomposition H= Hb+ Hsec. The term Hb, called the basic Hamiltonian, is a model of two degrees of freedom characterizing the short-period and synodic motions of a Trojan body. Also, it yields a constant ‘proper eccentricity’ allowing to define a third secular frequency connected to the body’s perihelion precession. Hsec contains all remaining secular perturbations due to the primary or to additional perturbing bodies. Here, we first investigate up to what extent the decomposition H= Hb+ Hsec provides a meaningful model. To this end, we produce numerical examples of surfaces of section under Hb and compare with those of the full model. We also discuss how secular perturbations alter the dynamics under Hb. Secondly, we explore the normal form approach introduced in Páez and Locatelli (2015) in order to find an ‘averaged over the fast angle’ model derived from Hb, circumventing the problem of the series’ limited convergence due to the collision singularity at the 1:1 MMR. Finally, using this averaged model, we compute semi-analytically the position of the most important secondary resonances and compare the results with those found by numerical stability maps in specific examples. We find a very good agreement between semi-analytical and numerical results in a domain whose border coincides with the transition to large-scale chaotic Trojan motions
On classical series expansions for quasi-periodic motions
No full textWe reconsider the problem of convergence of classical
expansions in a parameter for quasiperiodic motions on
invariant tori in nearly integrable Hamiltonian systems. Using a
reformulation of the algorithm proposed by Kolmogorov, we show that if
the frequencies satisfy the nonresonance condition proposed by Bruno,
then one can construct a normal form such that the coefficient of
is a sum of terms each of which is bounded by
. This allows us to produce a direct proof of the classical
expansions. We also discuss some relations between our
expansions and the Lindstedt's ones
Sulla stabilità del problema planetario dei tre corpi
No full textSi discute se il celebre teorema di Kolmogorov sulla persistenza di moti quasi periodici si applichi al sistema solare, prendendo a modello il problema dei tre corpi nel caso in cui le masse ed i parametri orbitali siano quelli di Sole, Giove e Saturno. Facendo ricorso a metodi di manipolazione algebrica al calcolatore si mostra per la prima volta che almeno in questo caso il teorema si puo' applicare, e dunque che le orbite dei due pianeti maggiori, almeno nell'approssimazione del problema dei tre corpi, giacciono in prossimita' di tori invarianti di Kolmogorov. Si discutono poi brevemente le possibili estensioni di questo lavoro accennando alla possibilita' di affrontare il problema classico della stabilita' del sistema solare alla luce della teoria della stabilita' esponenziale alla Nekhoroshev
Kolmogorov theorem and classical perturbation theory
No full textWe reconsider the original proof of Kolmogorov's theorem in
the light of classical perturbation methods based on expansions in some
parameter. With a careful analysis of the accumulation of small divisors
we prove that their effect is bounded by a geometrically increasing
numerical sequence. This allows us to achieve the proof without using
the so called quadratic method.
Construction of Kolmogorov's normal form for a planetary system
No full textWe describe an algorithm constructing an invariant KAM torus for a class of planetary systems, such that the mutual
attractions, the eccentricities and the inclinations of the planets are small enough. By using computer algebra, we
explicitly implement this algorithm for approximating a KAM torus for the problem of three bodies in a case similar to
the Sun{Jupiter{Saturn system. We show that, by reducing the masses of the planets by a factor 10 and with a small
displacement of the orbits, our semianalytical construction of the torus turns out to be successful
A classical self-consistent proof of Kolmogorov's theorem on invariant tori
No full textThe celebrated theorem of Kolmogorov on persistence of
invariant tori of a nearly integrable Hamiltonian system is revisited
in the light of classical perturbation algorithm. It is shown that the
original Kolmogorov's algorithm can be given the form of a
constructive scheme based on expansion in a parameter. A careful
analysis of the accumulation of the small divisors shows
that it can be controlled geometrically. As a consequence, the proof
of convergence is based essentially on Cauchy's majorant's method,
with no use of the so called quadratic method. A short comparison with
Lindstedt's series is included
Canonical perturbation theory for nearly integrable systems
No full textThese lectures are devoted to the main results of classical
perturbation theory. We start by recalling the methods of
Hamiltonian dynamics, the problem of small divisors, the series of
Lindstedt and the method of normal form. Then we discuss the theorem
of Kolmogorov with an application to the Sun--Jupiter--Saturn problem
in Celestial Mechanics. Finally we discuss the problem of long--time
stability, by discussing the concept of exponential stability as
introduced by Moser and Littlewood and fully exploited by
Nekhoroshev. The phenomenon of superexponential stability is also
recalled
Invariant tori in the secular motions of the three-body planetary systems
No full textWe consider the problem of the applicability of KAM theorem to a realistic problem of
three bodies. In the framework of the averaged dynamics over the fast angles for the Sun–Jupiter–
Saturn system we can prove the perpetual stability of the orbit. The proof is based on semi-numerical
algorithms requiring both explicit algebraic manipulations of series and analytical estimates. The
proof is made rigorous by using interval arithmetics in order to control the numerical errors
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