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  • 1
    Electronic Resource
    Electronic Resource
    A Tribute to Jürgen Moser (2000)
    Springer
    Celestial mechanics and dynamical astronomy 77 (2000), S. 153-155 
    Details
    ISSN: 1572-9478
    Source: Springer Online Journal Archives 1860-2000
    Topics: Physics
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1023/A:1011158900147
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  • 2
    Electronic Resource
    Electronic Resource
    Invariant Tori in the Secular Motions of the Three-body Planetary Systems (2000)
    Springer
    Celestial mechanics and dynamical astronomy 78 (2000), S. 47-74 
    Details
    ISSN: 1572-9478
    Keywords: KAM theory ; stability in planetary motion ; three-body problem ; computer-assisted proofs
    Source: Springer Online Journal Archives 1860-2000
    Topics: Physics
    Notes: Abstract We 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.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1023/A:1011139523256
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  • 3
    Electronic Resource
    Electronic Resource
    Formal integrals for an autonomous Hamiltonian system near an equilibrium point (1978)
    Springer
    Celestial mechanics and dynamical astronomy 17 (1978), S. 267-280 
    Details
    ISSN: 1572-9478
    Source: Springer Online Journal Archives 1860-2000
    Topics: Physics
    Notes: Abstract In this paper we give a new method to construct formal integrals for an autonomous Hamiltonian system near an equilibrium point. Our construction is reminiscent of the algorithms introduced by Hori and Deprit; we show however how it presents itself quite naturally if one follows the line of attack of Whittaker, Cherry, and Contopoulos.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01232832
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  • 4
    Electronic Resource
    Electronic Resource
    Rigorous estimates for the series expansions of Hamiltonian perturbation theory (1985)
    Springer
    Celestial mechanics and dynamical astronomy 37 (1985), S. 95-112 
    Details
    ISSN: 1572-9478
    Source: Springer Online Journal Archives 1860-2000
    Topics: Physics
    Notes: Abstract In the present paper we prove a theorem giving rigorous estimates in the problem of bringing to normal form a nearly integrable Hamiltonian system, using methods of classical perturbation theory, i.e. series expansions in the “small parameter” ε. For any order of normalization, we give a lower bound ɛ * r for the convergence radius of the normalized Hamiltonian, and a greater bound for the remainder, i.e. the non normalized part of the Hamiltonian. As an application, we consider the case of weakly coupled harmonic oscillators with highly nonresonant frequencies and show how, by optimizing, for fixed ε, the orderr of normalization, one gets for the remainder a greater bound of the formAe −(ɛ*1/ɛ) a , with positive constantsA,a and ɛ 1 * exponential estimate of Nekhoroshev's type.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01230921
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  • 5
    Electronic Resource
    Electronic Resource
    On the dynamics in the asteroids belt. Part I : General theory (1989)
    Springer
    Celestial mechanics and dynamical astronomy 47 (1989), S. 145-172 
    Details
    ISSN: 1572-9478
    Keywords: resonance ; perturbation method ; Kirkwood gaps ; periodic orbits
    Source: Springer Online Journal Archives 1860-2000
    Topics: Physics
    Notes: Abstract The classical problem of the dynamics in the asteroids belt is revisited in the light of recently developed perturbation methods. We consider the spatial problem of three bodies both in the circular and in the elliptic case, looking for families of periodic or quasi periodic orbits. Some criteria for deciding the stability of these families are also indicated.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00051203
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  • 6
    Electronic Resource
    Electronic Resource
    On the stability of the lagrangian points in the spatial restricted problem of three bodies (1990)
    Springer
    Celestial mechanics and dynamical astronomy 50 (1990), S. 31-58 
    Details
    ISSN: 1572-9478
    Keywords: Three body problem ; Lagrangian points ; Nonlinear stability
    Source: Springer Online Journal Archives 1860-2000
    Topics: Physics
    Notes: Abstract The problem of stability of the Lagrangian equilibrium point of the circular restricted problem of three bodies is investigated in the light of Nekhoroshev-like theory. Looking for stability over a time interval of the order of the estimated age of the universe, we find a physically relevant stability region. An application of the method to the Sun-Jupiter and the Earth-Moon systems is made. Moreover, we try to compare the size of our stability region with that of the region where the Trojan asteroids are actually found; the result in such case is negative, thus leaving open the problem of the stability of these asteroids.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00048985
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  • 7
    Electronic Resource
    Electronic Resource
    Quantitative perturbation theory by successive elimination of harmonics (1993)
    Springer
    Celestial mechanics and dynamical astronomy 55 (1993), S. 131-159 
    Details
    ISSN: 1572-9478
    Keywords: Perturbation methods ; KAM theorem ; Nekhoroshev theorem ; action-angle variables
    Source: Springer Online Journal Archives 1860-2000
    Topics: Physics
    Notes: Abstract We revisit some results of perturbation theories by a method of successive elimination of harmonics inspired by some ideas of Delaunay. On the one hand, we give a connection between the KAM theorem and the Nekhoroshev theorem. On the other hand, we support in a quantitative fashion a semi-numerical method of analysis of a perturbed system recently introduced by one of the authors.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00692425
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  • 8
    Electronic Resource
    Electronic Resource
    Long Time Stability for the Main Problem of Artificial Satellites (1997)
    Springer
    Celestial mechanics and dynamical astronomy 69 (1997), S. 317-330 
    Details
    ISSN: 1572-9478
    Keywords: artificial satellite ; Nekhoroshev's theory ; normal form ; stability
    Source: Springer Online Journal Archives 1860-2000
    Topics: Physics
    Notes: Abstract We investigate the significance of long time stabilty predictions in the light of Nekhoroshev's theory by studying the orbits of artificial satellites. As a simplified model problem we consider the so-called J2problem for an earth's satellite, neglecting luni-solar perturbations and nonconservative effects. We consider a wide range of orbits, excluding those which are too close to the critical inclination. Most of the orbits turn out to be stable for times larger than the estimated age of the solar system, thus proving that, as far as dissipation can be neglected, stability in Nekhoroshev's sense may be effective for physically realistic systems.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1023/A:1008277122375
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  • 9
    Electronic Resource
    Electronic Resource
    A proof of Nekhoroshev's theorem for the stability times in nearly integrable Hamiltonian systems (1985)
    Springer
    Celestial mechanics and dynamical astronomy 37 (1985), S. 1-25 
    Details
    ISSN: 1572-9478
    Source: Springer Online Journal Archives 1860-2000
    Topics: Physics
    Notes: Abstract In the present paper we give a proof of Nekhoroshev's theorem, which is concerned with an exponential estimate for the stability times in nearly integrable Hamiltonian systems. At variance with the already published proof, which refers to the case of an unperturbed Hamiltonian having the generic property of steepness, we consider here the particular case of a convex unperturbed Hamiltonian. The corresponding simplification in the proof might be convenient for an introduction to the subject.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01230338
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  • 10
    Electronic Resource
    Electronic Resource
    Estimates for normal forms of differential equations near an equilibrium point (1988)
    Springer
    Zeitschrift für angewandte Mathematik und Physik 39 (1988), S. 713-732 
    Details
    ISSN: 1420-9039
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    Description / Table of Contents: Riassunto Si considera il problema di porre in forma normale un sistema di equazioni differenziali nell'intomo di un punto di equilibrio, e si danno in generale stime esplicite sia per la forma normale troncata ad un ordine finito che per i resti. Si fa uso dell'algoritmo della trasformata di Lie. Con questo metodo si riottengono i teoremi classici di Poincaré-Dulac, e si discute il problema della stabilità per tempi esponenzialmente lunghi di un sistema reversibile di oscillatori armonici accoppiati.
    Notes: Abstract We consider the problem of finding a normal form for differential equations in the neighbourhood of an equilibrium point, and produce general explicit estimates for both the normal form at a finite order and the remainder, using the method of Lie transforms. With such technique, the classical Poincaré-Dulac theorems are recovered, and the problem of the stability of a reversible system of coupled harmonic oscillators up to exponentially large times is discussed.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00948732
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