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


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Article Info

DATE

2014-08

AUTHORS

Antonio Giorgilli, Ugo Locatelli, Marco Sansottera

ABSTRACT

We give a constructive proof of the existence of elliptic lower dimensional tori in nearly integrable Hamiltonian systems. In particular we adapt the classical Kolmogorov 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-analytic work in our previous article (Sansottera et al., Celest Mech Dyn Astron 111:337–361, 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. More... »

PAGES

397-424

References to SciGraph publications

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  • 1995-03. Superexponential stability of KAM tori in JOURNAL OF STATISTICAL PHYSICS
  • 2013-10. On the extension of the Laplace-Lagrange secular theory to order two in the masses for extrasolar systems in CELESTIAL MECHANICS AND DYNAMICAL ASTRONOMY
  • 1999. A Classical Self-Contained Proof of Kolmogorov’s Theorem on Invariant Tori in HAMILTONIAN SYSTEMS WITH THREE OR MORE DEGREES OF FREEDOM
  • 2000-09. Invariant Tori in the Secular Motions of the Three-body Planetary Systems in CELESTIAL MECHANICS AND DYNAMICAL ASTRONOMY
  • 1971-07. Existence of quasi-periodic solutions to the three-body problem in CELESTIAL MECHANICS AND DYNAMICAL ASTRONOMY
  • 2003-11. Elliptic Two-Dimensional Invariant Tori for the Planetary Three-Body Problem in ARCHIVE FOR RATIONAL MECHANICS AND ANALYSIS
  • 2009-06. Kolmogorov and Nekhoroshev theory for the problem of three bodies in CELESTIAL MECHANICS AND DYNAMICAL ASTRONOMY
  • 1997-01. Invariant KAM tori and global stability for Hamiltonian systems in ZEITSCHRIFT FÜR ANGEWANDTE MATHEMATIK UND PHYSIK
  • 2011-11. A semi-analytic algorithm for constructing lower dimensional elliptic tori in planetary systems in CELESTIAL MECHANICS AND DYNAMICAL ASTRONOMY
  • 2011-08. Branching of Cantor Manifolds of Elliptic Tori and Applications to PDEs in COMMUNICATIONS IN MATHEMATICAL PHYSICS
  • 2015-08. Improved convergence estimates for the Schröder–Siegel problem in ANNALI DI MATEMATICA PURA ED APPLICATA (1923 -)
  • 1989-12. On elliptic lower dimensional tori in hamiltonian systems in MATHEMATISCHE ZEITSCHRIFT
  • 1997-03. Kolmogorov theorem and classical perturbation theory in ZEITSCHRIFT FÜR ANGEWANDTE MATHEMATIK UND PHYSIK
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    URI

    http://scigraph.springernature.com/pub.10.1007/s10569-014-9562-7

    DOI

    http://dx.doi.org/10.1007/s10569-014-9562-7

    DIMENSIONS

    https://app.dimensions.ai/details/publication/pub.1035910283


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