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 Regul. Chaotic Dyn., 2015, Volume 20, Issue 1, Pages 1–18 (Mi rcd54)

Numerical Verification of the Steepness of Three and Four Degrees of Freedom Hamiltonian Systems

Gabriella Schirinzia, Massimiliano Guzzob

a Università del Salento, Dipartimento di Matematica e Fisica, Via per Arnesano - 73100 Lecce, Italy
b Università degli Studi di Padova, Dipartimento di Matematica, Via Trieste, 63 - 35121 Padova, Italy

Abstract: We describe a new algorithm for the numerical verification of steepness, a necessary property for the application of Nekhoroshev’s theorem, of functions of three and four variables. Specifically, by analyzing the Taylor expansion of order four, the algorithm analyzes the steepness of functions whose Taylor expansion of order three is not steep. In this way, we provide numerical evidence of steepness of the Birkhoff normal form around the Lagrangian equilibrium points L4–L5 of the spatial restricted three-body problem (for the only value of the reduced mass for which the Nekhoroshev stability was still unknown), and of the four-degreesof-freedom Hamiltonian system obtained from the Fermi–Pasta–Ulam problem by setting the number of particles equal to four.

Keywords: Nekhoroshev’s theorem, steepness, three-body-problem, Fermi–Pasta–Ulam

 Funding Agency Grant Number PRIN 2010JJ4KPA_009 CaRiPaRo 11/2012 This research has been supported by the Italian project PRIN “Teorie geometriche e analitiche dei sistemi Hamiltoniani in dimensioni finite e infinite”. M.Guzzo has been also supported by CaRiPaRo “Nonlinear Partial Differential Equations: models, analysis, and control-theoretic problems” of the University of Padova.

DOI: https://doi.org/10.1134/S1560354715010013

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MSC: 70F15, 70H08, 37J40
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Citation: Gabriella Schirinzi, Massimiliano Guzzo, “Numerical Verification of the Steepness of Three and Four Degrees of Freedom Hamiltonian Systems”, Regul. Chaotic Dyn., 20:1 (2015), 1–18

Citation in format AMSBIB
\Bibitem{SchGuz15} \by Gabriella Schirinzi, Massimiliano Guzzo \paper Numerical Verification of the Steepness of Three and Four Degrees of Freedom Hamiltonian Systems \jour Regul. Chaotic Dyn. \yr 2015 \vol 20 \issue 1 \pages 1--18 \mathnet{http://mi.mathnet.ru/rcd54} \crossref{https://doi.org/10.1134/S1560354715010013} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=3304934} \zmath{https://zbmath.org/?q=an:06468419} \adsnasa{http://adsabs.harvard.edu/cgi-bin/bib_query?2015RCD....20....1S} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000349024900001} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-84944180419} 

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• http://mi.mathnet.ru/eng/rcd/v20/i1/p1

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. M. Guzzo, “The nekhoroshev theorem and long-term stabilities in the solar system”, Serb. Astron. J., 190 (2015), 1–10
2. Massimiliano Guzzo, Elena Lega, “The Nekhoroshev Theorem and the Observation of Long-term Diffusion in Hamiltonian Systems”, Regul. Chaotic Dyn., 21:6 (2016), 707–719
3. L. Chierchia, M. A. Faraggiana, M. Guzzo, “On steepness of 3-jet non-degenerate functions”, Ann. Mat. Pura Appl., 198:6 (2019), 2151–2165
4. Daniela Cárcamo-Díaz, Jesús F. Palacián, Claudio Vidal, Patricia Yanguas, “On the Nonlinear Stability of the Triangular Points in the Circular Spatial Restricted Three-body Problem”, Regul. Chaotic Dyn., 25:2 (2020), 131–148