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Annali di Matematica Pura ed Applicata (1923 -)

, Volume 198, Issue 6, pp 2151–2165 | Cite as

On steepness of 3-jet non-degenerate functions

  • L. ChierchiaEmail author
  • M. A. Faraggiana
  • M. Guzzo
Article
  • 51 Downloads

Abstract

We consider geometric properties of 3-jet non-degenerate functions in connection with Nekhoroshev theory. In particular, after showing that 3-jet non-degenerate functions are “almost quasi-convex”, we prove that they are steep and compute explicitly the steepness indices (which do not exceed 2) and the steepness coefficients.

Keywords

Steepness Steep functions 3-Jet non-degeneracy Nekhoroshev’s theorem Hamiltonian systems Steepness indices Exponential stability 

Mathematics Subject Classification

34D20 37J40 70H08 

Notes

References

  1. 1.
    Benettin, G., Fassò, F., Guzzo, M.: Nekhoroshev stability of L4 and L5 in the spatial restricted three body problem. Regul. Chaotic Dyn. 3, 56–72 (1998)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Benettin, G., Galgani, L., Giorgilli, A.: A proof of Nekhoroshev’s theorem for the stability times in nearly integrable Hamiltonian systems. Cel. Mech. 37, 1–25 (1985)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Benettin, G., Gallavotti, G.: Stability of motions near resonances in quasi-integrable Hamiltonian systems. J. Stat. Phys. 44, 293–338 (1985)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Bounemoura, A., Marco, J.P.: Improved exponential stability for near-integrable quasi-convex Hamiltonians. Nonlinearity 24(1), 97–112 (2011)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Gallavotti, G.: Quasi-integrable mechanical systems. In: Osterwalder, K., Stora, R. (eds.) Critical Phenomena, Random Systems, Gauge Theories. Les Houches, Session XLIII, 1984. North-Holland, Amsterdam (1986)Google Scholar
  6. 6.
    Guzzo, M., Chierchia, L., Benettin, G.: The steep Nekhoroshev’s theorem and optimal stability exponents (an announcement). Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. 25, 293–299 (2014)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Guzzo, M., Chierchia, L., Benettin, G.: The steep Nekhoroshev’s theorem. Commun. Math. Phys. 342, 569–601 (2016)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Guzzo, M., Lega, E., Froeschlé, C.: First numerical investigation of a conjecture by N.N. Nekhoroshev about stability in quasi-integrable systems. Chaos 21(3), 1–13 (2011). paper 033101MathSciNetCrossRefGoogle Scholar
  9. 9.
    Guzzo, M.: The Nekhoroshev theorem and long-term stabilities in the solar system. Serb. Astron. J. 190, 1–10 (2015)CrossRefGoogle Scholar
  10. 10.
    Lhotka, Ch., Efthymiopoulos, C., Dvorak, R.: Nekhoroshev stability at \(L_4\) or \(L_5\) in the elliptic-restricted three-body problem—application to Trojan asteroid. Mon. Not. R. Astron. Soc. 384(3), 1165–1177 (2008)CrossRefGoogle Scholar
  11. 11.
    Lochak, P.: Canonical perturbation theory via simultaneous approximations. Russ. Math. Surv. 47, 57–133 (1992)CrossRefGoogle Scholar
  12. 12.
    Lochak, P., Neishtadt, A.: Estimates in the theorem of N.N. Nekhoroshev for systems with quasi-convex Hamiltonian. Chaos 2, 495–499 (1992)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Morbidelli, A., Guzzo, M.: The Nekhoroshev theorem and the asteroid belt dynamical system. Celest. Mech. Dyn. Astron. 65, 107–136 (1997)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Nekhoroshev, N.N.: Behavior of Hamiltonian systems close to integrability. Funct. Anal. Appl. 5, 338–339 (1971)CrossRefGoogle Scholar
  15. 15.
    Nekhoroshev, N.N.: Behavior of Hamiltonian systems close to integrability. Funk. An. Ego Prilozheniya 5, 82–83 (1971)Google Scholar
  16. 16.
    Nekhoroshev, N.N.: Stable lower estimates for smooth mappings and for gradients of smooth functions. Math USSR Sbornik 19, 425–467 (1973)CrossRefGoogle Scholar
  17. 17.
    Nekhoroshev, N.N.: An exponential estimate of the time of stability of nearly-integrable Hamiltonian systems I. Uspekhi Mat. Nauk 32, 5–66 (1977)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Nekhoroshev, N.N.: An exponential estimate of the time of stability of nearly-integrable Hamiltonian systems I. Russ. Math. Surv. 32, 1–65 (1977)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Nekhoroshev, N.N.: An exponential estimate of the time of stability of nearly-integrable Hamiltonian systems II. Tr. Semin. Petrovsk. 5, 5–50 (1979)MathSciNetGoogle Scholar
  20. 20.
    Nekhoroshev N.N.: In: Oleinik, O.A. (ed.) Topics in Modern Mathematics, Petrovskii Seminar, No. 5. Consultant Bureau, New York (1985)Google Scholar
  21. 21.
    Niederman, L.: Hamiltonian stability and subanalytic geometry. Ann. Inst. Fourier (Grenoble) 56(3), 795–813 (2006)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Pavlovic, R., Guzzo, M.: Fulfillment of the conditions for the application of the Nekhoroshev theorem to the Koronis and Veritas asteroid families. Mon. Not. R. Astron. Soc. 384, 1575–1582 (2008)CrossRefGoogle Scholar
  23. 23.
    Pinzari, G.: Aspects of the planetary Birkhoff normal form. Regul. Chaotic Dyn. 18(6), 860–906 (2013)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Pöschel, J.: Nekhoroshev estimates for quasi-convex hamiltonian systems. Math. Z. 213, 187 (1993)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Sansottera, M., Locatelli, U., Giorgilli, A.: On the stability of the secular evolution of the planar Sun-Jupiter-Saturn-Uranus system. Math. Comput. Simul. 88, 1–14 (2013)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Schirinzi, G., Guzzo, M.: On the formulation of new explicit conditions for steepness from a former result of N.N. Nekhoroshev. J. Math. Phys. 54(072702), 1–22 (2013)MathSciNetzbMATHGoogle Scholar
  27. 27.
    Schirinzi, G., Guzzo, M.: Numerical verification of the steepness of three and four degrees of freedom hamiltonian systems. Regul. Chaotic Dyn. 20(1), 1–18 (2015)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Todorović, N., Guzzo, M., Lega, E., Froeschlé, Cl: A numerical study of the stabilization effect of steepness. Celest. Mech. Dyn. Astr. 110, 389–398 (2011)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Zhang, Ke: Speed of Arnold diffusion for analytic Hamiltonian systems. Invent. Math. 186(2), 255–290 (2011)MathSciNetCrossRefGoogle Scholar

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© Fondazione Annali di Matematica Pura ed Applicata and Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Dipartimento di Matematica e FisicaUniversità degli Studi Roma TreRomeItaly
  2. 2.UBS - ZürichZurichSwitzerland
  3. 3.Dipartimento di Matematica “Tullio Levi-Civita”Università degli Studi di PadovaPaduaItaly

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