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An Overview on the Nekhoroshev Theorem

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Topics in Gravitational Dynamics

Part of the book series: Lecture Notes in Physics ((LNP,volume 729))

Abstract

The Nekhoroshev theorem has provided in the last decades an important framework to study the long-term stability of quasi-integrable dynamical systems. While KAM theorem predicts the stability of invariant tori for non–resonant motions, the Nekhoroshev theorem provides exponential stability estimates for resonant motions. The ?rst part of this chapter reviews the mechanisms which are at the basis of the proof of Nekhoroshev theorem.

The application of Nekhoroshev theorem to systems of interest for Celestial Mechanics encounters di?culties due to the presence of the so–called proper degeneracy, or super–integrability, of the Kepler problem. For this reason, for concrete systems such as planetary systems or asteroids of the main belt, the application of the Nekhoroshev theorem requires modi?cations of the theory. At variance with the non-degenerate case, these systems can have chaotic di?usion in relatively short times. The second part of this chapter reviews the problems related to degenerate systems, and describes the mechanism of production of chaotic motions in short times.

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References

  1. Andoyer, H., 1923 Cours de Mécanique Céleste. Gauthier–Villars, Paris.

    MATH  Google Scholar 

  2. Arnol’d, V. I., 1976, Méethodes Mathéematiques de la Méechanique Classique. MIR, Moscow.

    Google Scholar 

  3. Arnol’d, V. I., 1963 Proof of a theorem by A. N. Kolmogorov on the invariance of quasi-periodic motions under small perturbations of the Hamiltonian. Russ. Math. Surv. 18(9).

    Google Scholar 

  4. Arnol’d, V. I., 1963, Small denominators and problems of stability of motion in classical and celestial mechanics. Russ. Math. Surveys 18, 85–191.

    Article  MATH  ADS  Google Scholar 

  5. Benettin, G. and Fassò, F., 1996, Fast rotations of the rigid body: A study by Hamiltonian perturbation theory. Part I. Nonlinearity 9, 137–186.

    Article  MATH  ADS  MathSciNet  Google Scholar 

  6. Benettin, G., Fassò, F., and Guzzo, M., 1997, Fast rotations of the rigid body: A study by Hamiltonian perturbation theory. Part II: Gyroscopic rotations. Nonlinearity 10, 1695–1717.

    Article  MATH  ADS  MathSciNet  Google Scholar 

  7. Benettin, G., Fassò, F., and Guzzo, M., 1998, Nekhoroshev-stability of L4 and L5 in the spatial restricted three–body problem. Regular Chaotic Dyn., 3(3).

    Google Scholar 

  8. Benettin, G., Cherubini, A. M., and Fassò F., 2002, Regular and chaotic motions of the fast rotating rigid body: A numerical study. Discrete Cont. Dyn. Syst., Series B 2, 521–540.

    Article  MATH  Google Scholar 

  9. Benettin, G., Fassò, F., and Guzzo, M., 2005, Long–term stability of proper rotations and local chaotic motions in the perturbed Euler rigid body. Regular Chaotic Mech., 11, 267–284.

    Google Scholar 

  10. Benettin, G., Galgani, L., and Giorgilli A., 1985, A proof of Nekhoroshev’s theorem for the stability times in nearly integrable Hamiltonian systems. Cel. Mech. 37, 1.

    Article  MATH  ADS  MathSciNet  Google Scholar 

  11. Benettin, G., and Gallavotti, G., 1986, Stability of motions near resonances in quasi–integrable Hamiltonian systems. J. Stat. Phys. 44, 293–338.

    Article  MATH  MathSciNet  ADS  Google Scholar 

  12. Bretagnon, P., 1974, Termesà longues périodes dans le système solaire. Astron. Astrophys. 30, 141–154.

    ADS  Google Scholar 

  13. Biasco, L., Chierchia, L., 2005, Exponential stability for the resonant d’Alembert model of celestial mechanics. Discrete Contin. Dyn. Syst. 12(4), 569–594.

    Google Scholar 

  14. Biasco, L., Chierchia, L., and Treschev, D., 2006, Stability of nearly integrable, degenerate Hamiltonian systems with two degrees of freedom. J. Nonlinear Sci. 16(1), 79–107.

    Article  MATH  ADS  MathSciNet  Google Scholar 

  15. Celletti A., and Chierchia L., 1995, A constructive theory of Lagrangian tori and computer-assisted applications. Dynamics Reported, New Series, Springer Verlag 4.

    Google Scholar 

  16. Celletti A., and Chierchia L., 1997, On the stability of realistic three-body problems. Comm. Math. Phys. 186(2),413–449.

    Article  MATH  ADS  MathSciNet  Google Scholar 

  17. Celletti, A., Giorgilli, A., and Locatelli U., 2000, Improved estimates on the existence of invariant tori for Hamiltonian systems. Nonlinearity 13(2),397–412.

    Article  MATH  ADS  MathSciNet  Google Scholar 

  18. Dazord, P., and Delzant, T., 1987, Le problème général des variables actions–angles. J. Diff. Geom. 26, 223–251.

    MATH  MathSciNet  Google Scholar 

  19. Deprit, A., 1967, Free rotations of a rigid body studied in phase plane. Am. J. Phys. 55, 424–428.

    Article  ADS  Google Scholar 

  20. Dullin, H. and Fassó F., 2004, An algorithm for detecting Directional Quasi-Convexity. BIT—Numer. Math. 44,571–584.

    MATH  Google Scholar 

  21. Fassò, F., 1995, Hamiltonian perturbation theory on a manifold. Cel. Mech. Dyn. Astron. 62, 43.

    Article  MATH  ADS  Google Scholar 

  22. Fassò, F., 2005, Superintegrable Hamiltonian systems: Geometry and perturbations. Acta Appl. Math. 87, 93–121.

    Article  MATH  MathSciNet  Google Scholar 

  23. Fassò, F., Guzzo, M., and Benettin, G., 1998, Nekhoroshev-Stability of Elliptic Equilibria in Hamiltonian Systems, Communications in Mathematical Physics, 197, 34–360.

    Google Scholar 

  24. Gallavotti, G., 1986, Quasi–integrable Mechanical systems. In Osterwalder, K. and Stora, R. (eds). Critical phenomena, andom systems, Gauge theories, North Olland, Amsterdam.

    Google Scholar 

  25. Giorgilli, A., Delshams, A., Fontich, E., Galgani, L., and Simó, C., 1989, Effective stability for a Hamiltonian system near an elliptic equilibrium point, with an application to the restricted three body problem. J. Diff. Eq. 77, 167–198.

    Article  MATH  Google Scholar 

  26. Giorgilli, A. and Skokos, C., 1997, On the stability of the Trojan asteroids. Astron. Astrophys. 17, 254–261.

    ADS  Google Scholar 

  27. Guzzo, M., Lega, E., and Froeschlé, C., 2006, Diffusion and stability in perturbed non-convex integrable systems. Nonlinearity 19, 1049–1067.

    Article  MathSciNet  Google Scholar 

  28. Guzzo, M., Knezevic, Z., and Milani, A., 2002, Probing the Nekhoroshev Stability of Asteroids. Celest. Mech. Dyn. Astroc. 831–4.

    Article  MathSciNet  Google Scholar 

  29. Guzzo, M. and Morbidelli, A., 1997, Construction of a Nekhoroshev like result for the asteroid belt dynamical system. Cel. Mech. Dyn. Astron. 66, 255–292.

    Google Scholar 

  30. Guzzo, M., 1999, Nekhoroshev stability of quasi–integrable degenerate Hamiltonian systems. Regular Chaotic Dyn. 4(2).

    Google Scholar 

  31. Guzzo, M., 2005, The web of three–planets resonances in the outer Solar System. Icarus 174(1), 273–284.

    Article  ADS  MathSciNet  Google Scholar 

  32. Guzzo, M. 2006, The web of three-planet resonances in the outer solar system II: A source of orbital instability for Uranus and Neptune. Icarus 181, 475–485.

    Article  ADS  Google Scholar 

  33. Kolmogorov A. N., 1954, On the conservation of conditionally periodic motions under small perturbation of the hamiltonian. Dokl. Akad. Nauk. SSSR 98, 524.

    MathSciNet  Google Scholar 

  34. Kozlov, V. V., 1974, Geometry of “action–angle” variables in Euler–Poinsot problem. Vestn. Moskov. Univ., Math. Mekh. 29, 74–79.

    MATH  Google Scholar 

  35. Laskar, J., 1990, The chaotic motion of the solar system. A numerical estimate of the size of the chaotic zones. Icarus 88, 266–291.

    Article  ADS  Google Scholar 

  36. Laskar, J., Froeschlé, C., and Celletti, A., 1992, The measure of chaos by the numerical analysis of the fundamental frequencies. Application to the standard mapping. Physica D 56,253.

    Article  ADS  MathSciNet  MATH  Google Scholar 

  37. Laskar, J., 1996, Large scale chaos and marginal stability in the Solar System. Celest. Mech. Dyn. Astron. 64(1/2),115–162.

    Article  MATH  ADS  MathSciNet  Google Scholar 

  38. Lochak, P., 1992, Canonical perturbation theory via simultaneous approximation. Russ. Math. Surv. 47, 57–133.

    Article  MathSciNet  Google Scholar 

  39. Locatelli, U. and Giorgilli, A., 2000, Invariant tori in the secular motions of the three–body planetary system. Cel. Mech. Dyn.Astron. 78(1–4), 47–74.

    Article  MATH  ADS  MathSciNet  Google Scholar 

  40. Mischenko, A. S. and Fomenko, A. T., 1978, Generalized Liouville method of integration of Hamiltonian systems. Funct. Anal. Appl. 12, 113–121.

    Article  Google Scholar 

  41. Morbidelli, A. and Guzzo, M., 1997, The Nekhoroshev theorem and the asteroid belt dynamical system. Cel. Mech. Dyn. Astron. 65, 107–136.

    Article  MATH  ADS  MathSciNet  Google Scholar 

  42. Moser, J., 1958, On invariant curves of area-preserving maps of an annulus. Comm. Pure Appl. Math. 11, 81–114.

    Article  MATH  MathSciNet  Google Scholar 

  43. Murray, N. and Holman, M., 1999, The origin of chaos in the Outer Solar System. Science 283, 1877–1881.

    Article  ADS  Google Scholar 

  44. Nekhoroshev, N. N., 1977, Exponential estimates of the stability time of near-integrable hamiltonian systems. Russ. Math. Surveys 32, 1–65.

    Article  MATH  ADS  Google Scholar 

  45. Nekhoroshev, N. N., 1979, Exponential estimates of the stability time of near–integrable Hamiltonian systems, 2. Trudy Sem. Petrovs. 5, 5–50.

    MATH  MathSciNet  Google Scholar 

  46. Nekhoroshev, N. N., 1972, Action–angle variables and their generalizations. Trudy Moskow Mat. Obsc. 26, 181–198 (Russian). English transl. in Moskow Math. Soc. 26, 180–198.

    Google Scholar 

  47. Neishtadt, A. I., 1987, Jumps of the adiabatic invariant on crossing the separatrix and the origin of the 3:1 Kirkwood gap. Dokl. Akad. Nauk SSSR 295, 47–50 (Russian). English transl. in Sov. Phys. Dokl. 32, 571–573.

    Google Scholar 

  48. Nesvorny, D. and Morbidelli, A., 1998, Three–body mean motion resonances and the chaotic structure of the asteroid belt. Astron. J. 116, 3029–3037.

    Article  ADS  Google Scholar 

  49. Nesvorny, D. and Morbidelli, A., 1998, An analytic model of three–body mean motion resonances. Celest. Mech. Dyn. Astron. 71, 243–271.

    Article  ADS  MathSciNet  Google Scholar 

  50. Neishtadt, A. I., 1987, On the change in the adiabatic invariant on crossing a separatrix in systems with two degrees of freedom. Prikl. Matem. Mekhan 51(5) 750–757. PMM USSR 51(5), 586–592.

    ADS  MathSciNet  Google Scholar 

  51. Neishtadt, A. I., and Sidorenko, V. V., 2004, Wisdom system: Dynamics in the adiabatic approximation. Cel. Mech. Dyn. Astron. 90(3–4), 307–330.

    Article  MATH  ADS  MathSciNet  Google Scholar 

  52. Niederman, L., 1996, Stability over exponentially long times in the planetary problem. Nonlinearity 9(6), 1703–1751.

    Article  MATH  ADS  MathSciNet  Google Scholar 

  53. Niederman, L., 2004, Exponential stability for small perturbations of steep integrable Hamiltonian systems. Ergodic Theory Dyn.Syst. 24(2), 593–608.

    Article  MATH  MathSciNet  Google Scholar 

  54. Niederman, L. 2007, Prevalence of exponential stability among nearly integrable Hamiltonian systems. Ergodic Theory Dyn. Syst 27, 905–928.

    Article  MATH  MathSciNet  Google Scholar 

  55. Niederman, L., 2006, Hamiltonian stability and subanalytic geometry. Ann. Inst. Fourier (Grenoble) 56(3), 795–813.

    MATH  MathSciNet  Google Scholar 

  56. Poschel, J., 1993, Nekhoroshev estimates for quasi–convex hamiltonian systems. Math. Z. 213, 187.

    Article  MathSciNet  Google Scholar 

  57. Poincaré, H. 1982, Les méthodes nouvelles de la mécanique celeste. Gauthier–Villars, Paris.

    Google Scholar 

  58. Sussman, G. J. and Wisdom, J., 1992, Chaotic evolution of the Solar System. Science 257, 56–62.

    Article  ADS  MathSciNet  Google Scholar 

  59. Williams, J. G., 1969, Secular perturbations in the solar system. Ph.D dissertation, University of California, Los Angeles.

    Google Scholar 

  60. Wisdom, J., 1983, Chaotic behavior and the origin of the 3/1 Kirkwood gap. Icarus 56, 51–74.

    Article  ADS  Google Scholar 

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Authors and Affiliations

  1. Dipartimento di Matematica Pura ed Applicata, Università degli Studi di Padova, via Trieste 63, 35121 Padova, Italy

    Massimiliano Guzzo

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  1. Massimiliano Guzzo
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  1. OCA - Nice, Observatoire de la Cote d’Azur, 06304 Nice CX 4, France

    Daniel Benest , Claude Froeschle  & Elena Lega ,  & 

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Guzzo, M. (2007). An Overview on the Nekhoroshev Theorem. In: Benest, D., Froeschle, C., Lega, E. (eds) Topics in Gravitational Dynamics. Lecture Notes in Physics, vol 729. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-72984-6_1

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