Abstract
We investigate the possibility of obtaining a Nekhoroshev like result for the dynamical system describing the motion of an asteroid in the main belt, From the mathematical point of view this is a new result since the problem is degenerate and we want to control also the motion of degenerate actions, We find that there are regions, such as the resonances of low order among the fast angles (mean motion resonances), where a Nekhoroshev like result cannot be proved a priori, Conversely, we are able to confine the motions in the mean motion resonances of logarithmically large order in the perturbation parameters, as well as in the non-resonant region, We discuss also the connection with the existence of invariant tori.
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References
Arnold, V. I.: 1963a, Proof of A. N. Kolmogorov's Theorem on the conservation of conditionally periodic motions with a small variation in the Hamiltonian, Russian Math. Surv., 18(5), 9–36.
Arnold, V. I.: 1963b, Small denominators and problems of stability of motion in classical and celestial mechanics. Russ. Math. Surveys, 18, 85–191.
Arnold, V. I.: 1963c, On a theorem of Liouville concerning integrable problems of dynamics, Sib. Mathem. Zh., 4(2).
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(l).
Benettin, G. and Gallavotti G.: 1986, Stability of motions near resonances in quasi-integrable Hamiltonian systems, J. Stat. Phys., 44, 293–338.
Benettin, G. and Fassb F.: 1995, Fast rotations of the rigid body: a study by Hamiltonian perturbation theory. Part I, Nonlinearity, 9, 137–186.
Benettin, G., Fassò F. and Guzzo M.: 1996, Fast rotations of the rigid body: a study by Hamiltonian perturbation theory, Part II: Gyroscopic Rotations preprint.
Bretagnon, P.: 1974, ‘Termes à longues périodes dans le système solaire’, Astron. Astrophys. 30, 141–154.
Fassò, F.: 1995, Hamiltonian perturbation theory on a manifold, Cel. Mech. Dyn. Astron. 62(43).
Gallavotti, G.: 1986, Quasi-Integrable Mechanical Systems, critical phenomena, random systems, gauge theories, K. Osterwalder and R. Stora (eds), (Amsterdam: North Holland).
Giorgilli, A. and Morbidelli A.: 1995, Invariant KAM tori and global stability for Hamiltonian systems, submitted to ZAMP.
Giorgilli, A. and Zehnder E.: 1992, Exponentially stability for time dependent potentials, ZAMP, 43, 827–855.
Kolmogorov, A. N.: 1954, On the preservation of conditionally periodic motions, Dokl. Akad. Nauk SSSR, 98, 527.
Kozai, Y.: 1962, Secular perturbations of asteroids with high inclination and eccentricities, Astron. J., 67, 591–598.
Laskar, J.: 1988, Secular evolution of the solar system over 10 million years, Astron. Astrophys., 198, 341–362.
Laskar, J.: 1990, The chaotic motion of the solar system: a numerical estimate of the size of the chaotic zones, Icarus, 88, 266–291.
Laskar, J.: 1995, Large scale chaos and marginal stability in the Solar System, XIth ICMP Colloquium, Paris.
Lochak, P.: 1992, Canonical perturbation theory via simultaneous approximations, Usp. Math. Nauk., 47, 59–140. English transl. in Russ. Math, Surv.
Morbidelli, A. and Henrard, J.: 1991, Secular resonances in the asteroid belt: theoretical perturbation approach and the problem of their location, Celest. Mech., 51, 131–167.
Morbidelli, A. and Giorgilli, A.: 1995, On a connection between KAM and Nekhoroshev Theorems, Physica D, 86, 514–516.
Morbidelli, A. and Guzzo, M.: 1996, The Nekhoroshev Theorem and the asteroid belt dynamical system, Celest. Mech., submitted.
Morbidelli A. and Froeschlé C.: 1996, On the relationship between Lyapunov times and macroscopic instability times Celest. Mech., in press.
Moser, J.: 1958, New aspects in the theory of stability of hamiltonian systems, Comm. on Pure and Appl. Math., 11, 81–114.
Nekhoroshev, N. N.: 1977, Exponential estimates of the stability time of near-integrable Hamiltonian systems, Russ. Math, Surveys, 32, 1–65.
Nekhoroshev, N. N.: 1979, Exponential estimates of the stability time of near-integrable Hamiltonian systems, 2, Trudy Sem. Petrovs., 5, 5–50.
Niederman, L.: 1993, Résonances et stabilité dans le problème planétaire, Ph.D. thesis, Univ. de Paris VI.
Niederman, L.: 1996, Stability over exponentially long times in the planetary problem, Nonlinearity, 9, 1703–1751.
Nobili, A., Milani, A. and Carpino, M.: 1989, Fundamental frequencies and small divisors in the orbits of the outer planets, Astron. Astrophys., 210, 313–336.
Pöschel, J.: 1993, Nekhoroshev's estimates for quasi-convex Hamiltonian systems, Math. Z., 213, 187.
Poincaré, H.: 1892, Les Méthodes Nouvelles de la Mécanique Céleste, Gauthier-Villars, Paris.
Robutel, P.: 1995, Stability of the planetary three-body problem, II: KAM theory and the existence of quasiperiodic motions, Celest. Mech., 62, 219–261.
Williams, J. G.: 1969, Secular perturbations in the solar system, Ph.D. dissertation, University of California, Los Angeles.
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Guzzo, M., Morbidelli, A. Construction of a Nekhoroshev like result for the asteroid belt dynamical system. Celestial Mech Dyn Astr 66, 255–292 (1996). https://doi.org/10.1007/BF00049382
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DOI: https://doi.org/10.1007/BF00049382