Stability of motions near resonances in quasi-integrable Hamiltonian systems. (English) Zbl 0636.70018

Summary: N. N. Nekhoroshev’s theorem on the stability of motions in quasi- integrable Hamiltonian systems [Usp. Mat. Nauk 32, No.6(198), 5-66 (1977; Zbl 0383.70023)] is revisited. At variance with the proofs already available in the literature, we explicitly consider the case of weakly perturbed harmonic oscillators, furthermore we prove the confinement of orbits in resonant regions, in the general case of nonisochronous systems, by using the elementary idea of energy conservation instead of more complicated mechanisms. An application of Nekhorosev’s theorem to the study of perturbed motions inside resonances is also provided.


70K30 Nonlinear resonances for nonlinear problems in mechanics
70H08 Nearly integrable Hamiltonian systems, KAM theory
70H14 Stability problems for problems in Hamiltonian and Lagrangian mechanics
37J40 Perturbations of finite-dimensional Hamiltonian systems, normal forms, small divisors, KAM theory, Arnol’d diffusion
37J25 Stability problems for finite-dimensional Hamiltonian and Lagrangian systems


Zbl 0383.70023
Full Text: DOI


[1] N. N. Nekhoroshev,Usp. Mat. Nauk 32:5 (1977) [Russ. Math. Surv. 32:1 (1977)];Tr. Sem. Petrows. No. 5, 5 (1979).
[2] G. Benettin, L. Galgani, and A. Giorgilli,Celest. Mech. 37:1 (1985). · Zbl 0602.58022 · doi:10.1007/BF01230338
[3] G. Gallavotti, lectures given at the 1984Les Hauches Summer School, to be published.
[4] C. E. Wayne,Commun. Math. Phys. 103:351 (1986);104:21 (1986). · Zbl 0596.58021 · doi:10.1007/BF01211753
[5] G. Benettin, L. Galgani, and A. Giorgilli,Nuovo Cimento B. 89:89 (1985). · doi:10.1007/BF02723539
[6] G. Benettin, L. Galgani, and A. Giorgilli,Nuovo Cimento B. 89:103 (1985). · doi:10.1007/BF02723540
[7] V. I. Arnold,Dokl. Akad. Nauk SSSR 156:9 (1964) [Sov. Math. Dokl. 6:581 (1964)].
[8] V. I. Arnold and A. Avez,Ergodic Problems of Classical Mechanics (Benjamin, New York, 1968). · Zbl 0167.22901
[9] H. Poincaré,Les Méthodes Nouvelles de la Méchanique Céleste, Vol. 3, Chapter 33 (Gauthier-Villars, Paris, 1899).
[10] G. Gallavotti,The Elements of Mechanics (Springer Verlag, Berlin 1983). · Zbl 0512.70001
[11] A. N. Kolmogorov,Dokl. Akad. Nauk. SSSR 98:527 (1954); English translation inLecture Notes in Physics, G. Casati and G. Ford, eds., No. 93 (Springer Verlag, Berlin, 1979).
[12] V. I. Arnold,Usp. Mat. Nauk 18:13 (1963) [Russ. Math. Sun. 18:9 (1963)];Usp. Mat. Nauk 18:91 (1963) [Russ. Math. Surv. 18:85 (1963)].
[13] J. Moser,Nachr. Akad. Wiss. Gottingen Math. Phys. Kl. 2:1 (1962).
[14] J. Poeschel,Commun. Pure Appl. Math. 35:653 (1982);Celest. Mech. 28:133 (1982). · Zbl 0542.58015 · doi:10.1002/cpa.3160350504
[15] L. Chierchia and G. Gallavotti,Nuovo Cimento B 67:277 (1982). · doi:10.1007/BF02721167
[16] V. K. Melnikov,Trans. Moscow Math. Soc. 12:1 (1963).
[17] J. Moser,Stable and Random Motions in Dynamical Systems (Princeton University Press, Princeton, 1973). · Zbl 0271.70009
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