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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.

MSC:

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

Citations:

Zbl 0383.70023
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References:

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