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Properly-degenerate KAM theory (following V. I. Arnold). (English) Zbl 1282.37033
Summary: V. I. Arnold’s “Fundamental Theorem” on properly-degenerate systems [Russ. Math. Surv. 18, No. 6, 85–191 (1963); translation from Usp. Mat. Nauk 18, No. 6(114), 91–192 (1963), Chapter IV] is revisited and improved with particular attention to the relation between the perturbative parameters and to the measure of the Kolmogorov set. Relations with the planetary many-body problem are shortly discussed.

##### MSC:
 37J40 Perturbations of finite-dimensional Hamiltonian systems, normal forms, small divisors, KAM theory, Arnol’d diffusion 70F10 $$n$$-body problems 70F15 Celestial mechanics 70H08 Nearly integrable Hamiltonian systems, KAM theory
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