Effective stability and KAM theory
This paper revisits two main stability results for nearly-integrable Hamiltonian systems. Nekhoroshev's theorem provides exponential lower bounds for the stability time. The theorem of Kolmogorov, Arnold and Moser (KAM), on the other hand, establishes conditions for the preservation of a majority of the nonresonant invariant tori. The former stability may be called ``effective stability'', and the latter ``perpetual stability''. To stress the relationships and interconnections between these two theorems, the authors provide proofs for both theorems using a common approach. The core of this approach consists of bringing the system to a normal form via Lie series methods. The authors' proof of Nekhoroshev's theorem provides an optimal stability exponent for quasi-convex systems. The authors' method provides a direct proof of the isoenergetic KAM theorem. In addition, a modification of this proof leads to the concept of the nearly-invariant torus, which connects the KAM and Nekhoroshev results more directly.
scientific article; zbMATH DE number 909999