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The Poincaré–Lyapounov–Nekhoroshev Theorem

✍ Scribed by Giuseppe Gaeta

Publisher
Elsevier Science
Year
2002
Tongue
English
Weight
169 KB
Volume
297
Category
Article
ISSN
0003-4916

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✦ Synopsis

We give a detailed and mainly geometric proof of a theorem by N. N. Nekhoroshev for hamiltonian systems in n degrees of freedom with k constants of motion in involution, where 1 ≤ k ≤ n. This state's persistence of k-dimensional invariant tori, and local existence of partial action-angle coordinates, under suitable nondegeneracy conditions. Thus it admits as special cases the Poincaré-Lyapounov theorem (corresponding to k = 1) and the Liouville-Arnold one (corresponding to k = n) and interpolates between them. The crucial tool for the proof is a generalization of the Poincaré map, also introduced by Nekhoroshev.

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