Normal form for lower dimensional elliptic tori in Hamiltonian systems

Abstract

We give a proof of the convergence of an algorithm for the construction of lower dimensional elliptic tori in nearly integrable Hamiltonian systems. The existence of such invariant tori is proved by leading the Hamiltonian to a suitable normal form. In particular, we adapt the procedure described in a previous work by Giorgilli and co-workers, where the construction was made so as to be used in the context of the planetary problem. We extend the proof of the convergence to the cases in which the two sets of frequencies, describing the motion along the torus and the transverse oscillations, have the same order of magnitude.

accepted for publication in "Mathematics in Engineering"

References in corpus (6)

• Kolmogorov and Nekhoroshev theory for the problem of three bodies
• Computer-assisted estimates for Birkhoff normal forms
• Methods of algebraic manipulation in perturbation theory
• Elliptic tori in FPU non-linear chains with a small number of nodes
• Improved convergence estimates for the Schröder-Siegel problem
• Kolmogorov variation: KAM with knobs (à la Kolmogorov)

Cited by in corpus (2)

• Librational KAM tori in the secular dynamics of the $\upsilon$ Andromedæ planetary system
• A numerical criterion evaluating the robustness of planetary architectures; applications to the $\upsilon$ Andromedæ system