Normal form for lower dimensional elliptic tori in Hamiltonian systems

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.

Automated summary

We provide a proof of convergence for an algorithm used to construct lower dimensional elliptic tori in near-integrable Hamiltonian systems. The existence of such invariant tori is demonstrated by transforming the Hamiltonian into a suitable normal form. We adapt a previously described procedure, originally used in the context of the planetary problem, and extend the proof of convergence to cases where the two sets of frequencies have similar magnitudes.