Normal form for lower dimensional elliptic tori in Hamiltonian systems

TL;DR

This paper proves the convergence of an algorithm to construct lower dimensional elliptic tori in nearly integrable Hamiltonian systems, extending previous methods to cases with comparable frequency magnitudes.

Contribution

It extends the convergence proof of a normal form algorithm for elliptic tori to cases with similar frequency scales, applicable to planetary problems.

Findings

Convergence of the algorithm is established for a broader class of frequency ratios.

The normal form approach successfully constructs invariant tori in complex Hamiltonian systems.

The method adapts previous procedures for new dynamical scenarios.

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.