On the convergence of an algorithm constructing the normal form for lower dimensional elliptic tori in planetary systems

TL;DR

This paper provides a rigorous constructive proof of the existence and convergence of lower dimensional elliptic tori in nearly integrable Hamiltonian systems, specifically tailored for planetary systems, using an adapted Kolmogorov normalization algorithm.

Contribution

It adapts Kolmogorov's normalization to planetary systems and offers rigorous convergence estimates supporting previous semi-analytical results.

Findings

Rigorous convergence estimates for elliptic tori in planetary systems

Explicit calculation of an invariant torus for a Sun-Jupiter-Saturn-Uranus model

Slight relaxation of non-resonance conditions on frequencies

Abstract

We give a constructive proof of the existence of lower dimensional elliptic tori in nearly integrable Hamiltonian systems. In particular we adapt the classical Kolmogorov's normalization algorithm to the case of planetary systems, for which elliptic tori may be used as replacements of elliptic keplerian orbits in Lagrange-Laplace theory. With this paper we support with rigorous convergence estimates the semi-analytical work in our previous article (2011), where an explicit calculation of an invariant torus for a planar model of the Sun-Jupiter-Saturn-Uranus system has been made. With respect to previous works on the same subject we exploit the characteristic of Lie series giving a precise control of all terms generated by our algorithm. This allows us to slightly relax the non-resonance conditions on the frequencies.