Invariant KAM tori: from theory to applications to exoplanetary systems

TL;DR

This paper develops computational algorithms based on Hamiltonian normal forms to construct invariant tori, applying them to analyze the secular dynamics of an exoplanetary system with two planets orbiting HD 4732.

Contribution

It introduces explicit algorithms for constructing Kolmogorov and lower-dimensional elliptic invariant tori using Lie series, with applications to celestial mechanics.

Findings

Algorithms successfully approximate the secular dynamics of the HD 4732 system.

Extension of normal form methods to lower-dimensional elliptic tori.

Application demonstrates practical relevance in exoplanetary system analysis.

Abstract

We consider the classical problem of the construction of invariant tori exploiting suitable Hamiltonian normal forms. This kind of approach can be translated by means of the Lie series method into explicit computational algorithms, which are particularly suitable for applications in the field of Celestial Mechanics. First, the algorithm constructing the Kolmogorov normal form is described in detail. Then, the extension to lower-dimensional elliptic tori is provided. We adopt the same formalism and notations in both cases, with the aim of making the latter easier to understand. Finally, they are both used in a combined way in order to approximate carefully the secular dynamics of the extrasolar system hosting two planets orbiting around the HD 4732 star.