The Hamiltonian Normal Form

TL;DR

This paper introduces the Hamiltonian normal form, a simplified normal form with poles along resonance loci, which improves upon the Birkhoff normal form for theoretical analysis and numerical approximations.

Contribution

It presents the Hamiltonian normal form in its simplest form, offering a potentially better approximation method than the Birkhoff normal form.

Findings

Expected to provide better approximations in numerical computations.

Simplifies the normal form used in the proof of the Herman invariant tori conjecture.

Builds on previous versions to clarify the Hamiltonian normal form.

Abstract

An important step in the proof of the Herman invariant tori conjecture was the introduction of a normal form with poles along the resonance loci, replacing the Birkhoff normal form, which we call the Hamiltonian normal form. This paper is extracted from previous versions (arXiv:1206.1245, arXiv:1909.06053) and aims to present this Hamiltonian normal form in its simplest form. It is expected that, not only theoretically but also in numerical computations, it will provide better approximations than the standard Birkhoff normal form.