Mould Calculus for Hamiltonian Vector Fields

TL;DR

This paper develops a mould calculus framework for Hamiltonian vector fields, enabling systematic normalization and proving a Kolmogorov theorem near Diophantine tori using these techniques.

Contribution

It adapts Ecalle's mould calculus to Hamiltonian systems, providing a universal, calculable method for formal normalization and proving a Kolmogorov theorem.

Findings

Mould calculus can produce successive canonical transformations for Hamiltonian normal forms.

The framework applies to Hamiltonian vector fields in Cartesian coordinates.

A Kolmogorov theorem is proved using mould techniques for systems near Diophantine tori.

Abstract

We present the general framework of \'Ecalle's moulds in the case of linearization of a formal vector field without and within resonances. We enlighten the power of moulds by their universality, and calculability. We modify then \'Ecalle's technique to fit in the seek of a formal normal form of a Hamiltonian vector field in cartesian coordinates. We prove that mould calculus can also produce successive canonical transformations to bring a Hamiltonian vector field into a normal form. We then prove a Kolmogorov theorem on Hamiltonian vector fields near a diophantine torus in action-angle coordinates using moulds techniques.