Mould Calculus for Hamiltonian Vector Fields

2008/01/18 by Cresson, Jacky, Morin, Guillaume
#17A50 #17B40 #17B66 #17B70 #37G05 #37J40 #Dynamical Systems (math.DS) #FOS: Mathematics

paper · doi:10.48550/arxiv.0801.2953

Abstract

We present the general framework of Écalle'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 Écalle'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.

Related

• Calcul Moulien
• Initiation to mould calculus through the example of saddle-node singularities
• Mould expansions for the saddle-node and resurgence monomials
• Normal forms for perturbations of systems possessing a Diophantine invariant torus
• Normal forms of ω-Hamiltonian vector fields with symmetries