Mould Calculus for Hamiltonian Vector Fields

Jacky Cresson (1, 2) , Guillaume Morin (2, 3)

1 LMAP - Laboratoire de Mathématiques et de leurs Applications [Pau]
2 IMCCE - Institut de Mécanique Céleste et de Calcul des Ephémérides
3 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision

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.

Keywords: normal form; continuous prenormal form; mould; mould calculus; Hamiltonian systems; Kolmogorov theorem

Domains: Dynamical Systems [math.DS]

Jacky Cresson, Guillaume Morin. Mould Calculus for Hamiltonian Vector Fields. 2008. ⟨hal-00207918⟩