Normal forms of vector fields on Poisson manifolds

Philippe Monnier,Nguyen Tien Zung

doi:10.5802/ambp.221

Abstract

We study formal and analytic normal forms of radial and Hamiltonian vector fields on Poisson manifolds near a singular point.

Highlights

We study formal and analytic normal forms of radial and Hamiltonian vector fields on Poisson manifolds near a singular point
This paper is devoted to the study of normal forms à la Poincaré-Birkhoff for analytic or formal vector fields on Poisson manifolds
Our motivation for studying radial vector fields comes from Jacobi structures [5], while the main motivation for studying Hamiltonian vector fields comes from Hamiltonian dynamics

Summary

Introduction

This paper is devoted to the study of normal forms à la Poincaré-Birkhoff for analytic or formal vector fields on Poisson manifolds. We will assume that our vector field X vanishes at a point, X(0) = 0, and that the linear part of Π or of its transverse structure at 0 corresponds to a semisimple Lie algebra. In this case, it is well known [12, 4] that Π admits a formal or analytic linearization in a neighborhood of 0. Let (Π, X) be an analytic homogeneous Poisson structure on Kn (where K is C or R) such that the linear part Π1 of Π corresponds to a semisimple Lie algebra. Analytic integrability means that all Hamiltonian functions and vector fields in question are analytic

Homogeneous Poisson structures

Formal linearization

Analytic linearization

Hamiltonian vector fields on Poisson manifolds

Formal Poincaré-Birkhoff normalization

Analytic normalization for integrable Hamiltonian systems