Compactification d'actions de [formula omitted]k et systèmes hamiltoniens de type torique

Abstract

Let ( M, ω, H) be an hamiltonian system endowed with a finite dimensional space A of first integrals. The center of A for the Poisson bracket defines an infinitesimal hamiltonian action of R k on M. If O is a compact orbit of this action, we give a necessary and sufficient condition (involving the principal parts of functions in A at a point of O ) for the system to admit a normal form of ‘toric type’ in a neighbourhood of O . This result, conjectured in [7], generalizes the theorems of Arnol'd-Liouville [1], Eliasson [4], Nekhoroshev [8] and its version with singularities [2]. The key of the proof is a criterium of ‘compactification’ for R k -actions which extends a result of J.-P. Dufour and P. Molino [3].