Convergence of Birkhoff normal forms for integrable systems
The object of this article is to give a contribution to the study of normal forms of Hamiltonian systems near an equilibrium point of a periodic motion. It is considered analytic (or real analytic) Hamiltonian systems near a non-resonant equilibrium point (there is a canonical transformation which takes the Hamiltonian into the form \(H=\sum_{i}\lambda_ ix_ iy_ i-...\), \(i=1,...,n\) where (1) \((x,y)=z\) are coordinates in a neighborhood of the origin which is supposed to be in equilibrium and (2) the term not written out explicitly in H denotes a power series containing terms of order \(\geq 3)\). The higher order terms of H can be normalized by a canonical transformation f, called Birkhoff. In general f is divergent. A Hamiltonian system is called integrable if the Birkhoff transformation is convergent. Inspired by a result of \textit{J. Vey} [Am. J. Math. 100, 591-614 (1978; Zbl 0384.58012)] the author shows that if the system is integrable near a non- resonant equilibrium point then there is an analytic (convergent) Birkhoff transformation. A similar result is obtained for the real analytic case.
scientific article; zbMATH DE number 4124653
scientific article