tailieunhanh - Đề tài " Convergence versus integrability in Birkhoff normal form "

We show that any analytically integrable Hamiltonian system near an equilibrium point admits a convergent Birkhoff normalization. The proof is based on a new, geometric approach to the topic. 1. Introduction Among the fundamental problems concerning analytic (real or complex) Hamiltonian systems near an equilibrium point, one may mention the following two: 1) Convergent Birkhoff. In this paper, by “convergent Birkhoff” we mean a normalization, ., a local analytic symplectic system of coordinates in which the Hamiltonian function will Poisson commute with the semisimple part of its quadratic part. . | Annals of Mathematics Convergence versus integrability in Birkhoff normal form By Nguyen Tien Zung Annals of Mathematics 161 2005 141 156 Convergence versus integrability in Birkhoff normal form By Ngwen Tien Zung Abstract We show that any analytically integrable Hamiltonian system near an equilibrium point admits a convergent Birkhoff normalization. The proof is based on a new geometric approach to the topic. 1. Introduction Among the fundamental problems concerning analytic real or complex Hamiltonian systems near an equilibrium point one may mention the following two 1 Convergent Birkhoff. In this paper by convergent Birkhoff we mean a normalization . a local analytic symplectic system of coordinates in which the Hamiltonian function will Poisson commute with the semisimple part of its quadratic part. 2 Analytic integrability. By analytic integrability we mean of a complete set of local analytic functionally independent first integrals in involution. These concepts have been studied by many classical and modern mathematicians including Poincare Birkhoff Siegel Moser Bruno etc. In this paper we will be concerned with the relations between the two. The starting point is that since both the Birkhoff normal form and the first integrals are ways to simplify and solve Hamiltonian systems these two must be very closely related. Indeed it was known to Birkhoff 2 that for nonresonant Hamiltonian systems convergent Birkhoff implies analytic integrability. The inverse is also true though much more difficult to prove 9 . What has been known to date concerning convergent Birkhoff vs. analytic integrability may be summarized in the following list. Denote by q q 0 the degree of resonance see Section 2 for a definition of an analytic Hamiltonian system at an equilibrium point. Then we have a When q 0 . for nonresonant systems convergent Birkhoff is equivalent to analytic integrability. The implication is straightforward. The inverse has been a difficult problem. Under an

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