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On the Birkhoff normal form of a completely integrable Hamiltonian system near a fixed point with resonance. (English) Zbl 0919.58030

This paper is concerned with the normal form of a completely integrable Hamiltonian system near an equilibrium point. For the purpose of classification it is useful to generalize the concept of Birkhoff normal form to Hamiltonian systems near a resonant fixed point and one might ask whether an integrable Hamiltonian system has a Birkhoff normal form near a resonant fixed point. The authors deal with the general case of a simple resonance. They show that if the Hamiltonian admits a semisimple Hessian at an elliptic fixed point, then there exists a real analytic change of coordinates which brings the Hamiltonian into normal form. Then, in the new coordinates, the level sets of the system are analyzed in terms of the nature of the simple resonance.

MSC:

37J35 Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests
37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
37C85 Dynamics induced by group actions other than \(\mathbb{Z}\) and \(\mathbb{R}\), and \(\mathbb{C}\)
37G05 Normal forms for dynamical systems
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