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Birkhoff normalization and superintegrability of Hamiltonian systems

Published online by Cambridge University Press:  02 March 2009

HIDEKAZU ITO
HIDEKAZU ITO*
Affiliation:
Division of Mathematics and Physics, Graduate School of Natural Science and Technology, Kanazawa University, Kakuma, Kanazawa, 920-1192, Japan (email: [email protected])
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Abstract

We study Birkhoff normalization in connection with superintegrability of an n-degree-of-freedom Hamiltonian system XH with holomorphic Hamiltonian H. Without assuming any Poisson commuting relation among integrals, we prove that, if the system XH has n+q holomorphic integrals near an equilibrium point of resonance degree q≥0, there exists a holomorphic Birkhoff transformation φ such that H∘φ becomes a holomorphic function of n−q variables and that XH∘φ can be solved explicitly. Furthermore, the Birkhoff normal form H∘φ is determined uniquely, independently of the choice of φ, as convergent power series. We also show that the system XH is superintegrable in the sense of Mischenko–Fomenko as well as Liouville-integrable near the equilibrium point.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 29 , Issue 6 , December 2009 , pp. 1853 - 1880
Copyright
Copyright © Cambridge University Press 2009

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