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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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##### References:
 [1] V.I. Arnold , ”Mathematical methods of classical mechanics” , 2 nd ed., Springer , 1989 . MR 997295 [2] V. I. ARNOLD (Ed.), ” Dynamical systems III , Encycl. Math. Sci. ”, vol. 3 , Springer , 1988 . MR 923953 [3] R.H. Cushman - L.M. Bates , ” Global Aspects of Classical Integrable Systems ”, Birkhäuser , 1997 . MR 1438060 | Zbl 0882.58023 · Zbl 0882.58023 [4] J.J. Duistermaat , On global action-angle coordinates , Comm. Pure Appl. Math. 33 ( 1980 ), 687 - 706 . MR 596430 | Zbl 0439.58014 · Zbl 0439.58014 [5] H. Eliasson , Normal form of Hamiltonian systems with Poisson commuting integrals- elliptic case , Comment. Math. Helv. 65 ( 1990 ), 4 - 35 . MR 1036125 | Zbl 0702.58024 · Zbl 0702.58024 [6] A.T. Fomenko (Ed.), ” Topological Classification of Integrable Systems ”, Adv. in Soviet Math. 6 AMS ( 1991 ). MR 1141218 | Zbl 0741.00026 · Zbl 0741.00026 [7] H. Grauert - R. Remmert , ” Theory of Stein Spaces ”, Grundl. Math. Wiss. 236 , Springer-Verlag , 1979 . MR 580152 | Zbl 0433.32007 · Zbl 0433.32007 [8] M. Hirsch , ” Differential Topology ”, Graduate Text in Mathematics , Springer-Verlag , 1976 . MR 448362 | Zbl 0356.57001 · Zbl 0356.57001 [9] H. Ito , Convergence of Birkhoff normal forms for integrable systems , Comment. Math. Helv. 64 ( 1989 ), 363 - 407 . MR 998858 | Zbl 0686.58021 · Zbl 0686.58021 [10] H. Ito , Integrability of Hamiltonian systems and Birkhoff normal forms in the simple resonance case , Math. Ann. 292 ( 1992 ), 411 - 444 . MR 1152944 | Zbl 0735.58022 · Zbl 0735.58022 [11] H. Ito , Action-angle coordinates at singularities for analytic integrable systems , Math. Z. 206 ( 1991 ), 363 - 407 . MR 1095762 | Zbl 0707.58026 · Zbl 0707.58026 [12] L. Kaup - B. Kaup , ”Holomorphic functions of several variables”, de Gruyter Studies in Mathematics 3 , Walter de Gruyter , Berlin - NewYork , 1983 . MR 716497 | Zbl 0528.32001 · Zbl 0528.32001 [13] E.J.N. Looijenga , ”Isolated singular points on complete intersections” , Cambridge University Press , 1984 . MR 747303 | Zbl 0552.14002 · Zbl 0552.14002 [14] M. Marcus , ”Finite dimensional multilinear algebra” , M. Dekker , New York , 1973 . MR 352112 · Zbl 0284.15024 [15] M. Marcus , ” Finite dimensional multilinear algebra, Part 2 ”, M. Dekker , New York , 1975 . MR 401796 | Zbl 0339.15003 · Zbl 0339.15003 [16] J. Moser , ”Lectures on Hamiltonian systems” , Memoir AMS , vol. 81 , 1968 . MR 230498 | Zbl 0172.11401 · Zbl 0172.11401 [17] Y. Sibuya , ”Linear differential equations in the complex domain: problems of analytic continuation” , vol. 82 , Translations of mathematical monographs AMS , 1990 . MR 1084379 | Zbl 00048899 · Zbl 1145.34378 [18] C.L. Siegel , On the integrals of canonical systems , Ann. of Math. 42 ( 1941 ), 806 - 822 . MR 5818 | Zbl 0025.26503 | JFM 67.0768.01 · Zbl 0025.26503 [19] J. Vey , Sur certain systèmes dynamiques séparables , Amer. J. Math. 100 ( 1978 ), 591 - 614 . MR 501141 | Zbl 0384.58012 · Zbl 0384.58012
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