Hostname: page-component-cd9895bd7-mkpzs Total loading time: 0 Render date: 2024-12-27T03:53:17.406Z Has data issue: false hasContentIssue false
  • English
  • Français

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])
Get access Rights & Permissions [Opens in a new window]

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Adler, M.. Some finite dimensional integrable systems and their scattering behavior. Comm. Math. Phys. 55 (1977), 195–230.CrossRefGoogle Scholar
[2]Arnol’d, V. I., Kozlov, V. V. and Neishtadt, A. I.. Mathematical Aspects of Classical and Celestial Mechanics, 3rd edn. (Encyclopaedia of Mathematical Sciences, 3). Springer, Berlin, 2006.CrossRefGoogle Scholar
[3]Bambusi, D. and Grébert, B.. Birkhoff normal form for partial differential equations with tame modulus. Duke. Math. J. 135 (2006), 507–567.CrossRefGoogle Scholar
[4]Bogoyavlenski, O. I.. Extended integrability and bi-Hamiltonian systems. Comm. Math. Phys. 196 (1998), 19–51.CrossRefGoogle Scholar
[5]Eliasson, H.. Normal forms for Hamiltonian systems with Poisson commuting integrals—elliptic case. Comment. Math. Helv. 65 (1990), 4–35.CrossRefGoogle Scholar
[6]Fassò, F.. Superintegrable Hamiltonian systems: geometry and perturbations. Acta Appl. Math. 87 (2005), 93–121.CrossRefGoogle Scholar
[7]Ito, H.. Convergence of Birkhoff normal forms for integrable systems. Comment. Math. Helv. 64 (1989), 412–461.CrossRefGoogle Scholar
[8]Ito, H.. Action-angle coordinates at singularities for analytic integrable systems. Math. Z. 206 (1991), 363–407.CrossRefGoogle Scholar
[9]Ito, H.. Integrability of Hamiltonian systems and Birkhoff normal forms in the simple resonance case. Math. Ann. 292 (1992), 411–444.CrossRefGoogle Scholar
[10]Kapeller, T., Kodama, Y. and Némethi, A.. On the Birkhoff normal form of a completely integrable Hamiltonian system near a fixed point with resonance. Ann. Scuola Norm. Sup. Pisa C1. Sci. (4) 26 (1998), 623–661.Google Scholar
[11]Kapeller, T. and Pöschel, J.. KdV & KAM. Springer, Berlin, 2003.CrossRefGoogle Scholar
[12]Karasev, M. V. and Maslov, B. P.. Nonlinear Poisson Brackets. Geometry and Quantization (Translations of Mathematical Monographs, 119). American Mathematical Society, Providence, RI, 1993.Google Scholar
[13]Miranda, E. and Zung, N. T.. Equivariant normal form for nondegenerate singular orbits of integrable Hamiltonian systems. Ann. Sci. École Norm. Sup. 37 (2004), 819–839.CrossRefGoogle Scholar
[14]Mischenko, A. S. and Fomenko, A. T.. Generalized Liouville method of integration of Hamiltonian systems. Funct. Anal. Appl. 12 (1978), 113–121.CrossRefGoogle Scholar
[15]Nekhoroshev, N. N.. Action-angle variables and their generalizations. Trans. Moscow Math. Soc. 26 (1972), 180–198.Google Scholar
[16]Pérez-Marco, R.. Convergence or generic divergence of the Birkhoff normal form. Ann. of Math. (2) 157 (2003), 557–574.CrossRefGoogle Scholar
[17]Rüssmann, H.. Über das Verhalten analytischer Hamiltonscher Differentialgleichungern in der Nähe einer Gleichgewichtslösung. Math. Ann. 154 (1964), 285–300.CrossRefGoogle Scholar
[18]Siegel, C. L.. Über die Existenz einer Normalform analytischer Hamiltonscher Differentialgleichungern in der Nähe einer Gleichgewichtslösung. Math. Ann. 128 (1954), 144–170.CrossRefGoogle Scholar
[19]Stolovitch, L.. Singular complete integrability. Publ. Math. Inst. Hautes Études Sci. 91 (2000), 133–210.CrossRefGoogle Scholar
[20]Stolovitch, L.. Normalisation holomorphe d’algèbres de type Cartan de champs de vecteurs holomorphes singuliers. Ann. of Math. (2) 161 (2005), 589–612.CrossRefGoogle Scholar
[21]Vey, J.. Sur certain systèmes dynamiques séparables. Amer. J. Math. 100 (1978), 591–614.CrossRefGoogle Scholar
[22]Zelditch, S.. The inverse spectral problem. Surv. Differ. Geom. IX (2004), 401–467.CrossRefGoogle Scholar
[23]Ziglin, S. L.. Branching of solutions and nonexistence of first integrals in Hamiltonian mechanics I. Funct. Anal. Appl. 16 (1983), 181–189.CrossRefGoogle Scholar
[24]Zung, N. T.. Convergence versus integrability in Birkhoff normal form. Ann. of Math. (2) 161 (2005), 141–156.CrossRefGoogle Scholar
[25]Zung, N. T.. Convergence versus integrability in Poincaré–Dulac normal form. Math. Res. Lett. 9 (2002), 217–228.CrossRefGoogle Scholar
[26]Zung, N. T.. Torus actions and integrable systems. Preprint, 2004, math/0407455v1.Google Scholar