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Small divisors with spatial structure in infinite dimensional Hamiltonian systems

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Abstract

A general perturbation theory of the Kolmogorov-Arnold-Moser type is described concerning the existence of infinite dimensional invariant tori in nearly integrable hamiltonian systems. The key idea is to consider hamiltonians with aspatial structure and to express all quantitative aspects of the theory in terms of rather general weight functions on such structures. This approach combines great flexibility with an effective control of the vrious interactions in infinite dimensional systems.

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References

  • [Arn-1] Arnold, V. I.: Proof of a theorem by A. N. Kolmogorov on the invariance of quasiperiodic motions under small perturbations of the Hamiltonian. Uspehi Math. Nauk18, 13–40 (1963); Russ. Math. Surv.18, 9–36 (1963)

    Google Scholar 

  • [Arn-2] Arnold, V. I.: Dynamical systems III, encyclopaedia of mathematical sciences, vol. 3. Berlin, Heidelberg, New York: Springer 1988

    Google Scholar 

  • [Brj] Brjuno, A. D.: Analytic form of differential equations. Trans. Moscow Math. Soc.25, 131–288 (1971)

    Google Scholar 

  • [FSW] Fröhlich, J., Spencer, T., Wayne, C. E.: Localization in disordered, nonlinear dynamical systems. J. Stat. Phys.42 247–274 (1986)

    Article  Google Scholar 

  • [Kol] Kolmogorov, A. N.: On the conservation of conditionally periodic motions for a small change in Hamilton's function. Dokl. Akad. Nauk SSSR98, 525–530 (1954)

    Google Scholar 

  • [Kuk] Kuksin, S. B.: Hamiltonian perturbations of infinite-dimensional linear systems with an imaginary spectrum. Funktsional. Anal. i Prilozhen.21, 22–37 (1987); Funct. Anal. Appl.21, 192–205 (1988)

    Google Scholar 

  • [Mos-1] Moser, J.: On invariant curves of area preserving mappings of an annulus. Nachr. Akad. Wiss. Gött. Math. Phys. K 1, 1–20 (1962)

    Google Scholar 

  • [Mos-2] Moser, J.: Convergent series expansions for quasi-periodic motions. Math. Ann.169, 136–176 (1967)

    Article  Google Scholar 

  • [Mos-3] Moser, J.: Stable and random motions in dynamical systems. Princeton, NJ: Princeton University Press 1973

    Google Scholar 

  • [Pös-1] Pöschel, J.: Über invariante Tori in differenzierbaren Hamiltonschen Systemen. Bonn. Math. Schr.120, 1–103 (1980)

    Google Scholar 

  • [Pös-2] Pöschel, J.: Integrability of hamiltonian systems on Cantor sets. Commun. Pure Appl. Math.35, 653–695 (1982)

    Google Scholar 

  • [Pös-3] Pöschel, J.: On invariant manifolds of complex analytic mappings near fixed points. Expo. Math.4, 97–109 (1986)

    Google Scholar 

  • [Pös-4] Pöschel, J.: A general infinite dimensional KAM-theorem. In: IXth International Congress on Mathematical Physics. Bristol, New York: Adam Hilger 1989

    Google Scholar 

  • [Pös-5] Pöschel, J.: On elliptic lower dimensional tori in hamiltonian systems. Math. Z. (to appear)

  • [Rüs] Rüssmann, H.: On the one-dimensional Schrödinger equation with a quasi-periodic potential. Ann. New York Acad. Sci.357, 90–107 (1980)

    Google Scholar 

  • [SM] Siegel, C.L., Moser, J.: Lectures on Celestial Mechanics. Berlin, Heidelberg, New York: Springer 1971

    Google Scholar 

  • [SZ] Salamon, D., Zehnder, E.: KAM theory in configuration space. Comment. Math. Helv.64, 84–132 (1989)

    Google Scholar 

  • [Vit] Vittot, M.: Théorie classique des perturbations et grand nombre de degres de liberté. Thèse de doctorat de l'université de Provence, 1985

  • [VB] Vittot, M., Bellissard, J.: Invariant tori for an infinite lattice of coupled calssical rotators. Preprint, CPT-Marseille, 1985

  • [Way-1] Wayne, C. E.: The KAM theory of systems with short range interactions, I and II. Commun. Math. Phys.96, 311–329 (1984) and 331–344

    Article  Google Scholar 

  • [Way-2] Wayne, C. E.: On the elimination of non-resonance harmonics. Commun. Math. Phys.103, 351–386 (1986)

    Article  Google Scholar 

  • [Way-3] Wayne, C. E.: Bounds on the trajectories of a system of weakly coupled rotators. Commun. Math. Phys.104, 21–36 (1986)

    Article  Google Scholar 

  • [Way-4] Wayne, C. E.: Periodic and quasi-periodic solutions of nonlinear wave equations via KAM theory. Preprint No. 88027, Penn. State University, 1988

  • [Yoc] Yoccoz, J. C.: Linéarisation des germes de difféomorphismes holomorphes de (C,0). C. R. Acad. Sci. Paris306, 55–58 (1988)

    Google Scholar 

  • [Zeh] Zehnder, E.: Generalized implicit function theorems with applications to some small divisor problems, I. Commun. Pure Appl. Math.28, 91–140 (1975); II. Commun. Pure Appl. Math.29, 49–111 (1976)

    Google Scholar 

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Authors and Affiliations

  1. Institut für Angewandte Mathematik, SFB 256, Universität Bonn, Wegelerstrasse 6, D-5300, Bonn 1, Federal Republic of Germany

    Jürgen Pöschel

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  1. Jürgen Pöschel
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Communicated by J. Fröhlich

Supported by Sonderforschungsbereich 256 at the University of Bonn

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Pöschel, J. Small divisors with spatial structure in infinite dimensional Hamiltonian systems. Commun.Math. Phys. 127, 351–393 (1990). https://doi.org/10.1007/BF02096763

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  • DOI: https://doi.org/10.1007/BF02096763

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