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Classical Mechanics and Renormalization Group

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Regular and Chaotic Motions in Dynamic Systems

Abstract

The theory of Kolmogorov-Arnold-Moser (KAM) is discussed in detail from the point of view of the “renormalization group approach”. Similarly we discuss some aspects of the problem of the existence of universal structures in the chaotic transition. The quasi-periodic Schroedinger equation in one dimension is discussed as a special case.

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References

  1. G. Gallavotti, “The Elements of Mechanics”, Springer-Verlag, New York, 1983

    Book  MATH  Google Scholar 

  2. V. Arnold, “Méthodes Mathématiques de la Mécanique Classique”, MIR, Moscow, 1978

    Google Scholar 

  3. H. Poincaré, “Méthodes Nouvelles de la Mécanique Celiste”, Gauthier-Villiers, Paris, 1897

    Google Scholar 

  4. J. Pöshel, Integrability of Hamiltonian Systems on Cantor Sets, Comm. Pure and Appl. Math., 35, 653, 1982

    Google Scholar 

  5. L. Chierchia, G. Gallavotti, Smooth Prime Integrals for Quasi-Integrable Hamiltonian Systems, Nuovo Cimento, 67B, 277, 1982

    Article  MathSciNet  Google Scholar 

  6. V. Arnold, Small Denominators and Problems of Stability of Motion in Classical and Celestial Mechanics, Russian Math. Surv., 18 (6), 85, 1963

    ADS  Google Scholar 

  7. J. Moser, Stable and Random Motions in Dynamical Systems, Princeton University Press, Princeton, 1963

    Google Scholar 

  8. E. Dinaburg, Ja. Sinai, The One-Dimensional Schroedinger Equation with a Quasi-Periodic Potential, Functional Anal. Appl. 9, 279, 1975

    MathSciNet  Google Scholar 

  9. H. Rüssmann, On the One Dimensional Schr8dinger Equation with a Quasi-Periodic Potential, Ann. New York Acad. Sci., 90, 197

    Google Scholar 

  10. G. Gallavotti, A Criterion of Integrability for Perturbed Nonresonant Harmonic Oscillators. “Wick Ordering” of the Perturbations in Classical Mechanics and Invariance of the Frequency Spectrum, Comm. Math. Phys. 87, 365, 1982

    MathSciNet  ADS  MATH  Google Scholar 

  11. The main result of this reference was actually essentially proved earlier by H. Rüssmann. However, the difference in notation and spirit might make it worthwhile for the reader to consult also my paper after the original reference of Rüssmann, see [13] below.

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  12. G. Gallavotti, Lectures on Hamiltonian Systems, to appear in the series, Progress in Physics, ed. J. Fröhlich, Birkhauser, Boston, in 1983.

    Google Scholar 

  13. S. Shenker, L. Kadanoff, Critical Behaviour of a KAM Surface: I, Empirical Results. J. Stat. Phys. 27, 631, 1982

    Article  MathSciNet  ADS  Google Scholar 

  14. M. Feigenbaum, L. Kadanoff, S. Shenker, Quasi-Periodicity in Dissipative Systems. A Renormalization Group Analysis, Physica 5D, 370, 1983

    MathSciNet  Google Scholar 

  15. R. McKay, A Renormalization Approach to Invariant Circles in Area Preserving Maps, Princeton Preprint, 1982; Other papers can be found by looking at the references of those listed above. See also R. McKay, Thesis, Princeton, 1982.

    Google Scholar 

  16. H. Rüssmann, Über die Normalform Analytischer Hamiltonscher Differentialgleichungen in der Nähe Einer Gleichgewichtlösung, Math. Annalen, 169, 55, 1967

    Article  MATH  Google Scholar 

  17. V. Arnold, A Proof of a Theorem of A. N. Kolmogorov on the Invariance of Quasi-Periodic Motions under Small Perturbations of the Hamiltonian, Russian Math. Surv., 18 (5), 9, 1963

    ADS  Google Scholar 

  18. J. P3shel, Examples of Discrete Schroedinger Operators with Pure Point Spectrum, Commun. Math. Phys. 88, 44, 7, 1983

    Google Scholar 

  19. D. Rand, S. Ostlund, J. Sethna, E. Siggia, Universal Transition from Quasi-Periodicity to Chaos in Dissipative Systems, Phys. Rev. Lett., 49, 132, 1982

    Article  MathSciNet  ADS  Google Scholar 

  20. M. Feigenbaum, Quantitative Universality for a Class of Nonlinear Transformations, J. Stat. Phys., 19, 25, 1978

    Article  MathSciNet  ADS  MATH  Google Scholar 

  21. M. Campanino, H. Epstein, D. Ruelle, On Feigenbaum’s Functional Equation g • g(Xx) + Xg(x) = 0, Topology, 21, 125, 1982

    Article  MathSciNet  Google Scholar 

  22. O. Lanford, A Computer Assisted Proof of Feigenbaum’s Conjecture, Preprint, Berkeley, 1981

    Google Scholar 

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Author information

Authors and Affiliations

  1. Istituto Matematico, Ia Università di Roma, 00185, Roma, Italia

    Giovanni Gallavotti

Authors
  1. Giovanni Gallavotti
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Editor information

Editors and Affiliations

  1. Institute of Physics, University of Bologna, Bologna, Italy

    G. Velo

  2. Princeton University, Princeton, New Jersey, USA

    A. S. Wightman

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© 1985 Plenum Press, New York

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Gallavotti, G. (1985). Classical Mechanics and Renormalization Group. In: Velo, G., Wightman, A.S. (eds) Regular and Chaotic Motions in Dynamic Systems. Springer, Boston, MA. https://doi.org/10.1007/978-1-4684-1221-5_5

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  • DOI: https://doi.org/10.1007/978-1-4684-1221-5_5

  • Publisher Name: Springer, Boston, MA

  • Print ISBN: 978-1-4684-1223-9

  • Online ISBN: 978-1-4684-1221-5

  • eBook Packages: Springer Book Archive

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