Skip to main content
SpringerLink
Log in

Recent Progress in Classical Mechanics

  • Conference paper
Mathematical Physics X

Abstract

The goal of this lecture is to review several developments in classical mechanics that have taken place in the last years, that will fit in the time of the talk and that I have become aware of. Unfortunately, the latter is a constraint more severe than what I would like and I apologize to the authors and the audience for many things that have been left out. In particular, I have left out topics such as “twist mappings”, “geometric quantization” or “geometric phases” and “quantum chaos” that are generating a great deal of activity in the literature. I would also like to recommend the excellent set of reviews [AANS] for the many developments that took place up to the time the books were published.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Amik, C., King, E.C.S., Kadanoff, L.P., Rom-Kedar, V.: Beyond all orders: singular perturbations in a mapping.

    Google Scholar 

  2. Angenent, S.: Lecture given at IMA conference on twist mappings. To appear

    Google Scholar 

  3. Anosov, D.V.: Geodesic flows on closed riemannian manifolds of negative curvature. Proc. Stekelov Inst. 90 (1967)

    Google Scholar 

  4. Anosov, D.V., Arnol’d, V.I., Novikov, S.P., Sinai, V.G. (eds.): Dynamical systems I–IV (Encyclopedia of Mathematical Sciences). Springer-Verlag, 1988

    Google Scholar 

  5. Arnol’d, V.I.: Instability of dynamical systems with several degrees of freedom. Soviet Math.Dokl. 5, 581–585 (1964)

    MATH  Google Scholar 

  6. Arnol’d, V.I.: Proof of a theorm of A.N. Kolmogorov on the invariance of quasi-periodic motions under small perturbations of the hamiltonian. Russian Math.Surveys 18, 9–36 (1963)

    Article  MATH  ADS  Google Scholar 

  7. Benettin, G., Galgani, L., Giorgilli, A.: A proof of Nelshoroshev’s theorem for the stability times in nearly integrable systems.

    Google Scholar 

  8. Bunimovich L.A., Sinai, Y.G.: Space time chaos in coupled map lattices. Nonlinearity 1, 491–516 (1988)

    Article  MathSciNet  MATH  ADS  Google Scholar 

  9. Barrar, R.: Convergence of the VonZeipel procedure. Celestial Mech. 2, 491–504 (1970)

    Article  MathSciNet  ADS  Google Scholar 

  10. Brjno, A.D.: Analytical form of differential equations. I. Trans.Moscow Math.Soc. 25, 131–288 (1971)

    Google Scholar 

  11. Brjno, A.D.: Analytical form of differential equations. II. Trans.Moscow Math.Soc. Ibid. 26, 199–239 (1972)

    Google Scholar 

  12. Bunimovitch S.: On the ergodic porperties of nowhere dispersing billiards. Comm.Math.Phys. 65, 295–312 (1979)

    Article  MathSciNet  ADS  Google Scholar 

  13. Celletti, A., Chierchia, L.: Construction of analytic KAM surfaces and effective stability bounds Comm.Math.Phys. 118, 119–161 (1988)

    Article  MathSciNet  MATH  ADS  Google Scholar 

  14. Celletti, A.,Chierchia, L.: Invariant curves for area-preserving twist maps far from integrable. J.Statist.Phys. to appear.

    Google Scholar 

  15. Celletti, A.: Analysis of resonances in the spin orbit problem in celestial mechanics. Part I., ZAMP. 41, 174–204 (1990);

    Article  MathSciNet  MATH  ADS  Google Scholar 

  16. Celletti, A.: Analysis of resonances in the spin orbit problem in celestial mechanics. Part II., ZAMP. 41, (1990); Ibid. 453–479

    Article  MathSciNet  MATH  ADS  Google Scholar 

  17. Cheng, Chong-quing, Sun, Yi-Sui: Existence of invariant tori in three-dimesional measure preserving maps. Cel.Mech.Dyn.Ast. 47, 275–292 (1990)

    Article  MathSciNet  MATH  ADS  Google Scholar 

  18. Chierchia, L., Gallavotti, G.: Smooth prime integrals for quasi-integrable hamiltonian systems. Nuovo Cimento B67, 277–295 (1982)

    Article  MathSciNet  ADS  Google Scholar 

  19. Collet, P., Epstein, H., Gallavotti, C.: Perturbations of geodesic flows on surfaces of constant negative curvature and their mixing properties. Comm.Math.Phys. 95, 61–112 (1984)

    Article  MathSciNet  MATH  ADS  Google Scholar 

  20. Delft, P., Li, L.-C., Tomei, C.: The Bidiagonal Singular Value Decomposition and Hamiltonian Mechanics.

    Google Scholar 

  21. Delshams, A., de la Llave, R.: Existence of quasi-periodic orbits and absence of transport for volume preserving transformation and flows. Preprint

    Google Scholar 

  22. Devaney, R.: Singularities in classical mechanical systems. In: Ergodic Theory and Dynamical Systems, I. Birkhauser, Boston, 1981

    Google Scholar 

  23. Donnay, V.: Geodesic flow on the two sphere, Part I, positive topological entropy. Ergodic Theory Dynamical Systems 8, 531–553 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  24. Donnay, V., Liverani, C.: Potentials on the two torus for which the Hamiltonian flow is ergodic. Comm.Math.Phys. 135, 367–303 (1991)

    Article  MathSciNet  Google Scholar 

  25. Douady, R.: Stabilité ou Instabilité des points fixes elliptiques. Ann.Sci. ENS 21, 1–46 (1988)

    MathSciNet  MATH  Google Scholar 

  26. Eliasson, L.H.: Perturbations of stable invariant tori for Hamiltonian systems. Publ.Sc.Norm.Piser 15, 115–147 (1988)

    MathSciNet  MATH  Google Scholar 

  27. Feingold, M., Kadanoff, L.P., Piro, O.: Passive scalars, three dimensional volume preserving maps and chaos. J.Statist.Phys. 50, 529–565 (1988)

    Article  MathSciNet  MATH  ADS  Google Scholar 

  28. Feldman, J., Ornstein, D.: Semi-rigidity of horocycle flows over compact surfaces of variable negative curvature. Ergodic Theory Dynamical Systems 7, 49–72 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  29. Fomenko, A.: The symplectic topology of completely integrable Hamiltonian systems. Russian Math.Surveys 44 (1989)

    Google Scholar 

  30. Fontich, E., Simo, C.: The splitting of separatices for analytic diffeomor-phisms. Ergodic Theory Dynamical Systems 10, 295–318 (1990)

    MathSciNet  MATH  Google Scholar 

  31. Frölich, J., Spencer, T., Wayne, C.E.: Localization in disordered nonlinear dynamical systems. J.Statist.Phys. 42, 247–274 (1986)

    Article  MathSciNet  ADS  Google Scholar 

  32. Gever, J.L.: The existence of pseudocollisions in the plane. To appear in J.Differential Equations

    Google Scholar 

  33. Ghys, E.: Flots d’Anosov dont les feulletages stables sont différentiales. Ann.Sci.ENS 20, 251–271 (1987)

    MathSciNet  MATH  Google Scholar 

  34. Guillemin, V., Kazhdan, D.: Some inverse spectral results for negatively curved 2-manifolds. Topology 19, 301–312 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  35. Guillemin, V., Kazhdan, D.: Some inverse spectral results for negatively curved n manifolds. AMS Proc.Sympos.Pure Math. 35, 153–180 (1982)

    Google Scholar 

  36. Gutkin, E.: Asymptotic velocities in classical mechanics. J.Math.Phys. 30, 1245–1249 (1989)

    Article  MathSciNet  MATH  ADS  Google Scholar 

  37. Hagedorn, G.A.: Proof of the Landau-Zenner formula in an adiabatic limit with small eigenvalue gaps. Comm.Math.Phys. 136, 433–451 (1991)

    Article  MathSciNet  MATH  ADS  Google Scholar 

  38. Hedlund, G.: The dynamics of geodesic flows. Bull.Amer.Math.Soc. 45, 241–260 (1969)

    Article  MathSciNet  Google Scholar 

  39. Herman, M.R.: Sur les courbes invariantes par les diffeomorphismes de l’anneau. I. Asterisque 103–104 (1983)

    MATH  Google Scholar 

  40. Herman, M.R.: Sur les courbes invariantes par les diffeomorphismes de l’anneau. II. Asterisque. Ibid. 144 (1986)

    MATH  Google Scholar 

  41. Herman, M.R.: Inégalités a priori pour des tores Lagrangunes invariants par des diffeomorphisms symplectiques. Publ.Math.IHES 63, 47–101 (1990)

    Google Scholar 

  42. Hopf, E.: Statistik der geodätischen Linien in Mannigfaltigkesten negativer Krümmung. Akad.Wiss.Leipzig 91, 261–304 (1939)

    MathSciNet  Google Scholar 

  43. Hubacher, A.: Classical scattering theory in 1 dimension. Preprint

    Google Scholar 

  44. Hurder, S.: Rigiditity for Anosov actions of higher rank lattices. Preprint

    Google Scholar 

  45. Hurder, S., Katok, A.: Differentiability, rigidity and Godbillon-Vey classes for Anosov flows. To appear in Publ.Math.IHES

    Google Scholar 

  46. Jaksic, V., Segert, J.: Exponential approach to the limit and the Landau-Zenner formula. Preprint

    Google Scholar 

  47. Jorba, A., Simo, C.: On the reducibility of linear differential equations with quasi-periodic coefficients. Preprint

    Google Scholar 

  48. Jungreis, I.: Ergodic Theory Dynamical Systems 11, 79–84 (1991)

    Article  MathSciNet  Google Scholar 

  49. Katok, A., Knieper, G., Weiss, H.: Formulas for the derivative and critical points of topological entropy for Anosov and geodesic flows. Comm.Math.Phys. 130, 19–33 (1991)

    Article  MathSciNet  ADS  Google Scholar 

  50. Krámli, A., Simányi, N., Szász, D.: A “transversal” fundamental theorem for semi dispersing billiards. Comm.Math.Phys. 129, 535–560 (1989);

    Article  Google Scholar 

  51. Krámli, A., Simányi, N., Szász, D.: A “transversal” fundamental theorem for semi dispersing billiards. Erratum Comm.Math.Phys. 138, 207 (1991)

    Article  MATH  ADS  Google Scholar 

  52. Kozlov, V.V.: Integrability nad non-integrability in Hamiltonian mechanics. Russian Math.Surveys 38, 1–76 (1983)

    Article  MATH  ADS  Google Scholar 

  53. Kuksin, S.B.: Perturbation theory for quasiperiodic solutions of infinite dimensional hamiltonian systems and its applications to the Korteweg-deVries equation. Math.USSR-Sb. 64, 397–413 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  54. Kuksin, S.B.: Perturbation theory for quasi-periodic solutions of infinite dimensional Hamiltonian systems, Parts I, II, III. Max Plank preprints

    Google Scholar 

  55. Lanford, O.E.: Computer assisted proofs in Analysis. Physica A 124, 465–470 (1984)

    Article  MathSciNet  MATH  ADS  Google Scholar 

  56. Lochak, P.: Stabilité en temps exponentiels des systèmes hamiltoniens proches de systèmes integrables: Resonances et orbites fermées. ENS preprint

    Google Scholar 

  57. de la Llave, R.: Invariants for smooth conjugacy of hyperbolic dynamical systems II. Comm.Math.Phys. 109, 369–378 (1987)

    Article  MathSciNet  MATH  ADS  Google Scholar 

  58. de la Llave, R.: Analytic regularity of solutions of Livsic’s cohomology equation and application to smooth conjugacy of hyperbolic dynamical systems. To appear in Ergodic Theory Dynamical Systems

    Google Scholar 

  59. de la Llave, R.: Smooth conjugacy and SRB measures for uniformly and non-uniformly hyperbolic systems. To appear in Comm.Math.Phys.

    Google Scholar 

  60. de la Llave, R.: A simple proof of a particular case of C Siegel’s center theorem. J.Math.Phys. 24, 2118–2121 (1983)

    Article  MathSciNet  ADS  Google Scholar 

  61. de la Llave, R., Marco, J.M., Moriyón, R.: Canonical perturbation theory for Anosov systems and regularity properties of Livsic’s cohomology equation. Ann.of Math. 123, 537–611 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  62. de la Llave, R., Moriyón, R.: Invariants for smooth conjugacy of hyperbolic dynamical systems IV. Comm.Math.Phys. 116, 185–192 (1988)

    Article  MathSciNet  MATH  ADS  Google Scholar 

  63. de la Llave, R., Rana D.: Accurate strategies for K.A.M. bounds and their implementation. In: Computer aided Proofs in Analysis. Meyer, K., Schmidt, D. (eds), Springer Verlag (1991)

    Google Scholar 

  64. de la Llave, R., Wayne, C.E.: Whiskered and low dimensional tori in nearly integrable dynamical systems. Preprint

    Google Scholar 

  65. Mather, J., McGehee, R.G.: Solutions of the four body problem which become unbounded in finite time. Lecture Notes in Phys. 38 (1975)

    Google Scholar 

  66. McKay R.S., Meiss J.D., Stark, J.: Converse KAM theory fro symplectic twist maps. Nonlinearity 2, 555–570 (1989)

    Article  MathSciNet  ADS  Google Scholar 

  67. McKay, R.S., Percival, I.C.: Converse K.A.M. theory and practice. Comm.Math.Phys. 98, 569–512 (1985)

    Google Scholar 

  68. Milojevic, P.S. (ed): Nonlinear functional analysis. Marcel Decker, New York, 1990

    MATH  Google Scholar 

  69. Moser, J.: Is the solar system stable? Math.Intelligencer 1, 65–71 (1978)

    Article  MathSciNet  ADS  Google Scholar 

  70. Morales, J.J.: Técnicas algebraicas para el estudio de la integrabilidad algebraica de sistemas hamiltonianos. Tesis U. Barcelona (1989)

    Google Scholar 

  71. Muldoon, M.: Gohsts of order on the frontier of chaos. PhD thesis, Caltech, 1990

    Google Scholar 

  72. Nehoroshev, A.N.: An exponential estimate for the time of stability of nearly integrable Hamiltonian systems. Russian Math.Surveys 32, no.6, 1–65 (1977)

    Article  ADS  Google Scholar 

  73. Neishdtadt, A.: The separation of motions in systems with a rapidly rotating phase. P.M.M. USSR 40, 133–139 (1984)

    Google Scholar 

  74. Nikolenko, N.V.: Invariant asymptotically stable tori of the perturbed Korteweg-deVries equation. Russian Math.Surveys 35, 139–207 (1980)

    Article  MathSciNet  MATH  ADS  Google Scholar 

  75. Otal, J.P.: Le spectre marqué des longeurs des surfaces à courbure negative. Ann.of Math. 131, 151–162 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  76. Palis, J., Yoccoz, J-C.: Rigidity of centralizers of diffeomorphisms. Ann.Sci.ENS 22, 81–99 (1989)

    MathSciNet  MATH  Google Scholar 

  77. Palis, J., Yoccoz, J-C.: Centralizers of Anosov diffeomorphisms on tori. Ann.Sci.ENS 22, 99–109 (1989)

    MathSciNet  MATH  Google Scholar 

  78. Perez Marco, R.: Centralizers I: Uncountable centralizers for non linearizable germs of (C,0) IHES preprint 91/23

    Google Scholar 

  79. Pesin, Y., Sinai, Y.G. Space time chaos in the system of weakly interactive hyperbolic systems. Preprint

    Google Scholar 

  80. Paluba, W.: The Lipschitz condition for the conjugacies of Feigenbaum-like mappings. Fund. Math. 132, 227–257 (1989)

    MathSciNet  MATH  Google Scholar 

  81. Pesin, Y.: Characteristic Lyapunov exponents and smooth ergodic theory. Russian Math.Surveys 32, 55–114 (1976)

    Article  MathSciNet  ADS  Google Scholar 

  82. Pesin, Y.: An example of a non-ergodic flow with non-zero characteristic exponents. Functional Anal.Appl. 8, 263–264 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  83. Pollicott, M.: C k rigidity for hyperbolic flows. To appear in Israel J.Math.

    Google Scholar 

  84. Pöschel, J.: On elliptic lower dimensional tori in hamiltonian systems. Math.Z. 202, 559–608 (1989)

    Article  Google Scholar 

  85. Pöschel, J.: Small divisors with spatial structure in infinite dimensional dynamical systems. Comm.Math.Phys. 127, 351–393 (1990)

    Article  MathSciNet  MATH  ADS  Google Scholar 

  86. Rana, D. Proof of accurate upper and lower bounds to stability domains in small denominator problems: Princeton thesis (1987)

    Google Scholar 

  87. Ramani, A., Gramatikos, B., Bountis, T.: The Panilevé Property and singularity analysis of integrable and non-integrable systems. Phys.Rep. 180 (1989)

    Google Scholar 

  88. Radin, C., Simon, B.: Invariant domains for the time- dependent Schrödinger equation. J.Differential Equations 29, 289–296 (1978)

    Article  MathSciNet  MATH  ADS  Google Scholar 

  89. Rand, D.: Global phase space universality, smooth conjugacies and renormalisations. The C1+α case. Nonlinearity 1, 181–202 (1988)

    Article  MathSciNet  MATH  ADS  Google Scholar 

  90. Rüssmann, H.: On a new proof of Moser’s twist mapping theorem. Celestial Mech. 14, 19–31 (1976)

    Article  MathSciNet  MATH  ADS  Google Scholar 

  91. Siegel, C.L., Moser, J.: Lectures on Celestial Mechanics. Springer-Verlag, 1971

    Book  MATH  Google Scholar 

  92. Sinai, Y.G., Chernoff, S.: Ergodic porperties of certain systems of two dimensional discs and three dimensional balls. Russian Math.Surveys 42, 181–207 (1987)

    Article  ADS  Google Scholar 

  93. Sinai, Ya.G.: Dynamical systems with elastic reflections. Ergodic properties of dispersing billiards. Russian Math.Surveys 25 no.2, 137–189 (1970)

    Article  MathSciNet  MATH  ADS  Google Scholar 

  94. Sullivan, D.: Bounds, quadratic differentials and renormalization conjectures. Preprint

    Google Scholar 

  95. Sunada, T.: Riemannian coverings and isospectral manifolds. Ann.of Math. 121, 169–186 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  96. Sun, Y.S.: On the measure preserving mappings with odd dimension. Celestial Mech. 30, 7–19 (1983)

    Article  MathSciNet  MATH  ADS  Google Scholar 

  97. Treshev, D.V.: The mechanisms of destruction of resonance tori of hamiltonian systems. Math.USSR-Sb. 68, 181–203 (1991)

    Article  MathSciNet  Google Scholar 

  98. Vitott, M. Théorie classique des perturbations en grand nombre de degrés de liberté. Univ. de Marseille, Thesis (1985)

    Google Scholar 

  99. Wojtkowski, M.: Principles for the design of Billiards with Non-vanishing Lyapunov exponents. Comm.Math.Phys. 105, 391–414 (1986)

    Article  MathSciNet  MATH  ADS  Google Scholar 

  100. Wayne, C.E.: The K.A.M. theory of systems with short range interactions. Part I, Comm.Math.Phys. 96, 311–329 (1984)

    Article  MathSciNet  MATH  ADS  Google Scholar 

  101. Wayne, C.E.: The K.A.M. theory of systems with short range interactions. Part II, Comm.Math.Phys. 96, (1984); ibid. 331–344

    Article  MathSciNet  MATH  ADS  Google Scholar 

  102. Wayne, C.E.: Periodic and quasi-periodic solutions of nonlinear wave equations via K.A.M. Theory. Comm.Math.Phys. 127, 479–528 (1990)

    Article  MathSciNet  MATH  ADS  Google Scholar 

  103. Wayne, C. E.: On the elimination of non-resonant harmonics. Comm. Math.Phys. 103, 351–386 (1986)

    Article  MathSciNet  MATH  ADS  Google Scholar 

  104. Wayne, C. E.: Uniform estimates on trajectories in hamiltonian systems with many degrees of freedom. Comm.Math.Phys. 104, 21–36 (1986)

    Article  MathSciNet  MATH  ADS  Google Scholar 

  105. Xia, Z.: The existence of the non-collision singularities in Newtonian systems. PhD thesis, Northwestern University, 1988

    Google Scholar 

  106. Yoccoz, J-C.: Théorème de Siegel, polynômes quadratiques et nombres de Brjuno. Preprint

    Google Scholar 

  107. Yoccoz, J-C.: Linéarization des germes de diffeomorphismes holomorphes de (C,0). C.R.Acad.Sci.Paris 306, 555–558 (1988)

    MathSciNet  Google Scholar 

  108. Zimmer, W.: Infinitesimal rigidity for smooth actions of discrete subgroups of Lie groups.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, The University of Texas at Austin, Austin, Texas, 78712, USA

    R. de la Llave

Authors
  1. R. de la Llave
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Fachbereich Mathematik, Universität Leipzig, Augustusplatz 10, O-7010, Leipzig, Germany

    Konrad Schmüdgen

Rights and permissions

Reprints and permissions

Copyright information

© 1992 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

de la Llave, R. (1992). Recent Progress in Classical Mechanics. In: Schmüdgen, K. (eds) Mathematical Physics X. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-77303-7_1

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-77303-7_1

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-77305-1

  • Online ISBN: 978-3-642-77303-7

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation