KAM theory: The legacy of Kolmogorov’s 1954 paper

Broer, Henk

doi:10.1090/S0273-0979-04-01009-2

KAM theory: The legacy of Kolmogorov’s 1954 paper
HTML articles powered by AMS MathViewer

by Henk W. Broer PDF

Bull. Amer. Math. Soc. 41 (2004), 507-521 Request permission

Abstract:

Kolmogorov-Arnold-Moser (or kam) theory was developed for conservative dynamical systems that are nearly integrable. Integrable systems in their phase space usually contain lots of invariant tori, and kam theory establishes persistence results for such tori, which carry quasi-periodic motions. We sketch this theory, which begins with Kolmogorov’s pioneering work.

References

V. I. Arnol′d, Proof of a theorem of A. N. Kolmogorov on the preservation of conditionally periodic motions under a small perturbation of the Hamiltonian, Uspehi Mat. Nauk 18 (1963), no. 5 (113), 13–40 (Russian). MR 0163025
V. I. Arnol′d, Small denominators and problems of stability of motion in classical and celestial mechanics, Uspehi Mat. Nauk 18 (1963), no. 6 (114), 91–192 (Russian). MR 0170705
V. I. Arnol′d, Instability of dynamical systems with many degrees of freedom, Dokl. Akad. Nauk SSSR 156 (1964), 9–12 (Russian). MR 0163026
V. I. Arnol′d, Small denominators. I. Mapping the circle onto itself, Izv. Akad. Nauk SSSR Ser. Mat. 25 (1961), 21–86 (Russian). MR 0140699
V. I. Arnol′d, Geometrical methods in the theory of ordinary differential equations, 2nd ed., Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 250, Springer-Verlag, New York, 1988. Translated from the Russian by Joseph Szücs [József M. Szűcs]. MR 947141, DOI 10.1007/978-1-4612-1037-5
S. Bergmann and J. Marcinkiewicz, Sur les fonctions analytiques de deux variables complexes, Fund. Math. 33 (1939), 75–94 (French). MR 57, DOI 10.4064/fm-33-1-75-94
V. I. Arnol′d and A. Avez, Ergodic problems of classical mechanics, W. A. Benjamin, Inc., New York-Amsterdam, 1968. Translated from the French by A. Avez. MR 0232910
J. Bricmont, Science of chaos or chaos in science?, The flight from science and reason (New York, 1995) Ann. New York Acad. Sci., vol. 775, New York Acad. Sci., New York, 1996, pp. 131–175. MR 1607505, DOI 10.1111/j.1749-6632.1996.tb23135.x
Jean Bricmont, Krzysztof Gawȩdzki, and Antti Kupiainen, KAM theorem and quantum field theory, Comm. Math. Phys. 201 (1999), no. 3, 699–727. MR 1685894, DOI 10.1007/s002200050573
Paul Blanchard, Complex analytic dynamics on the Riemann sphere, Bull. Amer. Math. Soc. (N.S.) 11 (1984), no. 1, 85–141. MR 741725, DOI 10.1090/S0273-0979-1984-15240-6

Bull. de la Societat Catalana de Matemàtiques

Nonlinearity

16

Preprint

H. W. Broer and G. B. Huitema, A proof of the isoenergetic KAM-theorem from the “ordinary” one, J. Differential Equations 90 (1991), no. 1, 52–60. MR 1094448, DOI 10.1016/0022-0396(91)90160-B
H. W. Broer and G. B. Huitema, Unfoldings of quasi-periodic tori in reversible systems, J. Dynam. Differential Equations 7 (1995), no. 1, 191–212. MR 1321710, DOI 10.1007/BF02218818
Hendrik W. Broer, George B. Huitema, and Mikhail B. Sevryuk, Quasi-periodic motions in families of dynamical systems, Lecture Notes in Mathematics, vol. 1645, Springer-Verlag, Berlin, 1996. Order amidst chaos. MR 1484969
H. W. Broer, G. B. Huitema, F. Takens, and B. L. J. Braaksma, Unfoldings and bifurcations of quasi-periodic tori, Mem. Amer. Math. Soc. 83 (1990), no. 421, viii+175. MR 1041003, DOI 10.1090/memo/0421

Commun. Math. Phys.

641

A. D. Brjuno, On convergence of transforms of differential equations to the normal form, Dokl. Akad. Nauk SSSR 165 (1965), 987–989 (Russian). MR 0192098
L. Chierchia and G. Gallavotti, Drift and diffusion in phase space, Ann. Inst. H. Poincaré Phys. Théor. 60 (1994), no. 1, 144 (English, with English and French summaries). MR 1259103

Ph.D. Thesis

kam

Peyresq Lectures on Geometric Mechanics and Symmetry,

306

Math. Ann.

98

Robert L. Devaney, An introduction to chaotic dynamical systems, 2nd ed., Addison-Wesley Studies in Nonlinearity, Addison-Wesley Publishing Company, Advanced Book Program, Redwood City, CA, 1989. MR 1046376
Florin Diacu and Philip Holmes, Celestial encounters, Princeton University Press, Princeton, NJ, 1996. The origins of chaos and stability. MR 1435973
E. I. Dinaburg and Ja. G. Sinaĭ, The one-dimensional Schrödinger equation with quasiperiodic potential, Funkcional. Anal. i Priložen. 9 (1975), no. 4, 8–21 (Russian). MR 0470318
L. H. Eliasson, S. B. Kuksin, S. Marmi, and J.-C. Yoccoz, Dynamical systems and small divisors, Lecture Notes in Mathematics, vol. 1784, Springer-Verlag, Berlin; Centro Internazionale Matematico Estivo (C.I.M.E.), Florence, 2002. Lectures from the C.I.M.E. Summer School held in Cetraro, June 13–20, 1998; Edited by Marmi and Yoccoz; Fondazione CIME/CIME Foundation Subseries. MR 1924909, DOI 10.1007/b83847
Jürg Fröhlich and Thomas Spencer, Absence of diffusion in the Anderson tight binding model for large disorder or low energy, Comm. Math. Phys. 88 (1983), no. 2, 151–184. MR 696803
Giovanni Gallavotti, Statistical mechanics, Texts and Monographs in Physics, Springer-Verlag, Berlin, 1999. A short treatise. MR 1707309, DOI 10.1007/978-3-662-03952-6
G. Gallavotti, G. Gentile, and V. Mastropietro, Field theory and KAM tori, Math. Phys. Electron. J. 1 (1995), Paper 5, approx. 13 pp.}, issn=1086-6655, review= MR 1359460,
Heinz Hanßmann, The quasi-periodic centre-saddle bifurcation, J. Differential Equations 142 (1998), no. 2, 305–370. MR 1601868, DOI 10.1006/jdeq.1997.3365
Michael-Robert Herman, Sur la conjugaison différentiable des difféomorphismes du cercle à des rotations, Inst. Hautes Études Sci. Publ. Math. 49 (1979), 5–233 (French). MR 538680
Morgan Ward, Ring homomorphisms which are also lattice homomorphisms, Amer. J. Math. 61 (1939), 783–787. MR 10, DOI 10.2307/2371336

Ph.D. Thesis

Àngel Jorba and Jordi Villanueva, On the normal behaviour of partially elliptic lower-dimensional tori of Hamiltonian systems, Nonlinearity 10 (1997), no. 4, 783–822. MR 1457746, DOI 10.1088/0951-7715/10/4/001

Ergebnisse der Mathematik und ihrer Grenzgebiete, 3. Folge

45

Tadasi Nakayama, On Frobeniusean algebras. I, Ann. of Math. (2) 40 (1939), 611–633. MR 16, DOI 10.2307/1968946
A. N. Kolmogorov, Théorie générale des systèmes dynamiques et mécanique classique, Proceedings of the International Congress of Mathematicians, Amsterdam, 1954, Vol. 1, Erven P. Noordhoff N. V., Groningen; North-Holland Publishing Co., Amsterdam, 1957, pp. 315–333 (French). MR 0097598
Sergej B. Kuksin, Nearly integrable infinite-dimensional Hamiltonian systems, Lecture Notes in Mathematics, vol. 1556, Springer-Verlag, Berlin, 1993. MR 1290785, DOI 10.1007/BFb0092243
S. Kuksin, V. Lazutkin, and J. Pöschel (eds.), Seminar on Dynamical Systems, Progress in Nonlinear Differential Equations and their Applications, vol. 12, Birkhäuser Verlag, Basel, 1994. Papers from the seminar held in St. Petersburg, October 14–25 and November 18–29, 1991. MR 1279384, DOI 10.1007/978-3-0348-7515-8
Albert Eagle, Series for all the roots of the equation $(z-a)^m=k(z-b)^n$, Amer. Math. Monthly 46 (1939), 425–428. MR 6, DOI 10.2307/2303037
L. D. Landau and E. M. Lifshitz, Fluid mechanics, Course of Theoretical Physics, Vol. 6, Pergamon Press, London-Paris-Frankfurt; Addison-Wesley Publishing Company, Inc., Reading, Mass., 1959. Translated from the Russian by J. B. Sykes and W. H. Reid. MR 0108121
Jacques Laskar, Large scale chaos and marginal stability in the solar system, XIth International Congress of Mathematical Physics (Paris, 1994) Int. Press, Cambridge, MA, 1995, pp. 75–120. MR 1370669
Rafael de la Llave, A tutorial on KAM theory, Smooth ergodic theory and its applications (Seattle, WA, 1999) Proc. Sympos. Pure Math., vol. 69, Amer. Math. Soc., Providence, RI, 2001, pp. 175–292. MR 1858536, DOI 10.1090/pspum/069/1858536
R. S. MacKay and S. Aubry, Proof of existence of breathers for time-reversible or Hamiltonian networks of weakly coupled oscillators, Nonlinearity 7 (1994), no. 6, 1623–1643. MR 1304442

An introduction to small divisor problems

J. Moser, On invariant curves of area-preserving mappings of an annulus, Nachr. Akad. Wiss. Göttingen Math.-Phys. Kl. II 1962 (1962), 1–20. MR 147741
J. Moser, On the theory of quasiperiodic motions, SIAM Rev. 8 (1966), 145–172. MR 203160, DOI 10.1137/1008035
Jürgen Moser, Convergent series expansions for quasi-periodic motions, Math. Ann. 169 (1967), 136–176. MR 208078, DOI 10.1007/BF01399536
Jürgen K. Moser, Lectures on Hamiltonian systems, Mem. Amer. Math. Soc. 81 (1968), 60. MR 0230498
Jürgen Moser, Stable and random motions in dynamical systems, Annals of Mathematics Studies, No. 77, Princeton University Press, Princeton, N. J.; University of Tokyo Press, Tokyo, 1973. With special emphasis on celestial mechanics; Hermann Weyl Lectures, the Institute for Advanced Study, Princeton, N. J. MR 0442980
N. N. Nehorošev, An exponential estimate of the time of stability of nearly integrable Hamiltonian systems, Uspehi Mat. Nauk 32 (1977), no. 6(198), 5–66, 287 (Russian). MR 0501140
S. Newhouse, D. Ruelle, and F. Takens, Occurrence of strange Axiom A attractors near quasiperiodic flows on $T^{m}$,$\,m\geq 3$, Comm. Math. Phys. 64 (1978/79), no. 1, 35–40. MR 516994
John C. Oxtoby, Measure and category, 2nd ed., Graduate Texts in Mathematics, vol. 2, Springer-Verlag, New York-Berlin, 1980. A survey of the analogies between topological and measure spaces. MR 584443
Jürgen Pöschel, Integrability of Hamiltonian systems on Cantor sets, Comm. Pure Appl. Math. 35 (1982), no. 5, 653–696. MR 668410, DOI 10.1002/cpa.3160350504
Jürgen Pöschel, On elliptic lower-dimensional tori in Hamiltonian systems, Math. Z. 202 (1989), no. 4, 559–608. MR 1022821, DOI 10.1007/BF01221590
Jürgen Pöschel, A lecture on the classical KAM theorem, Smooth ergodic theory and its applications (Seattle, WA, 1999) Proc. Sympos. Pure Math., vol. 69, Amer. Math. Soc., Providence, RI, 2001, pp. 707–732. MR 1858551, DOI 10.1090/pspum/069/1858551
David Ruelle, Elements of differentiable dynamics and bifurcation theory, Academic Press, Inc., Boston, MA, 1989. MR 982930
David Ruelle and Floris Takens, On the nature of turbulence, Comm. Math. Phys. 20 (1971), 167–192. MR 284067
Helmut Rüssmann, Konvergente Reihenentwicklungen in der Störungstheorie der Himmelsmechanik, Selecta Mathematica, V (German), Heidelberger Taschenbücher, vol. 201, Springer, Berlin-New York, 1979, pp. 93–260 (German). MR 597727
Mikhail B. Sevryuk, New results in the reversible KAM theory, Seminar on Dynamical Systems (St. Petersburg, 1991) Progr. Nonlinear Differential Equations Appl., vol. 12, Birkhäuser, Basel, 1994, pp. 184–199. MR 1279398, DOI 10.1007/978-3-0348-7515-8_{1}4
J. J. Corliss, Upper limits to the real roots of a real algebraic equation, Amer. Math. Monthly 46 (1939), 334–338. MR 4

Trans. Amer. Math. Soc.

36

J.-C. Yoccoz, $C^{1}$-conjugaison des difféomorphismes du cercle, Geometric dynamics (Rio de Janeiro, 1981) Lecture Notes in Math., vol. 1007, Springer, Berlin, 1983, pp. 814–827 (French). MR 730301, DOI 10.1007/BFb0061448
Jean-Christophe Yoccoz, Théorème de Siegel, nombres de Bruno et polynômes quadratiques, Astérisque 231 (1995), 3–88 (French). Petits diviseurs en dimension $1$. MR 1367353

Dynamical Systems and Small Divisors,

1784

E. Zehnder, An implicit function theorem for small divisor problems, Bull. Amer. Math. Soc. 80 (1974), 174–179. MR 339259, DOI 10.1090/S0002-9904-1974-13407-5
E. Zehnder, Generalized implicit function theorems with applications to some small divisor problems. II, Comm. Pure Appl. Math. 29 (1976), no. 1, 49–111. MR 426055, DOI 10.1002/cpa.3160290104