Hostname: page-component-84b7d79bbc-7nlkj Total loading time: 0 Render date: 2024-07-25T14:12:33.287Z Has data issue: false hasContentIssue false
  • English
  • Français

KAM theory for particles in periodic potentials

Published online by Cambridge University Press:  19 September 2008

Mark Levi
Mark Levi
Affiliation:
Department of Mathematics, Boston University, Boston, Mass. 02215, USA
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

It is shown that the system of the form x + V′ (x) = p (t) with periodic V and p and with (p) = 0 is near-integrable for large energies. In particular, most (in the sense of Lebesgue measure) fast solutions are quasiperiodic, provided V ∈ C(5) and p ∈ L1; furthermore, for any solution x(t) there exists a velocity bound c for all time: |x(t)| < c for all t ∈ R. For any real number r there exists a solution with that average velocity, and when r is rational, this solution can be chosen to be periodic.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 10 , Issue 4 , December 1990 , pp. 777 - 785
Copyright
Copyright © Cambridge University Press 1990

References

REFERENCES

[1]Aubry, S. & LeDaeron, P. Y.. The discrete Frenkel-Kontorova model and its extensions I: Exact results for the ground states. Physica 8D (1983), 381–422.Google Scholar
[2]Birkhoff, G. D.. Dynamical Systems, Amer. Math. Soc. Colloq. Publ. IX (1966), 165–169.Google Scholar
[3]Dieckerhoff, R. & Zehnder, E.. An ‘a priori’ estimate for oscillatory equation. Dyn. Sys. and Bifurcations, Groningen, 1984. LNM, 1125 Springer: Berlin-New York, 1985, pp. 9–14.CrossRefGoogle Scholar
[4]Dieckerhoff, R. & Zehnder, E.. Boundedness of solution via the twist-theorem. Preprint No. 22/1984 Ruhr-Universität Bochum.Google Scholar
[5]Franks, J.. Generalization of the Poincaré-Birkhoff Theorem. Ann. Math. To appear.Google Scholar
[6]Grüner, G. & Zettl, A.. CDW conduction: a novel collective transport phenomenon in solids. Phys. Rep. 119 No. 3 (03 1985), 119–232.CrossRefGoogle Scholar
[7]Hartman, P.. On boundary value problems for superlinear second order differential equations. J. Diff. Eq. 26(1977), 37–53.CrossRefGoogle Scholar
[8]Herman, M. R.. Sur les courbes invariantes par des diffeomorphismes de l'anneau. Asterisque 1 (1983), 103–104;Google Scholar
Asterisque 2 (1986), 144.Google Scholar
[9]Jacobowitz, H. & Struble, R. A.. Periodic solutions of x″ + f(x, t) = 0 via the Poincaré-Birkhoff theorem. J. Diff. Eq. 20 No. 1 (1976), 37–52.CrossRefGoogle Scholar
Corrigendum: The existence of the second fixed point: a correction to ‘Periodic solutions … ’ above. J. Diff. Eq. 25 No. 1 (1977), 148–149.CrossRefGoogle Scholar
[10]Katok, A.. Some remarks on the Birkhoff and Mather twist theorems. Ergod. Th. Dynam. Sys. 1 (1982), 183–194.Google Scholar
[11]Morris, G. R.. A case of boundedness in Littlewood's problem on oscillatory differential equation. Bull. Austr. Math. Soc. 14 (1976), 71–93.CrossRefGoogle Scholar
[12]Mather, J. N.. Existence of quasi-periodic orbits for twist homeomorphisms of the annulus. Topology 21 (1982), 457–476.CrossRefGoogle Scholar
[13]Moser, J. K.. On invariant curves of area-preserving mappings of annulus. Nachr. Acad. Wiss. Göttingen Math-Phys. KIII (1962), 1–20.Google Scholar
[14]Moser, J. K.. Break-down of stability. Led. Notes in Phys. 247, Jowett, J. M., Month, M. and Turner, S., eds., Springer: Berlin-New York, 1986, pp. 492–518.Google Scholar
[15]Moser, J. K.. Monotone twist mappings and the calculus of variations. Ergod. Th. Dynam. Sys. 6 (1986) 401–413.CrossRefGoogle Scholar
[16]Rüssmann, H.. Über invariante kurven differenzierbarer abbildungen eines kreisringes. Nachr. Akad. Wiss., Göttingen II, Math. Phys. Kl. (1970) 67–105.Google Scholar
[17]Moser, J. K.. Quasi-periodic solutions of nonlinear elliptic partial differential equations. Bol. Soc. Bras. Mat. 20 (1) (1989), 29–45.CrossRefGoogle Scholar