Hostname: page-component-cd9895bd7-7cvxr Total loading time: 0 Render date: 2024-12-27T04:34:42.506Z Has data issue: false hasContentIssue false
  • English
  • Français

Invariant curves and time-dependent potentials

Published online by Cambridge University Press:  19 September 2008

Stephane Laederich  and
Mark Lev
Stephane Laederich
Affiliation:
Department of Mathematics, Boston University, Boston, MA 02215, USA
Mark Lev
Affiliation:
Department of Mathematics, Boston University, Boston, MA 02215, USA
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper we prove the existence of invariant curves and thus stability for all time for a class of Hamiltonian systems with time-dependent potentials, namely, for systems of the form

where

is a superquadratic polynomial potential with periodic coefficients. As a limiting case, a proof of the stability of Ulam's problem of a particle bouncing between two periodicially moving walls is given.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 11 , Issue 2 , June 1991 , pp. 365 - 378
Copyright
Copyright © Cambridge University Press 1991

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1]Arnol'd, V. I.. On the behavior of an adiabatic invariant under a slow periodic change of the Hamiltonian. Dokl. Akad. Nauk. 142 (4) (1962), 758–761;Google Scholar
Translated as: Sov. Math. Dokl. 3 (1962), 136–139.Google Scholar
[2]Chiercia, L. & Zehnder, E.. On asymptotic expansion of quasiperiodic solutions. Preprint, ETH, Zürich, 1988.Google Scholar
[3]Dieckerhoff, R. & Zehnder, E.. An ‘a priori’ estimate for oscillatory equation. Dyn. Systems and Bifurcations, Groningen, 1984. Springer Lecture Notes in Mathematics 1125, Springer: Berlin, New York, 1985, pp. 9–14.CrossRefGoogle Scholar
[4]Dieckerhoff, R. & Zehnder, E.. Boundedness of solutions via the Twist theorem. Ann. Scuola Norm, Sup. Pisa 14(1) (1987), 79–85.Google Scholar
[5]Douady, R.. PhD Thesis, Ecole Polytechnique (1988).Google Scholar
[6]Herman, M. R.. Sur les Courbes Invariantes par les Difféomorphismes de l'Anneau. Vol. 1 Astérisque 1 (1983), 103–104;Google Scholar
and Astérisque 2 (1986), 144.Google Scholar
[7]Levi, M.. KAM Theory for particles in periodic potentials. Ergod. Th. & Dynam. Sys. 10 (1990), 777–785.CrossRefGoogle Scholar
[8]Levi, M.. On the Littlewood's counterexample of unbounded motions in superquadratic potentials. Preprint, ETH, Zürich, October 1988; to appear in Dynamics Reported.Google Scholar
[9]Littlewood, J. E.. Unbounded solutions of an equation ÿ−g(y) = p(t), with p(t) periodic and bounded and g(y)/y → ∞ as y → ±∞. J. London Math. Soc. 41 (1966), 497–507.CrossRefGoogle Scholar
[10]Morris, G. R.. A case of boundedness in Littlewood's problem on oscillatory differential equations. Bull. Austr. Math. Soc. 14 (1976) 71–93.CrossRefGoogle Scholar
[11]Moser, J. K.. On invariant curves of area preserving mappings of an annulus. Nachr. Akad. Wiss. Göttingen Math-Phys. Kl.II (1962), 1–20.Google Scholar
[12]Moser, J. K.. Quasi-periodic solutions of nonlinear elliptic partial differential equations. Bol. Soc. Bras. Mat. 20(1) (1989), 29–45.CrossRefGoogle Scholar
[13]Moser, J. K.. Stable and random motions in Dynamical Systems. Princeton University Press, 1973.Google Scholar
[14]Rüssman, H.. Über invariante Kurven differenzierbarer Abbildungen eines Kreisringes. Nachr. Akad. Wiss. Göttingen, Math. Phys. Kil. II (1970), 67–105.Google Scholar
[15]Rüssman, H.. On the existence of invariant curves of twist mapping of an annulus, in: Palis, J., ed., Geometric Dynamics, Springer Lecture Notes in Mathematics 1007 Springer: New York (1981), pp. 677–718.CrossRefGoogle Scholar