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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
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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

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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