KAM theory for particles in periodic potentials

Research output: Contribution to journalArticle

25 Citations (Scopus)

Abstract

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 L 1; 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.

Original languageEnglish (US)
Pages (from-to)777-785
Number of pages9
JournalErgodic Theory and Dynamical Systems
Volume10
Issue number4
DOIs
StatePublished - Jan 1 1990

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KAM Theory
Periodic Potential
Lebesgue Measure
Energy
Form

All Science Journal Classification (ASJC) codes

  • Mathematics(all)
  • Applied Mathematics

Cite this

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KAM theory for particles in periodic potentials. / Levi, Mark.

In: Ergodic Theory and Dynamical Systems, Vol. 10, No. 4, 01.01.1990, p. 777-785.

Research output: Contribution to journalArticle

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AB - 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 L 1; 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.

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