### 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 language | English (US) |
---|---|

Pages (from-to) | 777-785 |

Number of pages | 9 |

Journal | Ergodic Theory and Dynamical Systems |

Volume | 10 |

Issue number | 4 |

DOIs | |

State | Published - Jan 1 1990 |

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### All Science Journal Classification (ASJC) codes

- Mathematics(all)
- Applied Mathematics

### Cite this

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*Ergodic Theory and Dynamical Systems*, vol. 10, no. 4, pp. 777-785. https://doi.org/10.1017/S0143385700005897

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

Research output: Contribution to journal › Article

TY - JOUR

T1 - KAM theory for particles in periodic potentials

AU - Levi, Mark

PY - 1990/1/1

Y1 - 1990/1/1

N2 - 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.

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.

UR - http://www.scopus.com/inward/record.url?scp=84971946620&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84971946620&partnerID=8YFLogxK

U2 - 10.1017/S0143385700005897

DO - 10.1017/S0143385700005897

M3 - Article

AN - SCOPUS:84971946620

VL - 10

SP - 777

EP - 785

JO - Ergodic Theory and Dynamical Systems

JF - Ergodic Theory and Dynamical Systems

SN - 0143-3857

IS - 4

ER -