### Abstract

We consider initial value problems for nearly integrable Hamiltonian systems. We formulate a sufficient condition for each initial value to admit the quasi-periodic solution with a Diophantine frequency vector, without any nondegeneracy of the integrable part. We reconstruct the KAM theorem under Rssmann's nondegeneracy by the measure estimate for the set of initial values satisfying this sufficient condition. Our point-wise version is of the form analogous to the corresponding problems for the integrable case. We compare our framework with the standard KAM theorem through a brief review of the KAM theory.

Original language | English |
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Pages (from-to) | 3151-3161 |

Number of pages | 11 |

Journal | Nonlinear Analysis, Theory, Methods and Applications |

Volume | 73 |

Issue number | 10 |

DOIs | |

Publication status | Published - 2010 Nov 15 |

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### Keywords

- KAM theory
- Nearly integrable Hamiltonian systems
- Quasi-periodic motions

### ASJC Scopus subject areas

- Analysis
- Applied Mathematics