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On the elimination of non-resonance harmonics

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Abstract

Given a weakly coupled Hamiltonian system with short range, one dimensional interactions, andany initial conditions a canonical change of variables is constructed which yields a new Hamiltonian consisting of three parts—an integrable term, a resonant term whose effects are localized in those regions of the system which give small denominators in the Kolmogorov-Arnol'd-Moser iteration scheme and a non-resonant interaction term which is very small. (In particular, much, much smaller than our original interactions.) The conditions which allow such a transformation to be constructed are independent of the number of degrees of freedom in the system, as are the estimates on the size of the various terms. Thus, if the resonances are “sparsely” distributed through the system most of the sites in the transformed Hamiltonian behave essentially like an integrable system, at least for as long a time as the trajectory of the system lies within the region where the canonical transformation is defined. In subsequent work it is shown that this time is long, and once again independent of the number of degrees of freedom in the system.

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Authors and Affiliations

  1. Department of Mathematics, The Pennsylvania State University, 16802, University Park, Pennsylvania, USA

    C. Eugene Wayne

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  1. C. Eugene Wayne
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Additional information

Communicated by J. L. Lebowitz

Supported in part by NSF Grant DMS-8403664

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Wayne, C.E. On the elimination of non-resonance harmonics. Commun.Math. Phys. 103, 351–386 (1986). https://doi.org/10.1007/BF01211753

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  • DOI: https://doi.org/10.1007/BF01211753

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