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Bounds on the trajectories of a system of weakly coupled rotators

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Abstract

In this paper we study a classical mechanical system of weakly coupled rotators on a one-dimensional lattice. Such systems are of interest in statistical mechanics. We prove that for any site in the system there is a “large” set of initial conditions for which there exists a canonical change of variables such that the trajectory of that site, in the transformed system, is essentially indistinguishable from that of an integrable system for a long (but finite) time. Alternatively, the trajectory of this site lies very close to a torus in the phase space of the system for times very long in comparison with the typical period of the unperturbed rotators. All the estimates in this theory areindependent of the number of degrees of freedom in the system. We propose this mechanism as an explanation of certain numerical experiments.

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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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Communicated by J. L. Lebowitz

Supported in part by NSF Grant DMS-8403664

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Wayne, C.E. Bounds on the trajectories of a system of weakly coupled rotators. Commun.Math. Phys. 104, 21–36 (1986). https://doi.org/10.1007/BF01210790

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  • DOI: https://doi.org/10.1007/BF01210790

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