A New Proof of Anosov’s Averaging Theorem
A New Proof of Anosov’s Averaging Theorem
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DOI：
10.1007/978-1-4613-8448-9_8
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发表时间：
1995
影响因子：
1
通讯作者：
H. Dumas
中科院分区：
文献类型：
--
作者：
H. Dumas
介绍。我想描述和概述我最近与 François Golse 和 Pierre Lochak 一起获得的平均理论结果的证明 [6]。我们的结果包括 D.V. 的新证明和概括。阿诺索夫一般多相平均定理 [1]，适用于具有在光滑紧凑浸没流形上演化的慢速变量和在 Rman 中演化的快速变量的 ODE 系统。我们通过允许快速变量属于任意光滑紧致黎曼流形，并且向量场仅具有索博列夫正则来扩展阿诺索夫的工作。这是使用适用于 ODE 广义流 DiPerna-Lions 理论的稍微广义版本的范式技术来完成的 [5]。通过专门研究哈密顿量向量场的情况，我们获得了低正则性哈密顿量的有趣且有些令人惊讶的结果，以及将本文纳入这些程序的原因。
Introduction. I would like to describe and sketch the proof of a result in averaging theory I obtained recently with François Golse and Pierre Lochak [6]. Our result comprises both a new proof, as well as a generalization, of D.V. Anosov’s general multiphase averaging theorem [1] for systems of ODEs with slow variables evolving inRmand fast variables evolving on a smooth compact immersed manifold. We extend Anosov’s work by allowing the fast variables to belong to an arbitrary smooth compact Riemannian manifold, and the vector field to have only Sobolev regularity. This is accomplished using normal form techniques adapted to a slightly generalized version of the DiPerna-Lions theory of generalized flows for ODEs [5]. By specializing to the case of Hamiltonian vector fields, we obtain an interesting and somewhat surprising result for Hamiltonians of low regularity, as well as a reason for including this article in these proceedings.