近可积广义哈密顿系统的作用极小测度.docVIP

近可积广义哈密顿系统的作用极小测度 王楷植, 李勇 吉林大学数学学院，长春　 130012 摘要：本文研究一类近可积广义哈密顿系统的作用极小测度的存在性问题. 支撑作用极小测度 的不变集是不变环面的推广. 本文的证明方法本质上是变分的, 并且涉及由 Evans L. 引入的一 种泛函的极小化问题. 在给出极小化子梯度的先验估计和定义广义 Legendre 变换之后, 一种修 正的弱 KAM 技巧可被用于得到广义哈密顿系统的作用极小测度的存在性. 关键词：广义哈密顿系统, 作用极小测度, 弱 KAM 理论 中图分类号： O175.13 Action minimizing measures for nearly integrable generalized Hamiltonian systems WANG Kai-Zhi, LI Yong College of Mathematics, Jilin University, Changchun 130012 Abstract: In this article, we study the existence of action minimizing measures for a class of nearly integrable, generalized Hamiltonian systems. The invariant sets, which support the action minimizing measures, are generalizations of invariant tori. The proof in this paper is variational in nature, and involves the minimization of the functional introduced by Evans L.. After giving an a priori estimate of gradients of the minimizers and de?ning the generalized Legendre transformation, a modi?ed version of weak KAM techniques is available to obtain action minimizing measures for the generalized Hamiltonian system. Key words: generalized Hamiltonian systems, action minimizing measures, weak KAM theory 0 Introduction It is well known that Mather [1, 2] generalized the results—Aubry-Mather theory [3, 4] on minimal orbits of Hamiltonian systems in two degrees of freedom and of area preserving monotone twist di?eomorphisms to periodic Hamiltonian systems in more degrees of freedom. In the sort of the generalization, minimal orbits were replaced by action minimizing measures. An existence theorem and a regularity theorem for action minimizing measures were obtained 基金项目： Specialized Research Fund for the Doctoral Program of Higher Education (20100061120094, 20090061110001 ). 作者简介： Wang Kaizhi (1981-), male, associate professor, major research direction：Hamiltonian systems. Li Yong (1958-) male, professor, major research direction：Hamiltonian systems. Email: kaizhiwang@jlu.edu.cn, liyong@jlu.edu.cn -1- there. And the invariant sets, which support