Action minimizing invariant measures for positive definite Lagrangian systems
Action minimizing invariant measures for positive definite Lagrangian systems
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DOI：
10.1007/bf02571383
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发表时间：
1991-05
影响因子：
0.8
通讯作者：
J. Mather
中科院分区：
文献类型：
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作者：
J. Mather
近年来，一些作者研究了二自由度Hamilton系统的“极小”轨道和保面积单调扭同态的“极小”轨道。这里，“最小化”意味着动作最小化。这类轨道有许多有趣的性质，可以在Bangert的综述文章[4]中看到。这是自然的问，如果有任何推广这类轨道的哈密顿系统在更多的自由度。在这篇文章中，我们提出了一个推广的周期哈密顿系统在更多的自由度。然而，我们推广的不是极小轨道的概念，而是与之密切相关的极小测度的概念，我们在[18]中引入了这个概念。我们在这里得到了两个基本结果：极小测度的存在性定理，以及一个正则性定理，它断言极小测度可以表示为切丛的（部分定义的）Lipschitz截面。在我们这里所做的归纳中，一个主要的困难是找到正确的设置。我们在这里提出的设置有两个重要的特点：结果是有效的周期正定拉格朗日系统，结果制定不变的措施。我感谢J. Moser几年前向我指出，单自由度周期正定拉格朗日系统提供了一种设置，在这种设置中，有可能制定推广作者结果的结果[17]。（以及Aubry和Le Dacron [1]的密切相关的结果）和Hedlund [12]关于"A类"的结果黎曼流形上的测地线事实上，Moser已经证明了[20]，每一个twist双同态都是与一个适当的周期正定拉格朗日系统相关联的时间一映射。Denzler [10]在一个自由度上执行了Moser的程序。这句话的莫泽建议我，定期正定拉格朗日系统应提供正确的设置在更多的自由度。有一些早期的工作在本文件的方向。伯恩斯坦和Katok [6]利用与本文变分方法有关的变分方法得到了关于不变环面附近周期轨道的结果。
In recent years, several authors have studied" minimal" orbits of Hamiltonian systems in two degrees of freedom and of area preserving monotone twist diffeomorphisms. Here," minimal" means action minimizing. This class of orbits has many interesting properties, as may be seen in the survey article of Bangert [4]. It is natural to ask if there is any generalization of this class of orbits to Hamiltonian systems in more degrees of freedom. In this article, we propose a generalization to periodic Hamiltonian systems in more degrees of freedom. However, we generalize not the notion of minimal orbit, but the closely related notion of minimal measure, which we introduced in [18].We obtain two basic results here: an existence theorem for minimal measures, and a regularity theorem which asserts that the minimal measures can be expressed as (partially defined) Lipschitz sections of the tangent bundle. In the sort of generalization that we do here, a major difficulty is finding the right setting. The setting which we propose here has two important features: the results are valid for periodic positive definite Lagrangian systems, and the results are formulated in terms of invariant measures. I am indebted to J. Moser for pointing out to me several years ago that periodic positive definite Lagrangian systems in one degree of freedom provide a setting in which it is possible to formulate results which generalize both the author's results [17](and the closely related results of Aubry and Le Dacron [1]) and the results of Hedlund [12] concerning" class A" geodesics on a Riemannian manifold diffeomorphic to the 2-torus. Indeed, Moser has proved [20] that every twist diffeomorphism is the time one map associated to a suitable periodic positive definite Lagrangian system. Denzler [10] has carried out Moser's program in one degree of freedom. This remark of Moser suggested to me that periodic positive definite Lagrangian systems should provide the right setting in more degrees of freedom. There is some earlier work in the direction of this paper. Bernstein and Katok [6] obtained results concerning periodic orbits near invariant tori, using a variational method related to the variational method of this paper.