Particle Dynamics on Manifolds

本文概述了微分方程定性理论中三个相互联系的研究方向：(1)将经典天体力学n体问题自然推广到常曲率空间，即球和双曲球；(2)将这些方程推广到球和双曲球曲率变化的情况（即它们随时间膨胀或收缩）；(3)将恒星动力学的Vlasov-Poisson方程推广到球体和双曲球体。(2)和(3)都建立在(1)的基础上，但方向不同。虽然(2)扩展到一个非自治常微分方程系统，在宇宙学中具有潜在的应用，(3)在动力学理论的框架中使用(1)，并且可能有助于理解星系动力学的某些方面。对于(1)，要探索的主要概念是中心构型，我们最近通过探索欧几里得情况下的类似情况为该系统引入了中心构型。中心构型允许对相对平衡进行统一处理，相对平衡是点质量在时间上保持恒定相互距离的解。他们的研究是理解微分方程的基础。为此，我们将使用几何和代数方法来研究动力系统、微分和非欧几里得几何、拓扑学、几何力学以及李群和代数理论。我们将对中心构型的数量（Wintner-Smale猜想的推广）以及相对平衡的稳定性感兴趣。对于(2)，我们将依赖于我们最近得到的一个结果，该结果表明(1)的相对平衡可以用来获得与膨胀或收缩弯曲宇宙相对应的非自治运动方程的解。特别是，在哈勃定律下，我们可能能够理解与当前宇宙膨胀有关的一些动力学方面，并在经典力学和广义相对论之间找到新的联系。类似于上面描述的技术也可以用于这个问题。对于(3)，我们最近成功地证明了一些有趣的定性性质，如朗道阻尼，出现在球和双曲球上的线性化Vlasov-Poisson系统中。我们的目标是证明它们在一般情况下也会发生。为了实现这一目标，我们必须克服维拉尼、彭鲁斯、穆奥等人在欧几里得案例中使用的某些技术的扩展困难，考虑到问题的新背景，这是一项艰巨的任务。然而，我们已经有一些迹象表明这个计划是可以实现的。这项研究计划为本科生和研究生以及博士后提供了许多机会。

This proposal outlines the study of three connected research directions in the qualitative theory of differential equations: (1) the natural generalization of the classical N-body problem of celestial mechanics to spaces of constant curvature, namely spheres and hyperbolic spheres; (2) the extension of these equations to the case when the curvature of the spheres and hyperbolic spheres varies (i.e. they inflate or contract in time); (3) the generalization of the Vlasov-Poisson equations of stellar dynamics to spheres and hyperbolic spheres. Both (2) and (3) are built on (1), but in different directions. While (2) extends to a system of non-autonomous ordinary differential equations, with potential applications in cosmology, (3) uses (1) in the framework of kinetic theory, and may help understand some aspects of galactic dynamics. For (1), the main concept to be explored is that of central configurations, which we recently introduced for this system by exploring its analogue from the Euclidean case. Central configurations allow a unifying treatment of relative equilibria, which are solutions for which the point masses maintain constant mutual distances in time. Their study is fundamental for understanding the differential equations. For this purpose we will use geometric and algebraic methods in the study of dynamical systems, differential and non-Euclidean geometry, topology, geometric mechanics, as well as the theory of Lie groups and algebras. We will be interested in aspects related to the number of central configurations (a generalization of the Wintner-Smale conjecture) as well as in the stability of relative equilibria. For (2), we will rely on a recent result we obtained, which shows that the relative equilibria of (1) can be used to obtain solutions of the non-autonomous equations of motion that correspond to an expanding or contracting curved universe. In particular, we may be able to understand some dynamical aspects related to the current expansion of the universe under Hubble's law and find new connections between classical mechanics and general relativity. Techniques similar to the ones described above can be used in this problem too. For (3), we recently succeeded to prove that some interesting qualitative properties, such as Landau damping, occur in the linearized version of the Vlasov-Poisson system on spheres and hyperbolic spheres. We aim to show that they occur in the general case as well. To achieve this goal we will have to surmount the difficulties of extending certain techniques used in the Euclidean case by Villani, Penrouse, Mouhot, and others, a nontrivial task, given the new setting of the problem. Nevertheless, we already have some indication that this plan can be achieved. This research proposal offers many opportunities to engage undergraduates and graduate students as well as postdoctoral fellows.