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）将天体力学经典N体问题自然推广到常曲率空间，即球面和双曲球面;（2）将这些方程推广到球面和双曲球面曲率变化的情况（3）将恒星动力学的Vlasov-Poisson方程推广到球面和双曲球面。（2）和（3）都建立在（1）的基础上，但方向不同。虽然（2）扩展到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），并可能有助于理解银河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，我们最近通过探索它在欧几里德情况下的类似物，为这个系统引入了中心构型。中心配置允许相对平衡的统一处理，这是解决方案，其中点质量保持恒定的相互距离的时间。他们的研究是理解微分方程的基础。为了这个目的，我们将使用几何和代数方法在动力系统，微分和非欧几何，拓扑，几何力学，以及李群和代数理论的研究。我们将感兴趣的方面有关的中心配置的数量（温特纳-斯梅尔猜想的推广），以及在相对平衡的稳定性。对于（2），我们将依赖于我们最近获得的一个结果，该结果表明（1）的相对平衡可以用于获得对应于膨胀或收缩的弯曲宇宙的非自治运动方程的解。特别是，我们可能能够理解与哈勃定律下宇宙当前膨胀相关的一些动力学方面，并找到经典力学和广义相对论之间的新联系。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球面和双曲球面上的Vlasov-Poisson系统的线性化形式具有一些有趣的定性性质，如朗道阻尼.我们的目的是表明，它们也发生在一般情况下。为了实现这一目标，我们将不得不克服的困难，延长某些技术中使用的欧几里德的情况下，由维拉尼，彭罗斯，穆奥，和其他人，一个不平凡的任务，鉴于新的设置的问题。然而，我们已经有一些迹象表明，这一计划是可以实现的。这项研究计划提供了许多机会，从事本科生和研究生以及博士后研究员。

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.