RUI: Investigating Central Configurations in the N-Body and N-Vortex Problems

该项目的目标是进一步发展天体力学中一个重要而活跃的分支领域——中心构型的研究。中心结构是一组特殊的位置，在这些位置上，由于重力作用在每个物体上的力指向与该物体位置矢量（相对于质心）相反的方向。寻找中心位形涉及到求解一个复杂的非线性代数方程组。中心构型是重要的，因为它们在n -体和n -涡问题中都导致了同质和同质解族。本课题的具体目标包括对五体共圆中心构型进行分类，研究四涡和五涡问题的相对平衡，利用对称性简化对特殊质量选择的中心构型的研究，分析相应的相对平衡的线性稳定性，并描述某些对称族的解。这些主题将使用分析和现代计算代数几何的结合来探索。n体问题涉及天体（恒星、行星、小行星，甚至宇宙飞船）在相互引力作用下的运动。虽然本质上是一门数学学科，但在天文学和航天器运输方面的应用是丰富的。例如，最近在地球-太阳系统的“特洛伊点”发现了一颗小行星，这是在拉格朗日关于三体中心构型的数学工作之后的几个世纪。n涡问题是流体动力学中用于理解涡度演化的一个广泛使用的模型。这些问题中一些最重要的解决方案本质上是周期性的，在一段固定的时间后返回到它们的初始配置。在这类解中有简单的、刚性旋转的轨道，称为相对平衡，在整个运动过程中，构型的大小和形状是不变的。定位一个相对平衡需要首先找到一个中心构型。分析相对均衡的结构和稳定性可以更好地理解整个问题的复杂性。在天体力学中，这项研究对于绘制宇宙飞船的轨迹和发现探索空间的廉价方法是有用的。此外，找到稳定的解提供了关于天文学家和其他研究人员期望在宇宙中看到的轨道的关键信息。该项目的一个重要优先事项是指导、支持和与对该领域感兴趣的本科生研究人员合作。

The goal of this project is to further develop the study of central configurations, an important and active sub-field of celestial mechanics. A central configuration is a special set of positions where the force on each body due to gravity points in the opposite direction of that body's position vector (with respect to the center of mass). Finding a central configuration involves solving a complicated system of nonlinear algebraic equations. Central configurations are important since they lead to families of homographic and homothetic solutions in both the N-body and N-vortex problems. Specific aims of the project include classifying five-body co-circular central configurations, studying relative equilibria in the four and five-vortex problems, using symmetry to simplify the study of central configurations for special choices of the masses, analyzing the linear stability of the corresponding relative equilibria, and describing certain symmetric families of solutions. These topics will be explored using a combination of analysis and modern and computational algebraic geometry.The N-body problem concerns the motion of celestial bodies (stars, planets, asteroids, even spaceships) interacting under their mutual gravitational attraction. Although inherently a mathematical discipline, applications to astronomy and spacecraft transport are plentiful. For example, the recent astronomical discovery of an asteroid at a ``Trojan point'' in the Earth-Sun system came centuries after the mathematical work of Lagrange on three-body central configurations. The N-vortex problem is a widely used model for understanding vorticity evolution in fluid dynamics. Some of the most important types of solutions in these problems are periodic in nature, returning to their initial configuration after some fixed amount of time. Among this class of solutions are simple, rigidly rotating orbits, known as relative equilibria, where the size and shape of the configuration is unchanged throughout the motion. Locating a relative equilibrium requires first finding a central configuration. Analyzing the structure and stability of relative equilibria leads to a greater understanding of the complexities in the full problem. In celestial mechanics this study is useful for plotting spacecraft trajectories and for discovering inexpensive methods of exploring space. In addition, locating stable solutions provides key information about the orbits astronomers and other researchers expect to see in the universe. An important priority of the project is to mentor, support and collaborate with undergraduate researchers interested in the field.