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

RUI：研究 N 体和 N 涡问题中的中心配置

• 批准号: 1211675
• 负责人: Gareth Roberts
• 金额: $ 13.72万
• 依托单位: College of the Holy Cross
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2012
• 资助国家: 美国
• 起止时间: 2012-07-01 至 2016-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1211675&HistoricalAwards=false
• 关键词: RUI; Investigating; Central; Configurations; Body

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.

该项目的目标是进一步发展中心构型的研究，这是天体力学中一个重要而活跃的子领域。中心构型是一组特殊的位置，其中每个物体上的重力指向与该物体的位置向量(相对于质心)相反的方向。寻找中心构型涉及到求解复杂的非线性代数方程系统。中心构型很重要，因为它们导致了N体和N涡旋问题中的一族同形和同形解。该项目的具体目标包括对五体共圆中心构型进行分类，研究四旋涡和五旋涡问题中的相对平衡，利用对称性简化对特定质量选择的中心构型的研究，分析相应相对平衡的线性稳定性，并描述某些对称解族。这些主题将使用分析和现代计算代数几何相结合的方法来探索。N体问题涉及天体(恒星、行星、小行星，甚至宇宙飞船)在相互引力下相互作用的运动。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却很多。例如，在拉格朗日关于三体中心构型的数学工作几个世纪之后，最近在地球-太阳系统的“特洛伊点”上发现了一颗小行星。N涡问题是流体力学中广泛使用的涡度演化模型。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轨道，称为相对平衡，其中构型的大小和形状在整个运动过程中保持不变。要找到一个相对均衡，首先需要找到一个中心构型。通过分析相对平衡的结构和稳定性，可以更好地理解整个问题的复杂性。在天体力学中，这项研究有助于绘制航天器的轨迹，并发现探索太空的廉价方法。此外，定位稳定的解提供了关于天文学家和其他研究人员预期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。