RUI: Questions on Finiteness and Stability in Celestial Mechanics

RUI：天体力学的有限性和稳定性问题

• 批准号: 0708741
• 负责人: Gareth Roberts
• 金额: --
• 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: 2007
• 资助国家: 美国
• 起止时间: 2007-07-01 至 2011-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0708741&HistoricalAwards=false
• 关键词: RUI; Questions; Finiteness; Stability; Celestial

Proposal: DMS - 0708741 PI: Roberts, Gareth EInstitution: College of the Holy CrossTitle: RUI: QUESTIONS ON FINITENESS AND STABILITY IN CELESTIAL MECHANICSAbstractThis project investigates some well-known finiteness and stability questions in celestial mechanics. Particular attention will be given to the question on finiteness of relative equilibria equivalence classes, linear stability of relative equilibria and Saari's conjecture, that the only solutions in the n-body problem with a constant moment of inertia are relative equilibria. These questions will be approached using modern tools from algebraic geometry such as Grobner bases and BKK theory. Analytic and numerical techniques from the theory of differential equations and dynamical systems will also be employed. The problems considered here are easily generalizable to other fields of study such as geometric mechanics, the motion of point vortices, the motion of a generalized rigid body and power-law potential systems depending only on the mutual distances between bodies.The n-body problem concerns the motion of celestial bodies interacting through gravitational attraction. One of the most important types of solutions are periodic in nature, returning to their initial configuration after some fixed amount of time. Among this class of solutions, analyzing the structure and stability of simple, rigidly rotating orbits, known as relative equilibria, leads to a greater understanding of the complexities in the full problem. The study of relative equilibria is particularly useful for plotting spacecraft trajectory and discovering inexpensive methods of exploring space. Moreover, locating stable solutions provides key information pertaining to the kinds of orbits we expect to see in the universe. The educational impact of this project includes the continued mentoring of undergraduate researchers and the creation of a capstone seminar integrating the fields of celestial mechanics and algebraic geometry.

提案：DMS - 0708741 PI：Roberts, Gareth E 机构：圣十字学院 标题：RUI：天体力学中的有限性和稳定性问题摘要 该项目研究天体力学中一些众所周知的有限性和稳定性问题。将特别关注相对平衡等价类的有限性、相对平衡的线性稳定性和萨里猜想，即惯性矩恒定的n体问题的唯一解是相对平衡。这些问题将使用代数几何的现代工具（例如格罗布纳基和 BKK 理论）来解决。还将采用微分方程和动力系统理论的分析和数值技术。这里考虑的问题很容易推广到其他研究领域，例如几何力学、点涡运动、广义刚体的运动和仅取决于物体之间相互距离的幂律势系统。 n 体问题涉及通过万有引力相互作用的天体运动。最重要的解决方案类型之一本质上是周期性的，在一段固定的时间后返回到其初始配置。在此类解决方案中，分析简单、刚性旋转轨道（称为相对平衡）的结构和稳定性，可以更好地理解整个问题的复杂性。相对平衡的研究对于绘制航天器轨迹和发现探索太空的廉价方法特别有用。此外，找到稳定的解决方案提供了与我们期望在宇宙中看到的轨道类型有关的关键信息。该项目的教育影响包括继续指导本科生研究人员以及创建一个整合天体力学和代数几何领域的顶点研讨会。