RUI: Questions on Finiteness and Stability in Celestial Mechanics

建议：dms-0708741 Pi：Roberts，Gareth e Institute：圣十字学院标题：Ri：天体力学中的有限性和稳定性问题本项目研究天体力学中的一些著名的有限性和稳定性问题。将特别注意相对平衡等价类的有限性问题、相对平衡的线性稳定性问题以及Saari猜想，即具有恒定转动惯量的n体问题的唯一解是相对平衡。这些问题将使用代数几何中的现代工具来解决，如Grobner基和BKK理论。本课程还将使用微分方程式和动力系统理论中的分析和数值技术。这里讨论的问题很容易推广到其他研究领域，如几何力学、点涡旋运动、广义刚体的运动和仅依赖于物体之间相互距离的幂函数势系统。n体问题涉及通过引力相互作用的天体的运动。最重要的解决方案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。在这类解中，分析简单的刚性旋转轨道的结构和稳定性，即所谓的相对平衡，可以更好地理解整个问题的复杂性。对相对平衡的研究对于绘制航天器的轨迹和发现探索太空的廉价方法特别有用。此外，定位稳定的解提供了与我们预期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，并创建了一个整合了天体力学和代数几何领域的顶峰研讨会。

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.