RUI: Questions on Finiteness and Stability in Celestial Mechanics

项目申请：DMS - 0708741 PI: Roberts， Gareth e机构：the College of the Holy crostle: RUI：天体力学中的有限性和稳定性问题摘要本项目研究天体力学中一些著名的有限性和稳定性问题。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，以及关于具有常转动惯量的n体问题的唯一解是相对平衡的Saari猜想。这些问题将使用代数几何中的现代工具，如Grobner基和BKK理论来解决。从微分方程和动力系统的理论分析和数值技术也将被采用。这里考虑的问题很容易推广到其他研究领域，如几何力学、点涡的运动、广义刚体的运动和仅依赖于物体之间相互距离的幂律势系统。n体问题涉及天体通过万有引力相互作用的运动。最重要的一种解决方案本质上是周期性的，在一段固定的时间后恢复到初始配置。在这类解中，分析简单的刚性旋转轨道的结构和稳定性，即相对平衡，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空间的廉价方法特别有用。此外，找到稳定的解提供了与我们期望在宇宙中看到的各种轨道有关的关键信息。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。

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.