Dynamics, Stability and Symmetric minimization

动力学、稳定性和对称性最小化

• 批准号: 41872-2013
• 负责人: Offin, Daniel
• 金额: $ 0.8万
• 依托单位: Queen's University
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2015
• 资助国家: 加拿大
• 起止时间: 2015-01-01 至 2016-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=568869
• 关键词: Dynamics; Stability; Symmetric; minimization

The oldest problem in dynamics was uncovered by in the 18th C. by Laplace and Lagrange, namely the question of the stability of the solar system. The equations of Celestial mechanics have been studied for over 300 years, yet the introduction of new methods and techniques of global analysis have created a modern renaissance of results, applications and interconnected research programs. Modern applications include space mission design, molecular dynamics and considerations of dynamics and stability for exoplanet systems discovered recently outside our own solar system. Applications such as these require information about mixtures of stable and hyperbolic motion on invariant sets of the phase space which carry important dynamic properties. The investigation of these invariant sets requires new global geometric techniques. Symmetric minimization in dynamical settings including collision free solutions of the N-body problem is an important new method for studying global periodic orbits and their dynamic neighbourhoods. My research program in the period covered by this proposal will involve projects centred around the theme of characterizing stability or instability of symmetric minimizing periodic orbits. Additional projects using the methods and structures of global symplectic geometry such as using the Maslov index to characterize long period orbits in the neighbourhood of relative equilibria for the spatial restricted three body problem will involve and encourage the participation of HQP. With the help of funding provided by NSERC, my program will continue to provide opportunities for advanced learning and development for undergraduate and graduate students working together with me. My research program continues to be enhanced and deepened by research input and collaboration by current and future graduate students under my supervision. The impact of this fundamental research into the dynamics of the N-body problem will contribute to research groups in Canada and worldwide currently involved in development of fundamental understanding for this important topic.

动力学中最古老的问题是由18世纪的科学家发现的。由拉普拉斯和拉格朗日，即太阳系的稳定性问题。天体力学的方程已经研究了300多年，但全球分析的新方法和技术的引入已经创造了结果，应用和相互关联的研究计划的现代复兴。现代应用包括空间使命设计、分子动力学以及最近在我们太阳系外发现的系外行星系统的动力学和稳定性考虑。 应用程序，如这些需要的信息混合物的稳定和双曲运动的不变集的相空间进行重要的动态特性。这些不变集的调查需要新的全球几何技术。动力学中的对称极小化方法包括N体问题的无碰撞解，是研究全局周期轨道及其动力学邻域的一种重要新方法。 我的研究计划在此期间所涵盖的建议将涉及项目围绕的主题的特点稳定性或不稳定性的对称最小化周期轨道。使用全局辛几何的方法和结构的其他项目，如使用马斯洛夫指数来表征空间限制三体问题相对平衡附近的长周期轨道，将涉及并鼓励HQP的参与。在NSERC提供的资金帮助下，我的项目将继续为与我一起工作的本科生和研究生提供高级学习和发展的机会。我的研究项目将继续通过我监督下的当前和未来研究生的研究投入和合作来加强和深化。这一基础研究对N体问题动力学的影响将有助于加拿大和全世界目前参与发展对这一重要主题的基本理解的研究小组。