Dynamics, Stability and Symmetric minimization

动力学中最古老的问题是由拉普拉斯和拉格朗日在18世纪发现的，即太阳系的稳定性问题。天体力学方程的研究已经有300多年的历史了，然而全球分析的新方法和新技术的引入创造了结果、应用和相互关联的研究计划的现代复兴。现代应用包括空间任务设计、分子动力学以及最近在太阳系外发现的系外行星系统的动力学和稳定性考虑。 这样的应用需要关于相空间的不变集上的稳定运动和双曲运动的混合的信息，这些相空间具有重要的动力学性质。对这些不变集的研究需要新的全局几何技术。动力学环境下的对称极小化是研究全球周期轨道及其动态邻域的一种重要的新方法。我在这项提议所涵盖的期间的研究计划将涉及围绕对称最小化周期轨道的稳定性或不稳定性这一主题的项目。使用全局辛几何的方法和结构的其他项目，例如使用Maslov指数来表征空间受限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合作的本科生和研究生提供深造和发展的机会。我的研究项目在我的指导下，通过现在和未来的研究生的研究投入和合作，继续得到加强和深化。这项对N-Body问题动力学的基础研究的影响将有助于加拿大和世界各地目前参与发展对这一重要主题的基本理解的研究小组。

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.