Research Initiation Award: Central Configurations, New Variational Method and Periodic Solutions in Celestial Mechanics

研究启动奖为历史上黑人学院和大学的初级和中级职业教师提供支持，他们正在建立新的研究项目或重新定向和重建现有的研究项目。预计该奖项将有助于进一步提高教师的研究能力和效率，改善其所在机构的研究和教学，并使本科生参与研究经验。 授予弗吉尼亚州立大学的奖项在许多领域都有潜在的更广泛的影响。该项目的目标是发展新的变分方法来研究不寻常的星星系统的轨道，如四重星星系统。在天体力学领域，将开发具有符号计算等先进数学工具的新教材。该项目还将增强弗吉尼亚州立大学本科生的研究经验和培训。本项目主要是对天体力学中的中心构型、新的变分方法和结构规定边界条件（SPBC）以及一些新的周期轨道的存在性等问题进行数学求解。 在牛顿平面四体问题中，用变分法数值地发现了五角形的星星编排，并从理论上加以证明。该项目现在的目标是建立一个完整的数学和计算框架的新变分方法与SPBC的这个系统。本项目的第二个目标是利用变分方法寻找新的周期解。 应用该方法来寻找具有三个不等质量的系统的周期或准周期解的灵感来自于2011年发现存在一颗有两颗恒星的行星。数值解可以指导理解为什么存在这样一个不寻常的系统。然后，这些计算技术可以应用于搜索任何n体问题的周期和准周期解。在开发和应用新方法的结果承诺令人兴奋的例子和实用的技术，以促进天体力学的动力学行为的理解。

Research Initiation Awards provide support for junior and mid-career faculty at Historically Black Colleges and Universities who are building new research programs or redirecting and rebuilding existing research programs. It is expected that the award helps to further the faculty member's research capability and effectiveness, improves research and teaching at his home institution, and involves undergraduate students in research experiences. The award to Virginia State University has potential broader impact in a number of areas. The goal of the project is to develop new variational methods to investigate orbits of unusual star system, such as quadruple star systems. New teaching materials with advanced mathematical tools, such as symbolic computation, will be developed in the field of celestial mechanics. This project will also enhance the research experience and training of undergraduate students at Virginia State University. This project seeks mathematical solutions to some questions on the central configurations, the new variational method with Structural Prescribed Boundary Conditions (SPBC) and the existence of some new periodic orbits in celestial mechanics. The Star Pentagon choreography has been numerically discovered and theoretically proven by the variational method in the Newtonian planar four-body problem. The project now aims to build a complete mathematical and computational framework for the new variational method with SPBC for this system. General criteria will be established for SPBC which can generate classical solutions without singularities.The second goal of this project is to use the variational method with SPBC to search for new periodic solutions. The application of the method to search for periodic or quasi-periodic solutions to systems with three unequal masses was inspired by the 2011 discovery of the existence of a planet with two stars. The numerical solutions may guide the understanding of why such an unusual system exists. These computational techniques can then be applied to search for periodic and quasi-periodic solutions for any n-body problem. The results in developing and applying the new methods promise exciting examples and practical techniques to advance the understanding of the dynamical behavior in celestial mechanics.