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

研究启动奖为传统黑人学院和大学的初级和中期职业教师提供支持，他们正在建立新的研究项目或重新指导和重建现有的研究项目。期望该奖项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可能在许多领域产生更广泛的影响。该项目的目标是开发新的变分方法来研究不寻常的恒星系统，如四星系统的轨道。利用符号计算等先进数学工具开发天体力学新教材。该项目还将提高弗吉尼亚州立大学本科生的研究经验和培训。本课题对天体力学中中心位形、结构规定边界条件下的新变分方法和一些新周期轨道的存在性等问题寻求数学解。在牛顿平面四体问题中，用变分方法对星五边形编排进行了数值发现和理论证明。该项目目前的目标是为该系统的SPBC新变分方法建立一个完整的数学和计算框架。建立了SPBC的一般准则，该准则可以产生无奇点的经典解。本课题的第二个目标是利用SPBC的变分方法来寻找新的周期解。该方法应用于寻找具有三个不等质量的系统的周期或准周期解，灵感来自2011年发现的一颗有两颗恒星的行星的存在。数值解可以指导理解为什么这样一个不寻常的系统存在。这些计算技术可以应用于任何n体问题的周期和准周期解的搜索。这些新方法的发展和应用结果为促进对天体力学动力学行为的理解提供了令人兴奋的例子和实用技术。

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.