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

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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.