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[PAST EVENT] Mathematics Colloquium: Zhifu Xie, Virginia State University
October 25, 2013
2pm - 3pm
Abstract: This is a joint work with Tiancheng Ouyang. We develop a new variational method with structural prescribed boundary conditions (SPBC) to discover new periodic solutions and also to theoretically prove the existence of such solutions. In the past decades, the existence of many new interesting periodic orbits is proved by using variational method for the n-body problem. Most of them are found by minimizing the Lagrangian action on a symmetric loop space with some topological constraints. Our new variational method by minimizing the Lagrangian action on a path space with SPBC largely complements the current variational methods.
An exciting new stable choreographic solution which is called star pentagon choreography has not only been numerically discovered but also been theoretically proved by the new variational method with SPBC in the Newtonian planar four-body problem. This is the second stable choreographic solution after the publication of the famous figure-eight solution at the Annals of Mathematics by A. Chenciner and R. Montgomery in 2000. Many expertises attempt to study choreographic solutions and a large number of simple choreographic solutions have been discovered numerically but very few of them have rigorous existence proofs and none of them are stable. Significantly different from the remarkable figure-eight solution, we proved that the unequal-mass variants of the stable start pentagon are just as stable as the basic equal mass choreography. This fact makes the beautiful star pentagon orbit all the more remarkable because such periodic solutions actually have more chance to be seen in some quadruple star system. The result was mentioned at Scientific American {{http://blogs.scientificamerican.com/guest-blog/2013/07/30/5-gifs-of-n-body-orbits/, The result was mentioned at Scientific American.}}
An exciting new stable choreographic solution which is called star pentagon choreography has not only been numerically discovered but also been theoretically proved by the new variational method with SPBC in the Newtonian planar four-body problem. This is the second stable choreographic solution after the publication of the famous figure-eight solution at the Annals of Mathematics by A. Chenciner and R. Montgomery in 2000. Many expertises attempt to study choreographic solutions and a large number of simple choreographic solutions have been discovered numerically but very few of them have rigorous existence proofs and none of them are stable. Significantly different from the remarkable figure-eight solution, we proved that the unequal-mass variants of the stable start pentagon are just as stable as the basic equal mass choreography. This fact makes the beautiful star pentagon orbit all the more remarkable because such periodic solutions actually have more chance to be seen in some quadruple star system. The result was mentioned at Scientific American {{http://blogs.scientificamerican.com/guest-blog/2013/07/30/5-gifs-of-n-body-orbits/, The result was mentioned at Scientific American.}}
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Chi-Kwong Li