國立清華大學 數學系

Abstract: In recent years, variationalmethods have been applied to construct many periodic and relative periodic solutions for both the n-body and the n-center problem. Collision avoidance is often carried out by either local deformation or global estimate for the action functional, but not both. It would be interesting to find examples which require both methods, we will show that few-body-few-center systems are such examples. We present examples for which both techniques are needed. These examples are combinations of n-body and n-center problems, that is, the few-body-few-center problem. Another motivation for our study of few-body-few-center systems is due to the fact that n-body systems are generally much more complex than n-center systems, as the later is often served as simplification of the former, it would be interesting to understand the intermediate case. Due to reasons above, here we focus on the n1-body-n2-center system, meaning the system has n1free bodies and n2fixed ones. We want to study the motions of n1free bodies under the gravitational influence of the whole system. Comparing with n-center systems, some nice features of action minimizers are no longer valid in n1-body-n2-center systems, such as collision reflection property. We will show existence of some collision-free minimizers for such systems.