Date of Award
5-2020
Degree Type
Honors College Thesis
Department
Mathematics
First Advisor
Zhifu Xie, Ph.D
Advisor Department
Mathematics
Abstract
The N-body problem qualifies as the problem of the twenty-first century because of its fundamental importance and difficulty to solve [1]. A number of great mathematicians and physicists have tried but failed to come up with the general solution of the problem. Due to the complexity of the problem, even a partial result will help us in the understanding of the N-body problem. Central configurations play a ‘central’ role in the understanding of the N-body problem. The well known Euler and Lagrangian solutions are both generated from three-body central configurations. The existence and classifications of central configurations have attracted number of researchers in the past three hundred years. A number of results and papers have been published but the study of central configurations is still far from complete. We do not even know how to prove the finiteness of the number of central configurations for N > 5.
In this paper, we studied a special type of central configuration: twisted central configurations of the eight-body problem. Consider the eight-body problem where these bodies m1;m2; ;m8 are located on the vertices of two squares 1234 and 5678. We assume that both squares have the same centroid and they are symmetrical about the lines connecting m5, m7 and m6, m8 (see Figure 4.1). We show that the masses must satisfy m1 = m2 = m3 = m4 and m5 = m6 = m7 = m8 when the configuration forms a central configuration. When the two square configurations have a common centroid, the configuration can form a central configuration only if the ratio of the size of the two squares falls into one of three intervals. Moreover there are some numerical evidences that there are exactly three nested central configurations for each given mass ratio m1|m5 .
Copyright
Copyright for this thesis is owned by the author. It may be freely accessed by all users. However, any reuse or reproduction not covered by the exceptions of the Fair Use or Educational Use clauses of U.S. Copyright Law or without permission of the copyright holder may be a violation of federal law. Contact the administrator if you have additional questions.
Recommended Citation
Bhusal, Gokul, "Twisted Central Configurations of the Eight-Body problem" (2020). Honors Theses. 703.
https://aquila.usm.edu/honors_theses/703