Type, The Central Configurations Of Five Bodies, And The Nine-body Problem

Celestial mechanics is an old subject. It is a branch of astronomy, but it has intimate relationship with mathematics and physics. Among the modern celestial mechanics, n-body problem is an important field. N-body problem is a common field between celestial mechanics and general mechanics vs. applied mathematics. Central configurations have long history more than 200 years , and which is an important tool to study n-body problem, and the famous mathematician Stephen Smale once described it as one of the 18 important mathematics problems in 21 century.In this paper, we find the sufficient or necessary conditions for two special configurations to form central configurations under some conditions. One of the special configurations is"1+kite"configuration in five body problem. The other is the configuration in nine body problem that four bodies are at the vertices of a rhombus , four bodies are at the midpoint of the side of the rhombus and the last body is at the center.