On shape sphere and rotation of three-body motion

Abstract

For three-body problem, R.Montgomery [3] proved a reconstruction formula which calculates the overall rotation relating two similar triangle configurations if the initial triangular configuration is similar to the configuration formed at some later time. In this paper, we extend the formula so that it gives the angle of rotation for a single particle without requiring the similarity of initial and final configurations. The proof is different from that in [3] and uses fundamental calculus. Moreover, we answered a question proposed in [3].