On shape sphere and rotation of three-body motion

TL;DR

This paper extends Montgomery's reconstruction formula for the three-body problem to compute the rotation angle of a single particle without initial-final configuration similarity, using fundamental calculus and addressing an open question.

Contribution

It provides a new method to determine single-particle rotation angles in three-body motion, expanding the applicability of previous reconstruction formulas.

Findings

Extended the reconstruction formula to single particles

Provided a different proof using fundamental calculus

Answered an open question from previous work

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].