The Three-body problem and the shape sphere

TL;DR

This paper explores the geometry and dynamics of the three-body problem using the shape sphere, revealing new insights and theorems about the behavior of planar triangle configurations.

Contribution

It introduces a shape space perspective to derive and analyze the geometry and dynamics of the three-body problem, leading to new theorems.

Findings

Derived the geometry of shape space related to the three-body problem

Discovered two new theorems about three-body dynamics using shape space

Provided insights into the moduli space of triangles in the context of celestial mechanics

Abstract

[This is an expository article. I have submitted it to the American Mathematical Monthly.] The three-body problem defines a dynamics on the space of triangles in the plane. The shape sphere is the moduli space of oriented similarity classes of planar triangles and lies inside shape space, a Euclidean 3-space parametrizing oriented congruence classes of triangles. We derive and investigate the geometry and dynamics induced on these spaces by the three-body problem. We present two theorems concerning the three-body problem whose discovery was made through the shape space perspective