Natural dynamical reduction of the three-body problem

TL;DR

This paper introduces a natural and general dynamical reduction of the three-body problem by decomposing variables into geometry and orientation, revealing new insights into the problem's structure and extending to the four-body case.

Contribution

The paper presents a novel, symmetric reduction method that avoids previous limitations, providing a natural framework for analyzing three- and four-body problems.

Findings

Geometry variables describe motion in a curved 3D space.

Orientation variables follow Euler-like dynamics with geometry-dependent moments.

Applications include analyzing global features and exact solutions.

Abstract

The three-body problem is a fundamental long-standing open problem, with applications in all branches of physics, including astrophysics, nuclear physics and particle physics. In general, conserved quantities allow to reduce the formulation of a mechanical problem to fewer degrees of freedom, a process known as dynamical reduction. However, extant reductions are either non-general, or hide the problem's symmetry or include unexplained definitions. This paper presents a dynamical reduction that avoids these issues, and hence is general and natural. Any three-body configuration defines a triangle, and its orientation in space. Accordingly, we decompose the dynamical variables into the geometry (shape + size) and orientation of the triangle. The geometry variables are shown to describe the motion of an abstract point in a curved 3d space, subject to a potential-derived force and a magnetic-like force with a monopole charge. The orientation variables are shown to obey a dynamics analogous to the Euler equations for a rotating rigid body, only here the moments of inertia depend on the geometry variables, rather than being constant. The reduction rests on a novel symmetric solution to the center of mass constraint inspired by Lagrange's solution to the cubic. The formulation of the orientation variables is novel and rests on a little known generalization of the Euler-Lagrange equations to non-coordinate velocities. Applications to global features, to the statistical solution and to special exact solutions are presented. A generalization to the four-body problem is presented.