A New Set of Variables in the Three-Body Problem

Abstract

We propose a set of variables of the general three-body problem both for two-dimensional and three-dimensional cases. The variables are ($\lambda, \theta, \Lambda, \Theta, k, \omega$), or equivalently ($\lambda, \theta, L, \dot{I},k,\omega)$ for the two-dimensional problem, and ($\lambda, \theta, L, \dot{I}, k, \omega, \phi, \psi$) for the three-dimensional problem. Here, ($\lambda, \theta$) and ($\Lambda, \Theta)$ specify the positions in the shape spheres in the configuration and momentum spaces, $k$ is the virial ratio, $L$ is the total angular momentum, $\dot{I}$ is the time derivative of the moment of inertia, and $\omega, \phi$, and $\psi$ are the Euler angles to bring the momentum triangle from the nominal position to a given position. This set of variables defines a shape space of the three-body problem. This is also used as an initial-condition space. The initial condition of the so-called free-fall three-body problem is ($\lambda, \theta, k =$ 0, $L =$ 0, $\dot{I} =$ 0, $\omega =$ 0). We show that the hyper-surface $\dot{I} =$ 0 is a global surface of section.

Saari’s homographic conjecture for a planar equal-mass three-body problem under the Newton gravity

“…Actually, Hsiang and Straume [4,5], Chenciner and Montgomery [1], Montgomery [9] and Mockel [7] showed that this metric space is the 'shape sphere' and the distance ( 27) is the distance on the shape sphere. Kuwabara and Tanikawa also noted that the shape sphere is useful to investigate the equal-mass free-fall problem [6,12]. The map (25) makes the shape sphere 2π/3 rotation around the axis that connects the two Lagrange points.…”

Saari's homographic conjecture for general masses in planar three-body problem under Newton potential and a strong force potential

“…This sphere is exactly the Riemann sphere of the complex plane x + iy. This fact was first noticed by George Lemaître [12] and used by Hsiang and Straume [9,10], Chenciner and Montgomery [2], Montgomery and Mockel [16], Kuwabara and Tanikawa [11].…”

“…To describe the motion in shape, we used the shape variable ζ ∈  in equation (11), or η ∈  in equation ( 15), introduced by Moeckel and Montgomery. We wrote the Lagrangian in the size variable r, the rotation variable ϕ, and the shape variable η.…”