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. Variables are $(λ,θ,Λ, Θ,k,ω)$ or equivalently $(λ,θ,L,\dot{I},k,ω)$ for the two-dimensional problem, and $(λ,θ,L,\dot{I},k,ω,ϕ,ψ)$ for the three-dimensional problem. Here $(λ,θ)$ and $(Λ,Θ)$ specifies 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 $ω,ϕ$, and $ψ$ are the Euler angles to bring the momentum triangle from the nominal position to a given position. This set of variables defines a {\it 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 $(λ,θ,k=0,L=0,\dot{I}=0,ω=0)$. We show that the hyper-surface $\dot{I} = 0$ is a global surface of section.