A new set of variables in the three-body problem

We propose a set of variables of the general three-body problem both for two-dimensional and three-dimensional cases. 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 $(\Lambda,\Theta)$ 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 $\omega,\phi$, 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 {\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 $(\lambda,\theta,k=0,L=0,\dot{I}=0,\omega=0)$. We show that the hyper-surface $\dot{I} = 0$ is a global surface of section.