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. The variables are (lambda, theta, Lambda, Theta, k, omega), or equivalently (lambda, theta, L, (I) over dot, k, omega) for the two-dimensional problem, and (lambda, theta, L, (I) over dot, 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, (I) over dot 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, (I) over dot = 0, omega = 0). We show that the hyper-surface (I) over dot = 0 is a global surface of section.