The 3-body problem in Shape Space (1)

The zero energy, zero angular momentum 3-body problem has two main kinds of behaviors: the dynamical orbits that end up in a three-body collision (or a three-body escape), and the ones that end up with a tight Keplerian pair plus a third particle that flies away (hyperbolic-elliptic escape). Curiously, the three-body collisions (and the corresponding escapes to infinity), can only happen in one out of four special configurations of the particles: either they form an equilateral triangle, or they are collinear, with the relative distances in one of three paticular ratios. These are called Euler configurations. This fact can be easily explained with the equivalent formulation of the 3-body problem as a dissipative system on Shape Space S. 3-body Shape Space The scale-invariant Newtonian potential Vs has only five stationary points on S: the two equilateral triangles (one is the mirror image of the other), and the three Euler configurations. These points are unstable equilibria of the dynamics (the formers are maxima of Vs, the latters saddle points). Therefore there are some special dynamical curve on S that can end up in those points with zero momenta, and stay there forever. In the logarithmic time this can only happen in the infinite future of those dynamical curves, as the corresponding arrow of time always points towards those equilibria. The corresponding Newtonian description will be that of a three-body collision or explosion, depending on whether Newtonian time has the same arrow as the logarithmic time. A degenerate case is when the representative point in S just stays forever at one of the equilibria. The corresponding Newtonian description is that of a homothetic fall to the center or explosion, again according to the sign of t. All of the situations we have described so far are a subset of measure zero of the solutions of the problem. The vast majority of them end up in a hyperbolic-elliptic escape, which we'll describe now,

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