Abstract
We discuss the total collision singularities of the gravitational N-body problem on shape space. Shape space is the relational configuration space of the system obtained by quotienting ordinary configuration space with respect to the similarity group of total translations, rotations, and scalings. For the zero-energy gravitating N-body system, the dynamics on shape space can be constructed explicitly and the points of total collision, which are the points of central configuration and zero shape momenta, can be analyzed in detail. It turns out that, even on shape space where scale is not part of the description, the equations of motion diverge at (and only at) the points of total collision. We construct and study the stratified total-collision manifold and show that, at the points of total collision on shape space, the singularity is essential. There is, thus, no way to evolve solutions through these points. This mirrors closely the big bang singularity of general relativity, where the homogeneous-but-not-isotropic cosmological model of Bianchi IX shows an essential singularity at the big bang. A simple modification of the general-relativistic model (the addition of a stiff matter field) changes the system into one whose shape-dynamical description allows for a deterministic evolution through the singularity. We suspect that, similarly, some modification of the dynamics would be required in order to regularize the total collision singularity of the N-body model.
Highlights
Ever since Newton published the Principia in 1687, the gravitational N-body problem and, with it, the total collision singularity has been an object of intensive study among mathematicians, starting from Euler and Lagrange who found special solutions to the Newtonian 3-body problem, up to Poincaré who famously received the price of the King of Sweden for his proposal of a general solution to it—a work he had to withdraw due to errors.It was Sundman who solved the 3-body problem in 1907 [1,2]
As shown in [8,9], the dynamics of the N-body problem can be equivalently formulated as a non-autonomous system of ODEs on shape space, reducing the system to its irreducible core of physical degrees of freedom
In this formulation, as was shown in [17], the total-collision solutions can be characterized neatly as solutions that end at a central configuration with zero dilatational momentum and zero shape momenta
Summary
Ever since Newton published the Principia in 1687, the gravitational N-body problem and, with it, the total collision singularity has been an object of intensive study among mathematicians, starting from Euler and Lagrange who found special solutions to the Newtonian 3-body problem, up to Poincaré who famously received the price of the King of Sweden for his proposal of a general solution to it—a work he had to withdraw due to errors (still, it was a brilliant work and, in a revised form, became the foundations of chaos theory) It was Sundman who solved the 3-body problem in 1907 [1,2].
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