Total Collisions in the N-Body Shape Space

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.

Scaling symmetries, contact reduction and Poincaré’s dream

“…The corresponding scale-invariant equations of motion (equation (7), with Λ = −2) are exactly the usual McGehee blow-up equation ( 39) (e.g. section 4.2 of [43]), the collision manifold being given by the invariant level H = 0 (see as well [40]). Recalling the discussion in section 4.2, we conclude that these McGehee blow-up equations are a −2-Herglotz system with Lagrangian function given by ( 40) via Legendre transform of H .…”

Scaling Symmetries, Contact Reduction and Poincaré's dream

“…s := q/ρ 2 ∈ S nd−1 , y := ρp ∈ R nd . the collision manifold being given by the invariant level H = 0 (see as well [32]).…”

“…It should be clear that in any scale-invariant description of the physical world one would need one less piece of information with respect to the standard descriptions that include reference to some (unmeasurable) absolute scale, the datum being removed being precisely the one corresponding to the scale. Moreover, such scale reductions contain valuable information on certain singularities, eg collision singluarities, of the original systems and this has been exploited in several works, [28,31,33,38], to find that the scale-invariant description is freed from the singularities of the original dynamics (although care must be taken, since this is not always necessarily the case [32]). As well, one finds interesting dynamical features such as a dissipative-like behavior, which can provide a natural origin for the observed arrow of time [3,26,37].…”

Herglotz action for homogeneous cosmologies

“…The purpose of this paper is to provide an action formulation of cosmology which is compatible with the relationalist/Shape Dynamics viewpoint [5][6][7][8][9], from which the evolution of shape can be deduced without ever making reference to scale. This approach has met with great success in treating the initial singularity of cosmology [10][11][12], and by parallel work the central singularity of Schwarzschild black holes [13] (and see [14,15] for interesting potential consequences), as well as a number of other interesting physical systems [16,17]. The reduced descriptions appear frictional [18,19] which has implications for the interpretation of fundamental issues in physics and cosmology such as the existence of a gravitational arrow of time, and the status of the past hypothesis [20][21][22][23].…”