On the C8/3-regularisation of simultaneous binary collisions in the collinear 4-body problem

[摘要] A conjecture on the nature of the simultaneous binary collision is explored in the collinear 4-body problem. It is known that any attempt to remove the singularity via block regularisation will result in a regularised flow that is no more than C-8/3 differentiable with respect to initial conditions. Through a blow-up of the singularity, this loss of differentiability is investigated and a new proof of the C-8/3 regularity is provided. In the process, it is revealed that the collision manifold consists of two manifolds of normally hy-perbolic saddle singularities which are connected by a manifold of heteroclinics. By utilising recent work on transitions near such objects and their normal forms, an asymptotic series of the transition past the singular-ity is explicitly computed. It becomes remarkably apparent that the finite differentiability at 8/3 is due to the inability to construct a set of integrals local to the simultaneous binary collision. The finite differentiability is shown to be independent of the value of the masses. (c) 2020 Elsevier Inc. All rights reserved.

[发布日期] 2020-11-05