Abstract 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 hyperbolic 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 singularity 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.

中文翻译：

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关于共线四体问题中同时二元碰撞的 C8/3 正则化

摘要 在共线四体问题中探索了关于同时二元碰撞性质的猜想。众所周知，任何通过块正则化去除奇点的尝试都会导致正则化流相对于初始条件的微分不超过 C 8 / 3。通过奇点的爆炸，研究了这种可微性的损失，并提供了 C 8 / 3 规律性的新证明。在这个过程中，揭示了碰撞流形由通常双曲鞍奇点的两个流形组成，这些流形由异宿相流形连接。通过利用最近关于此类物体及其正常形式附近过渡的工作，明确计算了经过奇点的过渡的渐近系列。很明显，8/3 处的有限可微性是由于无法构造一组局部于同时发生的二元碰撞的积分。有限可微性被证明与质量值无关。