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arXiv · 2103.13004

On the $ C^{8/3} $-Regularisation of Simultaneous Binary Collisions in the Planar 4-Body Problem

Nathan Duignan·Holger R. Dullin
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Abstract

The dynamics of the 4-body problem allows for two binary collisions to occur simultaneously. It is known that in the collinear 4-body problem this simultaneous binary collision (SBC) can be block-regularised, but that the resulting block map is only $C^{8/3}$ differentiable. In this paper, it is proved that the $C^{8/3}$ differentiability persists for the SBC in the planar 4-body problem. The proof uses several geometric tools, namely, blow-up, normal forms, dynamics near normally hyperbolic manifolds of equilibrium points, and Dulac maps.

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Nathan Duignan, Holger R. Dullin. 2021-03-24. On the $ C^{8/3} $-Regularisation of Simultaneous Binary Collisions in the Planar 4-Body Problem. https://doi.org/10.1088/1361-6544/ac0127

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