The collision singularity in a perturbedN-body problem

Abstract

Consider the perturbedN-body problem

$$Z_k = - \gamma \sum\limits_{\mathop {j \ne k}\limits^{j = 1} }^N {\frac{{Z_k - Zj}}{{|Z_k - Zj|^3 }}} + P_k ,{\text{ }}k = 1,{\text{ }}...,N,$$

and assume that at the instantt ^* allN bodies collide, while theP _k remain bounded. It is shown that the motion prior to the instant of collision is essentially the same as in the unperturbedN-body problemP _k =0, i.e., the same asymptotic relations are valid in both cases ast→t ^*.