The degree of differentiability of the regularization of simultaneous binary collisions in some N-body problems

Simultaneous binary collisions in the four-body problem are studied in the cases which reduce to one-dimensional problems: the collinear, bi-isosceles and trapezoidal cases and the tetrahedron problem, some of them requiring certain symmetries of the masses. Tt is shown that in all of these cases th e regularization is differentiable but the map passing from initial to fina l conditions (in some suitable transversal sections) is exactly C-8/3. Then the result is extended to other symmetric N-body problems. AMS classificat ion scheme numbers: 70F16, 70F10.