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. It is shown that in all of these cases the regularization is differentiable but the map passing from initial to final conditions (in some suitable transversal sections) is exactly C8/3. Then the result is extended to other symmetric N-body problems.