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

Regina Mart\'{\i}nez, Departament de Matem\`atiques,
Universitat Aut\`onoma de Barcelona, Bellaterra, Barcelona, Spain

and 

Carles Sim\'o, Dept. de Matem\`atica Aplicada i An\`alisi, Univ. de
Barcelona, Gran Via 585, 08007 Barcelona, Spain

Abstract

Simultaneous binary collisions in the four-body problem are studied in cases 
which reduce to 1D problems: the collinear, biisosceles, trapezoidal cases and
tetrahedron problem, some of them requiring certain symmetries of the masses.
It is shown that in all these cases the regularization is differentiable but 
the map passing from initial to final conditions (in some suitable transversal
sections) has a regularity r < 8/3 but not 8/3. Then the result is extended to
other symmetric N-body problems.