Regularization of the two-body problem via smoothing the potential

We investigate the existence of global solutions for the two-body problem, when the particles interact with a potential of the form 1/r(alpha), for alpha > 0. Our solutions are pointwise limits of approximate solutions u(alpha)(epsilon(k,)nu(k)) which solve the equation of motion with the regularized potential 1/(r(2)+epsilon(k)(2))(alpha/2),and with an initial condition nu(k); (epsilon(k,)nu(k))(k) is a sequence converging to (0,(ν) over bar) as k --> +infinity, where (ν) over bar is an initial condition leading to collision in the non-regularized problem. We classify all the possible limits and we compare them with the already known solutions, in particular with those obtained in the paper [9] by McGehee using branch regularization and block regularization. It turns out that when alpha > 2 the double limit exist, therefore in this case the problem can be regularized according to a suitable definition.