Application of Morse index in weak force $N$-body problem

Due to collision singularities, the Lagrange action functional of the N-body problem in general is not differentiable. Because of this, the usual critical point theory can not be applied to this problem directly. Following ideas from \cite{BR91}, \cite{Tn93a} and \cite{ABT06}, we introduce a notion called weak critical point for such an action functional, as a generalization of the usual critical point. A corresponding definition of Morse index for such a weak critical point will also be given. Moreover it will be shown that the Morse index gives an upper bound of the number of possible binary collisions in a weak critical point of the $N$-body problem with weak force potentials including the Newtonian potential.