| Abstract: |
| Due to collision singularities, the Lagrange action functional of the N-body problem generally is not differentiable in the space of Sobolev paths. Because of this, the usual critical point theory can not be applied to this problem directly. In this paper, 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. |
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