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

TL;DR

This paper introduces a weak critical point concept and Morse index for the N-body problem with weak force potentials, providing bounds on binary collisions despite the non-differentiability caused by singularities.

Contribution

It generalizes critical point theory to handle collision singularities in the N-body problem by defining weak critical points and Morse index, offering new analytical tools.

Findings

Morse index bounds the number of binary collisions

Weak critical points extend variational methods to singular problems

Analysis includes Newtonian and other weak force potentials

Abstract

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.