# Global Existence and Singularity of the N-body Problem with Strong Force

## TL;DR

This paper characterizes the global existence and singularity of solutions in the N-body problem with strong force using concepts from nonlinear dispersive PDEs, introducing ground and excited states to establish a dichotomy based on energy constraints.

## Contribution

It introduces a novel PDE-inspired framework for analyzing the N-body problem, including the concepts of ground and excited states, and provides a new dichotomy criterion for solution behavior.

## Key findings

- Conditional dichotomy between global existence and singularity based on energy thresholds.
- Characterization of solutions for the two-body problem that parallels PDE results.
- Refined analysis for N≥3, including infinite sign transitions of a threshold function.

## Abstract

We use the idea of ground states and excited states in nonlinear dispersive equations (e.g. Klein-Gordon and Schr\"odinger equations) to characterize solutions in the N-body problem with strong force under some energy constraints. Indeed, relative equilibria of the N-body problem play a similar role as solitons in PDE. We introduce the ground state and excited energy for the N-body problem. {We are able to give a conditional dichotomy of the global existence and singularity below the excited energy in Theorem \ref{thm:dichotomy}, the proof of which seems original and simple. This dichotomy is given by the sign of a threshold function $K_\omega$}. The characterization for the two-body problem in this new perspective is non-conditional and it resembles the results in PDE nicely. For $N\geq3$, we will give some refinements of the characterization, in particular, we examine the situation where there are infinitely transitions for the sign of $K_\omega$.