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

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 $K_\omega$.