On weak KAM theory for N-body problems

TL;DR

This paper extends weak KAM theory to N-body problems with homogeneous potentials, establishing regularity, existence of solutions, and invariance properties, thereby advancing the mathematical understanding of celestial mechanics.

Contribution

It proves the weak KAM theorem for N-body problems with homogeneous potentials, including existence of fixed points and invariant solutions, using uniform action bounds and regularity estimates.

Findings

Established uniform upper bounds for minimal action paths.

Proved Hölder regularity of the critical action potential.

Demonstrated existence of fixed points and invariant solutions.

Abstract

We consider N-body problems with homogeneous potential $1/r^{2\kappa}$ where $\kappa\in(0,1)$, including the Newtonian case ($\kappa=1/2$). Given $R>0$ and $T>0$, we find a uniform upper bound for the minimal action of paths binding in time $T$ any two configurations which are contained in some ball of radius $R$. Using cluster partitions, we obtain from these estimates H\"{o}lder regularity of the critical action potential (i.e. of the minimal action of paths binding in free time two configurations). As an application, we establish the weak KAM theorem for these N-body problems, i.e. we prove the existence of fixed points of the Lax-Oleinik semigroup and we show that they are global viscosity solutions of the corresponding Hamilton-Jacobi equation. We also prove that there are invariant solutions for the action of isometries on the configuration space.