Minimizing configurations and Hamilton-Jacobi equations of homogeneous N-body problems

TL;DR

This paper explores special central configurations in homogeneous N-body problems, establishing the existence of homogeneous weak KAM solutions for certain potentials and linking them to viscosity solutions of Hamilton-Jacobi equations, with applications to the three-body problem.

Contribution

It introduces a new class of central configurations related to homothety reduction and proves the existence of homogeneous weak KAM solutions for potentials with lpha in (0,2).

Findings

Existence of homogeneous weak KAM solutions for lpha in (0,2)

Homogeneous solutions relate to viscosity solutions on the sphere

No smooth homogeneous solutions exist for the Newtonian three-body problem at the critical Hamilton-Jacobi equation

Abstract

For $N$-body problems with homogeneous potentials we define a special class of central configurations related with the reduction of homotheties in the study of homogeneous weak KAM solutions. For potentials in $1/r^\alpha$ with $\alpha\in (0,2)$ we prove the existence of homogeneous weak KAM solutions. We show that such solutions are related to viscosity solutions of another Hamilton-Jacobi equation in the sphere of normal configurations. As an application we prove for the Newtonian three body problem that there are no smooth homogeneous solutions to the critical Hamilton-Jacobi equation.