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

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.