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.