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 ^α with α ∈ (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.