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

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.

References in corpus (4)

• On the Existence of Collisionless Equivariant Minimizers for the Classical n-body Problem
• Globally minimizing parabolic motions in the Newtonian N-body problem
• On weak KAM theory for N-body problems
• Translation Invariance of weak KAM solutions of the Newtonian N-body problem

Cited by in corpus (2)

• Geodesic rays of the $N$-body problem
• Uniqueness of hyperbolic Busemann functions in the Newtonian N-body problem