Translation Invariance of weak KAM solutions of the Newtonian N-body problem

TL;DR

This paper investigates the translation invariance of weak KAM solutions in the Newtonian N-body problem, showing invariance at the critical value and existence of non-invariant solutions for super-critical cases.

Contribution

It proves that all weak KAM solutions at the critical value are translation invariant and demonstrates the existence of non-invariant solutions when the energy level exceeds the critical value.

Findings

Weak KAM solutions at c=0 are translation invariant.

Existence of non-invariant solutions for c>0.

Invariance under Euclidean translations for critical solutions.

Abstract

We consider in this note the Hamilton-Jacobi equation H(x, dx u) = c, where c \geq 0, of the classical N-body problem in an Euclidean space E of dimension k \geq 2. The fixed points of the Lax-Oleinik semigroup are global viscosity solutions for the critical value of the constant (c = 0) also called weak KAM solutions. We show that all these solutions are invariant under the action of E by translations on the space of configurations. We deduce the existence of non-invariant solutions for the super-critical equations (c > 0).