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

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We consider in this note the Hamilton-Jacobi equation H(x, dx u) = c, where c ≥ 0, of the classical N-body problem in an Euclidean space E of dimension k ≥ 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).