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

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).

arXiv comment9 pages

References in corpus (3)

• Action minimizing solutions of the Newtonian n-body problem: from homology to symmetry
• On weak KAM theory for N-body problems
• On the free time minimizers of the Newtonian N-body problem

Cited by in corpus (4)

• On the free time minimizers of the Newtonian N-body problem
• Free time minimizers for the planar three-body problem
• Minimizing configurations and Hamilton-Jacobi equations of homogeneous N-body problems
• Null Angular Momentum and Weak KAM Solutions of the Newtonian $N$-Body Problem