Viscosity solutions and hyperbolic motions: a new PDE method for the $N$-body problem

TL;DR

This paper introduces a novel PDE-based method using viscosity solutions of Hamilton-Jacobi equations to establish the existence of hyperbolic motions in the N-body problem with arbitrary shape, initial conditions, and energy levels.

Contribution

It presents a new approach employing viscosity solutions to prove hyperbolic motions in the N-body problem, expanding the application of PDE techniques beyond periodic orbit existence.

Findings

Existence of hyperbolic motions for any prescribed shape and initial configuration.

Construction of global viscosity solutions fixed points of the Lax-Oleinik semigroup.

Application of the method to extend Marchal's theorem.

Abstract

We prove for the $N$-body problem the existence of hyperbolic motions for any prescribed limit shape and any given initial configuration of the bodies. The energy level $h>0$ of the motion can also be chosen arbitrarily. Our approach is based on the construction of global viscosity solutions for the Hamilton-Jacobi equation $H(x,d_xu)=h$. We prove that these solutions are fixed points of the associated Lax-Oleinik semigroup. The presented results can also be viewed as a new application of Marchal's theorem, whose main use in recent literature has been to prove the existence of periodic orbits.