A N-body problem with weak force potential through Hamilton-Jacobi equation approach

TL;DR

This paper investigates the existence of hyperbolic motions in an N-body problem with a weak force potential using Hamilton-Jacobi equations, allowing for arbitrary limit shapes, initial configurations, and energy levels.

Contribution

It introduces a novel approach employing global viscosity solutions to Hamilton-Jacobi equations to analyze hyperbolic motions in weak force potential N-body problems.

Findings

Existence of hyperbolic motions for prescribed limit shapes

Applicability to any initial configuration

Flexibility in choosing energy levels h > 0

Abstract

This paper we consider for the N-body problem with potential 1/r{\alpha} (0 < {\alpha} < 1) the existence of hyperbolic motions for any prescribed limit shape and any given initial configuration of the bodies. Here E is the Euclidean space where the bodies moving and is the norm induced by the inner product. The energy level h > 0 of the motion can also be chosen arbitrarily. We use the global viscosity solutions for the Hamilton-Jacobi equation H (x, dxu) = h and geodesics.