Null Angular Momentum and Weak KAM Solutions of the Newtonian $N$-Body Problem

TL;DR

This paper extends previous results on the Newtonian N-body problem by showing the existence of completely parabolic motions with zero angular momentum, characterized through weak KAM solutions and free time minimizers.

Contribution

It proves that under certain conditions, completely parabolic motions with zero angular momentum exist and are characterized by rotation invariant weak KAM solutions.

Findings

Existence of zero angular momentum parabolic motions.

Characterization of weak KAM solutions as lamination by free time minimizers.

Extension of previous results to a broader class of motions.

Abstract

In [Arch. Ration. Mech. Anal. 213 (2014), 981-991] it has been proved that in the Newtonian $N$-body problem, given a minimal central configuration $a$ and an arbitrary configuration $x$, there exists a completely parabolic orbit starting on $x$ and asymptotic to the homothetic parabolic motion of $a$, furthermore such an orbit is a free time minimizer of the action functional. In this article we extend this result in abundance of completely parabolic motions by proving that under the same hypothesis it is possible to get that the completely parabolic motion starting at $x$ has zero angular momentum. We achieve this by characterizing the rotation invariant weak KAM solutions as those defining a lamination on the configuration space by free time minimizers with zero angular momentum.