On the free time minimizers of the Newtonian N-body problem

TL;DR

This paper investigates free time minimizers in the Newtonian N-body problem, proving they are parabolic and showing that no entire free time minimizers exist, implying the Mañé set is empty.

Contribution

It establishes the parabolic nature of free time minimizers and proves their non-existence on the entire real line in the Newtonian N-body problem.

Findings

Free time minimizers are completely parabolic.

No entire free time minimizers exist on R.

The Mañé set of the N-body problem is empty.

Abstract

In this paper we study the existence and the dynamics of a very special class of motions, which satisfy a strong global minimization property. More precisely, we call a free time minimizer a curve which satisfies the least action principle between any pair of its points without the constraint of time for the variations. An example of a free time minimizer defined on an unbounded interval is a parabolic homothetic motion by a minimal central confguration. The existence of a large amount of free time minimizers can be deduced from the weak KAM theorem. In particular, for any choice of x0, there should be at least one free time minimizer x(t) defined for all positive time and satisfying x(0)=x0. We prove that such motions are completely parabolic. Using Marchal's theorem we deduce as a corollary that there are no entire free time minimizers, i.e. defined on R. This means that the Ma\~n\'e set of the Newtonian N-body problem is empty.