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

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.