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

Published online by Cambridge University Press:  26 November 2013

ADRIANA DA LUZ  and
EZEQUIEL MADERNA
ADRIANA DA LUZ
Affiliation:
Centro de Matemática, Universidad de la Repüblica, Iguá 4225, CP 11400 Montevideo, Uruguay. e-mail: emaderna@cmat.edu.uy
EZEQUIEL MADERNA
Affiliation:
Centro de Matemática, Universidad de la Repüblica, Iguá 4225, CP 11400 Montevideo, Uruguay. e-mail: emaderna@cmat.edu.uy
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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 configuration. 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 t ≥ 0 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 $\mathbb{R}$. This means that the Mañé set of the Newtonian N-body problem is empty.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 156 , Issue 2 , March 2014 , pp. 209 - 227
Copyright
Copyright © Cambridge Philosophical Society 2013 

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