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Adriana Da Luz and Ezequiel Maderna (2014)

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

Math. Proc. Cambridge Philos. Soc., 156(2):209-227.

The Hamiltonian formulation of the Newtonian N-body problem assures that motions are characterized by the local minimization property of the Lagrangian action. In this paper we study 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. A simple 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 y in E^N , there should be at least one free time minimizer x on [0,+∞) which satisfies x(0)=y. We prove that such motions are completely parabolic meaning that the velocity of each body goes to zero as t goes to +∞. More precisely, we show that the energy of that motions is zero, that the moment of inertia grows like t^{4/3}, and that the potential energy decay is like t^{-2/3}. Using Marchal's theorem, which states that minimizers avoid collisions we can deduce as a corollary that there are no complete free time minimizers, i.e. defined on (-∞,+∞).
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