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On the free time minimizers of the Newtonian \(N\)-body problem. (English) Zbl 1331.70035

Summary: 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 \(x_0\), there should be at least one free time minimizer \(x(t)\) defined for all \(t\geq 0\) and satisfying \(x(0)=x_0\). 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.

MSC:

70F10 \(n\)-body problems
49K05 Optimality conditions for free problems in one independent variable
58E30 Variational principles in infinite-dimensional spaces
70H08 Nearly integrable Hamiltonian systems, KAM theory
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