Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 13 (2017), 068, 8 pages      arXiv:1611.01573

Null Angular Momentum and Weak KAM Solutions of the Newtonian $N$-Body Problem

Boris A. Percino-Figueroa
Facultad de Ciencias en Física y Matemáticas, Universidad Autónoma de Chiapas, México

Received November 09, 2016, in final form August 16, 2017; Published online August 24, 2017

In [Arch. Ration. Mech. Anal. 213 (2014), 981-991] it has been proved that in the Newtonian $N$-body problem, given a minimal central configuration $a$ and an arbitrary configuration $x$, there exists a completely parabolic orbit starting on $x$ and asymptotic to the homothetic parabolic motion of $a$, furthermore such an orbit is a free time minimizer of the action functional. In this article we extend this result in abundance of completely parabolic motions by proving that under the same hypothesis it is possible to get that the completely parabolic motion starting at $x$ has zero angular momentum. We achieve this by characterizing the rotation invariant weak KAM solutions as those defining a lamination on the configuration space by free time minimizers with zero angular momentum.

Key words: $N$-body problem; angular momentum; free time minimizer; Hamilton-Jacobi equation.

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