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

SIGMA 13 (2017), 068, 8 pages      arXiv:1611.01573      https://doi.org/10.3842/SIGMA.2017.068

### 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

Abstract
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.

pdf (309 kb)   tex (13 kb)

References

1. Abraham R., Marsden J.E., Raţiu T.S., Manifolds, tensor analysis, and applications, Global Analysis Pure and Applied: Series B, Vol. 2, Addison-Wesley Publishing Co., Reading, Mass., 1983.
2. Arnol'd V.I. (Editor), Dynamical systems. VIII, Encyclopaedia of Mathematical Sciences, Vol. 39, Springer-Verlag, Berlin, 1993.
3. Chenciner A., Action minimizing solutions of the Newtonian $n$-body problem: from homology to symmetry, in Proceedings of the International Congress of Mathematicians, Vol. III (Beijing, 2002), Higher Ed. Press, Beijing, 2002, 279-294.
4. Contreras G., Action potential and weak KAM solutions, Calc. Var. Partial Differential Equations 13 (2001), 427-458.
5. da Luz A., Maderna E., On the free time minimizers of the Newtonian $N$-body problem, Math. Proc. Cambridge Philos. Soc. 156 (2014), 209-227, arXiv:1301.7034.
6. Ferrario D.L., Terracini S., On the existence of collisionless equivariant minimizers for the classical $n$-body problem, Invent. Math. 155 (2004), 305-362, math-ph/0302022.
7. Maderna E., On weak KAM theory for $N$-body problems, Ergodic Theory Dynam. Systems 32 (2012), 1019-1041, arXiv:1502.06273.
8. Maderna E., Translation invariance of weak KAM solutions of the Newtonian $N$-body problem, Proc. Amer. Math. Soc. 141 (2013), 2809-2816, arXiv:1105.4484.
9. Marchal C., How the method of minimization of action avoids singularities, Celestial Mech. Dynam. Astronom. 83 (2002), 325-353.
10. Meyer K.R., Offin D.C., Introduction to Hamiltonian dynamical systems and the $N$-body problem, Applied Mathematical Sciences, Vol. 90, 3rd ed., Springer, Cham, 2017.
11. Percino B., Sánchez-Morgado H., Busemann functions for the $N$-body problem, Arch. Ration. Mech. Anal. 213 (2014), 981-991.
12. Saari D.G., Symmetry in $n$-particle systems, in Hamiltonian Dynamical Systems (Boulder, CO, 1987), Contemp. Math., Vol. 81, Amer. Math. Soc., Providence, RI, 1988, 23-42.