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

SIGMA 12 (2016), 104, 9 pages      arXiv:1510.05979      https://doi.org/10.3842/SIGMA.2016.104

### Continuous Choreographies as Limiting Solutions of $N$-body Type Problems with Weak Interaction

a) Instituto de Investigaciones en Matemáticas Aplicadas y en Sistemas, UNAM, México D.F. 04510, México
b) Instituto de Matemáticas, UNAM, México D.F. 04510, México

Received October 20, 2015, in final form October 29, 2016; Published online October 31, 2016

Abstract
We consider the limit $N\to +\infty$ of $N$-body type problems with weak interaction, equal masses and $-\sigma$-homogeneous potential, $0$<$\sigma$<$1$. We obtain the integro-differential equation that the motions must satisfy, with limit choreographic solutions corresponding to travelling waves of this equation. Such equation is the Euler-Lagrange equation of a corresponding limiting action functional. Our main result is that the circle is the absolute minimizer of the action functional among zero mean (travelling wave) loops of class $H^1$.

Key words: $N$-body problem; continuous coreography; Lagrangian action.

pdf (316 kb)   tex (12 kb)

References

1. Barutello V., Terracini S., Action minimizing orbits in the $n$-body problem with simple choreography constraint, Nonlinearity 17 (2004), 2015-2039, math.DS/0307088.
2. Buck G., Most smooth closed space curves contain approximate solutions of the $n$-body problem, Nature 395 (1998), 51-53.
3. Chenciner A., Desolneux N., Minima de l'intégrale d'action et équilibres relatifs de $n$ corps, C. R. Acad. Sci. Paris Sér. I Math. 326 (1998), 1209-1212.
4. Chenciner A., Gerver J., Montgomery R., Simó C., Simple choreographic motions of $N$ bodies: a preliminary study, in Geometry, Mechanics, and Dynamics, Springer, New York, 2002, 287-308.
5. Chenciner A., Montgomery R., A remarkable periodic solution of the three-body problem in the case of equal masses, Ann. of Math. 152 (2000), 881-901, math.DS/0011268.
6. Perko L.M., Walter E.L., Regular polygon solutions of the $N$-body problem, Proc. Amer. Math. Soc. 94 (1985), 301-309.
7. Xie Z., Zhang S., A simpler proof of regular polygon solutions of the $N$-body problem, Phys. Lett. A 277 (2000), 156-158.
8. Yu G., Simple choreography solutions of the Newtonian $N$-body problem, arXiv:1509.04999.