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

Reynaldo Castaneira a, Pablo Padilla a and Héctor Sánchez-Morgado b
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.

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