Geodesic
Continuous Choreographies as Limiting Solutions of $N$-body Type Problems with Weak Interaction
Symmetry, integrability and geometry: methods and applications, Tome 12 (2016) Cet article a éte moissonné depuis la source Math-Net.Ru

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We consider the limit $N\to +\infty$ of $N$-body type problems with weak interaction, equal masses and $-\sigma$-homogeneous potential, $0\sigma1$. 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 $H^1$.
Keywords: $N$-body problem; continuous coreography; Lagrangian action. 
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     author = {Reynaldo Castaneira and Pablo Padilla and H\'ector S\'anchez-Morgado},
     title = {Continuous {Choreographies} as {Limiting} {Solutions} of $N$-body {Type} {Problems} with {Weak} {Interaction}},
     journal = {Symmetry, integrability and geometry: methods and applications},
     year = {2016},
     volume = {12},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SIGMA_2016_12_a103/}
}
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Reynaldo Castaneira; Pablo Padilla; Héctor Sánchez-Morgado. Continuous Choreographies as Limiting Solutions of $N$-body Type Problems with Weak Interaction. Symmetry, integrability and geometry: methods and applications, Tome 12 (2016). http://geodesic.mathdoc.fr/item/SIGMA_2016_12_a103/

[1] Barutello V., Terracini S., “Action minimizing orbits in the $n$-body problem with simple choreography constraint”, Nonlinearity, 17 (2004), 2015–2039, arXiv: math.DS/0307088 | DOI | MR | Zbl

[2] Buck G., “Most smooth closed space curves contain approximate solutions of the $n$-body problem”, Nature, 395 (1998), 51–53 | DOI

[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 | DOI | MR | Zbl

[4] Chenciner A., Gerver J., Montgomery R., Simó C., “Simple choreographic motions of $N$ bodies: a preliminary study”, Geometry, Mechanics, and Dynamics, Springer, New York, 2002, 287–308 | DOI | MR | Zbl

[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, arXiv: math.DS/0011268 | DOI | MR | Zbl

[6] Perko L. M., Walter E. L., “Regular polygon solutions of the $N$-body problem”, Proc. Amer. Math. Soc., 94 (1985), 301–309 | DOI | MR | Zbl

[7] Xie Z., Zhang S., “A simpler proof of regular polygon solutions of the $N$-body problem”, Phys. Lett. A, 277 (2000), 156–158 | DOI | MR | Zbl

[8] Yu G., Simple choreography solutions of the Newtonian $N$-body problem, arXiv: 1509.04999