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

Symmetry Integrability and Geometry-Methods and Applications

article

Reynaldo Castaneira¹  Pablo Padilla¹  Héctor Sánchez-Morgado² 

[1] Instituto de Investigaciones en Matemáticas Aplicadas y en Sistemas;Instituto de Matemáticas

关键词: N-body problem;    continuous coreography;    Lagrangian action;   

DOI  :  10.3842/SIGMA.2016.104

来源: National Academy of Science of Ukraine

【 摘 要 】

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 $H^1$.

【 授权许可】

Unknown