Session S03 - Geometric and Variational Methods in Celestial Mechanics

Wednesday, July 14, 16:00 ~ 16:25 UTC-3

Symmetric bicircular central configurations of the $3n$--body problem

Montserrat Corbera

Universitat de Vic-Universitat central de Catalunya, Spain   -   This email address is being protected from spambots. You need JavaScript enabled to view it.

We consider symmetric central configurations of the planar $3n$--body problem consisting of $n$ equal masses at the vertices of a regular $n$-gon inscribed on a circle of radius $r$ and $2n$ masses at the vertices of a second concentric $2n$--gon inscribed in a circle of radius $ar$. We call this kind of central configurations bicircular central configurations of the $3n$--body problem. We analyze two different types of such configurations. In the first type, called regular bicircular central configurations of the $3n$--body problem, the second $2n$--gon is regular and the masses at the vertices of this $2n$--gon alternate values. In the second type, called semiregular bicircular central configurations of the $3n$--body problem, the second $2n$--gon is semiregular and the masses at its vertices are all of them equal. A semiregular $2n$--gon is a $2n$--gon having $n$ pairs of vertices symmetric by a reflection of an angle $\beta$ with respect to the axis of symmetry of the regular $n$--gon. Our aim is to analyze the set of values of the parameter $a$ providing regular bicircular central configurations of the $3n$--body problem and the set of values of the parameters $(a,\beta)$ providing semiregular bicircular central configurations of the $3n$--body problem. In particular, we prove analytically the existence of regular bicircular central configurations with $a$ sufficiently large for all $n\geq 2$ and with $a$ sufficiently close to the origin for all $n\geq 3$, and we compute numerically the entire set of values of $a$ providing regular bicircular central configurations for fixed values of $n$. Furthermore we prove analytically the existence of two families when $n=2$ and four families when $n\geq 2$ of semiregular bicircular central configurations with $\beta$ sufficiently close to $\pi/2n$ and the existence of one family when $n=2$ and two families when $n\geq 3$ of semiregular bicircular central configurations with $\beta$ sufficiently close to $\pi/n$. We also study numerically the entire families of semiregular bicircular central configurations for fixed values of $n$.

Joint work with Claudia Valls (Instituto Superior Técnico, Universidade de Lisboa).

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