Choreography in Physics (living in motion, moving polymers, superintegrability etc)
Alexander Turbiner
ICN-UNAM, Mexico and Stony Brook University, USA
Tue, Sep. 03rd 2019, 11:00-12:00
Salle Claude Itzykson, Bât. 774, Orme des Merisiers

By definition the choreography (dancing curve) is a closed trajectory on which $n$ classical bodies move chasing each other without collisions. The first choreography (the Remarkable Figure Eight) at zero angular momentum was discovered unexpectedly by C Moore (Santa Fe Institute) at 1993 for 3 equal masses in $R^3$ Newtonian gravity numerically. At the moment about 6,000 choreographies in $R^3$ Newtonian gravity are found, all numerically for different $n > 2$. A number of 3-body choreographies is known in $R^2$ Newtonian gravity, for Lennard-Jones potential (hence, relevant for molecular physics), and for some other potentials, again numerically; it might be proved their existence for quarkonia potential. \par Does exist (non)-Newtonian gravity for which dancing curve is known analytically? - Yes, a single example is known - it is algebraic lemniscate by Jacob Bernoulli (1694) - and it will be a concrete example of the talk. Astonishingly, $R^3$ Newtonian Figure Eight coincides with algebraic lemniscate with $\chi^2$ deviation $\sim 10^{-7}$. Both choreographies admit any odd numbers of bodies on them. Both 3-body choreographies define maximally superintegrable trajectory with 7 constants of motion. \par Talk will be accompanied by numerous animations.

Contact : Loic BERVAS