Seminario de geometría diferencial y sistemas dinámicos, CIMAT

Fecha y lugar: Lunes 17 febrero 2020, 4:45pm, Salón Diego Bricio

Speaker: Alexander Turbiner, ICN-UNAM, Mexico and Stony Brook University, USA

Title: Choreography in Nature (towards a theory of dancing curves, superintegrability)

Abstract: By definition, a choreography (or a "dancing curve") is a closed trajectory
along  which n classical bodies move, chasing each other without collisions.
The first choreography (the remarkable Figure Eight) at zero angular momentum was 
discovered numerically in physics unexpectedly by C Moore (Santa Fe Institute) in 
1993 for 3 equal masses in R^3 Newtonian gravity  and independently in mathematics 
by Chenciner-Montgomery in 2000. At the moment about 6,000 choreographies in R^3 
Newtonian gravity are known, all numerically, for various n > 2. Will General 
Relativity support such choreographies?

Does there exist a non-Newtonian gravity for which a dancing curve is known
analytically? - Yes, a single example is known - an algebraic curve, the 
lemniscate of  Jacob Bernoulli (1694) - and it will be a concrete example in this talk. 
Astonishingly, the R^3 Newtonian Figure Eight trajectory coincides with the lemniscate 
with a chi-squared deviation of about 10^{-7}. Both choreographies admit any odd number 
of bodies along them. Both 3-body choreographies admit a maximally superintegrable 
trajectory with 7 constants of motion.

The talk is directed to general mathematical public (also undergraduates). It will 
be accompanied by numerous animations.