# Choreography in Nature (towards theory of dancing curves, superintegrability)

## Document Type

Presentation Abstract

## Presentation Date

2-22-2021

## Abstract

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 so-called Remarkable Figure Eight) at zero angular momentum was discovered in physics *unexpectedly* by C Moore (Santa Fe Institute) in 1993 for 3 equal masses in *R*^{3} Newtonian gravity numerically and independently in mathematics by Chenciner (Paris)-Montgomery (Santa Cruz) in 2000. At the moment about 6,000 choreographies in *R*^{3} Newtonian gravity are found, all numerically, for different *n*>2. All of them are represented by transcendental curves. It manifests the major discovery in celestial mechanics, next after H Poincare chaotic nature of *n* body problem.

Does exist (non)-Newtonian gravity for which dancing curve is known analytically? - Yes, a single example is known - it is the algebraic lemniscate by Jacob Bernoulli (1694) - and it will be the subject of the talk. Astonishingly, the Figure Eight trajectory in *R*^{3} Newtonian gravity coincides with algebraic lemniscate with *χ*^{2} deviation ~ 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.

Talk will be accompanied by numerous animations.

## Recommended Citation

Turbiner, Alexander, "Choreography in Nature (towards theory of dancing curves, superintegrability)" (2021). *Colloquia of the Department of Mathematical Sciences*. 614.

https://scholarworks.umt.edu/mathcolloquia/614

## Additional Details

February 22, 2021 at 3:00 p.m. via Zoom