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 R3 Newtonian gravity numerically and independently in mathematics by Chenciner (Paris)-Montgomery (Santa Cruz) in 2000. At the moment about 6,000 choreographies in R3 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 R3 Newtonian gravity coincides with algebraic lemniscate with χ2 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.

Additional Details

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

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