Dr. Alexander Turbiner
(Instituto de Ciencias Nucleares UNAM, Mexico; Stony Brook Uni.; Centre de recherches mathématiques, Montreal )
Friday, August 26, 2022
1 – 2 PM
Venue: PG6 112


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 Remarkable Figure Eight) at zero angular momentum was discovered in physics unexpectedly by C Moore (Santa Fe Institute) at 1993 for 3 equal masses in R3 Newtonian gravity numerically and independently (but later) in mathematics by Chenciner (Paris)-Montgomery (Santa Cruz) in 2000. At the moment over 6,000 choreographies in R3 Newtonian gravity are found, all numerically for different n > 2. Will General Relativity support such choreographies? This is an open question. A number of 3-body choreographies is known in Rd Newtonian gravity at d = 2, 3,..., for Lennard-Jones type potentials (hence, relevant for molecular physics). Does a (non)-Newtonian gravity exist for which a dancing curve is known analytically? A single example is known - it is the algebraic lemniscate by Jacob Bernoulli (1694) - and it will be a concrete example through the talk. Astonishingly, R3 Newtonian Figure Eight trajectory coincides
with algebraic lemniscate with χ2 deviation ∼ 10−7. Both choreographies admit any odd numbers of bodies on them leading to a type of string with one self-intersection. Both 3-body choreographies define maximally superintegrable trajectory with 7 constants of motion and corresponding Liouville integrals.
The talk will be accompanied by numerous animations.
 

Short Bio: Dr. Turbiner graduated from the Moscow Institute for Physics and Technology and in 1978 obtained a PhD from the Institute for Theoretical and Experimental Physics in Moscow, where he was affiliated until 1990. He is an APS and IOP fellow, awarded for his work on the discovery of quasi-exactly-solvable Schroedinger equations and the author of more than 180 research papers. His research interests are in non-perturbative quantum mechanics and quantum field theory, AMO physics in strong magnetic fields and neutron star atmospheres.